Dsge - Stata Manual

STATA LINEARIZED DYNAMIC
STOCHASTIC GENERAL EQUILIBRIUM

REFERENCE MANUAL
RELEASE 15
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Contents
intro . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction 1
intro 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction to DSGEs and dsge 3
intro 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning the syntax 16
intro 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classic DSGE examples 21
intro 3a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Keynesian model 22
intro 3b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New Classical model 27
intro 3c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Financial frictions model 34
intro 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Writing a DSGE in a solvable form 39
intro 4a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Specifying a shock on a control variable 45
intro 4b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Including a lag of a control variable 47
intro 4c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Including a lag of a state variable 50
intro 4d . . . . . . Including an expectation of a control dated by more than one period ahead 53
intro 4e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Including a second-order lag of a control 56
intro 4f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Including an observed exogenous variable 59
intro 4g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Correlated state variables 62
intro 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Stability conditions 65
intro 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identification 71
intro 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convergence problems 76
intro 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wald tests vary with nonlinear transforms 82
dsge . . . . . . . . . . . . . . . . . . . . . . . Linearized dynamic stochastic general equilibrium models 89
dsge postestimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Postestimation tools for dsge 95
estat policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Display policy matrix 99
estat stable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Check stability of system 100
estat transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Display state transition matrix 102
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Subject and author index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
i
Cross-referencing the documentation
When reading this manual, you will find references to other Stata manuals. For example,
[U] 26 Overview of Stata estimation commands
[XT] xtabond
[D] reshape
The first example is a reference to chapter 26, Overview of Stata estimation commands, in the Users
Guide; the second is a reference to the xtabond entry in the Longitudinal-Data/Panel-Data Reference
Manual; and the third is a reference to the reshape entry in the Data Management Reference Manual.
All the manuals in the Stata Documentation have a shorthand notation:
[GSM] Getting Started with Stata for Mac

[GSU] Getting Started with Stata for Unix
[GSW] Getting Started with Stata for Windows
[U ] Stata Users Guide
[R] Stata Base Reference Manual
[BAYES] Stata Bayesian Analysis Reference Manual
[D] Stata Data Management Reference Manual
[ERM] Stata Extended Regression Models Reference Manual
[FMM] Stata Finite Mixture Models Reference Manual
[FN] Stata Functions Reference Manual
[G] Stata Graphics Reference Manual
[IRT] Stata Item Response Theory Reference Manual
[DSGE] Stata Linearized Dynamic Stochastic General Equilibrium
Reference Manual
[XT] Stata Longitudinal-Data/Panel-Data Reference Manual
[ME] Stata Multilevel Mixed-Effects Reference Manual
[MI] Stata Multiple-Imputation Reference Manual
[MV] Stata Multivariate Statistics Reference Manual
[PSS] Stata Power and Sample-Size Reference Manual
[P] Stata Programming Reference Manual
[SP] Stata Spatial Autoregressive Models Reference Manual
[SEM] Stata Structural Equation Modeling Reference Manual
[SVY] Stata Survey Data Reference Manual
[ST] Stata Survival Analysis Reference Manual
[TS] Stata Time-Series Reference Manual
[TE] Stata Treatment-Effects Reference Manual:
Potential Outcomes/Counterfactual Outcomes
[I] Stata Glossary and Index
[M] Mata Reference Manual
ii
Title
intro Introduction
Description Remarks and examples Also see
Description
DSGE stands for dynamic stochastic general equilibrium. DSGE models are multivariate time-series
models that are used in economics, in particular, macroeconomics, for policy analysis and forecasting.
These models are systems of equations that are typically derived from economic theory. As such,
the parameters are often directly interpretable based on theory. DSGE models are unique in that
equations in the system allow current values of variables to depend not only on past values but also
on expectations of future values.
The dsge command estimates parameters of linearized DSGE models.
Remarks and examples

We recommend that you read this manual beginning with [DSGE] intro 1 and then continue with
the remaining introductions. In these introductions, we will introduce DSGE models, show you how to
use the dsge command, walk you through worked examples of classic models, and present solutions
to common stumbling blocks.
[DSGE] intro 1 and [DSGE] intro 2 are essential reading. Read them first. Here you will find
an overview of DSGE models, descriptions of concepts used throughout the manual, discussion of
assumptions, a first worked example, and an introduction to the syntax.
[DSGE] intro 1 Introduction to DSGE models

[DSGE] intro 2 Learning the syntax
[DSGE] intro 3 focuses on classical DSGE models. It includes a series of examples that illustrate
model solution, model estimation, and postestimation procedures for simple variants of common
models.
[DSGE] intro 3 Classic DSGE examples

[DSGE] intro 3a New Keynesian model
[DSGE] intro 3b New Classical model
[DSGE] intro 3c Financial frictions model
[DSGE] intro 4 discusses some common problems and solutions for them. The structural equations
of the linearized DSGE model must have a specific structure so that the model can be solved. Often
DSGE models are written using intuitive forms that do not have this structure. These intuitive forms can
be rewritten in a logically equivalent form that has the structure required for solution. [DSGE] intro 4
provides an overview of this topic and examples demonstrating solutions.
1
2 intro Introduction
[DSGE] intro 4 Writing a DSGE in a solvable form

[DSGE] intro 4a Specifying a shock on a control variable
[DSGE] intro 4b Including a lag of a control variable
[DSGE] intro 4c Including a lag of a state variable
[DSGE] intro 4d Including an expectation of a control dated by more than
one period ahead
[DSGE] intro 4e Including a second-order lag of a control
[DSGE] intro 4f Including an observed exogenous variable
[DSGE] intro 4g Correlated state variables
[DSGE] intro 5[DSGE] intro 8 discuss technical issues. These introductions are essential reading,
even though they are last.
[DSGE] intro 5 Stability conditions

[DSGE] intro 6 Identification
[DSGE] intro 7 Convergence problems
[DSGE] intro 8 Wald tests vary with nonlinear transforms
The main command entries are references for syntax and implementation details. All the examples
are in the introductions discussed above.
[DSGE] dsge Linearized dynamic stochastic general equilibrium models

[DSGE] dsge postestimation Postestimation tools for dsge
[DSGE] estat policy Display policy matrix
[DSGE] estat stable Display stability results
[DSGE] estat transition Display state transition matrix
Also see
[DSGE] intro 1 Introduction to DSGEs and dsge
Title
intro 1 Introduction to DSGEs and dsge
Description Remarks and examples References Also see
Description
In this entry, we introduce DSGE models and the dsge command. We begin with an overview of
DSGE models. We then illustrate the complete process of DSGE modeling by doing an example from
start to finish. In this example, we demonstrate how to describe a model in its original nonlinear
form, write it in a corresponding linearized form, estimate the parameters of the linearized model, and
interpret the results. We conclude by showing how this example fits into the general DSGE modeling
framework.

Remarks are presented under the following headings:
Introduction to DSGE models
How to write down a DSGE
Using the dsge command
Writing down linearized DSGEs
Data preparation
Specifying the model to dsge
Parameter estimation and interpretation
Postestimation
Policy and transition matrices
Impulse responses
Forecasts
Structural and reduced forms of DSGE models
Introduction to DSGE models

DSGE models are models for multiple time series used in macroeconomics and finance. These
models are systems of equations that are motivated by economic theory and in which expectations
of future values of variables play an important role. Because these models come from theory, the
parameters of these models can typically be directly interpreted in terms of the motivating theory.
DSGE models are used for macroeconomic policy analysis and forecasting.
In DSGE models, individuals actions are summarized by decision rules that take the form of
nonlinear systems of dynamic equations. These decision rules often come from dynamic stochastic
optimization problems. A DSGE model consists of these decision rules, plus any aggregation conditions,
resource or budget constraints, and stochastic processes for exogenous variables.
Because the models equations are the solution to dynamic optimization problems, model equations
can feature expectations of future variables. These expectations are endogenous. In DSGE models,
expectations of future variables correspond to their conditional mean as implied by the model. In
other words, individuals expectations of future values are correct, on average. Such expectations are
said to be model-consistent expectations or rational expectations.
3
4 intro 1 Introduction to DSGEs and dsge
There are three kinds of variables in DSGE models: control variables, state variables, and shocks.
The terminology is taken from the state-space and optimal control literatures. In DSGE models, the
concepts of exogeneity and endogeneity are understood relative to a time period. A state variable is
fixed, or exogenous, in a given time period. The system of equations then determines the value of
the state variable one period in the future. On the other hand, the system of equations determines
the value of a control variable in the current time period. State variables evolve over time and may
depend on control variables. State variables can also be correlated. Control variables in a DSGE model
can be either observed or unobserved. State variables are always unobserved.
DSGE models can be written in multiple forms. The model based on economic theory may consist
of equations that are nonlinear both in the variables and in the parameters. A DSGE model is said to be
linear, or linearized, when the model equations are linear in the variables. DSGE models are commonly
linearized prior to analysis. After linearization, the model variables are expressed as deviations from
a steady state.
The DSGE model must be solved prior to estimation. Dynamic linear models are easier to solve
and fit than dynamic nonlinear models, particularly when future expectations are incorporated into
the model.
In any analysis of simultaneous equations systems, to solve a model means to write the models
endogenous variables as functions of its exogenous variables. In DSGE models, the analogous solution
concept is to write the models control variables in terms of its state variables. The models solution
consists of a system of equations relating the control variables to the state variables and a system
of equations describing the evolution of the state variables over time. The solution to a DSGE model
thus takes the form of a state-space model. The solution to a DSGE model is a crucial object for
both estimation and analyses after estimation. Both the likelihood function and the impulseresponse
functions are formed from the model solution.
dsge solves and estimates the parameters of linearized DSGE models.
General introductions to DSGE modeling are available in Ljungqvist and Sargent (2012) and
Woodford (2003). Canova (2007) and DeJong and Dave (2011) describe parameter estimation using
DSGE models.
How to write down a DSGE

Writing down a DSGE is a two-step process. First, we write down (potentially) nonlinear structural
equations that come from theory, usually economic theory. In many such theories, individuals do the
best that they can given their constraints. This idea is formalized in dynamic stochastic optimization
problems. The solutions to these problems are the nonlinear form of DSGE models. Second, we write
down the linearized model corresponding to the nonlinear structural model.
Consider the following nonlinear model, similar to that in Woodford (2003, chap. 4). The model
consists of equations that describe the behavior of households, firms, and a central bank. Interactions
among these actors produce a model of inflation, output growth, and the interest rate. Models of
this type are popular in academic and policy settings and are used to describe and analyze monetary
policy.
Household optimization generates an equation that relates current output Yt to the expected value
of a function of tomorrows output Yt+1 , tomorrows inflation t+1 , and the current nominal interest
rate Rt ,

1 1 Rt
= Et (1a)
Yt Yt+1 t+1
where is a parameter that captures households willingness to delay consumption.
intro 1 Introduction to DSGEs and dsge 5
Optimization by firms generates an equation that links the current deviation of inflation from its
steady state, t , to the expected value of the deviation of inflation from its steady state in the
future, Et (t+1 ), and to the ratio of actual output, Yt , to the natural level of output, Zt ,
1
Yt
(t ) + 1 = + Et (t+1 ) (2a)
Zt
where , , and are parameters linked to the pricing decision of firms. Firms are not affected by
inflation per se; they are affected only by deviations of inflation from its steady-state value.
Finally, there is an equation that describes central bank policy. The central bank adjusts the interest
rate in response to inflation and, perhaps, to other factors that we leave unmodeled. The equation for
the central bank policy is
1/
Rt t
= eut (3a)
R
where R is the steady-state value of the interest rate and ut is a state variable that captures all
movements in the interest rate that are not driven by inflation.
Using the dsge command

Before you can use dsge, you must make sure certain things about your data and your model are
true.
First, you must linearize the model by using deviations from steady state. dsge implements a
check for linearity and will complain if it detects nonlinearities. We do not discuss how to linearize
the equations in this manual. See, for example, DeJong and Dave (2011, chap. 2), which provides an
overview of the linearization process.
Second, the data must be tsset prior to using dsge. Using tsset on the data allows us to use
time-series operators.
Finally, the series in your model must have zero mean and be weakly stationary. dsge will demean
the series for you, but you need to ensure that your data are weakly stationary before specifying them
in the dsge command.
Writing down linearized DSGEs

The model in (1a)(3a) is nonlinear. We now need to write the model in its corresponding linearized
form, which we show below. Throughout this manual, we use lowercase letters to denote percentage
deviations of variables from the steady state. The linearized versions of the above equations are
yt = Et yt+1 (rt Et t+1 ) (1b)
t = Et t+1 + (yt zt ) (2b)
1
rt = t + ut (3b)

The new parameter is a complicated function of the underlying parameters from (2a). Those
underlying parameters cannot be separately identified in this model, but can be.
We have implicitly constrained some of the coefficients; for example, the coefficient on the interest
rate in the output equation is constrained to 1.
Note that we have not yet specified stochastic processes for zt and ut .
Data preparation
We have data on price levels and interest rates in rates.dta. These data were obtained from the
Federal Reserve Economic Database (FRED), which contains many macroeconomic and financial time
series; see [D] import fred.
. use http://www.stata-press.com/data/r15/rates2
(Federal Reserve Economic Data - St. Louis Fed, 2017-02-10)
. describe
Contains data from http://www.stata-press.com/data/r15/rates2.dta
obs: 281 Federal Reserve Economic Data -
St. Louis Fed, 2017-02-10
vars: 5 26 Apr 2017 21:22
size: 6,182
storage display value

variable name type format label variable label
datestr str10 %-10s Observation date

daten int %td Numeric (daily) date
gdpdef float %9.0g GDP deflator GDPDEF
r float %9.0g Federal funds rate FEDFUNDS
dateq int %tq Quarterly date
Sorted by: dateq
The dataset includes the price level and the interest rate. But the model is written in terms of
the inflation rate. For quarterly data, the inflation rate is conventionally obtained as 400 times the
difference in log of price variable. Therefore, we begin by generating an inflation rate variable p by
using the L. lag operator.
. generate p = 400*(ln(gdpdef) - ln(L.gdpdef))
(2 missing values generated)
. label variable p "Inflation rate"
We now have inflation and interest rates. They are not mean zero, but dsge will demean the data
for us.
Specifying the model to dsge
We make one final modification to the model equations before estimating the parameters. Wood-
ford (2003) rewrites the model in (1b)(3b) by defining xt = yt zt as the output gap. We do the
same. Substituting in xt gives us the three-equation system
xt = Et xt+1 (rt Et t+1 gt ) (4)

t = Et t+1 + xt (5)
1
rt = t + ut (6)

where gt = Et (zt+1 ) zt is a state variable. Equations (4)(6) are the structural form of how the
endogenous control variables xt , t , and rt evolve as functions of the exogenous state variables gt
and ut .
We complete the model by specifying processes for how the state variables evolve. Per standard
practice, both are modeled as first-order autoregressive processes.
ut+1 = u ut + t+1 (7)

gt+1 = g gt + t+1 (8)
The variables t+1 and t+1 are shocks to the state variables.
Equations (4)(6) are the linearized DSGE model derived from (1a)(3a). Together with (7)(8) for
the evolution of state variables, this is a complete model whose parameters can be estimated.
The command to estimate the parameters in the system of (4)(8) is
. dsge (p = {beta}*E(F.p) + {kappa}*x) ///
(x = E(F.x) -(r - E(F.p) - g), unobserved) ///
(r = (1/{beta})*p + u) ///
(F.u = {rhou}*u, state) ///
(F.g = {rhog}*g, state)
Each equation is bound in parentheses. The equations look almost identical to the system in (4)(8).
Because a model has as many variables as it has equations, each variable will appears on the left-hand
side of one and only one equation.
The equation options unobserved and state modify how dsge interprets the equations we supply.
Equations may be specified in any order; in this manual, we typically write control equations and
then state equations, but you dont have to follow that convention.
p, x, and r are our control variables pt , xt , and rt . Each appears on the left-hand side of one
equation and can appear on the right-hand side of as many equations as we like. There are two
state equations with shocks. The model must have the same number of shocks as observed control
variables. Therefore, we can treat only two of three control variables as observed. Of our three control
variables, the output gap is the most plausible to be unobserved. Thus we model the output gap as
unobserved, and we model the inflation rate and the interest rate as observed. To specify inflation
and the interest rate as observed, we only need to write their equations. We specify the output gap
as unobserved using the unobserved option.
u and g are the state variables ut and gt . Recall that state variables are fixed in the current period,
so we specify how they evolve through time by modeling the one-period leadhence, the F. on the
left-hand side of each state equation. The state equations specify how the state variable evolves as a
function of the current state variables and, possibly, the control variables.
The shocks t and t enter the system through the state equations of their corresponding variable.
By default, a shock is attached to each state equation. So, when we typed
. dsge ... (F.u = {rhou}*u, state) ...
the underlying equation is what we wrote in (7). If a state variable is treated as deterministic in
your model, then it will not have a shock. For example, capital accumulation is often treated as
deterministic. To include an equation for a state variable without a shock, we would include the
noshock option within the equation.
Expectations of future variables appear within the E() operator and use Statas F. operator. For
example, type E(F.x) to specify Et (xt+1 ).
The parameters we want to estimate are bound in braces.
For more details on the dsge syntax, see [DSGE] intro 2.
Parameter estimation and interpretation
We estimate the parameters of the model in (4)(8). These equations are much discussed in the
monetary economics literature. Equation (4) is known as the output-gap Euler equation. Equation (5)
is known as a New Keynesian Phillips Curve, and the parameter is known as the slope of the Phillips
curve. In New Keynesian models, prices depend on output, and is a measure of that dependence.
Equation (6) is known as a Taylor rule, after Taylor (1993). The coefficient on inflation in a Taylor rule
is a commonly discussed parameter. has two roles in the model above. It relates current inflation
deviations to expected future inflation deviations, and it relates interest rate deviations to inflation
deviations.
. dsge (p = {beta}*E(F.p) + {kappa}*x)
> (x = E(F.x) -(r - E(F.p) - g), unobserved)
> (r = (1/{beta})*p + u)
> (F.u = {rhou}*u, state)
> (F.g = {rhog}*g, state)
(setting technique to bfgs)
Iteration 0: log likelihood = -13931.564
Iteration 1: log likelihood = -1301.5118 (backed up)
(switching technique to nr)
DSGE model
Sample: 1954q3 - 2016q4 Number of obs = 250
Log likelihood = -768.09383
OIM
Coef. Std. Err. z P>|z| [95% Conf. Interval]
/structural
beta .5112881 .075791 6.75 0.000 .3627404 .6598359
kappa .1696296 .0475492 3.57 0.000 .076435 .2628243
rhou .6989189 .0449192 15.56 0.000 .6108789 .7869588
rhog .9556407 .0181342 52.70 0.000 .9200983 .9911831
sd(e.u) 2.317589 .2988024 1.731947 2.90323

sd(e.g) .6147348 .0973279 .4239757 .8054939
Two of the parameters have structural interpretations. The parameter kappa is the slope of the
Phillips curve. Theory predicts that this parameter will be positive, and indeed our estimate is positive.
The parameter beta is the inverse of the coefficient on inflation in the interest rate equation. We
can obtain an estimate of 1/ , which is interpreted as the degree to which the central bank responds
to movements in inflation, by using nlcom.
. nlcom 1/_b[beta]
_nl_1: 1/_b[beta]
_nl_1 1.955844 .2899255 6.75 0.000 1.387601 2.524088
A typical value for 1/ in the literature is 1.5. Our point estimate is around 2.
Postestimation

