The State-Space Representation

Abstract

In the following chapters we will be concerned with the statistical analysis of MS(M)-VAR(p) models. As a formal framework for these investigations we employ the state-space model which has been proven useful for the study of time series with unobservable states. In order to motivate the introduction of state-space representations for MS(M)-VAR(p) models it might be helpful to sketch its use for the three main tasks of statistical inference:

• 1. Filtering & smoothing of regime probabilities: Given the conditional density function p(yt|Yt-1, 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 full-sample 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 state-space form, prediction of the system is a straightforward task. See Chapter 4 and Section 8.5.