Panel Data - Memorial University of Newfoundland

ECON 4551 

Econometrics II 

Memorial University of Newfoundland 

Panel Data Models 

Adapted from Vera Tabakova’s notes

• 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 

Principles of Econometrics, 3rd Edition 

Slide 15-2

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 15-3

( , ) 

INV = f V K 

it it it 

(15.1) 

The data consist of T = 20 years of data (1935-1954) for 

N = 10 large firms. Value of stock, proxy for expected profits 

Let y it = INV it and x 2it = V it and x 3it = K it 

Capital stock, proxy for desired permanent 

Capital stock 

y =β +β x +β x + e 

it 1it 2it 2it 3it 3it it 

(15.2) 

Notice the subindices! 


Slide 15-4

INV =β +β V +β K + e t = 1, , 20 

GE, t 1 2 GE, t 3 GE, t GE, 

t 

INV =β +β V +β K + e t = 1, , 20 

WE, t 1 2 WE, t 3 WE, t WE, 

t 

(15.3a) 

yit =β 

1+β 2x2it +β 

3x3it + eit 

i= 1, 2; t = 1, , 20 

(15.3b) 

For simplicity we focus on only two firms 


GRETL: 

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

Slide 15-5

INV =β +β V +β K + e t = 1, , 20 

GE, t 1, GE 2, GE GE, t 3, GE GE, t GE, 

t 

INV =β +β V +β K + e t = 1, , 20 

WE, t 1, WE 2, WE WE, t 3, WE WE, t WE, 

t 

(15.4a) 

yit =β 

1i +β 

2ix2it +β 

3ix3it + eit 

i= 1, 2; t = 1, , 

20 

(15.4b) 


Slide 15-6

( ) ( ) 

2 

( ) 

E e = 0 var e =σ cov e , e = 0 

GE, t GE, t GE GE, t GE, 

s 

( ) ( ) 

2 

( ) 

E e = 0 var e =σ cov e , e = 0 

WE, t WE, t WE WE, t WE, 

s 

(15.5) 

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. 

2 2 

The two equations do have different error variances σ and σ . 

GE 

WE 


GRETL 

ols Inv const V K 

modtest –panel 

wrong in posted notes!!! 

Slide 15-7


Slide 15-8

• Let D i 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: 

INV =β +δ D +β V +δ D × V +β K +δ D × K + e 

it 1, GE 1 i 2, GE it 2 i it 3, GE it 3 i it it 

(15.6) 


Slide 15-9

So we have two separate stories 


Slide 15-10

( ) 

cov e , e = σ 

GE, t WE, t GE, 

WE 

(15.7) 

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 15-11

(i) 

(ii) 

Econometric software includes commands for SUR (or SURE) that 

carry out the following steps: 

Estimate the equations separately using least squares; 

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

σ , σ and σ 

2 2 

GE WE GE, 

WE 

; 

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

within a generalized least squares framework. 


Slide 15-12


Slide 15-13

* Open and summarize data from grunfeld2.gdt (which, luckily for us, is already in 

wide format!!!) 

open "c:\Program Files\gretl\data\poe\grunfeld2.gdt" 

system name="Grunfeld" 

equation inv_ge const v_ge k_ge 

equation inv_we const v_we k_we 

end system 

estimate "Grunfeld" method=sur --geomean 


Slide 15-14

In GRETL the restrict command can be used to impose the cross-equation 

restrictions on a system of equations that has been previously defined and 

named. 

The set of restrictions is started by restrict and terminated with end restrict. 

Each restriction in the set is expressed as an equation. Put the linear combination 

of parameters to be tested on the left-hand-side of the equality and a numeric 

value on the right. 

Parameters are referenced using b[i,j] where i refers to the equation number in 

the system, and j the parameter number. 


Slide 15-15

estrict "Grunfeld" 

b[1,1]-b[2,1]=0 

b[1,2]-b[2,2]=0 

b[1,3]-b[2,3]=0 

end restrict 


Slide 15-16

There are two situations where separate least squares estimation is 

just as good as the SUR technique : 

(i) 

(ii) 

when the equation errors are not contemporaneously correlated; 

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

each equation. 

