Part 6: MLE for RE Models [ 1/38] Econometric Analysis of Panel Data William Greene Department of Economics Stern School of Business

Part 6: MLE for RE Models [ 2/38] 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

Part 6: MLE for RE Models [ 3/38] Error Components Model Generalized Regression Model

Part 6: MLE for RE Models [ 4/38] Notation

Part 6: MLE for RE Models [ 5/38] Maximum Likelihood

Part 6: MLE for RE Models [ 6/38] MLE Panel Data Algebra (1)

Part 6: MLE for RE Models [ 7/38] MLE Panel Data Algebra (1, cont.)

Part 6: MLE for RE Models [ 8/38] MLE Panel Data Algebra (1, conc.)

Part 6: MLE for RE Models [ 9/38] Maximizing the Log Likelihood  Difficult: “Brute force” + some elegant theoretical results: See Baltagi, pp (Back and forth from GLS to  ε 2 and  u 2.)  Somewhat less difficult and more practical: At any iteration, given estimates of  ε 2 and  u 2 the estimator of  is GLS (of course), so we iterate back and forth between these. See Hsiao, pp

Part 6: MLE for RE Models [ 10/38] Direct Maximization of LogL

Part 6: MLE for RE Models [ 11/38] 

Part 6: MLE for RE Models [ 12/38]

Part 6: MLE for RE Models [ 13/38] Maximum Simulated Likelihood

Part 6: MLE for RE Models [ 14/38] Likelihood Function for Individual i

Part 6: MLE for RE Models [ 15/38] Log Likelihood Function

Part 6: MLE for RE Models [ 16/38] Computing the Expected LogL Example: Hermite Quadrature Nodes and Weights, H=5 Nodes: , , , , Weights: , , , , Applications usually use many more points, up to 96 and Much more accurate (more digits) representations.

Part 6: MLE for RE Models [ 17/38] Quadrature

Part 6: MLE for RE Models [ 18/38] Gauss-Hermite Quadrature

Part 6: MLE for RE Models [ 19/38] Simulation

Part 6: MLE for RE Models [ 20/38] Convergence Results

Part 6: MLE for RE Models [ 21/38] MSL vs. ML =

Part 6: MLE for RE Models [ 22/38] 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 for an estimate of σ u 2. What to use?

Part 6: MLE for RE Models [ 23/38] Balanced Nested Panel Data Z i,j,k,t = test score for student t, teacher k, school j, district i L = 2 school districts, i = 1,…,L M i = 3 schools in each district, j = 1,…,M i N ij = 4 teachers in each school, k = 1,…,N ij T ijk = 20 students in each class, t = 1,…,T ijk Antweiler, W., “Nested Random Effects Estimation in Unbalanced Panel Data,” Journal of Econometrics, 101, 2001, pp

Part 6: MLE for RE Models [ 24/38] Nested Effects Model

Part 6: MLE for RE Models [ 25/38] GLS with Nested Effects

Part 6: MLE for RE Models [ 26/38] Unbalanced Nested Data  With unbalanced panels, all the preceding results fall apart.  GLS, FGLS, even fixed effects become analytically intractable.  The log likelihood is very tractable  Note a collision of practicality with nonrobustness. (Normality must be assumed.)

Part 6: MLE for RE Models [ 27/38] Log Likelihood (1)

Part 6: MLE for RE Models [ 28/38] Log Likelihood (2)

Part 6: MLE for RE Models [ 29/38] Maximizing Log L  Antweiler provides analytic first derivatives for gradient methods of optimization. Ugly to program.  Numerical derivatives:

Part 6: MLE for RE Models [ 30/38] Asymptotic Covariance Matrix

Part 6: MLE for RE Models [ 31/38] An Appropriate Asymptotic Covariance Matrix

Part 6: MLE for RE Models [ 32/38] Some Observations  Assuming the wrong (e.g., nonnested) error structure Still consistent – GLS with the wrong weights Standard errors (apparently) biased downward (Moulton bias)  Adding “time” effects or other nonnested effects is “very challenging.” Perhaps do with “fixed” effects (dummy variables).

Part 6: MLE for RE Models [ 33/38] An Application  Y 1jkt = 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 spaced  Three levels, not 4 as in article.  X jkt =1,log(GDP/km 2 ),log(K/L),log(Income), Suburban, Rural,Communist,log(Oil price), average temperature, time trend.

Part 6: MLE for RE Models [ 34/38] Estimates

Part 6: MLE for RE Models [ 35/38] 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 numbered consecutively. The sample in period 1 consists of N, individuals. In period 2, a fraction, me t (0 < me 2 < N 1 ) of the sample in period 1 are replaced by mi 2 new individuals from the population. In period 3 another fraction of the sample in the period 2, me 2 (0 < me 2 < N 2 ) individuals are replaced by mi 3 new individuals and so on. Thus the sample size in period t is N t = {N t-1 - me t-1 + mi i }. The procedure of dropping me t-1 individuals selected in period t - 1 and replacing them by mi t individuals from the population in period t is called rotating sampling. In this framework total number of observations and individuals observed are Σ t N t and N 1 + Σ t=2 to T mi t respectively. Heshmati, A,“Efficiency measurement in rotating panel data,” Applied Economics, 30, 1998, pp

Part 6: MLE for RE Models [ 36/38] Rotating Panel-2 The outcome of the rotating sample for farms producing dairy products is given in Table 1. Each of the annual 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 The last subsample contains 74 newly included farms from the population. At the same time 85 farms are excluded from the sample in 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 = -11. This difference includes only the part of the sample where each farm is observed consecutively for four years, N rot. 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, N non in the total annual sample, N non varies from 11.2 to 22.6% with an average of 18.7 per cent.

Part 6: MLE for RE Models [ 37/38] Rotating Panels-3  Simply an unbalanced panel Treat with the familiar techniques Accounting is complicated  Time 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.

Part 6: MLE for RE Models [ 38/38] Pseudo Panels