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Econometric Analysis of Panel Data

5. Random Effects Linear Model

The Random Effects Model

 The random effects model

 c i is uncorrelated with x it for all t;

 E[c (^) i | X i ] = 0  E[εit| X i ,c (^) i ]=

it it i it i i i i i i i i i i i i Ni=1 i 1 2 N

y = +c +ε , observation for person i at time t = +c + , T observations in group i = + + , note (c , c ,...,c ) = + + , T observations in the sample c=( , ,... ) ,

′

= ′ Σ ′ ′ ′ ′

x β y X β i ε X β c ε c y Xβ c ε c c c ΣNi=1 T by 1 vectori

Notation

1 1 1 2 2 2 N N N i

u T observations u T observations u T observations = + + T observations = +

          (^) =   (^) +   (^) +          Σ

1 1 1 2 2 2 N N N Ni=

y X ε i y X (^) β ε i y X ε i Xβ ε u Xβ w In all that fo

    

′ ′

i it it

llows, except where explicitly noted, X, X and x contain a constant term as the first element. To avoid notational clutter, in those cases, x etc. will simply denote the counterpart without the constant term. Use of the symbol K for the number of variables will thus be context specific but will usually include the constant term.

Notation

(^2) u (^2) u (^2) u 2 u^2 2 u^2 u^2 i i u^2 u^2 2 u^2 (^2) u (^2) i i (^2) u 2 i 1 2 N

Var[ +u ]

= T T

=

Var[ | ]

 (^) +  = ^ +     + 

• ′ ×
• ′

 =  

i i

T T

ε i

I ii I ii Ω Ω 0 0 w X^0 Ω^0 0 0 Ω

      

      

ε ε ε ε ε

σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ σ

i

(Note these differ only in the dimension T )

   

Convergence of Moments

N^ Ni 1 i i 1 i N Ni 1 i i 1 i Ni 1 i (^) u Ni 1 i i

T f^ T a weighted sum of individual moment matrices

T f^ T a weighted sum of individual moment matrices = f (^) T f

Note asymptoti

=^ =

= = = =

′ (^) = Σ ′ = Σ ′ (^) = Σ ′ = Σ Σ ′^ + Σ ′

i i

i i i

(^2) ii i (^2) i i

X X X X

X ΩX X Ω X

X X (^) x x σ ε σ

i i

i

cs are with respect to N. Each matrix (^) T is the

moments for the T observations. Should be 'well behaved' in micro level data. The average of N such matrices should be likewise. T or T is assum

X X^ ′ ii i

ed to be fixed (and small).

Random vs. Fixed Effects

 Random Effects

 Small number of parameters

 Efficient estimation

 Objectionable orthogonality assumption (c i ⊥ X i)

 Fixed Effects

 Robust – generally consistent

 Large number of parameters

Estimating the Variance for OLS



1 1 Ni 1 (^) i Ni 1 (^) i Ni 1 (^) i Ni 1 i

N Ni 1 i i 1 i

N Ni 1 i i 1 i

Var[ | ] (^) T T T T

T f^ T , where^ =^ =E[^ |^ ] In the spirit of the White estimator, use f ˆ^ ˆ^ , ˆ T T

− − = = = =

= =

= =

= ^ ′^ ^ ^ ′^  ^ ′  Σ ^ Σ ^  Σ (^)  ^ Σ  ′ (^) = Σ ′ ′ Σ

′ (^) = Σ ′^ ′ Σ

i i i i i i i

i i i i i

b X^1 X X^ X^ ΩX^ X X X ΩX X^ Ω X Ω w w X

X ΩX X w w X w =

Hypothesis tests are then based on Wald statistics.

y - X b i i

THIS IS THE 'CLUSTER' ESTIMATOR

Mechanics

1 ( Ni 1 ) 1

i i

Est.Var[ | ] ˆ ˆ

ˆ = set of T OLS residuals for individual i. = T xK data on exogenous variable for individual i. ˆ = K x 1 vector of products ( ˆ^ )( ˆ )

− − = (^)  ′^  Σ= ′^ ′^  ′  ′

′ ′ ′ (^) =

i i i i i i i i i i i i

b X X X X w w X X X w X X w X w w X

( (^ ) (^ ))

Ni 1 Ni 1 Ni 1

KxK matrix (rank 1, outer product) ˆ ˆ (^) = sum of N rank 1 matrices. Rank K. We could compute this as ˆ^ ˆ = ˆ. Why not do it that way?