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

to see if the correlation between the errors is significantly different 

from zero is of interest. 


Slide 15-17

• (although text reads 0.729): 

r 

2 

( ) 

( )( ) 

2 

2 

σˆ GE, 

WE 

207.5871 

GE, WE 2 2 

σGEσWE 

= = = 

ˆ ˆ 777.4463 104.3079 

0.53139 

σ 

20 20 

1 1 

ˆ = ∑eˆ eˆ = ∑eˆ eˆ 

− 

GEWE , GEt , WEt , GEt , WEt , 

T −K t 1 T 3 

GE 

T −K WE = t= 

1 

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

K 

GE 

= K = 3. 

WE 


Slide 15-18

Testing for correlated errors for two equations: 

H : σ = 0 

0 GE, 

WE 

LM = Tr ∼χ under H . 

2 2 

GE, WE (1) 0 

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 15-19

Testing for correlated errors for three equations: 

H0 : σ 

12 

=σ 

13 

=σ 

23 

= 0 

( ) 

LM = T r + r + r χ 

2 2 2 2 

12 13 23 (3) 


Slide 15-20

Testing for correlated errors for M equations: 

M 

i−1 

LM T r 

= ∑∑ 

i= 2 j= 

1 

2 

ij 

Under the null hypothesis that there are no contemporaneous 

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

degrees of freedom, in large samples. 


Slide 15-21

H : β =β , β =β , β =β 

0 1, GE 1, WE 2, GE 2, WE 3, GE 3, WE 

(15.8) 

Most econometric software will perform an F-test and/or a Wald χ 2 –test; in 

the context of SUR equations both tests are large sample approximate tests. 

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

denominator degrees of freedom, where J is the number of hypotheses, M is 

the number of equations, and K is the total number of coefficients in the 

whole system, and T is the number of time series observations per equation. 

The χ 2 -statistic has J degrees of freedom. 


Slide 15-22

• SUR is OK when the panel is long and narrow, not when it is short and wide. 

Consider instead… 

y =β +β x +β x + e 

it 1it 2it 2it 3it 3it it 

(15.9) 

We cannot consistently estimate the 3×N×T parameters in (15.9) with 

only NT total observations. But we can impose some more 

structure… 

β =β , β =β , β =β 

1it 1i 2it 2 3it 

3 

(15.10) 

We consider only one-way effects and assume a common slope 

parameters across cross-sectional units 


Slide 15-23

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. 

y =β +β x +β x + e 

it 1i 2 2it 3 3it it 

(15.11) 


Slide 15-24

⎧1 i= 1 ⎧1 i= 2 ⎧1 i= 

3 

D1 i 

= ⎨ , D2i = ⎨ , D3i 

= ⎨ , etc. 

⎩0 otherwise ⎩0 otherwise ⎩0 otherwise 

INV =β D +β D + +β 

D +β V +β K + e 

it 11 1i 12 2i 1,10 10i 2 2it 3 3it it 

(15.12) 

This specification is sometimes called the least squares dummy 

variable model, or the fixed effects model. 


Slide 15-25


Slide 15-26

H 

H 

: β =β = =β 

0 11 12 1N 

: the β are not all equal 

1 1i 

(15.13) 

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 β 1 , then the restricted model is: 

INV =β +β V +β K + e 

it 1 2 it 3 it it 

So this is just OLS, the pooled model 


Slide 15-27

eg inv v k 


Slide 15-28

F 

= 

( R 

− 

U) 

SSE ( NT − K ) 

SSE SSE J 

U 

( − ) 

522855 ( 200 −12) 

1749128 522855 9 

= = 

48.99 

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 15-29

yit =β 

1i +β 

2x2it +β 

3x3it + eit 

t = 1, , 

T 

(15.14) 

1 T it 1i 2 2it 3 3it it 

t 1 

( y x x e ) 

∑ =β +β +β + 

T = 

T T T T 

1 1 1 1 

y = y =β +β x +β x + e 

∑ ∑ ∑ ∑ 

i it 1i 2 2it 3 3it it 

T t= 1 T t= 1 T t= 1 T t= 

1 

(15.15) 

=β +β x +β x + e 

1i 2 2i 3 3i i 


Slide 15-30

y =β +β x +β x + e 


− ( y =β +β x +β x + e) 

i 1i 2 2i 3 3i i 

(15.16) 

y − y =β ( x − x ) +β ( x − x ) + ( e −e) 

it i 2 2it 2i 3 3it 3i it i 

y =β x +β x + e 

it 2 it 3 it it 

(15.17) 


Slide 15-31


Slide 15-32

Usually, there is no interest in the intercepts…. 

