یک روش عملکرد کنترل برای آزمایش سودمندی متغیرهای نمودار در مدل های پیش بینی و رگرسیون خطی

Many predictors employed in forecasting macroeconomic and finance variables display a great deal of persistence. Tests for determining the usefulness of these predictors are typically oversized, overstating their importance. Similarly, hypothesis tests on cointegrating vectors will typically be oversized if there is not an exact unit root. This paper uses a control variable approach where adding stationary covariates with certain properties to the model can result in asymptotic normal inference for prediction regressions and cointegration vector estimates in the presence of possibly non-unit root trending covariates. The properties required for this result are derived and discussed.

A common problem in constructing forecasting regressions is understanding the sampling uncertainty of the coefficients in the prediction regression when the predictors appear to be persistent. Forecasters care about this sampling uncertainty because it is often unclear whether or not the predictors are useful for forecasting. Given the near unit root behavior of the predictors, standard asymptotic normal distributions have been shown to be very poor approximations to the sampling distributions of the coefficients in the prediction regression. Such approximations lead to oversized tests and are hence potentially misleading as guides as to whether or not the variables actually have any predictive usefulness. The classic example in the forecasting literature is the prediction of stock market returns with the dividend price ratio—the dividend price ratio appears to have a root on or near the unit circle. For this regression the problem extends to other popular predictors such as interest rate differentials and the earnings price ratio as well. Similar issues arise with forecasting exchange rates with the forward premium, forecasting changes in income or consumption with interest rate differentials and ratios of macroeconomic variables. A number of approaches have been taken to provide tests for inclusion of the predictive variables that control size even when the predictive variable has a unit root or near unit root. The common tool has been to use local to unity asymptotics — using limit theory for sequences of models where the largest root for the predictive regression remains in the neighborhood of one — to approximate distributions. The tests then differ in the precise statistic to be computed and how they handle the unknown root. One approach has been to use Bonferroni or related methods (Cavanagh et al., 1995, Lewellen, 2004 and Campbell and Yogo, 2006). Alternatively Moreira and Jansson (2006) condition on a sufficient statistic for the root to remove the dependence in their test. None of these methods dominates each other theoretically or in practice. The precise nature of the problem and these methods are reviewed in Section 2 of this paper. The methods we have so far have some limitations. First, for each of these methods extending the methods beyond the bivariate regression is extremely challenging, and no results exist in the literature for more than a single predictor in the regression. This is in part because the methods themselves are somewhat cumbersome to apply. This paper suggests a different approach to obtaining tests which control size. In the method presented here additional covariates are added to the regression. The problems of size distortion and inference are shown to depend on a convolution of the parameters of the trending process and nuisance parameters that describe the relationships between the shocks to the forecasting regression and the shocks to the variables used for forecasting. Judiciously chosen, the covariates have the potential to remove the dependence of the hypothesis tests on the nuisance parameters describing the trend, and so provide inference that is robust to lack of knowledge over the trending behavior of the data. The method is discussed in Section 2 in a simplified bivariate case and examined for very general models in Section 3. The method is applicable for a very wide set of problems. First, we allow for the inclusion of a general number of regressors in the prediction regression, hence we are not (as in the methods above) restricted to a single regressor. Second, for the prediction regression the method involves only the running of OLS regressions, and hence even for the general case is straightforward to apply. Third, inference is standard asymptotic (mixed) normal and so hypothesis tests are also straightforward to apply. We also show, in Section 4, that the method can also be used when examining ‘cointegrating’ regressions where there is uncertainty over a unit root. Such uncertainty is ubiquitous in the cointegration literature, hence the use of unit root and rank pretests. With use of additional stationary covariates, we show similar results as for the prediction equation problem—asymptotically standard (mixed) normal inference can still apply even when the roots are not exactly on the unit circle. The suggested method, as in the case in instrumental variables, requires finding additional data with particular properties. Such data are referred to as orthogonalizing covariates. In Section 5 we go into more detail regarding these properties. We suggest how such variables can be found in practice. We also show, in Theorem 5, that the use of such covariates that satisfy these properties have at minimum the same power (and generally better power) as would be obtained if we knew the size of the largest root of the predictor variable. Since all of the methods discussed in Section 2 cannot have better power than the case of the root known, this leads directly to the implication that none of the current methods can have better power against any alternative than the method suggested here when we indeed have orthogonalizing covariates. Monte Carlo results are presented in Section 6 to examine both size and power properties of the suggested method. We also examine the implications of Theorem 5 for the relationship between these methods and the popular Campbell and Yogo (2006) procedure in terms of power. Proofs of the results are contained in an Appendix.

This paper analyzes how control variables that are not persistent can be employed in linear regressions with persistent regressors to reduce or eliminate size distortions of hypothesis tests for the coefficients on these variables. In particular interest is focussed on inference on the coefficients on the persistent covariates. It is shown that in the context of both predictive regressions (where a persistent regressor is employed to forecast a variable that may or may not be persistent) and for near cointegrating regressions (where we suspect that the trends in each of the variables is annihilated by a linear combination but we are unsure if the trend is an exact unit root process) that the addition of control variables to the regression may result in standard asymptotic analysis for the coefficients on the trending regressors even when we do not know the precise specification of the trend itself. Further, when such orthogonalizing control variables are available, inference is improved above and beyond that available through ignoring the control variables but knowing the exact nature of the trend. Finding candidate orthogonalizing covariates — discussed in Section 5 — follows from economic theory and considering