Abstract
This paper investigates, by means of Monte Carlo simulation, the effects of different choices of order for autoregressive approximation on the fully efficient parameter estimates for autoregressive moving average models. Four order selection criteria, AIC, BIC, HQ and PKK, were compared and different model structures with varying sample sizes were used to contrast the performance of the criteria. Some asymptotic results which provide a useful guide for assessing the performance of these criteria are presented. The results of this comparison show that there are marked differences in the accuracy implied using these alternative criteria in small sample situations and that it is preferable to apply BIC criterion, which leads to greater precision of Gaussian likelihood estimates, in such cases. Implications of the findings of this study for the estimation of time series models are highlighted.
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References
Akaike, H. (1974). A new look at the statistical model identification, IEEE Trans. Autom. Contr., AC-19, 716–723.
Andersen, N. (1974). On the calculation of filter coefficients for maximum entropy spectral analysis, Geophysics, 39, 69–72.
Bhansali, R. J. (1978). Linear prediction by autoregressive model fitting in the time domain. Ann. Statist., 6, 224–231.
Berk, K. N. (1974). Consistent autoregressive spectral estimates, Ann. Statist., 2, 489–502.
Box, G.E.P, Jenkins, G.M. & G. C. Reinsel(1994). Time Series Analysis: Forecasting and control, 3rd ed., Prentice-Hall, New Jersey.
Burg, J. (1967). Maximum entropy spectral analysis, 37th Annual International Meeting, Soc. of Explor. Geophysics, Oklahoma city, Oklahoma.
De Gooijer, J. G. Abraham, B., Gould, A & L. Robinson (1985). Methods for determining the order of an autoregressive moving average process: A survey, Int. Statist. Rev., 53, 301–329.
Findley, D. F.(1984). On some ambiguities associated with the fitting of ARMA models to time series, J. Time Ser. Anal., 5, 213–225.
Friedlander, B.(1982). Lattice methods for spectral estimation, Proc. IEEE, 70, 990–1017.
Fuller, W. A. (1996). Introduction to statistical time series (2nd ed.), Wiley & sons, New York.
Hannan, E. J. & M. Deistler (1988). The statistical theory of linear systems, Wiley & sons, New York.
Hannan, E. J. & L. Kavalieris (1986). Regression, Autoregression models, J. Time Ser. Anal., 7, 27–49.
Hannan, E. J., McDougall, A.M. & D. S. Poskitt (1989). Recursive estimation for autoregressions, J. R. Statist. Soc., B51, 217–233.
Hannan, E. J. & B. G. Quinn (1979). The determination of the order of an autoregression, J. Roy. Statist. Soc., Ser. B, 40, 190–195.
Hannan, E. J. & J. Rissanen (1982). Recursive estimation of mixed autoregressive moving average order, Biometrika, 69, 81–94.
Kohn, R (1978). Asymptotic properties of time domain Gaussian estimators, Adv. Appl. Prob., 10, 339–359.
Lewis, R. & G. C. Reinsel (1985). Prediction of multivariate time series by autoregressive model fitting, J. Multiv. Anal., 16, 393–411.
Lütkepohl, H. (1985). Comparison of criteria for estimating the order of a vector autoregressive processes, J. Time Ser. Anal., 6, 35–52.
Marple, L. (1980). A new autoregressive spectrum analysis algorithm, IEEE Trans. Acoust speech signal process, 28, 4, 441–454.
Parzen, E. (1969). Multiple time series modeling, In P. R. Krishnaiah (ed.), Multivariate analysis II, 389–409, New York: Academic Press.
Poskitt, D. S. & M. O. Salau (1995). On the relationship between generalized least squares and Gaussian estimation of vector ARMA models, J. Time Ser. Anal., 16, 6, 617–645.
Pötscher, B. M. & S. Srinivasan (1994). A comparison of order estimation procedures for ARMA models, Statistica Sinica, 4, 29–50.
Pukkila, T., Koreisha, S. & A. Kallinen (1990). The identification of ARMA models, Biometrika, 77, 537–548
Rissanen, J. (1978). Modelling by shortest data description, Automatica, 14, 465–471.
Salau, M. O. (1995). On the numerical implementation of the generalised least squares procedure for ARMA estimation, Comm. Statist: Simul. Comput., 24, 1, 111–130.
Schwarz, G (1978). Estimating the dimension of a model, Ann. Statist., 461–464.
Shibata, R (1981). An optimal autoregressive spectral estimate. Ann. Statist., 9, 300–306.
Tjostheim, D. & J. Paulsen (1983). Bias of some commonly-used time series estimates, Biometrika, 70, 2, 389–399.
Tunnicliffe-Wilson, G. (1969). Factorization of the covariance generating functions of a pure moving-average process, SIAM J. Numer. Anal., 6, 1–7.
Tunicliffe-Wilson, G (1979). Some efficient computational procedures for high order ARMA models, J. Statist. Comp. Simul., 8, 301–309.
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Salau, M.O. The effects of different choices of order for autoregressive approximation on the Gaussian likelihood estimates for ARMA models. Statistical Papers 44, 89–105 (2003). https://doi.org/10.1007/s00362-002-0135-6
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DOI: https://doi.org/10.1007/s00362-002-0135-6