Autocorrelation in the squared residuals of Multivariate GARCH models Most papers use the Hosking (1980) to detect whether the multivariate GARCH model used captures all heteroskedasticity effects. However, Bauwens (2006)(p.101-102) do state shortcomings of it, when applied in this context, but they do also mention that it does provide a useful diagnostic in many situations.
So, I would like to ask you, whether a better test exists (and if it was available in R or Matlab would also be nice).
 A: I think Li-Mak (Li & Mak, 1994) (univariate) and Ling-Li (Ling & Li, 1997) (multivariate) tests are suitable candidates for error diagnostics in univariate and multivariate GARCH models, respectively. Unlike some other (popular) tests*, they account for the fact that standardized residuals from GARCH models are not equal to true standardized innovations but are mere estimates thereof and contain estimation errors**. However, I have not investigated their performance empirically nor do I remember studying any references doing that. Nevertheless, there is a paper (Wong & Ling, 2005) investigating some practical aspects of and containing tips on the use of these and other tests; I hope it can be useful.
*The deficiencies of some popular (univariate) tests are analyzed in Chen (2002), among other. The deficiencies extend from the univariate to the multivariate case as well, since taking residuals for actual innovations is a problem regardless of the dimension of the time series.
**This problem is characteristic not only to testing residuals of GARCH models, but also those of ARMA models; see the thread "Testing for autocorrelation: Ljung-Box versus Breusch-Godfrey" for why popular tests fail there.
References


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*Li, W. K. & Mak, T. K. (1994). On the squared residual autocorrelations in non-linear time series with conditional heteroscedasticity. Journal of Time Series Analysis 15, 627–36.

*Ling, S. & Li, W. K. (1997). Diagnostic checking of nonlinear multivariate time series with multivariate arch errors. Journal of Time Series Analysis 18, 447–64.

*Wong, H., & Ling, S. (2005). Mixed portmanteau tests for time‐series models. Journal of Time Series Analysis, 26(4), 569-579.

*Chen, Yi−Ting. (2002). On the Robustness of Ljung−Box and McLeod−Li Q Tests: A Simulation Study. Economics Bulletin, 3(17), 1−10.

