Pairwise-Difference Estimation of a Dynamic Optimization Model

Abstract

We develop a new estimation methodology for dynamic optimization models with unobserved shocks and deterministic accumulation of the observed state variables. Investment models are an important example of such models. Our pairwise-difference approach exploits two common features of these models: (1) the monotonicity of the agent's decision (policy) function in the shocks, conditional on the observed state variables; and (2) the state-contingent nature of optimal decision making which implies that, conditional on the observed state variables, the variation in observed choices across agents must be due to randomness in the shocks across agents. We illustrate our procedure by estimating a dynamic trading model for the milk production quota market in Ontario, Canada. Copyright © 2009 The Review of Economic Studies Limited.

Nonparametric identification of dynamic models with unobserved state variables

“…When X * t is continuous, however, our identification results require highlevel ''completeness'' assumptions which are difficult to verify in practice. One worked-out example (in Section 4.2) shows that these completeness assumptions are implied by independent initial conditions, in addition to other restrictions on the laws of 1 Subsequent methodological developments for CCP-based estimation include Mira (2002, 2007), Pesendorfer and Schmidt-Dengler (2008), Bajari et al (2007a), Pakes et al (2007), and Hong and Shum (2009). At the same time, Magnac and Thesmar (2002) and Bajari et al (2007b) use the CCP logic to provide identification results for dynamic discrete-choice models.…”

Pairwise Difference Estimation With Nonparametric Control Variables*

“…We will assume here that ξ it +1 is independent of all other covariates in the model. This model was analyzed in detail by Hong and Shum (2004), who assume a deterministic accumulation equation of the form x it +1 = x it + q it . Our assumptions imply conditional on μ( x it , q it ), x it +1 is independent of x it , q it .…”

“…In a stationary setting, agents' optimal policy functions solve the following Bellman equation for t = 1, 2, 3, … : Note that this function does not depend on ξ t +1 because E [ U ( x i τ , s i τ , q τ ;θ) | { q i τ } τ , x t +1 , s t +1 , ξ t +1 ]= E [ U ( x i τ , s i τ , q τ ;θ) |{ q i τ } τ , x t +1 , s t +1 ] for all τ≥ t + 1. We have so agent i 's optimal policy can be expressed compactly as As in Hong and Shum (2004), the optimal policy function q ( x t , s t ; θ, γ) will be nondecreasing in s t conditional on x t if U ( x , s , q ; θ) is supermodular in ( q , s ) given x . This is a useful result because it enables us to recover s it by inverting conditional quantiles of q it given x it .…”

Identification in Discrete Markov Decision Models

“…The linear structure that defines the policy value functions is a general property that is also present in other Markov decision problems and dynamic games outside the discrete choice framework. For instance, a closely related class of decision problems imposes increasing differences conditions on the payoff functions instead of additive separability, see Hong and Shum (2010) in a single agent setting, and Bajari et al (2007) and Srisuma (2013) for games. However, the lack of separability means we can no longer apply Hotz and Miller's inversion lemma, the parameterization of for the identification of the pseudomodel for these problems are therefore less tractable than the ones considered in this paper.…”