= = =

Σ ′^ ′ ≤ Σ ′^ ′^ Σ ′

i i i i i i i i i i i

X w w X X w w X X Ω X

OLS Results

+----------------------------------------------------+| Residuals Sum of squares = 522.2008 | || (^) Fit Standard errorR-squared of e == .3544712.4112099 || |+----------------------------------------------------+ Adjusted R-squared = .4100766 | +---------+--------------+----------------+--------+---------+----------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X| +---------+--------------+----------------+--------+---------+----------+Constant 5.40159723 .04838934 111.628. EXPEXPSQ (^) -.00068788.04084968 (^) .480428D-04.00218534 (^) -14.31818.693 .0000.0000 (^) 514.40504219. OCCSMSA -.13830480.14856267 .01480107.01206772 -9.34412.311 .0000.0000 .51116447. MSFEM (^) -.40020215.06798358 .02074599.02526118 (^) -15.8433.277 .0010.0000 .81440576. UNIONED .09409925.05812166 .01253203.00260039 (^) 22.3517.509 .0000.0000 (^) 12.8453782.

Alternative Variance Estimators

• +---------+--------------+----------------+--------+---------+Constant 5.40159723 .04838934 111.628. +---------+--------------+----------------+--------+---------+|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |
  • EXPEXPSQ -.00068788.04084968 .480428D-04.00218534 -14.31818.693 .0000.
  • OCCSMSA -.13830480.14856267 .01480107.01206772 -9.34412.311 .0000.
  • MSFEM -.40020215.06798358 .02074599.02526118 -15.8433.277 .0010.
  • UNIONED .09409925.05812166 .01253203.00260039 22.3517.509 .0000.
• RobustConstant 5.40159723 .10156038 53.186.
  • EXPEXPSQ -.00068788.04084968 .983981D-04.00432272 -6.9919.450 .0000.
  • OCCSMSA -.13830480.14856267 .02772631.02423668 -4.9886.130 .0000.
  • MSFEM -.40020215.06798358 .04382220.04961926 -8.0651.551 .1208.
  • UNIONED .09409925.05812166 .02422669.00555697 10.4593.884 .0001.

Panel Data Algebra (1) (^2) ε (^2) u i i (^2) ε (^2 2) u (^2) ε (^2) ε (^2) ε

(^2) ε

= σ +σ , depends on 'i' because it is T T = σ [ ], = σ / σ = σ [ ] = σ [ ], = , =. Using (A-66) in Greene (p. 822) 1 1 σ 1+ =

′ ×

ρ ′ ρ ρ ′^ + ′ ρ

= ^ ′ 

i i^2 i^2

-1i -1 (^) -1 -1 -

Ω I ii Ω I + ii Ω I + ii A bb A I b i

Ω A - (^) b A b A bb A

(^2 2) u (^2) ε i 2 2 ε 2 ε i 2 u

(^1 1) = 1 σ σ 1+T σ σ +Tσ

 (^) ρ ′   (^) ′  (^) ρ       

I - ii I - ii

Panel Data Algebra (2)

(^2) ε 2 u (^2) ε i 2 u (^2) ε i u 2 (^2) ε i 2 u (^2) ε i 2 u i 2 ε 2 ε i 2 u (^2) ε i 2 u (^2) ε

(Based on Wooldridge p. 286) σ +σ σ +Tσ σ +Tσ σ +Tσ ( ) (σ +Tσ )[ ( )], = σ /(σ +Tσ ) (σ +Tσ )[ ] (σ

= ′^ = ′^ ′= = − = + η − η = + η =

i -1^ Di iD Di^ Di Di^ Di

Ω I ii I i(i i) i I P I I M P I P P M i 2 u (^1) i 1 / 2 i i i i i a ai

+Tσ ) (1 / ) (Prove by multiplying. .) (1 / ) 1 , =1- 1 (Note )

− −

= + η = = + η = (^) − θ ^ − θ  θ η = + η

i i Di^ iD^ Di^ iD i Di^ Di^ Di i Di^ iD

S S P M P M 0

S P M I P

S P M

GLS (cont.)

it it i i it it i i

i (^2 ) i u 2

GLS is equivalent to OLS regression of

y * y y. on * .,

where 1 T

Asy.Var[ ˆ] [ ] [ ]

ε ε

ε

= − θ = − θ σ θ = − σ + σ

= ′^ -1^ -1^ = σ ′ -

x x x

β X Ω X X * X*

Estimators for the Variances

i i

it it i OLS Ni 1 Tt 1 (^) it 2 Ni 1 (^) t 1T 2 U 2 (^2 2 2) U i LSDV i

y u With a consistent estimator of , say , (y ) estimates ( ) Divide by something to estimate = With the LSDV estimates, a and ,

= = = = ε ε

=

= ′ + ε +

Σ Σ ′ Σ Σ σ + σ σ σ + σ

it

it

x β β b

- x b

b

N 1 Tt 1 i (^) it 2 Ni 1 Tt 1i 2 2 (^2) U (^2 2) U 2

(y a ) estimates Divide by something to estimate Estimate with ( ) - (^) ˆ.

= = = ε ε ε ε

Σ ′ Σ Σ σ σ σ σ + σ σ

- (^) i - x bit