Empirical Modelling of Japans Markup and Ination,

1976-2000

Takamitsu Kurita

Faculty of Economics, Fukuoka University

1 Objective

The objective of this paper is to model Japans markup and ination using
historical time series data covering the last quarter of the 20th century.

It is demonstrated that a cointegrated vector autoregressive (VAR) model re-

duces to a parsimonious data-congruent dynamic system subject to economic
interpretations.
Interactions between prices and wages have been of great interest for macro-
economists, and theoretical models for prices and wages have played a key role
in the development of modern macroeconomics.

Tobin (1972) presents a two-dimensional theoretical model encompassing a

markup equation, and the model is successful in describing dynamic interde-
pendent relationships of prices and wages.

Regarding applied work using time series data, Sargan (1964) is a seminal paper
in that he gives congruent econometric representations of wages and prices in
the UK based on an equilibrium correction approach, the approach intimately
linked to a cointegration analysis.

A recent empirical analysis of price and wage time series data was performed
by Brdsen, Jansen and Nymoen (2003), Marcellino and Mizon (2001), inter
alia.
Macroeconomic time series data often exhibit non-stationary behaviour, and
thus need to be treated as integrated processes rather than stationary.

The concept of cointegration introduced by Granger (1981) therefore plays

an important role in time series econometrics, and a cointegrated VAR model
developed by Johansen (1988, 1996) has become a major econometric tool for
macro and nancial economists.

The cointegrated VAR analysis is well tted in general-to-specic modelling

approach, due to the fact that the analysis usually starts with the investigation
of general unrestricted VAR models.

Hendry and Mizon (1993) discuss a model reduction procedure using the coin-
tegrated VAR model. See also Hendry and Doornik (1994), Kurita (2007)
for the general-to-specic modelling methodology using the cointegrated VAR
analysis.
Furthermore, the concept of weak exogeneity introduced by Engle, Hendry and
Richard (1983) also plays an important part in econometrics.

Weak exogeneity permits us to model a partial or conditional system alone,

instead of a full system, for the purpose of making e cient statistical inferences
on parameters of interest.

Weak exogeneity in the cointegrated VAR system is explored by Johansen

(1992) and Urbain (1992).

Cointegration and weak exogeneity provide a methodological basis for empirical

investigation pursued in this paper.
This paper aims to achieve a data-congruent parsimonious representation of
markup and ination using Japans historical time series data.

The empirical exploration sheds useful light on deeper understanding of the

Japanese economy in 1976 - 2000, a quarter-century period of Japans eco-
nomic turmoil, during which an asset-price bubble economy took place and
then collapsed, leading to Japans lost decade.

This paper adopts a cointegrated VAR approach to modelling the data of

markup, ination and various other macroeconomic series in Japan.

The analysis indicates the existence of a long-run economic linkage interpreted

as an empirical representation of countercyclical markup, see Rotemberg and
Woodford (1999), inter alia.
A set of variables in the cointegrated system, apart from markup and ination,
are judged to be weakly exogenous for parameters of interest, thereby allowing
us to estimate a partial model given the weakly exogenous variables.

The model reduction is then conducted so as to achieve a parsimonious dynamic

econometric system for markup and ination.

It is noteworthy that such a stable structure has been revealed from the analysis
of the data covering the period of Japans economic turmoil.
2 Outline

1. Countercyclical Markup and Ination Dynamics

2. Cointegrated VAR Model

3. Empirical Analysis of Japans Time Series Data

4. Conclusion
3 Countercyclical Markup and Ination Dynam-
ics

This section introduces a set of economic variables to be analysed in a cointe-

grated VAR model.

Since the aim of this paper is to estimate an empirical model for Japans
markup and ination, it is necessary to conceive a plausible long-run economic
relationship associated with these two variables

an interpretable economic linkage which may correspond to an empirical

cointegrating relation estimated from the data.
To this end, let us suppose that markup pricing is formulated as
!
Wt
Pt = t ; (1)
At
where Pt is the price level, Wt is the nominal wage, At is the labour productivity,
and t is the markup. Let us introduce the output Yt and dene yt = log Yt.

