Bayesian Estimation of Cost Functions with Stochastic or Exact Constraints on Parameters
研究在系统方程中施加经济理论约束（如对称性）的贝叶斯估计方法，分别处理随机约束和精确约束，并以成本函数估计为例展示应用。
Often economic theory implies cross-equation constraints on parameters in a system of behavioral equations. Some of these constraints are in the form of equality conditions on parameters of different equations, as, for example, symmetry conditions in consumer demand and input demand systems. In this paper we study the Bayesian estimation of a multivariate model with these kinds of constraints, using as an application the estimation of cost functions. We use two different approaches. First, a hierarchical model is used to impose stochastically the constraints implied by economic theory. The linear hierarchical model was suggested by Lindley and Smith [1972] and subsequently used by Kiefer [1977] to introduce symmetry and homogeneity constraints in a consumer expenditure system. Secondly, we examine alternative estimation methods of the model when the constraints hold exactly. Related work has been done in the Bayesian estimation of seemingly unrelated regression (SUR) models. As Zellner [1971] has shown, in SUR models the analytical derivation of marginal pdfs of the parameters is difficult. Each equation in the system has different explanatory variables; on the other hand, there are no restrictions on the parameters. Alternatively, as Dreze and Morales [1976] have shown, the SUR model can be reparameterized so that all equations have a common data matrix. However, this reparameterization implies that in each equation some parameters are constrained to be zeros. In contrast, we study a model where we have equality constraints on parameters across equations, but all equations have the same data matrix. For example, in demand systems, same price data appears in all demand equations and economic theory implies symmetry constraints. The structure of the paper is the following. Section 2 presents assumptions about the technology. We discuss estimation with stochastic constraints in Section 3 and estimation with exact constraints in Section 4. In Section 5, we apply the estimation methods to a cost function.