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Test Consistency with Varying Sampling Frequency

Published online by Cambridge University Press:  11 February 2009

Abstract

This paper considers the consistency property of some test statistics based on a time series of data. While the usual consistency criterion is based on keeping the sampling interval fixed, we let the sampling interval take any equispaced path as the sample size increases to infinity. We consider tests of the null hypotheses of the random walk and randomness against positive autocorrelation (stationary or explosive). We show that tests of the unit root hypothesis based on the first-order correlation coefficient of the original data are consistent as long as the span of the data is increasing. Tests of the same hypothesis based on the first-order correlation coefficient of the first-differenced data are consistent against stationary alternatives only if the span is increasing at a rate greater than T½, where T is the sample size. On the other hand, tests of the randomness hypothesis based on the first-order correlation coefficient applied to the original data are consistent as long as the span is not increasing too fast. We provide Monte Carlo evidence on the power, in finite samples, of the tests Studied allowing various combinations of span and sampling frequencies. It is found that the consistency properties summarize well the behavior of the power in finite samples. The power of tests for a unit root is more influenced by the span than the number of observations while tests of randomness are more powerful when a small sampling frequency is available.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1991

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References

REFERENCES

1. Aiyar, R.J. Asymptotic efficiency of rank tests of randomness against autocorrelation. Annals of the Institute of Statistical Mathematics 33 (1981): 255262.10.1007/BF02480939CrossRefGoogle Scholar
2. Bergstrom, A.R. Continuous time stochastic models and issues of aggregation over time. In Intriligator, M.D. and Griliches, Z. (eds.), Handbook of Econometrics, vol. 2, 11451212. Amsterdam: North-Holland, 1984.10.1016/S1573-4412(84)02012-2CrossRefGoogle Scholar
3. Dickey, D.A. Estimation and hypothesis testing for nonstationary time series. PhD Thesis, Iowa State University, Ames, IA, 1976.Google Scholar
4. Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (1979): 427431.Google Scholar
5. Evans, G.B.A. & Savin, N.E.. The calculation of the limiting distribution of the least squares estimator of the parameter in a random walk model. Annals of Statistics 9 (1981): 11141118.10.1214/aos/1176345591CrossRefGoogle Scholar
6. Fuller, W.A. Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
7. Kreiss, J.P. On adaptive estimation in stationary ARMA processes. Annals of Statistics 15 (1987): 112133.10.1214/aos/1176350256CrossRefGoogle Scholar
8. Mikulski, P.W. & Monsour, M.J.. Optimality of the maximum likelihood estimator in first-order autoregressive processes. Mimeo, University of Maryland, 1989.Google Scholar
9. Perron, P. A continuous time approximation to the unstable first-order autoregressive model: the case without an intercept. Econometrica 59 (1991): 211256.10.2307/2938247CrossRefGoogle Scholar
10. Perron, P. The calculation of the limiting distribution of the least squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241255.CrossRefGoogle Scholar
11. Perron, P. Testing for a random walk: A simulation experiment when the sampling interval is varied. In Raj, B. (ed.), Advances in Econometrics and Modeling, 4768. Dordrecht: Kluwer Academic Publisher, 1989.10.1007/978-94-015-7819-6_4CrossRefGoogle Scholar
12. Phillips, P.C.B. Time series regression with unit roots. Economelrica 55 (1987): 277301.CrossRefGoogle Scholar
13. Phillips, P.C.B. Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535547.CrossRefGoogle Scholar
14. Phillips, P.C.B. & Ouliaris, S.. Asymptotic properties of residual based tests for cointegration. Econometrica 58 (1990): 165193.CrossRefGoogle Scholar
15. Priestley, M.B. Spectral Analysis and Time Series. New York: Academic Press, 1981.Google Scholar
16. Rao, C.R. Linear Statistical Inference and Its Applications, 2nd ed. New York: Wiley, 1973.CrossRefGoogle Scholar
17. Roussas, G.G. Contiguity of Probability Measures. Cambridge: Cambridge University Press, 1972.CrossRefGoogle Scholar
18. Sargan, J.D. & Bhargava, A.. Testing residuals from least squares regression for being generated by the Gaussian random walk. Econometrica 51 (1983): 153174.CrossRefGoogle Scholar
19. Shiller, R.J. & Perron, P.. Testing the random walk hypothesis: Power versus frequency of observations. Economics Letters 18 (1985); 381386.CrossRefGoogle Scholar