Ch11 Slides
Ch11 Slides
Ch11 Slides
Panel Data
• Panel data, also known as longitudinal data, have both time series and cross-
sectional dimensions.
• They arise when we measure the same collection of people or objects over a
period of time.
• Econometrically, the setup is
yit xit uit
where yit is the dependent variable, is the intercept term, is a k 1 vector
of parameters to be estimated on the explanatory variables, xit; t = 1, …, T;
i = 1, …, N.
• The simplest way to deal with this data would be to estimate a single, pooled
regression on all the observations together.
• But pooling the data assumes that there is no heterogeneity – i.e. the same
relationship holds for all the data.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 2
The Advantages of using Panel Data
There are a number of advantages from using a full panel technique when a panel
of data is available.
• We can address a broader range of issues and tackle more complex problems
with panel data than would be possible with pure time series or pure cross-
sectional data alone.
• One approach to making more full use of the structure of the data would be
to use the SUR framework initially proposed by Zellner (1962). This has
been used widely in finance where the requirement is to model several
closely related variables over time.
• A SUR is so-called because the dependent variables may seem unrelated
across the equations at first sight, but a more careful consideration would
allow us to conclude that they are in fact related after all.
• Under the SUR approach, one would allow for the contemporaneous
relationships between the error terms in the equations by using a generalised
least squares (GLS) technique.
• The idea behind SUR is essentially to transform the model so that the error
terms become uncorrelated. If the correlations between the error terms in the
individual equations had been zero in the first place, then SUR on the system
of equations would have been equivalent to running separate OLS
regressions on each equation.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 4
Fixed and Random Effects Panel Estimators
• There are two main classes of panel techniques: the fixed effects estimator
and the random effects estimator.
• The fixed effects model for some variable yit may be written
yit xit i vit
• We can think of i as encapsulating all of the variables that affect yit cross-
sectionally but do not vary over time – for example, the sector that a firm
operates in, a person's gender, or the country where a bank has its
headquarters, etc. Thus we would capture the heterogeneity that is
encapsulated in i by a method that allows for different intercepts for each
cross sectional unit.
• This model could be estimated using dummy variables, which would be
termed the least squares dummy variable (LSDV) approach.
where D1i is a dummy variable that takes the value 1 for all observations on
the first entity (e.g., the first firm) in the sample and zero otherwise, D2i is a
dummy variable that takes the value 1 for all observations on the second
entity (e.g., the second firm) and zero otherwise, and so on.
• The LSDV can be seen as just a standard regression model and therefore it
can be estimated using OLS.
• Now the model given by the equation above has N+k parameters to estimate.
In order to avoid the necessity to estimate so many dummy variable
parameters, a transformation, known as the within transformation, is used to
simplify matters.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 7
The Within Transformation
• Time-variation in the intercept terms can be allowed for in exactly the same
way as with entity fixed effects. That is, a least squares dummy variable
model could be estimated
yit xit 1D1t 2 D2t ... T DTt vit
where D1t, for example, denotes a dummy variable that takes the value 1 for
the first time period and zero elsewhere, and so on.
• The only difference is that now, the dummy variables capture time variation
rather than cross-sectional variation. Similarly, in order to avoid estimating a
model containing all T dummies, a within transformation can be conducted to
subtract away the cross-sectional averages from each observation
• Finally, it is possible to allow for both entity fixed effects and time fixed
effects within the same model. Such a model would be termed a two-way
error component model, and the LSDV equivalent model would contain both
cross-sectional and time dummies
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 11
Investigating Banking Competition with a Fixed Effects
Model
• The UK banking sector is relatively concentrated and apparently extremely
profitable.
• It has been argued that competitive forces are not sufficiently strong and that
there are barriers to entry into the market.
• A study by Matthews, Murinde and Zhao (2007) investigates competitive
conditions in UK banking between 1980 and 2004 using the Panzar-Rosse
approach.
• The model posits that if the market is contestable, entry to and exit from the
market will be easy (even if the concentration of market share among firms is
high), so that prices will be set equal to marginal costs.
• The technique used to examine this conjecture is to derive testable
restrictions upon the firm's reduced form revenue equation.
• The model also includes several variables that capture time-varying bank-
specific effects on revenues and costs, and these are: RISKASS, the ratio of
provisions to total assets; ASSET is bank size, as measured by total assets; BR
is the ratio of the bank's number of branches to the total number of branches
for all banks.
