Nonparametric identification of dynamic models with unobserved state variables

We consider the identification of a Markov process {Wt, X ∗ t } for t = 1, 2, ..., T when only {Wt} for t = 1, 2, ..., T is observed. In structural dynamic models, Wt denotes the sequence of choice variables and observed state variables of an optimizing agent, while X∗ t denotes the sequence of serially correlated unobserved state variables. The Markov setting allows the distribution of the unobserved state variable X∗ t to depend on Wt−1 and X ∗ t−1. We show that the joint distribution fWt,X∗ t ,Wt−1,X∗ t−1 is identified from the observed distribution fWt+1,Wt,Wt−1,Wt−2,Wt−3 under reasonable assumptions. Identification of fWt,X∗ t ,Wt−1,X∗ t−1 is a crucial input in methodologies for estimating dynamic models based on the “conditional-choice-probability (CCP)” approach pioneered by Hotz and Miller.