# Hidden Markov model regression

#### Abstract

Hidden Markov Model Regression (HMMR) is an extension of the Hidden Markov Model (HMM) to regression analysis. The types of problems to which HMM regression applies can be easily understood by considering the following example: Assume that the net return per share of a corporation is a random variable linearly related to the market return. If the market transits between two states, say high risk and low risk market, and unobservable transitions follow a stationary Markov model then the problem falls into the HMM regression category. In general terms, we assume in HMM regression that at each time t the state $Q\sb{t}\in\{S\sb1,\...,S\sb{N}\}$ of an unobserved Markov chain determines the regression parameter values. We develop the HMM regression model for two situations, one where the covariate $X\sb{t}$ are fixed and the other where $X\sb{t}$ are random variables. For the fixed covariates model, we assume that $Y\sb{t}=X\sbsp{t}{\prime}\beta\sb{Q\sb{t}}+\sigma\sb {Q\sb{t}}\epsilon\sb{t}$ where $\beta\sb{Q\sb{t}}=\beta\sb{i}$ and $\sigma\sb{Q\sb{t}}=\sigma\sb{i}$ if $Q\sb{t}=S\sb{i}.$ The conditional distribution of the error term is assumed to be $N(0,\sigma\sbsp{i}{2}).$ For the random covariates model we assume that the vectors $(Y\sb{t},X\sbsp{t}{\prime})$ are conditionally independent given that $Q\sb{t}=S\sb{i}$ and multivariate normally distributed with parameters that vary with the states.^ The theory of HMM regression is quite new, but some of its development calls on the natural extension of the work by Baum, Petrie and Eagon. We consider the problem of maximum likelihood estimation of the HMMR parameters and develop an analog for the Baum-Welch and Forward-Backward algorithms used in HMM's for our regression case. Parallels are drawn to the EM algorithm and to various related models.^ The methodology developed is applied to the Capital Asset Pricing Model. Simulation studies indicate consistency and asymptotic normality of the suggested estimates. ^

Statistics

#### Recommended Citation

Moshe Fridman, "Hidden Markov model regression" (January 1, 1993). Dissertations available from ProQuest. Paper AAI9331776.
http://repository.upenn.edu/dissertations/AAI9331776

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