• Finally, GROWTHt is the rate of growth of GDP, which obviously varies
over time but is constant across banks at a given point in time; i is a bank-
specific fixed effects and vit is an idiosyncratic disturbance term. The
contestability parameter, H is given as 1 + 2 + 3
• Unfortunately, the Panzar-Rosse approach is only valid when applied to a
banking market in long-run equilibrium. Hence the authors also conduct a
test for this, which centres on the regression
ln ROAit 0 '1 ' ln PLit 2 ' ln PKit 3 ' ln PFit 1 ' ln RISKASSit
2 ' ln ASSETit 3 ' ln BRit 1 ' GROWTH t i wit
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Methodology (Cont’d)
• The explanatory variables for the equilibrium test regression are identical to
those of the contestability regression but the dependent variable is now the
log of the return on assets (lnROA).
• Equilibrium is argued to exist in the market if 1' + 2' + 3'
• Matthews et al. employ a fixed effects panel data model which allows for
differing intercepts across the banks, but assumes that these effects are fixed
over time.
• The fixed effects approach is a sensible one given the data analysed here
since there is an unusually large number of years (25) compared with the
number of banks (12), resulting in a total of 219 bank-years (observations).
• The data employed in the study are obtained from banks' annual reports and
the Annual Abstract of Banking Statistics from the British Bankers
Association. The analysis is conducted for the whole sample period, 1980-
2004, and for two sub-samples, 1980-1991 and 1992-2004.
• The null hypothesis that the bank fixed effects are jointly zero (H0: i = 0) is
rejected at the 1% significance level for the full sample and for the second
sub-sample but not at all for the first sub-sample.
• Overall, however, this indicates the usefulness of the fixed effects panel
model that allows for bank heterogeneity.
• The main focus of interest in the table on the previous slide is the equilibrium
test, and this shows slight evidence of disequilibrium (E is significantly
different from zero at the 10% level) for the whole sample, but not for either
of the individual sub-samples.
• Thus the conclusion is that the market appears to be sufficiently in a state of
equilibrium that it is valid to continue to investigate the extent of competition
using the Panzar-Rosse methodology. The results of this are presented on the
following slide.
• The value of the contestability parameter, H, which is the sum of the input
elasticities, falls in value from 0.78 in the first sub-sample to 0.46 in the
second, suggesting that the degree of competition in UK retail banking
weakened over the period.
• However, the results in the two rows above that show that the null
hypotheses that H = 0 and H = 1 can both be rejected at the 1% significance
level for both sub-samples, showing that the market is best characterised by
monopolistic competition.
• As for the equilibrium regressions, the null hypothesis that the fixed effects
dummies (i) are jointly zero is strongly rejected, vindicating the use of the
fixed effects panel approach and suggesting that the base levels of the
dependent variables differ.
• Finally, the additional bank control variables all appear to have intuitively
appealing signs. For example, the risk assets variable has a positive sign, so
that higher risks lead to higher revenue per unit of total assets; the asset
variable has a negative sign, and is statistically significant at the 5% level or
below in all three periods, suggesting that smaller banks are more profitable.
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The Random Effects Model
• Unlike the fixed effects model, there are no dummy variables to capture the
heterogeneity (variation) in the cross-sectional dimension.
• Instead, this occurs via the i terms.
• Note that this framework requires the assumptions that the new cross-
sectional error term, i, has zero mean, is independent of the individual
observation error term vit, has constant variance, and is independent of the
explanatory variables.
• The parameters ( and the vector) are estimated consistently but
inefficiently by OLS, and the conventional formulae would have to be
modified as a result of the cross-correlations between error terms for a given
cross-sectional unit at different points in time.
• Instead, a generalised least squares (GLS) procedure is usually used. The
transformation involved in this GLS procedure is to subtract a weighted mean
of the yit over time (i.e. part of the mean rather than the whole mean, as was
the case for fixed effects estimation).
• Define the ‘quasi-demeaned’ data as yit* yit yi and similarly for xit,
• will be a function of the variance of the observation error term, v2, and of
the variance of the entity-specific error term, 2:
v
1
T 2 v2
• This transformation will be precisely that required to ensure that there are no
cross-correlations in the error terms, but fortunately it should automatically
be implemented by standard software packages.
• Just as for the fixed effects model, with random effects, it is also
conceptually no more difficult to allow for time variation than it is to allow
for cross-sectional variation.
• In the case of time-variation, a time period-specific error term is included and
again, a two-way model could be envisaged to allow the intercepts to vary
both cross-sectionally and over time.
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Fixed or Random Effects?
• It is often said that the random effects model is more appropriate when the
entities in the sample can be thought of as having been randomly selected
from the population, but a fixed effect model is more plausible when the
entities in the sample effectively constitute the entire population.
• Also, since there are fewer parameters to be estimated with the random
effects model (no dummy variables or within transform to perform), and
therefore degrees of freedom are saved, the random effects model should
produce more efficient estimation than the fixed effects approach.
