Modeling Mean and Volatility Spillovers in GARCH Systems

# GARCH mean and volatility spillover R commands needed


# GARCH mean and volatility spillover R commands needed












I analyzed an MA(1)-GARCH(1,1) model in R, and now I want to test the conditional mean and volatility spillover effect between the two time series (exchange rates) (based on Hamao et al., 1990). Therefore I have to include the exogenous variable from the other stock market. Here is the model:

$R_{j,t-1}$ is the previous currency exchange rates and $\sigma^2_{j,t-1}$ is the squared residual derived from the MA(1)-GARCH(1,1) model applied to $R_{j,t-1}$. I understand the model, but I don’t know how to run this in R.

What are the R commands for that?

## Answer by Richard Hardy (score 1, accepted)

https://quant.stackexchange.com/a/33961

If you were interested in including exogenous1 or predetermined regressors in the conditional mean and variance specifications of a univariate time series, you could do that using the package "rugarch" in R. There you can specify your model with the function `ugarchspec` and estimate it with `ugarchfit`. The specification will use

- `external.regressors=cbind(x.mean)` inside `mean.model` where `x.mean` is the vector corresponding to $R_{j,t−1}$ (for conditional mean spillover);

- `external.regressors=cbind(x.variance)` inside `variance.model` where `x.variance` is the vector corresponding to $\varepsilon^2_{j,t−1}$ (for conditional variance spillover).

Now in your case the extra regressors both in the conditional mean and the conditional variance equation are endogenous (tricky!). To estimate the model consistently and efficiently, you would need to use a (restricted) VARMA-MGARCH model where the MGARCH part is a restricted BEKK-GARCH model, for example. However, the implementation can be quite problematic in practice as there are few software packages that accommodate this model (I think it is available in RATS, but not sure if anywhere else). I do not see an easy way around the problem unless you are willing to simplify the model (e.g. remove the MA components from the condiitonal mean model). For example, using two-stage modelling (first conditional mean, then conditional variance) is not really an option as that could affect consistency (let alone efficiency) of the estimators.

1 exogenous = determined outside the system being modelled