ECON 4551 Econometrics II Memorial University of...

Panel Data Models

Adapted from Vera Tabakova’s notes

ECON 4551 Econometrics II Memorial University of Newfoundland

15.1 Grunfeld’s Investment Data

15.2 Sets of Regression Equations

15.3 Seemingly Unrelated Regressions

15.4 The Fixed Effects Model

15.4 The Random Effects Model

Extensions RCM, dealing with endogeneity when we have

static variables

Slide 15-2 Principles of Econometrics, 3rd Edition

The different types of panel data sets can be described as:

“long and narrow,” with “long” time dimension and “narrow”, few

cross sectional units;

“short and wide,” many units observed over a short period of time;

“long and wide,” indicating that both N and T are relatively large.


The data consist of T = 20 years of data (1935-1954) for N = 10 large firms.

Let yit = INVit and x2it = Vit and x3it = Kit


(15.1)

(15.2)

( ),it it itINV f V K=

1 2 2 3 3it it it it it it ity x x e= β +β +β +

Notice the subindices!

Value of stock, proxy for expected profits Capital stock, proxy for desired permanent Capital stock

5 Principles of Econometrics, 3rd Edition

(15.3a)

(15.3b)

, 1 2 , 3 , ,

, 1 2 , 3 , ,

1, ,20

1, ,20

GE t GE t GE t GE t

WE t WE t WE t WE t

INV V K e t

INV V K e t

= β +β +β + =

= β +β +β + =

1 2 2 3 3 1, 2; 1, ,20it it it ity x x e i t= β +β +β + = =

For simplicity we focus on only two firms

GRETL: smpl firm = 3 || firm = 8 --restrict


(15.4a)

(15.4b)

, 1, 2, , 3, , ,

, 1, 2, , 3, , ,

1, ,20

1, ,20

GE t GE GE GE t GE GE t GE t

WE t WE WE WE t WE WE t WE t

INV V K e t

INV V K e t

= β +β +β + =

= β +β +β + =

1 2 2 3 3 1, 2; 1, ,20it i i it i it ity x x e i t= β +β +β + = =

Assumption (15.5) says that the errors in both investment functions

(i) have zero mean, (ii) are homoskedastic with constant variance, and (iii) are not correlated over time; autocorrelation does not exist. The two equations do have different error variances


(15.5) ( ) ( ) ( )( ) ( ) ( )

2, , , ,

2, , , ,

0 var cov , 0

0 var cov , 0GE t GE t GE GE t GE s

WE t WE t WE WE t WE s

E e e e e

E e e e e

= = σ =

= = σ =

2 2 and .GE WEσ σ

GRETL ols Inv const V K modtest –panel wrong in posted notes!!!

Let Di be a dummy variable equal to 1 for the Westinghouse

observations and 0 for the General Electric observations. If the

variances are the same for both firms then we can run:


(15.6) 1, 1 2, 2 3, 3it GE i GE it i it GE it i it itINV D V D V K D K e= β + δ +β + δ × +β + δ × +


So we have two separate stories

This assumption says that the error terms in the two equations, at the same point in time, are correlated. This kind of correlation is called a contemporaneous correlation.

Under this assumption, the joint regression would be better than the

separate simple OLS regressions


(15.7) ( ), , ,cov ,GE t WE t GE WEe e = σ

Econometric software includes commands for SUR (or SURE) that

carry out the following steps:

(i) Estimate the equations separately using least squares;

(ii) Use the least squares residuals from step (i) to estimate

;

(iii) Use the estimates from step (ii) to estimate the two equations jointly

within a generalized least squares framework.


2 2,, and GE WE GE WEσ σ σ


* Open and summarize data from grunfeld2.gdt (which, luckily for us, is already in wide format!!!) open "c:\Program Files\gretl\data\poe\grunfeld2.gdt" system name="Grunfeld" equation inv_ge const v_ge k_ge equation inv_we const v_we k_we end system estimate "Grunfeld" method=sur --geomean


In GRETL the restrict command can be used to impose the cross-equation restrictions on a system of equations that has been previously defined and named. The set of restrictions is started by restrict and terminated with end restrict. Each restriction in the set is expressed as an equation. Put the linear combination of parameters to be tested on the left-hand-side of the equality and a numeric value on the right. Parameters are referenced using b[i,j] where i refers to the equation number in the system, and j the parameter number.


restrict "Grunfeld" b[1,1]-b[2,1]=0 b[1,2]-b[2,2]=0 b[1,3]-b[2,3]=0 end restrict

There are two situations where separate least squares estimation is

just as good as the SUR technique :

(i) when the equation errors are not contemporaneously correlated;

(ii) when the same (the “very same”) explanatory variables appear in

each equation.

