MODELS FOR PANEL DATA

PANEL DATA REGRESSION• Double subscript on variables (observations)

i … households, individuals, firms, countriest … period (time-series dimension)

… scalar … vector K × 1 … vector of i,t th observation on K explanatory var.

T 1, ,

1, ,it it ity X u i N

t T

itX

ONE-WAY ERROR COMPONENT MODEL

• Utilized by most of the panel data applications

… denotes unobservable individual specific effect time-invariant

accounts for any individual-specific factor not included in the regression

… the remainder disturbance term… varies both with individual and in time

it i itu v

i

itv

• Potential extension

… denotes unobservable time effect individual-invariant

accounts for any time-specific effect that is not included in the regression

TWO-WAY ERROR COMPONENT MODEL

it i t itu v

t

PANEL DATA REGRESSION• vector form of the model

… vector of ones of dimension NT

stacked observationsthe slower index is index over INDIVIDUALS, the faster

index is over TIME

( 1)1 1 1 ( 1) 1 111 1 NT K NT KNT KNT

NTKNT NT

y X u Z u

NT

[ ]NTZ X

PANEL DATA REGRESSION• Vector of disturbances

… matrix of individual dummies

T11 1 21 2 1[ , , , , , , , , , ]T T N NTu u u u u u u

u Z v

NT NN TZ I

Z

T1[ , , ]N

T11 1 21 2 1[ , , , , , , , , , ]T T N NTv v v v v v v

Notes about matrices

… square matrix of ones of dimension T

• matrix P– the projection matrix on

– averages the observations across time for each individual

– generates individual means

TN TZ Z I J

TJ

ZT 1 T( ) N TP Z Z Z Z I J

1T TTJ J

Notes about matrices• matrix Q

– obtains deviations from individual means

• Application of P and Q:

NTQ I P

T1 1 2 2

times times times

( ) [ , , , , , , , , , ]N N

T T T

Pu u u u u u u

T11 1 1 1 1( ) [( ), ,( ), ,( ), ,( )]T N N NT NQu u u u u u u u u

11

T

i itT tu u

Properties of P and Q• Symmetric, idempotent

• P and Q are othogonal

• They sum to the identity matrix

TP P P P P TQ Q Q Q Q

rank( ) tr( )

rank( ) tr( ) ( 1)

P P N

Q Q N N

NTP Q I

0P Q

THE FIXED EFFECTS MODEL (FE)• i’s … assumed to be fixed parameters to be

estimated• • … assumed to be independent of the vit for all

i and t

• FE model is an appropriate specification if we are focusing on a specific set of N individuals (firms, countries,…)

• Inference is restricted to the behavior of these sets of individuals

2(0; )it vv iid

itx

THE FIXED EFFECTS MODEL (FE)• Model can be rewritten

• OLS can be used to obtain estimates of unknown parameters

• BUT! If N is large:– Too many individual dummies are included into the

model– Matrix to be inverted by OLS is of dimension (N+K)

NTy X Z v Z Z v

THE FIXED EFFECTS MODEL (FE)• LSDV (least squares dummy variable) estimator– The model is premultiplied by Q

Using the fact that: and

– OLS performed on the resulting transformed model– matrix Q wipes out the individual specific effects– LSDV involves the inversion of a K × K matrix

NTy X Z vQ

Qy QX Qv

y X

Q

Q

Q Q Q

v

0NTQ 0QZ

THE FIXED EFFECTS MODEL (FE)• LSDV (least squares dummy variable) estimator

• unbiased estimate of– residual sum of squares from LSDV regression

divided by (NT-N-K) – Not by (NT-K)!

T 1 T( )X QX X Qy

2 T 1var( ) ( )v X QX

2v

THE FIXED EFFECTS MODEL (FE)• Dummy variable trap (perfect multicolinearity)

Without additional restriction just (+i)’s are estimable, not and i ‘s separately

• Possible restrictions:1. 2.Particular3.

• Ad 3:

0 0i

10

N

ii

Ty X

Ti i iy X

T1, ,[ , , ]KX X X

T1, ,[ , , ]i i K iX X X

THE FIXED EFFECTS MODEL (FE)• Limitations:– FE is not feasible for large panels (N is large)

N-1 dummies included in the model large loss of degrees of freedom (extra N-1 parameters are to be

estimated)Too many dummies may aggravate the problem of multicollinearity

among regressors

– (LSDV) estimator cannot estimate the effect of any time-invariant variable (race, religion, sex,…)

Time-invariant variables are wiped out by the Q transformation

THE FIXED EFFECTS MODEL (FE)• Properties of LSDV estimator:

If is the true model:• LSDV is the BLUE as long as• as T LSDV is consistent• If T is fixed and N :

– LSDV estimator of is consistent– Estimators of the individual-specific effects (+i) are not

consistent (the number of parameters increases as N increases)

• OLS on (pooled OLS estimator) yields biased and inconsistent estimates (due to omission variable bias)

y Z Z v 2(0; )it vv iid

y Z u

THE FIXED EFFECTS MODEL (FE)• Testing for fixed effects:– test of the joint significance of the individual

dummies– H0:

– F-test:• Restricted model : model without individual dummies

• Unrestricted model : model with individual dummies (FE model)

1 2 1 0N

y Z Z v

y Z u

1( 1),( )

RRSS URSS aN

N NT N KURSSNT N K

F F