panel data models econ 6002 econometrics memorial university of newfoundland adapted from vera...
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Chapter 15
Panel Data Models
ECON 6002EconometricsMemorial University of Newfoundland
Adapted from Vera Tabakova’s notes
Chapter 15: Panel Data Models
15.1 Grunfeld’s Investment Data
15.2 Sets of Regression Equations
15.3 Seemingly Unrelated Regressions
15.4 The Fixed Effects Model
15.4 The Random Effects Model
Extensions RCM, dealing with endogeneity when we have
static variables
Slide 15-2Principles of Econometrics, 3rd Edition
Chapter 15: Panel Data Models
The different types of panel data sets can be described as:
“long and narrow,” with “long” time dimension and “narrow”, few
cross sectional units;
“short and wide,” many units observed over a short period of time;
“long and wide,” indicating that both N and T are relatively large.
Slide 15-3Principles of Econometrics, 3rd Edition
15.1 Grunfeld’s Investment Data
The data consist of T = 20 years of data (1935-1954) for N = 10 large firms.
Let yit = INVit and x2it = Vit and x3it = Kit
Slide 15-4Principles of Econometrics, 3rd Edition
(15.1)
(15.2)
,it it itINV f V K
1 2 2 3 3it it it it it it ity x x e
Notice the subindices!
Value of stock, proxy for expected profitsCapital stock, proxy for desired permanentCapital stock
15.2 Sets of Regression Equations
Slide 15-5Principles of Econometrics, 3rd Edition
(15.3a)
(15.3b)
, 1 2 , 3 , ,
, 1 2 , 3 , ,
1, ,20
1, ,20
GE t GE t GE t GE t
WE t WE t WE t WE t
INV V K e t
INV V K e t
1 2 2 3 3 1, 2; 1, ,20it it it ity x x e i t
For simplicity we focus on only two firmskeep if (i==3 | i==8) in STATA
15.2 Sets of Regression Equations
Slide 15-6Principles of Econometrics, 3rd Edition
(15.4a)
(15.4b)
, 1, 2, , 3, , ,
, 1, 2, , 3, , ,
1, ,20
1, ,20
GE t GE GE GE t GE GE t GE t
WE t WE WE WE t WE WE t WE t
INV V K e t
INV V K e t
1 2 2 3 3 1, 2; 1, ,20it i i it i it ity x x e i t
15.2 Sets of Regression Equations
Assumption (15.5) says that the errors in both investment functions (i) have zero mean, (ii) are homoskedastic with constant variance, and (iii) are not correlated over time; autocorrelation does not exist. The two equations do have different error variances
Slide 15-7Principles of Econometrics, 3rd Edition
(15.5)
2, , , ,
2, , , ,
0 var cov , 0
0 var cov , 0
GE t GE t GE GE t GE s
WE t WE t WE WE t WE s
E e e e e
E e e e e
2 2 and .GE WE
15.2 Sets of Regression Equations
Slide 15-8Principles of Econometrics, 3rd Edition
reg inv v k if i==3scalar sse_ge = e(rss)
reg inv v k if i==8scalar sse_we = e(rss)
15.2 Sets of Regression Equations
Let Di be a dummy variable equal to 1 for the Westinghouse
observations and 0 for the General Electric observations. If the
variances are the same for both firms then we can run:
Slide 15-9Principles of Econometrics, 3rd Edition
(15.6)1, 1 2, 2 3, 3it GE i GE it i it GE it i it itINV D V D V K D K e
* Create dummy variablegen d = (i == 8)gen dv = d*vgen dk = d*k
* Estimate dummy variable modelreg inv d v dv k dktest d dv dk
15.2 Sets of Regression Equations
Slide 15-10Principles of Econometrics, 3rd Edition
15.2 Sets of Regression Equations
Slide 15-11Principles of Econometrics, 3rd Edition
* Goldfeld-Quandt testscalar GQ = sse_ge/sse_wescalar fc95 = invFtail(17,17,.05)di "Goldfeld-Quandt Test statistic = " GQdi "F(17,17,.95) = " fc95
Goldfeld-Quandt Test statistic = 7.45338
F(17,17,.95) = 2.2718929
So we reject equality at the 5% level…=> we cannot really merge the two equations for now…
15.3 Seemingly Unrelated Regressions
This assumption says that the error terms in the two equations, at the same point in time, are correlated. This kind of correlation is called a contemporaneous correlation.
