Evaluating Vector Multiplicative Error Models with the Hosking–Ljung–Box Portmanteau Test and Kernel-Based Test Statistics

Abstract

Multivariate nonlinear time series models have experienced many developments for modeling data coming from financial applications. Several financial time series are realizations from nonnegative processes. An important class of models is composed of vector multiplicative error models (vMEM), which can describe contemporaneous correlations among innovations and the dynamic interdependencies among variables. Modeling and estimation issues have been addressed, but few diagnostic checking procedures are available. Here, new tests are proposed to check vMEM models. The asymptotic distributions of the popular Hosking–Ljung–Box (HLB) test statistics are found to converge in distribution to weighted sums of independent Chi-squared random variables under the null hypothesis of adequacy. A generalized HLB test statistic is motivated by comparing a vector spectral density estimator of the residuals with the spectral density calculated under the null hypothesis. Under general conditions, that kind of approach leads to consistent procedures and to powerful measures of lack of fit. To improve the finite sample properties, the spectral test statistics rely on the power transformation of Chen and Deo (J R Stat Soc Ser B 66:117–130, 2004). Appealing properties of the spectral procedures include the distribution-free property and the fact that they converge in distribution to convenient standard normal distributions under the null hypothesis. Simulation experiments are reported to appreciate the properties of the methods. An application using financial data previously analyzed by Cipollini et al. (J Appl Econom 28:1067–1086, 2013) illustrates the merits of our procedures.