Exponential class of dynamic binary choice panel data models with fixed effects

ABSTRACT

This paper proposes an exponential class of dynamic binary choice panel data models for the analysis of short T (time dimension) large N (cross section dimension) panel data sets that allow for unobserved heterogeneity (fixed effects) to be arbitrarily correlated with the covariates. The paper derives moment conditions that are invariant to the fixed effects which are then used to identify and estimate the parameters of the model. Accordingly, generalized method of moments (GMM) estimators are proposed that are consistent and asymptotically normally distributed at the root-N rate. We also study the conditional likelihood approach and show that under exponential specification, it can identify the effect of state dependence but not the effects of other covariates. Monte Carlo experiments show satisfactory finite sample performance for the proposed estimators and investigate their robustness to misspecification.

KEYWORDS:

• CMLE
• dynamic discrete choice
• fixed effects
• GMM
• panel data

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• C23
• C25

Acknowledgment

We are grateful to the Co-Editor and two referees for their constructive comments that have greatly improved the paper.

Notes

¹Strict exogeneity typically allows us to specify the likelihood of y_it conditional on c_i, x_it, and y_it−1. But by Bartolucci and Nigro’s (2010) specification, all periods’ observations of x_it must be taken into account. On the strict exogeneity assumption and the other approaches in the literature, see Wooldridge (2002) for a survey.

²See Remark 1 for more details.

³To be more precise, we prove that the general form of a function F that satisfies the factorization is given by F(z) = 1−Cexp(−Dz), for C and D>0. Since these two parameters are not identifiable, we set them both equal to 1. Similar rescaling and normalization is also used for the standard logit and probit models.

⁴See Chamberlain (2010) for identification in a two-period case and Magnac (2004) for more general identification results with the conditional likelihood approach, and also Magnac (2001) for an empirical application.

⁵Note that since 1−γ>0, then 1−γy_i,t−1≠0, noting that y_i,t−1 can only take the values of 0 and 1.

⁶In the appendix, we consider in detail the case of T = 3 and the single moment $E\left(e_{i3}{y}_{i1}\right)=0$. In this case, the GMM estimator has a closed-form solution.

⁷The assumptions we lay out here demonstrate the fact that while the asymptotic properties of GMM estimators such as consistency and asymptotic normality are established under high-level regularity conditions as by Hansen (1982), whether they are satisfied in a specific nonlinear model is often technically involved and has to be examined case by case. It is worth noting that in the literature where GMM estimators are proposed, the conventional approach has been to derive moment conditions of the model and then claim the GMM estimators based on these moment conditions are consistent and asymptotically normally distributed; implicitly assuming that the required regularity conditions are satisfied.

⁸The same caveat as mentioned earlier continues to hold. E(e_it) = 0 for (γ,β) = (0,0) and for $(\gamma ,\beta )=({\gamma }_0,{\beta }_0)$. Therefore, the constant should never be used as an instrument unless accompanied by at least one lagged variable as an additional instrument.

⁹We thank a referee for drawing our attention to this point.

¹⁰The full set of Monte Carlo results is available from the authors on request.

¹¹To simplify the computations, we first estimated γ and then estimated ρ as -ln(1−γ). See (11). This approach requires γ<1. In several experiments, we encountered estimates for γ that were inadmissible (namely they were larger than 1). This was particularly the case for small values of N. However, the likelihood of obtaining an inadmissible estimate decreased sharply with N. As a check, in the case of a few experiments we also estimated ρ directly and without any restrictions and overall found the results to be very similar to the ones based on the indirect approach.

¹²We also tried setting the threshold at 90%. This gets rid of too much information when T is small and does not help much for large T so it does not substantively change the main results of our experiments.

¹³It is also possible to match the transitions from 1 to 1 given $x_{it}={\bar{x}}_i$. This gives slightly different exponential fixed effects. But it does not change the general conclusion of this section. The results that condition on ${\bar{x}}_i$ are available from the authors on request.

¹⁴To save space, the results for the probit distribution are available in an online supplement.

¹⁵Since γ = 1−exp(−ρ), and because y_i,t−1 and y_i,t−2 take 0 and 1 values only, then it is easily verified that $\left(1-\gamma y_{i,t-2}\right)∕\left(1-\gamma y_{i,t-1}\right)$ and $e^{\rho \Delta y_{i,t-1}}$ give the same values for all admissible choices of y_i,t−1 and y_i,t−2.

¹⁶See the discussion by Harris and Mátyás (1999), pp. 14–17.