Fixed Effect model

For panels and pseudo panels, fixed effect models do not assume specific distribution of the unobserved heterogeneity ^✂_c. This unrestrictive feature makes fixed effect model particularly attractive. However, unlike the linear model, whose fixed effect can be eliminated by demeaning or differencing, it is not possible for the case of nonlinear panel model. Nevertheless, in the case of a limited number of individuals observed over many time periods, one can justify treating the cohort fixed effects as parameters to be estimated (Honore, 2002). More specifically, the maximum likelihood estimation of panel data fixed effect model is consistent when T →∞. For pseudo panel, if the measurement error problem can be ignored, FE model can also be estimated by maximum likelihood using cohort dummy variables.

However, when estimating discrete choice pseudo panel data, it is important that the data are weighted by the number of the observations in each cohort. For binary choice model, noting the proportion of decision makers in cohort c making Choice 1 in year t (y_it =1,∀i∈c) as rct, un-weighted MLE assumes that rct is from a distribution with variance r_ct(1−r_ct). However, the unconditional variance is in fact r_ct(1−r_ct)/n_ct, where n_ct is the number of observations in the cohort sample in year t, so the efficiency of maximum likelihood estimation using un-weighted data is underestimated (Greene, 1995). Furthermore, if the choice proportion r_ct is based on different numbers of observations, the variances will differ correspondingly, so the un-weighted model will not account for the inherent heteroskedasticity of the pseudo panel model. The maximum likelihood estimation of standard logit or probit model based on proportions data (the form of pseudo panel data), both weighted and unweighted, has been implemented in standard econometric software such as Limdep³¹.

The above discussion will become clear with the derivation of log likelihood function for the pseudo panel data. Note the number of individuals in the cohort sample making Choice 1 in year t as m_ct, we have m_ct =n_ct⋅r_ct, with r_ct and n_ct as defined above. The likelihood function can be expressed as:

∏∏

where Pct is the probability of the individuals in cohort c making Choice 1 in year t. In the current study of car ownership, it could take the form of equation (12).

Taking logarithm of expression (13) we have derived the log likelihood function of logit model based on proportions (pseudo panel) data:

∑∑

= =

Comparing the log likelihood function of (14) with that of binary choice model based on individual (discrete) data, it is clear that the only difference is the introduction of weighting n_ct. It should be noted that while the discussion here is based on binary choice model, it can be easily extended to the case of multinomial choice model.

However, the maximum likelihood estimator of the Fixed Effect model is only consistent when the number of time period is large, i.e. T →∞. This can be illustrated using the analysis of asymptotic variance. Rewrite the log likelihood function of (14) as:

∑∑

not known, is has to be estimated, and the estimator is a function of the maximum log likelihood estimator of ^✂c. As a result, the asymptotic variance of bML (maximum log likelihood estimator of ^✁) must also be of order T(c). In another word, the MLE of ^✁ is a function of a random variable which does not converge to a constant as C→∞ (Greene, 2001a; 2001b). This is the incidental parameter problems as identified in Neyman and Scott (1948). This problem can also be explained intuitively. For nonlinear panel model in general (and same for nonlinear pseudo panel), the incidental parameter ^✂_i can not be differenced away as in the case of linear model. Only new observations for individual i give new information about ^✂i; however, given a fixed T more individuals do not help with the estimation of ^✂i because they add more parameters to be estimated.

In some cases, the pseudo panel might be constructed in a way that the number of cohorts is large and the number of observations per cohort is small. If the number of time periods is also small, it would be appealing to consider asymptotic with large C and fixed T. However, further research is required to establish the consistent pseudo panel estimator under such asymptotic, as measurement error needs to be taken into account. For genuine panel, fixed T consistent estimator has been proposed in the literature. For example, although the nuisance parameter ^✂_i can not be differenced away,

∑

yit is the sufficient statistic for binary logit and other small class of models.

Conditional on the sufficient statistic

∑

yit , conditional maximum likelihood estimator would produce unbiased estimate of ^✁ (Andersen, 1970). The conditional maximum likelihood estimator is extended to the multinomial logit by Chamberlain (1980).

While

∑

yit is the sufficient statistic for panel data logit model, the existence and form of such statistic are difficult to establish for the case of pseudo panel. Furthermore, the conditional maximum likelihood approach has one significant drawback: it does not allow the calculation of the average effect of xit on the probability of yit =1 across the distribution of ^✂_i. The similar problem applies to other semi-parametric estimators such as the maximum score estimator of Manski (1987).

Another class of estimators does not attempt to be fixed T consistent; instead, the objective is to reduce the biases rather than to eliminate biases completely. One prime example is the modified concentrated likelihood estimator proposed by Arellano (2003), which has bias of order 1/T² rather than the maximum likelihood estimator of 1/T. The modified concentrated likelihood estimation has been extended to dynamic panel by Carro (2003), and both will be discussed in the next chapter.

Given that the maximum likelihood estimator of the fixed effect model is not consistent, it is important to establish the extent of biases. The discrete choice fixed effect estimator shows a substantial finite sample bias when the number of time period is very small. Hsiao (1986) found that for T = 2, the maximum likelihood estimator for a binary logit model is 100%. However, such large bias might only be of theoretical

importance, as the bias reduces rapidly as T increases to 3 or more. In another widely cited study, Heckman (1981b) found that the small sample bias of fixed effect estimator is surprisingly small even with moderate T. Using Monte Carlo simulation, the author showed that for the probit model with sample size of T_i = 8 and N = 100, the bias of the slope estimator is on the order of only 10%. In a more recent study, Greene (2002) found that the bias in the marginal effects is smaller than the bias in the slope parameters of MLE, which suggests that even when T = 2, the bias of 100% might also be overstated.

In the current study, 11 out of a total of 16 cohorts have pseudo panel observations for 19 years. As a result, the problem of small T bias for the MLE of fixed effect model might not be significant. In the empirical study of car ownership, the use of weighted MLE for the discrete choice pseudo panel model with fixed effects seems justified.

Im Dokument The Use of Pseudo Panel Data for Forecasting Car Ownership (Seite 113-116)