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Econometric Analysis of Panel Data. William Greene Department of Economics Stern School of Business. 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[ ε it | X i ,c i ]=0. Random vs. Fixed Effects.

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Econometric Analysis of Panel Data

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Econometric analysis of panel data l.jpg

Econometric Analysis of Panel Data

William Greene

Department of Economics

Stern School of Business


The random effects model l.jpg

The Random Effects Model

  • The random effects model

  • ci is uncorrelated with xit for all t;

    E[ci |Xi] = 0

    E[εit|Xi,ci]=0


Random vs fixed effects l.jpg

Random vs. Fixed Effects

  • Random Effects

    • Small number of parameters

    • Efficient estimation

    • Objectionable orthogonality assumption (ciXi)

  • Fixed Effects

    • Robust – generally consistent

    • Large number of parameters

    • More reasonable assumption

    • Precludes time invariant regressors 

  • Which is the more reasonable model?


Error components model l.jpg

Error Components Model

Generalized Regression Model


Notation l.jpg

Notation


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Notation


Regression model orthogonality l.jpg

Regression Model-Orthogonality


Convergence of moments l.jpg

Convergence of Moments


Ordinary least squares l.jpg

Ordinary Least Squares

  • Standard results for OLS in a GR model

    • Consistent

    • Unbiased

    • Inefficient

  • True Variance


Estimating the variance for ols l.jpg

Estimating the Variance for OLS


Mechanics l.jpg

Mechanics


Cornwell and rupert data l.jpg

Cornwell and Rupert Data

Cornwell and Rupert Returns to Schooling Data, 595 Individuals, 7 YearsVariables in the file are

EXP = work experience, EXPSQ = EXP2WKS = weeks workedOCC = occupation, 1 if blue collar, IND = 1 if manufacturing industrySOUTH = 1 if resides in southSMSA= 1 if resides in a city (SMSA)MS = 1 if marriedFEM = 1 if femaleUNION = 1 if wage set by unioin contractED = years of educationLWAGE = log of wage = dependent variable in regressions

These data were analyzed in Cornwell, C. and Rupert, P., "Efficient Estimation with Panel Data: An Empirical Comparison of Instrumental Variable Estimators," Journal of Applied Econometrics, 3, 1988, pp. 149-155.  See Baltagi, page 122 for further analysis.  The data were downloaded from the website for Baltagi's text.


Alternative ols variance estimators l.jpg

Alternative OLS 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 l.jpg

Generalized Least Squares


Panel data algebra 1 l.jpg

Panel Data Algebra (1)


Panel data algebra 2 l.jpg

Panel Data Algebra (2)


Panel data algebra 3 l.jpg

Panel Data Algebra (3)


Gls cont l.jpg

GLS (cont.)


Estimators for the variances l.jpg

Estimators for the Variances


Feasible gls l.jpg

Feasible GLS

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


Practical problems with fgls l.jpg

Practical Problems with FGLS


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Stata Variance Estimators


Computing variance estimators l.jpg

Computing Variance Estimators


Application l.jpg

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 an lm test l.jpg

Testing for Effects: An LM Test


Lm tests l.jpg

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 $


Testing for effects moments l.jpg

Testing for Effects: Moments


Testing 2 l.jpg

Testing (2)


Testing for effects l.jpg

Testing for Effects

? Obtain OLS residuals

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

? Vector of group sums of residuals

Calc; T = 7 ; Groups = 595

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

? Direct computation of LM statistic

Calc; list;lm=Groups*T/(2*(T-1))*

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

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

Create; e2=e*e$

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

Matrix; ri=dirp(tebar,tebar)-e2i ; sumri=ri'1$

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

LM = .37970675705025540D+04

Z2 = .16533465085356830D+03


Two way random effects model l.jpg

Two Way Random Effects Model


One way rem l.jpg

One Way REM


Two way rem l.jpg

Two Way REM

Note sum = .102705


Hausman test for fe vs re l.jpg

Hausman Test for FE vs. RE


Hausman test for effects l.jpg

Hausman Test for Effects

β does not contain the constant term in the preceding.


Computing the hausman statistic l.jpg

Computing the Hausman Statistic

β does not contain the constant term in the preceding.