• However, the random effects approach has a major drawback which arises
from the fact that it is valid only when the composite error term it is
uncorrelated with all of the explanatory variables.
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Fixed or Random Effects? (Cont’d)
• This assumption is more stringent than the corresponding one in the fixed
effects case, because with random effects we thus require both i and vit to be
independent of all of the xit.
• This can also be viewed as a consideration of whether any unobserved
omitted variables (that were allowed for by having different intercepts for
each entity) are uncorrelated with the included explanatory variables. If they
are uncorrelated, a random effects approach can be used; otherwise the fixed
effects model is preferable.
• A test for whether this assumption is valid for the random effects estimator is
based on a slightly more complex version of the Hausman test.
• If the assumption does not hold, the parameter estimates will be biased and
inconsistent.
• To see how this arises, suppose that we have only one explanatory variable,
x2it that varies positively with yit, and also with the error term, it. The
estimator will ascribe all of any increase in y to x when in reality some of it
arises from the error term, resulting in biased coefficients.
• There may also be differences in policies for credit provision dependent upon
the nature of the formation of the subsidiary abroad – i.e. whether the
subsidiary's existence results from a take-over of a domestic bank or from the
formation of an entirely new startup operation (a ‘greenfield investment’).
• The data cover the period 1993-2000 and are obtained from BankScope.
• These are: weakness parent bank, defined as loan loss provisions made by
the parent bank; solvency is the ratio of equity to total assets; liquidity is the
ratio of liquid assets / total assets; size is the ratio of total bank assets to total
banking assets in the given country; profitability is return on assets and
efficiency is net interest margin.
• and the 's are parameters (or vectors of parameters in the cases of 4 and
5), i is the unobserved random effect that varies across banks but not over
time, and it is an idiosyncratic error term.
• de Haas and van Lelyveld discuss the various techniques that could be
employed to estimate such a model.
• OLS is considered to be inappropriate since it does not allow for differences
in average credit market growth rates at the bank level.
• A model allowing for entity-specific effects (i.e. a fixed effects model that
effectively allowed for a different intercept for each bank) is ruled out on the
grounds that there are many more banks than time periods and thus too many
parameters would be required to be estimated.
• They also argue that these bank-specific effects are not of interest to the
problem at hand, which leads them to select the random effects panel model.
• This essentially allows for a different error structure for each bank. A
Hausman test is conducted, and shows that the random effects model is valid
since the bank-specific effects i are found “in most cases not to be
significantly correlated with the explanatory variables.”
• The main result is that during times of banking disasters, domestic banks
significantly reduce their credit growth rates (i.e. the parameter estimate on
the crisis variable is negative for domestic banks), while the parameter is
close to zero and not significant for foreign banks.
• This indicates that, as the authors expected, when foreign banks have fewer
viable lending opportunities in their own countries and hence a lower
opportunity cost for the loanable funds, they may switch their resources to
the host country.
• Lending rates, both at home and in the host country, have little impact on
credit market share growth.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 31
Analysis of Results (Cont’d)
• Overall, both home-related (‘push’) and host-related (‘pull’) factors are found
to be important in explaining foreign bank credit growth.
• Dickey-Fuller and Phillips-Perron unit root tests have low power, especially
for modest sample sizes
• It is believed that more powerful versions of the tests can be employed when
time-series and cross-sectional information is combined – as a result of the
increase in sample size
• We could increase the number of observations by increasing the sample
period, but this data may not be available, or may have structural breaks
• A valid application of the test statistics is much more complex for panels than
single series
• Two important issues to consider:
– The design and interpretation of the null and alternative hypotheses needs careful
thought
– There may be a problem of cross-sectional dependence in the errors across the
unit root testing regressions
• Early studies that assumed cross-sectional independence are sometimes
known as ‘first generation’ panel unit root tests
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The MADF Test
• We could run separate regressions over time for each series but to use
Zellner’s seemingly unrelated regression (SUR) approach, which we might
term the multivariate ADF (MADF) test
• This method can only be employed if T >> N, and Taylor and Sarno (1998)
provide an early application to tests for purchasing power parity
• However, that technique is now rarely used, researchers preferring instead to
make use of the full panel structure
• A key consideration is the dimensions of the panel – is the situation that T is
large or that N is large or both? If T is large and N small, the MADF
approach can be used
• But as Breitung and Pesaran (2008) note, in such a situation one may
question whether it is worthwhile to adopt a panel approach at all, since for
sufficiently large T, separate ADF tests ought to be reliable enough to render
the panel approach hardly worth the additional complexity.
