Econometric Analysis of Panel Data

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business

Econometric Analysis of Panel Data







5. Random Effects Linear Model

The Random Effects Model

 The random effects model

y it =x  β+c i +εit , observation for person i at time t

it



y i =X iβ+c ii+ε i , Ti observations in group i

=X iβ+c i +ε i , note c i  (c i , c i ,...,c i )

y =Xβ+c +ε , N Ti observations in the sample

i=1



 2

c=(c1 , c ,...c ), N Ti by 1 vector

N i=1





 ci is uncorrelated with xit for all t;

 E[ci |Xi] = 0

 E[εit|Xi,ci]=0

Error Components Model

Generalized Regression Model



y it  x  b+εit +ui

it



E[εit | X i ]  0  2   u

2

u2

u

2



 

u2

 2   u

2

u

2

Var[ε i +uii ]   

2 2

E[εit | X i ]  σ 

 

E[ui | X i ]  0  2

 u u   u 

2 2 2

2

E[ui2 | X i ]  σ u  



y i =X iβ+ε i +uii for Ti observations

Notation

 y1   X1   ε1   u1i1  T1 observations

y  X   ε   u i  T observations

 2

 2

β   2   2 2 2

       

       

 yN   XN   εN  uNiN  TN observations

= Xβ+ε+u N Ti observations

i=1



= Xβ+w

In all that follows, except where explicitly noted, X, X i

and x  contain a constant term as the first element.

it



To avoid notational clutter, in those cases, x  etc. will

it



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

u2

u

2



 

 u  2   u u

2 2 2

Var[ε i +uii ]  

 

 2

 u u  2   u 

2 2

 

=  2I Ti   u ii Ti  Ti

2





=  2I Ti   u ii

2





= Ωi

Ω1 0 0 

0 Ω2 0  (Note these differ only

Var[w | X ]   

  in the dimension Ti )

 

0

 0 ΩN 



Regression Model-Orthogonality

1

plim X'w  0

# observations

1 1

plim N i=1 X w i  plim N N X (ε i +uii)  0

N

i i=1 i

i1 Ti i1 Ti

1  N X ε i N X ii 

plim N i=1 Ti i

+ i=1 Tui i 

i

i1 Ti  Ti Ti 

 N X ε i N X ii  Ti

plim i=1 fi i

+ i=1 fi i

ui  , 0 z] | Mean of X|

+---------+--------------+----------------+--------+---------+----------+

Constant 5.40159723 .04838934 111.628 .0000

EXP .04084968 .00218534 18.693 .0000 19.8537815

EXPSQ -.00068788 .480428D-04 -14.318 .0000 514.405042

OCC -.13830480 .01480107 -9.344 .0000 .51116447

SMSA .14856267 .01206772 12.311 .0000 .65378151

MS .06798358 .02074599 3.277 .0010 .81440576

FEM -.40020215 .02526118 -15.843 .0000 .11260504

UNION .09409925 .01253203 7.509 .0000 .36398559

ED .05812166 .00260039 22.351 .0000 12.8453782

Alternative Variance Estimators

+---------+--------------+----------------+--------+---------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] |

+---------+--------------+----------------+--------+---------+

Constant 5.40159723 .04838934 111.628 .0000

EXP .04084968 .00218534 18.693 .0000

EXPSQ -.00068788 .480428D-04 -14.318 .0000

OCC -.13830480 .01480107 -9.344 .0000

SMSA .14856267 .01206772 12.311 .0000

MS .06798358 .02074599 3.277 .0010

FEM -.40020215 .02526118 -15.843 .0000

UNION .09409925 .01253203 7.509 .0000

ED .05812166 .00260039 22.351 .0000

Robust

Constant 5.40159723 .10156038 53.186 .0000

EXP .04084968 .00432272 9.450 .0000

EXPSQ -.00068788 .983981D-04 -6.991 .0000

OCC -.13830480 .02772631 -4.988 .0000

SMSA .14856267 .02423668 6.130 .0000

MS .06798358 .04382220 1.551 .1208

FEM -.40020215 .04961926 -8.065 .0000

UNION .09409925 .02422669 3.884 .0001

ED .05812166 .00555697 10.459 .0000

Generalized Least Squares



ˆ

β=[X Ω-1 X ]1 [X Ω-1 y ]