The matrix of parameters in the state-space form that specifies how the state variables affect the
control variables is known as the policy matrix. Each policy matrix parameter is the effect of a
one-unit shock to a state variable on a control variable.
Example 1: Obtaining the policy matrix

We use estat policy to view these results.
. estat policy
Policy matrix
Delta-method
p
u -.4170859 .0389324 -10.71 0.000 -.4933919 -.3407799
g .881884 .2330573 3.78 0.000 .4251001 1.338668
x
u -1.580153 .3926336 -4.02 0.000 -2.3497 -.8106049
g 2.658667 .9045286 2.94 0.003 .885823 4.43151
r
u .1842449 .056798 3.24 0.001 .072923 .2955669
g 1.724828 .2210259 7.80 0.000 1.291625 2.158031
Results are listed equation by equation. The first block is the policy equation for inflation p and
writes it as a function of the state variables alone. A unit shock to the state u reduces inflation by
an estimated 0.417, and a unit shock to g raises inflation by an estimated 0.882.
The matrix of parameters that specifies the dynamic process for the state variables is known as
the state transition matrix. The state transition equation relates the future values of the state variables
to their values in the current period. Each state transition matrix parameter is the effect of a one-unit
shock to a state variable on its one-period-ahead mean.
Example 2: Obtaining the transition matrix

. estat transition
Transition matrix of state variables
Delta-method
F.u
u .6989189 .0449192 15.56 0.000 .6108789 .7869588
g 1.11e-16 7.11e-12 0.00 1.000 -1.39e-11 1.39e-11
F.g
u 0 (omitted)
g .9556407 .0181342 52.70 0.000 .9200983 .9911831
Both state variables are modeled as autoregressive processes, so the results in estat transition
repeat the estimates of rhou and rhog from the dsge output. In this case, the other entries in the
state transition matrix are 0 or differ from 0 only because of lack of numerical precision. In more
complicated models, such as those in which a state equation depends on a control variable, the state
transition matrix will contain new information about that state variable.
Impulse responses
The state-space form allows us to trace the path of a control or state in response to a shock to a
state. This path is known as an impulseresponse function (IRF). irf after dsge estimates IRFs, and
it puts the named set of estimates into an .irf file, whose results can be displayed using irf graph
or irf table.
Example 3: Graphing an IRF

To graph the IRF, we first create the nkirf.irf file and set it as the active .irf file using the
irf set command.
. irf set nkirf.irf
(file nkirf.irf created)
(file nkirf.irf now active)
Next, we use irf create to estimate a complete set of impulse responses based on our dsge
command. A complete set of impulse responses is an impulse to each shock and the response of each
state and control variable to that impulse. For the model in this example, irf create generates an
impulse to e.u and e.g, then stores the response to that impulse on p, x, r, g, and u. The results
are stored in the nkirf.irf file.
. irf create model1
(file nkirf.irf updated)
We now use irf graph to plot the impulse responses. The impulse() and response() options
control which impulse and which responses are chosen. To view the response of p, x, r, and u to a
shock to u, we type
. irf graph irf, impulse(u) response(x p r u) byopts(yrescale)
model1, u, p model1, u, r
0 .6
.4
.5
.2
1
0
model1, u, u model1, u, x
3 0
2 2
1 4
0 6
0 2 4 6 8 0 2 4 6 8
step
95% CI impulseresponse function (irf)
Graphs by irfname, impulse, and response
The state variable u models movements in the interest rate that occur for reasons other than the
feedback between inflation and the interest rate. A shock to u is effectively a surprise increase in the
interest rate, and the IRF traces out how this shock causes temporary decreases to inflation (top-left
graph) and to the output gap (bottom-right graph).
Forecasts
The forecast suite of commands produces dynamic forecasts from the fitted model.
Example 4: Out-of-sample forecast

We first store the dsge estimation results.
. estimates store dsge_est
We use tsappend() to extend the dataset by 3 years, or 12 quarters.

. tsappend, add(12)
To set up a forecast, we perform three steps. forecast create initializes a new forecasting model,
which we name dsgemodel.
. forecast create dsgemodel
Forecast model dsgemodel started.
Next, we add the estimates from dsge to the forecasting model using forecast estimates.
. forecast estimates dsge_est
Added estimation results from dsge.
Forecast model dsgemodel now contains 2 endogenous variables.
This command adds the estimates stored in dsge est to the model dsgemodel. We now produce
dynamic forecasts beginning in the first quarter of 2017 using forecast solve. The prefix(d1 )
option specifies the d1 prefix that will be given to the variables created by forecast. We also
request that dynamic forecasts begin in the first quarter of 2017 with the begin(tq(2017q1)) option.
. forecast solve, prefix(d1_) begin(tq(2017q1))
Computing dynamic forecasts for model dsgemodel.
Starting period: 2017q1

Ending period: 2020q1
Forecast prefix: d1_
2017q1: ..............
2017q2: ..............
2017q3: ..............
2017q4: ..............
2018q1: ..............
2018q2: .............
2018q3: .............
2018q4: .............
2019q1: .............
2019q2: .............
2019q3: .............
2019q4: .............
2020q1: .............
Forecast 2 variables spanning 13 periods.
The dynamic forecast begins in the first quarter of 2017, so all forecasts are out of sample.
We can graph the forecast for inflation d1 p using tsline.
. tsline d1_p if tin(2010q1, 2021q1), tline(2017q1)
3
Inflation rate (dsgemodel d1_)
1 0 2
2010q1 2012q3 2015q1 2017q3 2020q1

Quarterly date
The model forecasts that inflation will smoothly return to its long-run value, the sample mean.
We can also begin the forecast during a time period for which observations are available.
Example 5: Within-sample forecast

Specifying the begin(tq(2014q1)) option produces dynamic forecasts beginning in the first
quarter of 2014, so we can compare the forecast for 20142016 with the actual observations over
that period.
. forecast solve, prefix(d2_) begin(tq(2014q1))
Computing dynamic forecasts for model dsgemodel.
Starting period: 2014q1

Ending period: 2020q1
Forecast prefix: d2_
2014q1: ..............
2014q2: ..............
2014q3: ..............
2014q4: ..............
2015q1: ..............
2015q2: ..............
2015q3: ..............
2015q4: .............
2016q1: .............
2016q2: .............
2016q3: .............
2016q4: .............
2017q1: .............
2017q2: .............
2017q3: .............
2017q4: .............
2018q1: .............
2018q2: .............
2018q3: .............
2018q4: .............
2019q1: .............
2019q2: .............
2019q3: .............
2019q4: .............
2020q1: .............
Forecast 2 variables spanning 25 periods.
. tsline p d2_p if tin(2010q1, 2021q1), tline(2014q1)
We plot both the observed inflation and the forecast.

3
2
1
0
2010q1 2012q3 2015q1 2017q3 2020q1

Quarterly date
Inflation rate Inflation rate (dsgemodel d2_)
The forecast captures the general upward trend in inflation from 20142016, but it does not predict
the variation in inflation around the upward trend.
Structural and reduced forms of DSGE models

Now that we have worked an example, we show how it fits in the more general formulation of
DSGE models. The model in (4)(8) is an example of a linearized DSGE model. In general, a linearized
DSGE model can be expressed as
A0 yt = A1 Et (yt+1 ) + A2 yt + A3 xt (9)
B0 xt+1 = B1 Et (yt+1 ) + B2 yt + B3 xt + Ct+1 (10)
where yt is a vector of control variables, xt is a vector of state variables, and t is a vector of

shocks. A0 through A3 and B0 through B3 are matrices of parameters. We require that A0 and
B0 be diagonal matrices. The entries in Ai and Bj are all functions of the structural parameters,
which we denote by vector . Economic theory places restrictions on the Ai and Bj matrices. C is
a selection matrix that determines which state variables are subject to shocks.
The state-space form of the model is given by
yt = Gxt (11)
xt+1 = Hxt + Mt+1 (12)

shocks. G is the policy matrix, and H is the state transition matrix. M is diagonal and contains the
standard deviations of the shocks.
yt is partitioned into observed and unobserved controls, yt = (y1,t , y2,t ). The observed control
variables are related to the control variables by the equation
y1,t = Dyt
where D is a selection matrix. Only observed control variables play a role estimation. The number
of observed control variables must be the same as the number of state equations that include shocks.
You specify a model of the form (9)(10) to dsge. Many models require some manipulation to
fit into the structure in (9) and (10); see [DSGE] intro 4 for details. Postestimation commands estat
policy and estat transition will display the policy and transition matrices in (11) and (12),
respectively.
References
Canova, F. 2007. Methods for Applied Macroeconomic Research. Princeton, NJ: Princeton University Press.
DeJong, D. N., and C. Dave. 2011. Structural Macroeconometrics. 2nd ed. Princeton, NJ: Princeton University Press.
Ljungqvist, L., and T. J. Sargent. 2012. Recursive Macroeconomic Theory. 3rd ed. Cambridge, MA: MIT Press.
Taylor, J. B. 1993. Discretion versus policy rules in practice. Carnegie-Rochester Conference Series on Public Policy
39: 195214.
Woodford, M. 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton
University Press.
Also see
[TS] forecast Econometric model forecasting
[TS] irf Create and analyze IRFs, dynamic-multiplier functions, and FEVDs
[TS] sspace State-space models
Title
intro 2 Learning the syntax
Description
In this introduction, we demonstrate how to specify a DSGE model using the dsge command. We
focus on two unique aspects of DSGEs that must be considered when writing the syntaxwriting the
system of equations and identifying each type of variable within those equations.

Preview of dsge syntax
Specifying the system of equations
Control variables
State variables and shocks
Expectations of future values of control variables
Specifying parameters using substitutable expressions
Preview of dsge syntax

If you have not read [DSGE] intro 1, we recommend that you read it first. In particular, see
Structural and reduced forms of DSGE models in [DSGE] intro 1, where we discuss the structural
form of a DSGE that is required by dsge. Below we assume that your model has this structural form,
and we discuss the syntax for specifying such a model with the dsge command.
As an example, if we wanted to fit the DSGE model
pt = Et (pt+1 ) + yt
yt = Et (yt+1 ) (rt Et (pt+1 ) zt )
rt = pt + ut
zt+1 = zt + t+1
ut+1 = ut + t+1
we would type
. dsge (p = {beta}*E(F.p) + {kappa}*y) ///
(y = E(F.y) -(r - E(F.p) - {rho}*z), unobserved) ///
({beta}*r = p + {beta}*u) ///
(F.z = {rho}*z, state) ///
(F.u = {delta}*u, state)
The syntax looks a lot like the equations, but it does require a little explanation. In what follows, we
discuss each element of the dsge syntax that you will need to specify the equations of your DSGE
model.
16
intro 2 Learning the syntax 17
In [DSGE] intro 1, we told you that three types of variablescontrol variables, state variables,
and shocksappear in DSGE models. We will demonstrate how to specify equations involving each
of these types of variables. We will also show you how to specify the required types of equations
that are linear in these variables and nonlinear in the parameters.
Specifying the system of equations

For a DSGE model, we specify an equation for each control variable and an equation for each
state variable. The equations are bound by parentheses. Therefore, the basic structure of the dsge
command is
. dsge (eq1) ///
(eq2) ///
..., ...
where eq1 and eq2 will be replaced with the syntax representing one of the equations in our model.
We use . . . to indicate that we can include more than two equations as well as options.
The basic form of an equation within the parentheses is
term = term [ + term [ + term [. . . ]]]
where each term includes a variable name. The variable name may be preceded by a parameter or
nonlinear combination of parameters. For instance, a valid equation might look something like
(y = {kappa}*z)
or
(y = {kappa}*z + {kappa}*{beta}*x)
or
(1/{beta}*y = {gamma}*z + x)
We will explain this later. For now, simply note that these equations are written in dsge syntax
much like we would write the math, but the parameters we want to estimate are offset with braces,
{}. If you have used any other Stata commands that work with substitutable expressions, you may
recognize this notation. In fact, dsge uses a special form of substitutable expressions.
Before we discuss how to use dsges substitutable expressions, we will focus on how to specify
each type of variable.
Control variables
Control variables can be modeled as a function of other control variables, expectations of the future
value of control variables, and state variables. The equations for control variables do not include
shocks.
We continue from the basic dsge command in the previous section, which was
. dsge (eq1) ///
(eq2) ///
..., ...
If we have an observed control variable y , and for eq1, we want to specify that yt be modeled as
a function of zt and xt , we can type this equation in our dsge command as
. dsge (y = pexp*z + pexp*x) ///
(eq2) ///
..., ...
where pexp is a possibly nonlinear expression of parameters in each term.
18 intro 2 Learning the syntax
Control variables can be observed or unobserved. Because we did not include any options within
this set of parentheses, y is assumed to be an observed control variable. If y were unobserved, we
would add the unobserved option as follows:
(y = pexp*z + pexp*x, unobserved)
Note that each control variable in the model must be included on the left-hand side of one, and
only one, equation.
State variables and shocks

To model a state variable, we specify an equation with the one-period lead of that state variable
on the left-hand side. On the right-hand side of the equation, we can include state variables, control
variables, expectations of the future value of control variables, and shocks.
When we specify an equation for a state variable, we include the state option within the
parentheses defining the equation.
Continuing with the syntax above, suppose eq2 is an equation for a state variable x, and we model
xt+1 as a function of xt . We expand our dsge command to
. dsge (y = pexp*z + pexp*x) ///
(F.x = pexp*x, state) ///
..., ...
We used the F. lead operator to specify the one-period lead of x as F.x; see [U] 11.4.4 Time-
series varlists. However, notice that the full list of time-series operators is not available here. We can
specify only equations for one-period leads of state variables; we could not replace F.x with F2.x
in the equation above. Note that this restriction does not limit the types of models we can fit; see
[DSGE] intro 4c.
By default, the equation for a state variable includes an unobserved shock. However, equations
for state variables are not required to include a shock. Within the system of equations, the number
of shocks should be equal to the number of observed control variables. If we did not wish to include
a shock in the equation for xt+1 , we could add the noshock option,
(F.x = pexp*x, state noshock)
Note that the one-period lead of each state variable must be included on the left-hand side of one,
and only one, equation.
Expectations of future values of control variables

Expectations of the one-period lead of control variables can appear in equations for both control
and state variables. Mathematically, we write these expectations as Et (). For instance, we write the
expectation of y in time t + 1 as Et (yt+1 ). In dsge, we write this expectation as E(F.y), where
E() represents an expectation and F.y is the one-period lead of y . Other than using the special E()
notation, expectations are included in the model in the same way as variables.
If we model yt as a function of Et (yt+1 ) in addition to zt and xt , we can expand our previous
dsge command to
. dsge (y = pexp*E(F.y) + pexp*z + pexp*x) ///
..., ...
intro 2 Learning the syntax 19
Expectations in E() are strictly for one-period leads of control variables. You cannot, for instance,
use E(F2.y) to include the expectation of y in time t + 2 in the model. This does not prevent you,
however, from including such terms in your model. See [DSGE] intro 4d for details of fitting models
including expectations of control variables more than one period in the future.
Specifying parameters using substitutable expressions

At this point, we know how to specify each type of variable that may arise in our DSGE model.
We now turn to specifying the parameters that we want to estimate.
Recall that the basic form of an equation is
term = term [ + term [ + term [. . . ]]]
where each term includes a variable name. The variable name may be preceded by a parameter or
a nonlinear combination of parameters. For terms on the right-hand side of the equation, variable
names can also be followed by a parameter specification.
Observed control variables are variables in your dataset. Unobserved control variables and state
variables are not variables in your dataset.
We specify the terms using a special type of substitutable expressions that we call scalar substitutable
expressions. In scalar substitutable expressions, parameters are enclosed in braces, {}, and may enter
the model either linearly or nonlinearly. The restriction that each term includes only one variable
implies that the variables must enter the equation linearly with these scalar substitutable expressions.
If our model for y in the equation above is
yt = Et (yt+1 ) + zt + xt
we could extend our previous dsge command as follows:

. dsge (y = {beta}*E(F.y) + {kappa}*z + {gamma}*x) ///
..., ...
However, the equation for y need not be linear in the parameters. If instead we wanted to model
y as
yt = (1/)Et (yt+1 ) + zt + (/)xt
we would change our command to

. dsge (y = 1/{beta}*E(F.y) + {kappa}*z + ({gamma}/{beta})*x) ///
..., ...
We can even include parameters on the left-hand side of the equation. For instance, we could
model y as
(1/)yt = Et (yt+1 ) + zt + (/)xt
and change our dsge command to

. dsge ((1/{beta})*y = E(F.y) + {kappa}*z + ({gamma}/{beta})*x) ///
..., ...
20 intro 2 Learning the syntax
Finally, there is an extension to the basic form of the equation that we should mention. Sometimes,
it is more convenient to write an equation so that a scalar value or a parameter or a nonlinear
combination of parameters is multiplied by a linear combination of terms. The dsge command allows
this. For instance, instead of
yt = (1/)Et (yt+1 ) + (/)xt + zt
we can write
yt = (1/){Et (yt+1 ) + xt } + zt
Similarly, we could write this equation in the dsge command as

(y = (1/{beta}) * (E(F.y) + {gamma}*x) + {kappa}*z)
Now you know the general syntax of dsge and are ready to fit your own DSGE model with dsge.
For examples of fitting classic DSGE models using this syntax, see [DSGE] intro 3. If you find
some of the rules of this syntax too restrictive for your modelsay you want a shock in an equation
for a control variable, or you need to include the lag of a state variablesee [DSGE] intro 4 for
examples of rewriting these types of models so that they can be fit using dsge.
For more details on the dsge syntax, see [DSGE] dsge.
Also see
Title
intro 3 Classic DSGE examples
Description
In this entry, we present three classic DSGE examplesthe New Keynesian model, the New
Classical model, and the financial frictions model.

