geneve chapitre1

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Specication tests and analysis of covarianceLinear unobserved eects panel data models

Random coecients models

Chapter 1. Linear Panel Models and HeterogeneityMaster of Science in Economics - University of Geneva

Christophe Hurlin, Universit dOrlans

Universit dOrlans

January 2010

C. Hurlin Panel Data Econometrics
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Introduction

The outline of the chapter:

1 Specication tests and analysis of covariance

2 Linear unobserved eects panel data models3 Fixed or random methods?

4 Fixed-Eects methods: Least Squares dummy variableapproach

5 Random eects methods6 Heterogeneous panels: random coecients models

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Specication testsApplication: strikes in OECD

Specication tests and analysis of covariance

Section 1.Specication tests and analysis ofcovariance

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A linear model commonly used to assess the eects of bothquantitative and qualitative factors is postulated as

yi,t=i,t+0i,txi,t+i,t

where8 i=1, .., N,8 t=1, .., Tit and = (1it,2it, ...,Kit) are 1 1 and 1Kvectors ofparameters that vary across i and t

,

xit= (x1it, ..., xKit) is a 1Kvector of exogenous variables,uit is the error term.

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Three aspects of the estimated regression coecients can betested:

1 the homogeneity of regression slope coecients

2 the homogeneity ofregression intercept coecients.

3 the time stability of parameters (slopes and constants). Wewill not consider this issue (not specic to panel data models)here.

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We assume that parameters are constant over time, but can varyacross individuals.

yi,t=i+0ixi,t+i,t

Three types of restrictions can be imposed on this model.Regression slope coecients are identical, and intercepts are not(model with individual / unobserved eects).

yi,t=i+0xi,t+i,t

Regression intercepts are the same, and slope coecients are not(unusual).

yi,t=+0ixi,t+i,t

Both slope and intercept coecients are the same (homogeneous/ pooled panel).

yi,t=+0xi,t+i,tC. Hurlin Panel Data Econometrics
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Denition

Anheterogeneous panel data model is a model in which all

parameters (constant and slope coecients) vary accrossindividuals.

Denition

Anhomogeneous panel data model (or pooled model) is amodel in which all parameters (constant and slope coecients) arecommon

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Specication Tests

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Figure: Hsiao (2003)

[ ]NiHTest ii ,1:1

0 ==

vraieH

1

0rejeteH

1

0

tititi xy ,,, ' ++=[ ]NiHTest i ,1:2

0 =

vraieH2

0rejeteH2

0

titiiiti xy ,,, ' ++= [ ]NiHTest i ,1:

3

0 =

vraieH3

0 rejeteH3

0

titiiti xy ,,, ' ++=tititi xy ,,, ' ++=

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LemmaUnder the assumption that theitare independently normallydistributed over i and t with mean zero and variance2 , F testscan be used to test the restrictions postulated

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First step (homogeneous/ pooled assumption)

Let us consider the general model

yi,t=i+0xi,t+i,t

The hypothesis of common intercept and slope can be viewed as aset of(K+1)(N 1) linear restrictions:

H10 : i= i =8 i2 [1, N]H1a :9 (i,j)2[1, N] / i6= j ou i6=j

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Under the alternative H1 , there are NKestimated slope

coecients for the N vectors i (K 1) and estimated Nconstants.

Under H1, the unrestricted residual sum of squares S1 dividedby2 has a chi-square distribution with NTN(K+1)degrees of freedom.



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Denition

Under the homogeneous assumption H10,

H10 : i(K

,

1)

=

(K,

1)

i=

8i

2[1, N]

the F statistic, denoted F1 ,and dened by:

F1 =(RSS1,cRSS1)/[(N 1) (K+1)]

RSS1/[NT

N(K+1)]

has a Fischer distribution with (N 1) (K+1) andNTN(K+1) degrees of freedom. RSS1 denotes the residualsum of squares of the model and RSS1,cthe residual sum ofsquares of the constrained model:


S f
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UnderH1 , the residual sum of squares is equal to the sum of the Nresidual sum of squares associated to the Nindividual regressions:

RSS1 =N

i=1

RSS1,i=N

i=1 Syy,i S

0xy,iS

1xx,iSxy,i

with

Syy,i =T

t=1

(yi,t yi)2 with xi = 1

T

T

t=1

xi,t and yi= 1

T

T

t=1

yi,t

Sxx,i =Tt=1

(xi,t xi) (xi,t xi)0

Sxy,i =T

t=1

(xi,t xi) (yi,t yi)0


S i i d l i f i


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Under H10, the model is :

yi,t=+0xi,t+i,t

the least-squares regression of the pooled model yields parameterestimates b=S1xx Sxy

Sxx =N

i=1

T

t=1

(xi,t

x) (xi,t

x)0 with x= 1

NT

N

i=1

T

t=1

xi,t

Sxy =N

i=1

T

t=1

(xi,t x) (yi,t y)0 with y= 1NT

N

i=1

T

t=1

yi,t


S i ti t t d l i f i
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Under H10, the overall RSS is dened by

SCR1,c=Syy S0xyS1xx Sxy

withSyy =

N

i=1

T

t=1

(yi,t yi)2

Sxx,i =N

i=1

T

t=1

(xi,t

xi) (xi,t

xi)0

Sxy,i=N

i=1

T

t=1



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Second step (individual/unobserved eects)

Let us consider the general model

yi,t=i+0ixi,t+i,t

The hypothesis of heterogeneous intercepts but homogeneousslopes can be reformulated as subject to (N 1)K linearrestrictions (non restrictions on i).

