chapter 15 panel data models

Principles of Econometrics, 4th Edition Chapter 15: Panel Data Models

Chapter 15Panel Data Models

Walter R. Paczkowski Rutgers University


15.1 A Microeconomic Panel15.2 A Pooled Model15.3 The Fixed Effects Model15.4 The Random Effects Model15.5 Comparing Fixed and Random Effects

Estimators15.6 The Hausman-Taylor Estimator15.7 Sets of Regression Equations

Chapter Contents


A panel of data consists of a group of cross-sectional units (people, households, firms, states, countries) who are observed over time– Denote the number of cross-sectional units

(individuals) by N– Denote the number of time periods in which we

observe them as T


Different ways of describing panel data sets:– Long and narrow• ‘‘Long’’ describes the time dimension and

‘‘narrow’’ implies a relatively small number of cross sectional units

– Short and wide• There are many individuals observed over a

relatively short period of time– Long and wide• Both N and T are relatively large


It is possible to have data that combines cross-sectional and time-series data which do not constitute a panel–We may collect a sample of data on individuals

from a population at several points in time, but the individuals are not the same in each time period• Such data can be used to analyze a ‘‘natural

experiment’’


15.1 A Microeconomic Panel


In microeconomic panels, the individuals are not always interviewed the same number of times, leading to an unbalanced panel in which the number of time series observations is different across individuals– In a balanced panel, each individual has the

same number of observations

15.1A Microeconomic

Panel


15.1A Microeconomic

Panel Table 15.1 Representative Observations from NLS Panel Data


15.2 A Microeconomic Panel


A pooled model is one where the data on different individuals are simply pooled together with no provision for individual differences that might lead to different coefficients

– Notice that the coefficients (β1, β2, β3) do not have i or t subscripts

15.2Pooled Model

1 2 2 3 3it it it ity x x e Eq. 15.1


The least squares estimator, when applied to a pooled model, is referred to as pooled least squares– The data for different individuals are pooled

together, and the equation is estimated using least squares

15.2Pooled Model


It is useful to write explicitly the error assumptions required for pooled least squares to be consistent and for the t and F statistics to be valid when computed using the usual least squares variance estimates and standard errors

15.2Pooled Model

2 2

2 3

0

var

cov , , 0 for or

cov , 0, cov , 0

it

it it e

it js it js

it it it it

E e

e E e

e e E e e i j t s

e x e x

Eq. 15.2

Eq. 15.3

Eq. 15.4

Eq. 15.5


Applying pooled least squares in a way that ignores the panel nature of the data is restrictive in a number of ways– The first unrealistic assumption that we

consider is the lack of correlation between errors corresponding to the same individual

15.2Pooled Model

15.2.1Cluster-Robust Standard Errors


To relax the assumption of zero error correlation over time for the same individual, we write:

– This also relaxes the assumption of homoskedasticity:

We continue to assume that the errors for different individuals are uncorrelated:

15.2Pooled Model


cov , ψit is tse e Eq. 15.6

cov , var =ψit it it tte e e

cov , 0 for it jse e i j


What are the consequences of using pooled least squares in the presence of the heteroskedasticity and correlation?– The least squares estimator is still consistent– Its standard errors are incorrect• This implies that hypothesis tests and

interval estimates based on these standard errors will be invalid–Typically, the standard errors will be too

small, overstating the reliability of the least squares estimator

15.2Pooled Model



Standard errors that are valid for the pooled least squares estimator under the assumption in Eq. 15.6 can be computed

• Various names are: –Panel-robust standard errors–Cluster-robust standard errors»The time series observations on

individuals are the clusters

15.2Pooled Model



15.2Pooled Model

15.2.2Pooled Least

Squares Estimates of Wage Equation

Table 15.2 Pooled Least Squares Estimates of Wage Equation


15.3 The Fixed Effects Model


We can extend the model in Eq. 15.1 to relax the assumption that all individuals have the same coefficients:

– An i subscript has been added to each of the subscripts, implying that (β1, β2, β3) can be different for each individual

15.3The Fixed Effects

Model

1 2 2 3 3it i i it i it ity x x e Eq. 15.7


A popular simplification is one where the intercepts β1i are different for different individuals but the slope coefficients β2 and β3 are assumed to be constant for all individuals:


Model

1 2 2 3 3it i it it ity x x e Eq. 15.8


All behavioral differences between individuals, referred to as individual heterogeneity, are assumed to be captured by the intercept– Individual intercepts are included to ‘‘control’’

for individual-specific, time-invariant characteristics.

