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Econometric Analysis of Panel Data

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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  1. Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

  2. 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

  3. 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?

  4. Error Components Model Generalized Regression Model

  5. Notation

  6. Notation

  7. Regression Model-Orthogonality

  8. Convergence of Moments

  9. Ordinary Least Squares • Standard results for OLS in a GR model • Consistent • Unbiased • Inefficient • True Variance

  10. Estimating the Variance for OLS

  11. Mechanics

  12. 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.

  13. 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

  14. Generalized Least Squares

  15. Panel Data Algebra (1)

  16. Panel Data Algebra (2)

  17. Panel Data Algebra (3)

  18. GLS (cont.)

  19. Estimators for the Variances

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

  21. Practical Problems with FGLS

  22. Stata Variance Estimators

  23. Computing Variance Estimators

  24. 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

  25. Testing for Effects: An LM Test

  26. 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 $

  27. Testing for Effects: Moments

  28. Testing (2)

  29. 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

  30. Two Way Random Effects Model

  31. One Way REM

  32. Two Way REM Note sum = .102705

  33. Hausman Test for FE vs. RE

  34. Hausman Test for Effects β does not contain the constant term in the preceding.

  35. Computing the Hausman Statistic β does not contain the constant term in the preceding.

  36. 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.

  37. 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.

  38. Variable Addition

  39. 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 $

  40. Means Added

  41. Wu (Variable Addition) Test

  42. 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 $

  43. Robust Standard Errors

  44. Fixed vs. Random Effects β does not contain the constant term in the preceding.

  45. Another Comparison (Baltagi p. 20)

  46. A Hierarchical Linear ModelInterpretation of the FE Model

  47. 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

  48. Evolution: Correlated Random Effects

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