Testing for Deterministic Components in Vector Seasonal Time Series

ABSTRACT

Certain locally optimal tests for deterministic components in vector time series have associated sampling distributions determined by a linear combination of Beta variates. Such distributions are nonstandard and must be tabulated by Monte Carlo simulation. In this paper, we provide closed form expressions for the mean and variance of several multivariate test statistics, moments that can be used to approximate unknown distributions. In particular, we find that the two-moment Inverse Gaussian approximation provides a simple and fast method to compute accurate quantiles and p-values in small and asymptotic samples. To illustrate the scope of this approximation we review some standard tests for deterministic trends and/or seasonal patterns in VARIMA and structural time series models.

Certain locally optimal tests for deterministic components in vector time series have associated sampling distributions determined by a linear combination of Beta variates. Such distributions are nonstandard and must be tabulated by Monte Carlo simulation. In this paper, we provide closed form expressions for the mean and variance of several multivariate test statistics, moments that can be used to approximate unknown distributions. In particular, we find that the two-moment Inverse Gaussian approximation provides a simple and fast method to compute accurate quantiles and p-values in small and asymptotic samples. To illustrate the scope of this approximation we review some standard tests for deterministic trends and/or seasonal patterns in VARIMA and structural time series models.

KEYWORDS

Vector Time Series, Deterministic Components, Parametric Stability, Non-Invertibility, Unit Roots

Vector Time Series, Deterministic Components, Parametric Stability, Non-Invertibility, Unit Roots

Cite this paper

nullJ. Gallego and C. Díaz, "Testing for Deterministic Components in Vector Seasonal Time Series,"*Open Journal of Statistics*, Vol. 1 No. 3, 2011, pp. 145-150. doi: 10.4236/ojs.2011.13017.

nullJ. Gallego and C. Díaz, "Testing for Deterministic Components in Vector Seasonal Time Series,"

References

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[3] J. Nyblom, “Testing for Deterministic Linear Trend in Time Series,” Journal of the American Statistical Association, Vol. 81, No. 394, 1986, pp. 545-549. doi:10.2307/2289247

[4] D. Kwiatkowski, P. C. B. Phillips, P. Schmidt and Y. Shin, “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root,” Journal of Econometrics, Vol. 54, No. 1-3, 1992, pp. 159-178. doi:10.1016/0304-4076(92)90104-Y

[5] J. Nyblom and A. Harvey. “Testing against Smooth Stochastic Trends,” Journal of Applied Econometrics, Vol. 16, 2001, pp. 415-429. doi:10.1002/jae.604

[6] F. Canova and B. E. Hansen, “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability,” Journal of Business and Economic Statistics, Vol. 13, No. 3, 1995, pp. 237-252. doi:10.2307/1392184

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[13] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3-4, 1961, pp. 419-426. doi:10.1093/biomet/48.3-4.419

[14] R. B. Davies, “Numerical Inversion of a Characteristic Function,” Biometrika, Vol. 60, No. 2, 1973, pp. 415-417. doi:10.1093/biomet/60.2.415

[15] T. W. Anderson and D. A. Darling, “Asymptotic Theory of Certain ‘Goodness of Fit’ Criteria Based on Stochastic Processes,” The Annals of Mathematical Statistics, Vol. 23, No. 2, 1952, pp. 193-212. doi:10.1214/aoms/1177729437

[16] F. Busetti, “Tests of Seasonal Integration and Cointegration in Multivariate Unobserved Component Models,” Journal of Applied Econometrics, Vol. 21, 2006, pp. 419- 438. doi:10.1002/jae.852

[17] J. Nyblom, “Invariant Tests for Covariance Structures in Multivariate Linear Model,” Journal of Multivariate Ana- lysis, Vol. 76, 2001, pp. 294-315. doi:10.1006/jmva.2000.1918

[18] J. MacKinnon, “Approximate Asymptotic Distribution Functions for Unit-Roots and Cointegration Tests,” Journal of Business and Economic Statistics, Vol. 12, 1994, pp. 167-176. doi:10.2307/1391481

[19] J. A. Doornik, “Approximation to the Asymptotic Distributions of Cointegration Tests,” Journal of Economic Surveys, Vol. 12, 1998, pp. 573-593. doi:10.1111/1467-6419.00068

[20] C. R. Rao, “Linear Statistical Inference and its Applications,” 2nd Edition, Wiley, New York, 1973. doi:10.1002/9780470316436

[21] J. Nyblom and A. Harvey, “Tests of Common Stochastic Trends,” Econometric Theory, Vol. 16, 2000, pp. 176-199. doi:10.1017/S0266466600162024

[22] A. M. R. Taylor, “LocallyOptimal Tests against Unit Roots in Seasonal Time Series Processes,” Journal of Time Series Analysis, Vol. 24, No. 5, 2003, pp. 591-612. doi:10.1111/1467-9892.00324

[23] P. C. B. Phillips and S. Jin, “The KPSS Test with Seasonal Dummies,” Economics Letters, Vol. 77, 2002, pp. 239-243. doi:10.1016/S0165-1765(02)00127-1

[24] B. M. Brown, “Cramèr-von Mises Distributions and Permutation Tests,” Biometrika, Vol. 69, No. 3, 1982, pp. 619-624.