If the explanatory variables in each equation are different, then a test

to see if the correlation between the errors is significantly different

from zero is of interest. Slide 15-17 Principles of Econometrics, 3rd Edition

(although text reads 0.729):

In this case we have 3 parameters in each equation so:


( )( )( )

22,2

, 2 2

ˆ 207.58710.53139

ˆ ˆ 777.4463 104.3079GE WE

GE WEGE WE

rσ

= = =σ σ

20 20

, , , , ,1 1

1 1ˆ ˆ ˆ ˆ ˆ3GE WE GE t WE t GE t WE t

t tGE WE

e e e eTT K T K = =

σ = =−− −

∑ ∑

3.GE WEK K= =

Testing for correlated errors for two equations:

LM = 10.628 > 3.84 (Breusch-Pagan test of independence: chi2(1))

Hence we reject the null hypothesis of no correlation between the

errors and conclude that there are potential efficiency gains from

estimating the two investment equations jointly using SUR.


0 ,: 0GE WEH σ =

2 2, (1) 0 under .GE WELM Tr H= ∼ χ

Testing for correlated errors for three equations:


0 12 13 23: 0H σ = σ = σ =

( )2 2 2 212 13 23 (3)LM T r r r= + + χ

Testing for correlated errors for M equations:

Under the null hypothesis that there are no contemporaneous

correlations, this LM statistic has a χ2-distribution with M(M–1)/2

degrees of freedom, in large samples.


12

2 1

M i

iji j

LM T r−

= == ∑∑

Most econometric software will perform an F-test and/or a Wald χ2–test; in the context of SUR equations both tests are large sample approximate tests.

The F-statistic has J numerator degrees of freedom and (MT−K)

denominator degrees of freedom, where J is the number of hypotheses, M is the number of equations, and K is the total number of coefficients in the whole system, and T is the number of time series observations per equation. The χ2-statistic has J degrees of freedom.


(15.8) 0 1, 1, 2, 2, 3, 3,: , ,GE WE GE WE GE WEH β = β β = β β = β

SUR is OK when the panel is long and narrow, not when it is short and wide. Consider instead…

We cannot consistently estimate the 3×N×T parameters in (15.9) with only NT total observations. But we can impose some more structure…


(15.9)

(15.10)

1 2 2 3 3it it it it it it ity x x e= β +β +β +

1 1 2 2 3 3, ,it i it itβ = β β = β β = β

We consider only one-way effects and assume a common slope parameters across cross-sectional units

All behavioral differences between individual firms and over time are

captured by the intercept. Individual intercepts are included to

“control” for these firm specific differences.


(15.11) 1 2 2 3 3it i it it ity x x e= β +β +β +

This specification is sometimes called the least squares dummy

variable model, or the fixed effects model.


(15.12)

1 2 3

1 1 1 2 1 3, , , etc.

0 otherwise 0 otherwise 0 otherwisei i i

i i iD D D

= = = = = =

11 1 12 2 1,10 10 2 2 3 3it i i i it it itINV D D D V K e= β +β + +β +β +β +

These N–1= 9 joint null hypotheses are tested using the usual F-test

statistic. In the restricted model all the intercept parameters are equal.

If we call their common value β1, then the restricted model is:


(15.13) 0 11 12 1

1 1

:

: the are not all equal

N

i

H

H

β = β = = β

β

1 2 3it it it itINV V K e= β +β +β +

So this is just OLS, the pooled model


reg inv v k

We reject the null hypothesis that the intercept parameters for all

firms are equal. We conclude that there are differences in firm

intercepts, and that the data should not be pooled into a single model

with a common intercept parameter.