– A model with these features is called a fixed effects model• The intercepts are called fixed effects


Model


We consider two methods for estimating Eq. 15.81. The least squares dummy variable estimator 2. The fixed effects estimator


Model


One way to estimate the model in Eq. 15.8 is to include an intercept dummy variable (indicator variable) for each individual

– If we have 10 individuals, we define 10 such dummies

Now we can write:


Model

15.3.1The Least Square Dummy Variable

Estimator for Small N

1 2 3

1 1 1 2 1 3

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 ity D D D V K e Eq. 15.9


If the error terms eit are uncorrelated with mean zero and constant variance σ2

e for all observations, then the best linear unbiased estimator of Eq. 15.9 is the least squares estimator– In a panel data context, it is called the least

squares dummy variable estimator


Model





Model



Table 15.3 Dummy Variable Estimation of Wage Equation for N = 10



Model



Table 15.4 Pooled Least Squares Estimates of Wage Equation for N = 10


We can test the estimates of the intercepts:


Model



0 11 12 1,10

1 1

:

: the are not all equali

H

H

Eq. 15.10


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 the pooled model:


Model



21 2 3

24 5

6

ln β β β

β β β

WAGE EXPER EXPER

TENURE TENUREUNION e


The F-statistic is:


Model



5.502466 2.667190 92.667190 50 15

4.134

R U

U

SSE SSE JF

SSE NT K


The value of the test statistic F = 4.134 yields a p-value of 0.0011–We reject the null hypothesis that the intercept

parameters for all individuals are equal. –We conclude that there are differences in

individual intercepts, and that the data should not be pooled into a single model with a common intercept parameter


Model




Using the dummy variable approach is not feasible when N is large– Another approach is necessary


Model

15.3.2The Fixed Effects

Estimator


Take the data on individual i:

– Average the data across time:


Model


Estimator

1 2 2 3 3 1, , it i it it ity x x e t T Eq. 15.11

1 2 2 3 31

1 T

it i it it itt

y x x eT


Using the fact that the parameters do not change over time, we can simplify this as:


Model


Estimator

Eq. 15.121 2 2 3 3

1 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


Now subtract Eq. 15.12 from Eq. 15.11, term by term, to obtain:

or


Model


Estimator

Eq. 15.13

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

2 3it it it ity x x e Eq. 15.14



Model


Estimator

Table 15.5 Data in Deviation from Individual Mean Form



Model

15.3.2aThe Fixed Effects Estimates of Wage

Equation for N = 10

Table 15.6 Fixed Effects Estimation of Wage Equation for N = 10


If we multiply the standard errors from estimating Eq. 15.14 by the correction factor

the resulting standard errors are identical to those in Table 15.3


Model


Equation for N = 10

5 5 45 35 1.133893NT NT N


Usually we are most interested in the coefficients of the explanatory variables and not the individual intercept parameters– These coefficients can be ‘‘recovered’’ by using

the fact that the least squares fitted regression passes through the point of the means

– That is:

– So that the fixed effects are:


Model


Equation for N = 10

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 Eq. 15.15



Model

15.3.3Fixed Effects

Estimates of Wage Equation from

Complete Panel

Table 15.7 Fixed Effects Estimates of Wage Equation for N = 716



Model

15.3.3Fixed Effects

Estimates of Wage Equation from

Complete Panel

Table 15.8 Percentage Marginal Effects on Wages


15.4 The Random Effects Model


In the random effects model we assume that all individual differences are captured by the intercept parameters– But we also recognize that the individuals in

our sample were randomly selected, and thus we treat the individual differences as random rather than fixed, as we did in the fixed-effects dummy variable model

15.4The Random Effects

Model


Random individual differences can be included in our model by specifying the intercept parameters to consist of a fixed part that represents the population average and random individual differences from the population average:

– The random individual differences ui are called random effects and have:


Model

1 1i iu Eq. 15.16

20, cov , 0, vari i j i uE u u u u Eq. 15.17


Substituting, we get:

– Rearranging:


Model

Eq. 15.18

Eq. 15.19

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

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


The combined error term is:

– The random effects error has two components:• One for the individual• One for the regression

– The random effects model is often called an error components model


Model

Eq. 15.20 it i itv u e


The combined error term has zero mean:

– And a constant, homoskedastic, variance:


Model

Eq. 15.21

15.4.1Error Term

Assumptions

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


There are several correlations that can be considered:1. The correlation between two individuals, i and

j, at the same point in time, t.