Under this assumption, the joint regression would be better than the separate simple OLS regressions
Slide 15-12Principles of Econometrics, 3rd Edition
(15.7) , , ,cov ,GE t WE t GE WEe e
15.3 Seemingly Unrelated Regressions
Econometric software includes commands for SUR (or SURE) that
carry out the following steps:
(i) Estimate the equations separately using least squares;
(ii) Use the least squares residuals from step (i) to estimate
;
(iii) Use the estimates from step (ii) to estimate the two equations jointly
within a generalized least squares framework.
Slide 15-13Principles of Econometrics, 3rd Edition
2 2,, and GE WE GE WE
15.3 Seemingly Unrelated Regressions
Slide 15-14Principles of Econometrics, 3rd Edition
15.3 Seemingly Unrelated Regressions
Slide 15-15Principles of Econometrics, 3rd Edition
* Open and summarize data (which is already in wide format!!!)use grunfeld2, clearsummarize
* SUR sureg ( inv_ge v_ge k_ge) ( inv_we v_we k_we), corrtest ([inv_ge]_cons = [inv_we]_cons) ([inv_ge]_b[v_ge] = [inv_we]_b[v_we]) ([inv_ge]_b[k_ge] = [inv_we]_b[k_we])
15.3.1 Separate or Joint Estimation?
There are two situations where separate least squares estimation is
just as good as the SUR technique :
(i) when the equation errors are not contemporaneously correlated;
(ii) when the same (the “very same”) explanatory variables appear in
each equation.
If the explanatory variables in each equation are different, then a test
to see if the correlation between the errors is significantly different
from zero is of interest.Slide 15-16Principles of Econometrics, 3rd Edition
15.3.1 Separate or Joint Estimation?
In this case we have 3 parameters in each equation so:
Slide 15-17Principles of Econometrics, 3rd Edition
22,2
, 2 2
ˆ 207.58710.53139
ˆ ˆ 777.4463 104.3079GE WE
GE WEGE WE
r
20 20
, , , , ,1 1
1 1ˆ ˆ ˆ ˆ ˆ
3GE WE GE t WE t GE t WE tt tGE WE
e e e eTT K T K
3.GE WEK K
15.3.1 Separate or Joint Estimation?
Testing for correlated errors for two equations:
LM = 10.628 > 3.84 (Breusch-Pagan test of independence: chi2(1))
Hence we reject the null hypothesis of no correlation between the
errors and conclude that there are potential efficiency gains from
estimating the two investment equations jointly using SUR.
Slide 15-18Principles of Econometrics, 3rd Edition
0 ,: 0GE WEH
2 2, (1) 0 under .GE WELM Tr H
15.3.1 Separate or Joint Estimation?
Testing for correlated errors for three equations:
Slide 15-19Principles of Econometrics, 3rd Edition
0 12 13 23: 0H
2 2 2 212 13 23 (3)LM T r r r
15.3.1 Separate or Joint Estimation?
Testing for correlated errors for M equations:
Under the null hypothesis that there are no contemporaneous
correlations, this LM statistic has a χ2-distribution with M(M–1)/2
degrees of freedom, in large samples.
Slide 15-20Principles of Econometrics, 3rd Edition
12
2 1
M i
iji j
LM T r
15.3.2 Testing Cross-Equation Hypotheses
Most econometric software will perform an F-test and/or a Wald χ2–test; in the context of SUR equations both tests are large sample approximate tests.
The F-statistic has J numerator degrees of freedom and (MTK) denominator degrees of freedom, where J is the number of hypotheses, M is the number of equations, and K is the total number of coefficients in the whole system, and T is the number of time series observations per equation. The χ2-statistic has J degrees of freedom.
Slide 15-21Principles of Econometrics, 3rd Edition
(15.8)0 1, 1, 2, 2, 3, 3,: , ,GE WE GE WE GE WEH
15.4 The Fixed Effects Model
SUR is OK when the panel is long and narrow, not when it is short and wide. Consider instead…
We cannot consistently estimate the 3×N×T parameters in (15.9) with only NT total observations. But we can impose some more structure…
Slide 15-22Principles of Econometrics, 3rd Edition
(15.9)
(15.10)
1 2 2 3 3it it it it it it ity x x e
1 1 2 2 3 3, ,it i it it
We consider only one-way effects and assume common slopeparameters across cross-sectional units
15.4 The Fixed Effects Model
All behavioral differences between individual firms and over time are
captured by the intercept. Individual intercepts are included to
“control” for these firm specific differences.