H20 : i = 8 i=1, ..N


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Denition

Under the assumption H20,

H20 : i = 8 i=1, ..N

the F statistic, denoted F2 , and dened by:

F2 =(RSS1,c0 RSS1)/[(N 1) K]

RSS1/[NTN(K+1)]has a Fischer distribution with (N 1) K et NTN(K+1)degrees of freedom under H20 . RSS1 denotes the residual sum ofsquares of the model and RSS1,c0 the residual sum of squares ofthe constrained model (model with individual eects):

yi,t=i+0xi,t+i,tC. Hurlin Panel Data Econometrics

Specication tests and analysis of covarianceS i i


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Under H20, the residual sum of squares is:

RSS1,c0 =N

i=1

Syy,i

N

i=1

Sxy,i

!0 N

i=1

Sxx,i

!1 N

i=1

Sxy,i

!

Syy,i =T

t=1

(yi,t yi)2 with xi= 1

T

T

t=1

xi,t yi= 1

T

T

t=1

yi,t

Sxx,i =

T

t=1 (xi,t xi) (xi,t xi)0

Sxy,i =T

t=1



Specication tests and analysis of covarianceS i ti t t
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Last step

IfH20 is not rejected, one can also apply a conditional test forhomogeneous intercepts (N

1 linear restrictions).

H30 : i= 8 i=1, .., N given i =

Under the null, the model is homogeneous (pooled) and the

restricted residual sum of squares is SCR1,

c.

.Under the alternative, the model is yi,t=i+

0xi,t+i,t,and there is NTKNdegrees of freedom


Specication tests and analysis of covarianceSpecication tests
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p yLinear unobserved eects panel data models




Denition

Under the assumption H30,

H30 : i=

8i=1, .., N given i =

the F statistic, denoted F3 , and dened by:

F3 = (RSS1,cRSS1,c0)/(N 1)

RSS1,c0/[N(T 1)K] (1)

has a Fischer distribution with N 1 and N(T 1)K degreesof freedom under H20 . RSS1,c0 denotes the residual sum of squaresof the model with individual eects and SCR1,cthe residual sum ofsquares of the pooled model previously dened.


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Linear unobserved eects panel data modelsRandom coecients models



Application: Strikes in OECD


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Example

Let us consider a simple panel regression model for the totalnumber of strikes days in OECD countries. We have a balancedpanel data set for 17 countries (N=17) and annual data form

1951 to 1985 (T =35). General idea: evaluate the link betweenstrikes and some macroeconomic factors (inatiion, unemploymentetc..)

s,t=i+iu,t+ipit+it8 i=1, .., 17

si,tthe number of strike days for 1000 workers for the countryiat time t.

uittthe unemployement rate

pi,t, the ination rate


Specication tests and analysis of covarianceLi b d l d d l

Specication tests
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Figure: Spcication tests with TSP 4.3A


Specication tests and analysis of covarianceLi b d t l d t d l

Specication tests
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pApplication: strikes in OECD



Specication tests
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pApplication: strikes in OECD



Specication tests
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Application: strikes in OECD


Conclusion of section 1.Fact

It is possible to test the heterogeneity / homogeneity of theparameters under some specic assumptions (normality of

residuals). More generaly, the assumption of heterogenity /homogeneity of the parameters (slope coecients and constants)has to be evaluated with an economic interpretation.

ExampleExample: It is reasonnable to assume that the slope parameters ofthe production function are the same accros countries? what doesit imply? Should I impose a common mean for the TFP for Franceand Germany?



DenitionsFixed and random eectsFixed eects methods
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Fixed eects methodsRandom eects methodsFixed or random methods?

Linear unobserved eects panel data models

Section 2.Linear unobserved/individual eectspanel data models



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2.1. Denitions



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pRandom coecients models Random eects methods

Fixed or random methods?


We now consider a model with individual eects and commonslope parameters.

Denition

A linear unobserved / individual eects panel data model is dened

as follows:yit=i+

0xit+it

where8 i=1, .., N,8 t=1, .., Ti and = (1 ,2 , ...,K)are 1 1 and 1Kvectors ofparameters,xit= (x1it, ..., xKit) is a 1Kvector of exogenous variables,it is the error term, assumed to be i.i.d., with8 i=1, .., N,8 t=1, .., T

E(it)=0 E2it=

2



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pRandom coecients models Random eects methods



Denition

There are many names for i: (1) unobserved eects, (2)

individual eects, (3) unobserved component, (4) latent variable(for random eects models), (5) individual heterogeneity.

Denition

The itare called the idiosyncratic errors oridiosyncraticdisturbances because these change accross tas well as accross i.



