19.3 MODELS WITH A SINGLE BREAK

19.3 MODELS WITH A SINGLE BREAK

For forecasting models with a single break to the parameters we can attempt to estimate the date and magnitude of the break. If the model truly has a single break, this approach has the potential—given a sufficiently accurate estimate of the break date—to provide a better forecasting model based on the post-break parameter values. Consider the linear regression

$$y_{t+1}=\{\begin{array}{ll}(\beta +d{)}^{\prime }{x}_t+{\epsilon }_{t+1},& t\le {\tau }^0,\\ {\beta }^{\prime }{x}_t+{\epsilon }_{t+1},& t>{\tau }^0,\end{array}t=1,\dots ,T-1.\text{\hspace{1em}(19.8)}$$

We could use the forecast ${\mathbf{f}}_{T+1}={\hat{\mathbf{\beta }}}^{\prime }{\mathbf{x}}_T$ , where $(\hat{\mathbf{\beta }},\hat{\mathbf{d}})$ are estimated from a regression that replaces the single (unknown) break point in (19.8) with the estimated break point, $\hat{\tau }$. To estimate the break point, note that for an arbitrary break date, $\tau$, the sum of squared residuals in (19.8) is

$$\mathrm{SSR}(\tau )=\sum _{t=1}^{T-1}(y_t-{\beta }^{\prime }{x}_{t-1}-d^{\prime }{x}_{t-1}1(t\le \tau ){)}^2.\text{\hspace{1em}(19.9)}$$

For each value of $\tau$ we can obtain estimates of $(\hat{\beta }(\tau ),\hat{d}(\tau ))$ by minimizing SSR($\tau$) in (19.9). The least squares estimator for $\tau$ is then given by

$$\hat{\tau }=\arg \underset{\tau \in ({\tau }_{\min },{\tau }_{\max })}{\min }\mathrm{SSR}(\tau ).\text{\hspace{1em}(19.10)}$$

Note that we “trim” the data by searching only for breaks between lower and upper limits, ${\tau }_{\min }$ and ${\tau }_{\max }$ . Some minimal amount of data is required, in part to make estimation feasible in both the pre- and post-break samples, and also for the asymptotic justifications to be reasonable. Typical choices for ${\tau }_{\min }$ and ${\tau }_{\max }$ exclude 10–15% of the sample at both ends.

In practice, estimates of the break date are often inaccurate. If the estimated break date exceeds the true value $\hat{\tau }>{\tau }^0$ the resulting parameter estimates are inefficient since we could have used more post-break observations to estimate the parameters. Conversely, if the estimated break date occurs prior to $\tau (i.e.,\hat{\tau }<{\tau }^0)$ the estimates will use pre-break data and hence be biased towards $(\beta +d)$

The location of the break date can be estimated by means of either least squares or maximum likelihood methods. Bai (1997) establishes conditions under which least squares estimates of the break proportion $T^{-1}\hat{\tau }$ are consistent for the true break proportion. This result does not imply that $\hat{\tau }\to P{\tau }^0$ , the true break date.5

Least squares methods are consistent for breaks that are an order of magnitude larger than those for which tests have local power, i.e., $\delta =T^{-1/2+r}$ σ d for $0<r<1/2$ . Yao (1987) establishes results for correctly specified maximum likelihood estimates for independently distributed data.

Pesaran and Timmermann (2002) propose an alternative approach to estimating the break point. Rather than directly testing for a break, Pesaran and Timmermann address how much historical data to use, viewed from the perspective of the forecasting date, T , at the end of the sample. The idea is to perform a sequence of break tests and then limit the estimation sample to post-break data. Pesaran and Timmermann first reverse the ordering of the data and then employ a reverse-ordered CUSUM squared (ROC) break test. To see how this works, let ${\dot{\tilde{Y}}}_t$ and ${\tilde{X}}_t$ denote the reverseordered data so that ${\tilde{Y}}_t=(y_t,y_{t-1},...,y_1)$ and ${\tilde{X}}_t=(x_t,x_{t-1},\dots ,x_1)$ . For linear forecasting models $y_{t+1}={\beta }^{\prime }{x}_t+{\epsilon }_{t+1}$ , the ROC method involves running a set of regressions using only post-break data $[\tau :T-1]$ for a sequence of break dates $\tau ={\tau }_{\min },{\tau }_{\min }-1,\dots ,2,1$ , where $\tau =1$ indicates using the full sample. Specifically, define ${\tilde{Y}}_{T,\tau }=(y_T,y_{T-1},\dots ,y_{\tau }{)}^{\prime }$ and ${\tilde{X}}_{T,\tau }=(x_T,x_{T-1},\dots ,x_{\tau }{)}^{\prime }$ . Then the (backward) recursive least squares estimate of $\dot{\mathbf{\mu }}$ is given by

$$\begin{array}{r}{\tilde{\beta }}_{\tau }=({\tilde{X}}_{T,\tau }^{\prime }{\tilde{X}}_{T,\tau }{)}^{-1}{\tilde{X}}_{T,\tau }^{\prime }{\tilde{Y}}_{T,\tau }.\end{array}\text{\hspace{1em}(19.11)}$$

Pesaran and Timmermann suggest restricting the first break date, $\tau ={\tau }_{\min }$ , to a certain multiple of the number of regressors to ensure that the parameter estimates associated with the shortest estimation window are not too imprecise.