Model

15.4.1Error Term

Assumptions

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


There are several correlations (Continued):2. The correlation between errors on the same

individual (i) at different points in time, t and s


Model

Eq. 15.22

15.4.1Error Term

Assumptions

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


There are several correlations (Continued):3. The correlation between errors for different

individuals in different time periods


Model

15.4.1Error Term

Assumptions

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


The errors vit = ui + eit are correlated over time for a given individual, but are otherwise uncorrelated– The correlation is caused by the component ui

that is common to all time periods– It is constant over time and, in contrast to the

AR(1) error model, it does not decline as the observations get further apart in time:


Model

15.4.1Error Term

Assumptions

2

2 2

cov( , )corr( , )

var( ) var( )it is u

it isu eit is

v vv v t s

v v

Eq. 15.23


In terms of the notation introduced to explain the assumptions that motivate the use of cluster-robust standard errors:


Model

15.4.1Error Term

Assumptions

2 2 2var( ) ψ and cov( , ) ψ it it u e it is is uv v v t s


Summary of the error term assumptions of the random effects model:


Model

15.4.1Error Term

Assumptions

2 2

2

2 3

2 3

0

var( )

cov( , ) cov( , ) 0

cov( , ) 0, cov( , ) 0cov( , ) 0, cov( , ) 0

it

it u e

it is u

it js

it it it it

i it i it

E v

v

v v t sv v i j

e x e xu x u x

Eq. 15.24

Eq. 15.25

Eq. 15.26

Eq. 15.27

Eq. 15.28

Eq. 15.29


We can test for the presence of heterogeneity by testing the null hypothesis H0: σ2

u = 0 against the alternative hypothesis H1: σ2

u > 0– If the null hypothesis is rejected, then we

conclude that there are random individual differences among sample members, and that the random effects model is appropriate

– If we fail to reject the null hypothesis, then we have no evidence to conclude that random effects are present


Model

15.4.2Testing for Random

Effects


The Lagrange multiplier (LM) principle for test construction is very convenient in this case– If the null hypothesis is true, then ui = 0 and the

random effects model in Eq. 15.19 reduces to:


Model


Effects

1 2 2 3 3it it it ity x x e


The test statistic is based on the least squares residuals:

– The test statistic for balanced panels is:


Model


Effects

2

1 1

2

1 1

ˆ1

2 1 ˆ

N T

iti t

N T

iti t

eNTLMT e

Eq. 15.30

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


If the null hypothesis H0: σ2u = 0 is true, then LM

~ N(0, 1) in large samples– Thus, we reject H0 at significance level α and

accept the alternative H1: σ2u > 0 if LM > z(1-α),

where z(1-α) is the 100(1–α) percentile of the standard normal distribution

– This critical value is 1.645 if α = 0.05 and 2.326 if α = 0.01

– Rejecting the null hypothesis leads us to conclude that random effects are present


Model


Effects


We can obtain the generalized least squares estimator in the random effects model by applying least squares to a transformed model:

where the transformed variables are:

and α is defined as


Model

15.4.3Estimation of the Random Effects

Model

* * * * *1 1 2 2 3 3it it it it ity x x x v Eq. 15.31

* * * *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 Eq. 15.32

2 21 e

u eT

Eq. 15.33


For α = 1, the random effects estimator is identical to the fixed effects estimatorFor α < 1, it can be shown that the random effects estimator is a ‘‘matrix-weighted average’’ of the fixed effects estimator that utilizes only within individual variation and a ‘‘between estimator’’ which utilizes variation between individuals


Model

15.4.3Estimation of the Random Effects

Model



Model

15.4.4Random Effects Estimation of the Wage Equation

Table 15.9 Random Effects Estimates of Wage Equation

The estimate of the transformation parameter α is:

2 2

ˆ 0.1951ˆ 1 1 0.74375 0.1083 0.0381ˆ ˆ

e

u eT


15.5 Comparing Fixed and Random

Effects Estimators


If random effects are present, then the random effects estimator is preferred for several reasons:1. The random effects estimator takes into

account the random sampling process by which the data were obtained

2. The random effects estimator permits us to estimate the effects of variables that are individually time-invariant

3. The random effects estimator is a generalized least squares estimation procedure, and the fixed effects estimator is a least squares estimator

15.5Comparing Fixed

and Random Effects Estimators


If the random error vit = ui + eit 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– The problem of endogenous regressors was

considered before– The problem is common in random effects

models, because the individual specific error component ui may well be correlated with some of the explanatory variables