Slide 15-23Principles of Econometrics, 3rd Edition
(15.11)1 2 2 3 3it i it it ity x x e
15.4.1 A Dummy Variable Model
This specification is sometimes called the least squares dummy
variable model, or the fixed effects model.
Slide 15-24Principles of Econometrics, 3rd Edition
(15.12)
1 2 3
1 1 1 2 1 3, , , etc.
0 otherwise 0 otherwise 0 otherwisei i i
i i iD D D
11 1 12 2 1,10 10 2 2 3 3it i i i it it itINV D D D V K e
15.4.1 A Dummy Variable Model
Slide 15-25Principles of Econometrics, 3rd Edition
15.4.1 A Dummy Variable Model
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:
Slide 15-26Principles of Econometrics, 3rd Edition
(15.13)0 11 12 1
1 1
:
: the are not all equal
N
i
H
H
1 2 3it it it itINV V K e
So this is just OLS, the pooled model
15.4.1 A Dummy Variable Model
Slide 15-27Principles of Econometrics, 3rd Edition
reg inv v k
15.4.1 A Dummy Variable Model
We reject the null hypothesis that the intercept parameters for all
firms are equal. We conclude that there are differences in firm
intercepts, and that the data should not be pooled into a single model
with a common intercept parameter.
Slide 15-28Principles of Econometrics, 3rd Edition
1749128 522855 948.99
522855 200 12
R U
U
SSE SSE JF
SSE NT K
15.4.2 The Fixed Effects Estimator
Slide 15-29Principles of Econometrics, 3rd Edition
(15.14)1 2 2 3 3 1, ,it i it it ity x x e t T
(15.15)
1 2 2 3 31
1 T
it i it it itt
y x x eT
1 2 2 3 31 1 1 1
1 2 2 3 3
1 1 1 1T T T T
i it i it it itt t t t
i i i i
y y x x eT T T T
x x e
15.4.2 The Fixed Effects Estimator
Slide 15-30Principles of Econometrics, 3rd Edition
(15.16)
1 2 2 3 3
1 2 2 3 3
2 2 2 3 3 3
( )
( ) ( ) ( )
it i it it it
i i i i i
it i it i it i it i
y x x e
y x x e
y y x x x x e e
(15.17)2 3it it it ity x x e
15.4.2 The Fixed Effects Estimator
Slide 15-31Principles of Econometrics, 3rd Edition
15.4.2 The Fixed Effects Estimator
Slide 15-32Principles of Econometrics, 3rd Edition
(15.18) .1098 .3106
(se*) (.0116) (.0169)
itit itINV V K
2*ˆ 2e SSE NT
2 2 198 188 1.02625NT NT N
15.4.2 The Fixed Effects Estimator
Slide 15-33Principles of Econometrics, 3rd Edition
15.4.2 The Fixed Effects Estimator
Slide 15-34Principles of Econometrics, 3rd Edition
(15.19)
1 2 2 3 3i i i iy b b x b x
1 2 2 3 3 1, ,i i i ib y b x b x i N
15.4.2 The Fixed Effects Estimator
Slide 15-35Principles of Econometrics, 3rd Edition
ONE PROBLEM: Even with the trick of using the within estimator, we still implicitly (even if no longer explicitly) include N-1 dummy variables in our model (not N, since we remove the intercept), so we use up N-1 degrees of freedom.
It might not be then the most efficient way to estimate the common slope
ANOTHER ONE. By using deviations from the means, the procedure wipes out all the static variables, whose effects might be of interest
In order to overcome this problem, we can consider the random effects/or error components model
15.5 The Random Effects Model
Slide 15-36Principles of Econometrics, 3rd Edition
(15.20)
(15.22)
1 1i iu
(15.21) 20, cov , 0, vari i j i uE u u u u
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
Randomness of the intercept
Usual error
Average intercept
15.5 The Random Effects Model
Because the random effects regression error has two components, one
for the individual and one for the regression, the random effects
model is often called an error components model.
Slide 15-37Principles of Econometrics, 3rd Edition
(15.23)
(15.24)
1 2 2 3 3
1 2 2 3 3
it it it it i
it it it
y x x e u
x x v
it i itv u e
a composite error
15.5.1 Error Term Assumptions
Slide 15-38Principles of Econometrics, 3rd Edition
(15.25)
0 0 0it i it i itE v E u e E u E e
2
2 2
var var
var var 2cov ,
v it i it
i it i it
u e
v u e
u e u e
v has zero mean
v has constant varianceIf there is no correlation betweenthe individual effects and the error term
15.5.1 Error Term Assumptions
Slide 15-39Principles of Econometrics, 3rd Edition
But now there are several correlations that can be considered.
