19.6 AD HOC METHODS FOR DEALING WITH BREAKS

19.6 AD HOC METHODS FOR DEALING WITH BREAKS

Each of the methods discussed in the previous section matches the estimation scheme to the model assumed for the break process. If the forecaster strongly believes that a particular break process generated the data, it makes sense to choose an estimation strategy that matches this particular break process. However, as noted above, it is often difficult to determine the break process most likely to have generated the data. Tests for a single break and tests for random walk breaks are almost equally likely to reject regardless of the break process. Rejections of parameter stability tests are therefore not an indication of the specific break process that generated the data.

In situations where the form of the break process is unknown, instead we might use ad hoc methods that do not relate directly to the type of break process. This section describes a variety of such approaches.

19.6.1 Weighting Schemes That Downweight Past Observations

When ${\beta }_t$ is time varying we generally try to approximate ${\beta }_T$ in the forecast regression through some “average” value for the parameters. A simple scheme is to put greater weight on recent observations than on past observations. This can be accomplished by adding weights to the objective function, i.e., by choosing parameters that minimize the weighted loss,

$${\hat{\beta }}_T=\arg \underset{\beta \in \mathcal{B}}{\min }(T-1{)}^{-1}\sum _{t=1}^{T-1}{\omega }_{t}L(f_{t+1\mid t}(\beta ),y_{t+1}).\text{\hspace{1em}(19.27)}$$

The forecast $f_{T+1|T}({\hat{\mathbf{\beta }}}_T)$ is then based on the estimates, ${\hat{\mathbf{\beta }}}_T$ . The conventional approach for the stationary case arises as a special case with ${\omega }_t=1$ for all t. As described in section 16.2, older observations can be downweighted in different ways. A prominent example is rolling regressions that set ${\omega }_t=1(T-\bar{\omega }\le t\le T-1)$ , i.e., use a rectangular window (equal weights) for the most recent ω¯ observations and 0 otherwise. As the sample expands, this approach includes one new observation and drops one old observation, thereby keeping the estimation window constant at $\bar{\omega }.$ Another approach is to let the estimation window be a fixed proportion, $\rho ,$ of the sample size, i.e., ${\omega }_t=1((1-\rho )T+1\le t\le T)$ . Discounted least squares sets ${\omega }_{\tau T}={\lambda }^{T-\tau }$ for $\lambda \in (0,1]$ for the MSE loss function, although clearly the same idea extends readily to the construction of forecasting models in more general situations. Estimation under discounted least squares objectives is straightforward. For linear regression models and MSE loss, it amounts to weighted least squares.

Rolling regressions employ an intuitive trade-off. Shorter estimation windows are likely to reduce the bias in the estimates due to the use of stale data that comes from a different data-generating process than the one that generates the data at the time the forecast is computed. This bias reduction is achieved at the cost of a decreased precision in the parameter estimates as less data get used. The hope is that the bias reduction more than makes up for the increased parameter estimation error. Discounted least squares estimation makes a similar trade-off, although this method downweights nearby observations and puts some weight on even the oldest data points, potentially mitigating the effects on the variance of the parameter estimates.