INV .1098 .3106 

it = V it + K 

(se*) (.0116) (.0169) 

it 

(15.18) 

2 

* 

( ) 

σ ˆ 

e 

= SSE NT −2 

( NT ) ( NT N ) 

−2 − − 2 = 198 188 = 1.02625 


Slide 15-33

Some software comes up with one sometimes though… 

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


Slide 15-34

y = b + bx + bx 

i 1i 2 2i 3 3i 

b1 i 

= yi −bx 2 2i − bx 

3 3i 

i= 1, , 

N 

(15.19) 


Slide 15-35

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 


Slide 15-36

• In the RE model, the individual firm differences are thought 

to represent a random variation about some average 

intercept for the individual in the sample 

• Rather than a separate fixed effect for each firm, we now 

estimate an overall intercept that represents this average 

• Implicitly, the regression function for the sample firms vary 

randomly around this average. 

• The variability of the individual effects is captured by a new 

parameter, which is the variance of the random effect. 

• The larger this parameter is, the more variation you find in 

the implicit regression functions for the firms. 

Principles of Econometrics, 3rd Edition

β =β+u 

1i 

1 

i 

Average intercept 

(15.20) 

( ) ( ) 

2 

= 0, cov , = 0, var ( ) = σ 

E u u u u 

i i j i u 

(15.21) 

y =β +β x +β x + e 


Randomness of the intercept 

( ) 

= β + u +β x +β x + e 

1 i 2 2it 3 3it it 

Usual error 

(15.22) 


Slide 15-38

( ) 

y =β +β x +β x + e + u 

it 1 2 2it 3 3it it i 

=β +β x +β x + v 

1 2 2it 3 3it it 

a composite error 

(15.23) 

vit = ui + eit 

(15.24) 

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 15-39

( ) = ( + ) = ( ) + ( ) = 0+ 0= 

0 

E v E u e E u E e 

it i it i it 

( v ) var ( u e ) 

σ = var = + 

2 

v it i it 

v has zero mean 

( u ) ( e ) ( u e ) 

= var + var + 2cov , 

=σ +σ 

2 2 

u e 

i it i it 

v has constant variance 

If there is no correlation between 

the individual effects and the 

error term 

(15.25) 


Slide 15-40

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 

( v v ) = Evv = E⎡( u+ e)( u + e ) 

cov , ( ) 

it jt it jt i it j jt 

⎣ 

( ) ( ) ( ) ( ) 

= E uu + E ue + E e u + E e e 

i j i jt it j it jt 

= 0+ 0+ 0+ 0= 

0 

⎤ 

⎦ 


Slide 15-41

• 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 

( v v ) E v v E ( u e )( u e ) 

cov 

it 

, 

is 

= ( 

it is) 

= ⎡ ⎣ i 

+ 

it i 

+ 

is 

⎤⎦ 

( 

2 

) i ( i is ) ( it i ) ( it is ) 

= E u + E ue + E e u + E e e 

(15.26) 

=σ + 0+ 0+ 0=σ 

2 2 

u 

u 


Slide 15-42

• The correlation between errors for different individuals in 

different time periods. The covariance for this case is 

( v v ) = Evv = E⎡( u+ e)( u + e ) 

cov , ( ) 

it js it js i it j js 

⎣ 

( ) ( ) ( ) ( ) 

= E uu + E ue + E e u + E e e 

i j i js it j it js 

= 0+ 0+ 0+ 0= 

0 

⎤ 

⎦ 


Slide 15-43

cov( v , v ) σ 

ρ= corr( vit 

, vis 

) = = 

var( v ) var( ) σ +σ 

it 

2 

it is u 

2 2 

vis 

u e 

(15.27) 

