Solved – Regression model with GARCH (1, 1) error term

Question

I am attempting to analyze the relationship between exchange rates and stock market prices using the regression model below.

What software package can I use to achieve this? How would I get the variables $a, \beta_1$ and $\varepsilon_t$?

Model: bivariate regression model with GARCH (1, 1) error term ($\varepsilon_t$):

$$\log (EX_t) = a + \beta_1 \log (SP_t) + \varepsilon_t.$$

The level exchange rate series is denoted by $EX$ and first difference data for $EX$ (denoted $EX_1$) is equal to $\log (EX_t/EX_{t-1})$. Level stock price series is denoted by $SP$ and first difference data for $SP$ (denoted $SP_1$) is equal to $\log (SP_t/SP_{t-1})$.

Answer 1

Try "rugarch" package for R. It allows specifying an ARMA-GARCH model with exogenous regressors in both the conditional mean and the conditional variance equations. You will need to select ARMA order of (0,0) and specify $\log(SP_t)$ as an exogenous regressor in the conditional mean. Use functions ugarchspec for model specification and then ugarchfit for model estimation. Something like:

spec = ugarchspec(mean.model = list(armaOrder = c(0, 0), external.regressors = cbind(log(SP)))
fit = ugarchfit(spec=spec, data=log(EX))

Note that your variables will likely be integrated, which is why you will need to use first differences instead of levels (I do not expect them to be cointegrated, in which case you might need to include an error correction term in the equation). Thus something like:

spec = ugarchspec(mean.model = list(armaOrder = c(0, 0), external.regressors = cbind(diff(log(SP))))
fit = ugarchfit(spec=spec, data=diff(log(EX)))

Also, consider whether $\log(SP_t)$ might be endogenous. If so, you will have the endogeneity bias, yielding biased and inconsistent estimates of $\beta_1$. Moreover, consider whether the effect of $\log(SP)$ on $\log(EX)$ might have a time lag.