ARMA model checking with data-driven portmanteau tests

Linear ARMA model fitting requires to select the order of the model as accurately as possible. Many past studies are based on portmanteau tests, originally derived for the white noise hypothesis, but applied to the autocorrelations of the estimated residuals, and employ sums of the scaled squares of the first d residual autocorrelations. An automatic choice of d was proposed by Escanciano and Lobato (J Econom 151:140–149, 2009) based on the largest (in absolute value) residual autocorrelation. Such maximal autocorrelation may be itself employed as a test statistic, and Baragona et al. (Test 31:675–698, 2022) proposed white noise tests based on a bivariate statistic consisting of both the portmanteau and the largest autocorrelation. However, the derivation of the asymptotic null distribution requires asymptotic independence of the autocorrelation estimates and this is not true when they are computed on the residuals of a fitted ARMA model. Therefore, we propose to use a linear transformation of the estimated residual autocorrelation in order to achieve independence, in the same spirit as recursive residuals in regression. Monte Carlo experiments are performed for comparing the effectiveness of our new method to the Escanciano Lobato and a more classical portmanteau test. We find that the two data-driven tests are approximately equivalent if non negligible residual autocorrelation is found only at the first lags, while our test is more powerful if autocorrelations at larger lags arises.