=[N 1 X Ωi-1 X i ]1 [N 1 X Ωi-1 y i ]

i i i i



1  2 

-1

Ωi  2 I Ti  2 

2

ii

    Tiu 

(note, depends on i only through Ti )

Panel Data Algebra (1)

Ωi = σ 2I+σ uii, depends on 'i' because it is Ti  Ti

ε

2





Ωi = σ 2 [I + 2ii], 2 = σ u / σ 2

ε

2

ε



Ωi = σ 2 [I + 2ii] = σ 2 [A  bb], A=I, b=i.

ε ε



Using (A-66) in Greene (p. 964)

1  -1 1 

Ωi-1   A - A -1bbA -1 

σ2

ε  1+bA -1b 

1  1  1  2

σu 

= 2 I - 2

 ii = 2 I - 2

2

2

ii

σε  1+Ti  σ ε  σ ε +Tσ u 

i

Panel Data Algebra (2)

(Based on Wooldridge p. 286)

Ωi  σ 2I+σ uii  σ 2I+Tσ ui(ii)-1 i  σ 2I+Tσ uPD

ε

2

ε i

2

ε i

2 i



2 i

 σ 2I+Tσ u (I  MD )

ε i



 (σ 2 +Tσ u )[PD  (I  PD )], i = σ 2 /(σ 2 +Tσ u )

ε i

2 i i

ε ε i

2



2 i i

 (σ 2 +Tσ u )[PD  MD ]

ε i

2

 (σ 2 +Tσ u )S i

ε i



S i1  PD  (1 / i )MD (Prove by multiplying. PDMD  0.)

i i i i





1

S i1 / 2  PD  (1 / i )MD 

i i

I  iPD  , i=1- i

i



1  i  

i i

(Note S ia  PD  iaMD )

Panel Data Algebra (2, cont.)



1

Ωi1 / 2  i i

(PD  (1 / i )MD )

2

2  Tiu





1 1

 I  iPD  ,



i



2  Tiu 1  i



2







2 

i =1- 2  2  1 

  Tiu 2

2  Tiu





1

Ωi1 / 2  i

[I  iPD ]



Var[Ωi1 / 2 (ε i  uii)]  2I





Ωi1 / 2 y i  (1 /  )( y i  i y i .i) for the GLS transformation.

GLS (cont.)



GLS is equivalent to OLS regression of

y it *  y it  i y i . on x it *  x it  i x i .,



where i  1 

2

2  Tiu





ˆ

Asy.Var[β]  [X Ω-1 X ]-1  2 [X  * X*]-1



Estimators for the Variances

y it  x  β  it  ui

it



With a consistent estimator of β, say bOLS ,

N 1 tT1 (y it - x  b)2 estimates N 1 tT1 (2  U )

i

i

it i

i



2



2

Divide by something to estimate 2 = 2  U





With the LSDV estimates, ai and bLSDV ,

N 1 tT1 (y it - ai - x  b)2 estimates N 1 tT12

i

i

it i

i





Divide by something to estimate 2



2 2

Estimate U with Est(2  U )- 2 .

 ˆ

Feasible GLS

2

Feasible GLS requires (only) consistent estimators of 2 and u .





Candidates:

2 N 1 tT1 (y it  ai  x  bLSDV )2

i



From the robust LSDV estimator:  

ˆ i

N

it



i1 Ti  K  N

N 1 tT1 (y it  aOLS  x  bOLS )2

2

i

2

From the pooled OLS estimator: Est(   )  i

 u

it



N 1 Ti  K  1

i



2 2N 1 (y it  a  x bMEANS )2

From the group means regression: Est( / T   )  i

 u

i



N  K 1

2 2 N 1 tT1 s  t 1 w it wis

i Ti

ˆ ˆ

(Wooldridge) Based on E[w it w is | X i ]  u if t  s, u 

ˆ i 1



N 1 Ti  K  N

i



There are many others.





x´ does not contain a constant term in the preceding.