The correlation between two individuals, i and j, at the same
point in time, t. The covariance for this case is given by
cov , ( )
0 0 0 0 0
it jt it jt i it j jt
i j i jt it j it jt
v v E v v E u e u e
E u u E u e E e u E e e
15.5.1 Error Term Assumptions
Slide 15-40Principles of Econometrics, 3rd Edition
The correlation between errors on the same individual (i) at
different points in time, t and s. The covariance for this case is
given by
(15.26)
2
2 2
cov , ( )
0 0 0
it is it is i it i is
i i is it i it is
u u
v v E v v E u e u e
E u E u e E e u E e e
15.5.1 Error Term Assumptions
Slide 15-41Principles of Econometrics, 3rd Edition
The correlation between errors for different individuals in
different time periods. The covariance for this case is
cov , ( )
0 0 0 0 0
it js it js i it j js
i j i js it j it js
v v E v v E u e u e
E u u E u e E e u E e e
15.5.1 Error Term Assumptions
Slide 15-42Principles of Econometrics, 3rd Edition
(15.27)
2
2 2
cov( , )corr( , )
var( ) var( )it is u
it isu eit is
v vv v
v v
The errors are correlated over time for a given individual, but are otherwiseuncorrelatedThis correlation does not dampen over time as in the AR1 model
15.5.2 Testing for Random Effects
Slide 15-43Principles of Econometrics, 3rd Edition
(15.28)
1 2 2 3 3it it it ity x x e
1 2 2 3 3it it it ite y b b x b x
2
1 1
2
1 1
ˆ1
2 1 ˆ
N T
iti t
N T
iti t
eNT
LMT e
This is xttest0 in Stata if H0 is not rejected you can use OLS
15.5.3 Estimation of the Random Effects Model
Slide 15-44Principles of Econometrics, 3rd Edition
(15.29)
(15.30)
* * * * *1 1 2 2 3 3it it it it ity x x x v
* * * *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
(15.31)2 21 e
u eT
Is the transformation parameter
15.5.4 An Example Using the NLS Data
Slide 15-45Principles of Econometrics, 3rd Edition
2 2
ˆ .1951ˆ 1 1 .7437
5 .1083 .0381ˆ ˆe
u eT
Is the transformation parameter
Summary for now
Pooled OLS vs different intercepts: test (use a Chow type, after FE or run RE and test if the variance of the intercept component of the error is zero (xttest0))
You cannot pool onto OLS? Then…
FE vs RE: test (Hausman type)
Different slopes too perhaps? => use SURE or RCM and test for equality of slopes across units
Summary for now
Note that there is within variation versus between variation
The OLS is an unweighted average of the between estimator and the within estimator
The RE is a weighted average of the between estimator and the within estimator
The FE is also a weighted average of the between estimator and the within estimator with zero as the weight for the between part
Summary for now
The RE is a weighted average of the between estimator and the within estimator
The FE is also a weighted average of the between estimator and the within estimator with zero as the weight for the between part
So now you see where the extra efficiency of RE comes from!...
Summary for now
The RE uses information from both the cross-sectional variation in the panel and the time series variation, so it mixes LR and SR effects
The FE uses only information from the time series variation, so it estimates SR* effects
Summary for now
With a panel, we can learn about dynamic effects from a short panel, while we need a long time series on a single cross-sectional unit, to learn about dynamics from a time series data set
15.5.5a Endogeneity in the Random Effects Model
If the random error is correlated with any of the right-hand side
explanatory variables in a random effects model then the least squares and
GLS estimators of the parameters are biased and inconsistent.
