Econometric Analysis of Panel Data William Greene Department of Economics Stern
Szoke, Marika, Executive Producer has reference to this Academic Journal, PHwiki organized this Journal Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business The R in addition to om Effects Model The r in addition to om effects model ci is uncorrelated with xit as long as all t; E[ci Xi] = 0 E[itXi,ci]=0 Error Components Model Generalized Regression Model
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Notation Maximum Likelihood MLE Panel Data Algebra Second term in LogLi
MLE Panel Data Algebra (1) Third term in LogLi MLE Panel Data Algebra (1) Combine terms in logLi Maximizing the Log Likelihood by Iterated FGLS Difficult: Brute as long as ce + some elegant theoretical results: See Baltagi, pp. 22-23. (Back in addition to as long as th from GLS to 2 in addition to u2.) Somewhat less difficult in addition to more practical: At any iteration, given estimates of 2 in addition to u2 the estimator of is GLS (of course), so we iterate back in addition to as long as th between these. See Hsiao, pp. 39-40.
Direct Maximization of LogL by Nonlinear Optimization FGLS vs. MLE. The estimates of u2 are quite different.
Maximum Simulated Likelihood Likelihood Function as long as Individual i Log Likelihood Function
Computing the Expected LogL Example: Hermite Quadrature Nodes in addition to Weights, H=5 Nodes: -2.02018,-0.95857, 0.00000, 0.95857, 2.02018 Weights: 1.99532,0.39362, 0.94531, 0.39362, 1.99532 Applications usually use many more points, up to 96 in addition to Much more accurate (more digits) representations. Quadrature 32 Point Hermite Quadrature Nodes 0.194840741569399326708741289532, 0.584978765435932448466957544011, 0.976500463589682838484704871982, 1.37037641095287183816170564864, 1.76765410946320160462767325853, 2.16949918360611217330570559502, 2.57724953773231745403092930114, 2.99249082500237420628549407606, 3.41716749281857073587392729564, 3.85375548547144464388787292109, 4.30554795335119844526348653193, 4.77716450350259639303579405689, 5.27555098651588012781906048140, 5.81222594951591383276596615366, 6.40949814926966041217376374153, 7.12581390983072757279520760342/ Weights 3.75238352592802392866818389D-1, 2.77458142302529898137698919D-1, 1.51269734076642482575147115D-1, 6.04581309559126141865857608D-2, 1.75534288315734303034378446D-2, 3.65489032665442807912565712D-3, 5.36268365527972045970238102D-4, 5.41658406181998255800193939D-5, 3.65058512956237605737032419D-6, 1.57416779254559402926869258D-7, 4.09883216477089661823504101D-9, 5.93329146339663861451156822D-11, 4.21501021132644757296944521D-13,1.19734401709284866582868190D-15, 9.23173653651829223349442007D-19,7.31067642738416239327427846D-23/
Compute the Integral by Simulation Convergence Results MSL vs. ML FGLS MLE MSL 2 .023119 .023534 .023779 u2 .102531 .708869 .576658
Two Level Panel Data Nested by construction Unbalanced panels No real obstacle to estimation Some inconvenient algebra. In 2 step FGLS of the RE, need 1/T to solve as long as an estimate of u2. What to use Balanced Nested Panel Data Zi,j,k,t = test score as long as student t, teacher k, school j, district i L = 2 school districts, i = 1, ,L Mi = 3 schools in each district, j = 1, ,Mi Nij = 4 teachers in each school, k = 1, ,Nij Tijk = 20 students in each class, t = 1, ,Tijk Antweiler, W., Nested R in addition to om Effects Estimation in Unbalanced Panel Data, Journal of Econometrics, 101, 2001, pp. 295-313. Nested Effects Model
GLS with Nested Effects Unbalanced Nested Data With unbalanced panels, all the preceding results fall apart. GLS, FGLS, even fixed effects become analytically intractable. (Unless you just compute all the dummy variables.) The log likelihood is very tractable Note a collision of practicality with nonrobustness. (Normality must be assumed.) Log Likelihood (1)
Log Likelihood (2) Maximizing Log L Antweiler provides analytic first derivatives as long as gradient methods of optimization. Ugly to program. Numerical derivatives: Asymptotic Covariance Matrix
Estimates Gauss-Hermite Quadrature Change of Variable
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