Chapter 15
Panel Data Models
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 1
Chapter Contents





15.1 A Microeconomic Panel
15.2 A Pooled Model
15.3 The Fixed Effects Model
15.4 The Random Effects Model
15.5 Comparing Fixed and Random Effects
Estimators
 15.6 The Hausman-Taylor Estimator
 15.7 Sets of Regression Equations
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 2
A panel of data consists of a group of crosssectional 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
Chapter 15: Panel Data Models
Page 3
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
Chapter 15: Panel Data Models
Page 4
It is possible to have data that combines crosssectional 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
Chapter 15: Panel Data Models
Page 5
15.1
A Microeconomic Panel
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 6
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 7
15.1
A Microeconomic
Panel
Table 15.1 Representative Observations from NLS Panel Data
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 8
15.2
A Microeconomic Panel
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 9
15.2
Pooled Model
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
yit  1  2 x2it  3 x3it  eit
Eq. 15.1
– Notice that the coefficients (β1, β2, β3) do not
have i or t subscripts
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 10
15.2
Pooled Model
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 11
15.2
Pooled 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
Eq. 15.2
E  eit   0
Eq. 15.3
var  eit   E eit2   e2
Eq. 15.4
cov  eit , e js
Eq. 15.5
cov  eit , x2it   0, cov  eit , x3it   0
Principles of Econometrics, 4th Edition
 
  E  e , e   0 for i  j or t  s
it
js
Chapter 15: Panel Data Models
Page 12
15.2
Pooled Model
15.2.1
Cluster-Robust
Standard Errors
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 13
15.2
Pooled Model
15.2.1
Cluster-Robust
Standard Errors
To relax the assumption of zero error correlation
over time for the same individual, we write:
cov  eit , eis   ψts
Eq. 15.6
– This also relaxes the assumption of
homoskedasticity:
cov  eit , eit   var  eit  =ψtt
We continue to assume that the errors for different
individuals are uncorrelated:
cov  eit , e js   0 for i  j
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 14
15.2
Pooled Model
15.2.1
Cluster-Robust
Standard Errors
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 15
15.2
Pooled Model
15.2.1
Cluster-Robust
Standard Errors
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 16
15.2
Pooled Model
Table 15.2 Pooled Least Squares Estimates of Wage Equation
15.2.2
Pooled Least
Squares Estimates of
Wage Equation
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 17
15.3
The Fixed Effects Model
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 18
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:
yit  1i  2i x2it  3i x3it  eit
Eq. 15.7
– An i subscript has been added to each of the
subscripts, implying that (β1, β2, β3) can be
different for each individual
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 19
15.3
The Fixed Effects
Model
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:
Eq. 15.8
Principles of Econometrics, 4th Edition
yit  1i  2 x2it  3 x3it  eit
Chapter 15: Panel Data Models
Page 20
15.3
The Fixed Effects
Model
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 21
15.3
The Fixed Effects
Model
We consider two methods for estimating Eq. 15.8
1. The least squares dummy variable estimator
2. The fixed effects estimator
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 22
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
One way to estimate the model in Eq. 15.8 is to
include an intercept dummy variable (indicator
variable) for each individual
1 i  1
1 i  2
1 i  3
D1i  
D2i  
D3i  
0 otherwise
0 otherwise
0 otherwise
– If we have 10 individuals, we define 10 such
dummies
Now we can write:
Eq. 15.9
yit  11D1i  12 D2i 
Principles of Econometrics, 4th Edition
 1,10 D10i  2V2it  3 K3it  eit
Chapter 15: Panel Data Models
Page 23
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
If the error terms eit are uncorrelated with mean
zero and constant variance σ2e 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 24
15.3
The Fixed Effects
Model
Table 15.3 Dummy Variable Estimation of Wage Equation for N = 10
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 25
15.3
The Fixed Effects
Model
Table 15.4 Pooled Least Squares Estimates of Wage Equation for N = 10
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 26
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
We can test the estimates of the intercepts:
H 0 : 11  12 
Eq. 15.10
Principles of Econometrics, 4th Edition
 1,10
H1 : the 1i are not all equal
Chapter 15: Panel Data Models
Page 27
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
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:
ln WAGE   β1  β2 EXPER  β3 EXPER 2
 β4TENURE  β5TENURE 2
 β6UNION  e
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 28
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
The F-statistic is:
SSER  SSEU  J

