19.4 MODELS WITH MULTIPLE BREAKS

19.4 MODELS WITH MULTIPLE BREAKS

The single break model in (19.8) can be extended to a model that breaks at multiple points in time:

$$y_{t+1}=\{\begin{array}{ll}(\beta +d_1{)}^{\prime }{x}_t+{\sigma }_1{\epsilon }_{t+1},& t\le {\tau }_1,\\ ⋮& ⋮\\ (\beta +d_k{)}^{\prime }{x}_t+{\sigma }_k{\epsilon }_{t+1},& {\tau }_{k-1}<t\le {\tau }_k,\\ {\beta }^{\prime }{x}_t+\sigma {\epsilon }_{t+1},& t>{\tau }_k,\end{array}\text{\hspace{1em}(19.18)}$$

where ${\epsilon }_t\sim (0,1)$ and $t=1,\dots ,T.$ . Assuming no breaks between periods T and $T+1_{:}$ , the ideal forecasting model under MSE loss for period $T+1$ is $f_{T+1}={\beta }^{\prime }{x}_T$ As in the previous section, we do not know the timing of the breaks. Note that (19.18) allows the variance of the regression residuals to change discretely at the time of the breaks. We can also augment (19.18) to include variables $\left({\nu }_t\right)$ with constant coefficients; this is known as a partial break model:

$$y_{t+1}=\{\begin{array}{ll}(\beta +d_1{)}^{\prime }{x}_t+{\lambda }^{\prime }{v}_t+{\sigma }_1{\epsilon }_{t+1},& t\le {\tau }_1,\\ ⋮& ⋮\\ (\beta +d_k{)}^{\prime }{x}_t+{\lambda }^{\prime }{v}_t+{\sigma }_k{\epsilon }_{t+1},& {\tau }_{k-1}<t\le {\tau }_k,\\ {\beta }^{\prime }{x}_t+{\lambda }^{\prime }{v}_t+\sigma {\epsilon }_{t+1},& t>{\tau }_k.\end{array}$$

Bai and Perron (1998) provide a procedure for estimating multiple break dates $\{{\tau }_1,\dots ,{\tau }_k\}$ in linear regressions when $k,$ the number of breaks, is unknown and y is univariate.6 For a given number of breaks, Bai and Perron’s procedure chooses the break dates by minimizing the sum of squared residuals across potential break dates. This is a direct extension of the method discussed previously for the case with a single break. Their procedure estimates an entire set of break dates $\{{\hat{\tau }}_1,\dots ,{\hat{\tau }}_k\}$ . Given this set of break dates $\{{\hat{\tau }}_1,\dots ,{\hat{\tau }}_k\}$ , the parameters of (19.18) can be estimated by OLS and the sum of squared residuals can be computed:

$$\begin{array}{rl}& \mathrm{SSR}({\tau }_1,\dots ,{\tau }_k)=\sum _{t=1}^{T-1}\sum _{j=1}^k{\left(y_{t+1}-{\beta }_j^{\prime }{x}_t-d_j^{\prime }{x}_{t}1({\tau }_{j-1}\le t<{\tau }_j)\right)}^2,\\ & {\hat{\tau }}_1,\dots ,{\hat{\tau }}_k=\arg \min \mathrm{SSR}({\tau }_1,\dots ,{\tau }_k),\end{array}\text{\hspace{1em}(19.19)}$$

subject to the constraint that ${\tau }_1<{\tau }_2<\cdots <{\tau }_k$ . As in the case with a single break, some trimming of the data is required—now not only at the endpoints but also to ensure a sufficient distance between the individual breaks to allow consistent estimation of the break parameters.

Under relatively standard assumptions on the predictors and the residuals, and assuming that the break points are asymptotically distinct and exist, Bai and Perron show that their method consistently estimates the true $\mathbf{p}$ break proportions,7 i.e., they consistently estimate $T^{-1}{\hat{\tau }}_i$ for $i=1,\dots ,k$

To overcome the difficulty that the number of breaks is unknown, Bai and Perron suggest a sequential procedure that tests for k versus $k+1$ breaks until no further breaks are detected. With k breaks, there are k + 1 periods for which the parameters are potentially stable. Each of these periods is then tested for a break using a largest F test (SupWald) statistic. Critical values for this test depend on the number of breaks under the null. For details on algorithms to undertake these methods, see Bai and Perron (2003).8 The need to estimate the break date adds to the risk function of the OLS approach so this method again relies on having a good estimate of the final break date. This is most likely to be true in situations with large breaks that are easily identified in the data.

Noting that the last break of the sequence, ${\tau }_k,$ is most important when it comes to generating out-of-sample forecasts, the reverse CUSUM test discussed in section 19.3 could also be used. This method can also provide estimates of the remaining break dates, although these are of course not required.

19.5 FORECASTS THAT MODEL THE BREAK PROCESS

Suppose we suspect that there are breaks to the parameters, or, alternatively, that we have rejected the null of no breaks in a pre-test. We can then consider constructing models of the break process that would allow us to construct better forecasts than if we ignore model instability. One approach is to select and estimate a particular parametric specification for the break process with the hope that this provides a good approximation of the break process and improves forecasts. Different break processes give rise to different estimation methods. This section examines the properties of some popular break processes.

19.5.1 Markov Switching Models

The first-order Markov switching VAR, MSVAR for short, takes the form

$$y_{t+1}={\mu }_{s_{t+1}}+A_{s_{t+1}}{y}_t+{\epsilon }_{t+1},{\epsilon }_{t+1}\sim \mathrm{N}(0,{\Sigma }_{s_{t+1}}).\text{\hspace{1em}(19.20)}$$

Here $s_{t+1}$ is a scalar state variable. Typically it is assumed that $s_{t+1}$ moves around between one of K different values, $s_{t+1}\in \{1,2,\dots ,K\}$ , where $K\ge 2$ is a fixed number. Most empirical applications set $K=2$ . Transitions between states are often assumed to be governed by a homogenous first-order Markov process with state transitions $\mathrm{Pr}(s_{t+1}=i|s_t=j)=p_{ij}$ . The key assumption here is that the same K states repeat so that we can learn something from previous visits to the current state and other states (such as their means, variances and transition probabilities). However, as shown in section 8.3, the optimal forecast is a complex nonlinear function of all the data even under MSE loss.