The errors are correlated over time for a given individual, but are otherwise 

uncorrelated 

This correlation does not dampen over time as in the AR1 model 


Slide 15-44

y =β +β x +β x + e 

it 1 2 2it 3 3it it 

LM 


e = y −b−bx −bx 

ˆit it 1 2 2 it 3 3 it 

2 

⎧ N T 

⎛ 

ˆ 

⎞ ⎫ 

∑∑eit 

NT ⎪ 

⎜ ⎟ 

i= 1 t= 

1 ⎪ 

= 

⎝ ⎠ 

⎨ −1 

N T ⎬ 

2( T −1) 

⎪ 

2 

∑∑eˆ 

it ⎪ 

⎪ i= 1 t= 

1 

⎩ 

⎪ 

⎭ 

(15.28) 

GRETL shows this Breusch and Pagan 

Lagrange multiplier test for random effects by 

default 

Slide 15-45

• GRETL shows by default this Breusch and Pagan 

Lagrangian multiplier test for RE with the null of no 

variation about a mean (effects are fixed) in the 

individual effects. 

• This is xttest0 in Stata… 

• If H0 is not rejected you can use pooled OLS if the 

effects are common and the FE if they differ by 

group 


GRETL shows by default this Breusch and Pagan Lagrangian multiplier test for RE 

with the null of no variation about a mean (effects are fixed) in the individual 

effects. 


• GRETL also shows the Hausman test of the null 

hypothesis that the random effects are indeed 

random. 

• If they are random, then they should not be correlated 

with any of your other regressors. 

• If they are correlated with other regressors, then you 

should use the FE estimator to obtain consistent 

parameter estimates of your slopes 


y =β x +β x +β x + v 

* * * * * 

it 1 1it 2 2it 3 3it it 

(15.29) 

y = y −α y , x = 1 −α , x = x −α x , x = x −αx 

* * * * 

it it i 1it 2it 2it 2i 3it 3it 3i 

(15.30) 

α= − 

σ 

e 

1 

2 2 

Tσ u 

+σe 

(15.31) 

Is the transformation parameter 


Slide 15-49

σˆ e 

.1951 

α= ˆ 1− = 1 − = .7437 

Tσ ˆ +σˆ 

5 .1083 + .0381 

2 2 

u e 

Is the transformation parameter 

( ) 


Slide 15-50

• There are different ways to calculate FE (some 

packages will calculate an intercept, some 

won’t) 

• There are different ways to calculate sigma-sq 

(STATA in textbook and GRETL will give you 

slightly different results!) 


• Pooled OLS vs different intercepts: test (use a Chow 

type, after FE or run RE and test if the variance of 

the intercept component of the error is zero (Breusch- 

Pagan test (xttest0 in STATA)) 

• You cannot pool onto OLS? Then… 

• Choose between FE vs RE: (Hausman test) 

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

• Different slopes too perhaps? => use SURE or RCM 

and test for equality of slopes across units

• Note that there is within variation versus 

between variation 

• The OLS is an unweighted average of the 

between estimator and the within estimator 

• The RE is a weighted average of the between 

estimator and the within estimator 

• The FE is also a weighted average of the 


with zero as the weight for the between part

• The RE is a weighted average of the between 

estimator and the within estimator 

• The FE is also a weighted average of the 


with zero as the weight for the between part 

• So now you see where the extra efficiency of 

RE comes from!...

