19.8 CONCLUSION

The processes generating most economic and financial time series are likely to be changing over time. This raises some important issues. First, can model instability explain some of the empirical findings reported throughout the forecasting literature such as the good empirical performance of forecast combinations and the tendency of forecasting performance to worsen out-of-sample? Given the pervasiveness of model instability found in empirical research, it is natural to suspect that such instability is a key source of model misspecification.

Second, should a forecaster try to detect and model such instabilities and exploit them to generate more accurate forecasts? A variety of approaches have been used here, ranging from fully parametric methods that model the process giving rise to the change in parameters over time to more ad hoc, adaptive methods. A key challenge when deciding which method to use is that tests for model instability usually are not very informative about the nature of any detected instability, i.e., whether it takes the form of frequent small breaks to the parameters or, alternatively, rare but large breaks to the parameters which may even alternate between a few discrete states.

Third, how do we come up with robust forecasting approaches whose performance is not critically linked to getting the break point model exactly right? Model combination (covered in chapter 14) offers one way to achieve this. While using a rolling estimation window or a discounted least squares estimator offer some degree of robustness by putting more weight on recent data than on older data, these approaches are often not optimal and can amplify the effect of estimation error on the forecast which can lead to worse forecasting performance (compared to methods based on an expanding estimation window) in situations where parameter uncertainty is important.

Trending Variables and Forecasting

that standard laws of large numbers apply, so that sums of random variables suitably scaled (usually by the square root of the sample size) converge to normal distributions. Many variables of interest in forecasting—macro variables such as income or money growth, financial variables such as stock prices and trading volume, even weather variables such as temperature—appear to be sufficiently persistent that we may not be confident in such assumptions. We refer to such variables as trending variables, by which we mean that the unconditional mean or variance of the predicted variable diverges over time. For example, if ${\gamma }_t$ has a unit root, then its variance increases linearly in time. If ${\gamma }_t$ is dominated by a time trend, then it has a mean that diverges over time.

In many ways, this persistence issue has little or no effect on many of the questions discussed so far. For example, there are no real issues for loss functions or the choice of loss functions. Expected loss still exists under well-defined assumptions, so the results on loss functions in chapter 2 still apply. Similarly, many of the issues discussed in chapters 4 and 5 continue to apply. For example, uncertainty over parameters remains of the same order in the sample size. Some things do change, however. Many of the trade-offs appearing in the expression for risk will now be different from those discussed earlier, both in magnitude and in terms of the techniques required to understand them. Persistent dynamics in the predicted variable therefore has implications for model building, model selection and comparison, and evaluation of risk.

Suppose the nature of the trending behavior is precisely known—for example we may know that the trend is a deterministic function of time or that the variable has one root on the unit circle and all others well outside the unit circle. In these cases, the complications that arise in the presence of trending variables are relatively straightforward. For example, if there is a single unit root driving the variables, methods such as differencing or cointegration can be applied to model the variables. If there is a deterministic time trend, this can simply be added to the model.

Chapter 3 discussed plug-in estimators and estimators for the parameters of the forecasting model that minimize average loss. We established general conditions under which such estimators converge to the pseudo-true value that minimizes the expected risk. Estimation of the parameters of the forecasting model that minimize average loss require the data to be either strictly stationary (for the average loss with estimated parameters to converge to the expected loss at time T) or weakly stationary (for the average loss with estimated parameters to converge to an average of expected losses). Similarly general results are not available when the variables have unit roots or exhibit other forms of trending behavior; specific results hinge on the locations of the unit roots, the number of unit roots among the variables and also on whether the forecasting model is correctly specified in the sense of being “balanced,” ensuring that the trend in the predicted variable is correctly captured by the forecasting model.

Provided that the forecasting model for ${\gamma }_{T+h}$ is linear and the loss function is quadratic, some general points apply. First, if the trending behavior is correctly specified by the forecasting model, estimation error is unconditionally of the same order as in the ergodic case, i.e., of order $T^{-1}$ . This is true both in- and out-of-sample and holds despite the fact that coefficients in models with unit roots (or time trends) converge at a rate faster than $T^{1/2}$ . Second, when the trend is correctly specified the precise magnitude of the order $T^{-1}$ term differs from model to model. In the ergodic case this term is equal to the number of coefficients estimated; in the nonergodic case this term will depend on the precise form of the trend.

Section 20.1 discusses expected loss in the presence of trending variables. Section 20.2 focuses the problem on univariate forecasting models, while section 20.3 covers multivariate forecasting models. Section 20.4 covers the case with highly persistent predictor variables and section 20.5 discusses how forecast evaluation is affected by the presence of trending or persistent variables. Section 20.6 concludes.