R – How to Estimate GARCH with Exogenous Variables in the Mean Equation?

estimationgarchrregressiontime series

What I'm trying to do is estimate the following GARCH(1,1) model in R with the garchFit function from the fGarch package:

Mean equation:

$Y_t = a + bX_t + e_t$

$e_t = z_th_t^{0.5}$

$e_t \sim N(0,h_t)$, $z_t \sim N(0,1)$

Variance equation:

$h_t = \omega + k_0e^2_{t-1} + k_1h_{t-1}$

In the above, $Y_t$ is the response of my mean equation and $X_t$ is the predictor. I will estimate this equation within the GARCH framework because of heteroscedasticity of residuals.

I want estimates of both the mean equation and the variance equation (similar to what EViews would give). However, the garchFit function has two inputs that I'm concerned about:

  • formula – ARMA(m,n) + GARCH/APARCH(p,q) mean and variance specification

As you can see in my equations, my mean equation is not of the ARMA(m,n) form.

  • data – any univariate time series which can be converted into a timeSeries using the generic function as.timeSeries

I'm modeling both $Y_t$ and $X_t$ together; thus it shouldn't be univariate.

If I put this model through EViews, It'll give me parameter estimates of $a$ and $b$. This is what I want, but in R. In R, I can do garchFit(data=Y) (where Y is the column of $Y_t$), and it'll do a GARCH(1,1) over $Y_t = \mu(Y) + e_t$; but it doesn't give me $a$ or $b$ in the mean equation that I want!

Best Answer

The solution is to use another package called rugarch. 

install.packages("rugarch")
require(rugarch)

Let's construct a (Tx2) matrix called data, where column 1 is $Y_t$ and column 2 is $X_t$. 

data <- cbind(rnorm(1000),rnorm(1000))

We can then compute the GARCH(1,1) model that I described in my question:

spec <- ugarchspec(variance.model = list(model = "sGARCH", garchOrder = c(1, 1), 
submodel = NULL, external.regressors = NULL, variance.targeting = FALSE), 
mean.model = list(armaOrder = c(0, 0), external.regressors = matrix(data[,2]), 
distribution.model = "norm", start.pars = list(), fixed.pars = list()))

garch <- ugarchfit(spec=spec,data=data[,1],solver.control=list(trace=0))

Comparing this to EViews: The R coefficient estimates that are greater than $0.01$ differ to the EViews estimates by approximately $1-2\%$, which is acceptable in my mind. $t$-statistics differ by more but it doesn't effect inference.

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