1、Econometric Analysis of Panel Data,William GreeneDepartment of EconomicsStern School of Business,Econometric Analysis of Panel Data,6. Maximum Likelihood Estimation of the Random Effects Linear Model,The Random Effects Model,The random effects modelci is uncorrelated with xit for all t; Eci |Xi = 0E
2、it|Xi,ci=0,Error Components Model,Generalized Regression Model,Notation,Maximum Likelihood,Panel Data Algebra (3),Panel Data Algebra (3, cont.),Panel Data Algebra (3, conc.),Maximizing the Likelihood,Difficult: “Brute force” + some elegant theoretical results: See Baltagi, pp. 20-21. (Back and forth
3、 from GLS to 2 and u2.)Somewhat less difficult and more practical: At any iteration, given estimates of 2 and u2 the estimator of is GLS (of course), so we iterate back and forth between these. See Hsiao, pp. 39-40.,Direct Maximization of LogL,Application ML vs. FGLS,Maximum Simulated Likelihood,Lik
4、elihood Function for Individual i,Log Likelihood Function,Computing the Expected LogL,Example: Hermite Quadrature Nodes and Weights, H=5Nodes: -2.02018,-0.95857, 0.00000, 0.95857, 2.02018Weights: 1.99532,0.39362, 0.94531, 0.39362, 1.99532Applications usually use many more points, up to 96 andMuch mo
5、re accurate (more digits) representations.,Quadrature,Gauss-Hermite Quadrature,Simulation,Convergence Results,MSL vs. ML - Application,Two Level Panel Data,Nested by constructionUnbalanced panelsNo real obstacle to estimationSome inconvenient algebra.In 2 step FGLS of the RE, need “1/T” to solve for
6、 an estimate of u2. What to use?,Balanced Nested Panel Data,Zi,j,k,t = test score for student t, teacher k, school j, district iL = 2 school districts, i = 1,LMi = 3 schools in each district, j = 1,MiNij = 4 teachers in each school, k = 1,NijTijk = 20 students in each class, t = 1,Tijk,Antweiler, W.
7、, “Nested Random 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 intractab
8、le.The log likelihood is very tractableNote a collision of practicality with nonrobustness. (Normality must be assumed.),Log Likelihood (1),Log Likelihood (2),Maximizing Log L,Antweiler provides analytic first derivatives for gradient methods of optimization. Ugly to program.Numerical derivatives:,A
9、symptotic Covariance Matrix,An Appropriate Asymptotic Covariance Matrix,Some Observations,Assuming the wrong (e.g., nonnested) error structureStill consistent GLS with the wrong weightsStandard errors (apparently) biased downward (Moulton bias)Adding “time” effects or other nonnested effects is “ver
10、y challenging.” Perhaps do with “fixed” effects (dummy variables).,An Application,Y1jkt = log of atmospheric sulfur dioxide concentration at observation station k at time t, in country i.H = 2621, 293 stations, 44 countries, various numbers of observations, not equally spacedThree levels, not 4 as i
11、n article.Xjkt =1,log(GDP/km2),log(K/L),log(Income), Suburban, Rural,Communist,log(Oil price), average temperature, time trend.,Estimates,Rotating Panel-1,The structure of the sample and selection of individuals in a rotating sampling design are as follows: Let all individuals in the population be n
12、umbered consecutively. The sample in period 1 consists of N, individuals. In period 2, a fraction, met (0 me2 N1) of the sample in period 1 are replaced by mi2 new individuals from the population. In period 3 another fraction of the sample in the period 2, me2 (0 me2 N2) individuals are replaced by
13、mi3 new individuals and so on. Thus the sample size in period t is Nt = Nt-1 - met-1 + mii . The procedure of dropping met-1 individuals selected in period t - 1 and replacing them by mit individuals from the population in period t is called rotating sampling. In this framework total number of obser
14、vations and individuals observed are tNt and N1 + t=2 to Tmit respectively.Heshmati, A,“Efficiency measurement in rotating panel data,” Applied Economics, 30, 1998, pp. 919-930,Rotating Panel-2,The outcome of the rotating sample for farms producing dairy products is given in Table 1. Each of the ann
15、ual sample is composed of four parts or subsamples. For example, in 1980 the sample contains 79, 62, 98, and 74 farms. The first three parts (79, 62, and 98) are those not replaced during the transition from 1979 to 1980. The last subsample contains 74 newly included farms from the population. At th
16、e same time 85 farms are excluded from the sample in 1979. The difference between the excluded part (85) and the included part (74) corresponds to the changes in the rotating sample size between these two periods, i.e. 313-324 = -11. This difference includes only the part of the sample where each fa
17、rm is observed consecutively for four years, Nrot. The difference in the non-rotating part, N, is due to those farms which are not observed consecutively. The proportion of farms not observed consecutively, Nnon in the total annual sample, Nnon varies from 11.2 to 22.6% with an average of 18.7 per c
18、ent.,Rotating Panels-3,Simply an unbalanced panel Treat with the familiar techniquesAccounting is complicatedTime effects may be complicated.Biorn and Jansen (Scand. J. E., 1983) households cohort 1 has T = 1976,1977 while cohort 2 has T=1977,1978.But, “Time in sample bias” may require special treatment. Mexican labor survey has 3 periods rotation. Some families in 1 or 2 or 3 periods.,Pseudo Panels,
