Simulation Based. Inference for. Dynamic Multinomial. Choice Models
multinomial choice histories and partially observed payoffs. Many general surveys of simulation methods are now available (see Geweke, 1996; Monfort, Van Dijk, and Brown, 1995; and Gilks, Richardson, and Spiegelhalter, 1996), so in our view a detailed illustration of how to implement such methods in a specific case has greater marginal value than an additional broad survey. Moreover, the techniques we describe are directly applicable to a general class of models that includes static discrete choice models, the Heckman (1976) selection model, and all of the Heckman (1981) models (such as static and dynamic Bernoulli models, Markov models, and renewal processes). The particular procedure that we describe derives from a suggestion by Geweke and Keane (1999a), and has the advantages that it does not require the econometrician to solve the agents’ dynamic optimization problem, or to make strong assumptions about the way individuals form expectations.
This chapter focuses on Bayesian inference for dynamic multinomial choice models via the MCMC method. Originally, we also hoped to discuss classical estimation of such models, so that readers could compare the two approaches. But, when we attempted to estimate the model developed below using SML it proved infeasible. The high dimension of the parameter vector caused an iterative search for the maximum of the simulated likelihood function via standard gradient based methods to fail rather dismally. In fact, unless the initial parameter values were set very close to the true values, the search algorithm would quickly stall. In contrast, the MCMC procedure was computationally feasible and robust to initial conditions. We concluded that Bayesian inference via MCMC has an important advantage over SML for high dimensional problems because it does not require a search for the optimum of the likelihood.
We consider dynamic, stochastic, parametric models with intertemporally additively separable preferences and a finite time horizon. Suppose that in each period t = 1,…, T (T < ro) each agent chooses among a finite set At of mutually exclusive alternatives. Let be the date-t state space, where kt is a positive integer. Choosing alternative at Є At in state It Є leads to period payoff R(It, at; 0), where 0 is a finite-vector denoting the model’s structural parameters.
The value to choosing alternative at in state It, denoted by Vt(It, at), depends on the period payoff and on the way agents expect that choice to affect future payoffs. For instance, in the familiar case when agents have rational expectations, alternative specific values can be expressed:
Vt(It, at) = R(It, at; 0) + 5Ef max^AMVt+1(It+1, | It, at) (t = 1,…, T) (22.1)
Vtu(-) – 0 (22.2)
I+1 = H(It, at; 0) (22.3)
where 5 is the constant rate of time preference, H(It, at; 0) is a stochastic law of motion that provides an intertemporal link between choices and states, and Et is the date-t mathematical expectations operator so that expectations are taken with respect to the true distribution of the state variables PH(It+11 It, at; 0) as generated
by H(-). Individuals choose alternative a* if and only if a*) > Vt(It, at) Vat Є
At, at, Ф a*. See Eckstein and Wolpin (1989) for a description of many alternative structural models that fit into this framework.
The econometrician is interested in drawing inferences about 9, the vector of structural parameters. One econometric procedure to accomplish this (see Rust, 1987 or Wolpin, 1984) requires using dynamic programming to solve system
(22.1) -(22.3) at many trial parameter vectors. At each parameter vector, the solution to the system is used as input to evaluate a prespecified econometric objective function. The parameter space is systematically searched until a vector that "optimizes" the objective function is found. A potential drawback of this procedure is that, in general, solving system (22.1)-(22.3) with dynamic programming is extremely computationally burdensome. The reason is that the mathematical expectations that appear on the right-hand side of (22.1) are often impossible to compute analytically, and very time consuming to approximate well numerically. Hence, as a practical matter, this estimation procedure is useful only under very special circumstances (for instance, when there are a small number of state variables). Consequently, a literature has arisen that suggests alternative approaches to inference in dynamic multinomial choice models.
Some recently developed techniques for estimation of the system (22.1)-(22.3) focus on circumventing the need for dynamic programming. Several good surveys of this literature already exist, and we will not attempt one here (see Rust,
1994) . Instead, we simply note that the idea underlying the more well known of these approaches, i. e., Hotz and Miller (1993) and Manski (1993), is to use choice and payoff data to draw inferences about the values of the expectations on the right-hand side of (22.1). A key limitation of these procedures is that, in order to learn about expectations, each requires the data to satisfy a strict form of stationarity in order to rule out cohort effects.