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Random coecients models Random eects methodsFixed or random methods?


Writting in vector form. Let us denote

yi(T,1) = 0BB@yi,1yi,2...

yi,T1CCA Xi(T,K) = 0BB@

x1,i,1 x2,i,1 ... xK,i,1x1,i,2 x2,i,2 ... xK,i,2... ... ... ...

x1,i,T x2,i,T ... xK,i,T1CCA

Let us denote ea unit vector and i the vector of errors:

e(T,1)

=

0BB@ 11...1

1CCA i(T,1)

=

0BB@ i,1

i,2...

i,T

1CCAC. Hurlin Panel Data Econometrics


R d i d l

DenitionsFixed and random eectsFixed eects methodsR d h d
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Denition

For a given individual i, the linear unobserved / individual

eects panel data model is dened as follows:

yi =ei+Xi+i 8 i=1, .., N

E(i)=0

Ei0i=2 ITE

i0

j

= 0

(T,T)if i6=j



R d i t d l

DenitionsFixed and random eectsFixed eects methodsR d t th d
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Example

Let us consider the case of a Cobb Douglas production function inlog, as dened previously, for the case T =3. We have:

yi,t=i+kki,t+nni,t+i,t 8 i, 8 t2[1, 3]

or in a vectorial form for a country i :

0@ yi,1yi,2yi,3

1A= 0@ 111

1A i+0@ ki,1 ni,1ki,2 ni,2ki,3 ni,3

1A kn

+0@ i,1i,2i,3

1A




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It is also possible to stackle all this vectors/matrix as follows

Y =eee+X+Y

(TN,1)= 0BB@

y1

y2...

yN

1CCA X(TN,K) = 0BB@X1

X2...

XN

1CCA (TN,1) = 0BB@1

2...

N

1CCAwhere 0T is the null vector (T, 1).

ee(TN,N)

=

0BB@

e 0T ... 0T0T e ... 0T... ... ... 0T0T 0T ... e

1CCA

e(N,1)

=

0BB@

12...

N






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Example

Consider the case of the production function with T=3 andN=2

0BBBBBBB@

y1,1y1,2y1,3

y2,1

y2,

2y2,3

1CCCCCCCA

=

0BBBBBB@

1 01 01 00 1

0 10 1

1CCCCCCA

12

+

0BBBBBBB@

k11 n11k12 n12k13 n13

k21 n21

k22 n22k

,3 n23

1CCCCCCCA

kn

+

0BBBBBBB@

111

2,

22

()Y =

ee

e+X+




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2.2. Fixed and random eects






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Especially in methodological papers, but also in applications, oneoften sees a discussion about whether iwill be treated as a

random eect or a xed eect.Denition

In the traditional approach to panel data models, i is called arandom eect when it is treated as a random variable and axed eect when it is treated as a parameter to be estimatedfor each cross section observation i.




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a do coe c e ts ode s a do e ects et odsFixed or random methods?


Fact

For Wooldridge (2003), these discussions about whether thei

should be treated as random variables or as parameters to beestimated are wrongheaded for microeconometric panel dataapplications. With a large number of random draws from the crosssection, it almost always makes sense to treat the unobservedeects, i, as random draws from the population, along with

yitand xit. This approach is certainly appropriate from an omittedvariables or neglected heterogeneity perspective.




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Fact

As suggested by Mundlak (1978), the key issue involvingi iswhether or not it is uncorrelated with the observed explanatoryvariables xit, for t=1, .., T .

Mundlak Y. (1978), On the Pooling of Time Series and Cross

Section Data, Econometrica, 46, 69-85






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Denition

In modern econometric parlance, random eect is synonymouswith zero correlation between the observed explanatory variablesand the unobserved eect:

cov(xit, i)=0, t=1, .., T






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Actually, a stronger conditional mean independence assumption,

E( ij xi1 , .., xiT)=0will be needed to fully justify statistical inference. In appliedpapers, when i is referred to as, say, an individual randomeect, then iis probably being assumed to be uncorrelated with

the xit.




DenitionsFixed and random eectsFixed eects methodsRandom eects methodsFi d d h d ?


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Denition

In microeconometric applications, the term xed eect does notusually mean that iis being treated as nonrandom; rather, itmeans that one is allowing for arbitrary correlation between theunobserved eect iand the observed explanatory variables xit.




DenitionsFixed and random eectsFixed eects methodsRandom eects methodsFi d d th d ?


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Wooldridge (2003) avoids referring to ias a random eect or a

xed eect. Instead, we will refer to ias unobserved eect,unobserved heterogeneity, and so on. Nevertheless, later we willlabel two dierent estimation methods random eectsestimation and xed eects estimation. This terminology is soingrained that it is pointless to try to change it now.




