This paper is concerned with the development and application of diagnostic checks for vector linear time series models. A hypothesis testing procedure based upon the score, or Lagrangean multiplier, principle is advocated and the distributions of the test statistic both under the null hypothesis and under a Pitman sequence of alternatives are discussed. Consideration of alternative models with singular sensitivity matrices when the null hypothesis is true leads to an interpretation of the score test as a pure significance test and to a notion of an equivalence class of local alternatives. Portmanteau tests of model adequacy are also investigated and are seen to be equivalent to score tests.
D. S. Poskitt. A. R. Tremayne. "Diagnostic Tests for Multiple Time Series Models." Ann. Statist. 10 (1) 114 - 120, March, 1982. https://doi.org/10.1214/aos/1176345694