In order to map (1) to an observable relationship subject to an empirical inves-

tigation, the following specications of At and t are assumed:

log t = yt + ut and log At = t + t; (2)

where ut and t represent stationary error terms capturing unspecied dynam-
ics.

The specication of t is based on the fact that the markup tends to be mod-
erately countercyclical with economic growth.
See Blanchard and Fisher (1989, Ch.9), Solon, Barsky and Parker (1994),
Rotemberg and Woodford (1999), inter alia.

The productivity At is specied to be an exponential function of determinis-

tic trend in (2), based on the assumption that productivity growth should be
approximated to a stable upward trending path.

Substituting (2) into (1) and taking logs of both sides can lead to

pt (w t t) + yt stationary; (3)
for pt = log Pt and wt = log Wt. Equation (3) is a candidate for a cointe-
grating relationship based on the notion of countercyclical markup.

Let us dene productivity-adjusted wage wt = wt t for future reference.

An ination process should be dependent on the markup, possibly adjusting to
disequilibrium errors represented by (3).
It is therefore reasonable to conceive the following bivariate equilibrium cor-
rection model (ECM) capturing countercyclical markup and ination dynamics
simultaneously:
" # " # " #
(pt wt ) h i
= 1 + 1 pt 1 wt 1 + yt 1 (4)
2p
t 2 2
lX1 " #" # " #
11;i 12;i (pt wt ) 1;t
+ 2p + ;
i=1 21;i 22;i t 2;t

where j;t for j = 1; 2 is a set of stationary processes consisting of omitted

short-run dynamic terms and innovations.

Such a bivariate system as (4) may be seen as an empirical representation of

Tobins wage-price model (Tobin, 1972).
This paper is interested in estimating an empirical dynamic model for markup
and ination, thus the ECM given by (4) is seen as a target representation which
should be achieved as result of general-to-specic econometric modelling.

Furthermore, it is often pointed out that the spread between the long and
short term interest rates contains information about expected future economic
growth.

See Stock and Watson (1989), Bernard and Gerlach (1998), Hamilton and Kim
(2002), and Ichiue (2004), inter alia.

The interest rate dierential or yield spread, denoted by rst, may play a signif-
icant role in the short-run dynamics, or i;t, of the ECM.
4 Cointegrated VAR Model

The argument so far allows us to introduce an I (1) cointegrated VAR(k) model

encompassing (4), formulated as

Xt = (pt wt ; pt; yt; rst)0 ; (5)

and
kX1
0; Xt 1
Xt = + i Xt i + + "t; for t = 1; :::; T; (6)
t i=1
where a sequence of innovations "t has independent and identical normal
N (0; ) distributions conditional on X k+1; :::; X0, and ; 2 R4 r for
r < 4, 2 Rr 1 , 2 R4 1 and i 2 R4 4. Let 0 = 0; and
0
Xt 1 = Xt0 1; t for future reference.
Johansen (1996) demonstrates details of likelihood-based inference for these
parameters. In equation (6) is referred to as adjustment vectors, while 0
is called cointegrating vectors.

In practice, there may be a case where both pt and wt exhibit I (2)-type non-
stationary behaviour.

Considering the presence of a markup relation between pt and wt, it is reason-

able to conjecture that pt wt is seen as an I (1) process as a result of the
removal of the common I (2) stochastic trend or nominal-to-real transformation
(see Kongsted, 2005).

In addition, there is a possibility that yt may be seen as a stationary process

rather than I (1).
As shown in Johansen (1996, page 74), it is possible to include stationary
variables in the VAR model as long as they are relevant in terms of economic
theory and insight.

As the cointegrating rank r is unknown in practice, the rank needs to be deter-

mined based on the data analysis. A log-likelihood ratio (log LR) test statistic
consists of the null hypothesis of r cointegration rank H (r) against the alter-
native hypothesis H (p), and its asymptotic quantiles are provided by Johansen
(1996, Ch.15).