15.5Comparing Fixed


15.5.1Endogeneity in the

Random Effects Model


The panel data regression Eq. 15.19, including the error component ui, is:

Average the observations for each individual over time:

15.5Comparing Fixed



Estimator in a Random Effects

Model

1 2 2 3 3 ( )it it it i ity x x u e Eq. 15.34

1 2 2 3 31 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

Eq. 15.35


Subtract:

15.5Comparing Fixed



Estimator in a Random Effects

Model

Eq. 15.36

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


To check for any correlation between the error component ui and the regressors in a random effects model, we can use a Hausman test– The Hausman test can be carried out for

specific coefficients, using a t-test, or jointly, using an F-test or a chi-square test

15.5Comparing Fixed


15.5.3The Hausman Test


Let the parameter of interest be βk

– Denote the fixed effects estimate as bFE,k and the random effects estimate as bRE,k

– The t-statistic for testing that there is no difference between the estimators is:

15.5Comparing Fixed



, , , ,

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

Eq. 15.37


We expect to find:

– Also:

because Hausman proved that:

15.5Comparing Fixed



, ,var var 0FE 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 , varFE k RE k RE kb b b


Applying the t-test to the SOUTH we get:

– 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 that we should use the fixed effects estimator, or should attempt to improve the model specification

15.5Comparing Fixed



, ,

1 2 1 22 2 2 2

, ,

0.01632 ( 0.08181) 2.310.03615 0.02241se se

FE k RE k

FE k RE k

b bt

b b


The form of the Hausman test in Eq. 15.37 and its χ2 equivalent are not valid for cluster robust standard errors, because under these more general assumptions, it is no longer true that:

15.5Comparing Fixed



, , , ,var var varFE k RE k FE k RE kb b b b


15.6 The Hausman-Taylor Estimator


The Hausman-Taylor estimator is an instrumental variables estimator applied to the random effects model to overcome the problem of inconsistency caused by correlation between the random effects and some of the explanatory variables

15.6The Hausman-

Taylor Estimator


Consider the regression model:

with:xit,exog :exogenous variables that vary over time and individualsxit,endog: endogenous variables that vary over time and individualswi,exog: time-invariant exogenous variableswi,endog: time-invariant endogenous variables

15.6The Hausman-

Taylor Estimator

1 2 , 3 , 4 , 5 ,β β β β βit it exog it endog i exog i endog i ity x x w w u e Eq. 15.38


A slightly modify set is applied to the transformed generalized least squares model from Eq. 15.31:

15.6The Hausman-

Taylor Estimator

* * * * * *1 2 , 3 , 4 , 5 ,β β β β βit it exog it endog i exog i endog ity x x w w v Eq. 15.39


15.6The Hausman-

Taylor Estimator Table 15.10 Hausman-Taylor Estimates of Wage Equation


15.7 Sets of Regression Equations


Consider procedures for a panel that is long and narrow: T is large relative to N– If the number of time series observations is

sufficiently large, and N is small, we can estimate separate equations for each individual

– These separate equations can be specified as

15.7Sets of Regression

Equations

1 2 2 3 3it i i it i it ity x x e Eq. 15.40


An economic model for describing gross firm investment for the ith firm in the tth time period, denoted INVit, may be expressed as:


Equations

Eq. 15.41

15.7.1Grunfeld’s

Investment Data

,it it itINV f V K


We specify the following two equations for General Electric and Westinghouse:


Equations

Eq. 15.42

15.7.1Grunfeld’s

Investment Data

, 1 2 , 3 , ,

, 1 2 , 3 , ,

1935, ,1954

1935, ,1954

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


The choice of estimator depends on what assumptions we make about the coefficients and the error terms:1. Are the GE coefficients equal to the WE

coefficients?2. Do the equation errors eGE,t and eWE,t have the

same variance?3. Are the equation errors eGE,t and eWE,t

correlated?