DenitionsFixed and random eectsFixed eects methodsRandom eects methodsFixed or random methods?
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Example (Production function)

Let us consider the simple example of the Cobb Douglassproduction function.

yi,t=iki,t+ini,t+i+vi,t

In this case, icorresponds to the unobserved eect on TFP due toscountry specic omitted factor (climate, institutions, organiszationetc..). In this case, we might expect that the the level of factors

are positivily correlated with this component of TFP: the more acountry is productive, the more it invests in capital for instance.

cov(i, ki,t) > 0 cov(i, ni,t) > 0






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Example (Patents and R&D)

Hausman, Hall, and Griliches (1984) estimate (nonlinear)distributed lag models to study the relationship between patentsawarded to a rm and current and past levels of R&D spending. A

linear version of their model is:

patentsit=t+ zit + 0RDit+ 1RDi,t1+ .. + 5RDi,t5+ i+ vi,t

where RDit is spending on R&D for rm iat time t and zit

contains other explanatory variables. i is a rm heterogeneityterm that may inuence patentsitand that may be correlated withcurrent, past, and future R&D expenditures.

cov(i, RDi,tk)6=0 8kC. Hurlin Panel Data Econometrics





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Denition

The Hausmans test (1978), is a test of the null hypothesis

cov(xit, i)=0, t=1, .., T

and is generally presented as test of specication (xed orrandom) of the unobserved eects (see later, section 5.).




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2.3. Fixed eects methods




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Let us consider the following model:yi =ei+Xi+i 8 i=1, .., N

where i is a constant term,0 =(12 ....K)2 RK and:

yi(T,1)

= i,1 i,2 ... i,T 0i

(T,1)=

i,1 i,2 ... i,T0

e

(T,

1)

= 1 1 ... 1 0

Xi(T,K)

=

0BB@

x1,i,1 x2,i,1 ... xK,i,1x1,i,2 x2,i,2 ... xK,i,2... ... ... ...

x1,i,T x2,i,T ... xK,i,T




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Assumtions (H1) Let us assume that errors terms i,t arei.i.d.8 i2 [1, N],8 t2[1, T] with:

E(i,t)=0

E(i,ti,s)= 20 t=s8t6=s , or E(i0i)=2 ITwhere Itdenotes the identity matrix (T, T).

E(i,tj,s)=0, 8j6=i,8 (t, s), orEi0

j=0where 0 denotes the null matrix (T, T).




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Theorem

Under assumption H1 , the ordinary-least-squares (OLS) estimatorof is the best linear unbiased estimator (BLUE).

Denition

In this context, the OLS estimator

b is called the least-squares

dummy-variable (LSDV) ofFixed Eect (FE) estimator,

because the observed values of the variable for the coecient itakes the form of dummy variables.






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OLS estimators ofi and and are obtained by minimizing

nbi,bLSDVo arg minfi,gNi=1

S =N

i=1

0ii

=N

i=1

(yi eiXi)0(yi eiXi)




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FOC1 (with respect to i):

bi =yib0LSDVxiwith

xi= 1

T

T

t=1

xi,t yi= 1

T

T

t=1

yi,t




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Given the second FOC (with respect to ) and the previous result,we have:

Denition

Under assumptionH1,

the xed eect estimator or LSDV estimatorof parameter is dened by:

bLSDV =

" N

i=1

T

t=1

(xi,t xi) (xi,t xi)0#1

" Ni=1

T

t=1

(xi,t xi) (yi,t yi)#




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1 The computational procedure for estimating the slopeparameters in this model does not require that the dummyvariables for the individual (and/or time) eects actually beincluded in the matrix of explanatory variables.

2 We need only nd the means of time-series observationsseparately for each cross-sectional unit, transform theobserved variables by subtracting out the appropriate

time-series means, and then apply the leastsquares method tothe transformed data.






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The foregoing procedure is equivalent to premultiplying the ith

equationyi=ei+Xi+i

by a TTidempotent (covariance) transformation matrix(within operator)

Q=IT 1T

ee0

to sweep out the individual eect i so that individual

observations are measured as deviations from individual means(over time).




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Qyi and QXicorrespond to the observations are measured asdeviations from individual means :

Qyi = IT 1Tee0 yi= yi e

1

Te0yi

= 0BB@yi,1

yi,2...

yi,T

1CCA1T Tt=1 yi,t!0BB@1

1...

1

1CCA




Denitions

Fixed and random eectsFixed eects methodsRandom eects methodsFixed or random methods?
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QXi = Xi 1T

ee0Xi

=0BB@

x1,i,1 x2,i,1 ... xK,i,1x1,i,2 x2,i,2 ... xK,i,2... ... ... ...

x1,i,T x2,i,T ... xK,i,T

1CCA

1T0BB@

1

1...

1

1CCA Tt=1x1,i,t Tt=1x2,i,t ... Tt=1xK,i,t




Denitions

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Finally, when the transformation Q is applied to a vector ofconstant (or a time invariant variable), it lead to a null vector.