F
SSEU  NT  K 
5.502466  2.667190 


2.667190  50  15 
9
 4.134
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 29
15.3
The Fixed Effects
Model
15.3.1
The Least Square
Dummy Variable
Estimator for Small
N
The value of the test statistic F = 4.134 yields a pvalue 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 30
15.3
The Fixed Effects
Model
15.3.2
The Fixed Effects
Estimator
Using the dummy variable approach is not feasible
when N is large
– Another approach is necessary
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 31
15.3
The Fixed Effects
Model
15.3.2
The Fixed Effects
Estimator
Take the data on individual i:
Eq. 15.11
yit  1i  2 x2it  3 x3it  eit
t  1,
,T
– Average the data across time:
1 T
 yit  1i  2 x2it  3 x3it  eit 

T t 1
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 32
15.3
The Fixed Effects
Model
15.3.2
The Fixed Effects
Estimator
Using the fact that the parameters do not change
over time, we can simplify this as:
Eq. 15.12
1 T
1 T
1 T
1 T
yi   yit  1i  2  x2it  3  x3it   eit
T t 1
T t 1
T t 1
T t 1
 1i  2 x2i  3 x3i  ei
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 33
15.3
The Fixed Effects
Model
15.3.2
The Fixed Effects
Estimator
Now subtract Eq. 15.12 from Eq. 15.11, term by
term, to obtain:
yit  1i  2 x2it  3 x3it  eit
 ( yi  1i  2 x2 i  3 x3i  ei )
Eq. 15.13
yit  yi  2 ( x2it  x2i )  3 ( x3it  x3i )  (eit  ei )
or
Eq. 15.14
Principles of Econometrics, 4th Edition
yit  2 xit  3 xit  eit
Chapter 15: Panel Data Models
Page 34
15.3
The Fixed Effects
Model
Table 15.5 Data in Deviation from Individual Mean Form
15.3.2
The Fixed Effects
Estimator
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 35
15.3
The Fixed Effects
Model
Table 15.6 Fixed Effects Estimation of Wage Equation for N = 10
15.3.2a
The Fixed Effects
Estimates of Wage
Equation for N = 10
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 36
15.3
The Fixed Effects
Model
15.3.2a
The Fixed Effects
Estimates of Wage
Equation for N = 10
If we multiply the standard errors from estimating
Eq. 15.14 by the correction factor
 NT  5  NT  N  5 
45 35  1.133893
the resulting standard errors are identical to those
in Table 15.3
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 37
15.3
The Fixed Effects
Model
15.3.2a
The Fixed Effects
Estimates of Wage
Equation for N = 10
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:
yi  b1i  b2 x2i  b3 x3i
– So that the fixed effects are:
Eq. 15.15
Principles of Econometrics, 4th Edition
b1i  yi  b2 x2i  b3 x3i
Chapter 15: Panel Data Models
i  1,
,N
Page 38
15.3
The Fixed Effects
Model
Table 15.7 Fixed Effects Estimates of Wage Equation for N = 716
15.3.3
Fixed Effects
Estimates of Wage
Equation from
Complete Panel
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 39
15.3
The Fixed Effects
Model
Table 15.8 Percentage Marginal Effects on Wages
15.3.3
Fixed Effects
Estimates of Wage
Equation from
Complete Panel
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 40
15.4
The Random Effects Model
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 41
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 42
15.4
The 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:
1i  1  ui
Eq. 15.16
– The random individual differences ui are called
random effects and have:
Eq. 15.17
E  ui   0, cov  ui , u j   0, var  ui   
Principles of Econometrics, 4th Edition
2
u
Chapter 15: Panel Data Models
Page 43
15.4
The Random Effects
Model
Substituting, we get:
