On Diagnostic Checking of Vector ARMA-GARCH Models with Gaussian and Student-t Innovations

Abstract

This paper focuses on the diagnostic checking of vector ARMA (VARMA) models with multivariate GARCH errors. For a fitted VARMA-GARCH model with Gaussian or Student-t innovations, we derive the asymptotic distributions of autocorrelation matrices of the cross-product vector of standardized residuals. This is different from the traditional approach that employs only the squared series of standardized residuals. We then study two portmanteau statistics, called Q1(M) and Q2(M), for model checking. A residual-based bootstrap method is provided and demonstrated as an effective way to approximate the diagnostic checking statistics. Simulations are used to compare the performance of the proposed statistics with other methods available in the literature. In addition, we also investigate the effect of GARCH shocks on checking a fitted VARMA model. Empirical sizes and powers of the proposed statistics are investigated and the results suggest a procedure of using jointly Q1(M) and Q2(M) in diagnostic checking. The bivariate time series of FTSE 100 and DAX index returns is used to illustrate the performance of the proposed portmanteau statistics. The results show that it is important to consider the cross-product series of standardized residuals and GARCH effects in model checking

Goodness‐of‐fit tests for the multivariate Student‐t distribution based on i.i.d. data, and for GARCH observations

“…The only problem is the extra randomness introduced in the weight function, which will be discussed in the next paragraph. Alternatively, and as far as practical applications are concerned, the test may be applied with a few fixed degrees of freedom, and the appropriate model can then be decided on the basis of likelihood or other model-choice criteria performed on non-rejected models; see for instance Hafner et al (2020) and Wang and Tsay (2013). For a rigorous model selection procedure based on the CF we refer to Jiménez-Gamero et al (2016).…”

“…We propose an extension of the suggested method to test the validity of a Student‐ t innovation distribution in a given (multivariate) GARCH model, against general alternative distributions. In this connection, a wide variety of diagnostic procedures as well as other informal tests for certain modeling aspects within GARCH models have been applied, including likelihood criteria and information criteria (Rossi and Spazzini, 2010, Creal et al ., 2011), Ljung‐Box, LM and portmanteau tests (Tsay, 2006; Bauwens et al ., 2006; Wang and Tsay, 2013), individual Q‐Q plots and probability integral transform plots (Pesaran and Pesaran, 2007; Dube et al ., 2016) as well as sample autocorrelations (Zheng et al ., 2018). We also refer to the methods of Francq and Zakoïan (2022) for testing assumptions on specific characteristics of the innovation distribution, such as quantiles, moments, and asymmetry.…”

“…More recently however, models with time-varying volatility have been employed in finance, with prime examples being several versions of generalized autoregressive conditional heteroskedasticity (GARCH) models with increasing flexibility, and in this regard applications of StDs with such models are numerous; for reviews of multivariate GARCH models the reader is referred to Bauwens et al (2006), Tsay (2006), Silvennoinen and Teräsvirta (2009), and Boudt et al (2019). In most of these reviews as well as in many other studies, the multivariate StD is consistently suggested as innovation distribution; see for instance, Harvey et al (1992), Pesaran and Pesaran (2007), Santos et al (2013), Rossi and Spazzini (2010), Diamantopoulos and Vrontos (2010), Creal et al (2011), Wang and Tsay (2013), Asai and So (2015), Dube et al (2016), Zheng et al (2018), Chib and Zeng (2020), Chen and Gerlach (2021) and Hafner et al (2020), among others.…”

“…(2013), Rossi and Spazzini (2010), Diamantopoulos and Vrontos (2010), Creal et al . (2011), Wang and Tsay (2013), Asai and So (2015), Dube et al . (2016), Zheng et al .…”

Portmanteau test for a class of multivariate asymmetric power GARCH model

“…Duchesne (2004) (see also Duchesne, 2010) introduced the test that is a direct generalization of the portmanteau test of Li and Mak (1994) to the VEC‐GARCH model. Wang and Tsay (2013) extend Duchesne's approach to the case of multi‐variate GARCH models with Student innovations. Recently, Ke et al .…”