In [DSGE] intro 3a, [DSGE] intro 3b, and [DSGE] intro 3c, we fit simple variants of common DSGE
models. Through these examples, we demonstrate model solution, estimation, and interpretation.
[DSGE] intro 3a demonstrates how to fit a New Keynesian model. In this example, we interpret
structural parameters, policy matrix parameters, and state transition matrix parameters. We also predict
values of both observed control variables and unobserved states.
[DSGE] intro 3b illustrates how to solve a New Classical model and plot the IRFs to compare the
models theoretical predictions under different sets of parameter values.
[DSGE] intro 3c fits a financial frictions model. In this example, we also estimate parameters of
the policy matrix and evaluate the IRFs.
Also see
21
Title
intro 3a New Keynesian model
Description
This introduction estimates and interprets the parameters of a simple New Keynesian model. In
this entry, we demonstrate how to constrain parameters in the model and how to interpret structural
parameters, policy matrix parameters, and state transition matrix parameters. We also predict values
of both observed control variables and unobserved states.

The model
Parameter estimation
One-step ahead predictions
Estimating an unobserved state
The model
Equations (1)(5) specify a canonical New Keynesian model of inflation pt , the output gap xt , and
the interest rate rt . These are the linearized equations; the models nonlinear equations are similar to
those in How to write down a DSGE in [DSGE] intro 1.
pt = Et (pt+1 ) + xt (1)
xt = Et (xt+1 ) {rt Et (pt+1 ) gt } (2)
rt = pt + ut (3)
ut+1 = u ut + t+1 (4)
gt+1 = g gt + t+1 (5)
Equation (1) specifies inflation as a linear combination of expected future inflation and the output gap.
Equation (2) specifies the output gap as a linear combination of the expected future output gap, the real
interest rate, and a state variable gt . Equation (3) specifies the interest rate as a linear combination
of inflation and a state variable ut . The state variables are modeled as first-order autoregressive
processes. The state variable ut is the deviation of rt from its equilibrium value of pt . The state
variable gt is also the deviation of xt from its equilibrium value.
Three of the parameters have structural interpretation. The parameter is known as the slope
of the Phillips curve and is predicted to be positive. The parameter is the discount factor that
represents the degree to which agents discount the future relative to the current period. The parameter
measures the degree to which interest rates react to movements in inflation.
22
intro 3a New Keynesian model 23
Not all model parameters are identified. We constrain to be 0.96, a common value in the
literature. The remaining parameters are identified.
. use http://www.stata-press.com/data/r15/usmacro2
. constraint 1 _b[beta]=0.96

> (x = E(F.x) -(r - E(f.p) - g), unobserved)
> (r = {psi}*p + u)
> (F.g = {rhog}*g, state),
> from(psi=1.5) constraint(1)
Iteration 6: log likelihood = -766.17852 (not concave)
DSGE model
( 1) [/structural]beta = .96
OIM
/structural
beta .96 (constrained)
kappa .0849633 .0287693 2.95 0.003 .0285764 .1413502
psi 1.943007 .2957876 6.57 0.000 1.363274 2.52274
rhou .7005481 .0452603 15.48 0.000 .6118395 .7892568
rhog .9545256 .0186424 51.20 0.000 .9179872 .991064
sd(e.u) 2.318207 .3047443 1.720919 2.915495

sd(e.g) .5689901 .0982974 .3763308 .7616494
The slope of the Phillips curve, kappa, is estimated to be positive. The coefficient on inflation in
the interest rate equation is estimated to be almost 1.94, meaning that interest rates are expected to
rise almost two for one with increases in inflation.
24 intro 3a New Keynesian model

Elements of the policy matrix represent the response of a control variable to a one-unit increase
in a state variable.
. estat policy
Policy matrix
Delta-method
p
u -.4172517 .039361 -10.60 0.000 -.4943978 -.3401056
g .9678148 .2777441 3.48 0.000 .4234464 1.512183
x
u -1.608212 .4050524 -3.97 0.000 -2.4021 -.8143236
g .9529166 .481391 1.98 0.048 .0094076 1.896426
r
u .1892771 .059166 3.20 0.001 .0733139 .3052402
g 1.880471 .2615988 7.19 0.000 1.367746 2.393195
An increase in u decreases the extent to which the inflation is above its short-run equilibrium
value. This change decreases the output gap and increases interest rate.
An increase in g increases the extent to which the inflation is above its short-run equilibrium value.
This change also increases output gap and interest rates.
Because the states are uncorrelated with each other in this example, the elements of the state
transition matrix are just the persistence parameters in the model.
. estat transition
Delta-method
F.u
u .7005481 .0452603 15.48 0.000 .6118395 .7892568
g 1.67e-16 9.03e-12 0.00 1.000 -1.77e-11 1.77e-11
F.g
u 0 (omitted)
g .9545256 .0186424 51.20 0.000 .9179872 .991064
See [DSGE] intro 4g for an example in which the state variables depend on each other. The states
can also depend on each other when a state variable is specified to depend on control variables, as in
[DSGE] intro 3b, or when the state vector includes lagged control variables, as in [DSGE] intro 4a.
One-step ahead predictions

Predictions after dsge depend on the estimated state-space parameters. Below we obtain one-step
ahead predictions for each of the two observed control variables in the model, and we graph the
actual and predicted inflation rate.
intro 3a New Keynesian model 25
. predict dep*
(option xb assumed; fitted values)
. tsline p dep1, legend(col(1))
15
10
5
0
1955q3 1970q3 1985q3 2000q3 2015q3

Date (quarters)
Growth rate of prices (GDPDEF)

xb prediction, p, onestep
The graph shows that the one-step ahead predictions closely follow realized inflation.
Estimating an unobserved state

The observed control variables are driven by two unobserved state variables. We can use predict
with the state option to estimate the state variables.
Here we estimate the unobserved state u and plot it.
. predict state1, state
. tsline state1
5
xb states prediction, u, onestep
5 10 0
1955q3 1970q3 1985q3 2000q3 2015q3

Date (quarters)
The loose monetary policy in the mid 1970s where predicted values of ut are negative and the
subsequent Volcker contraction are apparent in this plot.
26 intro 3a New Keynesian model
Also see
Title
intro 3b New Classical model
Description Remarks and examples Reference Also see
Description
In this example, we solve a New Classical model similar to the one in King and Rebelo (1999). We
also demonstrate how to compare a models theoretical predictions under different parameter values
using IRFs.

The model
Solving the model
Impulse responses
Sensitivity analysis
The model
In this model, output, consumption, investment, employment, and other variables are driven by state
variables linked to production and demand. The model is similar to the one in King and Rebelo (1999)
and is referred to as a real business cycle model.
The nonlinear form of the model is

1 1
= Et (1 + Rt+1 ) (1a)
Ct Ct+1
Wt
Ht = (2a)
Ct
Yt = Ct + Xt + Gt (3a)
Yt = Kt (Zt Ht )1 (4a)
Yt
Wt = (1 ) (5a)
Ht
Yt
Rt = (6a)
Kt
Kt+1 = (1 )Kt + Xt (7a)
The first equation specifies consumption Ct as a function of expected future consumption and the
expected future interest rate Et (Rt+1 ). Equation (2a) specifies labor hours Ht as a function of the wage
Wt and consumption; it is a labor supply equation. Equation (3a) is the national income accounting
identity for a closed economy, specifying output Yt as the sum of consumption, investment Xt , and
government spending Gt . Equation (4a) is a production function that specifies output as a function of
labor input Ht , capital input Kt , and productivity Zt . Equations (5a) and (6a) specify labor demand
and capital demand, respectively. Equation (7a) specifies the equation for capital accumulation. The
model is completed when we add state transition equations for Zt and Gt . These state transition
equations are conventionally specified after the model has been linearized.
27
28 intro 3b New Classical model
The linearized form of the model is
ct = Et (ct+1 ) (1 + )Et (rt+1 )

ht = wt ct
1 xt = yt 2 ct gt
yt = (1 )(zt + ht ) + kt
wt = yt ht
rt = yt kt
kt+1 = xt + (1 )kt
zt+1 = z zt + t+1
gt+1 = g gt + t+1
The model has six control variables and three state variables. Two of the state variables, zt+1 and
gt+1 , are modeled as first-order autoregressive processes. The state equation for kt+1 depends on the
current value of a control variable, namely, xt .
Solving the model

The solve option of dsge places the model in state-space form without estimating parameters; it
is similar to iterate(0) but is faster because it does not calculate standard errors. Using solve for
different parameter values of your model is a useful way to explore the models theoretical properties.
The parameter values used here are similar to those used in King and Rebelo (1999). Each has
an interpretation. (1 alpha) is labors share of national income. delta is the depreciation rate of
capital. eta is the slope of the labor supply curve. phi1 and phi2 are share parameters related to
investments share of national income and consumptions share of national income, respectively. rhoz
and rhog are autoregressive parameters on the state variables.
. dsge (c = E(F.c) - (1-{beta}+{beta}*{delta})*E(F.r), unobserved)
> ({eta}*h = w - c, unobserved)
> ({phi1}*x = y - {phi2}*c - g, unobserved)
> (y = (1-{alpha})*(z+h) + {alpha}*k)
> (w = y - h, unobserved)
> (r = y - k, unobserved)
> (F.k = {delta}*x+ (1-{delta})*k, state noshock)
> (F.z = {rhoz}*z, state)
> from(beta=0.96 eta=1 alpha=0.3 delta=0.025 phi1=0.2 phi2=0.6 rhoz=0.8
> rhog=0.3) solve noidencheck
intro 3b New Classical model 29
DSGE model
OIM
y Coef. Std. Err. z P>|z| [95% Conf. Interval]
/structural
beta .96 . . . . .
delta .025 . . . . .
eta 1 . . . . .
phi1 .2 . . . . .
phi2 .6 . . . . .
alpha .3 . . . . .
rhoz .8 . . . . .
rhog .3 . . . . .
sd(e.z) 1 . . .
sd(e.g) 1 . . .
Note: Skipped identification check.

Note: Model solved at specified parameters.
The solve option solves the model at the specified values in from(). We skip the identification
check with noidencheck. Simply solving the model does not involve any reference to the data or
any estimation. Still we can explore what these parameters imply.

After solving, we can use many of the postestimation commands, though standard errors will be
missing throughout.
The state transition matrix shows how the state vector in the next period is related to the state
vector in the current period. Some state variables are specified as first-order autoregressive processes,
and their transition equations will simply repeat information that is already available in the estimation
table. However, if any state variable equation contains control variables, then that state variables
transition equation will depend on the other state variables.
. estat transition
Delta-method
F.k
k .9256785 . . . . .
z .1078282 . . . . .
g -.1070547 . . . . .
F.z
k 0 (omitted)
z .8 . . . . .
g 2.22e-16 . . . . .
F.g
k 0 (omitted)
z 0 (omitted)
g .3 . . . . .
The value of the state variables z and g in the next period depends only on their value in the
current period, but the value of the capital stock k in the next period depends on the current value
of all three state variables. This feature means that, for example, a shock to the z state variable has
two effects: it increases future values of z, because z is autoregressive, but it also increases future
values of k. Interrelationships among the state variables can generate more interesting patterns in the
IRFs than the AR(1) dynamics that we saw in [DSGE] intro 3a.
Impulse responses
One way to compare two parameter sets is to graph the impulse response of model variables to a
shock under each parameter set. We first set the impulseresponse file with irf set and then add
impulse responses named persistent to the file with irf create.
. irf set rbcirf
(file rbcirf.irf created)
(file rbcirf.irf now active)
. irf create persistent

(file rbcirf.irf updated)
The response of model variables to a shock to z is graphed by typing

. irf graph irf, irf(persistent) impulse(z) response(y c x h w z)
> byopts(yrescale)
persistent, z, c persistent, z, h persistent, z, w

.3 .4 .6
.3 .5
.25 .2
.4
.1
.3
.2 0
.2
persistent, z, x persistent, z, y persistent, z, z

1 1
4
.8 .8
3
.6 .6
2
.4 .4
1
.2 .2
0
0 2 4 6 8 0 2 4 6 8 0 2 4 6 8
step
Each graph is labeled with the IRF name, the impulse variable, and the response variable. For
instance, the top-left graph shows the response of consumption to a shock to the zt state variable. The
bottom-right graph shows the response of the state variable zt itself. The state is persistent, which is
not surprising: we set the autoregressive parameter in the zt equation to 0.8.
In the top-left graph, we see that consumption c rises over time before returning to steady state. The
time unit is quarters, so a value of about 0.27, 4 periods after the shock, indicates that consumption
is 0.27% above its steady-state value one year after the shock. Hours worked h are shown in the
top center graph and rise initially before falling below steady state. The real wage w, output y, and
investment x all rise.
Sensitivity analysis
The responses of variables to a shock to z are persistent. Some variables, like consumption and
the wage, show dynamics beyond the simple autoregressive behavior of z itself. How much of this
behavior is due to the intrinsic propagation mechanisms of the model, and how much is due to the
persistence of z?
To evaluate this, we rerun the dsge command. This time we set the persistence of z to a small
value of 0.01.
. dsge (c = E(F.c) - E(F.r), unobserved)
> ({eta}*h = w - c, unobserved)
> ({phi1}*x = y - {phi2}*c - g, unobserved)
> (y = (1-{alpha})*(z+h) + {alpha}*k)
> (w = y - h, unobserved)
> (r = y - k, unobserved)
> (F.k = {delta}*x+ (1-{delta})*k, state noshock)
> from(eta=1 alpha=.3 delta=0.025 phi1=0.2 phi2=0.6 rhoz=0.01 rhog=0.3)
> solve noidencheck
DSGE model
OIM
y Coef. Std. Err. z P>|z| [95% Conf. Interval]
/structural
eta 1 . . . . .
phi1 .2 . . . . .
phi2 .6 . . . . .
alpha .3 . . . . .
delta .025 . . . . .
rhoz .01 . . . . .
rhog .3 . . . . .
sd(e.z) 1 . . .
sd(e.g) 1 . . .

The only change in the parameter set is that rhoz has been set to 0.01 from its earlier setting of
0.9. We can add the impulse responses of this model to the irf file with the name transitory,
. irf create transitory, replace
irfname transitory not found in rbcirf.irf
(file rbcirf.irf updated)
and graph them.

. irf graph irf, irf(transitory) impulse(z) response(y c x h w z)
> byopts(yrescale)
transitory, z, c transitory, z, h transitory, z, w

.4 .6
.2
.3 .1
.4
.2 0
.2
.1 .1
0 .2 0
transitory, z, x transitory, z, y transitory, z, z

3 1
.8
2 .6
1 .4 .5
0 .2
0
1
0
0 2 4 6 8 0 2 4 6 8 0 2 4 6 8
step
Model variables are much less persistent. All the impulse responses have changed shape. We can
use irf ograph to overlay the IRF for a variable under the two calibrations. This way we can view
the differences across calibrations directly.
. irf ograph (persistent z c irf) (transitory z c irf)
.4
.3
.2
.1
0
0 2 4 6 8
step
persistent: irf of z > c

transitory: irf of z > c
When the shock itself is persistent, consumption responds persistently. When the shock is transitory,
consumption returns to its steady-state value quickly.
Reference
King, R. G., and S. T. Rebelo. 1999. Resuscitating real business cycles. In Handbook of Macroeconomics: Volume
1A, ed. J. B. Taylor and M. Woodford, 9271007. New York: Elsevier.
Also see
Title
intro 3c Financial frictions model
Description
This introduction estimates and interprets the parameters of a model that incorporates financial
frictions. The model is an extension of the one in [DSGE] intro 3a.

The model
Impulse responses
The model
Equations (1)(7) specify a model of financial frictions. The simplest such model places a wedge
between two interest rates: the safe interest rate set by the central bank and the market interest rate
that consumers and producers use.
t = Et t+1 + xt (1)
xt = Et xt+1 (it Et t+1 gt ) (2)
rt = t + ut (3)
it = rt + et (4)
gt+1 = g gt + t+1 (5)
ut+1 = u ut + t+1 (6)
et+1 = e et + t+1 (7)
These are the linearized equations. The nonlinear equations are similar to those in How to write down
a DSGE in [DSGE] intro 1.
This model is an extension to the one worked in [DSGE] intro 1. It has an additional equation for
the interest rate spread. As before, (1) specifies the inflation equation (a Phillips curve), (2) specifies
the output gap equation (an Euler equation), and (3) specifies the equation for the safe interest rate (a
Taylor rule). Equation (3) can be thought of as an equation that specifies the safe interest rate rt . The
new element is (4), which specifies an equation for the market interest rate it that enters the output
gap equation. The market interest rate it is a function of the safe interest rate and a state variable et .
The state variable et controls the interest rate spread and can be thought of as representing the state
of the financial system. A large realization of et represents a large interest rate spread, indicating
financial distress.
34
intro 3c Financial frictions model 35
We estimate the parameters of the model in (1)(7) using U.S. data on the Federal funds rate (safe
rate set by the central bank), the high-grade corporate bond interest rate (a measure of market interest
rates), and the inflation rate.
As is typical in these models, we constrain the parameter beta to 0.96.
. constraint 1 _b[beta]=0.96
> (x = E(F.x) -(i - E(f.p) - g), unobserved)
> (i = {chi}*r + e)
> (r = {psi}*p + u)
> (F.e = {rhoe}*e, state)
> (F.g = {rhoz}*g, state),
> from(psi=2 chi=0.8) constraint(1)
36 intro 3c Financial frictions model
DSGE model
OIM
/structural
kappa .0503238 .0184922 2.72 0.007 .0140796 .0865679
chi .9067346 .1249748 7.26 0.000 .6617885 1.151681
psi 6.332415 2.56766 2.47 0.014 1.299893 11.36494
rhoe .8478235 .0301215 28.15 0.000 .7887865 .9068605
rhou .8153493 .0332519 24.52 0.000 .7501768 .8805218
rhoz .9861866 .0099767 98.85 0.000 .9666327 1.005741
sd(e.e) .8857022 .1343 .622479 1.148925

sd(e.u) 7.16076 2.933846 1.410528 12.91099
sd(e.g) .3911864 .0358355 .3209501 .4614226
The persistence of the financial shock rhoe is estimated to be 0.85. The slope of the Phillips
curve, kappa, is somewhat flatter in this model than in the one in [DSGE] intro 1. The coefficient
on inflation in the interest rate equation is 6.3, and indicates that the central bank increases interest
rates much more than one for one in response to movements in inflation.