yit  1i  2 x2it  3 x3it  eit
Eq. 15.18
  1  ui   2 x2it  3 x3it  eit
– Rearranging:
yit  1   2 x2it  3 x3it   eit  ui 
Eq. 15.19
 1   2 x2it  3 x3it  vit
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 44
15.4
The Random Effects
Model
The combined error term is:
vit  ui  eit
Eq. 15.20
– 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 45
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
The combined error term has zero mean:
E  vit   E ui  eit   E ui   E  eit   0  0  0
– And a constant, homoskedastic, variance:
v2  var  vit   var  ui  eit 
Eq. 15.21
 var  ui   var  eit   2cov  ui , eit 
 u2  e2
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 46
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
There are several correlations that can be
considered:
1. The correlation between two individuals, i and
j, at the same point in time, t.
cov  vit , v jt   E (vit v jt )  E  ui  eit   u j  e jt  
 E  ui u j   E  ui e jt   E  eit u j   E  eit e jt 
00000
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 47
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
There are several correlations (Continued):
2. The correlation between errors on the same
individual (i) at different points in time, t and
s
cov  vit , vis   E (vit vis )  E  ui  eit  ui  eis  
 E  ui2   E  ui eis   E  eit ui   E  eit eis 
Eq. 15.22
 u2  0  0  0
 u2
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 48
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
There are several correlations (Continued):
3. The correlation between errors for different
individuals in different time periods
cov  vit , v js   E (vit v js )  E  ui  eit   u j  e js  
 E  ui u j   E  ui e js   E  eit u j   E  eit e js 
 0000 0
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 49
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
Eq. 15.23
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:
u2
  corr(vit , vis ) 
 2
2
var(vit ) var(vis ) u  e
Principles of Econometrics, 4th Edition
cov(vit , vis )
Chapter 15: Panel Data Models
ts
Page 50
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
In terms of the notation introduced to explain the
assumptions that motivate the use of cluster-robust
standard errors:
var(vit )  ψit  u2  e2 and cov(vit , vis )  ψis  u2 t  s
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 51
15.4
The Random Effects
Model
15.4.1
Error Term
Assumptions
Summary of the error term assumptions of the
random effects model:
Eq. 15.24
E  vit   0
Eq. 15.25
var(vit )  u2   e2
Eq. 15.26
cov(vit , vis )  u2 t  s
Eq. 15.27
cov(vit , v js )  0 i  j
Eq. 15.28
cov(eit , x2it )  0, cov(eit , x3it )  0
Eq. 15.29
cov(ui , x2it )  0, cov(ui , x3it )  0
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 52
15.4
The Random Effects
Model
15.4.2
Testing for Random
Effects
We can test for the presence of heterogeneity by
testing the null hypothesis H0: σ2u = 0 against the
alternative hypothesis H1: σ2u > 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 53
15.4
The Random Effects
Model
15.4.2
Testing 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:
yit  1  2 x2it  3 x3it  eit
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 54
15.4
The Random Effects
Model
15.4.2
Testing for Random
Effects
The test statistic is based on the least squares
residuals:
eˆit  yit  b1  b2 x2it  b3 x3it
– The test statistic for balanced panels is:
Eq. 15.30
Principles of Econometrics, 4th Edition
 N  T 2 
    eˆit 

