chapter 15 panel data models
Post on 22-Feb-2016
203 Views
Preview:
DESCRIPTION
TRANSCRIPT
Principles of Econometrics, 4th Edition Page 1Chapter 15: Panel Data Models
Chapter 15Panel Data Models
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition Page 2Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 3Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 4Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 5Chapter 15: Panel Data Models
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’’
Principles of Econometrics, 4th Edition Page 6Chapter 15: Panel Data Models
15.1 A Microeconomic Panel
Principles of Econometrics, 4th Edition Page 7Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 8Chapter 15: Panel Data Models
15.1A Microeconomic
Panel Table 15.1 Representative Observations from NLS Panel Data
Principles of Econometrics, 4th Edition Page 9Chapter 15: Panel Data Models
15.2 A Microeconomic Panel
Principles of Econometrics, 4th Edition Page 10Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 11Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 12Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 13Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 14Chapter 15: Panel Data Models
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
15.2.1Cluster-Robust Standard Errors
cov , ψit is tse e Eq. 15.6
cov , var =ψit it it tte e e
cov , 0 for it jse e i j
Principles of Econometrics, 4th Edition Page 15Chapter 15: Panel Data Models
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
15.2.1Cluster-Robust Standard Errors
Principles of Econometrics, 4th Edition Page 16Chapter 15: Panel Data Models
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
15.2.1Cluster-Robust Standard Errors
Principles of Econometrics, 4th Edition Page 17Chapter 15: Panel Data Models
15.2Pooled Model
15.2.2Pooled Least
Squares Estimates of Wage Equation
Table 15.2 Pooled Least Squares Estimates of Wage Equation
Principles of Econometrics, 4th Edition Page 18Chapter 15: Panel Data Models
15.3 The Fixed Effects Model
Principles of Econometrics, 4th Edition Page 19Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 20Chapter 15: Panel Data Models
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:
15.3The Fixed Effects
Model
1 2 2 3 3it i it it ity x x e Eq. 15.8
Principles of Econometrics, 4th Edition Page 21Chapter 15: Panel Data Models
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
15.3The Fixed Effects
Model
Principles of Econometrics, 4th Edition Page 22Chapter 15: Panel Data Models
We consider two methods for estimating Eq. 15.81. The least squares dummy variable estimator 2. The fixed effects estimator
15.3The Fixed Effects
Model
Principles of Econometrics, 4th Edition Page 23Chapter 15: Panel Data Models
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:
15.3The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 24Chapter 15: Panel Data Models
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
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
Principles of Econometrics, 4th Edition Page 25Chapter 15: Panel Data Models
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
Table 15.3 Dummy Variable Estimation of Wage Equation for N = 10
Principles of Econometrics, 4th Edition Page 26Chapter 15: Panel Data Models
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
Table 15.4 Pooled Least Squares Estimates of Wage Equation for N = 10
Principles of Econometrics, 4th Edition Page 27Chapter 15: Panel Data Models
We can test the estimates of the intercepts:
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
0 11 12 1,10
1 1
:
: the are not all equali
H
H
Eq. 15.10
Principles of Econometrics, 4th Edition Page 28Chapter 15: Panel Data Models
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:
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
21 2 3
24 5
6
ln β β β
β β β
WAGE EXPER EXPER
TENURE TENUREUNION e
Principles of Econometrics, 4th Edition Page 29Chapter 15: Panel Data Models
The F-statistic is:
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
5.502466 2.667190 92.667190 50 15
4.134
R U
U
SSE SSE JF
SSE NT K
Principles of Econometrics, 4th Edition Page 30Chapter 15: Panel Data Models
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
15.3The Fixed Effects
Model
15.3.1The Least Square Dummy Variable
Estimator for Small N