We can read off the impact effect of shocks using the policy matrix. The response to the financial
shock e will be of most interest.
. estat policy
Policy matrix
Delta-method
p
e -.1832602 .0740955 -2.47 0.013 -.3284848 -.0380357
u -.1584184 .0641356 -2.47 0.014 -.2841219 -.032715
g .2096325 .1003838 2.09 0.037 .0128838 .4063812
x
e -.6776679 .3498926 -1.94 0.053 -1.363445 .0081091
u -.6839457 .2783497 -2.46 0.014 -1.229501 -.1383903
g .2218675 .1284759 1.73 0.084 -.0299406 .4736756
i
e -.0522472 .040321 -1.30 0.195 -.1312748 .0267805
u -.0028753 .0041244 -0.70 0.486 -.0109589 .0052083
g 1.203672 .0979354 12.29 0.000 1.011722 1.395622
r
e -1.16048 .1361163 -8.53 0.000 -1.427263 -.8936968
u -.003171 .0045309 -0.70 0.484 -.0120515 .0057094
g 1.32748 .2224889 5.97 0.000 .8914098 1.76355
intro 3c Financial frictions model 37
Importantly, the financial shock causes r to fall by more than i, indicating an increase in the
interest rate spread. Inflation also falls, as does the output gap. It is the fall in the inflation rate that
causes the central banks interest rate to fall on impact of the shock.
Impulse responses
We use the irf set command to set finirf.irf as the active IRF file and then use irf create
to create and store the impulse responses under the name param.
. irf set finirf
(file finirf.irf created)
(file finirf.irf now active)
. irf create param1

(file finirf.irf updated)
Finally, we graph the impulse response to a financial shock.

. irf graph irf, irf(param1) impulse(e) response(e x p i r) byopts(yrescale)
param1, e, e param1, e, i param1, e, p

1.5 .05 0
0
1 .1
.05
.5 .2
.1
0 .15 .3
0 2 4 6 8
param1, e, r param1, e, x
0 0
.5 .5
1 1
0 2 4 6 8 0 2 4 6 8
step
The top middle panel shows the effect of the shock to the e equation on e itself. The shock
represents a persistent increase in the spread between the two interest rates. The increased interest rate
spread causes both the output gap and inflation rate to fall. The safe interest rate r falls dramatically,
and the combined effect of the increased spread e and reduced safe interest rate r causes the market
interest rate i to be little changed.
38 intro 3c Financial frictions model
Also see
Title
intro 4 Writing a DSGE in a solvable form
Description
Many DSGE models, when written in their natural representation based on economic theory, include
problematic terms that do not fit into the algebraic form required to solve a DSGE model. This entry
shows how to accommodate these problematic terms by defining a new state variable or a new control
variable and then rewriting the equations so that they have the required form.
We assume that you are already familiar with the elements of a DSGE; see [DSGE] intro 1.

Introduction
Shocks to a control equation
Including a lag of a control variable
Including a lag of a state variable
Including an expectation of a control dated by more than one period ahead
Including a second-order lag of a control variable
Including an observed exogenous variable
Introduction
We can estimate the parameters of a DSGE model only if we can solve for its state-space form,
and we can solve a DSGE model for its state-space form only if the equations for control variables
and state variables have specific structures. Unfortunately, many DSGE models contain problematic
terms that do not fit into these structures without some manipulation. Fortunately, the manipulation
is easy, once you know the tricks discussed below.
As we discussed in Structural and reduced forms of DSGE models in [DSGE] intro 1, we can only
solve a DSGE whose structural form can be written as
A0 yt = A1 Et (yt+1 ) + A2 yt + A3 xt (1)
shocks. A0 through A3 and B0 through B3 are matrices of parameters. The entries in Ai and Bj
are all functions of the structural parameters, which we denote by vector . A0 and B0 are diagonal
matrices, and A2 has zeros on its diagonal. C is a selection matrix that determines which state
variables are subject to shocks.
Equation (1) specifies the required structure for control variable equations, and (2) specifies the
required structure for state variable equations. Per (1), a control can depend on only three types
of termsexpected values of control variables in the next period, current values of other control
variables, and current values of state variables. All other terms are problematic. Per (2), the values of
state variables in the next period can depend on only four types of termsexpected values of control
variables in the next period, current values of control variables, current values of state variables, and
shocks. All other terms are problematic.
39
40 intro 4 Writing a DSGE in a solvable form
When written based on economic theory, the structural forms of many DSGE models include
problematic terms that do not fit into the required form. These problematic terms are handled by
defining a new state variable or a new control variable and rewriting the equations to have the required
form. We define a new state variable when the problematic term involves an exogenous, also known
as a predetermined, variable. We define a new control variable when the problematic term includes
a new endogenous variable. The rewriting process replaces the variable in the problematic term with
the new state or the new control variable.
Problematic terms usually violate one of the following four restrictions:
1. Equations for control variables may not contain shocks.

2. Equations may not contain lagged control variables.
3. Equations may not contain lagged state variables.
4. Equations may not contain expectations of control variables dated by more than one period
ahead.
All of these restrictions can be overcome by adding a new state variable or a new control variable
and rewriting the model so that it has the required form. Here are five examples of the four restrictions
and solutions to each one.
1. If you have an equation for a control variable that contains a shock, make this shock a new
state variable; see Shocks to a control equation below.
2. If you have an equation for a control variable that contains a lagged control variable, make
the lagged control variable a new state variable; see Including a lag of a control variable
below.
3. If you have an equation for a state variable that contains a lag of a state variable, make the
lagged state variable a new state variable; see Including a lag of a state variable below.
4. If you have an equation for a control variable that contains an expectation of a control
variable dated by more than one period ahead, make it a new control variable. We use a
new control variable instead of a new state variable because the far-future expectation is
endogenous, not predetermined. See Including an expectation of a control dated by more
than one period ahead below for details.
5. If you have an equation for a control variable that contains a second-order lag of a control
variable, make the second-order lagged control variable a new state variable. You will also
need to create a new state variable for the first-order lag. See Including a second-order lag
of a control variable below for details.
A nuanced issue not covered by restrictions 14 is that all the observed variables in a DSGE must
be endogenous control variables. This structure is usually not a problem, because DSGE models are
usually generated by theories that treat everything observed as endogenous. Occasionally, however,
models include an observed variable that is exogenous. For example, a small, open economy might
not be able to affect its real exchange rate. The solution for this problem is to model the observed
variable as a control variable that is equal to an exogenous state variable. See Including an observed
exogenous variable below for details.
For the most part, this entry discusses how to make models with problematic terms fit into
the structure of (1) and (2). Note that correlated state variables already fit into this structure. This
correlation is modeled by off-diagonal elements in B3 , not by elements in C. See [DSGE] intro 4g
for an example.
intro 4 Writing a DSGE in a solvable form 41
Shocks to a control equation

Sometimes, we only observe a control variable with error. Suppose that the equation for the control
variable consumption is
ct = Et ct+1 rt + t (3)
where ct is consumption, rt is the interest rate, and t is a shock. Shocks to a control variable are
not allowed, so t is a problematic term.
The solution entails three steps. First, we define a new state variable ut = t . We define a new
state instead of a new control because the shock is exogenous. Second, we write a state equation for
ut . Recall that the state equation specifies tomorrows state value in terms of todays state value and
tomorrows shock. The state equation for ut is
ut+1 = t+1
Third, we rewrite (3), substituting ut for t . The consumption equation becomes

ct = Et ct+1 rt + ut
which is in the required form. In the dsge command, this is expressed as
. dsge ... ///
(c = E(F.c) - r + u) ///
(F.u = , state) ///
...
See [DSGE] intro 4a for a complete example.
Including a lag of a control variable

Models that mix adaptive and rational expectations contain a lag of the control variable and the
expected future value of the control variable. Suppose our model had the following equation for the
control variable yt ,
yt = yt1 + (1 )Et (yt+1 ) rt (4)
where yt is output and rt is the interest rate. The problematic term is yt1 because it involves a
lag of the control variable.
The solution entails three steps. First, we define a new state variable Lyt = yt1 . We define a new
state instead of new control because the lagged control is predetermined, which makes it exogenous.
Second, we write its state equation as
Lyt+1 = yt
Third, we rewrite (4), substituting Lyt for yt1 ,

yt = Lyt + (1 )Et (yt+1 ) rt
which is in the required form. To do this using the dsge command, we would type
. dsge ... ///
(y = {alpha}*Ly + (1-{alpha})*E(F.y) - r) ///
(F.Ly = y, state noshock) ///
...
See [DSGE] intro 4b for a complete example.

Including a lag of a state variable

Lagged state variables are handled in a manner analogous to lagged control variables. Suppose we
have an equation for a state variable zt that contains a lag of itself. For example, this occurs when zt
is a second-order autoregressive process instead of a first-order autoregressive process. Specifically,
our model contains the equation
zt+1 = 1 zt + 2 zt1 + t+1 (5)
The problematic term is 2 zt1 because it violates the form required by (2).
The solution entails three steps. First, we define a new state variable Lzt = zt1 . We define a
new state instead of new control because the lagged state is exogenous. Second, we write its state
equation as
Lzt+1 = zt
Third, we rewrite (5), substituting Lzt for zt1 ,

zt+1 = 1 zt + 2 Lzt + t+1
which is in the required form. In the dsge command, we would type
. dsge ... ///
(F.z = {rho1}*z + {rho2}*Lz, state) ///
(F.Lz = z, state noshock) ///
...
See [DSGE] intro 4c for a complete example.
Including an expectation of a control dated by more than one period ahead

Many models include an expectation of a control variable dated by more than one period ahead.
For example, suppose that the equation for the control variable consumption growth is
ct = (1 h)wt + hEt ct+2 + rt (6)
where ct is consumption growth, wt is wage growth, and rt is the interest rate. Only expectations of
control variables dated one period ahead are allowed, so Et (ct+2 ) is a problematic term.
The solution entails three steps. First, we define a new control variable F ct = Et (ct+1 ). The
new variable is a control instead of a state because Et (ct+1 ) is endogenous instead of exogenous.
Second, we include this new control equation,
F ct = Et (ct+1 )
Third, we substitute F ct+1 for ct+2 in (6). The equation becomes
ct = (1 h)wt + hEt (F ct+1 ) + rt
which is in the required form. Using the dsge command, we would type
. dsge ... ///
(c = (1-{h})*w + {h}*E(F.Fc) + r) ///
(Fc = E(F.c), unobserved) ///
...
See [DSGE] intro 4d for a complete example.

intro 4 Writing a DSGE in a solvable form 43
Including a second-order lag of a control variable

Higher-order lags require defining more than one new state variable, as we illustrate here.
Equations for the state variable capital frequently involve time to build, so that the control
variable investment made in time t becomes available only as new capital several periods in the future.
Suppose it takes three periods to convert investment xt into capital. We denote current capital by kt
and write this three-period relationship as
kt+1 = (1 )kt + xt2 (7)
which specifies that the capital stock tomorrow is the sum of investment made two periods ago and
the capital today that has not depreciated at rate .
The term xt2 is problematic because it is a second-order lag of the state variable xt , which
violates the form required by (2).
The solution entails three steps. First, we define the new state variables L2xt = xt2 and
Lxt = xt1 . Second, we write their state equations as
L2xt+1 = Lxt
and
Lxt+1 = xt
Third, we rewrite (7) as
kt+1 = (1 )kt + L2xt
With the dsge command, we could fit this model as
. dsge ... ///
(F.k = (1-{delta})*k + L2x) ///
(F.L2x = Lx, state noshock) ///
(F.Lx = x, state noshock) ///
...
For a complete example, see [DSGE] intro 4e.

In general, if the lag t k appears anywhere in any of your equations, you will need k new state
equations.
Including an observed exogenous variable

Sometimes, we want to treat an observed variable as exogenous. Suppose we want to treat the
changes in an observed exchange rate et as exogenous. The problem is that all the observed variables
in a DSGE model must be modeled as endogenous control variables. The solution is to define a control
variable that is equal to a state variable that models the exogenous process. So we could set the
observed exchange rate et to equal the exchange rate state, est ,
et = est
and specify, say, a first-order autoregressive process for est ,
est+1 = e est + t+1
To fit this type of model using the dsge command, we would type
. dsge ... ///
(e = es) ///
(F.es = {rhoe}*es, state) ///
...
See [DSGE] intro 4f for a complete example.

Also see
Title
intro 4a Specifying a shock on a control variable
Description
A shock to a control variable is problematic because it does not fit into the form of a structural
model that is required to solve for the state-space form. This entry shows how to solve this problem
by defining a new state variable and rewriting the equations.

The model
The model
Equations (1)(3) specify a model of consumption growth and growth in hours worked. The
equation for consumption growth is subject to an additive shock.
ct = (1 h)wt + hEt ct+1 + t (1)

nt = wt ct (2)
wt+1 = wt + t+1 (3)
Equation (1) specifies that consumption growth ct is a linear combination of wage growth wt ,
expected future consumption growth Et ct+1 , and a consumption shock t . Equation (2) specifies that
the growth rate of hours worked nt depends on wage growth and consumption growth. Equation (3)
specifies an autoregressive process for wage growth. The structural parameter h controls the weights
on wage growth and expected future consumption growth in the consumption equation.
One cannot solve the model in (1)(3) for the state-space form because the shock t is added
to the control equation ct in (1). Shocks to control variables must be reparameterized as new state
variables so that the model can be solved for its state-space form. We define the shocks as new state
variables instead of new control variables because the shocks are exogenous. We rewrite the model
in (1)(3) so that the additive shock is a new state variable in (4)(7). Specifically, we define a new
state variable zt such that zt = t and write the model as
ct = (1 h)wt + hEt ct+1 + zt (4)

nt = wt ct (5)
wt+1 = wt + t+1 (6)
zt+1 = t+1 (7)
That we replaced t with the new state variable zt is the only difference between (4) and (1).
Equations (5) and (6) are identical to their counterparts, (2) and (3). Equation (7) specifies the
evolution of the new state variable zt ; it is similar to (6), except that it lacks an autoregressive
component.
45
46 intro 4a Specifying a shock on a control variable
We estimate the parameters of the model in (4)(7) using U.S. data on consumption growth and
growth in hours worked. We specify w and z as state variables. We specify c and n as control
variables. Both c and n are treated as observed. We specify that w and z are subject to shocks.
. dsge (c = (1-{h})*(w) + {h}*E(F.c) + z)
> (n = w - {gamma}*c)
> (F.w = {rho}*w, state)
> (F.z = , state)
DSGE model
OIM
/structural
h .7642505 .0360272 21.21 0.000 .6936386 .8348625
gamma .2726181 .1059468 2.57 0.010 .0649663 .4802699
rho .6545212 .0485627 13.48 0.000 .55934 .7497023
sd(e.w) 2.904762 .1958651 2.520873 3.28865

sd(e.z) 2.077219 .1400659 1.802695 2.351743
The estimate of h is 0.76. Being greater than 0.5, this indicates that consumption growth depends
more strongly on expected future consumption growth than on current wage growth.
Also see
Title
intro 4b Including a lag of a control variable
Description
dsge does not allow lags of control variables to be included in the model. The structural form of
the model that is required so that the model can be solved for its state-space form does not include
lags of control variables. This entry shows how to fit a DSGE model that involves a lag of control
variables by defining a new state variable and rewriting the equations.

A model with a lagged endogenous variable
A model with a lagged endogenous variable

The model in (1)(5) is similar to many in monetary economics in that it includes inertia in the
interest rate because the interest rate depends on its lagged value.
pt = Et pt+1 + yt (1)
yt = Et yt+1 (rt Et pt+1 z zt ) (2)

1
rt = r rt1 + (1 r ) pt + ut (3)

zt+1 = z zt + t+1 (4)
ut+1 = u ut + t+1 (5)
Equation (1) specifies the structural equation for inflation pt . Inflation is a linear combination
of expected future inflation Et (pt+1 ) and output growth yt . Equation (2) specifies the structural
equation for output growth. Output growth is a linear combination of expected future output growth
Et (yt+1 ), the interest rate rt , expected future inflation, and the state zt . The state zt is the first-order
autoregressive process that drives output growth. Equation (3) specifies the structural equation for the
interest rate. The interest rate depends on its own lagged value, the inflation rate, and the state ut .
The state ut is the first-order autoregressive process that drives the interest rate. The control variables
in this model are pt , yt , and rt . The state variables are ut and zt .
The term involving rt1 in (3) is problematic because the lag of a control variable does not fit
into the structure required to solve the model for the state-space form. We accommodate this term
by defining a new state variable Lrt that equals rt1 and replacing rt1 in (3) with this new state
variable. We define a new state instead of new control because the lagged control is predetermined,
which makes it exogenous. These changes yield the model in (6)(11).
47
48 intro 4b Including a lag of a control variable
pt = Et pt+1 + yt (6)
yt = Et yt+1 (rt Et pt+1 z zt ) (7)

1
rt = r Lrt + (1 r ) pt + ut (8)

Lrt+1 = rt (9)
zt+1 = z zt + t+1 (10)
ut+1 = u ut + t+1 (11)
We will estimate the parameters in the model in (6)(11) using data on U.S. interest rate r and
inflation p.
. dsge (p = {beta}*E(F.p) + {kappa}*y)
> (y = E(F.y) -(r - E(f.p) - {rhoz}*z), unobserved)
> (r = {rhor}*lr + (1-{rhor})*((1/{beta})*p + u))
> (F.lr = r, state noshock)
intro 4b Including a lag of a control variable 49
DSGE model
OIM
/structural
beta .5026369 .0791865 6.35 0.000 .3474342 .6578397
kappa .1760193 .0511049 3.44 0.001 .0758554 .2761831
rhoz .9591408 .0181341 52.89 0.000 .9235986 .994683
rhor -.0939028 .0939661 -1.00 0.318 -.278073 .0902673
rhou .7094625 .0447807 15.84 0.000 .6216939 .7972311
sd(e.u) 2.324311 .3236236 1.69002 2.958602

sd(e.z) .6111539 .1084981 .3985014 .8238063
Ironically, the inertia term for the interest rate rhor was not needed in this model. The autoregressive
state ut is probably modeling both the inertia and the persistence in the shocks to the interest rate.
Also see
Title
intro 4c Including a lag of a state variable
Description
Lags of state variables are not allowed to be included in the structural model specified in the
dsge command. These lags do not fit into the form of a structural model that can be solved for the
state-space form. However, this restriction does not prevent us from fitting models that involve the
lag of a state variable. This entry shows how to define a new state variable and rewrite the equations
in the required structural form.