The technique proposed by Geweke and Keane (1999a) for structural inference in dynamic multinomial choice models also circumvents the need for dynamic programming. A unique advantage of their method is that it does not require the econometrician to make strong assumptions about the way people form expectations. Moreover, their procedure is not hampered by strong data requirements. It can be implemented when the data include only partially observed payoffs from a single cohort of agents observed over only part of their lifecycle.
To develop the Geweke-Keane approach, it is useful to express the value function (22.1) as:
Vt(It, a) = R(It, at; 9) + FH(It, a), (22.4)
where FH(It, at) = 5E tmaxaf+1 ЄAf+1 Vt+1(at+1, H(It, at)). Geweke and Keane (1999a) observed that the definition of FH(), henceforth referred to as the "future component", makes sense independent of the meaning of E. If, as assumed above, E is the mathematical expectations operator then FH() is the rational expectations future component. On the other hand, if Et is the zero operator, then future payoffs do not enter the individuals’ decision rules, and FH() is identically zero. In general, the functional form of the future component FH() will vary with the way people form expectations. Unfortunately, in most circumstances the way people form expectations is unknown. Accordingly, the correct specification of the future component FH(-) is also unknown.
There are, therefore, two important reasons why an econometrician may prefer not to impose strong assumptions about the way people form expectations, or, equivalently, on the admissible forms of the future component. First, such assumptions may lead to an intractable econometric model. Second, the econometrician may see some advantage to taking a less dogmatic stance with respect to behaviors about which very little, if any, a priori information is available.
When the econometrician is either unwilling or unable to make strong assumptions about the way people form expectations, Geweke and Keane (1999a) suggest that the future component FH() be represented by a parameterized flexible functional form such as a high-order polynomial. The resulting value function can be written
Vt(It, af) = R(It, at; 0) + FH(It, at | n) (22.5)
where n is a vector of polynomial coefficients that characterize expectation formation. Given functional forms for the contemporaneous payoff functions, and under the condition that 0 and n are jointly identified, it is possible to draw inferences both about the parameters of the payoff functions and the structure of expectations.
This chapter focuses on an important case in which key structural and expectations parameters are jointly identified. We consider a model where an alterna – five’s payoff is partially observed if and only if that alternative is chosen. In this case, after substitution of a flexible polynomial function for the future component as in (22.5), the model takes on a form similar to a static Roy (1951) model augmented to include influences on choice other than the current payoffs, as in Heckman and Sedlacek (1986). The key difference is that FH() incorporates overidentifying restrictions on the non-payoff component of the value function that are implied by (22.1)-(22.3) and that are not typically invoked in the estimation of static selection models. Specifically, the parameters of the non-payoff component of the value function are constant across alternatives, and the arguments of the non-payoff component vary in a systematic way across alternatives that is determined by the law of motion H() for the state variables.
The structural model (22.1)-(22.3) also implies restrictions on the nature of the future component’s arguments. For instance, if H() and R() jointly imply that the model’s payoffs are path-independent, then the future component should be specified so that path-dependent expectation formation is ruled out.2 Similarly, contemporaneous realizations of serially independent stochastic variables contain no information relevant for forecasting future outcomes, so they should not enter the arguments of the flexible functional form. Without such coherency conditions one might obtain results inconsistent with the logic of the model’s specification.
A finite order polynomial will in general provide only an approximation to the true future component. Hence, it is important to investigate the extent to which misspecification of the future component may affect inference for the model’s structural parameters. Below we report the outcome of some Monte Carlo experiments that shed light on this issue. The experiments are conducted under both correctly and incorrectly specified future components. We find that the Geweke – Keane approach performs extremely well when FH() is correctly specified, and still very well under a misspecified future component. In particular, we find that assuming the future component is a polynomial when it is actually generated by rational expectations leads to only "second order" difficulties in two senses. First, it has a small effect on inferences with regard to the structural parameters of the payoff functions.3 Second, the decision rules inferred from the data in the misspecified model are very close to the optimal rule in the sense that agents using the suboptimal rule incur "small" lifetime payoff losses.
The remainder of this chapter is organized as follows. Section 2 describes the application, and Section 3 details the Gibbs sampling algorithm. Section 4 reviews our experimental design and results, and Section 5 concludes.