Principles of Econometrics, 4th Edition Page 31Chapter 15: Panel Data Models
Using the dummy variable approach is not feasible when N is large– Another approach is necessary
15.3The Fixed Effects
Model
15.3.2The Fixed Effects
Estimator
Principles of Econometrics, 4th Edition Page 32Chapter 15: Panel Data Models
Take the data on individual i:
– Average the data across time:
15.3The Fixed Effects
Model
15.3.2The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 33Chapter 15: Panel Data Models
Using the fact that the parameters do not change over time, we can simplify this as:
15.3The Fixed Effects
Model
15.3.2The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 34Chapter 15: Panel Data Models
Now subtract Eq. 15.12 from Eq. 15.11, term by term, to obtain:
or
15.3The Fixed Effects
Model
15.3.2The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 35Chapter 15: Panel Data Models
15.3The Fixed Effects
Model
15.3.2The Fixed Effects
Estimator
Table 15.5 Data in Deviation from Individual Mean Form
Principles of Econometrics, 4th Edition Page 36Chapter 15: Panel Data Models
15.3The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 37Chapter 15: Panel Data Models
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
15.3The Fixed Effects
Model
15.3.2aThe Fixed Effects Estimates of Wage
Equation for N = 10
5 5 45 35 1.133893NT NT N
Principles of Econometrics, 4th Edition Page 38Chapter 15: Panel Data Models
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:
15.3The Fixed Effects
Model
15.3.2aThe Fixed Effects Estimates of Wage
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
Principles of Econometrics, 4th Edition Page 39Chapter 15: Panel Data Models
15.3The Fixed Effects
Model
15.3.3Fixed Effects
Estimates of Wage Equation from
Complete Panel
Table 15.7 Fixed Effects Estimates of Wage Equation for N = 716
Principles of Econometrics, 4th Edition Page 40Chapter 15: Panel Data Models
15.3The Fixed Effects
Model
15.3.3Fixed Effects
Estimates of Wage Equation from
Complete Panel
Table 15.8 Percentage Marginal Effects on Wages
Principles of Econometrics, 4th Edition Page 41Chapter 15: Panel Data Models
15.4 The Random Effects Model
Principles of Econometrics, 4th Edition Page 42Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 43Chapter 15: Panel Data Models
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:
15.4The Random Effects
Model
1 1i iu Eq. 15.16
20, cov , 0, vari i j i uE u u u u Eq. 15.17
Principles of Econometrics, 4th Edition Page 44Chapter 15: Panel Data Models
Substituting, we get:
– Rearranging:
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 45Chapter 15: Panel Data Models
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
15.4The Random Effects
Model
Eq. 15.20 it i itv u e
Principles of Econometrics, 4th Edition Page 46Chapter 15: Panel Data Models
The combined error term has zero mean:
– And a constant, homoskedastic, variance:
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 47Chapter 15: Panel Data Models
There are several correlations that can be considered:1. The correlation between two individuals, i and
j, at the same point in time, t.
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 48Chapter 15: Panel Data Models
There are several correlations (Continued):2. The correlation between errors on the same
individual (i) at different points in time, t and s
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 49Chapter 15: Panel Data Models
There are several correlations (Continued):3. The correlation between errors for different
individuals in different time periods
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 50Chapter 15: Panel Data Models
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:
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 51Chapter 15: Panel Data Models
In terms of the notation introduced to explain the assumptions that motivate the use of cluster-robust standard errors:
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 52Chapter 15: Panel Data Models
Summary of the error term assumptions of the random effects model:
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 53Chapter 15: Panel Data Models
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
15.4The Random Effects
Model
15.4.2Testing for Random
Effects
Principles of Econometrics, 4th Edition Page 54Chapter 15: Panel Data Models
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:
15.4The Random Effects
Model
15.4.2Testing for Random
Effects
1 2 2 3 3it it it ity x x e
Principles of Econometrics, 4th Edition Page 55Chapter 15: Panel Data Models
The test statistic is based on the least squares residuals:
– The test statistic for balanced panels is:
15.4The Random Effects
Model
15.4.2Testing for Random
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
Principles of Econometrics, 4th Edition Page 56Chapter 15: Panel Data Models
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
15.4The Random Effects
Model
15.4.2Testing for Random
Effects
Principles of Econometrics, 4th Edition Page 57Chapter 15: Panel Data Models
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
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 58Chapter 15: Panel Data Models
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
15.4The Random Effects
Model
15.4.3Estimation of the Random Effects
Model
Principles of Econometrics, 4th Edition Page 59Chapter 15: Panel Data Models
15.4The Random Effects
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
Principles of Econometrics, 4th Edition Page 60Chapter 15: Panel Data Models
15.5 Comparing Fixed and Random
Effects Estimators
Principles of Econometrics, 4th Edition Page 61Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 62Chapter 15: Panel Data Models
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
and Random Effects Estimators
15.5.1Endogeneity in the
Random Effects Model
Principles of Econometrics, 4th Edition Page 63Chapter 15: Panel Data Models
The panel data regression Eq. 15.19, including the error component ui, is:
Average the observations for each individual over time:
15.5Comparing Fixed
and Random Effects Estimators
15.5.2The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 64Chapter 15: Panel Data Models
Subtract:
15.5Comparing Fixed
and Random Effects Estimators
15.5.2The Fixed Effects
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
Principles of Econometrics, 4th Edition Page 65Chapter 15: Panel Data Models
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
and Random Effects Estimators
15.5.3The Hausman Test
Principles of Econometrics, 4th Edition Page 66Chapter 15: Panel Data Models
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
and Random Effects Estimators
15.5.3The Hausman Test
, , , ,
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
Principles of Econometrics, 4th Edition Page 67Chapter 15: Panel Data Models
We expect to find:
– Also:
because Hausman proved that:
15.5Comparing Fixed
and Random Effects Estimators
15.5.3The Hausman Test
, ,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
Principles of Econometrics, 4th Edition Page 68Chapter 15: Panel Data Models
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
and Random Effects Estimators
15.5.3The Hausman Test
, ,
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
Principles of Econometrics, 4th Edition Page 69Chapter 15: Panel Data Models
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
and Random Effects Estimators
15.5.3The Hausman Test
, , , ,var var varFE k RE k FE k RE kb b b b
Principles of Econometrics, 4th Edition Page 70Chapter 15: Panel Data Models
15.6 The Hausman-Taylor Estimator
Principles of Econometrics, 4th Edition Page 71Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 72Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 73Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 74Chapter 15: Panel Data Models
15.6The Hausman-
Taylor Estimator Table 15.10 Hausman-Taylor Estimates of Wage Equation
Principles of Econometrics, 4th Edition Page 75Chapter 15: Panel Data Models
15.7 Sets of Regression Equations
Principles of Econometrics, 4th Edition Page 76Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 77Chapter 15: Panel Data Models
An economic model for describing gross firm investment for the ith firm in the tth time period, denoted INVit, may be expressed as:
15.7Sets of Regression
Equations
Eq. 15.41
15.7.1Grunfeld’s
Investment Data
,it it itINV f V K
Principles of Econometrics, 4th Edition Page 78Chapter 15: Panel Data Models
We specify the following two equations for General Electric and Westinghouse:
15.7Sets of Regression
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
Principles of Econometrics, 4th Edition Page 79Chapter 15: Panel Data Models
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?