( )( )

( )( )

1749128 522855 948.99

522855 200 12

R U

U

SSE SSE JF

SSE NT K−

=−

−= =

−


(15.14) 1 2 2 3 3 1, ,it i it it ity x x e t T= β +β +β + =

(15.15)

( )1 2 2 3 31

1 T

it i it it itt

y x x eT =

= β +β +β +∑

1 2 2 3 31 1 1 1

1 2 2 3 3

1 1 1 1T T T T

i it i it it itt t t t

i i i i

y y x x eT T T T

x x e

= = = == = β +β +β +

= β +β +β +

∑ ∑ ∑ ∑


(15.16)

1 2 2 3 3

1 2 2 3 3

2 2 2 3 3 3

( )

( ) ( ) ( )

it i it it it

i i i i i

it i it i it i it i

y x x e

y x x e

y y x x x x e e

= β +β +β +

− = β +β +β +

− = β − +β − + −

(15.17) 2 3it it it ity x x e= β +β +


(15.18) .1098 .3106(se*) (.0116) (.0169)

itit itINV V K= +

( )2*ˆ 2e SSE NTσ = −

( ) ( )2 2 198 188 1.02625NT NT N− − − = =

Usually, there is no interest in the intercepts….


Some software comes up with one sometimes though…

Or if wanted you should be able to retrieve the individual ones


(15.19)

1 2 2 3 3i i i iy b b x b x= + +

1 2 2 3 3 1, ,i i i ib y b x b x i N= − − =


ONE PROBLEM: Even with the trick of using the within estimator, we still implicitly (even if no longer explicitly) include N-1 dummy variables in our model (not N, since we remove the intercept), so we use up N-1 degrees of freedom. It might not be then the most efficient way to estimate the common slope ANOTHER ONE. By using deviations from the means, the procedure wipes out all the static variables, whose effects might be of interest In order to overcome this problem, we can consider the random effects/or error components model

In the RE model, the individual firm differences are thought to represent a random variation about some average intercept for the individual in the sample

Rather than a separate fixed effect for each firm, we now estimate an overall intercept that represents this average

Implicitly, the regression function for the sample firms vary randomly around this average.

The variability of the individual effects is captured by a new parameter, which is the variance of the random effect.

The larger this parameter is, the more variation you find in the implicit regression functions for the firms.

Principles of Econometrics, 3rd Edition


(15.20)

(15.22)

1 1i iuβ = β +

(15.21) ( ) ( ) ( ) 20, cov , 0, vari i j i uE u u u u= = = σ

( )

1 2 2 3 3

1 2 2 3 3

it i it it it

i it it it

y x x e

u x x e

= β +β +β +

= β + +β +β +Randomness of the intercept

Usual error

Average intercept

Because the random effects regression error has two components, one

for the individual and one for the regression, the random effects

model is often called an error components model.


(15.23)

(15.24)

( )1 2 2 3 3

1 2 2 3 3

it it it it i

it it it

y x x e u

x x v

= β +β +β + +

= β +β +β +

it i itv u e= +

a composite error


(15.25)

( ) ( ) ( ) ( ) 0 0 0it i it i itE v E u e E u E e= + = + = + =

( ) ( )

( ) ( ) ( )

2

2 2

var var

var var 2cov ,

v it i it

i it i it

u e

v u e

u e u e

σ = = +

= + +

= σ +σ

v has zero mean

v has constant variance If there is no correlation between the individual effects and the error term


But now there are several correlations that can be considered.

The correlation between two individuals, i and j, at the same

point in time, t. The covariance for this case is given by

( ) ( )( )

( ) ( ) ( ) ( )

cov , ( )

0 0 0 0 0

it jt it jt i it j jt

i j i jt it j it jt

v v E v v E u e u e

E u u E u e E e u E e e

= = + +

= + + +

= + + + =


The correlation between errors on the same individual (i) at

different points in time, t and s. The covariance for this case is

given by

(15.26)

( ) ( )( )

( ) ( ) ( ) ( )2

2 2

cov , ( )

0 0 0

it is it is i it i is

i i is it i it is

u u

v v E v v E u e u e

E u E u e E e u E e e

= = + +

= + + +

= σ + + + = σ


The correlation between errors for different individuals in

different time periods. The covariance for this case is

( ) ( )( )

( ) ( ) ( ) ( )

cov , ( )