Equations

15.7.1Grunfeld’s

Investment Data


The assumption that both firms have the same coefficients and the same error variances can be written as:


Equations

15.7.2Estimation: Equal

Coefficients, Equal Error Variances

2 21,GE 1,WE 2,GE 2,WE 3,GE 3,WE GE WEβ β β β β β σ σ Eq. 15.43


Let Di be an indicator variable equal to one for the Westinghouse observations and zero for the General Electric observations– Specify a model with slope and intercept

indicator variables:


Equations

15.7.3Estimation:

Different Coefficients, Equal

Error Variances

Eq. 15.44 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



Equations

15.7.3Estimation:


Error Variances

Table 15.12 Least Squares Estimates from the Dummy Variable Model


Using the Chow test, we get:

where NT - NK is the total number of degrees of freedom in the unrestricted model– The p-value for an F(3,34)-distribution is 0.328,

implying that the null hypothesis of equal coefficients cannot be rejected


Equations

15.7.3Estimation:


Error Variances

Eq. 15.45

16563.00 14989.82 3

1.18914989.82 40 6

R U

U

SSE SSE JF

SSE NT NK


When both the coefficients and the error variances of the two equations differ, and in the absence of contemporaneous correlation that we introduce in the next section, there is no connection between the two equations, and the best we can do is apply least squares to each equation separately


Equations

15.7.4Estimation:

Different Coefficients,

Different Error Variances



Equations

15.7.4Estimation:

Different Coefficients,

Different Error Variances

Table 15.13 Least Squares Estimates of Separate Investment Equations


Consider the following assumption:

– The error terms in the two equations, at the same point in time, are correlated• This kind of correlation is called

contemporaneous correlation


Equations

15.7.5Seemingly Unrelated

Regressions

, , , ,cov , 0GE t WE t GE WE GE WEe e Eq. 15.46


The dummy-variable model Eq. 15.44 represents a way to ‘‘stack’’ the 40 observations for the GE and WE equations into one regression– To improve the precision of the dummy variable

model estimates, we use seemingly unrelated regressions (SUR) estimation, which is a generalized least squares estimation procedure• It estimates the two investment equations jointly,

accounting for the fact that the variances of the error terms are different for the two equations and accounting for the contemporaneous correlation between the errors of the GE and WE equations


Equations


Regressions


Three stages in the SUR estimation procedure:1. Estimate the equations separately using OLS2. Use the OLS residuals from (1) to estimate

σ2GE, σ2

WE and σGE,WE

• The estimated covariance is given by:

3. Use the estimates from (2) to estimate the two equations jointly within a generalized least squares framework


Equations


Regressions

20 20

, , , , ,1 1

1 1ˆ ˆ ˆ ˆ ˆσ3

207.587

GE WE GE t WE t GE t WE tt tGE WE

e e e eTT K T K


The SUR estimation procedure is optimal under the contemporaneous correlation assumption, so no standard error adjustment is necessary


Equations


Regressions



Equations


Regressions

Table 15.14 SUR Estimates of Investment Equations


Two situations in which separate least squares estimation is just as good as the SUR technique1. The equation errors are not

contemporaneously correlated• If the errors are not contemporaneously

correlated, there is nothing linking the two equations, and separate estimation cannot be improved upon

2. Least squares and SUR give identical estimates when the same explanatory variables appear in each equation


Equations

15.7.5aSeparate or Joint

Estimation


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– Compute the squared correlation:


Equations


Estimation

22,2

, 2 2

ˆ 207.58710.5314

ˆ ˆ 777.4463 104.3079GE WE

GE WEGE WE

r


To check the statistical significance of r2GE,WE, test

the null hypothesis H0: σGE,WE = 0– If σGE,WE = 0, then LM = T x r2

GE,WE is a Lagrange Multiplier test statistic that is distributed as a χ2

(1) random variable in large samples

– The 5% critical value of a χ2-distribution with one degree of freedom is 3.841

– The value of the test statistic is LM = 10.628–We reject the null hypothesis of no correlation


Equations


Estimation


If we are testing for the existence of correlated errors for more than two equations, the relevant test statistic is equal to T times the sum of squares of all the correlations– The probability distribution under H0 is a χ2-

distribution with degrees of freedom equal to the number of correlations


Equations


Estimation


With three equations, denoted by subscripts 1, 2 and 3, the null hypothesis is:

– The χ2(3) test statistic is:

–With M equations:

with M(M – 1)/2 degrees of freedom


Equations


Estimation

0 12 13 23: 0H

2 2 212 13 23LM T r r r

12

2 1

M i

iji j

LM T r


We previously used the dummy variable model and the Chow test to test whether the two equations had identical coefficients:

– It is also possible to test hypotheses such as Eq 15.47 when the more general error assumptions of the SUR model are relevant• Because of the complicated nature of the model,

the test statistic can no longer be calculated simply as an F-test statistic based on residuals from restricted and unrestricted models