A model with a lagged state variable
A model with a lagged state variable

The model in (1)(5) is a standard monetary model in which the state driving output growth yt
is a second-order autoregressive AR(2) process instead of a first-order autoregressive process.
pt = Et (pt+1 ) + yt (1)
yt = Et (yt+1 ) {rt Et (pt+1 ) zt } (2)
1
rt = pt + ut (3)

zt+1 = z1 zt + z2 zt1 + t+1 (4)
ut+1 = u ut + t+1 (5)
Equation (1) specifies the structural equation for inflation pt . Inflation is a linear combination of
expected future inflation Et (pt+1 ) and output growth yt . Equation (2) specifies the structural equation
for output growth. Output growth is a linear combination of expected future output growth Et (yt+1 ),
the interest rate rt , expected future inflation, and the state zt . The state zt is an AR(2) process that
drives output growth. Equation (3) specifies the structural equation for the interest rate. The interest
rate depends on the inflation rate and the state ut . The state ut is an AR(1) process that drives the
interest rate. The control variables in this model are pt , yt , and rt . The state variables are ut and zt .
The AR(2) term in (4) is a problematic term because a lag of a state variable does not fit into the
structure required to solve the model for the state-space form. We accommodate this term by defining
a new state variable Lzt that equals zt1 and replacing zt1 in (4) with this new state variable. We
define a new state instead of new control because the lagged state is exogenous. These changes yield
the model in (6)(11).
50
intro 4c Including a lag of a state variable 51
pt = Et (pt+1 ) + yt (6)
yt = Et (yt+1 ) {rt Et (pt+1 ) zt } (7)
1
rt = pt + ut (8)

zt+1 = z1 zt + z2 Lzt + t+1 (9)
ut+1 = u ut + t+1 (10)
Lzt+1 = zt (11)
We will estimate the parameters in the model in (6)(11).
We estimate the parameters in the model in (6)(11) using data on U.S. interest rate r and
inflation p.
> (y = E(F.y) -(r - E(f.p) - z), unobserved)
> (r = (1/{beta})*p + u)
> (F.z = {rhoz1}*z + {rhoz2}*Lz, state)
> (F.Lz = z, state noshock)
52 intro 4c Including a lag of a state variable
DSGE model
OIM
/structural
beta .5154874 .0770496 6.69 0.000 .364473 .6665019
kappa .1662425 .0468473 3.55 0.000 .0744234 .2580616
rhou .6979877 .0452071 15.44 0.000 .6093834 .7865921
rhoz1 .6735462 .2666028 2.53 0.012 .1510143 1.196078
rhoz2 .2710021 .2564596 1.06 0.291 -.2316495 .7736537
sd(e.u) 2.315262 .2992183 1.728805 2.901719

sd(e.z) .7720678 .1716214 .4356961 1.10844
Looking at the confidence interval on rhoz2, we find no evidence that the AR(2) term is needed
in this model.
Also see
Title
intro 4d Including an expectation of a control dated by more than one period ahead
Description
Expectations of control variables dated by more than one period ahead are not allowed in the
equations specified in dsge because they do not appear in the required form of a structural model that
can be solved for the state-space form. In this entry, we demonstrate how to fit models with these
types of expectations by defining a new variable and rewriting our equations. Because the expectation
dated more than one period ahead is endogenous, the new variable that we define is a new control
variable.

The model
The model
Equations (1)(4) specify a model of consumption growth and growth in hours worked.
ct = (1 h)wt + hEt ct+2 + rt (1)

nt = wt ct (2)
wt+1 = w wt + t+1 (3)
rt+1 = r rt + t+1 (4)
Equation (1) specifies that consumption growth ct is a linear combination of wage growth wt , the
expected value of consumption growth two periods ahead Et ct+2 , and the interest rate rt . Equation (2)
specifies that the growth rate of hours worked nt depends on wage growth and consumption growth.
Equations (3) and (4) specify a first-order autoregressive process for wage growth and for the interest
rate, respectively. The control variables are ct and nt , and the state variables are wt and rt .
The expected value of consumption growth two periods ahead in (1) is a problematic term because
it does not fit into the structure required to solve for the state-space form. The structure requires
that expectations be only of one-period ahead values. We accommodate this term by defining a new
control variable F ct that equals Et (ct+1 ) and replacing ct+1 in (1) with this new control variable. We
define a new control variable instead of a new state variable because the expected future consumption
is endogenous instead of exogenous. These changes yield the model in (5)(9).
53
54 intro 4d Including an expectation of a control dated by more than one period ahead
ct = (1 h)wt + hEt (F ct+1 ) + rt (5)

nt = wt ct (6)
F ct = Et (ct+1 ) (7)
wt+1 = w wt + t+1 (8)
rt+1 = r rt + t+1 (9)
The new (7) defines the new control variable F ct as the expected value of next periods consumption.
Equation (5) is (1) but with ct+2 replaced with the new control F ct+1 .
There are now three control variables and only two shocks, so we treat the new control variable F ct
as unobserved. There is a logic to this process. The one-step ahead structure of state-space models
makes it impossible to solve for terms like Et (ct+2 ). In contrast, it is possible to solve for terms
like Et {Et (ct+1 )} as long as the unobserved Et (ct+1 ) is determined by another equation, which in
this case is (7). In fact, this example illustrates a part of the recursive dynamic modeling approach
at the heart of the DSGE approach to macroeconomics.
We estimate the parameters of the model in (5)(9) using U.S. data on consumption growth and
growth in hours worked.
. dsge (c = (1-{h})*(w) + {h}*E(F.fc) + r)
> (n = w - {gamma}*c)
> (fc = E(F.c), unobserved)
> (F.w = {rho_w}*w, state)
> (F.r = {rho_r}*r, state)
intro 4d Including an expectation of a control dated by more than one period ahead 55
DSGE model
OIM
/structural
h .661791 .0427917 15.47 0.000 .5779208 .7456611
gamma .273381 .1120734 2.44 0.015 .0537212 .4930408
rho_w .6545228 .0485628 13.48 0.000 .5593415 .7497041
rho_r .1050387 .0638171 1.65 0.100 -.0200406 .230118
sd(e.w) 2.905807 .2023204 2.509267 3.302348

sd(e.r) 2.049915 .1437862 1.7681 2.331731
The estimate of h is greater than 0.5, so we conclude that slightly more of the unobserved wage state
is allocated to the expected growth in consumption two periods ahead than to current consumption.
Also see
Title
intro 4e Including a second-order lag of a control
Description
Some DSGE models capture delayed effects by including a second-order lag of a control variable
and excluding the first-order lag. The second-order lag is a problematic term that does not fit into
the form required to solve a structural model for its state-space form. This entry shows how to solve
this problem by defining new state variables and rewriting the equations.

The model
The model
Consider a model in which changes in hours worked take two periods to adjust because next
periods hours have already been budgeted. In this model, the second-order lag of changes in hours
worked is included and the first-order lag is excluded. Equations (1)(4) specify such a model of
growth in hours worked and of consumption growth.
nt = b1 nt2 + wt ct (1)
ct = (1 h)wt + hEt ct+1 + rt (2)
wt+1 = wt + t+1 (3)
rt+1 = t+1 (4)
Equation (1) specifies that the growth rate of hours worked nt depends on a second-order lag of
itself, wage growth wt , and consumption growth ct . Equation (2) specifies that consumption growth
is a linear combination of wage growth, expected future consumption growth Et ct+1 , and the interest
rate rt . Equation (3) specifies an autoregressive process for wage growth. Equation (4) specifies that
interest rate is just a shock. The control variables are nt and ct . The state variables are wt and rt .
One cannot solve the model in (1)(4) for the state-space form because the problematic term b1 nt2
does not fit into the required form. To accommodate this term, we define two new state variables, one
for nt1 and one for nt2 . We define new state variables instead of new control variables because
lags of the control are predetermined and thus exogenous. The model with new state variables is
56
intro 4e Including a second-order lag of a control 57
nt = b1 L2nt + wt ct (5)
ct = (1 h)wt + hEt ct+1 + rt (6)
wt+1 = wt + t+1 (7)
rt+1 = t+1 (8)
Lnt+1 = nt (9)
L2nt+1 = Lnt (10)
Equation (9) defines the new state for nt1 , and (10) defines L2nt to be the new state for nt2 .
The L2nt in (5) replaces nt2 in (1).
We specify n and t as observed control equations. We specify w, r, Ln, and L2n as state equations.
We specify that w and r are subject to shocks; the new states to accommodate nt2 are not subject
to shocks.
. dsge (n = {b1}*L2n + w - {gamma}*c)
> (c = (1-{h})*w + {h}*E(F.c) + r)
> (F.w = {rho}*w, state)
> (F.r = , state)
> (F.L2n = Ln, state noshock)
> (F.Ln = n, state noshock)
DSGE model
OIM
/structural
b1 .1320845 .0608727 2.17 0.030 .0127763 .2513927
gamma .360925 .1298387 2.78 0.005 .1064457 .6154042
h .7238121 .0406724 17.80 0.000 .6440957 .8035285
rho .6177969 .0533568 11.58 0.000 .5132195 .7223743
sd(e.w) 3.033799 .2423856 2.558732 3.508866

sd(e.r) 1.970288 .1574527 1.661687 2.27889
58 intro 4e Including a second-order lag of a control
Looking at the confidence interval for b1, we conclude that the second-order lag of hours growth
impacts current hours growth.
Also see
Title
intro 4f Including an observed exogenous variable
Description
All the observed variables in a DSGE model must be modeled as endogenous control variables.
This requirement implies that there is no reduced form for the endogenous variables as a function of
observed exogenous variables. Any variable that is theoretically exogenous must be modeled.
Mechanically, the solution is to define a control variable that is equal to a state variable that
models the exogenous process. We clarify this issue by discussing a model in which the price of oil
is theoretically exogenous.

The model
The model
We model an economy in which the growth rate of the trade-weighted exchange rate is exogenous
and in which it affects inflation. Henceforth, we call the trade-weighted exchange rate just the exchange
rate. We begin by adding the growth rate of the exchange rate to the inflation equation in the New
Keynesian model of [DSGE] intro 1.
xt = Et (xt+1 ) {rt Et (pt+1 ) gt } (1)

1
rt = pt + ut (2)

pt = Et (pt+1 ) + xt + et (3)
ut+1 = u ut + t+1 (4)
gt+1 = g gt + t+1 (5)
xt is the output gap, which is modeled as an unobserved control variable. rt is the interest rate, which
is modeled as an observed control variable. pt is the inflation rate, which is modeled as an observed
control variable. gt and ut are first-order autoregressive state variables. et is the growth rate of the
exchange rate, which we want to model as an observed exogenous variable.
The model in (1)(5) cannot be solved because there is no equation for how the observed et
evolves over time. A first-order autoregressive process [AR(1)] is a standard approximation to an
exogenous variable, like the growth in exchange rate. We complete the model by adding an AR(1)
for the unobserved state variable est and an equation linking the unobserved est to the observed et .
59
60 intro 4f Including an observed exogenous variable
xt = Et (xt+1 ) {rt Et (pt+1 ) gt } (6)

1
rt = pt + ut (7)

pt = Et (pt+1 ) + xt + est (8)
et = est (8a)
ut+1 = u ut + t+1 (9)
gt+1 = g gt + t+1 (10)
est+1 = e est + t+1 (11)
New (8a) specifies how the unobserved state est is transformed into the observed control variable
et . Equation (11) specifies that the unobserved state est is an AR(1) process. Other equations are
unchanged.
We fit this model using data on the U.S. interest rate r, inflation rate p, and the growth rate of
the exchange rate e. The equation (e = es) links the observed variable e to the unobserved state
variable es.
. dsge (x = E(F.x) -(r - E(f.p) - g), unobserved)
> (r = 1/{beta}*p + u)
> (p = {beta}*E(F.p) + {kappa}*x + {psi}*es)
> (e = es)
> (F.u = {rho_u}*u, state)
> (F.g = {rho_g}*g, state)
> (F.es = {rho_e}*es, state)
intro 4f Including an observed exogenous variable 61
DSGE model
OIM
/structural
beta .5106735 .1022178 5.00 0.000 .3103303 .7110167
kappa .0901563 .0334284 2.70 0.007 .0246378 .1556747
psi .01373 .0037684 3.64 0.000 .006344 .021116
rho_u .7789341 .047395 16.43 0.000 .6860415 .8718266
rho_g .9597842 .0206193 46.55 0.000 .9193712 1.000197
rho_e .2372214 .0742225 3.20 0.001 .0917479 .3826948
sd(e.u) 2.206087 .3466166 1.526731 2.885443

sd(e.g) .703933 .1545767 .4009683 1.006898
sd(e.es) 10.48932 .5671986 9.37763 11.60101
From the estimates for psi, it appears that the growth of the exchange rate impacts inflation.
Also see
Title
intro 4g Correlated state variables
Description
Many models include correlated state variables. We illustrate how to specify correlated state
variables in a model of output growth yt and inflation pt .

The model
The model
We model output growth yt and inflation pt as functions of domestic and international factors. The
domestic factor gt that drives output growth is a first-order autoregressive process that is also affected
by the international factor zt . The international factor that drives inflation is a simple first-order
autoregressive process.
Mathematically, the model is
yt = Et yt+1 + pt + gt (1)
pt = zt (2)
gt+1 = g gt + gz zt + t+1 (3)
zt+1 = z zt + t+1 (4)
Equation (1) specifies that output growth depends on expected future output growth Et yt+1 , inflation
pt , and the domestic factor gt . This equation has the form of an aggregate demand curve, and
the parameter is referred to as the slope of the aggregate demand curve. It should be negative.
Equation (2) specifies that inflation is entirely driven by the international factor zt . Equations (3)
and (4) specify that the factors gt and zt follow a first-order vector autoregressive process with
parameters g , gz and z and with shocks t+1 and t+1 . The factors zt and gt are the state
variables, and pt and yt are the observed control variables.
62
intro 4g Correlated state variables 63
From the U.S. macroeconomic data, we use the GDP-growth data in y and the inflation data in p
and estimate the parameters of this model.
. dsge (y = E(F.y) + {alpha}*p + g)
> (p = z)
> (F.g = {rho_g}*g + {rho_gz}*z, state)
> (F.z = {rho_z}*z, state)
DSGE model
OIM
/structural
alpha -.1130023 .1554398 -0.73 0.467 -.4176587 .1916541
rho_g .335777 .0610763 5.50 0.000 .2160696 .4554843
rho_gz .0443504 .09013 0.49 0.623 -.1323013 .221002
rho_z .8626566 .0319007 27.04 0.000 .8001324 .9251807
sd(e.g) 2.184805 .2241105 1.745557 2.624054

sd(e.z) 1.146947 .0519234 1.045179 1.248715
The slope of the aggregate demand curve, , is estimated to be negative as we expected, but the
confidence interval is wide and includes zero. The imprecision in rho gz has caused imprecision in
the estimate of .
64 intro 4g Correlated state variables
Also see
Title
intro 5 Stability conditions
Description
We can estimate only the parameters of models that we can solve, and we can solve only models
that satisfy a stability condition. This entry discusses this condition, how it affects estimation in
practice, and how to find initial values that satisfy it.

Why we care about stability
What if the initial values are not saddle-path stable?
Why we care about stability

DSGE models are dynamic systems subject to random shocks. DSGE models that do not spiral out
of control or converge to a single point when shocked are said to be saddle-path stable. We can
solve only saddle-path stable DSGE models, and we can estimate only the parameters of models we
can solve.
The parameter values determine whether a DSGE model is saddle-path stable. Most DSGE models
are saddle-path stable for some parameter values and not for other values. As we discussed in How
to write down a DSGE in [DSGE] intro 1, the structural form of a DSGE model is
A0 yt = A1 Et (yt+1 ) + A2 yt + A3 xt (1)
Given this structural form, the values in the Ai and Bj matrices determine saddle-path stability.
In the process of solving the structural form in (1) and (2) for the state space, the Klein (2000)
solver computes the generalized eigenvalues of a matrix formed from the values in the Ai and Bj
matrices. An eigenvalue is said to be stable when its absolute value is less than 1. The model is
saddle-path stable when the number of stable eigenvalues equals the number of states in the model.
Once the maximization algorithm gets started, saddle-path stability usually is not an issue. Problems
with saddle-path stability usually occur when trying to get the maximization process started. The
initial values for the maximization algorithm must imply saddle-path stability because we cannot
solve a DSGE model for parameter values that do not imply saddle-path stability. When the default
initial values do not imply saddle-path stability, dsge searches for saddle-path-stable parameters, but
there is no guarantee that it will find any.
dsge uses a series of small positive numbers as the default initial values for the structural parameters
that appear in the A and B matrices. If you know that your model is not saddle-path stable when a
parameter, say, , is set to 0.1, you should use the from() option to specify an initial value that does
imply saddle-path stability. Alternatively, you could reparameterize the model so that = 0.1 does
imply saddle-path stability, but this solution tends to be more work than changing the initial value.
In rare cases, when the true parameters are close to values that are not saddle-path stable, the
maximum likelihood estimator will suffer from convergence problems. Better initial values can help
find the solution in these cases.
65
66 intro 5 Stability conditions
What if the initial values are not saddle-path stable?

If the initial values do not imply a saddle-path stable solution and dsge cannot find any such
values, you will see
. dsge ...
model is not saddle-path stable at current parameter values
The number of stable eigenvalues is greater than the number of state
variables.
(output omitted )
or
. dsge ...
The number of stable eigenvalues is less than the number of state
variables.
(output omitted )
In either case, you should specify the solve and noidencheck options and subsequently use
estat stable to find the problem. solve specifies that dsge only solve the model and not proceed
with estimation, and noidencheck suppresses the check of whether the parameters are identified.
estat stable will display the eigenvalues so that you can determine which ones are saddle-path
stable and which ones are not. The trick is to change the starting values so that the number of stable
eigenvalues equals the number of states in the model. Once you have determined which parameters
are problematic, you can use the from() option to change the initial values to ones that do not have
this problem.
Textbooks like those by Canova (2007), DeJong and Dave (2011), Ljungqvist and Sargent (2012),
and Woodford (2003) are full of examples in which some structural parameters must be either greater
than 1 or less than 1 for the model to be saddle-path stable. Similarly, some parameters must be either
greater than or less than 0 for the model to be saddle-path stable. Moving one structural parameter
over one of these boundaries and looking at how the change affects the number of stable eigenvalues
can help find a vector of parameters that yields a saddle-path stable solution.
That the model that you are trying to fit is likely related to other models in the literature or in the
textbooks suggests another strategy. Any such references probably contain some discussion of the set
of values that yield saddle-path stable solutions.
In especially difficult cases, you can use the strategy of solving the problem for a smaller model
and building back up. You temporarily remove equations from the model, find a parameter vector that
yields a saddle-path stable solution for the smaller model, and then progressively add back equations
as you find initial values that yield saddle-path stable solutions.
intro 5 Stability conditions 67
Example 1: Finding saddle-path stable initial values

Equations (1)(6) model the observed control variable (inflation) pt , the unobserved control variable
(output gap) yt , and the observed control variable (interest rate) rt as functions of the states ut and
zt . Lzt+1 is an auxiliary state that allows zt to be a second-order process instead of a first-order
process.
pt = (1/)Et (pt+1 ) + yt (3)

yt = Et (yt+1 ) {rt Et (pt+1 ) zt } (4)
rt = pt + ut (5)
ut+1 = u ut + t+1 (6)
zt+1 = z1 zt + z2 Lzt + t+1 (7)
Lzt+1 = zt (8)
We try to fit this model using default starting values below.