• The RE uses information from both the crosssectional 

variation in the panel and the time 

series variation, so it mixes LR and SR effects 

• The FE uses only information from the time 

series variation, so it estimates SR* effects

• With a panel, we can learn about dynamic 

effects from a short panel, while we need a 

long time series on a single cross-sectional 

unit, to learn about dynamics from a time 

series data set

If the random error 

v = u + e 

it i it 

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 15-57

yit =β 

1+β 2x2it +β 

3x3 it 

+ ( ui + eit 

) 

(15.32) 

T T T T T 

1 1 1 1 1 

y = y =β +β x +β x + u + e 

∑ ∑ ∑ ∑ ∑ 

i it 1 2 2it 3 3it i it 

T t= 1 T t= 1 T t= 1 T t= 1 T t= 

1 

(15.33) 

=β +β x +β x + u + e 

1 2 2i 3 3i i i 


Slide 15-58

y =β +β x +β x + u + e 

it 1 2 2it 3 3it i it 

− ( y =β +β x +β x + u + e) 

i 1 2 2i 3 3i i i 

(15.34) 

y − y =β ( x − x ) +β ( x − x ) + ( e −e) 



Slide 15-59

t 

bFE, k 

−bRE, k 

bFE, k 

−bRE, 

k 

= = 

 

2 2 

⎡var 

( b 

, ) var ( , ) se( FE, k ) se 

FE k 

− b ⎤ ⎡ b − 

RE k 

( bRE, 

k ) 

⎤ 

⎢⎣ 

⎥⎦ 

⎢⎣ ⎥⎦ 

12 12 

(15.35) 

We expect to find 

 

var − var b > 0. 

( b ) 

( ) 

FE, k 

RE, 

k 

( b − b ) = ( b ) + ( b ) − ( b b ) 

var var var 2cov , 

FE, k RE, k FE, k RE, k FE, k RE, 

k 

( b ) var ( b ) 

= var − 

FE, k 

RE, 

k 

because Hausman proved that 

( b b ) = ( b ) 

cov , var . 

FE, k RE, k RE, 

k 


Slide 15-60

The test statistic to the coefficient of SOUTH is: 

t 

b 

− b 

−.0163 −(.0818) 

FE, k RE, 

k 

= = = 

2 2 

12 

2 2 

12 

⎡se 

⎤ ⎡ ⎤ 

⎢ FE, k 

RE, 

k 

⎣ 

⎥⎦ 

⎣ 

⎦ 

( b ) − se( b ) (.0361) − (.0224) 

2.3137 

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 15-61

If the random error 

v = u + e 

it i it 

is correlated with any of the righthand 

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 15-62

Further issues 

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 15-63


Unit root tests and Cointegration in panels 

Dynamics in panels 


Slide 15-64


• 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 15-65


• 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 15-66


• 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 15-67


• Another criterion to choose between FE and RE 

• If the panel include all the relevant cross-sectional units, use FE, if 

only a random sample from a population, RE is more appropriate (as 

long as it is valid) 

• 


Slide 15-68

Readings 

Wooldridge’s book on panel data 

Baltagi’s book on panel data 

Greene’s coverage is also good 


Slide 15-69

• 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 

Slide 15-70


Slide 15-71

yit =β 

1+β 2x2it +β 

3x3 it 

+ ( ui + eit 

) 

(15A.1) 

y − y =β ( x − x ) +β ( x − x ) + ( e −e) 


(15A.2) 

σ ˆ = 

2 

e 

SSEDV 

NT −N −K 

slopes 

(15A.3) 

Principles of Econometrics, 3rd Edition Slide 15-72

yi =β 

1+β 2x2i +β 

3x3i + ui + ei 

i= 1, , 

N 

(15A.4) 

⎛ ⎞ 

var ( u + e ) = var ( u ) + var ( e ) = var ( u ) + var ⎜ e T⎟ 

⎠ 

T 

2 

2 1 ⎛ ⎞ 2 Tσ 

2 var 

e 

=σ 

u 

+ ⎜∑eit 

⎟=σ u 

+ 

2 

T t= 

1 T 

i i 

T 

i i i ∑ 

⎝ t= 

1 

it 

2 σ 

=σ 

u 

+ 

T 

2 

e 

⎝ 

⎠ 

(15A.5) 


2 

2 σe SSEBE 

u 

T N − KBE 

σ + = 

(15A.6) 

2 2 

2 2 σ ˆ 

e 

σe SSEBE SSEDV 

σ ˆ 

u 

=σ 

u 

+ − = − 

T T N − K T NT −N −K 

BE 

( ) 

slopes 

(15A.7)

Panel Data - Memorial University of Newfoundland

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