This bias creeps in through the between variation, of course, so the FE model
will avoid it
Slide 15-51Principles of Econometrics, 3rd Edition
it i itv u e
15.5.5b The Fixed Effects Estimator in a Random Effects Model
Slide 15-52Principles of Econometrics, 3rd Edition
(15.32)
(15.33)1 2 2 3 3
1 1 1 1 1
1 2 2 3 3
1 1 1 1 1T T T T T
i it it it i itt t t t t
i i i i
y y x x u eT T T T T
x x u e
1 2 2 3 3 ( )it it it i ity x x u e
15.5.5b The Fixed Effects Estimator in a Random Effects Model
Slide 15-53Principles of Econometrics, 3rd Edition
(15.34)
1 2 2 3 3
1 2 2 3 3
2 2 2 3 3 3
( )
( ) ( ) ( )
it it it i it
i i i i i
it i it i it i it i
y x x u e
y x x u e
y y x x x x e e
15.5.5c A Hausman Test
We expect to find
because Hausman proved that
Slide 15-54Principles of Econometrics, 3rd Edition
(15.35) , , , ,
1 2 1 22 2
, ,, ,se sevar var
FE k RE k FE k RE k
FE k RE kFE k RE k
b b b bt
b bb b
, ,var var 0.FE k RE kb b
, , , , , ,
, ,
var var var 2cov ,
var var
FE k RE k FE k RE k FE k RE k
FE k RE k
b b b b b b
b b
, , ,cov , var .FE k RE k RE kb b b
15.5.5c A Hausman Test
The test statistic to the coefficient of SOUTH is:
Using the standard 5% large sample critical value of 1.96, we reject the hypothesis that the estimators yield identical results. Our conclusion is that the random effects estimator is inconsistent, and we should use the fixed effects estimator, or we should attempt to improve the model specification.
Slide 15-55Principles of Econometrics, 3rd Edition
, ,
1 2 1 22 2 2 2
, ,
.0163 (.0818) 2.3137
.0361 .0224se se
FE k RE k
FE k RE k
b bt
b b
15.5.5c A Hausman Test
The Hausman test assumes that the RE estimator used in the comparison is fully efficient, which requires that the unobserved effect and the idiosyncratic error are both i.i.d. (Cameron & Trivedi MMA page 719)
often not the case => the hausman command yields incorrect statistic
Example: If the error terms are cluster, (e.g. due to autocorrelation across time for an individual, then the RE estimator is not efficient)
Solutions: do a panel bootstrap of the Hausman test or use the Wooldridge (2002) robust version of Hausman test.
Slide 15-56Principles of Econometrics, 3rd Edition
Test for gamma =0 in:
To run in Stata, generate the RE differences and the mean differences
Principles of Econometrics, 3rd Edition
To run in Stata, generate the RE differences and the mean differences manually
See an example here: pages 267-268 of Cameron&Trivedi’s MUS book
Principles of Econometrics, 3rd Edition
15.5.5a Endogeneity in the Random Effects Model
If the random error is correlated with any of the right-
hand side explanatory variables in a random effects model then the
least squares and GLS estimators of the parameters are biased and
inconsistent.
Then we would have to use the FE model
But with FE we lose the static variables?
Solutions? HT, AM, BMS, instrumental variables models could help
Slide 15-59Principles of Econometrics, 3rd Edition
it i itv u e
We can generalise the random effects idea and allow for different
slopes too: Random Coefficients Model
Again, the now it is the slope parameters that differ, but as in RE
model, they are drawn from a common distribution
The RCM in a way is to the RE model what the SURE model is to the
FE model
Slide 15-60Principles of Econometrics, 3rd Edition
Further issues
Unit root tests and Cointegration in panels
Dynamics in panels
Slide 15-61Principles of Econometrics, 3rd Edition
Further issues
Of course it is not necessary that one of the dimensions of the panel
is time as such Example: i are students and t is for each quiz they take
Of course we could have a one-way effect model on the time
dimension instead
Or a two-way model
Or a three way model! But things get a bit more complicated there…
Slide 15-62Principles of Econometrics, 3rd Edition
Further issues
Another way to have more fun with panel data is to consider
dependent variables that are not continuous
Logit, probit, count data can be considered
STATA has commands for these
Based on maximum likelihood and other estimation techniques we
have not yet considered
Slide 15-63Principles of Econometrics, 3rd Edition
Further issues
Another extension is to consider mixed linear models (Cameron&Trivedi MUS
page 305)
Stata’s xtmixed fits linear mixed models. From Stata;s help:
Mixed models contain both fixed effects and random effects.
The fixed effects are analogous to standard regression coefficients and are
estimated directly
Slide 15-64Principles of Econometrics, 3rd Edition
Further issues
The random effects are not directly estimated but are summarized according to their estimated variances and covariances
Although random effects are not directly estimated, you can form best linear unbiased predictions (BLUPs) of them (and standard errors) by using predict after xtmixed
Random effects may take the form of either random intercepts or random coefficients, and the grouping structure of the data may consist of multiple levels of nested groups.