Practical Problems with FGLS

2

All of the preceding regularly produce negative estimates of u .

Estimation is made very complicated in unbalanced panels.

A bulletproof solution (originally used in TSP, now LIMDEP and others).

N 1 tT1 (y it  ai  x  bLSDV )2

i

2

From the robust LSDV estimator:  

ˆ i



it



N 1 Ti

i



N 1 tT1 (y it  aOLS  x  bOLS )2

2

i

2

From the pooled OLS estimator: Est(   )  i

 u

it

 2

ˆ

N 1 Ti

i



2 N 1 tT1 (y it  aOLS  x  bOLS )2  N 1 tT1 (y it  ai  x  bLSDV )2

i i



 

ˆu

i it

N

i it

0

i1 Ti









x´ does not contain a constant term in the preceding.

Stata Variance Estimators



2 N 1 tT1 (y it  ai  x  bLSDV )2

i



 

ˆ i

N

it

based on FE estimates

i1 Ti  K  N

2  SSE(group means) (N  K)2  ˆ

u  Max 0,

ˆ  

 N A (N  A)T 

2

where A = K or if u is negative,

ˆ

A=trace of a matrix that somewhat resembles IK .

Many other adjustments exist. None guaranteed to be

positive. No optimality properties or even guaranteed consistency.

Computing Variance Estimators



Using full list of variables (FEM and ED are time invariant)

OLS sum of squares = 522.2008.

2

Est(2 +u ) = 522.2008 / (4165 - 9) = 0.12565.





Using full list of variables and a generalized inverse (same

as dropping FEM and ED), LSDV sum of squares = 82.34912.

2 = 82.34912 / (4165 - 8-595) = 0.023119.

ˆ

2

u  0.12565 - 0.023119 = 0.10253

ˆ

2

Both estimators are positive. We can stop here. If u were

ˆ

negative, we would use estimators without DF corrections.

Application

+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .231188D-01 |

| Var[u] = .102531D+00 |

| Corr[v(i,t),v(i,s)] = .816006 |

| (High (low) values of H favor FEM (REM).) |

| Sum of Squares .141124D+04 |

| R-squared -.591198D+00 |

+--------------------------------------------------+

+---------+--------------+----------------+--------+---------+----------+

|Variable | Coefficient | Standard Error |b/St.Er.|P[|Z|>z] | Mean of X|

+---------+--------------+----------------+--------+---------+----------+

EXP .08819204 .00224823 39.227 .0000 19.8537815

EXPSQ -.00076604 .496074D-04 -15.442 .0000 514.405042

OCC -.04243576 .01298466 -3.268 .0011 .51116447

SMSA -.03404260 .01620508 -2.101 .0357 .65378151

MS -.06708159 .01794516 -3.738 .0002 .81440576

FEM -.34346104 .04536453 -7.571 .0000 .11260504

UNION .05752770 .01350031 4.261 .0000 .36398559

ED .11028379 .00510008 21.624 .0000 12.8453782

Constant 4.01913257 .07724830 52.029 .0000

Testing for Effects: LM Test

Breusch and Pagan Lagrange Multiplier statistic

Assuming normality (and for convenience now, a

balanced panel)

2 2

NT   (Tei )

N 2

 NT   [(Tei )  eei ] 

N 2

LM= 

i1

 1  

i1 i



2(T-1)    tT1eit

N

i1

2

 2(T-1)  N 1eei

i i 

Converges to chi-squared[1] under the null hypothesis

of no common effects. (For unbalanced panels, the

scale in front becomes (N 1 Ti )2 /[2N 1 Ti (Ti  1)].)

i i



Many adjustments for unbalanced panels and "better small

sample performance," e.g., Baltagi and Li in NLOGIT.