NT  i 1  t 1 

LM 
 1
 N T
2 T  1 
2

ˆ
e

it
 i 1 t 1

Chapter 15: Panel Data Models
Page 55
15.4
The Random Effects
Model
15.4.2
Testing for Random
Effects
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 56
15.4
The Random Effects
Model
15.4.3
Estimation of the
Random Effects
Model
We can obtain the generalized least squares
estimator in the random effects model by applying
least squares to a transformed model:
yit*  1x1*it  2 x2*it  3 x3*it  vit*
Eq. 15.31
where the transformed variables are:
Eq. 15.32
yit*  yit   yi , x1*it  1  ,
x2*it  x2it   x2i , x3*it  x3it   x3i
and α is defined as
Eq. 15.33
Principles of Econometrics, 4th Edition
  1
e
T u2  e2
Chapter 15: Panel Data Models
Page 57
15.4
The Random Effects
Model
15.4.3
Estimation of the
Random Effects
Model
For α = 1, the random effects estimator is identical
to the fixed effects estimator
For 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 58
15.4
The Random Effects
Model
Table 15.9 Random Effects Estimates of Wage Equation
15.4.4
Random Effects
Estimation of the
Wage Equation
The estimate of the transformation parameter α is:
ˆ  1 
Principles of Econometrics, 4th Edition
ˆ e
T ˆ  ˆ
2
u
2
e
1
0.1951
5  0.1083  0.0381
Chapter 15: Panel Data Models
 0.7437
Page 59
15.5
Comparing Fixed and Random
Effects Estimators
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 60
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 61
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.1
Endogeneity in the
Random Effects
Model
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 62
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.2
The Fixed Effects
Estimator in a
Random Effects
Model
The panel data regression Eq. 15.19, including the
error component ui, is:
yit  1  2 x2it  3 x3it  (ui  eit )
Eq. 15.34
Average the observations for each individual over
time:
Eq. 15.35
1 T
1 T
1 T
1 T
1 T
yi   yit  1  2  x2it  3  x3it   ui   eit
T t 1
T t 1
T t 1
T t 1
T t 1
 1  2 x2i  3 x3i  ui  ei
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 63
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.2
The Fixed Effects
Estimator in a
Random Effects
Model
Subtract:
yit  1  2 x2it  3 x3it  ui  eit
Eq. 15.36
 ( yi  1  2 x2i  3 x3i  ui  ei )
yit  yi  2 ( x2it  x2i )  3 ( x3it  x3i )  (eit  ei )
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 64
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.3
The Hausman Test
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 65
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.3
The 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:
Eq. 15.37
t
bFE ,k  bRE ,k
12
 var  b   var  b 
FE ,k
RE ,k 


Principles of Econometrics, 4th Edition

Chapter 15: Panel Data Models
bFE ,k  bRE ,k
se  b   se  b  
FE ,k
RE ,k


2 12
2
Page 66
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.3
The Hausman Test
We expect to find:
var  bFE ,k   var  bRE ,k   0
– Also:
var  bFE , k  bRE , k   var  bFE , k   var  bRE ,k   2cov  bFE ,k , bRE ,k 
 var  bFE ,k   var  bRE ,k 
because Hausman proved that:
cov bFE,k , bRE,k   var bRE,k 
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 67
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.3
The Hausman Test
Applying the t-test to the SOUTH we get:
t
bFE ,k  bRE ,k
se  b   se  b  
FE , k
RE , k


