Forecasting with Trending Data
This chapter examines the problems of dealing with trending type data when there is uncertainty over whether or not we really have unit roots in the data. This uncertainty is practical - for many macroeconomic and financial variables theory does not imply a unit root in the data however unit root tests fail to reject. This means that there may be a unit root or roots close to the unit circle. We first examine the differences between results using stationary predictors and nonstationary or near nonstationary predictors. Unconditionally, the contribution of parameter estimation error to expected loss is of the same order for stationary and nonstationary variables despite the faster convergence of the parameter estimates. However expected losses depend on true parameter values. We then review univariate and multivariate forecasting in a framework where there is uncertainty over the trend. In univariate models we examine trade-offs between estimators in the short and long run. Estimation of parameters for most models dominates imposing a unit root. It is for these models that the effects of nuisance parameters in the models is clearest. For multivariate models we examine forecasting from cointegrating models as well as examine the effects of erroneously assuming cointegration. It is shown that inconclusive theoretical implications arise from the dependence of forecast performance on nuisance parameters. Depending on these nuisance parameters imposing cointegration can be more or less useful for different horizons. The problem of forecasting variables with trending regressors - for example, forecasting stock returns with the dividend-price ratio - is evaluated analytically. The literature on distortion in inference in such models is reviewed. Finally, forecast evaluation for these problems is discussed.
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