Equations

15.7.5bTesting Cross-

Equation Hypotheses

Eq. 15.47 0 1, 1, 2, 2, 3, 3,: GE WE GE WE GE WEH


Most econometric software will perform an F-test and/or a Wald χ2-test in a multi-equation framework such as we have here– In the context of SUR equations both tests are

large sample approximate tests


Equations


Equation Hypotheses


The equality of coefficients is not the only cross-equation hypothesis that can be tested– Any restrictions on parameters in different

equations can be tested– Tests for hypotheses involving coefficients within

each equation are valid whether done on each equation separately or using the SUR framework

– However, tests involving cross-equation hypotheses need to be carried out within an SUR framework if contemporaneous correlation exists


Equations


Equation Hypotheses


Key Words


Keywords

Balanced panelCluster-robust standard errorsContemporaneous correlationCross-equation hypothesesDeviations from individual meansEndogeneityError components modelFixed effects estimator

Fixed effects modelHausman testHausman-Taylor estimatorHeterogeneityInstrumental variablesLeast squares dummy variable modelLM testPanel corrected standard errors

Pooled least squaresPooled modelRandom effects estimatorRandom effects modelSeemingly unrelated regressionsTime-invariant variablesTime-varying variablesUnbalanced panel


Appendices


Consider a simple regression model for cross sectional data:

– The variance of b2, in the presence of heteroskedasticity, is given by:

15ACluster-Robust

Standard Errors:Some Details

1 2β βi i iy x e

22

1 1 1 1

2

1

2 2

1

var var var 2 cov ,

var

N N N N

i i i i i j i ji i i j i

N

i ii

N

i ii

b w e w e w w e e

w e

w


Now suppose we have a panel simple regression model:

with the assumptions:

15ACluster-Robust


1 2β βit it ity x e

cov , ψ and cov , 0 for it is ts it jse e e e i j

Eq. 15A.1


The pooled least squares estimator for β2 is:

where

with

15ACluster-Robust


Eq. 15A.2 2 21 1

βN T

it iti t

b w e

2

1 1

itit N T

iti t

x xw

x x

1 1N T

iti tx x NT


The variance of the pooled least squares estimator b2is given by:

with

15ACluster-Robust


Eq. 15A.3 21 1 1

var var varN T N

it it ii t i

b w e g

1

T

i it itt

g w e


We can now write:

15ACluster-Robust


Eq. 15A.4

21

1 1 1

1

var var

var 2cov ,

var

N

ii

N N N

i i ji i j i

N

ii

b g

g g g

g


To find var(gi), suppose for the moment that T = 2, then:

15ACluster-Robust


2

1

2 21 1 2 2 1 2 1 2

2 21 11 2 22 1 2 12

2 2

1 1

var var

var var 2 cov ,

ψ ψ 2 ψ

ψ

i it itt

i i i i i i i i

i i i i

it is tst s

g w e

w e w e w w e e

w w w w

w w


For T > 2,– Substituting:

15ACluster-Robust


1 1

var ψT T

i it is tst s

g w w

21 1 1

1 1 12

2

1 1

var ψ

ψ

N T T

it is tsi t s

N T T

it is tsi t s

N T

iti t

b w w

x x x x

x x

Eq. 15A.5


A cluster-robust standard error for b2 is given by the square root of:

15ACluster-Robust


1 1 1

2 22

1 1

ˆ ˆvar

N T T

it is it isi t s

N T

iti t

x x x x e eb

x x

Eq. 15A.6


The random effects model is:

We transform the panel data regression into ‘‘deviation about the individual mean’’ form:

15BEstimation of

Error Components

2 2 2 3 3 3β βit i it i it i it iy y x x x x e e Eq. 15B.2

Eq. 15B.1 1 2 2 3 3 ( )it it it i ity x x u e


A consistent estimator of σ2e is:

15BEstimation of

Error Components

Eq. 15B.32ˆ DVe

slopes

SSENT N K


The estimator of σ2u requires a bit more work

–Write:

– This estimator is called the between estimator• It uses variation between individuals as a

basis for estimating the regression parameters• This estimator is unbiased and consistent, but

not minimum variance under the error assumptions of the random effects model

15BEstimation of

Error Components

Eq. 15B.4 1 2 2 3 3 1, ,i i i i iy x x u e i N


The error term has homoskedastic variance:

15BEstimation of

Error Components

Eq. 15B.5

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


An estimate of the variance is:

– Therefore:

15BEstimation of

Error Components

Eq. 15B.6 2

2 e BEu

BE

SSET N K

2 2

2 2 ˆˆ e e BE DVu u

BE slopes

SSE SSET T N K T NT N K

Eq. 15B.7

chapter 15 panel data models

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