Matstat seminarium fredagen 8 december 2000


Tobias Ryden, Lund


TITLE: Maximum-likelihood estimation in hidden
Markov models and Markov-switching autoregressions

ABSTRACT: By a hidden Markov model (HMM) is meant a bivariate process
$(X, Y)=\{(X_k,Y_k)\}$ such that $X$ is a Markov chain and, informally
speaking, $Y_n$ is given by $X_n$ and noise. Similarly, a
Markov-switching autoregression (MSAR) is a process for which $Y_n$
is given by lagged $Y$'’s, $X_n$ and noise. In either case only $Y$ is
observed. HMMs and MSARs have been applied in a wide range of
areas, including finance and econometrics. Asymptotic analysis of 
the MLE in HMMs and MSARs involves the non-homogeneous Markov
chain $X|Y$. In this talk we present a new deterministic bound on the
mixing rate of this chain, valid if the transition probabilities 
of $X$ are all positive. We show how this bound may be used to 
obtain asymptotic properties of the MLE such as consistency and
asymptotic normality. For MSARs, to the best of our knowledge, 
these results are the first ones of their kind.