Solution of HMM

Question

Consider the following hidden markov model:

$X_t =F X_{t-1} + e_t$; $Z_t = H X_t + v_t$; $X_t \in \mathbb{R}^n, Z_t \in \mathbb{R}^m, e_t \sim N(0,Q), v_t \sim N(0,R)$.

Suppose that the process $X_t$ has unit roots. How does the presence of unit roots in the true Data-Generating Process affects evaluation of likelihood by Kalman filter and estimation of parameters $\theta$ ($F(\theta), H(\theta), R(\theta), Q(\theta)$)? Could you please be more specific in your answer and/or give reference to relevant literature regarding issues of consistency, rate of convergence, possible multimodality and etc. Thank you in advance.

Answer 1

Here's a couple of quotes from this paper. The first:

Under regularity conditions, the maximum likelihood estimators of $\theta$ are consistent and asymptotically normal, with covariance matrix equal to the inverse of the asymptotic Fisher information matrix (see Caines, 1988). Besides the technical conditions regarding the existence of derivatives and their continuity about the true parameter, regularity requires that the model is identifiable and the true parameter values do not lie on the boundary of the parameter space.

Number two:

Pagan (1980) has derived sufficient conditions for asymptotic identifiability in stationary models and sufficient conditions for consistency and asymptotic normality of the maximum likelihood estimators in non stationary but asymptotically identifiable models. Strong consistency of the maximum likelihood estimator in the general case of a non compact parameter space is proved in Hannan and Deistler (1988). Recently, full asymptotic theory for maximum likelihood estimation of nonstationary state space models has been provided by Chang, Miller and Park (2009).

Your model isn't identifiable. This issue came up on an earlier thread here.