19.5.2 Change Point Models

19.5.2 Change Point Models

The change point models considered by Chib (1998) provide an alternative to this class of models. Change point models allow the number of states to increase over time and so do not impose that the states are drawn repeatedly from the same fixed set of values, or that “history repeats.” An example from this class of models is

$$y_{t+1}={\mu }_i+{\Sigma }_i{\epsilon }_{t+1},{\tau }_i\le t\le {\tau }_{i+1},\text{\hspace{1em}(19.21)}$$

for $i=1,\dots ,K$ and assuming K states in the sample up to time T . Assuming that the probability of remaining within a particular state is constant, but possibly statespecific, the transitions for this class of models are

$$P=\left(\begin{array}{cccccc}{p}_{11}& p_{12}& 0& \cdots & 0& \\ 0& p_{22}& p_{23}& \cdots & 0& \\ ⋮& ⋮& ⋮& ⋮& ⋮& \\ 0& \cdots & 0& p_{K-1,K-1}& p_{K-1,K}& \\ 0& 0& \cdots & 0& 0& 1\end{array}\right).\text{\hspace{1em}(19.22)}$$

Here $p_{i,i+1}=1-p_{ii}$ is the probability of exiting from state i to state $i+1$ . According to this model the state variable either remains in the current state or moves to a new one. Usually the model is estimated under the assumption of k regime shifts occurring during the historical sample $t=1,\dots ,T$ . However, the implication of nonrepeated states is that the number of unique states can be expected to grow as the sample size expands.

Because the process could shift from state K at time T to a new state $K+1$ at time $T+1$ , forecasts generated by this model require us to make assumptions about the distribution from which future parameter values, ${\mu }_{k+1}$ and ${\Sigma }_{k+1}$ , are drawn in case of a break. Moreover, if forecasting two or more periods ahead, one also needs assumptions about the distribution from which $\varphi \mathbf{k}+1,\mathbf{k}+1$ and future values are drawn. Pesaran, Pettenuzzo, and Timmermann (2006) propose ways to model both the magnitude of shifts in parameters and the duration of new regimes. Given the paucity of detectable break points for most economic variables, in practice this approach requires combining prior information with empirical evidence on the frequency and magnitude of breaks. Pesaran, Pettenuzzo, and Timmermann (2006) and Koop and Potter (2007) are examples of this latter approach.

Bauwens, Korobilis, and Koop (2011) find extensive evidence of structural breaks for the majority of series they investigate; more than three-quarters of their series display evidence of one or more breaks. They compare the performance of a range of approaches that formally model the instability process such as the Bayesian discrete break models of Pesaran, Pettenuzzo, and Timmermann (2006) and Koop and Potter (2007) versus ad hoc approaches based on rolling window methods. Empirically, no particular approach is found to be dominant across all variables, as the results depend on the particular variable being considered.