 # Structural Estimation Using Parametric Mathematical Programming with Equilibrium Constraints and Homotopy Path Continuation

Müller, Philipp; Reich, Gregor Philipp (2019). Structural Estimation Using Parametric Mathematical Programming with Equilibrium Constraints and Homotopy Path Continuation. Econometrics: Econometric & Statistical Methods - General eJournal - CMBO 12, University of Zurich.

## Abstract

In this paper, we formulate the likelihood function of structural models as a parametric optimization problem, where the model equations enter as constraints, forming a mathematical program with equilibrium constraints (Su and Judd, 2012). We trace the solution to its first-order conditions in dependence on a controlled parameter using homotopy continuation, delivering a relation from the controlled parameter to the corresponding maximum likelihood estimates and their confidence intervals. This enables us to estimate models with identification issues, multiplicity of equilibria, etc. As applications, we first trace the parameter estimates of the bus engine replacement model of Rust (1987), a dynamic discrete choice model, in dependence of the discount factor β. Using relative value iteration, we find that β is well identified and statistically significantly larger than 1. Second, for a simple static binary choice model, we demonstrate how the effects of multiplicity of equilibria and a lack of identification can be mitigated by the tracing method.

## Abstract

In this paper, we formulate the likelihood function of structural models as a parametric optimization problem, where the model equations enter as constraints, forming a mathematical program with equilibrium constraints (Su and Judd, 2012). We trace the solution to its first-order conditions in dependence on a controlled parameter using homotopy continuation, delivering a relation from the controlled parameter to the corresponding maximum likelihood estimates and their confidence intervals. This enables us to estimate models with identification issues, multiplicity of equilibria, etc. As applications, we first trace the parameter estimates of the bus engine replacement model of Rust (1987), a dynamic discrete choice model, in dependence of the discount factor β. Using relative value iteration, we find that β is well identified and statistically significantly larger than 1. Second, for a simple static binary choice model, we demonstrate how the effects of multiplicity of equilibria and a lack of identification can be mitigated by the tracing method.

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