LM Tests

+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) | Unbalanced Panel

| Estimates: Var[e] = .216794D+02 | #(T=1) = 1525

| Var[u] = .958560D+01 | #(T=2) = 1079

| Corr[v(i,t),v(i,s)] = .306592 | #(T=3) = 825

| Lagrange Multiplier Test vs. Model (3) = 4419.33 | #(T=4) = 926

| ( 1 df, prob value = .000000) | #(T=5) = 1051

| (High values of LM favor FEM/REM over CR model.) | #(T=6) = 1200

| Baltagi-Li form of LM Statistic = 1618.75 | #(T=7) = 887

+--------------------------------------------------+

+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .210257D+02 | Balanced Panel

| Var[u] = .860646D+01 | T = 7

| Corr[v(i,t),v(i,s)] = .290444 |

| Lagrange Multiplier Test vs. Model (3) = 1561.57 |

| ( 1 df, prob value = .000000) |

| (High values of LM favor FEM/REM over CR model.) |

| Baltagi-Li form of LM Statistic = 1561.57 |

+--------------------------------------------------+

REGRESS ; Lhs=docvis ; Rhs=one,hhninc,age,female,educ ; panel ; pds=_groupti $

Testing for Effects: Moments



Wooldridge (page 265) suggests based on the off diagonal elements

T-1 T

N  t=1s=t+1eit eis

Z= i=1





 

2

N T-1 T

 i=1 t=1  e eis

s=t+1 it



which converges to standard normal. ("We are not assuming any

particular distribution for the it . Instead, we derive a similar test that

has the advantage of being valid for any distribution...") It's convenient

to examine Z 2 which, by the Slutsky theorem converges (also) to chi-

squared with one degree of freedom.

Testing (2)

T-1 T

 t=1 s=t+1eit eis = 1/2 of the sum of all off diagonal elements of

of eie or 1/2 the sum of all the elements minus the diagonal elements.

i



 t=1 s=t+1eit eis =1/2[i(eie)i  eei ]. But, iei = T ei . So,

T-1 T

i i



 t=1 s=t+1eit eis = (1/2)[(T ei )2  eei ]

T-1 T

i





 

2



2

N 1 [(T ei )2  eei ]

i i 2(T  1) [N 1eei ]2

Z   LM  i i

N 1 [(T ei )2  eei ]2

i i NT i1 [(T ei )  eei ]2

N 2

i



Note, also

N 1ri N r

Z= i

 , where ri  (T ei )2  eei .

i

N 1ri2

i

sr

The claim that one function of [ei , eei ]i1,...,N is more valid than the other

i



seems a little dubious.

Application: Cornwell-Rupert

Testing for Effects

? Obtain OLS residuals

Regress; lhs=lwage;rhs=fixedx,varyingx;res=e$

? Vector of group sums of residuals

Matrix ; tebar=7*gxbr(e,person)$

? Direct computation of LM statistic

Calc ; list;lm=595*7/(2*(7-1))*

(tebar'tebar/sumsqdev - 1)^2$

? Wooldridge chi squared (N(0,1) squared)

Create ; e2=e*e$

Matrix ; e2i=7*gxbr(e2,person)$

Matrix ; ri=dirp(tebar,tebar)-e2i$

Matrix ; sumri=ri'1$

Calc ; list;z2=(sumri)^2/ri'ri$



LM = .37970675705025540D+04

Z2 = .16533465085356830D+03

Two Way Random Effects Model

y it  x    ui  v t  it

it



How to estimate the variance components?

(1) Two way FEM residual variance estimates 2



2

(2) Simple OLS residual variance estimates 2  u  2

 v



(3) There are numerous ways to get a third equation.

E.g., the one way FEM residual variance in either dimension

One way FEM based on groups estimates (2  2 ) /(1  1 / T)

 v



E.g., the group mean regressions in either dimension.

2

Based on group means estimates u  (2  2 )/T

 v



(Period means regression may have a tiny number of observations.)

(And a whole library of others - see Baltagi, sec. 3.3.)

