A simple multiple variance ratio test. (English) Zbl 0776.62086
Summary: Empirical applications of the variance ratio (VR) test frequently employ multiple VR estimates to examine the random walk hypotheses against stationary alternatives. Failing to control the joint test size for these estimates results in very large Type I errors.
This manuscript extends the A. Lo and A. C. MacKinlay [Rev. Financial Stud. 1, 41-66 (1988)] methodology and provides a simple modification for testing multiple variance ratios. Monte Carlo results indicate that the size of our test is close to its nominal size and that it is as reliable as the Dickey-Fuller [D. A. Dickey and W. A. Fuller, Econometrika 49, 1057-1072 (1981; Zbl 0471.62090)] and the Phillips-Perron [P. C. B. Phillips and P. Perron, Biometrika 75, No. 2, 335-346 (1988; Zbl 0644.62094)] unit root tests. For a stationary AR(1) alternative, our test is comparable to both the D-F and P-P tests and seems to be more powerful than these tests against two unit root alternatives, an ARIMA(1,1,1) and an ARIMA(1,1,0).
This manuscript extends the A. Lo and A. C. MacKinlay [Rev. Financial Stud. 1, 41-66 (1988)] methodology and provides a simple modification for testing multiple variance ratios. Monte Carlo results indicate that the size of our test is close to its nominal size and that it is as reliable as the Dickey-Fuller [D. A. Dickey and W. A. Fuller, Econometrika 49, 1057-1072 (1981; Zbl 0471.62090)] and the Phillips-Perron [P. C. B. Phillips and P. Perron, Biometrika 75, No. 2, 335-346 (1988; Zbl 0644.62094)] unit root tests. For a stationary AR(1) alternative, our test is comparable to both the D-F and P-P tests and seems to be more powerful than these tests against two unit root alternatives, an ARIMA(1,1,1) and an ARIMA(1,1,0).
MSC:
| 62P20 | Applications of statistics to economics |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
Keywords:
random walk hypotheses; stationary alternatives; large Type I errors; testing multiple variance ratios; Monte Carlo results; unit root tests; stationary AR(1) alternative; ARIMA(1,1,1); ARIMA(1,1,0)References:
| [1] | 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, 427-431 (1979) · Zbl 0413.62075 |
| [2] | Dickey, D. A.; Fuller, W. A.: Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057-1072 (1981) · Zbl 0471.62090 |
| [3] | Fama, E.; French, K.: Permanent and temporary components of stock prices. Journal of political economy 96, 246-273 (1988) |
| [4] | Hahn, G. R.; Hendrickson, R. W.: A table of percentage points of the distribution of large absolute value of k student t variates and its applications. Biometrika 58, 323-332 (1971) · Zbl 0226.62063 |
| [5] | Hochberg, Y.: Some generalizations of T-method in simultaneous inference. Journal of multivariate analysis 4, 224-234 (1974) · Zbl 0282.62063 |
| [6] | Ibbottson, R.; Sinquefield, R.: Stock, bonds, and inflation: the past and the future. (1989) |
| [7] | Jegadeesh, N.: Evidence of predictable behavior of security returns. Journal of finance 45, 881-898 (1990) |
| [8] | Jegadeesh, N.: Seasonality in stock price mean reversion: evidence from the U.S. And UK. Journal of finance 46, 1427-1444 (1991) |
| [9] | Kim, M.; Nelson, C.; Startz, R.: Mean reversion in stock prices: A reappraisal of the empirical evidence. Review of economic studies 58, 515-528 (1991) |
| [10] | Lo, A.: Long-term memory in stock market prices. Econometrica 59, 1279-1313 (1991) · Zbl 0781.90023 |
| [11] | Lo, A.; Mackinlay, A. C.: Stock market prices do not follow random walks: evidence from a simple specification test. Review of financial studies 1, 41-66 (1988) |
| [12] | Lo, A.; Mackinlay, A. C.: The size and power of the variance ratio test in finite samples. Journal of econometrics 40, 203-238 (1989) |
| [13] | Miller, R. G.: Simultaneous statistical inference. (1966) · Zbl 0192.25702 |
| [14] | Miller, R. G.: Developments in multiple comparison, 1966–1976. Journal of the American statistical association 72, 779-788 (1977) |
| [15] | Perron, P.: Trends and random walks in macroeconomic time series: further evidence from a new approach. Journal of economic dynamics and control 12, 297-332 (1988) · Zbl 0659.62128 |
| [16] | Phillips, P.: Time series regression with a unit root. Econometrica 55, 277-301 (1987) · Zbl 0613.62109 |
| [17] | Phillips, P.; Perron, P.: Testing for a unit root in time series regression. Biometrika 75, 335-346 (1988) · Zbl 0644.62094 |
| [18] | Poterba, J.; Summers, L.: Mean reversion in stock returns: evidence and implications. Journal of financial economics 22, 27-59 (1988) |
| [19] | Richmond, J.: A general method for constructing simultaneous confidence intervals. Journal of the American statistical association 77, 455-460 (1982) · Zbl 0501.62021 |
| [20] | Savin, N. E.: Multiple hypothesis testing. Handbook of economics (1984) · Zbl 0591.62095 |
| [21] | Sedak, Z.: Rectangular confidence regions for the means of multivariate normal distributions. Journal of the American statistical association 62, 626-633 (1967) · Zbl 0158.17705 |
| [22] | Shiller, R. J.; Perron, P.: Testing the random walk hypothesis: power versus frequency of observations. Economics letters 18, 381-386 (1985) · Zbl 1273.91383 |
| [23] | Stolin, M. R.; Ury, H. K.: Tables of the studentized maximum modulus distribution and an application to multiple comparisons among means. Technometrics 21, 87-93 (1979) · Zbl 0426.62049 |
| [24] | Summers, L. H.: Does the stock market rationally reflect fundamental values?. Journal of finance 41, 591-600 (1986) |
| [25] | White, H.: A heteroscedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity. Econometrica 48, 817-838 (1980) · Zbl 0459.62051 |
| [26] | White, H.; Domowitz, I.: Nonlinear regression with dependent observations. Econometrica 52, 143-161 (1984) · Zbl 0533.62055 |
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