How do I estimate a GARCH model after estimating a VAR model?

Question

I've seen examples where people have estimated GARCH models after ARMA or ARIMA models, but how would you estimate volatility of a VAR model? For example, suppose I have the following data for three stock market indices:

ssc install getsymbols
getsymbols ^FTSE ^GSPC ^N225, ya fy(2015) freq(w)

And suppose after cleaning the data I'm left with ln_FTSE, ln_GSPC, ln_N225 representing the weekly closing prices. Then after estimating a var model, for example:

var d.ln_FTSE d.ln_GSPC d.ln_N225, lags(1)

How would I go about estimating a volatility model like GARCH? Also, how would I check clustering volatility of the VAR model?

Answer 1

As Jonas_Dim wrote in a comment, for a multivariate time series you use multivariate GARCH-models that describe the conditional covariance matrix. Some examples of such models are BEKK-GARCH, DCC-GARCH, GOGARCH and copula GARCH.

Regarding how to check volatility clustering of the VAR model, see "How should I test for multivariate ARCH effects in R?", "No ARCH effect in the Univariate case but there is an ARCH effect in the Multivariate case?" and "VAR - ARCH LM Test results are conflicting".

Regarding how to choose between alternative models, you could compare the models' performance out of sample; you could compare their AIC values; or you could do the diagnostics (test for remaining ARCH effects, see if the true error distribution matches the assumed error distribution). This logic is not really specific to GARCH but is based on the general ideas in model selection.