Determining the cointegrating rank in the VAR model allows us to test various
restrictions on , and in order to pursue the adjustment structure and
cointegrating relationships subject to economic interpretation.

Cointegrating relationships, embodied by 0Xt 1, correspond to a set of sta-

tionary linear combinations, acting as equilibrium correction mechanisms in the
VAR model.
Thus it is necessary to check if the theoretical relationship (3) belongs to a
class of estimated cointegrating relationships.

Next, in order to derive the bivariate system (4) from the cointegrated VAR
model (6) as a partial or conditional data-representation, let the process be
0
decomposed as Xt = 0 ; X0
X1t 2t for X1t = (pt wt ; pt)0 and X2t =
( yt; rst)0 :

The parameters and error terms appearing in (6) are also expressed as
! ! ! ! !
1 1;i 1 "1;t 11 12
= , i= ; = ; "t = ; = :
2 2;i 2 "2;t 21 22

We are interested in estimating (4), or a bivariate system for X1t conditional

on X2t with no loss of information.
Suppose 2 = 0, then (6) is decomposed into a model for X1t conditional on
X2t and a marginal model for X2t as follows:
kX1
X1t = ! X2t + 1 0X + e Xt i + e 1 + "e1;t; (7)
t 1 1;i
i=1
kX1
X2t = 2;i Xt i + 2 + "2;t; (8)
i=1
where
1 e
!= 12 22 ; 1;i = 1;i ! 2;i; e 1 = 1 ! 2; "e1;t = "1;t !"2;t;
and
! " ! !#
"e1;t 0 11:2 0
=N ; ;
"2;t 0 0 22
for
1
11:2 = 11 12 22 21 :
Note that 0X is not embedded in the marginal model (8).
t 1

It is then possible to say that (4) may correspond to the conditional model (7).

Under the condition of 2 = 0, X2t is said to be weakly exogenous for the

following parameters of interest:

1; ; !; e 1;1; :::; e 1;k 1; e 1; and 11:2 : (9)

The parameters in (9) correspond to those appearing in (4), and may therefore
be treated as the set of parameters of interest.

As long as the condition for weak exogeneity 2 = 0 is satised, the parameters

of interest (9) can be estimated from the conditional model (7) alone without
loss of information, with no need for the estimation of the marginal model (8).
5 Empirical Analysis of Japans Time Series Data

5.1 An Overview of the Data

An overview of Japans time series data for

Xt = (pt wt ; pt; yt; rst)0

is presented below.
 0.3
p w (a) p (b)
t t t
0.2 0.02

0.1
0.01

0.0

0.00
-0.1

1975 1980 1985 1990 1995 2000 1975 1980 1985 1990 1995 2000

(p w )* y y* (p w )* rs (d)
t t t t (c) t t t

0.02 0.02

0.00 0.00

-0.02 -0.02

1975 1980 1985 1990 1995 2000 1975 1980 1985 1990 1995 2000
5.2 Estimating the Unrestricted VAR Model

The empirical analysis commences with a general unrestricted VAR(5) model

incorporating two deterministic terms, a constant and linear trend.

The inclusion of deterministic trend can be justied as it is treated as a local

approximation to productivity; see the argument above.

F-tests for the lag order determination indicate that variables at lag length 5
seem to be irrelevant so the VAR(5) model reduces to the VAR(4) model.

Some evidence is then found for signicance of a variable with lag length 4,
suggesting that further model reduction is likely to be inappropriate.
The VAR(4) model is chosen for further analysis, so that the sample period
eective for estimation is 1976.3 - 2000.4 and the number of observations is
98.

The unrestricted VAR model is a purely statistical representation, so the esti-

mated coe cients are not necessarily subject to economic interpretation.

After identifying the cointegrating relations and conducting the model reduc-
tion, it is possible to pursue such interpretation.

