UC Riverside

Out-of-sample tests for equal predictive accuracy have been widely used in economics and finance and are regarded as the "ultimate test of a forecasting model". When two non-nested models are compared, Diebold and Mariano (DM 1995) point out that the t-statistic of the mean squared-error loss-differential is asymptotically standard normal. When two models are nested, however, Clark and McCracken (CM 2001, 2005, 2009) point out that due to the parameter prediction error (PEE), the statistics will result in non-standard distribution. Further more Clark and West (CW 2006, 2007) point out that the DM statistic for testing the equal predictive accuracy of two nested mean regression models gives a favor to a smaller (nested) model, because the DM statistic tends to be negative under the null hypothesis, penalizing the bigger (nesting) model for the finite sample parameter estimation sampling error. They point out that the negative bias can be corrected by adding a non-negative adjustment term. The adjusted DM statistics (DM plus the adjustment term) is equivalent to the "encompassing test".

The thesis consists of three chapters: The first chapter is comparing predictive accuracy and model combination using encompassing test for Nested Quantile Models, we consider using the quantile model and check loss function. We show that the adjusted DM statistics is asymptotically standard normal when out-of-sample to in-sample ratio goes to infinity. The second chapter is comparing nested predictive regression models with persistent predictors, in which we introduce a persistent estimator in the second model. We show that the adjusted DM statistics will still be asymptotically standard normal due to the faster convergence rate of the second model. The third chapter is encompassing test for nested predictive regression models with near unit root and drift, the big model contains a persistent estimator with drift. We show regardless whether drift term (deterministic trend) or the coefficient of autoregressive process of the predictor (stochastic trend) dominates the model, due to the higher than root-n convergence rate of the coefficient in the second model, the adjusted DM statistics is asymptotically standard normal.