Hausman test l.jpg

Hausman Test?

What went wrong?

The matrix is not positive definite.

It has a negative characteristic root.

The matrix is indefinite.

(Software such as Stata and NLOGIT find this problem and refuse to proceed.)

Properly, the statistic cannot be computed.

The naïve calculation came out positive by the luck of the draw.


A variable addition test l.jpg

A Variable Addition Test

  • Asymptotically 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.


Variable addition l.jpg

Variable Addition


Application wu test l.jpg

Application: Wu Test

NAMELIST ; XV = exp,expsq,wks,occ,ind,south,smsa,ms,union,ed,fem$

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

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

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 ; msb = groupmean(ms,pds=7) $

namelist ; xmeans = expb,expsqb,wksb,occb,indb,southb,smsab,msb,

unionb $

REGRESS ; Lhs = lwage ; Rhs = xmeans,Xv,one ; panel ; random $

MATRIX ; bmean = b(1:9) ; vmean = varb(1:9,1:9) $

MATRIX ; List ; Wu = bmean'<vmean>bmean $


Means added l.jpg

Means Added


Wu variable addition test l.jpg

Wu (Variable Addition) Test


Basing wu test on a robust vc l.jpg

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 = <Xwu'Xwu>*Gmmw(xwu,ewu,_stratum)*<XwU'xWU>

; Stat(b,RobustVc,Xwu) $

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

; List ; RobustW=Means'<Vmeans>Means $


Slide45 l.jpg

Robust Standard Errors


Fixed vs random effects l.jpg

Fixed vs. Random Effects

β does not contain the constant term in the preceding.


Another comparison baltagi p 20 l.jpg

Another Comparison (Baltagi p. 20)


A hierarchical linear model interpretation of the fe model l.jpg

A Hierarchical Linear ModelInterpretation of the FE Model


Hierarchical linear model as rem l.jpg

Hierarchical Linear Model as REM

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

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

| Estimates: Var[e] = .235368D-01 |

| Var[u] = .110254D+00 |

| Corr[v(i,t),v(i,s)] = .824078 |

| Sigma(u) = 0.3303 |

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

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

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

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

OCC | -.03908144 .01298962 -3.009 .0026 .51116447

SMSA | -.03881553 .01645862 -2.358 .0184 .65378151

MS | -.06557030 .01815465 -3.612 .0003 .81440576

EXP | .05737298 .00088467 64.852 .0000 19.8537815

FEM | -.34715010 .04681514 -7.415 .0000 .11260504

ED | .11120152 .00525209 21.173 .0000 12.8453782

Constant| 4.24669585 .07763394 54.702 .0000


Evolution correlated random effects l.jpg

Evolution: Correlated Random Effects


Mundlak s estimator l.jpg

Mundlak’s Estimator

Mundlak, Y., “On the Pooling of Time Series and Cross Section Data, Econometrica, 46, 1978, pp. 69-85.


Correlated random effects l.jpg

Correlated Random Effects


Mundlak s approach for an fe model with time invariant variables l.jpg

Mundlak’s Approach for an FE Model with Time Invariant Variables


Mundlak form of fe model l.jpg

Mundlak Form of FE Model

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

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

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

x(i,t)=================================================================

OCC | -.02021384 .01375165 -1.470 .1416 .51116447

SMSA | -.04250645 .01951727 -2.178 .0294 .65378151

MS | -.02946444 .01915264 -1.538 .1240 .81440576

EXP | .09665711 .00119262 81.046 .0000 19.8537815

z(i)===================================================================

FEM | -.34322129 .05725632 -5.994 .0000 .11260504

ED | .05099781 .00575551 8.861 .0000 12.8453782

Means of x(i,t) and constant===========================================

Constant| 5.72655261 .10300460 55.595 .0000

OCCB | -.10850252 .03635921 -2.984 .0028 .51116447

SMSAB | .22934020 .03282197 6.987 .0000 .65378151

MSB | .20453332 .05329948 3.837 .0001 .81440576

EXPB | -.08988632 .00165025 -54.468 .0000 19.8537815

Variance Estimates=====================================================

Var[e]| .0235632

Var[u]| .0773825


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