شناسایی نیمه پارامتری و ناهمگونی در انتخاب گسسته مدل های برنامه ریزی پویا

Empirical discrete choice dynamic programming models have become important empirical tools. A question that arises in estimation and interpretation of the results from these specifications is which combination of data and assumptions are needed to overcome problems of heterogeneity, selection, and omited variables bias. This paper addresses this question by considering nonparametric identification of a version of the model that allows for quite general forms of unobservable and information structures. I show that the model can be identified under conditions similar to a static polychotomous choice model. Using a stochastic version of an ‘identification of infinity’ argument, utility can be identified up to a monotonic transformation of the observables under strong support conditions and two types of exclusion restriction. The first type is similar to a standard static exclusion restriction: a variable that influences the first period decision, but does not enter the second period decision directly. The second type requires a variable that does not affect the utility of the first option directly, but is known during the first period, and has predictive power on the choice during the second. I also provide two specifications under which the full error structure can be identified. This requires the additional assumption of stochastic innovations in the observables. I then use the model to estimate schooling decisions in which students deciding whether to drop out of high school account for the option value of attending college.

Empirical discrete choice dynamic programming models have become important empirical tools. In some applications of these models, problems of substantial heterogeneity/selection/omitted variable bias arise (see, e.g. Keane and Wolpin (1997) or Eckstein and Wolpin (1997)). The source of these biases is potentially more complex in dynamic models than static ones in that agents may have heterogeneity not only in outcomes, but also in expectations about future outcomes. A question that arises in estimation and interpretation of the results in these cases is which combination of data and assumptions are needed to overcome these problems. This paper addresses this question by considering nonparametric identification of a version of the model that allows for quite general forms of unobservable and information structures. Despite the added complexity of the model, I show that it can be identified under conditions similar to a static polychotomous choice model. Using a stochastic version of an ‘identification of infinity’ argument, utility can be nonparametrically identified up to a monotonic transformation of the observables under strong support conditions and two types of exclusion restriction. The first type is similar to a standard static exclusion restriction: a variable that influences the first period decision, but does not enter the second period decision directly. The second type requires a variable that does not affect the utility of the first option directly, but is known during the first period and has predictive power on the choice during the second. I also provide two specifications under which the full error structure can be identified. This requires the additional assumption of stochastic innovations in the X's: a variable known at time one that helps predict the second period decision, but conditional on second period observables, has no influence on the decision. The specification I develop is a generalization of a dynamic ‘Roy’ type model, and I focus on schooling decisions. In deciding whether to drop out of high school a student takes into account both the direct value of graduating from high school as well as the value of the option to attend college. While making this decision, a student does not know whether he will attend college. Heterogeneity bias is likely to be important in that students with high returns or tastes for high school are also likely to have high returns or tastes for college. While there is a substantial literature addressing the selection/heterogeneity issue in schooling models, the previous work has typically ignored the complexity of the heterogeneity. The problem is not just that the returns to college are likely to be correlated with returns to high school, but also that agents may have additional information about their own private returns to college which is unobservable to the econometrician. For example, a high school student may know that he has excellent teaching skills. While this information may be correlated with the returns to high school, since teachers must have a college degree it is much more important for the decision about whether to attend college. Accounting for this type of heterogeneity in information requires a more complex information structure about unobservables than is often used in empirical work. This leads to two important questions (1) can an information structure such as this be identified? and (2) if not, can other important structural parameters be identified without this information? I provide a set of conditions under which the coefficients can be identified allowing for these forms of unobserved heterogeneity in information about the unobservables. While we can not identify an arbitrarily complicated information structure under standard conditions, I provide two specifications under which we can. Discrete choice dynamic programming models have been applied to a large range of topics. Examples include patent renewal (Pakes, 1986), bus engine replacement (Rust, 1987), job search (Wolpin, 1987), fertility (Hotz and Miller, 1993), life cycle earnings (Keane and Wolpin, 1997), and schooling (Taber, 1998); a survey can be found in Eckstein and Wolpin (1989) or Rust (1994). The main goal of this paper is to establish identification of these models under fairly weak assumptions about the distribution of the error and information structure. These results are useful for two reasons. First, they take a first step towards semiparametric estimation of this class of models by establishing sufficient conditions for their identification. To facilitate estimation, this work typically imposes strong parametric restrictions on the distribution of the unobservables and on the information structure that agents use to form their expectations. These assumptions are typically chosen out of mathematical convenience rather than as implications of the models themselves so it is important to check the sensitivity of the model to these assumptions. Secondly, and perhaps more importantly given current computational problems, they demonstrate the ideal data set under which these models can be identified without parametric restrictions. Solving the heterogeneity bias problem can typically be achieved by imposing functional forms on the distribution of the error terms. However, it is preferable to find data that can solve the problems. In practice the perfect data set rarely exists, so identification is achieved through a combination of data and assumptions. Nevertheless, this type of identification exercise is potentially useful both for understanding the trade-off between assumptions and data and for illuminating which type of data one should use when estimating these models. While much work has been done on semiparametric identification of other discrete choice models, it has not been systematically discussed in dynamic programming problems. There have been a few papers that focus on specific points, often with negative results. Flinn and Heckman (1982) consider identification of job search models. They show that these models are nonparametrically underidentified as one essentially can not distinguish high reservation wages from low arrival rates. Rust (1994) also shows a form of non-identification in a more general model. As I discuss below, this problem can be addressed fairly easily in a finite time model, but is a more serious concern in infinite time models. The most closely related work is by Pakes and Simpson (1989). They provide a sketch of identification for a finite period model of patent renewal that could be written as a special case of mine. They also use exclusion restrictions and essentially a similar identification at infinite argument.1 I extend this model into a broader framework by allowing for a more general form for unobservables and information, and a more general process for the observables. Cameron and Heckman (1998) also consider identification of schooling models, but the form of their models are quite different in that they do not use this dynamic programming framework. This paper extends the work on identification of discrete selection models in static cases to incorporate dynamics. As in this paper, most of the previous work generalizes the ideas behind the semiparametric identification of the binary choice model, equation(1) d=1(g(X,θ)+ε>0). The function g is assumed known up to parameter θ, but the distribution of ε is unspecified. Identification of this simple model is presented in Cosslett (1983) and Manski 1975 and Manski 1988. Extensions that allow for multiple choices or multiple periods include Manski (1987), Thompson (1989), Cameron and Heckman (1998), and Cameron and Taber (1994). Matzkin 1990, Matzkin 1992 and Matzkin 1993 follows another line. She extends the semiparametric identification to nonparametric identification. For instance in the binary choice model (1) she allows g(X,θ)=g(X) and provides conditions under which the function g is identified. I describe the model in Section 2. I provide identification of various components of the model in 3 and 4. In Section 5 I demonstrate how these results can be used by estimating a version of the model as a schooling model where a student first decides whether to graduate from high school and then conditional on high school graduation decides whether to attend college. Section 6 presents some conclusions.