0 0 0 0 0

it js it js i it j js

i j i js it j it js

v v E v v E u e u e

E u u E u e E e u E e e

= = + +

= + + +

= + + + =


(15.27) 2

2 2cov( , )corr( , )

var( ) var( )it is u

it isu eit is

v vv vv v

σρ = = =

σ +σ

The errors are correlated over time for a given individual, but are otherwise uncorrelated This correlation does not dampen over time as in the AR1 model


(15.28)

1 2 2 3 3it it it ity x x e= β +β +β +

1 2 2 3 3it it it ite y b b x b x= − − −

( )

2

1 1

2

1 1

ˆ1

2 1 ˆ

N T

iti t

N T

iti t

eNTLMT e

= =

= =

= − −

∑ ∑

∑∑

GRETL shows this Breusch and Pagan Lagrange multiplier test for random effects by default


GRETL shows by default this Breusch and Pagan Lagrangian multiplier test for RE with the null of no variation about a mean (effects are fixed) in the individual effects.

This is xttest0 in Stata…

If H0 is not rejected you can use pooled OLS if the effects are common and the FE if they differ by group


GRETL shows by default this Breusch and Pagan Lagrangian multiplier test for RE with the null of no variation about a mean (effects are fixed) in the individual effects.


GRETL also shows the Hausman test of the null hypothesis that the random effects are indeed random.

If they are random, then they should not be correlated with any of your other regressors.

If they are correlated with other regressors, then you should use the FE estimator to obtain consistent parameter estimates of your slopes


(15.29)

(15.30)

* * * * *1 1 2 2 3 3it it it it ity x x x v= β +β +β +

* * * *1 2 2 2 3 3 3, 1 , ,it it i it it it i it it iy y y x x x x x x x= −α = −α = −α = −α

(15.31) 2 21 e

u eTσ

α = −σ +σ

Is the transformation parameter


( )2 2

ˆ .1951ˆ 1 1 .74375 .1083 .0381ˆ ˆ

e

u eTσ

α = − = − =+σ +σ

Is the transformation parameter

There are different ways to calculate FE (some packages will calculate an intercept, some won’t)

There are different ways to calculate sigma-sq (STATA in textbook and GRETL will give you slightly different results!)


Pooled OLS vs different intercepts: test (use a Chow type, after FE or run RE and test if the variance of the intercept component of the error is zero (Breusch-Pagan test (xttest0 in STATA))

You cannot pool onto OLS? Then…

Choose between FE vs RE: (Hausman test)

GRETL summary tests: panel Inv const V K --pooled

Different slopes too perhaps? => use SURE or RCM and test for equality of slopes across units

Note that there is within variation versus between variation

The OLS is an unweighted average of the between estimator and the within estimator

The RE is a weighted average of the between estimator and the within estimator

The FE is also a weighted average of the between estimator and the within estimator with zero as the weight for the between part

The RE is a weighted average of the between

estimator and the within estimator The FE is also a weighted average of the

between estimator and the within estimator with zero as the weight for the between part

So now you see where the extra efficiency of RE comes from!...

The RE uses information from both the cross-

sectional variation in the panel and the time series variation, so it mixes LR and SR effects

The FE uses only information from the time series variation, so it estimates SR* effects

With a panel, we can learn about dynamic

effects from a short panel, while we need a long time series on a single cross-sectional unit, to learn about dynamics from a time series data set

If the random error is correlated with any of the right-hand side explanatory variables in a random effects model then the least squares and GLS estimators of the parameters are biased and inconsistent.

This bias creeps in through the between variation, of course, so the FE model will avoid it


it i itv u e= +


(15.32)

(15.33) 1 2 2 3 3

1 1 1 1 1

1 2 2 3 3

1 1 1 1 1T T T T T

i it it it i itt t t t t

i i i i

y y x x u eT T T T T

x x u e

= = = = == = β +β +β + +

= β +β +β + +

∑ ∑ ∑ ∑ ∑

1 2 2 3 3 ( )it it it i ity x x u e= β +β +β + +


(15.34)

1 2 2 3 3

1 2 2 3 3

2 2 2 3 3 3

( )

( ) ( ) ( )

it it it i it

i i i i i

it i it i it i it i

y x x u e

y x x u e

y y x x x x e e

= β +β +β + +

− = β +β +β + +

− = β − +β − + −

We expect to find because Hausman proved that


(15.35) ( ) ( ) ( ) ( ), , , ,

1 2 1 22 2, ,, , se sevar var

FE k RE k FE k RE k

FE k RE kFE k RE k

b b b bt

b bb b

− −= = −−

( ) ( ), ,var var 0.FE k RE kb b− >

( ) ( ) ( ) ( )

