Asymptotic Inference for the Weak Stationary Double AR(1) Model

Affiliation(s)

Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada. M3J 1P3.

Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada. M3J 1P3.

ABSTRACT

An AR(1) model with ARCH(1) error structure is known as the first-order double autoregressive (DAR(1)) model. In this paper, a conditional likelihood based method is proposed to obtain inference for the two scalar parameters of interest of the DAR(1) model. Theoretically, the proposed method has rate of convergence O(n^{-3/2}). Applying the proposed method to a real-life data set shows that the results obtained by the proposed method can be quite different from the results obtained by the existing methods. Results from Monte Carlo simulation studies illustrate the supreme accuracy of the proposed method even when the sample size is small.

An AR(1) model with ARCH(1) error structure is known as the first-order double autoregressive (DAR(1)) model. In this paper, a conditional likelihood based method is proposed to obtain inference for the two scalar parameters of interest of the DAR(1) model. Theoretically, the proposed method has rate of convergence O(n

Cite this paper

F. Chang, A. Wong and Y. Wu, "Asymptotic Inference for the Weak Stationary Double AR(1) Model,"*Open Journal of Statistics*, Vol. 2 No. 2, 2012, pp. 141-152. doi: 10.4236/ojs.2012.22016.

F. Chang, A. Wong and Y. Wu, "Asymptotic Inference for the Weak Stationary Double AR(1) Model,"

References

[1] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of Inflationary Expectations,” Econometrica, Vol. 50, 1982, pp. 987-1007. doi:10.2307/1912773

[2] A. A. Weiss, “ARMA Models with ARCH Errors,” Journal of Time Series Analysis, Vol. 3, 1984, pp. 129- 143. doi:10.1111/j.1467-9892.1984.tb00382.x

[3] D. Guégan and J. Diebolt, “Probabilistic Properties of the β-ARCH-Model,” Statistica Sininca, Vol. 4, 1994, pp. 71-87.

[4] M. Borkovec and C. Klüppelberg, “The Tail of the Stationary Distribution of an Autoregressive Process with ARCH(1) Errors,” Annals of Applied Probability, Vol. 11, No. 4, 1998, pp. 1220-1241.

[5] S. Ling, “Estimation and Testing Stationarity for Double-Auto Regressive Models,” Journal of the Royal Statistical Society Series B, Vol. 66, No. 1, 2004, pp. 63-78. doi:10.1111/j.1467-9868.2004.00432.x

[6] A. Wald, “Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large,” Transactions of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3

[7] T. Di Ciccio, C. Field and D. A. S. Fraser, “Approxmation of Marginal Tail Probabilities and Inference for Scalar Parameters,” Biometrika, Vol. 77, No. 1, 1990, pp. 77-95. doi:10.1093/biomet/77.1.77

[8] O. E. Barndorff-Nielsen, “Inference on Full and Partial Parameters Based on the Standardized Signed Log-like- lihood Ratio,” Biometrika, Vol. 73, 1986, pp. 307-322.

[9] O. E. Barndorff-Nielsen, “Modified Signed Log-Like- lihood Ratio Statistic,” Biometrika, Vol. 78, No. 3, 1991, pp. 557-563. doi:10.1093/biomet/78.3.557

[10] D. A. S. Fraser and N. Reid, “Ancillaries and Third-Order Significance,” Utilitas Mathematica, Vol. 47, 1995, pp. 33-53.

[11] D. Ling, “A Double AR(p) Model: Structure and Estimation,” Statistica Sinica, Vol. 17, 2007, pp. 161-175.

[12] S. Ling and D. Li, “Asymptotic Inference for a Nonstationary Double AR(1) Model,” Biometrika, Vol. 95, No. 1, 2008, pp. 257-263. doi:10.1093/biomet/asm084

[13] O. D. Anderson, “Time Series Analysis and Forecasting: the Box-Jenkins Approach,” Butterworth, 1976.

[1] R. F. Engle, “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of Inflationary Expectations,” Econometrica, Vol. 50, 1982, pp. 987-1007. doi:10.2307/1912773

[2] A. A. Weiss, “ARMA Models with ARCH Errors,” Journal of Time Series Analysis, Vol. 3, 1984, pp. 129- 143. doi:10.1111/j.1467-9892.1984.tb00382.x

[3] D. Guégan and J. Diebolt, “Probabilistic Properties of the β-ARCH-Model,” Statistica Sininca, Vol. 4, 1994, pp. 71-87.

[4] M. Borkovec and C. Klüppelberg, “The Tail of the Stationary Distribution of an Autoregressive Process with ARCH(1) Errors,” Annals of Applied Probability, Vol. 11, No. 4, 1998, pp. 1220-1241.

[5] S. Ling, “Estimation and Testing Stationarity for Double-Auto Regressive Models,” Journal of the Royal Statistical Society Series B, Vol. 66, No. 1, 2004, pp. 63-78. doi:10.1111/j.1467-9868.2004.00432.x

[6] A. Wald, “Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large,” Transactions of the American Mathematical Society, Vol. 54, No. 3, 1943, pp. 426-482. doi:10.1090/S0002-9947-1943-0012401-3

[7] T. Di Ciccio, C. Field and D. A. S. Fraser, “Approxmation of Marginal Tail Probabilities and Inference for Scalar Parameters,” Biometrika, Vol. 77, No. 1, 1990, pp. 77-95. doi:10.1093/biomet/77.1.77

[8] O. E. Barndorff-Nielsen, “Inference on Full and Partial Parameters Based on the Standardized Signed Log-like- lihood Ratio,” Biometrika, Vol. 73, 1986, pp. 307-322.

[9] O. E. Barndorff-Nielsen, “Modified Signed Log-Like- lihood Ratio Statistic,” Biometrika, Vol. 78, No. 3, 1991, pp. 557-563. doi:10.1093/biomet/78.3.557

[10] D. A. S. Fraser and N. Reid, “Ancillaries and Third-Order Significance,” Utilitas Mathematica, Vol. 47, 1995, pp. 33-53.

[11] D. Ling, “A Double AR(p) Model: Structure and Estimation,” Statistica Sinica, Vol. 17, 2007, pp. 161-175.

[12] S. Ling and D. Li, “Asymptotic Inference for a Nonstationary Double AR(1) Model,” Biometrika, Vol. 95, No. 1, 2008, pp. 257-263. doi:10.1093/biomet/asm084

[13] O. D. Anderson, “Time Series Analysis and Forecasting: the Box-Jenkins Approach,” Butterworth, 1976.