15.7Sets of Regression
Equations
15.7.1Grunfeld’s
Investment Data
Principles of Econometrics, 4th Edition Page 80Chapter 15: Panel Data Models
The assumption that both firms have the same coefficients and the same error variances can be written as:
15.7Sets of Regression
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
Principles of Econometrics, 4th Edition Page 81Chapter 15: Panel Data Models
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:
15.7Sets of Regression
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
Principles of Econometrics, 4th Edition Page 82Chapter 15: Panel Data Models
15.7Sets of Regression
Equations
15.7.3Estimation:
Different Coefficients, Equal
Error Variances
Table 15.12 Least Squares Estimates from the Dummy Variable Model
Principles of Econometrics, 4th Edition Page 83Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.3Estimation:
Different Coefficients, Equal
Error Variances
Eq. 15.45
16563.00 14989.82 3
1.18914989.82 40 6
R U
U
SSE SSE JF
SSE NT NK
Principles of Econometrics, 4th Edition Page 84Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.4Estimation:
Different Coefficients,
Different Error Variances
Principles of Econometrics, 4th Edition Page 85Chapter 15: Panel Data Models
15.7Sets of Regression
Equations
15.7.4Estimation:
Different Coefficients,
Different Error Variances
Table 15.13 Least Squares Estimates of Separate Investment Equations
Principles of Econometrics, 4th Edition Page 86Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5Seemingly Unrelated
Regressions
, , , ,cov , 0GE t WE t GE WE GE WEe e Eq. 15.46
Principles of Econometrics, 4th Edition Page 87Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5Seemingly Unrelated
Regressions
Principles of Econometrics, 4th Edition Page 88Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5Seemingly Unrelated
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
Principles of Econometrics, 4th Edition Page 89Chapter 15: Panel Data Models
The SUR estimation procedure is optimal under the contemporaneous correlation assumption, so no standard error adjustment is necessary
15.7Sets of Regression
Equations
15.7.5Seemingly Unrelated
Regressions
Principles of Econometrics, 4th Edition Page 90Chapter 15: Panel Data Models
15.7Sets of Regression
Equations
15.7.5Seemingly Unrelated
Regressions
Table 15.14 SUR Estimates of Investment Equations
Principles of Econometrics, 4th Edition Page 91Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5aSeparate or Joint
Estimation
Principles of Econometrics, 4th Edition Page 92Chapter 15: Panel Data Models
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:
15.7Sets of Regression
Equations
15.7.5aSeparate or Joint
Estimation
22,2
, 2 2
ˆ 207.58710.5314
ˆ ˆ 777.4463 104.3079GE WE
GE WEGE WE
r
Principles of Econometrics, 4th Edition Page 93Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5aSeparate or Joint
Estimation
Principles of Econometrics, 4th Edition Page 94Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5aSeparate or Joint
Estimation
Principles of Econometrics, 4th Edition Page 95Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5aSeparate or Joint
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
Principles of Econometrics, 4th Edition Page 96Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5bTesting Cross-
Equation Hypotheses
Eq. 15.47 0 1, 1, 2, 2, 3, 3,: GE WE GE WE GE WEH
Principles of Econometrics, 4th Edition Page 97Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5bTesting Cross-
Equation Hypotheses
Principles of Econometrics, 4th Edition Page 98Chapter 15: Panel Data Models
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
15.7Sets of Regression
Equations
15.7.5bTesting Cross-
Equation Hypotheses
Principles of Econometrics, 4th Edition Page 99Chapter 15: Panel Data Models
Key Words
Principles of Econometrics, 4th Edition Page 100Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 101Chapter 15: Panel Data Models
Appendices
Principles of Econometrics, 4th Edition Page 102Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 103Chapter 15: Panel Data Models
Now suppose we have a panel simple regression model:
with the assumptions:
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 104Chapter 15: Panel Data Models
The pooled least squares estimator for β2 is:
where
with
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 105Chapter 15: Panel Data Models
The variance of the pooled least squares estimator b2is given by:
with
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 106Chapter 15: Panel Data Models
We can now write:
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 107Chapter 15: Panel Data Models
To find var(gi), suppose for the moment that T = 2, then:
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 108Chapter 15: Panel Data Models
For T > 2,– Substituting:
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 109Chapter 15: Panel Data Models
A cluster-robust standard error for b2 is given by the square root of:
15ACluster-Robust
Standard Errors:Some Details
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
Principles of Econometrics, 4th Edition Page 110Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 111Chapter 15: Panel Data Models
A consistent estimator of σ2e is:
15BEstimation of
Error Components
Eq. 15B.32ˆ DVe
slopes
SSENT N K
Principles of Econometrics, 4th Edition Page 112Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 113Chapter 15: Panel Data Models
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
Principles of Econometrics, 4th Edition Page 114Chapter 15: Panel Data Models
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
top related