Mixed models are also known as multilevel models and hierarchical linear models
Quite rare in the econometric literature
Mixed Linear Models
Undergraduate Econometrics, 3rd Edition
Principles of Econometrics, 3rd Edition
Some particular specifications of the mixed linear models result in more standard models
OLS, RE are special cases of mixed linear models
Another one is known as the Random Coefficients Model
RCM also allows groupwise heteroskedasticity rather than imposing homoskedasticity like its mixed linear model equivalent
Random Coefficients Model
Undergraduate Econometrics, 3rd Edition
Principles of Econometrics, 3rd Edition
Example in Cameron & Trivedi MUS page 310
Random Coefficients Model
Undergraduate Econometrics, 3rd Edition
Principles of Econometrics, 3rd Edition
Data (available through Cameron & Trivedi’s MUS textbook ancillary files) :
mus08psidextract.dta (PSID wage data 1976-82 from Baltagi and
Khanti-Akom (1990))
I cut for you the first 994 observations mus08psidextract994
Random Coefficients Model
Undergraduate Econometrics, 3rd Edition
Principles of Econometrics, 3rd Edition
Random Coefficients Model
Principles of Econometrics, 3rd Edition
Test of parameter constancy: chi2(423) = 68970.73 Prob > chi2 = 0.0000
_cons 4.525957 .2825222 16.02 0.000 3.972224 5.07969
wks .0032176 .0050025 0.64 0.520 -.0065872 .0130223
exp .0973225 .0041396 23.51 0.000 .0892091 .1054359
lwage Coef. Std. Err. z P>|z| [95% Conf. Interval]
Prob > chi2 = 0.0000
Wald chi2(2) = 553.05
max = 7
avg = 7.0
Obs per group: min = 7
Group variable: id Number of groups = 142
Random-coefficients regression Number of obs = 994
. xtrc lwage exp wks, i(id)
DO we have a name for this test? Econometrics, 3rd Edition
You can understand the use of the FE model as a solution to omitted variable bias
If the unmeasured variables left in the error model are not correlated
with the ones in the model, we would not have a bias in OLS, so we
can safely use RE
If the unmeasured variables left in the error model are correlated with
the ones in the model, we would have a bias in OLS, so we cannot
use RE, we should not leave them out and we should use FE, which
bundles them together in each cross-sectional dummy
Slide 15-70Principles of Econometrics, 3rd Edition
Further issues
Another criterion to choose between FE and RE
If the panel includes all the relevant cross-sectional units, use FE, if only a random sample from a population, RE is more appropriate (as long as it is valid)
Slide 15-71Principles of Econometrics, 3rd Edition
Further issues
Wooldridge’s book on panel data
Baltagi’s book on panel data
Greene’s coverage is also good
Slide 15-72Principles of Econometrics, 3rd Edition
Readings
Keywords
Slide 15-73Principles of Econometrics, 3rd Edition
Balanced panel Breusch-Pagan test Cluster corrected standard errors Contemporaneous correlation Endogeneity Error components model Fixed effects estimator Fixed effects model Hausman test Heterogeneity Least squares dummy variable
model LM test Panel corrected standard errors Pooled panel data regression
Pooled regression Random effects estimator Random effects model Seemingly unrelated regressions Unbalanced panel
Chapter 15 Appendix
Slide 15-74Principles of Econometrics, 3rd Edition
Appendix 15A Estimation of Error Components
Appendix 15A Estimation of Error Components
Principles of Econometrics, 3rd Edition Slide 15-75
(15A.1)
(15A.2)
(15A.3)
1 2 2 3 3 ( )it it it i ity x x u e
2 2 2 3 3 3( ) ( ) ( )it i it i it i it iy y x x x x e e
2ˆ DVe
slopes
SSE
NT N K
Appendix 15A Estimation of Error Components
Principles of Econometrics, 3rd Edition Slide 15-76
(15A.4)
(15A.5)
1 2 2 3 3 1, ,i i i i iy x x u e i N
1
22 2
2 21
22
var var var var var
1var
T
i i i i i itt
Te
u it ut
eu
u e u e u e T
Te
T T
T
Appendix 15A Estimation of Error Components
Principles of Econometrics, 3rd Edition Slide 15-77
(15A.6)
(15A.7)
22 e BEu
BE
SSE
T N K
2 2
2 2 ˆˆ e e BE DV
u uBE slopes
SSE SSE
T T N K T NT N K