. dsge (p = (1/{gamma})*E(F.p) + {kappa}*y)
> (r = {gamma}*p + u)
> (F.z = {rho_z1}*z + {rho_z2}*Lz, state)
The number of stable eigenvalues is greater than the number of state
variables.
r(498);
The default initial values specify a model that is not saddle-path stable. We specify options solve
and noidencheck, which cause the command to attempt only to solve the model and to skip the
identification check.

> (r = {gamma}*p + u)
> (F.Lz = z, state noshock),
> solve noidencheck
DSGE model
Log likelihood = .
OIM
/structural
gamma .21 . . . . .
kappa .22 . . . . .
rho_u .23 . . . . .
rho_z1 .24 . . . . .
rho_z2 .25 . . . . .
sd(e.u) 1 . . .
sd(e.z) 1 . . .

Warning: Model cannot be solved at current parameter values. Current values
imply a model that is not saddle-path stable.
dsge displays the initial values used in the attempted solution, and it displays a warning message
indicating that the model cannot be solved at these values because they imply a model that is not
saddle-path stable.
We use estat stable to look at the eigenvalues implied by the initial values.
. estat stable
Stability results
Eigenvalues
stable -.3942
stable .6342
stable .21
stable .23
unstable 2.605e+16
unstable 1.046
The process is not saddle-path stable.

The process is saddle-path stable when there are 3 stable eigenvalues and 3
unstable eigenvalues.
The model is saddle-path stable when the number of stable eigenvalues equals the number of states
in the model. The initial values yield four stable eigenvalues, but there are only three states in the
model. So the model is not saddle-path stable at these values.
The initial value for gamma in the output from the failed solution attempt was 0.21. From (3), we
see that the coefficient on expected future inflation is greater than one when gamma is less than one.
We suspect that this relationship could be causing the lack of saddle-path stability. Below we use
option from() to specify 1.2 as an initial value for gamma and attempt to solve the model.
intro 5 Stability conditions 69

> (r = {gamma}*p + u)
> (F.Lz = z, state noshock),
> solve noidencheck from(gamma=1.2)
DSGE model
OIM
/structural
gamma 1.2 . . . . .
kappa .22 . . . . .
rho_u .23 . . . . .
rho_z1 .24 . . . . .
rho_z2 .25 . . . . .
sd(e.u) 1 . . .
sd(e.z) 1 . . .

There is no warning message, so the parameter values yielded a saddle-path stable solution. As a
final illustration of this point, we display the eigenvalues below.
. estat stable
Stability results
Eigenvalues
stable -.3942
stable .6342
stable .23
unstable 5.380e+16
unstable 1.2
unstable 1.264
The process is saddle-path stable.
We could now proceed with estimation.
References
Canova, F. 2007. Methods for Applied Macroeconomic Research. Princeton, NJ: Princeton University Press.
DeJong, D. N., and C. Dave. 2011. Structural Macroeconometrics. 2nd ed. Princeton, NJ: Princeton University Press.
Klein, P. 2000. Using the generalized Schur form to solve a multivariate linear rational expectations model. Journal
of Economic Dynamics and Control 24: 14051423.
Ljungqvist, L., and T. J. Sargent. 2012. Recursive Macroeconomic Theory. 3rd ed. Cambridge, MA: MIT Press.
Woodford, M. 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. Princeton, NJ: Princeton
University Press.
Also see
[DSGE] estat stable Check stability of system
Title
intro 6 Identification
Description Remarks and examples Reference Also see
Description
We can estimate only the parameters of DSGE models whose parameters are identified. We discuss
how to find the identification problems and how to fix them.

The parameters that maximize the likelihood function are the maximum likelihood (ML) estimates
of ML estimators like the one in dsge. When there is a unique vector of parameters that maximizes
the likelihood function, the parameters are identified and we can estimate them. When there is more
than one parameter vector that produces the same maximum value of the likelihood function, the
parameters are not identified, and we cannot estimate them because there is no way to choose among
the sets of equally best parameters.
dsge and all other commands that implement ML estimators check for identification at the solution
by checking that the matrix of second derivatives, known as the Hessian, is full rank at the maximum.
This test works only when the maximization algorithm arrives at a candidate maximum. In addition,
it provides no information about which parameters are not identified.
For many models, when the parameters are not identified, the only output produced by a standard
ML estimator is an ever growing series of not concave messages. This issue is especially relevant
to DSGE models because it is easy to write down theoretical DSGE models whose parameters are not
identified.
dsge uses the Iskrev (2010) diagnostic to resolve this issue. The Iskrev diagnostic detects lack of
parameter identification and reports which parameters are not identified at specific parameter values.
By default, dsge performs the Iskrev diagnostic at the initial values and at the solution values.
Instead of looking at the rank of the Hessian, the Iskrev (2010) diagnostic checks that all the
parameters affect the autocovariances of the observed control variables. When there is a one-to-one
relationship between the autocovariances for the observed control variables and the model parameters,
the parameters are identified. Intuitively, this one-to-one relationship allows us to back out the model
parameters from the autocovariances that we can always estimate from the data. When some of the
parameters do not affect the autocovariances of the observed control variables, there is no one-to-one
relationship and the model parameters are not identified.
Simply adding restrictions to parameters can make the parameters of an unidentified model identified.
This fact is frustrating but intuitive. When the parameters are not identified, they are not uniquely
determined by the data generated by the true model. Only adding restrictions, either by defining
constraints or by changing the equations, can identify the parameters.
71
72 intro 6 Identification
Example 1: A model with two unidentified parameters

Consider the following model of inflation pt , output growth yt , and the interest rate rt .
pt = Et (pt+1 ) + yt
yt = Et (yt+1 ) {rt Et (pt+1 ) zt }
rt = pt + ut
zt+1 = zt + t+1
ut+1 = ut + t+1
This model specifies how the observed control variables pt and rt depend on the unobserved control
variable yt , on the state variables zt and ut , and on the shocks t and t .
Lets try to estimate the parameters of this model.
> (y = E(F.y) -{gamma}*(r - E(F.p) - {rhoz}*z), unobserved)
> (r = (1/{beta})*p + u)
identification failure at starting values
Constrain some parameters or specify option noidencheck. Likely source of
identification failure: {kappa} {gamma}
r(498);
The error message indicates that kappa and gamma do not influence the autocovariances indepen-
dently; they are linearly dependent. If a list of linearly dependent parameters is supplied, then you
must constrain all but one of them prior to estimation. We constrain gamma.
. constraint 2 _b[gamma]=1
And we apply the constraint to the model:

> (r = (1/{beta})*p + u)
> (F.z = {rhoz}*z, state),
> constraint(2)
intro 6 Identification 73
DSGE model
( 1) [/structural]gamma = 1
OIM
/structural
beta .5146687 .078349 6.57 0.000 .3611076 .6682298
kappa .1659038 .0474064 3.50 0.000 .0729889 .2588187
gamma 1 (constrained)
rhoz .9545255 .0186424 51.20 0.000 .9179871 .9910639
rhou .7005496 .0452601 15.48 0.000 .6118414 .7892577
sd(e.u) 2.318197 .3047421 1.720914 2.915481

sd(e.z) .650711 .1123848 .4304409 .8709811
The remaining free parameters of the model are identified and can be estimated.
Keep in mind that the Iskrev (2010) diagnostic is performed at specific parameter values. Sometimes,
the parameters of a model cannot be identified at a vector of parameter values but can be identified
at practically any other nearby vector. This problem arises when simple initial values cause terms to
cancel out. If you suspect that the identification diagnostic is finding a problem because of initial
values, you can either specify noidencheck or specify other seemingly random initial values.
Example 2: Suppressing the identification check

We suppress the identification check in an attempt to estimate the parameters without imposing
additional constraints. We specify the iterate(50) option in case the parameters are not identified.
> (r = (1/{beta})*p + u)
> noidencheck iterate(50)
74 intro 6 Identification

convergence not achieved
DSGE model
OIM
/structural
beta .5146672 .078349 6.57 0.000 .3611059 .6682285
kappa .424358 . . . . .
gamma .3909567 .1117155 3.50 0.000 .1719983 .609915
rhoz .9545256 .0186424 51.20 0.000 .9179872 .991064
rhou .7005485 .0452604 15.48 0.000 .6118397 .7892573
sd(e.u) 2.318203 .3047446 1.720915 2.915492

sd(e.z) .6507122 .1123847 .4304422 .8709822
Note: Parameters not identified at reported values.
We see a long list of not concave messages, indicating that the parameters are indeed not identified
without additional constraints. We could remove the option limiting the number of iterations, and the
command should produce not concave iterations indefinitely.
intro 6 Identification 75
Reference
Iskrev, N. 2010. Local identification in DSGE models. Journal of Monetary Economics 57: 189202.
Also see
Title
intro 7 Convergence problems
Description
Fitting DSGE models is notoriously difficult. We discuss the causes of convergence problems and
some solutions.

When the iteration process never gets started or when it never ends are two types of convergence
problems. When it never gets started, you will see something like
. dsge ...
model could not be solved at starting values
equilibrium is locally indeterminate
(output omitted )
This error occurs when the initial values specify a model that violates the stability conditions required
for DSGE estimation; see [DSGE] intro 5 for a discussion of this issue and some solutions.
When the iteration process never ends, the cause is either that there is no unique solution or that
the optimizer cannot find the unique solution. When there is no unique solution, the parameters are
said to be not identified. In this case, you see (not concave) after each iteration, like
. dsge ...
(output omitted )
.
.
.
See [DSGE] intro 6 for a discussion of this issue and some solutions.
Even when the iterations get started and the parameters are identified, convergence problems are
still common. In these cases, there are usually many parameters.
Changing the optimization technique is the simplest method to overcome a convergence problem,
and it can be surprisingly effective. By default, dsge uses the BroydenFletcherGoldfarbShanno
(BFGS) algorithm for five iterations before switching to a modified NewtonRaphson (NR) algorithm.
The BFGS algorithm is especially effective at finding a maximum of the DSGE log-likelihood function
from poor initial values. In some cases, simply specifying technique(bfgs 200 nr) can overcome
the convergence problem.
The best way to overcome a convergence problem is to provide better initial values. The optimizer
used by dsge will almost always converge when it is provided with good initial values and the
parameters are identified. Specifying good initial values that come from theory or previous empirical
work can solve your convergence problem. You can specify initial values inside the scalar substitutable
expressions or by using the from() option; see examples 1 and 2 for details.
76
intro 7 Convergence problems 77
Sometimes, you have a convergence problem and do not have good initial values. One way to get
initial values is to estimate a series of progressively less constrained models. This process usually
produces convergence when the parameters are identified. See example 3 for details.
Example 1: Specifying starting values in scalar substitutable expressions

Models with lagged state variables and other models that allow for more persistent processes can
exhibit convergence problems. Equations (1)(6) model the observed control variable (inflation) pt ,
the unobserved control variable (output gap) yt , and the observed control variable (interest rate) rt
as functions of the states ut and zt . Lzt+1 is an auxiliary state that allows zt to be a second-order
process instead of a first-order process.
pt = Et (pt+1 ) + yt (1)
yt = Et (yt+1 ) {rt Et (pt+1 ) zt } (2)
1
rt = pt + ut (3)

ut+1 = u ut + t+1 (4)
zt+1 = z1 zt + z2 Lzt + t+1 (5)
Lzt+1 = zt (6)
Suppose that previous empirical work suggests the initial values of = 0.5, = 0.2, u = 0.7,
z1 = 0.7, and z2 = 0.2. Inside a scalar substitutable expression, we can specify an initial value
by typing {parameter = value}. We specify these initial values for the parameters in the scalar
substitutable expressions in our model specification below.
. dsge (p = {beta=.5}*E(F.p) + {kappa=.2}*y)
> (r = (1/{beta})*p + u)
> (F.u = {rho_u=.7}*u, state)
> (F.z = {rho_z1=.7}*z + {rho_z2=.2}*Lz, state)
78 intro 7 Convergence problems
DSGE model
OIM
/structural
beta .5154863 .0770497 6.69 0.000 .3644717 .666501
kappa .1662428 .0468474 3.55 0.000 .0744236 .258062
rho_u .6979879 .0452072 15.44 0.000 .6093834 .7865923
rho_z1 .6735493 .2666448 2.53 0.012 .150935 1.196164
rho_z2 .270999 .2564998 1.06 0.291 -.2317313 .7737294
sd(e.u) 2.315267 .2992202 1.728806 2.901727

sd(e.z) .7720681 .1716399 .43566 1.108476
Note that the output includes results for the standard deviations of the shocks, for which there are
no scalar substitutable expressions. In example 2, we illustrate how to specify initial values for all
the parameters using the from() option.
Example 2: Specifying starting values using from()

In example 1, we specified initial values inside the scalar substitutable expressions. You cannot
specify initial values for the standard deviations of the shocks this way, because the shocks do not
appear in any scalar substitutable expressions. We can specify starting values for any, or all, of the
parameters using the from() option.
from() has several syntaxes, but the vector of starting values is most frequently used in DSGE
applications. In this syntax, we specify a row vector containing the initial values = 0.5, = 0.2,
u = 0.7, z1 = 0.7, z2 = 0.2, e.u = 2.3, and e.z = 0.7. We begin by defining and listing the
vector of initial values.
. matrix ivalues = (.5, .2, .7, .7, .2, 2.3, .7)
. matrix list ivalues
ivalues[1,7]
c1 c2 c3 c4 c5 c6 c7
r1 .5 .2 .7 .7 .2 2.3 .7
Although we could specify and use the column names of the vector to assign elements of the vector
to parameters, we use froms suboption copy to make the assignment based on order of appearance
in the vector. Specifying from(, copy) implies that the first element in the vector specifies the initial
value for the first model parameter, the second element in the vector specifies the initial value for
the second model parameter, and so on.

> (r = (1/{beta})*p + u)
> (F.Lz = z, state noshock), from(ivalues, copy)
DSGE model
OIM
/structural
beta .5154871 .0770492 6.69 0.000 .3644734 .6665008
kappa .1662427 .0468473 3.55 0.000 .0744237 .2580616
rho_u .6979877 .0452072 15.44 0.000 .6093833 .7865921
rho_z1 .6735477 .2666002 2.53 0.012 .1510209 1.196074
rho_z2 .2710007 .2564571 1.06 0.291 -.2316459 .7736473
sd(e.u) 2.315264 .2992175 1.728808 2.901719

sd(e.z) .7720676 .1716211 .4356965 1.108439
Example 3: Starting with a more constrained model to obtain convergence

Suppose that we wanted to estimate the parameters of the model in example 1 but that the only
good initial value was that theory predicted that should be about 0.5. Further suppose that we were
having convergence problems fitting this model. In this case, one strategy would be to constrain to
0.5 and to constrain to a small, nonzero value. The small value minimizes the interaction between p
and y. Making the value nonzero can prevent problems when solving for the state-space form of the
model. The remaining parameters determine the autoregressive processes for the states. Given that
we suspect that the second-order lag is part of the problem, we also constrain it to a small, nonzero
value. We try to estimate the remaining parameters in the output below.
80 intro 7 Convergence problems
. constraint define 1 _b[beta] = 0.5

. constraint define 2 _b[kappa] = 0.1
. constraint define 3 _b[rho_z2] = 0.01
> (r = (1/{beta})*p + u)
> (F.Lz = z, state noshock), constraints(1 2 3)
DSGE model
( 2) [/structural]kappa = .1
( 3) [/structural]rho_z2 = .01
OIM
/structural
kappa .1 (constrained)
rho_u .7906612 .0102382 77.23 0.000 .7705947 .8107277
rho_z1 .9440872 .0188166 50.17 0.000 .9072073 .980967
rho_z2 .01 (constrained)
sd(e.u) 2.391372 .1083319 2.179045 2.603699

sd(e.z) .697013 .0731968 .5535498 .8404761
The estimator converged. We will want to use these estimates as initial values, so we put them
into the matrix b and list b to confirm it has the desired values.
. matrix b = e(b)
. matrix list b
b[1,7]
/structural: /structural: /structural: /structural: /structural:
beta kappa rho_u rho_z1 rho_z2
y1 .5 .1 .79066117 .94408715 .01
/: /:
sd(e.u) sd(e.z)
y1 2.3913719 .69701298
We use the previous estimates as initial values, drop the constraints, and try to estimate all the
parameters.
> (r = (1/{beta})*p + u)
> (F.Lz = z, state noshock), from(b, copy)
DSGE model
OIM
/structural
beta .5154872 .0770492 6.69 0.000 .3644736 .6665009
kappa .1662428 .0468472 3.55 0.000 .0744239 .2580617
rho_u .6979875 .0452071 15.44 0.000 .6093832 .7865918
rho_z1 .6735459 .266585 2.53 0.012 .1510489 1.196043
rho_z2 .2710024 .2564424 1.06 0.291 -.2316156 .7736203
sd(e.u) 2.315263 .2992172 1.728808 2.901718

sd(e.z) .7720683 .1716153 .4357085 1.108428
The estimator converged.

In larger problems, more steps are sometimes required. In these steps, you use the previous
estimates as initial values, but you drop only some of the constraints at each step.
Also see
Title
intro 8 Wald tests vary with nonlinear transforms
Description
After fitting a DSGE model, we often perform tests of structural parameters, and these tests often
place nonlinear restrictions on the parameters. The values and rejection rates of a Wald test for
different nonlinear expressions of the same null hypothesis are different. We illustrate this issue, show
that likelihood-ratio (LR) tests do not have this problem, and illustrate that you can parameterize your
model in terms of invertible transforms of each parameter.

