GJR-GARCH with $\alpha = 0$ as parameter estimate

Question

I am estimating a GJR-GARCH(1,1) model with variance targeting in R. As data I am using returns on some stock indices. While calculating the GARCH models I obtain $\alpha=0$ for some indices. From what I understand this means that volatility is constant. The code I am using for GJR-GARCH estimation is as follows and is based on the rugarch package:

garch.spec <- ugarchspec(
    variance.model = list(model="gjrGARCH", 
                          garchOrder=c(1,1), 
                          variance.targeting=TRUE), 
    mean.model = list(armaOrder=c(0,0)))
garch.fit <- ugarchfit(
    spec=garch.spec, 
    data=data, 
    solver="nlminb", 
    solver.control=list(trace=0))

And an example of my results:

           mu        alpha1         beta1        gamma1         omega 
-0.0057893647  0.0000000000  0.8666747910  0.1641368776  0.0002181445

Could you please give some advice whether such results are plausible or should I be worried? Of course I can provide the data that causes problems. And obviously I am running univariate estimations so I am taking only one series of index returns at a time.

edit: using a different solver algorithm I was able to obtain different results, however, $\alpha$ still seems to be extremely low for this model.

      mu       alpha1        beta1       gamma1        omega 
3.432135e-04 8.508012e-08 8.607153e-01 2.113815e-01 1.727337e-05 

What is the reasoning behind such low values of $\alpha$, since I am obtaining very similar results in R and in Matlab, so I doubt there is a mistake in the coefficient estimation.

Answer 1

$\alpha=0$ does not imply constant volatility. Consider just a simple Garch(1,1):

$$\sigma^2_t = \omega + \alpha \eta_t^2 + \beta \sigma^2_{t-1}$$

Note that:

$$\sigma^2_t = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1})$$

Now add $\eta_{t+1}^2$ to both sides:

$$\eta_{t+1}^2 = \omega + (\alpha + \beta) \eta_t^2 - \beta (\eta_t^2- \sigma^2_{t-1}) + (\eta_{t+1}^2 - \sigma^2_t).$$

So this is an ARMA(1,1) for $\eta_{t+1}^2$ with shocks: $\eta_{t+1}^2 - \sigma^2_t$.

So even if $\alpha=0$ volatility is not constant.