The unrestricted VAR model provides a starting point towards a parsimonious

representation, and should therefore pass a set of residual diagnostic tests such
as non-autocorrelation and normality.
 Single equation tests pt wt pt yt rst
Autocorr. [Far (6,74)] 1.48 [0.20] 1.47 [0.20] 0.50 [0.81] 1.37 [0.24]
ARCH [Farch(6,68)] 0.18 [0.98] 0.09 [1.00] 1.09 [0.38] 0.79 [0.58]
Hetero.[Fhet(34,45)] 0.61 [0.93] 0.72 [0.84] 0.87 [0.67] 0.49 [0.98]
Normality [ 2nd(2)] 3.16 [0.21] 3.02 [0.22] 3.66 [0.16] 2.22 [0.33]

Vector tests
Autocorr.[Far (16,223)] 1.22 [0.25] Hetero.[Fhet(340,374)] 0.66 [1.00]
- [Far (96,212)] 1.04 [0.41] Normality [ 2nd(8)] 7.17 [0.52]

N ote: Figures in brackets are p-values.

The diagnostic tests of the VAR(4) model are given in the table above.

Most of the test results are given in the form Fj (k; T l), which means an
approximate F-test against the alternative hypothesis j :
kth-order serial correlation (Far : see Godfrey, 1978; Nielsen, 2007), kth-order
autoregressive conditional heteroscedasticity (Farch: see Engle, 1982), het-
eroscedasticity (Fhet: see White, 1980), and a chi squared test for normality
( 2nd: see Doornik and Hansen, 1994).

The diagnostic test statistics are all insignicant, thereby allowing us to con-
clude that this model is subject to the subsequent likelihood-based cointegration
analysis and model reduction.
5.3 Choosing the Cointegrating Rank

The table below presents two types of log LR test statistics for the choice of
r, so-called trace test statistics (trace) and maximum eigenvalue test statistics
(max:eigen:). The modulus of the six largest roots of a companion matrix for
the VAR model are also provided in the table.

r=0 r 1 r 2 r 3
trace 64:92[0:04] 40:51[0:08] 23:77[0:09] 9:66[0:15]
max:eigen: 24:41[0:32] 16:73[0:48] 14:12[0:25] 9:66[0:15]

mod (r = 1) 1:00 1:00 1:00 0:78 0:78 0:76

mod (r = 2) 1:00 1:00 0:83 0:83 0:79 0:78

N ote: * denotes signicance at the 5% level.

According to the rst panel, the trace test rejects the null hypothesis of r = 0
and does not reject the remaining ones at the 5% signicance level, hence
supporting r = 1. In contrast, the maximum eigenvalue test does not reject
any null hypotheses at the same level.

As discussed by Juselius (2006, Ch.8), the choice of the cointegrating rank is a

very di cult task and so we should make use of as much additional information
as possible in order to determine the rank.

The second panel, motivated by the results of the trace test in the rst panel,
provides modulus (denoted mod) of the six largest eigenvalues of a companion
matrix of the VAR model restricted with r = 1 or with r = 2.

No eigenvalue over 1.0 suggests that the model does not include any explosive
root, and all the eigenvalues apart from the imposed unit roots are distinct
from unity.
Judging from these outcomes, the restriction of r = 1 seems to be appropriate
for the description of the data.

In order to consolidate the argument for r = 1, the gure below presents time
series plots of the estimated cointegrating combination under the restriction of
r = 1.

The gure displays recursive plots of some of the trace test statistics.
0.35 (a)
Cointegrating Relation

0.30

0.25
1980 1985 1990 1995 2000
trace (r=0) trace (r<=1) (b)
Critical Value (r=0) Critical Value (r<=1)
70

60

50

40

30

1985 1990 1995 2000

5.4 Identifying the Long-Run Economic Relationship

This sub-section explores valid restrictions on the adjustment and cointegrating

space in the I (1) cointegrated VAR system. The determination of the cointe-
grating rank, or r = 1, enables us to inspect such restrictions using a standard
2 -based asymptotic inference.

Weak exogeneity plays an important role in the model reduction procedure.

Testing weak exogeneity in the I (1) cointegrated system corresponds to check-
ing zero restrictions on elements of .