Wald tests vary with nonlinear transforms
LR tests do not vary with nonlinear transforms
Wald tests vary with nonlinear transforms

Performing a statistical test of whether a structural parameter in a DSGE has a specific value is one
of the most frequent forms of inference after dsge estimation. The null hypothesis in one of these
tests frequently places nonlinear restrictions on the underlying parameters. Two different nonlinear
expressions of the same null hypothesis produce different Wald test statistics in finite samples and
have different rejection rates. In other words, the Wald test is not invariant to nonlinear transforms
of the null hypothesis. The LR test, on the other hand, is invariant to nonlinear transforms of the null
hypothesis.
Example 1: Different values from logically equivalent Wald tests

Equations (1)(5) specify how the observed control variable inflation pt , the unobserved control
variable output growth yt , and the observed control variable (interest rate) rt depend on the states zt
and ut , given the shocks t and t .
pt = Et (pt+1 ) + yt (1)
yt = Et (yt+1 ) {rt Et (pt+1 ) zt } (2)
rt = (1/)pt + ut (3)
zt+1 = zt + t+1 (4)
ut+1 = ut + t+1 (5)
82
intro 8 Wald tests vary with nonlinear transforms 83
We estimate the parameters of this model using the macroeconomic data for the United States in
usmacro2.dta.
> (y = E(F.y) -(r - E(F.p) - {rhoz}*z), unobserved)
> (r = (1/{beta})*p + u)
DSGE model
OIM
/structural
beta .5146674 .0783489 6.57 0.000 .3611064 .6682284
kappa .1659057 .0474074 3.50 0.000 .0729889 .2588224
rhoz .9545256 .0186424 51.20 0.000 .9179872 .991064
rhou .7005483 .0452604 15.48 0.000 .6118395 .7892571
sd(e.u) 2.318203 .3047438 1.720916 2.915489

sd(e.z) .6507118 .1123845 .4304422 .8709814
The interest rate (3) links the nominal interest rate to the inflation rate. The coefficient on inflation
is 1/ . We test whether this parameter is 1.5, a common benchmark value in the literature.
. testnl 1/_b[beta] = 1.5
(1) 1/_b[beta] = 1.5
chi2(1) = 2.24
Prob > chi2 = 0.1342
We do not reject the null hypothesis that 1/ is 1.5.

84 intro 8 Wald tests vary with nonlinear transforms
If we test the logically equivalent hypothesis that = 2/3, the statistic and p-value change.
. test _b[beta] =2/3
chi2( 1) = 3.76
Prob > chi2 = 0.0524
The values of these two logically equivalent Wald tests differ because Wald tests are not invariant
to nonlinear transformation. This issue is well known in the literature; see Gregory and Veall (1985)
and Phillips and Park (1988) for details. In this example, the inference of failing to reject the null
hypothesis remains the same when using a 5% significance level, but this is not true in general.
Different formulations of Wald tests can lead to different inferences.
LR tests do not vary with nonlinear transforms

LR tests are invariant to nonlinear transforms.
Example 2: LR tests are invariant to nonlinear transforms

We illustrate this feature by performing LR tests that = 2/3 and that 1/ = 1.5. The cur-
rent estimates are those of the unconstrained model. We repeat these results and store them as
unconstrained.
. dsge
DSGE model
OIM
/structural
beta .5146674 .0783489 6.57 0.000 .3611064 .6682284
kappa .1659057 .0474074 3.50 0.000 .0729889 .2588224
rhoz .9545256 .0186424 51.20 0.000 .9179872 .991064
rhou .7005483 .0452604 15.48 0.000 .6118395 .7892571
sd(e.u) 2.318203 .3047438 1.720916 2.915489

sd(e.z) .6507118 .1123845 .4304422 .8709814
. estimates store unconstrained

Now, we estimate the parameters of the constrained model in which = 2/3, store the results as
constrained, and perform an LR test of the null hypothesis that = 2/3.
. constraint 1 _b[beta] = 2/3
> (r = (1/{beta})*p + u)
> constraint(1)
DSGE model
OIM
/structural
kappa .1076807 .0276892 3.89 0.000 .0534109 .1619506
rhoz .9538519 .0187789 50.79 0.000 .9170458 .9906579
rhou .7214333 .043967 16.41 0.000 .6352595 .807607
sd(e.u) 1.915459 .0867103 1.74551 2.085408

sd(e.z) .4936815 .0805138 .3358773 .6514856
. estimates store constrained

. lrtest unconstrained constrained
Likelihood-ratio test LR chi2(1) = 3.08
(Assumption: constrained nested in unconstrained) Prob > chi2 = 0.0794
Note that the value of the LR statistic is 3.08. We now illustrate an LR of the null hypothesis that
1/ = 1.5 produces the same value.
We cannot impose nonlinear restrictions on parameters, so we must begin by reparameterizing
the unconstrained model by replacing {beta} with 1/{beta}. To avoid having {beta} mean two
different things, we write the model in terms of {gamma}=1/{beta} and estimate the parameters:
. dsge (p = 1/{gamma}*E(F.p) + {kappa}*y)

> (r = ({gamma})*p + u)
> from(gamma=2 kappa=0.15 rhou=0.75 rhoz=0.95)
DSGE model
OIM
/structural
gamma 1.943005 .2957866 6.57 0.000 1.363274 2.522736
kappa .1659061 .0474073 3.50 0.000 .0729895 .2588226
rhoz .9545256 .0186424 51.20 0.000 .9179872 .991064
rhou .7005481 .0452604 15.48 0.000 .6118393 .7892568
sd(e.u) 2.318205 .3047432 1.720919 2.91549

sd(e.z) .6507123 .1123842 .4304434 .8709812
. estimates store unconstrained2
The estimates of the parameters other than gamma and the value of the log likelihood are nearly
the same as those for the unconstrained model. The value for gamma = 1.94 is the same as
1/beta = 1/0.514 = 1.95. By tightening the convergence tolerance, we could make these values
exactly the same. These values are nearly the same because this example is an instance of a general
property of maximum likelihood estimators. Transforming a parameter by an invertible function does
not change the log likelihood or the other parameter estimates. In other words, maximum likelihood
estimators are invariant to invertible transformations of the parameters; see Casella and Berger (2002,
319) for details.
Having stored the estimates from the unconstrained model, we now estimate the parameters of the
constrained model and store these results in constrained2.
. constraint 2 _b[gamma] = 1.5
. dsge (p = 1/{gamma}*E(F.p) + {kappa}*y)
> (r = ({gamma})*p + u)
> constraint(2)
DSGE model
( 1) [/structural]gamma = 1.5
OIM
/structural
gamma 1.5 (constrained)
kappa .1076807 .0276892 3.89 0.000 .0534109 .1619506
rhoz .9538519 .0187789 50.79 0.000 .9170458 .9906579
rhou .7214333 .043967 16.41 0.000 .6352595 .807607
sd(e.u) 1.915459 .0867103 1.74551 2.085408

sd(e.z) .4936815 .0805138 .3358773 .6514856
. estimates store constrained2
The estimates of the parameters other than gamma and the value of the log likelihood are the same
as those for the constrained1 model. This is another instance of the invariance of the maximum
likelihood estimator to invertible transformations of the parameters.
Having stored the log likelihoods from the constrained and unconstrained model, we now perform
an LR of the null hypothesis that = 1.5.
. lrtest unconstrained2 constrained2
Likelihood-ratio test LR chi2(1) = 3.08
(Assumption: constrained2 nested in unconstrained2) Prob > chi2 = 0.0794
The LR test statistic and its p-value are the same as those reported for the test against the null
hypothesis that = 2/3, which illustrates that LR tests are invariant to nonlinear transforms.
References
Casella, G., and R. L. Berger. 2002. Statistical Inference. 2nd ed. Pacific Grove, CA: Duxbury.
Gregory, A. W., and M. R. Veall. 1985. Formulating Wald tests of nonlinear restrictions. Econometrica 53: 14651468.
Phillips, P. C. B., and J. Y. Park. 1988. On the formulation of Wald tests of nonlinear restrictions. Econometrica 56:
10651083.
Also see
[R] lrtest Likelihood-ratio test after estimation
[R] test Test linear hypotheses after estimation
Title
dsge Linearized dynamic stochastic general equilibrium models
Description Menu Syntax Options Remarks

Stored results Methods and formulas References Also see
Description
dsge fits linearized dynamic stochastic general equilibrium (DSGE) models to multiple time series.
DSGE models are systems of equations that are motivated by economic theory. In these systems,
expectations of future values of variables can affect the current values. The parameters of these
models are often directly interpretable in terms of economic theory.
Menu
Statistics > Multivariate time series > Dynamic stochastic general equilibrium (DSGE) models
Syntax

dsge eqlist if in , options
options Description
Main
constraints(constraints) apply specified linear constraints
noidencheck do not check for parameter identification
solve return model solution at initial values
SE/Robust
vce(vcetype) vcetype may be oim or robust
Reporting
level(#) set confidence level; default is level(95)
nocnsreport do not display constraints
display options control columns and column formats and line width
Maximization
maximize options control the maximization process
Advanced
nodemean do not demean data prior to estimation
post force posting of estimation results in the event of errors
caused by lack of identification or stability
idtolerance(#) set tolerance used for identification check; seldom used
lintolerance(#) set tolerance used for linearity check; seldom used
coeflegend display legend instead of statistics
89
90 dsge Linearized dynamic stochastic general equilibrium models
You must tsset your data before using dsge; see [TS] tsset.
coeflegend does not appear in the dialog box.
See [U] 20 Estimation and postestimation commands for more capabilities of estimation commands.
Below we present the full specification of eqlist. You may prefer to start with the syntax discussion
in [DSGE] intro 2.
eqlist is
eq
eq eqlist
eq is
cntrl eq
state eq
cntrl eq contains
(ocntrl var = termlist)
(ucntrl var = termlist, unobserved)
(parm exp * ocntrl var = termlist)
(parm exp * ucntrl var = termlist, unobserved)
state eq contains
(F.state var = , state)

(F.state var = termlist, state noshock )

(parm exp * F.state var = termlist, state noshock )
ocntrl var is Stata variable to be treated as an observed control in the system
ucntrl var is name to be treated as an unobserved control in the system
state var is name to be treated as an unobserved state in the system
If there happens to be a Stata variable with the same name as ucntrl var or state var,
the variable is ignored and plays no role in the estimation.
termlist is
term
term + termlist
term - termlist
term is
rhs var
parm exp * rhs var
parm exp * (termlist)
rhs var is
ocntrl var
ucntrl var
state var
E(F.ocntrl var)
E(F.ucntrl var)
dsge Linearized dynamic stochastic general equilibrium models 91
parm exp is scalar substitutable expression;

parm spec elements are allowed and expected
rhs var are not allowed
parm spec is
{parm name}
{parm name=initial val}
parm name is name used to identify model parameter
initial val is numeric literal; specifies an initial value
Options

Main
constraints(constraints); see [R] estimation options.
noidencheck skips the check that the parameters are identified at the initial values. Models that
are not structurally identified can still converge, thereby producing meaningless results that only
appear to have meaning; thus care should be taken in specifying this option. See [DSGE] intro 6
for details.
solve puts the model into state-space form at the initial parameter values. No standard errors are
produced.

SE/Robust
vce(vcetype) specifies the type of standard error reported, which includes types that are derived
from asymptotic theory (oim) and that are robust to some kinds of misspecification (robust); see
[R] vce option.

Reporting
level(#), nocnsreport; see [R] estimation options.
display options: noci, nopvalues, cformat(% fmt), pformat(% fmt), sformat(% fmt), and nol-
stretch; see [R] estimation options.

Maximization

maximize options: difficult, technique(algorithm spec), iterate(#), no log, trace,
gradient, showstep, hessian, showtolerance, tolerance(#), ltolerance(#),
nrtolerance(#), and from(init specs); see [R] maximize.

Advanced
nodemean prevents dsge from removing the mean of the observed control variables prior to estimation.
post causes dsge to post the parameter vector into e(), even in the event of errors that arise from
checking stability conditions or identification.
idtolerance(#) specifies the tolerance used in the identification diagnostic. The default is
idtolerance(1e-6).
lintolerance(#) specifies the tolerance used in the linearity diagnostic. The default is
lintolerance(1e-12).
The following option is available with dsge but is not shown in the dialog box:
coeflegend; see [R] estimation options.
Remarks
For an introduction to what dsge can do and how to use it, see [DSGE] intro 1. It is highly
recommended that you read the introduction first.
For examples of dsge, see the examples of classic DSGE models in [DSGE] intro 3a, [DSGE] intro 3b,
and [DSGE] intro 3c. Additional examples are presented in [DSGE] intro 4a[DSGE] intro 4g. See
[DSGE] intro 4 for an overview of these examples.
Stored results
dsge stores the following in e():
Scalars
e(N) number of observations
e(k) number of parameters
e(k aux) number of auxiliary parameters
e(k eq) number of equations in e(b)
e(k dv) number of dependent variables
e(k state) number of state equations
e(k control) number of control equations
e(k shock) number of shocks
e(k observed) number of observed control equations
e(k stable) number of stable eigenvalues
e(ll) log likelihood
e(tmin) minimum time in sample
e(tmax) maximum time in sample
e(rank) rank of VCE
e(ic) number of iterations
e(rc) return code
e(converged) 1 if converged, 0 otherwise
Macros
e(cmd) dsge
e(cmdline) command as typed
e(depvar) unoperated names of dependent variables
e(state) unoperated names of state variables
e(control) unoperated names of control variables
e(observed) unoperated names of observed control variables
e(title) title in estimation output
e(tvar) variable denoting time within groups
e(tmins) formatted minimum time
e(tmaxs) formatted maximum time
e(vce) vcetype specified in vce()
e(vcetype) title used to label Std. Err.
e(opt) type of optimization
e(method) likelihood method
e(idencheck) passed, failed, or skipped
e(technique) maximization technique
e(properties) b V
e(estat cmd) program used to implement estat
e(predict) program used to implement predict
e(marginsnotok) predictions disallowed by margins
dsge Linearized dynamic stochastic general equilibrium models 93
Matrices
e(b) parameter vector
e(Cns) constraints matrix
e(ilog) iteration log (up to 20 iterations)
e(eigenvalues) generalized eigenvalues
e(gradient) gradient vector
e(shock coeff) estimated shock coefficient matrix
e(transition) estimated state transition matrix
e(policy) estimated policy matrix
e(V) variancecovariance matrix of the estimators
e(V modelbased) model-based variance
Functions
e(sample) marks estimation sample
Methods and formulas

A DSGE model is a system of equations in which the values of the control variables yt and of
the state variables xt depend on their values in previous periods and their one-period-ahead rational
expectations; see [DSGE] intro 1 for an introduction to control variables and state variables. The
structural form of a DSGE model is
A0 yt = A1 Et (yt+1 ) + A2 yt + A3 xt (1)
where the entries in Ai and Bi are functions of the parameter vector and C is a selector matrix of
zeros and ones that puts shocks in the state equations that include them. The A0 and B0 matrices are
diagonal, and A2 has zeros on its diagonal. See [DSGE] intro 1 for details about how the structural
parameters get mapped into entries in Ai and Bi matrices.
dsge estimates the structural parameters by maximum likelihood. The remainder of this entry
provides some details about this process.
We can estimate only the parameters of structural models like (1) and (2) that we can solve for the
state-space form that expresses the control variables as functions of the state variables and expresses
the state variables as first-order autoregressive processes, as in (3) and (4),
yt = G()xt (3)
xt+1 = H()xt + M()t+1 (4)
where G is known as the policy matrix and H is known as the state transition matrix. The elements
in the policy and state transition matrices are functions of the structural parameters that appear in the
matrices A and B.
dsge implements the solution technique described in Klein (2000), which obtains the state-space
form in (3) and (4) from the structural model in (1) and (2).
We can only solve a structural model for its state-space form when i) the structural form has
the structure implied by (1) and (2) and when ii) the values of the structural parameters imply that
the model neither spirals out of control nor converges to a point, a condition known as saddle-path
stability.
Condition (i) seems trivial, but some minor manipulation is frequently required to make it true.
Many DSGE models include a problematic term like a shock to a control variable or a lagged state
variable. These problematic terms do not fit into the structure required by (1) and (2), but the model
is easily rewritten to accommodate these problematic terms; see [DSGE] intro 4 for details.
Condition (ii), the saddle-path stability condition, depends on the values of the parameters. We
can only solve models for parameter values that imply a saddle-path-stable model; see [DSGE] intro 5
for details.
Assuming that the shocks t are mean-zero, independent, and normally distributed implies that
(3) and (4) form a linear state-space model in which the coefficients are nonlinear functions of the
structural parameters . This structure allows dsge to estimate the parameters and the standard
deviations of the shocks by maximum likelihood.
The Kalman filter is used to form the log-likelihood function. The Kalman filter is a method for
recursively obtaining linear, least-squares forecasts conditional on past information. These forecasts are
used to construct the log likelihood, assuming normality and stationarity. See Methods and formulas
in [TS] sspace for details of the Kalman filter.
When the shocks are independent and identically distributed, but not normally distributed, the
maximum likelihood estimator is consistent for , but vce(robust) standard errors must be used.
(The process of using a maximum likelihood estimator under weaker distributional assumptions and
correcting the standard errors is known as a quasimaximum likelihood estimator in the literature.)
One final caveat is that only a subset y1,t of the controls is treated as observed in some models.
This caveat is handled by augmenting the state-space systems in (3) and (4) with the observation
equation
y1,t = Dyt
where D is a selection matrix that picks out the rows in G corresponding to elements in the control
vector that are observed.
Asymptotic standard errors for postestimation objects such as the entries in the state-space matrices
and the impulse responses are obtained using the delta method. See Serfling (1980, sec. 3.3) for a
discussion of the delta method.
References
Serfling, R. J. 1980. Approximation Theorems of Mathematical Statistics. New York: Wiley.
Also see
[TS] sspace State-space models
[TS] tsset Declare data to be time-series data
[TS] var Vector autoregressive models
[U] 20 Estimation and postestimation commands
Title
dsge postestimation Postestimation tools for dsge
Postestimation commands predict Remarks and examples Methods and formulas

Also see
Postestimation commands
The following postestimation commands are of special interest after dsge:
Command Description
estat policy display policy matrix of estimated model
estat stable assess stability of the system
estat transition display transition matrix of estimated model
irf create and analyze IRFs and FEVDs
The following standard postestimation commands are also available:

Command Description
estat ic Akaikes and Schwarzs Bayesian information criteria (AIC and BIC)
estat summarize summary statistics for the estimation sample
estat vce variancecovariance matrix of the estimators (VCE)
estimates cataloging estimation results
forecast dynamic forecasts and simulations
lincom point estimates, standard errors, testing, and inference for linear combinations
of coefficients
lrtest likelihood-ratio test
nlcom point estimates, standard errors, testing, and inference for nonlinear
combinations of coefficients
predict one-step-ahead predictions, prediction errors, and other diagnostic measures
predictnl point estimates, standard errors, testing, and inference for generalized
predictions
test Wald tests of simple and composite linear hypotheses
testnl Wald tests of nonlinear hypotheses
95
96 dsge postestimation Postestimation tools for dsge
predict
Description for predict

predict creates new variables containing predictions such as expected values. Predictions are
available as static one-step-ahead predictions or as dynamic multistep predictions, and you can control
when dynamic predictions begin.
Menu for predict

Statistics > Postestimation
Syntax for predict

predict type { stub* | newvarlist } if in , statistic options
statistic Description
Main
xb linear prediction for observed variables
states linear prediction for latent state variables
options Description
Options
rmse(stub* | newvarlist)
put estimated root mean squared errors of predicted statistics in new
variables
dynamic(time constant) begin dynamic forecast at specified time
Advanced
smethod(method) method for predicting unobserved states
method Description
onestep predict using past information
filter predict using past and contemporaneous information
Options for predict

Main
xb, the default, calculates the linear predictions of the observed variables.
states calculates the linear predictions of the latent state variables.
dsge postestimation Postestimation tools for dsge 97

Options
rmse(stub* | newvarlist) puts the root mean squared errors of the predicted statistics into the specified
new variables. The root mean squared errors measure the variances due to the disturbances but do
not account for estimation error.
dynamic(time constant) specifies when predict starts producing dynamic forecasts. The specified
time constant must be in the scale of the time variable specified in tsset, and the time constant
must be inside a sample for which observations on the dependent variables are available. For
example, dynamic(tq(2008q4)) causes dynamic predictions to begin in the fourth quarter of
2008, assuming that your time variable is quarterly; see [D] datetime. If the model contains
exogenous variables, they must be present for the whole predicted sample. dynamic() may not
be specified with rstandard, residuals, or smethod(smooth).

