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It is known that the block-based version of the bootstrap method can be used for distributional parameter estimation of dependent data. One of the advantages of this method is that it improves mean square errors. The paper makes two contributions. First, we consider the moving blocking bootstrap method for estimation of parameters of the autoregressive model. For each block, the parameters are estimated based on the modified maximum likelihood method. Second, we provide a method for model selection, Vuong’s test and tracking interval, i.e. for selecting the optimal model for the innovation’s distribution. Our analysis provides analytic results on the asymptotic distribution of the bootstrap estimators and also computational results via simulations. Some properties of the moving blocking bootstrap method are investigated through Monte Carlo simulation. This simulation study shows that, sometimes, Vuong’s test based on the modified maximum likelihood method is not able to distinguish between the two models; Vuong’s test based on the moving blocking bootstrap selects one of the competing models as optimal model. We have studied real data, the S&P500 data, and select optimal model for this data based on the theoretical results.

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The sample ACF is the most common basic tool in analyzing time-series data. This paper provides a theoretical proof that, under some regularity conditions, sample ACF of a given stationary time series is not absolutely summable. Furthermore, it shows that under some mild conditions, the number of positive and negative sample ACFs and their absolute summation tend to infinity as the length of time series increases. The theoretical results are supported by practical evidence from a simulation study.