Semiparametric Instrumental Variables Estimation and Its Application to Dynamic Oligopoly
This paper considers a semiparametric regression model in which the error term is correlated with the nonparametric part. An example of this regression model can be found in structural models of dynamic oligopoly. Dynamic oligopoly is a situation in which firms' price-settings (or quantity-settings) are strategically interdependent and have durable effects on the stream of their profits. Dynamic oligopoly fits many industries characterized by the significance of network externalities, learning-by-doing, and informational product differentiation. For a dynamic structural model of the representative agent, the Euler-equation-based estimation technique is usually employed. However, the Euler equations cannot be generally obtained in dynamic oligopoly. As an alternative, we can consider an estimation procedure as follows. Under some regularity conditions, a firm's optimal pricing (or quantity-setting) in dynamic oligopoly can be formulated as a continuous Markov decision problem (MDP). Then we may apply an estimation procedure similar to the nested fixed point algorithm: using ad hoc assumptions for stochastic specification of the evolution of state variables, we may calculate each firm's value functions in equilibrium for each candidate value of the parameter vector and then search for the value of the parameter vector that maximizes the (log) likelihood function or minimizes some distance. It is, however, impractical to implement this estimation procedure in the case of dynamic oligopoly. Most of all, it will result in a prohibitive computational burden. It is well known that continuous MDPs have the problem of Bellman's curse of dimensionality. Even with some simple discretization assumptions and a stochastic algorithm to break the curse of dimensionality, the computational burden to calculate the equilibrium value functions for just one candidate value of the parameter vector is usually huge. In addition, the complexity of the estimation problem usually makes it difficult to determine the robustness of the conclusions to the ad hoc stochastic assumptions. Furthermore, if the stochastic process is misspecified, the estimator for the parameter vector is generally inconsistent. The estimation procedure suggested in this paper, however, enables us to semiparametrically estimate a class of structural models of dynamic oligopoly. It will be shown that first-order profit maximization conditions of dynamic oligopoly may lead to our generic semiparametric regression model. A technical difficulty of this semiparametric regression model, however, is that we can not eliminate the nonparametric part in the two-step estimation procedure of a typical semiparametric regression model. Yet, we can still obtain a semiparametric estimator, called a semiparametric instrumental variables (SIV) estimator, with consistency and asymptotic normality if there exist two sets of instrumental variables (IVs) satisfying both an identification condition and an orthogonality condition. Our estimation plan is as follows. In order to eliminate the nonparametric part, we first filter the nonparametric part by the first set of IVs. For identification, we need the second set of IVs which is not a function of the first set of IVs and must be orthogonal to the filtering error. The paper provides two generic examples in which we can construct these two sets of IVs and then discusses an empirical example of the application of the SIV estimation procedure to estimate network effects in the U.S. home VCR market during the years 1981 - 1988.
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