Abstract
We consider Markov-switching regression models, i.e. models for time series regression analyses where the functional relationship between covariates and response is subject to regime switching controlled by an unobservable Markov chain. Building on the powerful hidden Markov model machinery and the methods for penalized B-splines routinely used in regression analyses, we develop a framework for nonparametrically estimating the functional form of the effect of the covariates in such a regression model, assuming an additive structure of the predictor. The resulting class of Markov-switching generalized additive models is immensely flexible, and contains as special cases the common parametric Markov-switching regression models and also generalized additive and generalized linear models. The feasibility of the suggested maximum penalized likelihood approach is demonstrated by simulation. We further illustrate the approach using two real data applications, modelling (i) how sales data depend on advertising spending and (ii) how energy price in Spain depends on the Euro/Dollar exchange rate.
Highlights
In regression scenarios where the data have a time series structure, there is often parameter instability with respect to time (Kim et al 2008)
The AICbased smoothing parameter selection led to mean integrated squared error (MISE) estimates that overall were marginally lower than their counterparts obtained when using cross-validation (0.555 compared to 0.565, averaged over all four functions being estimated), so again in the following we report the results obtained based on the Akaike Information Criterion (AIC)-type criterion
We fitted the following Markov-switching generalized additive models (MS-generalized additive models (GAMs)) to the advertising data: yt = β0(st ) + f(xt ) + β1(st ) yt−1 + σst t, t=1908, . . . , 1960, again with t i∼id N (0, 1). This formulation is a semiparametric version of the general MS-GAM formulation, where we nonparametrically model the effect of the advertising expenditure xt but assume a simple linear effect of the previous year’s sales, yt−1, on the current year’s sales, yt
Summary
In regression scenarios where the data have a time series structure, there is often parameter instability with respect to time (Kim et al 2008). The aim of the present work is to provide effective and accessible methods for a nonparametric estimation of the functional form of the predictor These build on a) the strengths of the hidden Markov model (HMM) machinery (Zucchini and MacDonald 2009), in particular the forward algorithm, which allows for a simple and fast evaluation of the likelihood of a Markov-switching regression model (parametric or nonparametric), and b) the general advantages of penalized B-splines, i.e. P-splines (Eilers and Marx 1996), which we employ to obtain almost arbitrarily flexible functional estimators of the relationship between target variable and covariate(s). The flexibility of the HMM machinery allows for the consideration of models from a much bigger class, which we term Markov-switching generalized additive models (MS-GAMs) These are generalized additive models (GAMs) with an additional time component, where the predictor—including additive smooth functions of covariates, parametric terms and error terms—is subject to regime changes controlled by an underlying Markov chain, analogously to (1).
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