Advanced Econometrics Takeshi Amemiya
Models with Heterogeneity and True State Dependence
In this subsection we shall develop a generalization of model (9.7.2) that can incorporate both heterogeneity and true state dependence.
Following Heckman (1981a), we assume that there is an unobservable continuous variable y* that determines the outcome of a binary variable yit by the rule
У и= 1 if yt> 0 (9.7.10)
= 0 otherwise.
In a very general model, y* would depend on independent variables, lagged values of yg, lagged values of yt„ and an error term that can be variously specified. We shall analyze a model that is simple enough to be computationally feasible and yet general enough to contain most of the interesting features of this type of model: Let
Уи = х'иР + УУи-і + vu=Vi, + vtt, (9.7.11)
where for each i, vu is serially correlated in general. We define true state dependence to mean уф 0 and heterogeneityto mean serial correlation of{y„}.
Model (9.7.2) results from (9.7.11) if we put у = 0 and vit = ut + eu, where (ea) are serially independent. Thus we see that model (9.7.2) is restrictive not only because it assumes no true state dependence but also because it assumes a special form of heterogeneity. This form of heterogeneity belongs to a so- called one-factor model and will be studied further in Section 9.7.3.
Here we shall assume that for each i, {vu} are serially correlated in a general way and derive the likelihood function of the model. Because {yu} are assumed to be independent over i, the likelihood function is the product of individual likelihood functions. Therefore we suppress і and consider the likelihood function of a typical individual.
Define Г-vectors у, у/, and v the fth elements of which are y„ y/„ and vtb respectively, and assume v ~ N(0,2). Then the joint probability of у (hence, a typical individual likelihood function) conditional on y0 can be concisely expressed as
P(y) = F[y, * (2y -1); 2 * (2y - l)(2y -1)'], (9.7.12)
where * denotes the Hadamard product (see Theorem 23 of Appendix 1), 1 is a Г-vector of ones, and F(x; 2) denotes the distribution function of N(0,2) evaluated at x. Note that in deriving (9.7.12) we assumed that the conditional distribution of v given y0 is N(0,2). This, however, may not be a reasonable assumption.
As with continuous variable panel data discussed in Section 6.6.3, the specification of the initial conditions ую (reintroducing subscript i) is an important problem in model (9.7.11), where Tis typically small and N is large. The initial conditions ую can be treated in two ways. (We do assume they are observable. Treating them as N unknown parameters is not a good idea when N is large and Г is small.)
1. Treat ую as known constants, 1 or 0. Then the likelihood function is as given in (9.7.12).
2. Assume that ую is a random variable with the probability distribution РІУю = 1) = Ф(Хдаа), where a is a vector of unknown parameters.
