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ARIMA models, Ljung-Box statistic, Time series residuals.
We derive the exact joint asymptotic distribution for multiple Box-Pierce statistics, and use these results to determine appropriate critical values in joint testing problems of time series goodness-of-fit. A novel alpha-rationing scheme, motivated by the sequence of conditional probabilities for the statistical tests, is developed and implemented. This method can be used to produce critical values and p-values both for each step of the sequential testing procedure, and for the procedure as a whole. Efficient computational algorithms are discussed. Simulation studies assess the impact of finite samples on the real Type I error. It is also demonstrated empirically that the conventional x-squared critical values for the Box-Pierce statistics are too small, with a Type I error rate greater than the nominal; the new method does not suffer from this defect, and allows for more rigorous model-building.
Tucker McElroy and Brian Monsell. (2012). The Multiple Testing Problem for Box-Pierce Statistics. Center for Statistical Research & Methodology Research Report Series (Statistics #2012-15). U.S. Census Bureau. Available online at <http://www.census.gov/srd/papers/pdf/rrs2012-15.pdf>.
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