How can I estimate a GARCH(1,1) model with control variables like this: $$Y_t=a_0+a_1X_t+e_t$$ where$$ e_t\sim N(0,h_t)$$ $$h_t=b_0+b_1e_{t-1}^2+b_3h_{t-1}+b_3Z_t$$ I've checked some packages but can't fix it. Hope you guys can shed some lights!

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    $\begingroup$ I'm voting to close this question as off-topic because it seems better suited for Cross Validated SE. $\endgroup$ – LocalVolatility Nov 23 '17 at 11:00
  • $\begingroup$ @LocalVolatility, this is clearly off topic on Cross Validated because Cross Validated does not deal with software implementation of this kind. $\endgroup$ – Richard Hardy Nov 24 '17 at 19:19

Here is how you do that in R with library "rugarch":

#n=1e3; set.seed(1); x=rnorm(n); set.seed(2); y=rnorm(n); set.seed(3); z=rnorm(n)
spec = ugarchspec(variance.model = list(external.regressors = cbind(z)), mean.model = list(armaOrder = c(0, 0), external.regressors = cbind(x)))
fit = ugarchfit(spec = spec, data = y)

The object fit contains the fitted model. If you uncomment the second line, you can try it out with randomly generated data.

(I suppose you meant $b_2$ -- not $b_3$ -- before $h_{t-1}$.)

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