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Abstract
We present a decomposition of the correlation coefficient between xt and xt−k into three terms that include the partial and inverse autocorrelations. The first term accounts for the portion of the autocorrelation that is explained by the inner variables {xt−1 , xt−2 , …, x t− k+1}, the second one measures the portion explained by the outer variables {x t+1, x t+2, } ∪ {x t−k−1, x t−k−2,…} and the third term measures the correlation between x t and xt−k given all other variables. These terms, squared and summed, can form the basis of three portmanteau-type tests that are able to detect both deviation from white noise and lack of fit of an entertained model. Quantiles of their asymptotic sample distributions are complicated to derive at an adequate level of accuracy, so they are approximated using the Monte Carlo method. A simulation experiment is carried out to investigate significance levels and power of each test, and compare them to the portmanteau test.