Each of the regressions based on a given estimation window for the reverseordered data produces a standardized recursive residual,

$${\hat{v}}_{\tau }=\frac{(y_{\tau }-{\hat{\beta }}_{\tau -1}{x}_{\tau -1})}{1+x_{\tau -1}^{\prime }{\left({\tilde{X}}_{T,\tau -1}^{\prime }{\tilde{X}}_{T,\tau -1}\right)}^{-1}{x}_{\tau -1}}.\text{\hspace{1em}(19.12)}$$

Repeating this calculation for each τ results in a sequence $\{{\hat{v}}_{\tau }\},\tau ={\tau }_{\min },$ ${\tau }_{\min -1},\dots ,1$ . The associated sequence of reverse ordered CUSUM squared test statistics $\{{\mathrm{ROC}}_{\tau ,T}\}$ is

\mathrm {ROC} _ { \tau , T } = \frac { \sum _ {j = \tau _ {\mathrm {m i n}}} } ^ { \tau} \hat {v} _ {j} ^ { 2} } { \sum _ {j = \tau _ {\mathrm {m in}}} } ^ { T} \hat {v} _ {j} ^ { 2} } .\tag{19.13}

Under fairly stringent assumptions—regressors fixed in repeated samples and i.i.d. normal errors—Brown, Durbin, and Evans (1975) provide critical values for this sequence. Values for ${\mathrm{ROC}}_{\tau ,T}$ outside the critical value bands indicate a break; Pesaran and Timmermann suggest using the first value of τ for which the test rejects the null of constant parameters as the beginning of the post-break sample. This sample can then be used to estimate the forecasting model. Note that this procedure does not yield a consistent estimate of the break date.

Instead of only including data after the estimated break date, τˆ , Pesaran and Timmermann (2007) note that the expected loss can sometimes be reduced by adding pre-break observations to the post-break observations used to estimate the parameters of the linear regression model. Adding such pre-break observations introduces a bias in the forecast but can also reduce the variance of the estimator. Provided that the magnitude of the break is small enough that the bias does not dominate and provided that the break occurs late enough in the sample that additional observations reduce the variance of the estimates by a large enough amount, the MSE of the forecast can be reduced by incorporating pre-break data points.

Pesaran and Timmermann (2007) characterize the size of the trade-offs analytically for a single break to the coefficient of a strictly exogenous regressor:

$$y_{t+1}=\left(\beta +d\times 1(t\le {\tau }^{\ast })\right)x_t+{\epsilon }_{t+1},\text{\hspace{1em}(19.14)}$$

$$\left(\begin{array}{c}{\epsilon }_{t+1}\\ x_t\end{array}\right)\sim \mathrm{ind}\mathrm{N}\left[\left(\begin{array}{c}0\\ 0\end{array}\right),\left(\begin{array}{cc}{\sigma }^2& 0\\ 0& {\sigma }_x^2\end{array}\right)\right].$$

Suppose that the starting point of the estimation window is $\tau ,$ while ${\tau }^{\ast }$ is the time of the break. Assuming that ${\tau }^{\ast }$ is known, it will never be efficient to ignore postbreak information and so $\tau \le {\tau }^{\ast }+1$ . Let ${\omega }_{\mathrm{pre}}={\tau }^{\ast }-\tau +1$ and ${\omega }_{\mathrm{post}}=T-{\tau }^{\ast }$ be the number of pre-break and post-break observations, respectively, so the total length of the estimation window equals $\omega ={\omega }_{\mathrm{pre}}+{\omega }_{\mathrm{post}}=T-\tau +1$ . Pesaran and Timmermann show that the unconditional MSE associated with an estimation window that starts at $\tau \le {\tau }^{\ast }+1$ is given by

$$E\left[e_{T+1}^2(\tau )\right]={\sigma }^2+{\sigma }_x^2{d}^2\left(\frac{{\omega }_{\mathrm{pre}}({\omega }_{\mathrm{pre}}+2)}{\omega (\omega +2)}\right)+\frac{{\sigma }^2}{\omega -2}.\text{\hspace{1em}(19.15)}$$