Qe = IT 1

Tee0 e

= e 1T

ee0e

= e e=0

sincee0e=

1 .. 1

0@ 1..1

1A=TC. Hurlin Panel Data Econometrics



Denitions

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So, we have:yi=ei+Xi+i

()Qyi =Qei+QXi+Qi

()Qyi=QXi+Qi




Denitions


Li b d l d d l
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Denition

Under assumptionH1 , the xed eect estimator or LSDV estimatorof parameter is dened by:

bLSDV ="

N

i=1

X0iQXi

#1 " N

i=1

X0iQyi

#

where

Q=IT 1T

ee0




Denitions


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ExampleLet us consider a simple panel regression model for the totalnumber of strikes days in OECD countries. We have a balancedpanel data set for 17 countries (N=17) and annual data form

1951 to 1985 (T =35). General idea: evaluate the link betweenstrikes and some macroeconomic factors (inatiion, unemploymentetc..)








Denitions

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Denitions

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Denitions


Li b d t l d t d l
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Theorem

The LSDV estimator

b is unbiased and consistent when either N or

T or both tend to innity. Its variancecovariance matrix is:

VbLSDV=2

" N

i=1

X0iQXi

#1

bLSDV p!NT!




Denitions


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Theorem

However, the estimator for the intercept,bi, although unbiased, isconsistent only when T! .

bi

p!T!

i




Denitions




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2.4. Random eects methods




Denitions


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Denition

The random specication of unobserved eects corresponds to a

particular case of variance-component or error-componentmodel, in which the residual is assumed to consist of threecomponents

yi,t=0xit+i,t 8 i8t

i,

t=i+t+vi,

t




Denitions




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Assumptions (H2) Let us assume that error terms i,t=i+t+vi,t arei.i.d. with8 i=1, .., N,8 t=1, ., T E(i)=E(t)=E(vi,t)=0 E(it)=E(tvi,t)=E(ivi,t)=0 E(ij)=

2

0i=j8i6=j

E(ts)= 2

0t=s

8t

6=s

E(vi,tvj,s)= 2v0 t=s, i=j8t6=s,8i6=j Eix0i,t=Etx0i,t=Evi,tx0i,t=0




Denitions


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Denition

As suggested by Wooldridge (2001), the "xed eect"specication can be viewed as a case in which iisa random

parameter with covi, x0i,t6=0, whereas the "random eectmodel" correspond to the situation in which covi, x0i,t=0.Denition

The variance ofyitconditional on xitis the sum of threecomponents:

2y =2+

2+

2v




Denitions




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Denition

Sometimes, the individual eects iare supposed to have a nonzero mean, with E(i)=, then we can dened a individual

eects iwith zero mean. A linear unobserved eects panel datamodels is then dened as follows:

yi =e+Xi+i

i(T,1)

= e(T,1) i(T,1) + vi

(T,1)




Denitions


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Denition

The vectorial expression of the individual eects model is then:

yi(T,1) = eXi(T,K+1) (K+1,1) + i(T,1)i

(T,1)= e

(T,1)i

(T,1)+ vi

(T,1)

with eXi =(e :Xi) and 0 = : 0


Specication tests and analysis of covarianceLinear unobserved eects panel data modelsRandom coecients models

Denitions


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Denition

Under assumptions H2 , the variance-covariance matrix ofi isequal to:

V =E

i0i

=E

(ie+vi) (ie+vi)

0=2ee0+2vITIts inverse is:

V1 = 12v IT 2

2v+T

2 ee0



Denitions




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The presence ofiproduces a correlation among residuals of thesame crosssectional unit, though the residuals from dierentcross-sectional units are independent

V =Ei0i=2ee0+2vITV =

0BB@

2+2v

2 ...

2

2+2v ...

2

...

2

2+

2v

1CCA



Denitions




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However, regardless of whether the iare treated as xed or asrandom,the individual-specic eects for a given sample can beswept out by the idempotent (covariance) transformation matrix Q

Qyi=Qe+QXi+Qei+Qvi

Since Qe=

IT T1ee0

e=0, we have

Qyi=Qe+QXi+Qvi



Denitions


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p

Theorem

Under assumptions H2 , when iare treated as random, the LSDV

estimator is unbiased and consistent either N or T or both tend toinnity. However, whereas the LSDV is the BLUE under theassumption thatiare xed constants, it is not the BLUE in nitesamples when iare assumed random. The BLUE in the lattercase is the generalized-least-squares (GLS) estimator.



Denitions


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p

Fact

Moreover, if the explanatory variables contain some time-invariantvariables zi, their coecients cannot be estimated by LSDV,because the covariance transformation eliminates zi.



Denitions


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p

Let us consider the model

yi=eXi+i 8 i=1, .., Nwhere i=ie+ vi,

eXi =(e, Xi) and 0 =

,0

. The variance

covariance matrix V =E(i0i) is known.

Denition

If the variance covariance matrix V is known, the GLS estimator ofthe vector, denoted

bGLS, is dened by:

bGLS = " Ni=1

eX0iV1 eXi#1 " Ni=1

eX0iV1yi#Under assumptions H2 , this estimaor is BLUE.



