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

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.

## 1 Answer

$\alpha=0$ 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.

• I understand your point and I was basing the constant volatility statement on a presentation I read online. But can you tell me whether such a result in case of index returns is plausible or have some consequences? I have not encountered such a result before and I'm a little lost . – Masher Dec 28 '15 at 2:30
• Can you provide the link for that presentation? – phdstudent Dec 28 '15 at 12:22
• The presentation can be found here: halweb.uc3m.es/esp/Personal/personas/cbreto/files/teaching/… The beta in my case is not equal to 1, but the presentation covers the basic GARCH(1,1), while I am working with GJR-GARCH(1,1). – Masher Dec 28 '15 at 14:37
• Ha! Notice that the presentation says $\alpha=0$ and $\beta=1$. On that case volatility is constant. That should be straightforward to see from the last equality. But let me know if you have any issues. – phdstudent Jan 18 '16 at 11:48
• The presentation you link makes some inaccurate statements. For example the common mistake that the condition $\alpha+\beta=1$ implies non-stationarity, something that Nelson showed is wrong since 1990 (the correct is that $\alpha+\beta<1$ implies cov stationarity, nec and suff, cov stationarity tighter than stationarity, correct nec and suff condition for stationarity $E[\ln(\alpha u_t^2+\beta)]<0$). – Kiwiakos Mar 13 '16 at 0:49