The second term in (19.15) captures bias from using pre-break information in the estimation. It is equal to $0\mathrm{if}{\omega }_{\mathrm{pre}}=0$ , and otherwise is positive and increasing in ${\omega }_{\mathrm{pre}}.$ The third term represents estimation error and so is decreasing in the total length of the estimation window, $\omega .$ From this expression it follows that, in this simplest possible case, the optimal pre-break window that minimizes the MSE is longer the smaller the $R^2$ (lower ${\sigma }_x^2/{\sigma }^2)$ , the smaller the size of the break, $d,$ and the shorter the post-break window, ${\omega }_{\mathrm{post}}.$ This result is very intuitive: pre-break observations are more useful for estimation of the parameters of the forecasting model when the postbreak parameters are imprecisely estimated (low $R^2$ and short post-break window, ${\omega }_{\mathrm{post}})$ and this gain in estimation precision is not overwhelmed by the bias effect (small $d)$

Pesaran and Timmermann (2007) also allow the variance of the residuals to break at the same point as the coefficients. This too impacts the usefulness of prebreak observations in minimizing the MSE. A larger post-break variance provides an additional incentive to include pre-break observations because of the greater estimation error in the post-break estimates.

To implement these ideas, Pesaran and Timmermann (2007) suggest the following stopping rule approach. First, estimate the time at which a break may have occurred, $\hat{\tau }.$ If no break is detected, use all the data. If a break is found, proceed by estimating the MSE using only data after the break date, τˆ , $t=\hat{\tau }+1,...,T$ . Next, compute the MSE by including an additional observation, i.e., by using sample observations from $\hat{\tau }$ to $T.$ If the estimate suggests that this reduces the MSE, then continue by adding an additional data point $(\hat{\tau }-1)$ to the sample and again compute the MSE. This method is repeated until the data suggest that including additional pre-break data no longer reduces the MSE. Notice that this method produces an estimation window whose length is time varying: as the distance to the break gets larger, more (post-break) observations are used, while fewer observations are included closer to the estimated break date, τˆ .

Other methods can be used to select the estimation window when model instability is suspected. For example, if the last $T_P$ observations are used for crossvalidation purposes, one can choose the length of the rolling estimation window, ω, that minimizes the following pseudo out-of-sample criterion:

$$\mathrm{MSE}(\omega ,T_p,T)=T_P^{-1}\sum _{t=T-T_P}^{T-1}(y_{t+1}-x_t^{\prime }{\hat{\beta }}_{t-\omega +1:t}{)}^2,\text{\hspace{1em}(19.16)}$$

where ${\hat{\mathbf{\beta }}}_{t-\omega +1:t}$ is the OLS estimate of $\beta$ that uses observations $[t-\omega +1:t]$ . A limitation of this approach is that it requires a sufficiently long evaluation window, $T_p$ , to yield precise MSE estimates. Oftentimes, this will be difficult to obtain, particularly if the time of the break point is close to the end of the sample. As a result, this cross-validation procedure may not yield precise estimates of the optimal estimation window.

A very different procedure that partially handles this issue is to use model averaging to deal with the underlying uncertainty surrounding the selection of the estimation window. Pesaran and Timmermann (2007) propose a simple procedure that combines forecasts associated with estimation windows $\omega \in [{\omega }_0,{\omega }_1]$

$${\hat{y}}_{T+1|T}=\frac{{\sum }_{\omega ={\omega }_0}^{{\omega }_1}(x_T^{\prime }{\hat{\beta }}_{T-\omega +1:T})\mathrm{MSE}(\omega ,T_p,T{)}^{-1}}{{\sum }_{\omega ={\omega }_0}^{{\omega }_1}\mathrm{MSE}(\omega ,T_p,T{)}^{-1}}.\text{\hspace{1em}(19.17)}$$

As in chapter 14, the combination weights are proportional to the inverse of the MSE of the forecasts. If the break is very large, models that start the estimation sample after the break will receive greater weight than models that include pre-break data and thus get affected by a large (squared) bias term.

Procedures that base the forecasting model on an estimator for the break point might not work well if the breaks are small and the date of the break is imprecisely estimated. The results of Bai (1997) noted above show that breaks that are local in magnitude (i.e., of order $T^{-1/2})$ are not consistently estimated by least squares methods. To justify conditioning on the break estimates, breaks therefore need to be large enough that tests reject with probability 1. Elliott and Müller (2007) show that confidence intervals for the break date can be very wide for breaks small enough (local) that tests will detect them only some of the time. Of course, if the breaks are sufficiently small, ignoring them will not be too harmful to the forecast. For intermediate values of the break size, ignoring breaks will result in a larger MSE while estimating the break yields sufficiently imprecise values to also strongly impact the MSE of the forecast and so there are no dominant forecasting strategies for this case.