Negative estimators of common variances are common.

Solutions are complicated.

One Way REM

+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .0668628 |

| Var[u] = .0726096 |

| Corr[v(i,t),v(i,s)] = .520602 |

| Lagrange Multiplier Test vs. Model (3) = 3125.58 |

| ( 1 df, prob value = .000000) |

+--------------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|

+--------+--------------+----------------+--------+--------+----------+

WKS | .00159346 .00097991 1.626 .1039 46.8115246

OCC | -.11950533 .01947798 -6.135 .0000 .51116447

IND | .05312497 .01877477 2.830 .0047 .39543818

SOUTH | -.08108168 .02447416 -3.313 .0009 .29027611

SMSA | .06763259 .02070533 3.266 .0011 .65378151

MS | -.00957703 .02703049 -.354 .7231 .81440576

FEM | -.44918001 .04512643 -9.954 .0000 .11260504

UNION | .07865185 .01904248 4.130 .0000 .36398559

ED | .05247613 .00488996 10.731 .0000 12.8453782

BLK | -.13774497 .04737233 -2.908 .0036 .07226891

Constant| 5.98678270 .08881086 67.410 .0000

Two Way REM

+----------------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) + w(t) |

| Estimates: Var[e] = .0232838 (.0668628) |

| Var[u] = .0618548 (.0726096) |

| Var[w] = .0543338 (.0000000) |

| Corr[v(i,t),v(i,s)] = .726519 |

| Corr[v(i,t),v(j,t)] = .700019 |

| Lagrange Multiplier Test vs. Model (3) =93416.98 |

+----------------------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|

+--------+--------------+----------------+--------+--------+----------+

WKS | .00116797 .00059199 1.973 .0485 46.8115246

OCC | -.05385925 .01264036 -4.261 .0000 .51116447

IND | .03775548 .01300713 2.903 .0037 .39543818

SOUTH | -.07093664 .01957160 -3.624 .0003 .29027611

SMSA | .05803681 .01500299 3.868 .0001 .65378151

MS | -.00242497 .01748279 -.139 .8897 .81440576

FEM | -.44343656 .03740289 -11.856 .0000 .11260504

UNION | .04577568 .01290976 3.546 .0004 .36398559

ED | .05809106 .00412928 14.068 .0000 12.8453782

BLK | -.13918280 .04201095 -3.313 .0009 .07226891

Constant| 5.91602424 .11053610 53.521 .0000

Hausman Test for FE vs. RE





Estimator Random Effects Fixed Effects

E[ci|Xi] = 0 E[ci|Xi] ≠ 0

FGLS Consistent and Inconsistent

(Random Effects) Efficient

LSDV Consistent Consistent

(Fixed Effects) Inefficient Possibly Efficient

Hausman Test for Effects

ˆ ˆ

Basis for the test, βFE - βRE

ˆ ˆ ˆ

Wald Criterion: q = βFE - βRE ; W = q[Var(q)]-1q

ˆ ˆ ˆ

A lemma (Hausman (1978)): Under the null hypothesis (RE)

ˆ d



nT[βRE - β]  N[0,VRE ] (efficient)

ˆ d



nT[βFE - β]  N[0,VFE ] (inefficient)

ˆ ˆ ˆ

Note: q = (βFE - β)-(βRE  β). The lemma states that in the

ˆ

joint limiting distribution of nT[ βRE - β] and ˆ

nT q , the

limiting covariance, C Q,RE is 0. But, C Q,RE = CFE,RE - VRE . Then,

Var[q] = VFE + VRE - CFE,RE - C . Using the lemma, CFE,RE = VRE.

FE,RE



It follows that Var[q]=VFE - VRE. Based on the preceding

ˆ ˆ ˆ ˆ ˆ ˆ

H=(βFE - βRE ) [Est.Var(βFE ) - Est.Var(βRE )]-1 (βFE - βRE )





β does not contain the constant term in the preceding.

