Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-05T07:48:04.742Z Has data issue: false hasContentIssue false
  • English
  • Français

THE VARIANCE RATIO STATISTIC AT LARGE HORIZONS

Published online by Cambridge University Press:  09 February 2006

Willa W. Chen  and
Rohit S. Deo
Willa W. Chen
Affiliation:
Texas A&M University
Rohit S. Deo
Affiliation:
New York University
Get access Rights & Permissions [Opens in a new window]

Abstract

We make three contributions to using the variance ratio statistic at large horizons. Allowing for general heteroskedasticity in the data, we obtain the asymptotic distribution of the statistic when the horizon k is increasing with the sample size n but at a slower rate so that k/n → 0. The test is shown to be consistent against a variety of relevant mean reverting alternatives when k/n → 0. This is in contrast to the case when k/n → δ > 0, where the statistic has been recently shown to be inconsistent against such alternatives. Second, we provide and justify a simple power transformation of the statistic that yields almost perfectly normally distributed statistics in finite samples, solving the well-known right skewness problem. Third, we provide a more powerful way of pooling information from different horizons to test for mean reverting alternatives. Monte Carlo simulations illustrate the theoretical improvements provided.The authors thank Bruce Hansen and the referees for useful suggestions and comments that greatly improved the paper. The first author's research was supported by NSF grant DMS-0306726.

Type
Research Article
Information
Econometric Theory , Volume 22 , Issue 2 , April 2006 , pp. 206 - 234
Copyright
© 2006 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anderson, T.W. (1994) The Statistical Analysis of Time Series, Wiley Classics edition. Wiley.
Bollerslev, T. (1986) Generalized autoregressive conditional heteroscedasticity. Journal of Econometrics 31, 307–327.CrossRefGoogle Scholar
Bougerol, P. & N. Picard (1992) Stationarity of GARCH processes and of some non-negative time series. Journal of Econometrics 52, 115–127.CrossRefGoogle Scholar
Brockwell, P. & R. Davis (1996) Time Series: Theory and Methods, 2nd ed. Springer.
Campbell, J., A. Lo, & A. MacKinlay (1997) The Econometrics of Financial Markets. Princeton University Press.
Chen, W. & R. Deo (2004) Power transformations to induce normality and their applications. Journal of the Royal Statistical Society, Series B 66, 117–130.CrossRefGoogle Scholar
Deo, R. (2000) Spectral tests of the martingale hypothesis under conditional heteroscedasticity. Journal of Econometrics 99, 291–315.CrossRefGoogle Scholar
Deo, R. & M. Richardson (2003) On the asymptotic power of the variance ratio test. Econometric Theory 19, 231–239.CrossRefGoogle Scholar
Fama, E. & K. French (1988) Permanent and temporary components of stock prices. Journal of Political Economy 96, 246–273.CrossRefGoogle Scholar
Faust, J. (1992) When are variance ratio tests for serial dependence optimal? Econometrica 60, 1215–1226.Google Scholar
Follmann, D. (1996) A simple multivariate test for one-sided alternatives. Journal of the American Statistical Association 91, 854–861.CrossRefGoogle Scholar
Fuller, W. (1996) Introduction to Time Series Analysis, 2nd ed. Wiley.
Lo, A. & A.C. MacKinlay (1988) Stock market prices do not follow random walks: Evidence from a simple specification test. Review of Financial Studies 1, 41–66.CrossRefGoogle Scholar
Lo, A. & A.C. MacKinlay (1989) The size and power of the variance ratio test in finite samples: A Monte Carlo investigation. Journal of Econometrics 40, 203–238.CrossRefGoogle Scholar
Nelson, D. (1990) Stationarity and persistence in the GARCH(1,1) model. Econometric Theory 6, 318–334.CrossRefGoogle Scholar
Poterba, J. & L. Summers (1988) Mean reversion in stock returns: Evidence and implications. Journal of Financial Economics 22, 27–60.CrossRefGoogle Scholar
Richardson, M. & T. Smith (1991) Tests of financial models with the presence of overlapping observations. Review of Financial Studies 4, 227–254.CrossRefGoogle Scholar
Richardson, M. & J. Stock (1989) Drawing inferences from statistics based on multi-year asset returns. Journal of Financial Economics 25, 323–348.CrossRefGoogle Scholar
Shephard, N. (1996) Statistical aspects of ARCH and stochastic volatility. In D.R. Cox, D.V. Hinkley, & O.E. Barndorff-Nielsen (eds.), Time Series Models in Econometrics, Finance and Other Fields. Chapman and Hall.
Stout, W. (1974) Almost Sure Convergence. Academic Press.