Chapter 7 Dynamic Model and Model with Saturation
7.2 Consistent Estimation of Dynamic Model
7.2.2 Estimation Methods Proposed for the Current Study
The literature review shows that the consistent estimation of discrete choice model with state dependence is a complex issue and different approaches have their advantages and limitations. For fixed effect models, it is not necessary to specify any distribution of the unobserved heterogeneity conditional on the explanatory variables, thus avoiding the strong orthogonality assumptions. For this reason, the fixed effect model should be the
39 This condition determines the explanatory variables as predetermined (with feedback effect from the errors in the previous periods) rather than strictly exogenous.
40 Another reason preventing us adopting semi-parametric approach is that it seems all but impossible to
preferred choice. However, treating the unobserved effects as parameters and estimating the model using maximum likelihood method produce biased estimation when the number of time period T is small. In the case of dynamic model, the bias appears to be larger with the presence of lagged dependent variables. Furthermore, the marginal effects in discrete choice model depend on the nuisance parameters, so eliminating these parameters, as in the case of certain semi-parametric models, does not seem to be the solution here.
For this reason, the approach of Arellano (2003) and Carro (2003), which aims at reducing the order of bias rather than achieving fixed T consistency, appears to be the most attractive. Because their approach is based on modifying the concentrated likelihood, there is no reason why this approach can not be extended to the pseudo panel model by adapting the log likelihood function to one based on proportions data rather than discrete data. However, it has proved very complicated in implementation and we have to leave it to future research. In the empirical work of the current project, we would rely on a rather optimistic result of the fixed effect estimator, i.e. consistency under the asymptotic of large T41. This action is partially justifiable on the grounds that we have relatively long sample periods of up to 19 years.
Regarding the random effect model, the main criticism is the strong orthogonality assumption between the unobserved effects and the explanatory variables. Furthermore, the relationship between the first lagged dependent variable and the unobserved heterogeneity is usually undefined, which leads to the so-called initial condition problem. Two solutions to the initial condition problem have been proposed in the literature, i.e. assumption on a separate distribution of the initial conditions in Heckman (1981b), or assumption on the distribution of the unobserved effects conditional on the initial conditions in Wooldridge (2005). However, neither approach is the definitive answer, as both have the real danger of modelling the distributions that are inconsistent with the data generating process.
41 While consistency is established for genuine panel, it is likely to carry over to pseudo panel when
In the current study, we propose a slightly different approach, i.e. random parameter models. Unlike the random effect model, where a constant term is assumed to capture the unobserved heterogeneity, such unobserved effects are captured by a randomly distributed parameter vector in a random parameter model. Adding a subscript i (i=1,...,n) to recognize heterogeneity in the parameter vector γ =(α,β′)′, Model (4) can be re-written as:
yit = 1 ((yi,t−1,x′it)γi +εit >0) (16)
where, γi =γ +Γνi is a vector of random parameter with mean ✄ and variance ΓΓ′; Γ is a diagonal parameter matrix42;
νi is a random vector with zero vector mean and covariance matrix I.
Any fixed (non-random) parameters in γi can be specified by constraining the corresponding rows in Γ to be zero, and it is easy to see the random effect model is a special case of (16), where only the constant term is random. Recall the problem of random effect specification for dynamic discrete choice model. The distribution of the random effect is specified conditional on the explanatory variables, and it requires orthogonality assumption between the unobserved effect and the explanatory variables.
With the presence of lagged dependent variable, the orthogonality assumption would usually be violated except for some special circumstances (e.g. the initial conditions are strictly exogenous and the unobserved effect is conditional on such initial conditions).
However, in a random parameter model, the orthogonality condition becomes moot, as the individual specific heterogeneity is embodied in the marginal responses (parameters) of the model43 (Greene, 2001a).
If one assumes the error term εit in (16) follows an iid logistic distribution, the underlying probability model has a logit form and (16) becomes a special (binary choice) version of mixed logit model. As shown in McFadden and Train (2000), mixed logit is a highly flexible model that can be used to approximate any random utility model. It alleviates the three limitation of standard logit by allowing for random taste
42 The assumption of diagonal Γ makes our model a specific version of the general model in Greene (2004).
43 Random parameter model also partially incorporate the idea of Heckman (1981b). Assuming the element corresponding to yi,t−1 in the random vector νi is distributed as N(0,1), Model (16) would
variation, unrestricted substitution patterns and correlation in unobserved factors over time (Train, 2003). In recent years, mixed logit model has been applied to many empirical studies based on panel data (see, for example, Revelt and Train, 1998; Bhat, 2000; Mohammadian and Miller, 2003; Leth-Petersen and Bjorner, 2005, with the last two study being dynamic mixed logit model of car ownership) as well as cross sectional data (e.g. Bhat, 1998; Brownstone and Train, 1999; Hess and Polak, 2005;
Hess et al. 2006; various mixed logic models of car ownership including Brownstone et al., 2000; Whelan, 2003; Golounov el al., 2004). One contribution of this study is to extend the application of mixed logit model to pseudo panel data.
For panel data model, dependence of the Ti observations for individual i results from dynamic discrete choice model, as lagged dependent variables have been readily accommodated in the utility function in a given period to represent lagged response behaviour. Conditional on ✄i, the only remaining random terms in the mixed logit are the it’s, which are independent over time, so the lagged dependent variable in the utility function is uncorrelated with these remaining error terms for period t (Train, 2003). In another word, the lagged dependent variable can be added to the mixed logit model without having to change the estimation procedure.
If we only have repeated cross sectional data rather than genuine panel data, transformation similar to those described in section 7.1.4 of this chapter has to be applied. After aggregating individual observation into cohorts, the utility of choosing
Option 1 by a particular individual in cohort c in year t can be expressed as the sample average (deterministic) utility Vct plus various error terms. When the sample size is sufficiently large for each cohort, the measurement errors ηct can be ignored.
Deviation of individual deterministic utility from cohort mean, θi(t),t =Vi(t),t −Vct, can be merged with the residual error term εi′(t),t. The only difference from Section 7.1.4 is the treatment of unobserved heterogeneity, λc, which will be absorbed into the random parameter vector γc in the mixed logit model (Note γc =(αc,βc′)′=γ +Γνc and assume that xct contains a constant term). Conditional on random vector γc, the sample average deterministic utility for cohort c will become the argument to the exponential function in (17), and the probability of any individual within cohort c choosing Option 1 over T period can be expressed as a product of logit formulas Λγctc (raised to the
Similar to the case of genuine panel, the unconditional probability Pct is the integral of (19) over all values of γc. The mixed logit model of (19) can be estimated most conveniently using the method of Maximum Simulated Likelihood (MMSL), which is based on the simulated logit probability within the right hand side of equation (19):
∑
= proportions data, it gives a simulated log likelihood function:∑∑
= =where nct is the sample size for cohort c and rct is the proportion of household in cohort c choosing Option 1 in year t.
For empirical application, the random parameter model for pseudo panel based on equation (19) to (21) has been implemented in Gauss (Aptech Systems, 1996). The code was adapted from a mixed logit program of Revelt and Train (1998), and various checks have been applied to ensure the correct implementation of the model. Appendix 2 is simplified version of the Gauss code we have used.