Computing the Hausman Statistic

1

 

ˆ ]  2 N X   I  1 ii  X 

Est.Var[βFE ˆ  i1 i   i



  Ti   

-1

 N  ˆi  

 Tiu

ˆ

2

ˆ

Est.Var[βRE ]   i1 X   I  ii  X i  , 0  ˆi = 2

ˆ

2

  1

i 2



  Ti      Tiu

ˆ ˆ

2 ˆ ˆ

As long as 2 and u are consistent, as N  , Est.Var[βFE ]  Est.Var[βRE ]

ˆ ˆ

will be nonnegative definite. In a finite sample, to ensure this, both must

be computed using the same estimate of 2 . The one based on LSDV will

ˆ

generally be the better choice.





ˆ

Note that columns of zeros will appear in Est.Var[βFE ] if there are time

invariant variables in X.



β does not contain the constant term in the preceding.

Hausman Test



+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .235236D-01 |

| Var[u] = .133156D+00 |

| Corr[v(i,t),v(i,s)] = .849862 |

| Lagrange Multiplier Test vs. Model (3) = 4061.11 |

| ( 1 df, prob value = .000000) |

| (High values of LM favor FEM/REM over CR model.) |

| Fixed vs. Random Effects (Hausman) = 2632.34 |

| ( 4 df, prob value = .000000) |

| (High (low) values of H favor FEM (REM).) |

+--------------------------------------------------+

A Variable Addition Test

 Asymptotic equivalent to Hausman

 Also equivalent to Mundlak formulation

 In the random effects model, using FGLS

 Only applies to time varying variables

 Add expanded group means to the regression (i.e.,

observation i,t gets same group means for all t.

 Use standard F or Wald test to test for coefficients

on means equal to 0. Large F or chi-squared weighs

against random effects specification.

Application

Sample ; all$

? Fixed Effects Regression. Note, ED,BLK,FEM are time invariant.

regress; lhs = lwage

; rhs = wks,occ,ind,south,smsa,union,exp,expsq,ed,blk,fem

; pds = 7;panel;fixed$

? Pick up coefficients and variance from fixed effects model.

matrix; bf=b(1:8) ; vf=varb(1:8,1:8)$

? Random Effects Regression. Computed automatically.

