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Asymptotic laws of successive least squares estimates for seasonal ARIMA models and application. (English) Zbl 1117.62098
The authors consider a nonstationary seasonal ARIMA model slightly more general than that of S. Hylleberg, R. F. Engle, C. W. J. Granger and B. S. Yoo [J. Econ. 44, No. 1/2, 215–238 (1990; Zbl 0709.62102)]. For detecting stochastic non-stationarity of these models instead of testing the non-invertibility of the seasonal MA part, they adopt an alternative approach based on successive autoregressions $$x_t-\varphi_{i,1}x_{t-1}-\cdots-\varphi_{i,i}x_{t-i}=\varepsilon_{i,t}$$ for successive values $$i=1,2,\dots,$$ with $$\varepsilon_{i,t}$$ being the error terms. The estimates $$\widehat{\varphi}_{i,k}$$, $$i=1,2,\dots$$; $$k=1,\dots,i,$$ are called successive least squares estimates. The known results on the asymptotic distribution of the estimates $$\widehat{\varphi}_{i,k}$$, $$i=1,2,\dots$$; $$k=1,\dots,i,$$ are concerned with the special case where the autoregression order $$i$$ is exactly equal to the AR order of the process.
In this paper, successive Yule-Walker estimates are considered for general seasonal ARIMA models and their asymptotic laws are obtained. This extends results known on least squares estimates for stable-unstable ARMA. The results, combined with a simulation study, indicate that successive autoregressions provide a very useful tool both for identifying seasonal ARIMA processes and for distinguishing between stochastic and deterministic seasonal behaviours.
##### MSC:
 62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH) 62E20 Asymptotic distribution theory in statistics 62P20 Applications of statistics to economics
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##### References:
 [1] DOI: 10.1016/0167-7152(93)90151-8 · Zbl 0764.62070 · doi:10.1016/0167-7152(93)90151-8 [2] Ahtola J., J. Time Series Anal. 8 pp 1– (1987) [3] Arellano C., J. Time Series Anal. 16 pp 147– (1995) [4] Chan N. H., Ann. Statist. 16 pp 367– (1988) [5] Davis R. A., J. Probab. and Mathematical Statistics. 15 pp 227– (1995) [6] Dickey D. A., J. Amer. Statist. Assoc. 74 pp 427– (1979) [7] Dickey D. A., J. Amer. Statist. Assoc. 79 pp 355– (1984) [8] W. A. Fuller (1976 ) Introduction to Statistical time series . New York: John Wiley. · Zbl 0353.62050 [9] DOI: 10.1016/0304-4076(94)90030-2 · Zbl 04520314 · doi:10.1016/0304-4076(94)90030-2 [10] Gourieroux C., Cours de Series Temporelles (1983) [11] Hasza D. P., Ann. Statist. 10 pp 1209– (1982) [12] Hillmer S. C., Applied Time Series Analysis of Economic Data pp 74– (1982) [13] DOI: 10.1016/0304-4076(90)90080-D · Zbl 0709.62102 · doi:10.1016/0304-4076(90)90080-D [14] Leybourne S. J., Journal of Business and Economic Statistics 157 (1994) [15] Li W. K., Biometrika 78 pp 381– (1991) [16] DOI: 10.1214/aos/1030563979 · Zbl 0932.62103 · doi:10.1214/aos/1030563979 [17] Phillips P. C. B., Econometrica 55 pp 277– (1987) [18] Said S. E., Biometrika 71 pp 599– (1984) [19] Saikkonen P., J. Amer. Statist. Assoc. 88 pp 596– (1993) [20] Econometric Theory 9 pp 343– (1993) [21] Tam W., J. Amer. Statist. Assoc. 92 pp 725– (1997) [22] Tanaka K., Econometric theory 6 pp 433– (1990) [23] Tiao G. C., Ann. Statist. 11 pp 856– (1983) [24] Touati A., Ann. Inst. H. Poincarre 26 pp 549– (1990) [25] Truong-Van B., Statistics 28 pp 307– (1996) [26] Tsay R. S., Business and economic statistics 11 pp 225– (1993) [27] Tiao G. C., J. Amer. Statist. Assoc. 79 pp 84– (1984) [28] Ann. Statist. 18 pp 220– (1990) [29] White J. S., Ann. Math. Statist. 29 pp 1188– (1958)
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