Denitions


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p

Denition

Following Maddala (1971), we write V as:

V1 = 12vQ+1

Tee0

where Q=(IT ee0/T) and where parameter is dened by:

= 2v2v+T2



Denitions




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Given this denition ofV1 , we have:

bGLS = " Ni=1 eX0i Q+1Tee0eXi#1

" Ni=1 eX0i Q+1Tee0 yi#

bGLS =

" N

i=1 e

X0iQeXi+

1

T

N

i=1 e

X0iee0

eXi

#1 " N

i=1 e

X0iQyi+1

T

N

i=1 e

X0iee0yi

#witheXi =(e Xi) and 0 = 0



Denitions




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bGLSbGLS =2664

NT TN

i=1

x0i

T Ni=1

xi Ni=1

X0iQXi+T Ni=1

xix0i

37751

24

NT yN

i=1

X0iQyi+TN

i=1

xiyi

35

Using the formula of the partitioned inverse, we can derivebGLS.C. Hurlin Panel Data Econometrics


Denitions


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Denition

If the variance covariance matrix V is known, the GLS estimator ofvector is dened by:

bGLS = "1T Ni=1 X0iQXi+N

i=1

(xi x) (xi x)0#1

"1

T

N

i=1

X0iQyi+N

i=1

(xi x) (yi y)#where is a constant dened by =2v

2v+T

2

1C. Hurlin Panel Data Econometrics


Denitions


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This estimator can be expressed as a weigthed average of theLSDV (OLS) estimator and the between estimator.

Denition

The between-group estimator or between estimator,denoted

bBE,

corresponds to the OLS estimator obtained in the model:

yi=c+0xi+i 8 i=1, .., N

bBE = " N

i=1 (xi x) (xi x)0#1

" N

i=1 (xi x) (yi y)#The estimatorbBE is called the between-group estimator becauseit ignores variation within the group



Denitions


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Denition

The pooled estimator,denotedbpooled, corresponds to the OLSestimator obtained in the model:

yit=+0xit+it 8 i=1, .., N 8 t=1, .., T

bpooled =

" T

t=1

N

i=1

(xi,t x) (xi x)0#1

" Tt=1

N

i=1

(xit x) (yit y)#

C Hurlin Panel Data Econometrics


Denitions


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TheoremUnder assumptions H2, the GLS estimatorbGLS is a weightedaverage of the between-groupbBEand the within-group (LSDV)estimators

bLSDV.

bGLS = bBE+(IK )bLSDVwhere denotes a weigth matrix dened by:

= T" Ni=1

X0iQXi+TN

i=1

(xi x) (xi x)0#1" N

i=1

(xi x) (xi x)0#



Denitions


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1 If!1, then GLS converges to the OLS pooled estimator.

bGLS p!!1bpooled2 If!0, the GLS estimator converges to LSDV estimator.

bGLSp!

!0bLSDV



Denitions


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Fact

The constant measures the weight given to the between-groupvariation.

In the LSDV (or xed-eects model) procedure, this source ofvariation is completely ignored.

The OLS procedure corresponds to=1. The between-group and

within-group variations are just added up.



Denitions


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Fact

The procedure of treating ias random provides a solutionintermediate between treating them all as dierent andtreating them all as equal.





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Given the denition of, we have:

=

2v

2v+T2

lim

T!=0

FactWhen T tends to innity, the GLS estimator converges to theLSDV estimator:

bGLS!T!

bLSDV

This is because when T! , we have an innite number ofobservations for each i.Therefore, we can consider each i as arandom variable which has been drawn once and forever, so thatfor each i we can pretend that they are just like xed parameters.





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Fact

Computation of the GLS estimator can be simplied by introducinga transformation matrix P such that

P= hIT 1 1/2 (1/T) ee0iWe have

V1 = 1

2vP0P

Premultiplying the model by the transformation matrix P, weobtain the GLS estimator by applying the least-squares method tothe transformed model (Theil (1971, Chapter 6)).





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This is equivalent to rst transforming the data by subtracting a

fraction 1 1/2 of individual means yi and xi from theircorresponding yit and xit, then regressing yit 1 1/2 yi onaconstant and xit

1 1/2 xi.





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We can show that:

varbGLS=2v "

N

i=1

X0iQXi+TN

i=1

(xi

x) (xi

x)0#

1

Because > 0, we see immediately that the dierence betweenthe covariance matrices of

bLSDV and

bGLS is a positive

semidenite matrix. For K=1, we have:

varbGLSvarbLSDV







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Denition

If the variance components 2 and 2 are unknown, we can use

two-stepGLS estimation.

1 In the rst step, we estimate the variance components usingsome consistent estimators.

2 In the second step, we substitute their estimated values into

bGLS = " Ni=1 eX0ibV1 eXi#1

" Ni=1 eX0ibV1yi#or its equivalent form.







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LemmaWhen the sample size is large (in the sense of either N! ,orT! ), the two-step GLS estimator will have the sameasymptotic eciency as the GLS procedure with known variancecomponents.