regress; lhs = lwage

; rhs = wks,occ,ind,south,smsa,union,exp,expsq,ed,blk,fem

; pds = 7;panel$

? Pick up REM coefficients and variance then compute Hausman statistic

matrix; br=b(1:8) ; vr=varb(1:8,1:8)$

matrix ; db = bf - br ; dv = vf - vr $

matrix ; list ; h =db'db$

Fixed Effects

+----------------------------------------------------+

| Panel:Groups Empty 0, Valid data 595 |

| Smallest 7, Largest 7 |

| Average group size 7.00 |

| There are 3 vars. with no within group variation. |

| ED BLK FEM |

| Look for huge standard errors and fixed parameters.|

| F.E. results are based on a generalized inverse. |

| They will be highly erratic. (Problematic model.) |

| Unable to compute std.errors for dummy var. coeffs.|

+----------------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|

+--------+--------------+----------------+--------+--------+----------+

|WKS | .00083 .00060003 1.381 .1672 46.811525|

|OCC | -.02157 .01379216 -1.564 .1178 .5111645|

|IND | .01888 .01545450 1.221 .2219 .3954382|

|SOUTH | .00039 .03429053 .011 .9909 .2902761|

|SMSA | -.04451** .01939659 -2.295 .0217 .6537815|

|UNION | .03274** .01493217 2.192 .0283 .3639856|

|EXP | .11327*** .00247221 45.819 .0000 19.853782|

|EXPSQ | -.00042*** .546283D-04 -7.664 .0000 514.40504|

|ED | .000 ......(Fixed Parameter)....... |

|BLK | .000 ......(Fixed Parameter)....... |

|FEM | .000 ......(Fixed Parameter)....... |

+--------+------------------------------------------------------------+

Random Effects

+--------------------------------------------------+

| Random Effects Model: v(i,t) = e(i,t) + u(i) |

| Estimates: Var[e] = .231312D-01 |

| Var[u] = .990559D-01 |

| Corr[v(i,t),v(i,s)] = .810690 |

| Lagrange Multiplier Test vs. Model (3) = 3543.36 |

| ( 1 df, prob value = .000000) |

| (High values of LM favor FEM/REM over CR model.) |

+--------------------------------------------------+

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|

+--------+--------------+----------------+--------+--------+----------+

|WKS | .00094 .00059308 1.586 .1128 46.811525|

|OCC | -.04367*** .01299206 -3.361 .0008 .5111645|

|IND | .00271 .01373256 .197 .8434 .3954382|

|SOUTH | -.00664 .02246416 -.295 .7677 .2902761|

|SMSA | -.03117* .01615455 -1.930 .0536 .6537815|

|UNION | .05802*** .01349982 4.298 .0000 .3639856|

|EXP | .08744*** .00224705 38.913 .0000 19.853782|

|EXPSQ | -.00076*** .495876D-04 -15.411 .0000 514.40504|

|ED | .10724*** .00511463 20.967 .0000 12.845378|

|BLK | -.21178*** .05252013 -4.032 .0001 .0722689|

|FEM | -.24786*** .04283536 -5.786 .0000 .1126050|

|Constant| 3.97756*** .08178139 48.637 .0000 |

+--------+------------------------------------------------------------+

Hausman Test

--> matrix; br=b(1:8) ; vr=varb(1:8,1:8)$

--> matrix ; db = bf - br ; dv = vf - vr $

--> matrix ; list ; h =db'db$



Matrix H has 1 rows and 1 columns.

1

+--------------

1| 2523.64910



--> calc;list;ctb(.95,8)$

+------------------------------------+

| Listed Calculator Results |

+------------------------------------+

Result = 15.507313

Application: Wu Test

? Compute group means for Wu's variable addition test.

create ; wksb=groupmean(wks,pds=7)$

create ; occb=groupmean(occ,pds=7)$

create ; indb=groupmean(ind,pds=7)$

create ; southb=groupmean(south,pds=7)$

create ; smsab=groupmean(smsa,pds=7)$

create ; unionb=groupmean(union,pds=7)$

create ; expb=groupmean(exp,pds=7)$

create ; expsqb=groupmean(expsq,pds=7)$

? Random effects model with group means of time varying variables

regress; lhs = lwage

; rhs = wks,occ,ind,south,smsa,union,exp,expsq,ed,blk,fem,

wksb,occb,indb,southb,smsab,unionb,expb,expsqb

; pds=7;panel$

? Wald test of hypothesis that coefficients on the means are zero.

