19.6.2 Intercept Correction

19.6.2 Intercept Correction

Forecast models whose parameters are based on all observations in a historical sample typically provide biased forecasts in situations where the model parameters are time varying. Consider the forecasting model $f_{T+1|T}(\beta )$ , where $\beta$ is estimated over some sample $t=1,\dots ,T-1$ , resulting in the forecasting model $f_{T+1|T}({\hat{\mathbf{\beta }}}_T)$ . If the parameters change over time, typically $E[y_{T+1}-f_{T+1|T}(z_T,{\hat{\beta }}_T)]\ne 0.$ . Intercept corrections attempt to estimate this bias. Provided that the parameters do not break by a large amount every period, an estimator of the bias is the forecast error in the previous period, $y_T-f_{T|T-1}({\hat{\beta }}_{T-1})$ . Hence, we can use the intercept corrected forecast $f_{T+1|T}^{\mathrm{IC}}(\hat{\beta })=f_{T+1|T}({\hat{\beta }}_T)+[y_T-f_{T|T-1}({\hat{\beta }}_{T-1})]$ . The bias of the intercept corrected forecast is

$$\begin{array}{rl}& E\left[y_{T+1}-f_{T+1|T}({\hat{\beta }}_T)-(y_T-f_{T|T-1}({\hat{\beta }}_{T-1}))\right]\\ & \\ & =E\left[y_{T+1}-f_{T+1|T}({\hat{\beta }}_T)\right]-E\left[(y_T-f_{T|T-1}({\hat{\beta }}_{T-1}))\right].\end{array}$$

If the two terms are roughly similar—the bias in the forecasts does not change quickly—the bias will be close to 0.

Example 19.6.1 (Linear regression with a single break, continued). Consider the model in (19.1) with a single break ${\beta }_t=\beta +d1(t\le \tau )$ . For this model the bias in the forecast is $E[y_{T+1}-{\hat{\beta }}_T{x}_T]=T^{-1}\tau d^{\prime }E{x}_T$ and hence the bias of the intercept corrected forecast is $d^{\prime }E[T^{-1}\tau x_T-(T-1)\tau x_{T-1}]$ . Assuming that $x_t$ is covariance stationary, this is $\tau d^{\prime }E{x}_T{T}^{-1}(T-1{)}^{-1}$ . This will be close to 0 for large values of T.

The intercept correction approach has been popularized and analyzed by Clements and Hendry (1996, 1998). Analytical results in these papers do not consider the impact of estimation error. For local breaks, the size of the bias is of the same order as the estimation error, so an understanding of this approach needs to consider both effects jointly. In essence though, the intercept correction approach can indeed reduce or remove the bias from estimating a model subject to infrequent breaks. However, this bias reduction comes at the cost of an increase in the variance of the forecast. Hence, the value of this approach depends on the size of the bias to begin with, and the method will be most useful when this bias is relatively large so that the bias correction is large enough to offset the increased variance.

The last line shows that the effect of adding the intercept correction to the forecast is to roughly double the variance of the forecast error relative to the case with no correction. This happens because there are now two forecast errors (one at time $T+1$ , another at time T ) and the variance of this is the sum of the variances minus twice their covariance. Forecast errors are typically either uncorrelated (if we have a very good model) or weakly positively correlated due to omitted (persistent) terms.

Example 19.6.2 (Linear regression with a single break, continued). Consider again the linear regression with a single local break to the parameters ${\beta }_t=\beta +d1(t\le \tau )$

After considerable calculation along the lines of Examples 19.1.1 and 19.1.2, we have

$$\begin{array}{rl}& (y_{T+1}-{\hat{\beta }}_T{x}_T,y_T-{\hat{\beta }}_{T-1}{x}_{T-1}{)}^{\prime }\\ & \sim [\left(\begin{array}{c}{\Sigma }_{X,T-1}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_T\\ {\Sigma }_{X,T-2}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_{T-1}\end{array}\right),\\ & {\sigma }^2\left(\genfrac{}{}{0ex}{}{1+x_T^{\prime }{\Sigma }_{X,T-1}{x}_T}{0}1+x_{T-1}^{\prime }{\Sigma }_{X,T-2}{x}_{T-1}\right)],\end{array}$$

and so

$$\begin{array}{rl}& \left(1-1\right)\left(\begin{array}{l}{y}_{T+1}-{\hat{\beta }}_T^{\prime }{x}_T\\ y_T-{\hat{\beta }}_{T-1}^{\prime }{x}_{T-1}\end{array}\right)\\ & \sim [({\Sigma }_{X,T-1}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_T-{\Sigma }_{X,T-2}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_{T-1}),\\ & {\sigma }^2(2+x_T^{\prime }{\Sigma }_{X,T-1}{x}_T+x_{T-1}^{\prime }{\Sigma }_{X,T-2}{x}_{T-1})].\end{array}$$

The MSE for the intercept corrections approach is

$$\begin{array}{rl}& \mathrm{MSE}=({\Sigma }_{X,T-1}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_T-{\Sigma }_{X,T-2}^{-1}{\Sigma }_{X,\tau }{d}^{\prime }{x}_{T-1}{)}^2\\ & +{\sigma }^2(2+x_T^{\prime }{\Sigma }_{X,T-1}{x}_T+x_{T-1}^{\prime }{\Sigma }_{X,T-2}{x}_{T-1}).\end{array}$$

Notice that we get the bias reduction, but at the cost of essentially doubling the variance component.

These theoretical results show that breaks need to be very large for the gains from the intercept correction method to outweigh the increased variance of the forecast error—a situation unlikely to occur in practice. Such results appear to be borne out in practice. Examining a large number of variables, Rossi (2013a) finds that intercept corrections almost never improve upon univariate autoregressive forecasts.