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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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