Lemma

Even for moderate sample size [for T 3, N (K+1)9; forT 2, N (K+1)10], the two-step procedure is still moreecient than the covariance (or within-group) estimator in thesense that the dierence between the covariance matrices of thecovariance estimator and the two-step estimator is nonnegativedenite





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Noting that yi =i+0xi+i and

(yit yi)=(xit xi)+(vit vi), we can use the within- andbetween-group residuals to estimate 2 and

2 by

b2v =N

i=1

T

t=1h(yi,t yi)b0LSDV(xi,t xi)i2

N(T 1)K

b2 =N

i=1 yib0LSDVxi

2

NK 1 b2v







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Then, we have an estimate of and V1

b= b2vb2v+Tb2!

bV1 =

1

b2v

Q+

b

1

Tee0







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ExampleLet us consider a simple panel regression model for the totalnumber of strikes days in OECD countries. We have a balancedpanel data set for 17 countries (N=17) and annual data form

1951 to 1985 (T =35).

s,t=i+u,t+pit+it8 i=1, .., 17

si,tthe number of strike days for 1000 workers for the country

iat time t.



C H rlin Panel Data Econometrics




Fi R d t th d
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Figure: Random eects method

C H li P l D t E t i



DenitionsFixed and random eects


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2.5. Fixed or random methods?

C H li P l D t E t i





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Fact

Whether to treat the eects as xed or random makes no

dierence when T is large, because both the LSDV estimator andthe generalized least-squares estimator become the same estimator:

bGLS!T!

bLSDV






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WhenT is nite and Nis large, whether to treat the eects as

xed or random is not an easy question to answer. It can make asurprising amount of dierence in the estimates of the parameters.








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Example

For example, Hausman (1978) found that using a xed-eectsspecication produced signicantly dierent results from arandom-eects specication when estimating a wage equationusing a sample of 629 high school graduates followed over six yearsby the Michigan income dynamics study. The explanatory variablesin the Hausman wage equation include a piecewise-linearrepresentation of age, the presence of unemployment or poor

health in the previous year, and dummy variables forself-employment, living in the South, or living in a rural area.






Figure: Hausman (1978)
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Fact

In the random-eects framework, there are two fundamentalassumptions. One is that the unobserved individual eectsi are

random draws from a common population. The other is that theexplanatory variables are strictly exogenous. That is, the errorterms are uncorrelated with (or orthogonal to) the past, current,and future values of the regressors:

E ( itj xi1 , .., xiK)= E ( ij xi1 , .., xiK)= E ( vitj xi1 , .., xiK)=0








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What happens when this condition is violated?

E

( ijx

i1, .., x

iK)6=0 or E ix0i,t6=0

1 The Mundlaks specication (1978)

2 The Hausman test






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a. The Mundlaks specication

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The properties of various estimators we have discussed thus fardepend on the existence and extent of the relations between the

Xs and the eects. Therefore, we have to consider the jointdistribution of these variables. However, i are unobservable.

=) Mundlak (1978a) suggested that we approximateE(ixi,t) by a linear function.






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Denition

Let us assume that the individual eects satisfy:

i =x0ia

+iwith a2 RK and E(ix0it)=0. Under this assumption, the modelis:

yi,t=+0xi,t+x0ia+i,t

i,t=i +vi,t






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Assumptions H4 The error term i,t= i +vi,t satises,8 i2 [1, N] ,8 t2[1, T] E(i) =E(vi,t)=0

E(ivi,t)=0

Eij= 20 i=j8i6=j E(vi,tvj,s)=

2v

0t=s, i=j

8t6=s,8i6=j Evi,tx0i,t=Eix0i,t=0








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The model can be rewritten as follows:

yi(T,1)

=

eXi

(T,K+1)

(K+1,1)

+ i(T,1)

8 i=1, .., N

with i =ie+vi,eXi =(ex0i, e, Xi) and 0 = a, ,0 .E

i0

j

= E

h(ie+vi)

je+vj

0i= 2ee0+2vIT =V0 i=ji6=j






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Utilizing the expression for the inverse of a partitioned matrix, weobtain the GLS of (, , a ) as:

bGLS =y x0bBEbaGLS =bBEbLSDVb

GLS = b

BE+(IK )bLSDV






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Then, the GLS estimator

bGLS is unbiased and we have:

bGLS!T!bLSDVbGLS !NT!






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Now let us assume that the DGP corresponds to the Mundlaksmodel

i =x0ia+

i

and we apply GLS to the initial model:

yi,t=+0xi,t+i,t

i,t=i+vi,t






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In general, we have:

bGLS =

bBE+(IK )

bLSDV

In this case, we can show that:

E

bBE= +a bBE p!N! +aE bLSDV=






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Theorem

So, ifi=x0ia+i , the GLS is biaised when T is xed. More

precisely:

E bGLS= + abGLS p!N! + aAs usual, the GLS is asymptotically unbiased:

bGLS p!T!






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WhenT is xed and Ntends to innity, the GLS is biased if thereis a correlation between the individual eects and the expanatoryvariables:

plimN!bGLS =eplim

N!bBE+ IKeplim

N!bLSDV

=

e (+a)+

IK

e

= +eawithe = plim

N!.






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Summary

E ( ij

xi1 , .., xiK)=0 E ( ij

xi1 , .., xiK)6=0

LSDV GLS LSDV GLS

T Fixed, N! Unbiased Unbiased BiasedT

! and N

! Unbiased BLUE Unbiased Unbiased






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b. The Hausmans specication test






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Hausman (1978) proposes a general test of specication, that canbe applied in the specic context of linear panel models to the

issue of specication of individual eects (xed or random).

