Nonparametric identification of dynamic models with unobserved state variables

We consider the identification of a Markov process {W t , X t *} for t=1,2,...,T when only {W t } for t=1, 2,..,T is observed. In structural dynamic models, W t denotes the sequence of choice variables and observed state variables of an optimizing agent, while X t * denotes the sequence of serially correlated state variables. The Markov setting allows the distribution of the unobserved state variable X t * to depend on W t-1 and X t-1 *. We show that the joint distribution of (W t , X t *, W t-1 , X t-1 *) is identified from the observed distribution of (W t+1 , W t , W t-1 , W t-2 , W t-3 ) under reasonable assumptions. Identification of the joint distribution of (W t , X t *, W t-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.