Advanced
smethod(method) specifies the method for predicting the unobserved states, smethod(onestep)
and smethod(filter), and causes different amounts of information on the dependent variables
to be used in predicting the states at each time period.
smethod(onestep), the default, causes predict to estimate the states at each time period using
previous information on the dependent variables. The Kalman filter is performed on previous
periods, but only the one-step predictions are made for the current period.
smethod(filter) causes predict to estimate the states at each time period using previous and
contemporaneous data by the Kalman filter. The Kalman filter is performed on previous periods
and the current period. smethod(filter) may be specified only with states.

For examples of estat policy, see [DSGE] intro 1, [DSGE] intro 3a, and [DSGE] intro 3c.
For examples of estat transition, see [DSGE] intro 1, [DSGE] intro 3a, and [DSGE] intro 3b.
For an example of estat stable, see [DSGE] intro 5.
For examples of irf after dsge, see [DSGE] intro 1, [DSGE] intro 3b, and [DSGE] intro 3c.
For an example of forecast after dsge, see [DSGE] intro 1.
For examples of predict after dsge, see [DSGE] intro 3a.

Estimating the unobserved states is the key to predicting the observed variables.
By default and with the smethod(onestep) option, predict estimates the state in each period
by applying the Kalman filter to all previous periods and only making the one-step prediction to the
current period.
With the smethod(filter) option, predict estimates the states in each period by applying
the Kalman filter on all previous periods and the current period. The computational difference be-
tween smethod(onestep) and smethod(filter) is that smethod(filter) performs the update
step on the current period while smethod(onestep) does not. The statistical difference between
smethod(onestep) and smethod(filter) is that smethod(filter) uses contemporaneous in-
formation on the observed variables while smethod(onestep) does not.
The observed control variables are predicted by plugging in the estimated states.
98 dsge postestimation Postestimation tools for dsge
Also see
[DSGE] estat policy Display policy matrix
[DSGE] estat stable Check stability of system
[DSGE] estat transition Display state transition matrix
[TS] forecast Econometric model forecasting
[TS] irf Create and analyze IRFs, dynamic-multiplier functions, and FEVDs
[U] 20 Estimation and postestimation commands
Title
estat policy Display policy matrix
Description Menu for estat Syntax Options

Remarks and examples Methods and formulas Also see
Description
estat policy displays the estimated policy matrix of the state-space form of a DSGE model.
Menu for estat

Syntax

estat policy , level(#) display options
Options
level(#) specifies the confidence level, as a percentage, for confidence intervals. The default is
level(95) or as set by set level; see [U] 20.8 Specifying the width of confidence intervals.

The policy matrix is part of the state-space form of a DSGE model. It specifies the models control
variables as a function of the models state variables.
For examples, see [DSGE] intro 1, [DSGE] intro 3a, and [DSGE] intro 3c.

Entries in the policy matrix G are functions of the structural parameter vector . Standard errors
for G
b are calculated using the delta method.
Also see
99
Title
estat stable Check stability of system
Description Menu for estat Syntax Remarks and examples

Methods and formulas Reference Also see
Description
estat stable displays functions of the model parameters that indicate whether the model is
saddle-path stable at specific parameter values. These results can help you find initial values for
which the model is saddle-path stable. Saddle-path stability is required for solving and estimating the
parameters of DSGE models.
Menu for estat

Syntax
estat stable

DSGE models are dynamic systems that are subject to random shocks. DSGE models that do not
spiral out of control or converge to a single point when shocked are said to be saddle-path stable.
We can solve for the state-space form of DSGE models only if they are saddle-path stable, and we
can estimate the parameters of models only if we can solve for the state-space form.
The structural parameter values determine whether a DSGE model is saddle-path stable. In the
process of solving the structural form for the state-space form, the Klein (2000) solver computes the
generalized eigenvalues of a matrix formed from structural parameter values. An eigenvalue is said
to be stable when its absolute value is less than 1. The model is saddle-path stable when the number
of stable eigenvalues equals the number of states in the model.
estat stable displays the generalized eigenvalues implied by the parameter values in e(b),
indicates which are stable and which are unstable, and displays a note indicating whether the model
is saddle-path stable at these parameter values. estat stable can help you find initial values for
the maximization routine when the default values imply a model that is not saddle-path stable; see
[DSGE] intro 5 for details.

estat stable displays the generalized eigenvalues computed by the Klein (2000) solver. Values
less than 1 are labeled as stable; values greater than or equal to 1 are labeled as unstable.
100
estat stable Check stability of system 101
Reference
Also see
Title
estat transition Display state transition matrix
Description Menu for estat Syntax Options

Remarks and examples Methods and formulas Also see
Description
estat transition displays the estimated state transition matrix of the state-space form of a
DSGE model.
Menu for estat

Syntax

estat transition , level(#) display options
Options
level(#) specifies the confidence level, as a percentage, for confidence intervals. The default is
level(95) or as set by set level; see [U] 20.8 Specifying the width of confidence intervals.

The state transition matrix is part of the state-space form of a DSGE model. It specifies the transition
matrix of the models state variables.
For examples, see [DSGE] intro 1, [DSGE] intro 3a, and [DSGE] intro 3b.

Entries in the state transition matrix H are functions of the structural parameter vector . Standard
errors for entries in H
b are calculated using the delta method.
Also see
102
Glossary
autoregressive process. An autoregressive process is a time series in which the current value of a
variable is a linear function of its own past values and a white-noise error term. A first-order
autoregressive process, denoted as an AR(1) process, is yt = yt1 + t . An AR(p) model contains
p lagged values of the dependent variable.
conditional mean. A conditional mean expresses the mean of one variable as a function of some
other variables. A regression function is a conditional mean.
control variable. A control variable is an endogenous variable. Control variables can be observed or
unobserved.
In a structural DSGE model, the current value of a control variable depends on the current value of
other control variables, on the expected future values of any control variable, and on the current
values of state variables. The current value of a control variable is found by solving the model
for the state-space form.
covariance stationary. A covariance stationary process is a weakly stationary process.
dynamic forecast. A dynamic forecast uses forecasted values wherever lagged values of the endogenous
variables appear in the model, allowing one to forecast multiple periods into the future.
dynamic stochastic general equilibrium model. A dynamic stochastic general equilibrium model is
a multivariate time-series model that specifies the structural relationship between state variables
and control variables and is typically derived from economic theory.
endogenous variable. An endogenous variable is a variable whose values are determined by the
equilibrium of a structural model. The values of an endogenous variable are determined inside the
system.
equilibrium. The equilibrium values of variables in a model are the values that satisfy all the models
equations simultaneously.
exogenous variable. An exogenous variable is one whose values change independently of the other
variables in a structural model. The values of an exogenous variable are determined outside the
system. In a time-series model, an exogenous variable is also a predetermined variable.
expected future value. An expected future value is a forecast of the value of a variable in the future
based on current information. In DSGE models, expected future values are computed under rational
expectations.
Under rational expectations, Et (yt+1 ) is the condition mean of yt+1 conditional on the complete
history of all variables in the model and the structure of the model itself.
forward operator. The forward operator F denotes the value of a variable at time t + 1. Formally,
F yt = yt+1 , and F 2 yt = F yt+1 = yt+2 .
identified. Identified is a condition required to estimate the parameters of a model. In other words,
only identified parameters can be estimated.
In DSGE models, the parameters are identified when there is a unique parameter vector that
maximizes the likelihood function. For a discussion of identification, see [DSGE] intro 6.
impulseresponse function. An impulseresponse function (IRF) measures the effect of a shock to an
endogenous variable on itself or another endogenous variable. The k th impulseresponse function
of variable i on variable j measures the effect on variable j in period t + k in response to a
one-unit shock to variable i in period t, holding everything else constant.
103
104 Glossary
independent and identically distributed. A series of observations is independent and identically

distributed (i.i.d.) if each observation is an independent realization from the same underlying
distribution. In some contexts, the definition is relaxed to mean only that the observations are
independent and have identical means and variances; see Davidson and MacKinnon (1993, 42).
initial values. Initial values specify the starting place for the iterative maximization algorithm used
by DSGE.
Kalman filter. The Kalman filter is a recursive procedure for predicting the state vector in a state-space
model.
lag operator. The lag operator L denotes the value of a variable at time t 1. Formally, Lyt = yt1 ,
and L2 yt = Lyt1 = yt2 .
likelihood-ratio (LR) test. The LR test is a classical testing procedure used to compare the fit of
two models, one of which, the constrained model, is nested within the full (unconstrained) model.
Under the null hypothesis, the constrained model fits the data as well as the full model. The LR test
requires one to determine the maximal value of the log-likelihood function for both the constrained
and the full models.
linearized model. A linearized model is an approximation to a model that is nonlinear in the variables
and nonlinear in the parameters. The approximation is linear in variables but potentially nonlinear
in the parameters.
model solution. A model solution is a function for the endogenous variables in terms of the exogenous
variables. A model solution is also known as the reduced form of a model.
In DSGE terminology, a model solution expresses the control variables as a function of the state
variables alone and expresses the state variables as a function of their values in the previous period
and shocks. The reduced form of a DSGE model is also known as the state-space form of the DSGE
model.
model-consistent expectation. A model-consistent expectation is the conditional mean of a variable
implied by the model under consideration.
For example, under rational expectations the model-consistent expectation of Et (yt+1 ) is the mean
of yt+1 implied by the model, conditional on the realizations of variables dated time t or previously.
nonpredetermined variable. A nonpredetermined variable is a variable whose value at time t is
determined by the system of equations in the model. Contrast with predetermined variable.
null hypothesis. In hypothesis testing, the null hypothesis typically represents the conjecture that one
is attempting to disprove. Often the null hypothesis is that a parameter is zero or that a statistic
is equal across populations.
one-step-ahead forecast. See static forecast.
policy matrix. The policy matrix in the reduced form of a DSGE model is the matrix that expresses
control variables as a function of state variables.
predetermined variable. A predetermined variable is a variable whose value is fixed at time t, given
everything that has occurred previously. More technically, the value of a predetermined variable
is fixed, given the realizations of all observed and unobserved variables at times t 1, t 2, . . . .
Glossary 105
rational expectations. A rational expectation of a variable does not deviate from the mean of that
variable in a predictable way. More technically, a rational expectation of a variable is the conditional
mean of the variable implied by the model.
realization. The realization of a random variable is the value it takes on when drawn.
reduced form. The reduced form of a model expresses the endogenous variables as functions of the
exogenous variables.
The reduced form of a DSGE model expresses the control variables as a function of the state
variables alone and expresses the state variables as a function of their values in the previous period
and shocks. The reduced form of a DSGE model is a state-space model.
saddle-path stable. A saddle-path stable model is a structural model that can be solved for its
state-space form. The existence of a saddle-path stable solution depends on the parameter values
of the model. For a discussion of saddle-path stability, see [DSGE] intro 5.
shock variable. A shock variable is a random variable whose value is specified as an independently
and identically distributed (i.i.d.) random variable. The maximum likelihood estimator is derived
under normally distributed shocks but remains consistent under i.i.d. shocks. Robust standard errors
must be specified when the errors are i.i.d. but not normally distributed.
state transition matrix. The state transition matrix in the reduced form of a DSGE model is the matrix
that expresses how the future values of state variables depend on their current values.
state variable. A state variable is an unobserved exogenous variable.
In DSGE models, a state variable is an unobserved exogenous variable that may depend on its own
previous value, the previous values of other state variables, and shocks.
state-space model. A state-space model describes the relationship between an observed time series
and an unobservable state vector that represents the state of the world. The measurement equation
expresses the observed series as a function of the state vector, and the transition equation describes
how the unobserved state vector evolves over time. By defining the parameters of the measurement
and transition equations appropriately, one can write a wide variety of time-series models in the
state-space form.
For DSGE models, the state-space form is the reduced form of the structural model.
The DSGE framework changes the jargon and the structure of state-space models. The measurement
equation is the vector of equations for the control variables, and the transition equation is the
vector of equations for the state variables. In contrast to the standard state-space model, DSGE
models allow a control variable to be unobserved.
static forecast. A static forecast uses actual values wherever lagged values of the endogenous variables
appear in the model. As a result, static forecasts perform at least as well as dynamic forecasts,
but static forecasts cannot produce forecasts into the future when lags of the endogenous variables
appear in the model.
Because actual values will be missing beyond the last historical time period in the dataset, static
forecasts can forecast only one period into the future (assuming only first lags appear in the model);
thus they are often called one-step-ahead forecasts.
steady-state equilibrium. A steady-state equilibrium is a time-invariant rest point of a dynamic
system.
More technically, a steady-state equilibrium is a set of values for the endogenous variables to
which the dynamic system will return after an exogenous variable is changed or a random shock
occurs. This set of values is time invariant in that it does not depend on the time period in which
the change or shock occurs. Multistep dynamic forecasts converge to these values. A steady-state
106 Glossary
equilibrium is also known as a long-run equilibrium because it specifies time-invariant values for
the endogenous variables to which the dynamic system will return, if left unshocked.
stochastic equation. A stochastic equation, in contrast to an identity, is an equation in a forecast model
that includes a random component, most often in the form of an additive error term. Stochastic
equations include parameters that must be estimated from historical data.
stochastic trend. A stochastic trend is a nonstationary random process. Unit-root process and random
coefficients on time are two common stochastic trends. See [TS] ucm for examples and discussions
of more commonly applied stochastic trends.
strict stationarity. A process is strictly stationary if the joint distribution of y1 , . . . , yk is the same
as the joint distribution of y1+ , . . . , yk+ for all k and . Intuitively, shifting the origin of the
series by units has no effect on the joint distributions.
structural model. A structural model specifies the theoretical relationship among a set of variables.
Structural models contain both endogenous variables and exogenous variables. Parameter estimation
and interpretation require that structural models be solved for a reduced form.
trend. The trend specifies the long-run behavior in a time series. The trend can be deterministic or
stochastic. Many economic, biological, health, and social time series have long-run tendencies to
increase or decrease. Before the 1980s, most time-series analysis specified the long-run tendencies
as deterministic functions of time. Since the 1980s, the stochastic trends implied by unit-root
processes have become a standard part of the toolkit.
Wald test. A Wald test is a classical testing procedure used to compare the fit of two models, one
of which, the constrained model, is nested within the full (unconstrained) model. Under the null
hypothesis, the constrained model fits the data as well as the full model. The Wald test requires
one to fit the full model but does not require one to fit the constrained model.
weakly stationary. A process is weakly stationary if the mean of the process is finite and independent
of t, the unconditional variance of the process is finite and independent of t, and the covariance
between periods t and t s is finite and depends on t s but not on t or s themselves.
Weakly-stationary processes are also known as covariance stationary processes.
white noise. A variable ut represents a white-noise process if the mean of ut is zero, the variance
of ut is 2 , and the covariance between ut and us is zero for all s 6= t.
Reference
Davidson, R., and J. G. MacKinnon. 1993. Estimation and Inference in Econometrics. New York: Oxford University
Press.
Subject and author index
See the combined subject index and the combined author index in the Glossary and Index.
107

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STATA LINEARIZED DYNAMIC

STOCHASTIC GENERAL EQUILIBRIUM

A Stata Press Publication

The suggested citation for this software is

[GSM] Getting Started with Stata for Mac

[M] Mata Reference Manual

Description Remarks and examples Also see

Remarks and examples

[DSGE] intro 1 Introduction to DSGE models

[DSGE] intro 3 Classic DSGE examples

[DSGE] intro 4 Writing a DSGE in a solvable form

[DSGE] intro 5 Stability conditions

[DSGE] dsge Linearized dynamic stochastic general equilibrium models

Description Remarks and examples References Also see

Remarks and examples

Introduction to DSGE models

How to write down a DSGE

Using the dsge command

Writing down linearized DSGEs

storage display value

datestr str10 %-10s Observation date

Sorted by: dateq

Specifying the model to dsge

xt = Et xt+1 (rt Et t+1 gt ) (4)

ut+1 = u ut + t+1 (7)

Parameter estimation and interpretation

sd(e.u) 2.317589 .2988024 1.731947 2.90323

Coef. Std. Err. z P>|z| [95% Conf. Interval]

_nl_1 1.955844 .2899255 6.75 0.000 1.387601 2.524088

Policy and transition matrices

Example 1: Obtaining the policy matrix

Example 2: Obtaining the transition matrix

Example 3: Graphing an IRF

. irf graph irf, impulse(u) response(x p r u) byopts(yrescale)

Example 4: Out-of-sample forecast

We use tsappend() to extend the dataset by 3 years, or 12 quarters.

Starting period: 2017q1

2010q1 2012q3 2015q1 2017q3 2020q1

Example 5: Within-sample forecast

Starting period: 2014q1

. tsline p d2_p if tin(2010q1, 2021q1), tline(2014q1)

We plot both the observed inflation and the forecast.

2010q1 2012q3 2015q1 2017q3 2020q1

Inflation rate Inflation rate (dsgemodel d2_)

Structural and reduced forms of DSGE models

where yt is a vector of control variables, xt is a vector of state variables, and t is a vector of

where yt is a vector of control variables, xt is a vector of state variables, and t is a vector of

Description Remarks and examples Also see

Remarks and examples

Preview of dsge syntax

Specifying the system of equations

State variables and shocks

Expectations of future values of control variables

Specifying parameters using substitutable expressions

we could extend our previous dsge command as follows:

we would change our command to

(1/)yt = Et (yt+1 ) + zt + (/)xt

and change our dsge command to

yt = (1/)Et (yt+1 ) + (/)xt + zt

ut+1 = u ut + t+1 (7)

where yt is a vector of control variables, xt is a vector of state variables, and t is a vector of

where yt is a vector of control variables, xt is a vector of state variables, and t is a vector of

. dsge (p = {beta}E(F.p) + {kappa}x)

Third, we rewrite (3), substituting ut for t . The consumption equation becomes

ct = (1 h)wt + hEt ct+1 + t (1)