ARMA Autocorrelation Analysis: Parameter Estimation and Goodness of Fit Test

Document Type : Original Article

Author
Ahmad Reza Soltani
Department of Statistics and Operations Research, Faculty of Science, Kuwait University.
10.22034/jirss.2022.706993
Abstract
The celebrated Ljung-Box residual analysis is a widely used method in time series for the parameter estimation and the goodness of fit test for the ARMA time series models. The question is whether the autocorrelation function of the fitted ARMA(p,q) model for an observed time series, at different lags, in the Ljung-Box estimation method, is close to the correlogram of observed series. The answer indeed is not affirmative. In this article, firstly, we present a new procedure in solving the Yule-Walker equations for the exact computation of the autocorrelation functions of ARMA(p,q) models. Secondly, we provided a goodness of fit procedure using the limiting distribution of the the sample correlation function. Thirdly, we establish a new parameters estimation method based on examining the model autocorrelation function against the series autocorrelation coefficients. The effectiveness of the procedure is brought into sight using simulated data.
Keywords
Time Series ARMA(p
q) Models
Yule-Walker Equations
Autocorrelation Function
Parameters Estimation
Goodness of Fit
Model Diagnosing
Subjects
Basrak, B., R. A. Davis, and T. Mikosch (1999). The Sample ACF of a Simple Bilinear Process, Stoch. Process. Appl. 83 1-14.
Box, G. E. P. and D. A. Pierce, (1970). Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models. J. American Statist. Asso. 65 (332) 1509–1526.
Brockwell, P. J. and Davis, R. A. (1991). Time Series: Theory and Methods. Second edition, Springer-Verlag, Berlin Chatfield, Chris (1999). Introduction to Analysis of Time Series. Fifth Edition, CRC Press, Boca Raton, USA; first edition (1975), Chapman & Hall, London.
Chien-Fu Wu (1981). Asymptotic theory of nonlinear least squares estimation. Ann. Statist., 9, 3, 501-513.
Khmaladze E. (1998). Goodness of fit tests for "Chimeric" alternatives. Statist. Neerlandica, 52, 1, 90-111.
Khmaladze E, and H. L. Koul (2009). Goodness-of-fit problem for errors in nonparametric regression: distribution free approach. Ann. Statist., 37, 6, 3165-3185.
Koul, H. L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models, 2nd Eds. Springer Lecture Notes in Statistics, 166, Berlin.
Ljung, G. M. and G. E. P. Box (1978). On a Measure of a Lack of Fit in Time Series Models. Biometrika. 65 (2) 297–303.
Pourahmadi, M. (2001). Foundations of Time Series Analysis and Prediction Theory. John Wiley, New York.
Pourahmadi, M. (2003). High-Dimensional Covariance Estimation. John Wiley, New York.
Richardson, G. D. and Bhattacharyya, B. B. (1986). Consistent estimators in nonlinear regression for a noncompact parameter spaces. Ann. Statist., 14, 4, 1591-1596.
Shumway, R. H. and Stoffer, D. S. (2017). Time Series Analysis and Its Applications. Fourth Edition, Springer-Verlag, New York, Berlin.
Soltani, A. R. (2019). A pivot function and its limiting distribution: applications in goodness of fit and testing hypothesis. Statistics, 53, 6, 1386-1395.
Mathematica, Version 12.1, (2020). Wolfram Research, Inc. Champaign, IL.
Tsay, R. S. and M. Pourahmadi (2017). Modeling structured correlation matrices. Biometrika, 104, 237-242.
Journal of the Iranian Statistical Society
Volume 21, Issue 2
December 2022
Pages 1-20

  • Receive Date 20 December 2022
  • Revise Date 06 July 2023
  • Accept Date 15 August 2023