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Testing Covariance Stationarity Under Moment Condition Failure with an Application to Common Stock Returns

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Abstract

This paper studies tests for covariance stationarity under conditions which permit failure in the existence of fourth order moments. The problem is important because many econometric diagnostics such as tests for parameter constancy, constant variance and ARCH and GARCH effects routinely rely on fourth moment conditions. Moreover, such tests have recently been extensively employed with financial and commodity market data, where fourth moment conditions may well be quite tenuous and are usually untested. This paper considers several tests for covariance stationarity including sample split prediction tests, cusum of squares tests and modified scaled range tests. When fourth moment conditions fail we show how the asymptotic theory for these tests involves functionals of an asymmetric stable Levy process, in place of conventional standard normal or Brownian bridge asymptotics. An interesting outcome of the new asymptotics is that the power of these tests depends critically on the tail thickness in the data. Thus, for data with no finite second moment, the above mentioned tests are inconsistent. Some new tests for heterogeneity are suggested that are consistent in the infinite variance case. These are easily implemented and rely on standard normal asymptotics. A consistent estimator of the maximal moment exponent of a distribution is also proposed. Again this estimator is easily implemented, has standard normal asymptotics and leads to a simple test for the existence of moments up to a given order. An empirical application of these methods to the monthly stock return data recently studied in Pagan and Schwert (1989a, 1989b) and the daily returns of the Standard and Poors 500 stock index is presented.

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File URL: http://cowles.econ.yale.edu/P/cd/d09a/d0947.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 947.

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Length: 77 pages
Date of creation: Jul 1990
Date of revision:
Publication status: Published in Journal of Empirical Finance (1994), 1: 211-248
Handle: RePEc:cwl:cwldpp:947

Note: CFP 866.
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Keywords: Asymmetric stable process; characteristic exponent; covariance stationarity; cusum of squares test; maximal moment exponent; sample split prediction test; scaled range; stable Levy bridge; stock returns;

References

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  1. G. William Schwert, 1990. "Business Cycles, Financial Crises, and Stock Volatility," NBER Working Papers 2957, National Bureau of Economic Research, Inc.
  2. Davis, Richard & Resnick, Sidney, 1985. "More limit theory for the sample correlation function of moving averages," Stochastic Processes and their Applications, Elsevier, vol. 20(2), pages 257-279, September.
  3. Hamilton, James D., 1988. "Rational-expectations econometric analysis of changes in regime : An investigation of the term structure of interest rates," Journal of Economic Dynamics and Control, Elsevier, vol. 12(2-3), pages 385-423.
  4. Joseph G. Haubrich & Andrew W. Lo, 1991. "The sources and nature of long-term memory in the business cycle," Working Paper 9116, Federal Reserve Bank of Cleveland.
  5. Tim Bollerslev, 1986. "Generalized autoregressive conditional heteroskedasticity," EERI Research Paper Series EERI RP 1986/01, Economics and Econometrics Research Institute (EERI), Brussels.
  6. Blattberg, Robert C & Gonedes, Nicholas J, 1974. "A Comparison of the Stable and Student Distributions as Statistical Models for Stock Prices," The Journal of Business, University of Chicago Press, vol. 47(2), pages 244-80, April.
  7. Phillips, P.C.B., 1990. "Time Series Regression With a Unit Root and Infinite-Variance Errors," Econometric Theory, Cambridge University Press, vol. 6(01), pages 44-62, March.
  8. Engle, Robert F, 1982. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Econometric Society, vol. 50(4), pages 987-1007, July.
  9. Newey, Whitney & West, Kenneth, 2014. "A simple, positive semi-definite, heteroscedasticity and autocorrelation consistent covariance matrix," Applied Econometrics, Publishing House "SINERGIA PRESS", vol. 33(1), pages 125-132.
  10. Peter C.B. Phillips & Vassilis A. Hajivassiliou, 1987. "Bimodal t-Ratios," Cowles Foundation Discussion Papers 842, Cowles Foundation for Research in Economics, Yale University.
  11. Donald W.K. Andrews, 1986. "On the Performance of Least Squares in Linear Regression with Undefined Error Means," Cowles Foundation Discussion Papers 798, Cowles Foundation for Research in Economics, Yale University.
  12. Hall, Joyce A. & Brorsen, B. Wade & Irwin, Scott H., 1989. "The Distribution of Futures Prices: A Test of the Stable Paretian and Mixture of Normals Hypotheses," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(01), pages 105-116, March.
  13. Peter C.B. Phillips & Victor Solo, 1989. "Asymptotics for Linear Processes," Cowles Foundation Discussion Papers 932, Cowles Foundation for Research in Economics, Yale University.
  14. Avram, Florin & Taqqu, Murad S., 1989. "Probability bounds for M-Skorohod oscillations," Stochastic Processes and their Applications, Elsevier, vol. 33(1), pages 63-72, October.
  15. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-84, March.
  16. Akgiray, Vedat & Booth, G Geoffrey, 1988. "The Stable-Law Model of Stock Returns," Journal of Business & Economic Statistics, American Statistical Association, vol. 6(1), pages 51-57, January.
  17. Ploberger, Werner & Kramer, Walter, 1986. "On studentizing a test for structural change," Economics Letters, Elsevier, vol. 20(4), pages 341-344.
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Citations

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Cited by:
  1. Xu, Ke-Li & Phillips, Peter C.B., 2008. "Adaptive estimation of autoregressive models with time-varying variances," Journal of Econometrics, Elsevier, vol. 142(1), pages 265-280, January.
  2. Carmela E. Quintos & Zhenhong Fan & Peter C.B. Phillips, 2000. "Structural Change in Tail Behavior and the Asian Financial Crisis," Cowles Foundation Discussion Papers 1283, Cowles Foundation for Research in Economics, Yale University.
  3. Candelon, Bertrand & Straetmans, Stefan, 2006. "Testing for multiple regimes in the tail behavior of emerging currency returns," Journal of International Money and Finance, Elsevier, vol. 25(7), pages 1187-1205, November.
  4. Marc Sáez & Robert M. Kunst, 1995. "ARCH patterns in cointegrated systems," Economics Working Papers 110, Department of Economics and Business, Universitat Pompeu Fabra.

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