If rst and yt are judged to be weakly exogenous for parameters of interest,

one has only to model the conditional system for the variables pt wt and pt
in order to conduct inferences with no loss of information.
With regard to restrictions on , it is of interest to identify the long-run em-
pirical relationship interpreted as the markup equation given by (3).

It is expected that the ination process could play a little or no role in the
long-run relationship, so the exclusion of pt from the long-run cointegrating
relation should be examined.

Thus, the following restrictions on and are jointly investigated:

1. zero restrictions on the elements of corresponding to rst and yt ,

2. a zero restriction on the element of corresponding to pt.

A set of restricted estimates is given in the table below, together with the
corresponding log LR test statistic and p-value.

pt wt pt yt rst t 2 (3)
b0 0:19 0:19 0 0 4:76[0:19]
(0:05) (0:05) ( ) ( )
b 0 1 0 1:12 0:002 0:00389
( ) ( ) (0:51) (0:004) (1:51e-04)

N ote: The gures in the parentheses are standard errors.

The table shows that the set of hypotheses is not rejected at the 5% level,
indicating that rst and yt are weakly exogenous for the parameters of interest
and pt can be excluded from the cointegrating space.

It turns out that the coe cient for rst in the cointegrating space is insigni-
cant and that for yt is close to unity. Thus, it would be worthwhile to test
additional restrictions as follows:
 1. a zero restriction on the element of corresponding to rst;

2. a unitary restriction on the element of corresponding to yt :

The table below presents a set of restricted estimates, together with the corre-
sponding log LR test statistic and p-value.

pt wt pt yt rst t 2 (5)
b0 0:18 0:19 0 0 5:01[0:41]
(0:04) (0:04) ( ) ( )
b 0 1 0 1 0 0:00381
( ) ( ) ( ) ( ) (7:15e-05)

N ote: The gures in the parentheses are standard errors.

Again, the set of hypotheses is not rejected at the 5% level, indicating that the
long-run economic relationship is given by

pt (w t 0:00381t) + yt : (10)
Interpreting the linear trend as an approximation of labour productivity growth,
this cointegrating relation indicates that a markup over productivity-adjusted
wages tends to move in the opposite direction to the real output growth.

Relationship (10) is therefore interpreted as (3), consistent with the accepted

view of countercyclical markup, discussed in the literature of macro and labour
economics.

See Blanchard and Fisher, 1989, Ch.9; Solon, Barsky and Parker, 1994; Rotem-
berg and Woodford, 1999; Romer, 2001, Ch.5, inter alia.
Various theories have been developed to explain this pattern of countercyclical
markup such as collusion in imperfect competition and kinked demand curves,
see the above references.

Under these economic interpretations, (10) can be justied as the representation

of a meaningful long-run economic relationship.
5.5 A Parsimonious Equilibrium Correction Model

We are now in a position to achieve an equilibrium correction model for Japans

markup and ination, corresponding to the bivariate system (4) above.

The concept of productivity-adjusted wages is used to provide appropriate em-

pirical markup, that is, an adjusted wage index is dened as wt = wt
0:00381t such that markup is represented by pt wt .

As a result of this adjustment, the sample mean of (pt wt ) appears to be

around zero rather than a negative value.

The starting point of the analysis is to map the data to the I (0) space by
dierencing and using the restricted cointegrating combination.
We then estimate a two-dimensional I (0) VAR system for (pt wt ) and
2 p conditional on rs and 2 y .
t t t

A set of insignicant terms, 2pt 2, 2pt 3, 2yt 1, 2y

t 2 and rst 3,
are dropped so as to reach a parsimonious VAR model.

Short-run dynamics with large standard errors then continue to be removed,

that is, pt 2 wt 2 from the equation for (pt wt ), and rst 1
and rst 2 from the 2pt equation.