• Levin, Lin and Chu (2002) – LLC – develop a test based on the equation:
• The model is very general since it allows for both entity-specific and time-
specific effects through αi and θt respectively as well as separate
deterministic trends in each series through δit, and the lag structure to mop
up autocorrelation in Δy
• One of the reasons that unit root testing is more complex in the panel
framework in practice is due to the plethora of ‘nuisance parameters’ in the
equation which are necessary to allow for the fixed effects (i.e. αi, θt, δit)
• These nuisance parameters will affect the asymptotic distribution of the test
statistics and hence LLC propose that two auxiliary regressions are run to
remove their impacts
• Breitung (2000) develops a modified version of the LLC test which does
not include the deterministic terms and which standardises the residuals
from the auxiliary regression in a more sophisticated fashion.
• Under the LLC and Breitung approaches, only evidence against the non-
stationary null in one series is required before the joint null will be rejected
• Breitung and Pesaran (2008) suggest that the appropriate conclusion when
the null is rejected is that ‘a significant proportion of the cross-sectional
units are stationary’
• Especially in the context of large N, this might not be very helpful since no
information is provided on how many of the N series are stationary
• This difficulty led Im, Pesaran and Shin (2003) – IPS – to propose an
alternative approach where the null and alternative hypotheses are now H0:
ρi = 0 ∀ i and H1: ρi < 0, i = 1, 2, . . . , N1; ρi = 0, i = N1 + 1, N1 + 2, . . . , N
• So the null hypothesis still specifies all series in the panel as nonstationary,
but under the alternative, a proportion of the series (N1/N) are stationary,
and the remaining proportion ((N − N1)/N) are nonstationary
• No restriction where all of the ρ are identical is imposed
• The statistic in this case is constructed by conducting separate unit root
tests for each series in the panel, calculating the ADF t-statistic for each one
in the standard fashion, and then taking their cross-sectional average
• This average is then transformed into a standard normal variate under the
null hypothesis of a unit root in all the series
• While IPS’s heterogeneous panel unit root tests are superior to the
homogeneous case when N is modest relative to T, they may not be
sufficiently powerful when N is large and T is small, in which case the LLC
approach may be preferable.
‘Introductory Econometrics for Finance’ © Chris Brooks 2013 38
The Maddala and Wu (1999) and Choi (2001) Tests
• Maddala and Wu (1999) and Choi (2001) developed a slight variant on the
IPS approach based on an idea dating back to Fisher (1932)
• Unit root tests are conducted separately on each series in the panel and the
p-values associated with the test statistics are then combined
• If we call these p-values pvi, i = 1, 2, . . . ,N, under the null hypothesis of a
unit root in each series, pvi will be distributed uniformly over the [0,1]
interval and hence the following will hold for given N as T → ∞
• Testing for cointegration in panels is complex since one must consider the
possibility of cointegration across groups of variables (what we might term
‘cross-sectional cointegration’) as well as within the groups
• Most of the work so far has relied upon a generalisation of the single
equation methods of the Engle-Granger type following the pioneering work
by Pedroni (1999, 2004)
• For a set of M variables yit and xm,i,t that are individually integrated of order
one and thought to be cointegrated, the model is
• The residuals from this regression are then subjected to separate Dickey-
Fuller or augmented Dickey-Fuller type regressions for each group
• The null hypothesis is that the residuals from all of the test regressions are
unit root processes and therefore that there is no cointegration.
• To what extent are economic growth and the sophistication of the country’s
financial markets linked?
• Excessive government regulations may impede the development of the
financial markets and consequently economic growth will be slower
• On the other hand, if economic agents are able to borrow at reasonable rates
of interest or raise funding easily on the capital markets, this can increase
the viability of real investment opportunities
• Given that long time-series are typically unavailable developing economies,
traditional unit root and cointegration tests that examine the link between
these two variables suffer from low power
• This provides a strong motivation for the use of panel techniques as in the
study by Chrisopoulos and Tsionas (2004)
• The results, are much stronger than for the single series unit root tests and
show conclusively that all four series are non-stationary in levels but
stationary in differences:
• The LLC approach is used along with the Harris-Tzavalis technique, which
is broadly the same as LLC but has slightly different correction factors in
the limiting distribution
• These techniques are based on a unit root test on the residuals from the
potentially cointegrating regression
• Christopoulos and Tsionis investigate the use of panel cointegration tests
with fixed effects, and with both fixed effects and a deterministic trend in
the test regressions
• These are applied to the regressions both with y, and separately F, as the
dependent variables
• The results quite strongly demonstrate that when the dependent variable is
output, the LLC approach rejects the null hypothesis of a unit root in the
potentially cointegrating regression residuals when fixed effects only are
included in the test regression, but not when a trend is also included.