MarkovSwitching Vector Autoregressions pp 2946  Cite as
The StateSpace Representation
Abstract
 1.
Filtering & smoothing of regime probabilities: Given the conditional density function p(ytYt1, ξt), the discrete Markovian chain as regime generating process ξt, and some assumptions about the initial state \({y_0} = {\left( {{{y'}_0},...,{{y'}_{1  p}}} \right)^\prime }\) of the observed variables and the unobservable initial state ξ0 of the Markov chain, the complete density function p(ξ, Y) is specified. The statistical tools to provide inference for ξt given a specified observation set Yτ, τ ≤ T are the filter and smoother recursions which reconstruct the time path of the regime, \(\left\{ {{\xi _t}} \right\}_{t = 1}^T\) under alternative information sets:
 $${\hat \xi _{t\left \tau \right.}},\quad \tau < t\quad predicted\quad regime\,probabilities.$$
 $${\hat \xi _{t\left \tau \right.}},\quad \tau = t\quad filtered\quad regime\,probabilities,$$
 $${\hat \xi _{t\left \tau \right.}},\quad t < \tau \leqslant T\quad smoothed\quad regime\,probabilities.$$

In the following, mainly the filtered regime probabilities, \({\hat \xi _{t\left t \right.}}\) and fullsample smoothed regime probabilities, \({\hat \xi _{t\left T \right.}}\), are considered. See Chapter 5.
 2.
Parameter estimation & testing: If the parameters of the model are un known, classical Maximum Likelihood as well as Bayesian estimation methods are feasible. Here, the filter and smoother recursions provide the analytical tool to construct and evaluate the likelihood function. See Chapters 6–9.
 3.
Forecasting: Given the statespace form, prediction of the system is a straightforward task. See Chapter 4 and Section 8.5.
Keywords
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Reference
 1.Hamilton [1994a] considers MSIA(M)AR(p) and MSM(M)AR(p) models. A similar approach is taken in Hall & Sola [1993a], Hall & Sola [1993b] and Funke et al. [1994].Google Scholar
 2.Some information about the necessary updates of filtering and estimation procedures under nonnormality of u_{t} are provided by Holst et al. [1994].Google Scholar