2
2 12

0.01632  (0.08181)
2 12
 0.03615   0.02241 


2
 2.31
– 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 68
15.5
Comparing Fixed
and Random Effects
Estimators
15.5.3
The Hausman Test
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:
var bFE,k  bRE,k   var bFE,k   var bRE,k 
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 69
15.6
The Hausman-Taylor Estimator
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 70
15.6
The HausmanTaylor 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 71
15.6
The HausmanTaylor Estimator
Consider the regression model:
Eq. 15.38
yit  β1  β2 xit ,exog  β3 xit ,endog  β4 wi,exog  β5 wi,endog  ui  eit
with:
xit,exog :exogenous variables that vary over time and
individuals
xit,endog: endogenous variables that vary over time
and individuals
wi,exog: time-invariant exogenous variables
wi,endog: time-invariant endogenous variables
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 72
15.6
The HausmanTaylor Estimator
A slightly modify set is applied to the transformed
generalized least squares model from Eq. 15.31:
Eq. 15.39
yit*  β1  β2 xit*,exog  β3 xit*,endog  β4 wi*,exog  β5 wi*,endog  vit*
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 73
15.6
The HausmanTaylor Estimator
Table 15.10 Hausman-Taylor Estimates of Wage Equation
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 74
15.7
Sets of Regression Equations
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 75
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
Eq. 15.40
Principles of Econometrics, 4th Edition
yit  1i  2i x2it  3i x3it  eit
Chapter 15: Panel Data Models
Page 76
15.7
Sets of Regression
Equations
15.7.1
Grunfeld’s
Investment Data
An economic model for describing gross firm
investment for the ith firm in the tth time period,
denoted INVit, may be expressed as:
Eq. 15.41
Principles of Econometrics, 4th Edition
INVit  f Vit , Kit 
Chapter 15: Panel Data Models
Page 77
15.7
Sets of Regression
Equations
15.7.1
Grunfeld’s
Investment Data
We specify the following two equations for
General Electric and Westinghouse:
INVGE ,t  1  2VGE ,t  3 KGE ,t  eGE ,t
t  1935, ,1954
INVWE ,t  1  2VWE ,t  3 KWE ,t  eWE ,t
t  1935, ,1954
Eq. 15.42
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 78
15.7
Sets of Regression
Equations
15.7.1
Grunfeld’s
Investment Data
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?
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 79
15.7
Sets of Regression
Equations
15.7.2
Estimation: Equal
Coefficients, Equal
Error Variances
The assumption that both firms have the same
coefficients and the same error variances can be
written as:
Eq. 15.43
β1,GE  β1,WE
Principles of Econometrics, 4th Edition
β2,GE  β2,WE
Chapter 15: Panel Data Models
β3,GE  β3,WE
2
σGE
 σ2WE
Page 80
15.7
Sets of Regression
Equations
15.7.3
Estimation: Different
Coefficients, Equal
Error Variances
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:
Eq. 15.44
INVit  1,GE  1Di  2,GEVit  2  Di Vit   3,GE Kit  3  Di  Kit   eit
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 81
15.7
Sets of Regression
Equations
Table 15.12 Least Squares Estimates from the Dummy Variable Model
15.7.3
Estimation: Different
Coefficients, Equal
Error Variances
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 82
15.7
Sets of Regression
Equations
15.7.3
Estimation: Different
Coefficients, Equal
Error Variances
Eq. 15.45
Using the Chow test, we get:
SSER  SSEU  J 16563.00  14989.82

F

SSEU  NT  NK 
14989.82  40  6
3
 1.189
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 83
15.7
Sets of Regression
Equations
15.7.4
Estimation: Different
Coefficients,
Different Error
Variances
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 84
15.7
Sets of Regression
Equations
Table 15.13 Least Squares Estimates of Separate Investment Equations
15.7.4
Estimation: Different
Coefficients,
Different Error
Variances
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 85
15.7
Sets of Regression
Equations
15.7.5
Seemingly Unrelated
Regressions
Consider the following assumption:
cov  eGE,t , eWE,t   GE,WE GE ,WE  0
Eq. 15.46
– The error terms in the two equations, at the
same point in time, are correlated
• This kind of correlation is called
contemporaneous correlation
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 86
15.7
Sets of Regression
Equations
15.7.5
Seemingly Unrelated
Regressions
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 87
15.7
Sets of Regression
Equations
15.7.5
Seemingly Unrelated
Regressions
Three stages in the SUR estimation procedure:
1. Estimate the equations separately using OLS
2. Use the OLS residuals from (1) to estimate
σ2GE, σ2WE and σGE,WE
• The estimated covariance is given by:
σˆ GE ,WE 
1
T  KGE T  KWE
20
 eˆ
t 1
eˆ
GE ,t WE ,t
1 20

