Some Monte Carlo Evidence

This section presents results from a limited set of Monte Carlo experiments. In each experiment with fixed effect estimators, 25 samples of 100 individuals (/ = 100) are selected for 8 periods (T = 8). Results are presented for both a probit model with only exogenous explanatory variables and the first-order Markov model. The exogenous variables are assumed to follow a Nerlove process (Nerlove 1971):

Z{i,t) = 0.1? + 0.5Z(i,t — 1) + co{i,t), where co(i, t) is a uniform random variable with mean zero and range —1/2 to 1/2. This process well approximates the age-trended variables found in many microdata panel sets, especially in labor market analysis.

Results for the multivariate probit model with fixed effect but without lagged dummy variables are presented in table 4.1. Maximum likelihood fixed effect estimates are presented in the first three rows. The model generating the data is given at the bottom of the table. Samples are generated from a normal random number generator. The variance of U(i, t) is set at one. The variance of the fixed effect, of, is changed for different experiments. For = 0.1 the fixed effect estimator does well. The estimated value (denoted 0) comes very close to the true value. For /? = — 1, or /? = 1, the estimator does not perform as well, but the bias is never more than 10 percent and is always toward zero. As the variance in the fixed effects decreases, so does the bias.12

These results are consistent with the findings of Wright and Douglas (1976) who use Monte Carlo methods to investigate the performance of the fixed effect logit estimator for the Rasch-Andersen model. In a study with panels of length T = 20 per person, they find that the fixed effect logit

12. One unusual feature of these experiments is the consistent finding of a bias toward zero when a bias occurs. Andersen's two-period analysis would suggest an upward bias. The exogenous variables in the model investigated in the text have a much more complex character than the simple treatment effect variable used by Andersen.

estimator is virtually unbiased and its distribution is well described by a limiting normal distribution with variance-covariance matrix based on the estimated information matrix.

To judge the importance of the bias, one requires a benchmark. The benchmark selected in this chapter is a random effect estimator that integrates out the fixed effect. This is a multivariate probit model with random effect as presented in Heckman and Willis (1975) or chapter 3, section 3.5.

For each set of parameter values 25 samples with 100 observations of 3 periods are generated. The random effect estimator with T = 3 costs roughly the same to compute as the fixed effect estimator with T — 8. In terms of computational cost, the two estimators are equivalent.

The results with this model are presented in the final two rows of table 4.1. For a variance of of = 3, the random effect estimator displays more bias than the fixed effect estimator. For of = 1, the two estimators do about equally well. These experiments suggest that there is no clear ranking of the two estimators.

Test statistics for the random effect estimator (not given in the table) based on the estimated information matrix lead to rejection of the false null hypothesis that /? = 0 far more often than test statistics based on the information matrix for the fixed effect estimator. On the basis of this limited evidence, if the estimators are to be used to make inference, the random effect estimator seems preferable.

Next consider some Monte Carlo experiments with the fixed effect estimator for a first-order Markov process. The results from these experiments are displayed in the first part of table 4.2. The same Nerlove process that generates the exogenous variables used in the preceding experiments is used in these experiments. The process operates for 25 periods before samples of 8 periods for each of the 100 individuals used in the 25 samples for each parameter set are selected.

The fixed effect probit estimator performs badly. The bias is greatest for large values of the variance in person effects (erf) and when there are no exogenous variables in the model. But even the smallest bias reported in the table is still bad. The t statistics based on the estimated information matrix result in misleading inferences. From experimental results not reported in the table, one does not reject the false null hypothesis of >'=/?= 0 in the vast majority of samples.

Note that estimates of y are downward biased and estimates of /? are upward biased. These results are very similar to Nerlove's Monte Carlo

Table 4.1

Monte Carlo results for models without lagged variables3

Values of $ for the fixed effect probit model6 P = 1 0=-O.l ctt2 =3 0.90 -0.10 -0.94

Values of fi for the random effect probit model' 0=1 0=-l

"The model generating the data is Y(i,t) = Z(i,t)p + T(i)+ U(i,t), z"=l,... ,/; if Y(i,t) > 0, d(i,t) = 1, t = 1, . . . , T, otherwise, d(i,t) = 0. Z(i,t) is generated by the Nerlove (1971) process,

Z(i,t) = 0:lr + 0.5Z(i,r - 1) + ca(i,t) w(i,t)~ i/[ —0.5,0.5]. b/= 100, T= 8. c/= 100, T= 3.

results in a linear equation model analogue of the Markov model (Nerlove 1971). Fixed effect estimators generate a downward-biased estimate of the coefficient of the lagged value of the endogenous variable in that model Just as they do for the state dependence coefficient in the Markov model.

In view of the poor performance of the fixed effect estimator as a solution to the problem of initial conditions, it is of some interest to examine the performance of some alternative estimators. The middle section of table 4.2 reports the results of a limited Monte Carlo study of the approximate random effect estimator proposed in section 4.3. The samples used to generate these estimates are the first three periods of the data utilized in the samples of the Monte Carlo study of the Markov model estimated by the fixed effect probit scheme. The first-period marginal probability is assumed to depend solely on first-period values of the exogenous variables. The proposed approximate random effect estimator discussed in section 4.3 does somewhat better than the fixed effect estimator. The y consistently overstates the true y and 0 understates the true /?. As of declines, so does the bias in the estimator. In results not reported here, t statistics are much more reliable in this model than in the fixed effect probit model since they lead to correct inference in a greater proportion of the samples.13

As in the discussion of the fixed effect probit model with strictly exogenous explanatory variables, it is natural to seek a suitable benchmark with which to compare the performance of the proposed estimators. One benchmark that provides an ideal case is a model with known non-stochastic initial conditions. Twenty-five samples with 100 observations of 3 periods are generated for each set of parameter values. Random effect maximum likelihood estimates for this model are presented in the final section of table 4.2. While this estimator is less biased than the approximate estimator, it is nonetheless biased. The difference between the results for the approximate random effect estimator for the case of stochastic initial conditions and the results for the estimator with known initial conditions

13. Another ad hoc estimator was tried. This estimator fits a linear probability function to predict the marginal probability of the first sample period state. The predicted value is substituted in place of the actual value, and the y parameter associated with the state in the first period is permitted to be distinct from the y parameter for the sample transition. In terms of bias the performance of this estimator is intermediate between the fixed effect estimator and the proposed approximate random effect estimator given in the text, even when several years of presample data on the exogenous variables are used to predict the probability of the first sample period state. The estimator works well for the special problem of testing the null hypothesis of no state dependence (7 =0).

Table 4.2

Monte Carlo results for models with lagged variables3

Values of y and fi for the fixed effects estimator"

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