Hausman J.A., (1978) Specication Tests in Econometrics,Econometrica, 46, 1251-1271






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The general idea of the an Hausmans test is the following.Let usconsider a particular model y=f (x;)+ and particularhypothesis H0 on this model (parameter, error term etc.).Let usconsider two estimators of the K-vector , denotedb1 andb2 ,both consistent under H0 and asymptotically normally distributed.

1 Under H0 , the estimatorb1 attains the asymptoticCramerRao bound.

2 Under H1 , the estimatorb2 is biased.






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By examing the "distance" between

b1 and

b2 , it is possible to

conclude about H0 :

1 If the "distance" is small, H0 can not be rejected.

2 If the "distance" is large, H0 can be rejected.








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This distance is naturally dened as follows:

H=

b2

b1

0

hVar

b2

b1

i1

b2

b1

However, the issue is to compute the variance-covariancematrix Var

b2 b1 of the dierence between bothestimators. Generaly we know V

b2

and V

b1

, but not

Varb2 b1 .







L (H )
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Lemma (Hausman, 1978)Based on a sample of N observations, consider two estimatesb1andb2 that are both consistent and asymptotically normallydistributed, with

b1 attaining the asymptotic CramerRao bound

so thatpNb1 is asymptotically normally distributed withvariancecovariance matrix V1. Suppose

pNb2 is

asymptotically normally distributed, with mean zero andvariancecovariance matrix V2. Letb

q=

b2 b

1. Then the

limiting distribution [under the null] ofpNb1 andpNbqhas zero covariance:

E(b1q0) =0KC. Hurlin Panel Data Econometrics






Th
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Theorem

From this lemma, it follows that

Var

b2 b

1=Var

b2

Varb

1Thus, Hausman suggests using the statisticH=

b

2

b1

0 hVarb

2

Var

b

1

i1 b

2

b1

or equivalently

H=bq0[Var(bq)]1bqC. Hurlin Panel Data Econometrics





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Under the null hypothesis, this statistic is distributedasymptotically as central chisquare, with Kdegrees of freedom.

H HO

!N!

2 (K)

Under the alternative, it has a noncentral chi-square distributionwith noncentrality parameter

eq0[Var(

bq)]1

eq, where

eq is dened

as follows:

eq= plimH1/N! b2 b1C. Hurlin Panel Data Econometrics







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Problem

Now, apply the Hausmans test to discriminate between xedeects methods and random eects methods. We assume thati

are random variable and the key assumption tested is here denedas:

H0 : E( ijXi)=0H1 : E( i

jXi)6=0






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This test can be interpreted as a specication test between "xedeect methods" and "random eect methods".

1 If the null is rejected, the correlation between individualeects and the explicative variables induces a bias in the GLSestimates. So, a standard LSDV approach (xed eectmethod) has to be privilegiated.

2 If the null is not rejected, we can use a GLS estilmator(random eect method) and specify the individual eects as

random variables (random eects model).






Ho to implement this test? Let s consider the standard model
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How to implement this test? Let us consider the standard modelwith random eects (=0):

yi =Xi+ei+vi

1 Under H0 (and assumptions H2) we know thatbLSDV andbGLSare consistent and asymptotically normally distributed.2 Under H0 ,

bGLS is BLUE and attains asymptotic CramerRao

bound.

3 Under H1 ,bGLS is biased.C. Hurlin Panel Data Econometrics





According to the Hausmans lemma we have:
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According to the Hausman s lemma, we have:

covbGLS, bLSDVbGLS =0()covbLSDV,bGLS=VarbGL

Since,

VarbLSDVbGLS=VarbLSDV+ VarbGLS2covbLSDV,bGL

We have:

VarbLSDVbGLS=VarbLSDV VarbGLSC. Hurlin Panel Data Econometrics






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Denition

The Hausmans specication test statistic of individual eect canbe dened as follows:

H= bLSDVbGLS0 hVarbLSDVVarbGLSi1 bLSDVbGLSUnder H0 , we have:

H HO

!NT!

2 (K)






b b


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1 WhenN is xed and T tends to innity,GLS andMCGbecome identical. However, it was shown by Ahn and Moon(2001) that the numerator and denominator ofHapproachzero at the same speed. Therefore the ratio remains

chi-square distributed. However, in this situation thexed-eects and random-eects models becomeindistinguishable for all practical purposes.

2 The more typical case in practice is that Nis large relative to

T, so that dierences between the two estimators or twoapproaches are important problems.






Example
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ExampleLet us consider a simple panel regression model for the totalnumber of strikes days in OECD countries. We have a balancedpanel data set for 17 countries (N=17) and annual data form1951 to 1985 (T =35).









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Variable coecient modelRandom coecient modelMixed xed and random coecient model

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Section 3.





Variable coecient modelRandom coecient modelMixed xed and random coecient model

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There are cases in which there are changing economic structures ordierent socioeconomic and demographic background factors thatimply that the response parameters may be varying over timeand/or may be dierent for dierent crosssectional units.



geneve chapitre1

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