[1] M. L. King and G. H. Hillier, “Locally Best Invariant Tests of the Error Covariance Matrix of the Linear Regression Model,” Journal of the Royal Statistical Society, Series B (Methodological), Vol. 47, No. 1, 1985, pp. 98- 102.

[2] J. Nyblom and T. M?kel?inen, “Comparisons of Tests for the Presence of Random Walk Coefficients in a Simple Linear Model,” Journal of the American Statistical Association, Vol. 78, No. 384, 1983, pp. 856-864. doi:10.2307/2288196

[3] J. Nyblom, “Testing for Deterministic Linear Trend in Time Series,” Journal of the American Statistical Association, Vol. 81, No. 394, 1986, pp. 545-549. doi:10.2307/2289247

[4] D. Kwiatkowski, P. C. B. Phillips, P. Schmidt and Y. Shin, “Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root,” Journal of Econometrics, Vol. 54, No. 1-3, 1992, pp. 159-178. doi:10.1016/0304-4076(92)90104-Y

[5] J. Nyblom and A. Harvey. “Testing against Smooth Stochastic Trends,” Journal of Applied Econometrics, Vol. 16, 2001, pp. 415-429. doi:10.1002/jae.604

[6] F. Canova and B. E. Hansen, “Are Seasonal Patterns Constant over Time? A Test for Seasonal Stability,” Journal of Business and Economic Statistics, Vol. 13, No. 3, 1995, pp. 237-252. doi:10.2307/1392184

[7] M. Caner, “A Locally Optimal Seasonal Unit-Root Test,” Journal of Business and Economic Statistics, Vol. 16, No. 3, 1998, pp. 349-356. doi:10.2307/1392511

[8] F. Busetti and A. Harvey, “Seasonality Tests,” Journal of Business and Economic Statistics, Vol. 21, No. 3, 2003, pp. 420-436. doi:10.1198/073500103288619061

[9] K. Tanaka, “Testing for a Moving Average Unit Root,” Econometric Theory, Vol. 6, No. 4, 1990, pp. 433-444. doi:10.1017/S0266466600005442

[10] P. Saikkonen and R. Luukkonen, “Testing for a Moving Average Unit Root in Autoregressive Integrated Moving Average Models,” Journal of the American Statistical Association, Vol. 88, No. 422, 1993, pp. 596-601. doi:10.2307/2290341

[11] W. Tam and G. C. Reinsel, “Tests for Seasonal Moving Average Unit Root in ARIMA Models,” Journal of the American Statistical Association, Vol. 92, No. 438, 1997, pp. 725-738. doi:10.2307/2965721

[12] W. Tam and G. C. Reinsel, “Seasonal Moving-Average Unit Root tests in the Presence of a Linear Trend,” Journal of Time Series Analysis, Vol. 19, No. 5, 1998, pp. 609-625. doi:10.1111/1467-9892.00112

[13] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3-4, 1961, pp. 419-426. doi:10.1093/biomet/48.3-4.419

[14] R. B. Davies, “Numerical Inversion of a Characteristic Function,” Biometrika, Vol. 60, No. 2, 1973, pp. 415-417. doi:10.1093/biomet/60.2.415

[15] T. W. Anderson and D. A. Darling, “Asymptotic Theory of Certain ‘Goodness of Fit’ Criteria Based on Stochastic Processes,” The Annals of Mathematical Statistics, Vol. 23, No. 2, 1952, pp. 193-212. doi:10.1214/aoms/1177729437

[16] F. Busetti, “Tests of Seasonal Integration and Cointegration in Multivariate Unobserved Component Models,” Journal of Applied Econometrics, Vol. 21, 2006, pp. 419- 438. doi:10.1002/jae.852

[17] J. Nyblom, “Invariant Tests for Covariance Structures in Multivariate Linear Model,” Journal of Multivariate Ana- lysis, Vol. 76, 2001, pp. 294-315. doi:10.1006/jmva.2000.1918

[18] J. MacKinnon, “Approximate Asymptotic Distribution Functions for Unit-Roots and Cointegration Tests,” Journal of Business and Economic Statistics, Vol. 12, 1994, pp. 167-176. doi:10.2307/1391481

[19] J. A. Doornik, “Approximation to the Asymptotic Distributions of Cointegration Tests,” Journal of Economic Surveys, Vol. 12, 1998, pp. 573-593. doi:10.1111/1467-6419.00068

[20] C. R. Rao, “Linear Statistical Inference and its Applications,” 2nd Edition, Wiley, New York, 1973. doi:10.1002/9780470316436

[21] J. Nyblom and A. Harvey, “Tests of Common Stochastic Trends,” Econometric Theory, Vol. 16, 2000, pp. 176-199. doi:10.1017/S0266466600162024

[22] A. M. R. Taylor, “LocallyOptimal Tests against Unit Roots in Seasonal Time Series Processes,” Journal of Time Series Analysis, Vol. 24, No. 5, 2003, pp. 591-612. doi:10.1111/1467-9892.00324

[23] P. C. B. Phillips and S. Jin, “The KPSS Test with Seasonal Dummies,” Economics Letters, Vol. 77, 2002, pp. 239-243. doi:10.1016/S0165-1765(02)00127-1

[24] B. M. Brown, “Cramèr-von Mises Distributions and Permutation Tests,” Biometrika, Vol. 69, No. 3, 1982, pp. 619-624.