( ) ( )

, , , , , ,

, ,

var var var 2cov ,

var var

FE k RE k FE k RE k FE k RE k

FE k RE k

b b b b b b

b b

− = + −

= −

( ) ( ), , ,cov , var .FE k RE k RE kb b b=

The test statistic to the coefficient of SOUTH is: Using the standard 5% large sample critical value of 1.96, we reject

the hypothesis that the estimators yield identical results. Our conclusion is that the random effects estimator is inconsistent, and we should use the fixed effects estimator, or we should attempt to improve the model specification.


( ) ( ) ( ) ( ), ,

1 2 1 22 2 2 2, ,

.0163 (.0818) 2.3137.0361 .0224se se

FE k RE k

FE k RE k

b bt

b b

− − −= = = −−

If the random error is correlated with any of the right-hand side explanatory variables in a random effects model then the least squares and GLS estimators of the parameters are biased and inconsistent.

Then we would have to use the FE model

But with FE we lose the static variables?

Solutions? HT, AM, BMS, instrumental variables models could help


it i itv u e= +

We can generalise the random effects idea and allow for different

slopes too: Random Coefficients Model

Again, the now it is the slope parameters that differ, but as in RE

model, they are drawn from a common distribution

The RCM in a way is to the RE model what the SURE model is to the

FE model


Further issues

Unit root tests and Cointegration in panels

Dynamics in panels


Further issues

Of course it is not necessary that one of the dimensions of the panel is time

as such Example: i are students and t is for each quiz they take

Of course we could have a one-way effect model on the time dimension

instead

Or a two-way model

Or a three way model! But things get a bit more complicated there…


Further issues

Another way to have more fun with panel data is to consider dependent variables that are not continuous

Logit, probit, count data can be considered

STATA has commands for these

Based on maximum likelihood and other estimation techniques we have not yet considered


Further issues

You can understand the use of the FE model as a solution to omitted variable bias

If the unmeasured variables left in the error model are not correlated

with the ones in the model, we would not have a bias in OLS, so we

can safely use RE

If the unmeasured variables left in the error model are correlated with

the ones in the model, we would have a bias in OLS, so we cannot

use RE, we should not leave them out and we should use FE, which

bundles them together in each cross-sectional dummy


Further issues

Another criterion to choose between FE and RE

If the panel include all the relevant cross-sectional units, use FE, if only a random sample from a population, RE is more appropriate (as long as it is valid)


Further issues

Wooldridge’s book on panel data

Baltagi’s book on panel data

Greene’s coverage is also good


Readings


Balanced panel Breusch-Pagan test Cluster corrected standard errors Contemporaneous correlation Endogeneity Error components model Fixed effects estimator Fixed effects model Hausman test Heterogeneity Least squares dummy variable

model LM test Panel corrected standard errors Pooled panel data regression

Pooled regression Random effects estimator Random effects model Seemingly unrelated regressions Unbalanced panel

Principles of Econometrics, 3rd Edition Slide 15-72

(15A.1)

(15A.2)

(15A.3)

1 2 2 3 3 ( )it it it i ity x x u e= β +β +β + +

2 2 2 3 3 3( ) ( ) ( )it i it i it i it iy y x x x x e e− = β − +β − + −

2ˆ DVe

slopes

SSENT N K

σ =− −


(15A.4)

(15A.5)

1 2 2 3 3 1, ,i i i i iy x x u e i N= β +β +β + + =

( ) ( ) ( ) ( )1

22 2

2 21

22

var var var var var

1 var

T

i i i i i itt

Te

u it ut

eu

u e u e u e T

TeT T

T

=

=

+ = + = +

σ = σ + = σ +

σ= σ +

∑

∑


(15A.6)

(15A.7)

22 e BEu

BE

SSET N Kσ

σ + =−

( )2 2

2 2 ˆˆ e e BE DVu u

BE slopes

SSE SSET T N K T NT N Kσ σ

σ = σ + − = −− − −

ECON 4551 Econometrics II Memorial University of...

Documents

Transcript of ECON 4551 Econometrics II Memorial University of...