matrix ; bm=b(12:19);vm=varb(12:19,12:19)$

matrix ; list ; wu = bm'bm $

Means Added

+--------+--------------+----------------+--------+--------+----------+

|Variable| Coefficient | Standard Error |b/St.Er.|P[|Z|>z]| Mean of X|

+--------+--------------+----------------+--------+--------+----------+

|WKS | .00083 .00060070 1.380 .1677 46.811525|

|OCC | -.02157 .01380769 -1.562 .1182 .5111645|

|IND | .01888 .01547189 1.220 .2224 .3954382|

|SOUTH | .00039 .03432914 .011 .9909 .2902761|

|SMSA | -.04451** .01941842 -2.292 .0219 .6537815|

|UNION | .03274** .01494898 2.190 .0285 .3639856|

|EXP | .11327*** .00247500 45.768 .0000 19.853782|

|EXPSQ | -.00042*** .546898D-04 -7.655 .0000 514.40504|

|ED | .05199*** .00552893 9.404 .0000 12.845378|

|BLK | -.16983*** .04456572 -3.811 .0001 .0722689|

|FEM | -.41306*** .03732204 -11.067 .0000 .1126050|

|WKSB | .00863** .00363907 2.371 .0177 46.811525|

|OCCB | -.14656*** .03640885 -4.025 .0001 .5111645|

|INDB | .04142 .02976363 1.392 .1640 .3954382|

|SOUTHB | -.05551 .04297816 -1.292 .1965 .2902761|

|SMSAB | .21607*** .03213205 6.724 .0000 .6537815|

|UNIONB | .08152** .03266438 2.496 .0126 .3639856|

|EXPB | -.08005*** .00533603 -15.002 .0000 19.853782|

|EXPSQB | -.00017 .00011763 -1.416 .1567 514.40504|

|Constant| 5.19036*** .20147201 25.762 .0000 |

+--------+------------------------------------------------------------+

Wu (Variable Addition) Test



--> matrix ; bm=b(12:19);vm=varb(12:19,12:19)$

--> matrix ; list ; wu = bm'bm $



Matrix WU has 1 rows and 1 columns.

1

+--------------

1| 3004.38076

Basing Wu Test on a Robust VC

? Robust Covariance matrix for REM

Namelist ; XWU = wks,occ,ind,south,smsa,union,exp,expsq,ed,blk,fem,

wksb,occb,indb,southb,smsab,unionb,expb,expsqb,one $

Create ; ewu = lwage - xwu'b $

Matrix ; Robustvc = *Gmmw(xwu,ewu,_stratum)*

; Stat(b,RobustVc,Xwu) $

Matrix ; Means = b(12:19);Vmeans=RobustVC(12:19,12:19)

; List ; RobustW=Means'Means $

Robust Standard Errors

+--------+--------------+----------------+ +--------+--------------+

|Variable| Coefficient | Standard Error | |Variable|Standard Error|

+--------+--------------+----------------+ +--------+--------------+

|WKS | .00083 .00060070 |WKS |.00086319

|OCC | -.02157 .01380769 |OCC |.01877775 Matrix ROBUSTW has

|IND | .01888 .01547189 |IND |.02268776

1 rows and 1

|SOUTH | .00039 .03432914 |SOUTH |.08889931

|SMSA | -.04451** .01941842 |SMSA |.02947711 columns.

|UNION | .03274** .01494898 |UNION |.02500498 2253.97002

|EXP | .11327*** .00247500 |EXP |.00404305

|EXPSQ | -.00042*** .546898D-04 |EXPSQ |.823659D-04

|ED | .05199*** .00552893 |ED |.00587385

|BLK | -.16983*** .04456572 |BLK |.04488235

|FEM | -.41306*** .03732204 |FEM |.03062920

|WKSB | .00863** .00363907 |WKSB |.00359870

|OCCB | -.14656*** .03640885 |OCCB |.03800966

|INDB | .04142 .02976363 |INDB |.03503648

|SOUTHB | -.05551 .04297816 |SOUTHB |.09321915

|SMSAB | .21607*** .03213205 |SMSAB |.03863529

|UNIONB | .08152** .03266438 |UNIONB |.03857305

|EXPB | -.08005*** .00533603 |EXPB |.00616803

|EXPSQB | -.00017 .00011763 |EXPSQB |.00013248

|Constant| 5.19036*** .20147201 |Constant|.20601042

+--------+------------------------------ +--------+----------

Fixed vs. Random Effects

-1

  

ˆ     

ˆ  

ˆ

βModel  N 1 X   I  i,Model ii  X i  N 1 X   I  i,Model ii  y i 

i i i i



  Ti       Ti   



ˆModel  1 for fixed effects.

2

Tiu

ˆ



ˆi,Model  2 2

for random effects.

  Tiu

ˆ ˆ



As Ti  , ˆi,RE  1, random effects becomes fixed effects

2



As u  0, ˆi,RE  0, random effects becomes OLS (of course)

ˆ

2



As u  , ˆi,RE  1, random effects becomes fixed effects

ˆ

2

For the C&R application, u =.133156, 2 =.0235231, ˆ  .975384.

ˆ ˆ 

Looks like a fixed effects model. Note the Hausman statistic agrees.





β does not contain the constant term in the preceding.