Abstracts
Abstract
We review several exact sign-based tests that have been recently proposed for testing orthogonality between random variables in the context of linear and nonlinear regression models. The sign tests are very useful when the data at the hands contain few observations, are robust against heteroskedasticity of unknown form, and can be used in the presence of non-Gaussian errors. These tests are also flexible since they do not require the existence of moments for the dependent variable and there is no need to specify the nature of the feedback between the dependent variable and the current and future values of the independent variable. Finally, we discuss several applications where the sign-based tests can be used to test for multi-horizon predictability of stock returns and for the market efficiency.
Appendices
Bibliography
- Bahadur, R., and L. J. Savage (1956). “The Nonexistence of Certain Statistical Procedures in Nonparametric Problems.’’ Annals of Mathematical Statistics, 27: 1115-1122.
- Boldin, M. V., G. I. Simonova, and Y. N. Tyurin (1997). “Sign-based Methods in Linear Statistical Models.’’ in Translations of Mathematical Monographs, Volume 162. Maryland: American Mathematical Society..’’
- Campbell, B., and J. M. Dufour (1991). “Overrejections in Rational Expectations Models: A Non-parametric Approach to the Mankiw-Shapiro Problem.’’ Economic Letters, 35: 285-290.
- Campbell, B., and J. M. Dufour (1995). “Exact Nonparametric Orthogonality and Random Walk Tests.’’ Review of Economics and Statistics, 77: 1-16.
- Campbell, B., and J. M. Dufour (1997). “Exact Nonparametric Tests of Orthogonality and Random Walk in the Presence of a Drift Parameter.’’ International Economic Review, 38: 151-173.
- Campbell, J., and M. Yogo (2006). “Efficient Tests of Stock Return Predictability.’’ Journal of Financial Economics, 81: 27-60.
- Cavanagh, C. L., G. Elliott, and J. H. Stock (1995). “Inference in Models with Nearly Integrated Regressors.’’ Econometric Theory, 11: 1131-1147.
- Cochrane, J. (1991). “Volatility Tests and Efficient Markets: A Review Essay.’’ Journal of Monetary Economics, 27: 463-487.
- Coudin, E., and J. M. Dufour (2009). “Finite-sample Distribution-free Inference in Linear Median Regressions under Heteroscedasticity and Non-linear Dependence of Unknown Form.’’ The Econometrics Journal, 12: S19-S49.
- Dufour, J.-M. (1990). “Exact Tests and Confidence Sets in Linear Regressions with Autocorrelated Errors.’’ Econometrica, 58: 475-494.
- Dufour, J.-M. (2003). “Identification, Weak Instruments and Statistical Inference in Econometrics.’’ Canadian Journal of Economics, 36: 767-808.
- Dufour, J.-M., and M. Hallin (1991). “Nonuniform Bounds for Nonparametric Tests.’’ Econometric Theory, 7: 253-263.
- Dufour, J.-M., F. Jouneau, and O. Torrès (2008). “On (non)-testability: Applications to Linear and NonlinearSemiparametric and Nonparametric Regression Models.’’ Technical report, Department of Economics and CIREQ, McGill University, and GREMARS, Université de Lillie 3.
- Dufour, J. M., and A. Taamouti (2010). “Exact Optimal Inference in Regression Models under Heteroskedasticity and Non-normality of Unknown Form.’’ Computational Statistics & Data Analysis, 54: 2532-2553.
- Gibbons, M. R., S. A. Ross, and J. Shanken (1989). “A Test of the Efficiency of a Given Portfolio.’’ Econometrica, 57: 1121-1152.
- Gungor, S., and R. Luger (2009). “Exact Distribution-free Tests of Mean-variance Efficiency.’’ Journal of Empirical Finance, 16: 816-829.
- Gungor, S., and R. Luger (2013). “Testing Linear Factor Pricing Models with Large Cross Sections: A Distribution-free Approach.’’ Journal of Business and Economic Statistics, 31: 66-77.
- Jansson, M., and M. J. Moreira (2006). “Optimal Inference in Regression Models with Nearly Integrated Regressors.’’ Econometrica, 74: 681-715.
- Jegadeesh, N. (1991). “Seasonality in Stock Price Mean Reversion: Evidence from the U.S., and the U.K.’’ Journal of Finance, 46: 1427-1444.
- Koenker, R., and G. Bassett, Jr. (1982). “Tests of Linear Hypotheses and L1 Estimation.’’ Econometrica, 50: 1577-84.
- Lehmann, E. L. (1959). Testing Statistical Hypotheses. New York: John Wiley.
- Lehmann, E., and C. Stein (1949). “On the Theory of some Non-parametric Hypotheses.’’ Annals of Mathematical Statistics, 20: 28-45.
- Liu, W., and A. S. Maynard (2007). “ A New Application of Exact Nonparametric Methods to Long-horizon Predictability Tests.’’ Studies in Nonlinear Dynamics & Econometrics, 11: 163-199.
- Luger, R. (2003). “Exact Non-parametric Tests for a Random Walk with Unknown Drift under Conditional Heteroscedasticity.’’ Journal of Econometrics, 115: 259-276.
- Mankiw, N. G., and M. Shapiro (1986). “Do we Reject too often? Small Sample Properties of Tests of Rational Expectations Models.’’ Economic Letters, 20: 139-145.
- Pratt, J., and J. Gibbons (1981). Concepts of Nonparametric Theory. New York: Springer-Verlag.
- Rousseeuw, P. J., and A. Leroy (1987). Robust Regression and Outlier Detection, Wiley Series in Probability and Mathematical Statistics. Wiley, New York.
- Rousseeuw, P. J., and V. Yohai (1984). “Robust Regression by Means of S-Estimators.’’ in Robust and Nonlinear Time Series Analysis, edited by J. Franke, W. Hardle, and D. Martin, Lecture Notes in Statistics No. 26, Springer Verlag, Berlin.
- Sidak, Z. (1967). “Rectangular Confidence Regions for the Means of Multivariate Normal Distributions.’’ Journal of the American Statistical Association, 62: 626-633.
- Stambaugh, R. (1999). “Predictive Regressions.’’ Journal of Financial Economics, 54: 375-421.
- Yohai, V. J., and R. Zamar (1988). “High Breakdown Point Estimates of Regression by Means of the Minimization of an Efficient Scale.’’ Journal of the American Statistical Association, 83: 406-413.
- White, H. (1980). “A Heteroskedasticity-consistent Covariance Matrix Estimator and a Direct Test for Heteroskedasticity.’’ Econometrica, 48: 817-838.