Imposing constraints on some of the coe cients which have similar size in order
to seek a parsimonious representation, an empirical price-wage mechanism is
attained as follows:
 pt dwt = 0:12 a rs
t 0:08 2a y
t + 0:27 (pt 1 wt 1 )
(0:05) (0:02) (0:09)
0:14 2p 0:19 rst 1 + 0:11 pt 3 wt 3
t 1
(0:08) (0:06) (0:07)
0:17 ecmt 1 + 0:05 ;
(0:03) (0:01)
b = 0.0038; Far (6,81) = 1.97[0.08];
2 (2) = 2.75[0.25]; F
nd arch (6,78) = 0.35[0.91]; Fhet(20,69) = 1.02[0.46];

2p
bt = 0:12 rst 0:09 2y 0:64 2p 0:17 a (p wt 1 )
t t 1 t 1
(0:06) (0:03) (0:09) (0:07)
+ 0:4 (pt 2 wt 2 ) 0:16 ecmt 1 + 0:05 ;
(0:08) (0:03) (0:01)
b = 0.0038; Far (6,81) = 2.18[0.053];
2 (2) = 2.73[0.26]; F
nd arch(6,78) = 0.33[0.92]; Fhet(20,69) = 1.02[0.45];

V ector T ests : Far (24,154) = 1.18[0.27]; 2nd(4) = 4.69[0.32]; Fhet(60,200) = 0.90[0.68];

where
ecmt = pt wt + yt ;
and
a rs = rst + rst 2; 2a y= 2y + 2 yt 3 ;
t t t
a (p wt 1) = (pt 1 wt 1 ) (pt 3 wt 3):
t 1

The method of constrained full-information maximum likelihood was used to

estimate the model above.

The standard errors on coe cients are given in parentheses.

The test statistics for a single equation are reported under each equation, and
the statistics for the whole system (the vector tests) are reported under the two
equations.
0.015 (pt-w*t) Fitted 1.0
(a)
(b) (pt-w*t)

0.010
0.5

0.005

0.0
0.000

-0.005 -0.5

-0.010

1980 1985 1990 1995 2000 0 5 10

2
pt Fitted 1.0 (d) 2
(c) pt
0.01
0.5

0.00 0.0

-0.5

-0.01

1980 1985 1990 1995 2000 0 5 10

 2
2.5 (pt-w*t) (scaled) 2.5 pt (scaled)
(a) (b)

0.0 0.0

-2.5 -2.5

1980 1985 1990 1995 2000 1980 1985 1990 1995 2000
2
(pt-w*t) 0.01 pt (scaled)
0.01 (d)
(c)

0.00 0.00

-0.01
1985 1990 1995 2000 1985 1990 1995 2000

1.0 1.0
(e) (f) 2
(pt-w*t) 1% pt 1%

0.5 0.5

1985 1990 1995 2000 1985 1990 1995 2000

None of the diagnostic test statistics are signicant, suggesting that the parsi-
monious system is a data-congruent representation.

Next, let us consider interpretation of the parsimonious model.

A change in the interest rate dierential has a negative eect on the markup
and ination growth; this may reect information on expected economic growth
and ination contained in the dierential.

In line with the interest rate dierential, an acceleration of the real output
growth also has a negative inuence on the markup and ination growth.

This is consistent with the countercyclical behaviour found in the cointegrating

relation, and could be interpreted as a short-run reection of this.
6 Summary and Conclusion

This paper, using a cointegrated VAR methodology, estimates a data-congruent

econometric model for Japans markup and ination.

The analysis provides evidence for the presence of a long-run economic relation-
ship in the data, interpreted as an empirical representation of countercyclical
markup.

A set of variables in the cointegrated system except markup and ination are
judged to be weakly exogenous for parameters of interest, thereby enabling us
to estimate a partial system given the weakly exogenous variables.

The model reduction is then conducted in order to attain a parsimonious dy-

namic econometric system.
The preferred parsimonious system has passed a battery of diagnostic tests,
thereby being judged to be a data-congruent representation of countercyclical
markup and ination dynamics.

It should be noted that such a stable model has been estimated from the analysis
of the data covering the period of Japans economic turmoil.

The empirical exploration sheds useful light on deeper understanding of the

Japanese economy in the last quarter of the 20th century.