Using Out-of-Sample Mean Squared Prediction Errors to Test the Martingale Difference Hypothesis

We consider using out-of-sample mean squared prediction errors (MSPEs) to evaluate the null that a given series follows a zero mean martingale difference against the alternative that it is linearly predictable. Under the null of zero predictability, the population MSPE of the null "no change" model equals that of the linear alternative. We show analytically and via simulations that despite this equality, the alternative model's sample MSPE is expected to be greater than the null's. We propose and evaluate an asymptotically normal test that properly accounts for the upward shift of the sample MSPE of the alternative model. Our simulations indicate that our proposed procedure works well.

[1]  F. Diebold,et al.  Comparing Predictive Accuracy , 1994, Business Cycles.

[2]  Mark P. Taylor,et al.  The Term Structure of Forward Exchange Premiums and the Forecastability of Spot Exchange Rates: Correcting the Errors , 1993, Review of Economics and Statistics.

[3]  R. MacDonald.  The economics of exchange rates , 1991 .

[4]  Todd E. Clark,et al.  Evaluating Long – Horizon Forecasts ∗ , 2004 .

[5]  R. Hodrick.  Dividend Yields and Expected Stock Returns: Alternative Procedures for Interference and Measurement , 1991 .

[6]  Yangru Wu,et al.  Understanding spot and forward exchange rate regressions , 1997 .

[7]  W. Newey,et al.  Automatic Lag Selection in Covariance Matrix Estimation , 1994 .

[8]  M. Shapiro,et al.  Do We Reject Too Often? Small Sample Properties of Tests of Rational Expectations Models , 1985 .

[9]  C. Engel,et al.  The Forward Discount Anomaly and the Risk Premium: A Survey of Recent Evidence , 1995 .

[10]  Yin-Wong Cheung,et al.  Empirical Exchange Rate Models of the Nineties: Are Any Fit to Survive? , 2002 .

[11]  R. Shiller,et al.  Stock Prices, Earnings and Expected Dividends , 1988 .

[12]  D. Andrews.  Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimation , 1991 .

[13]  Naoto Kunitomo,et al.  Properties of Predictors in Misspecified Autoregressive Time Series Models , 1985 .

[14]  Michael W. McCracken,et al.  Evaluating the Predictability of Exchange Rates Using Long-Horizon Regressions: Mind Your p 's and q 's! , 2005 .

[15]  George Tauchen.  The bias of tests for a risk premium in forward exchange rates , 2001 .

[16]  Norman R. Swanson,et al.  OUT-OF-SAMPLE TESTS FOR GRANGER CAUSALITY , 2001, Macroeconomic Dynamics.

[17]  K. West,et al.  Asymptotic Inference about Predictive Ability , 1996 .

[18]  Richard Schmalensee,et al.  Advertising and aggregate consumption: an analysis of causality , 1980 .

[19]  K. West,et al.  A Utility Based Comparison of Some Models of Exchange Rate Volatility , 1992 .

[20]  Lawrence J. Christiano.  P*: not the inflation forecaster's holy grail , 1989 .

[21]  Chris I. Telmer,et al.  Affine Term Structure Models and the Forward Premium Anomaly , 2001 .

[22]  C. Nelson,et al.  Predictable Stock Returns: The Role of Small Sample Bias , 1993 .

[23]  John Y. Campbell,et al.  Why Long Horizons: A Study of Power Against Persistent Alternatives , 1993 .

[24]  Kenneth D West,et al.  Tests for Forecasts Encompassing When Forecasts Depend on Estimated Regression Parameters , 2001 .

[25]  Todd E. Clark,et al.  Tests of Equal Forecast Accuracy and Encompassing for Nested Models , 1999 .

[26]  K. West.  Another Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator , 1995 .

[27]  R. Stambaugh,et al.  Predictive Regressions , 1999 .

[28]  R. Shiller,et al.  The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors , 1986 .

[29]  Halbert White,et al.  Tests of Conditional Predictive Ability , 2003 .

[30]  Michael W. McCracken.  Asymptotics for Out of Sample Tests of Causality , 1998 .

[31]  P. Newbold,et al.  Tests for Forecast Encompassing , 1998 .

[32]  Kenneth S. Rogoff,et al.  Exchange rate models of the seventies. Do they fit out of sample , 1983 .

[33]  Ching-Kang Ing,et al.  MULTISTEP PREDICTION IN AUTOREGRESSIVE PROCESSES , 2003, Econometric Theory.