This paper develops a simple discrete choice dynamic programming model with a quite general form for unobservables and agent's information sets. The goal is to uncover what type of data can solve the selection problem induced by this structure. As in static models, I show that with strong support conditions and exclusion restrictions the model is identified. While these support conditions are strong, it is very difficult to avoid them. Essentially two types of exclusion restriction are required. The first is a variable that influences the first period decision, but does not enter the second period decision directly. The second type requires a variable that does not affect the utility of the first option directly, but is known during the first period and has predictive power on the choice during the second. I also provide two specifications under which the full error structure can be identified. This requires the additional assumption of stochastic innovations in the X's: a variable known at time one that helps predict the second period decision, but conditional on second period observables, has no influence on the decision. While the model presented here is special, generalizing these results to more complicated finite time models is straight forward. I estimate a schooling version of the model in which students first decide whether to graduate from high school and then decide whether to attend college. This procedure has only limited success. The model does not show signs of forward looking behavior and reliable standard errors could not be obtained. Part of the problem may be that the exclusion restrictions are weaker than one may hope, and they do not have large support. One possible direction for future research on dynamic schooling models is to obtain more powerful exclusion restrictions which may solve the problems, although this may prove difficult. More generally this paper has suggested that certain types of exclusion restrictions with strong support conditions should help solve the dynamic selection problem. This should be a useful input for empiricists who face this problem.