Testing the Predictive Ability of Possibly Persistent Variables under Asymmetric Loss

Tests of no predictability under an asymmetric power loss function are considered. While this task does not pose difficulties for stationary predictors, non-standard limiting distributions may arise for standard inferential tools when the putative predictors are endogenous (i.e. there is contemporaneous dependence between the shocks of the regressor and of the dependent variable) and of high persistence (i.e. the predictor is reverting slowly to its long-run mean, if at all). It is argued that endogeneity should be interpreted in relation to the relevant loss-function; thus, no endogeneity under MSE loss does not imply, and is not implied by, lack of endogeneity under an asymmetric loss function. To deal with other loss functions than the MSE loss, an overidentified instrumental variable-based test is proposed. The test statistic uses an instrument of high persistence, yet exogenous, and a possibly endogenous one, yet less persistent. The statistic follows a limiting null chi-squared distribution irrespective of the actual degree of persistence of the predictor. The proposed methodology is applied with the forward premium puzzle by providing evidence that asymmetric losses are of empirical relevance and by subsequently conducting robust inference of the rational expectations hypothesis.