eˆGE ,t eˆWE ,t

T  3 t 1
 207.587
3. Use the estimates from (2) to estimate the two
equations jointly within a generalized least
squares framework
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 88
15.7
Sets of Regression
Equations
15.7.5
Seemingly Unrelated
Regressions
The SUR estimation procedure is optimal under
the contemporaneous correlation assumption, so
no standard error adjustment is necessary
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 89
15.7
Sets of Regression
Equations
Table 15.14 SUR Estimates of Investment Equations
15.7.5
Seemingly Unrelated
Regressions
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 90
15.7
Sets of Regression
Equations
15.7.5a
Separate or Joint
Estimation
Two situations in which separate least squares
estimation is just as good as the SUR technique
1. 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 91
15.7
Sets of Regression
Equations
15.7.5a
Separate 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:
2
GE ,WE
r
Principles of Econometrics, 4th Edition
ˆ

ˆ ˆ
2
GE ,WE
2
2
GE WE
207.5871


 0.5314
 777.4463104.3079
Chapter 15: Panel Data Models
2
Page 92
15.7
Sets of Regression
Equations
15.7.5a
Separate or Joint
Estimation
To check the statistical significance of r2GE,WE, test
the null hypothesis H0: σGE,WE = 0
– If σGE,WE = 0, then LM = T x r2GE,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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 93
15.7
Sets of Regression
Equations
15.7.5a
Separate or Joint
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 χ2distribution with degrees of freedom equal to
the number of correlations
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 94
15.7
Sets of Regression
Equations
15.7.5a
Separate or Joint
Estimation
With three equations, denoted by subscripts 1, 2
and 3, the null hypothesis is:
H0 : 12  13  23  0
– The χ2(3) test statistic is:
LM  T  r122  r132  r232 
– With M equations:
M i 1
LM  T  rij2
i  2 j 1
with M(M – 1)/2 degrees of freedom
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 95
15.7
Sets of Regression
Equations
15.7.5b
Testing CrossEquation
Hypotheses
Eq. 15.47
We previously used the dummy variable model and
the Chow test to test whether the two equations had
identical coefficients:
H0 : 1,GE  1,WE
2,GE  2,WE
3,GE  3,WE
– 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 96
15.7
Sets of Regression
Equations
15.7.5b
Testing CrossEquation
Hypotheses
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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 97
15.7
Sets of Regression
Equations
15.7.5b
Testing CrossEquation
Hypotheses
The equality of coefficients is not the only crossequation 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 98
Key Words
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 99
Keywords
Balanced panel
Cluster-robust
standard errors
Contemporaneous
correlation
Cross-equation
hypotheses
Deviations from
individual means
Endogeneity
Error components
model
Fixed effects
estimator
Principles of Econometrics, 4th Edition
Fixed effects
model
Hausman test
Hausman-Taylor
estimator
Heterogeneity
Instrumental
variables
Least squares
dummy variable
model
LM test
Panel corrected
standard errors
Chapter 15: Panel Data Models
Pooled least squares
Pooled model
Random effects
estimator
Random effects
model
Seemingly
unrelated
regressions
Time-invariant
variables
Time-varying
variables
Unbalanced panel
Page 100
Appendices
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 101
15A
Cluster-Robust
Standard Errors:
Some Details
Consider a simple regression model for cross
sectional data:
yi  β1  β2 xi  ei
– The variance of b2, in the presence of
heteroskedasticity, is given by:
N
N
N
 N 2
var  b2   var   wi ei    wi var  ei     2wi w j cov  ei , e j 
i 1 j  i 1
 i 1
 i 1
N
  wi2 var  ei 
i 1
N
  wi2 i2
i 1
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 102
15A
Cluster-Robust
Standard Errors:
Some Details
Now suppose we have a panel simple regression
model:
yit  β1  β2 xit  eit
Eq. 15A.1
with the assumptions:
cov  eit , eis   ψts and cov  eit , e js   0 for i  j
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 103
15A
Cluster-Robust
Standard Errors:
Some Details
The pooled least squares estimator for β2 is:
N
T
b2  β 2    wit eit
Eq. 15A.2
i 1 t 1
where
wit 
with
xit  x
  xit  x 
N
T
2
i 1 t 1
x  i 1 t 1 xit NT
N
Principles of Econometrics, 4th Edition
T
Chapter 15: Panel Data Models
Page 104
15A
Cluster-Robust
Standard Errors:
Some Details
The variance of the pooled least squares estimator
b2is given by:
N T

N 
var  b2   var   wit eit   var   gi 
 i 1 t 1

 i 1 
Eq. 15A.3
with
T
g i   wit eit
t 1
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 105
15A
Cluster-Robust
Standard Errors:
Some Details
We can now write:
N 
var  b2   var   gi 
 i 1 
Eq. 15A.4
 2cov  gi , g j 
N
N
i 1
i 1 j  i 1
  var  gi   
N
N
  var  gi 
i 1
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 106
15A
Cluster-Robust
Standard Errors:
Some Details
To find var(gi), suppose for the moment that T = 2,
then:
 2

var  gi   var   wit eit 
 t 1

 wi21 var  ei1   wi22 var  ei 2   2wi1wi 2 cov  ei1 , ei 2 
 wi21ψ11  wi22 ψ 22  2wi1wi 2 ψ12
2
2
  wit wis ψts
t 1 s 1
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 107
15A
Cluster-Robust
Standard Errors:
Some Details
T
T
For T > 2, var  gi    wit wis ψts
t 1 s 1
– Substituting:
N
T
T
var  b2    wit wis ψts
i 1 t 1 s 1
  xit  x  xis  x  ψts
N
Eq. 15A.5

Principles of Econometrics, 4th Edition
T
T
i 1 t 1 s 1
2

   xit  x  
 i 1 t 1

N
T
Chapter 15: Panel Data Models
2
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15A
Cluster-Robust
Standard Errors:
Some Details
A cluster-robust standard error for b2 is given by
the square root of:
  xit  x  xis  x  eˆit eˆis
N
Eq. 15A.6
Principles of Econometrics, 4th Edition
var  b2  
T
T
i 1 t 1 s 1
2

   xit  x  
 i 1 t 1

N
T
Chapter 15: Panel Data Models
2
Page 109
15B
Estimation of Error
Components
The random effects model is:
yit  1  2 x2it  3 x3it  (ui  eit )
Eq. 15B.1
We transform the panel data regression into
‘‘deviation about the individual mean’’ form:
Eq. 15B.2
yit  yi  β2  x2it  x2i   β3  x3it  x3i    eit  ei 
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 110
15B
Estimation of Error
Components
A consistent estimator of σ2e is:
Eq. 15B.3
Principles of Econometrics, 4th Edition
SSEDV
ˆ 
NT  N  K slopes
2
e
Chapter 15: Panel Data Models
Page 111
15B
Estimation of Error
Components
The estimator of σ2u requires a bit more work
– Write:
yi  1  2 x2i  3 x3i  ui  ei
Eq. 15B.4
i  1,
,N
– 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
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 112
15B
Estimation of Error
Components
The error term has homoskedastic variance:
Eq. 15B.5
T

var  ui  ei   var  ui   var  ei   var  ui   var   eit T 
 t 1

1
T e2
T 
2
2
 u  2 var   eit   u  2
T
T
 t 1 
e2
 
T
2
u
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 113
15B
Estimation of Error
Components
An estimate of the variance is:
2

SSEBE
2
e
u 

T N  K BE
Eq. 15B.6
– Therefore:
Eq. 15B.7
2
2
ˆ


SSEBE
SSEDV
2
2
e
e
ˆ u  u 



T
T
N  K BE T  NT  N  K slopes 
Principles of Econometrics, 4th Edition
Chapter 15: Panel Data Models
Page 114