$y_t = b_0 + b_1 y_{t-1} + u_t$
$u_t = \sqrt{h_t} v_t$
$h_t = a_1 u^2_{t-1} + c_0 c_3 h_{t-1}$
where $v_t$ is a white noise process, $\mathbb{E}(v_t)=0$.
How would we go about testing the hypothesis $H_0\colon \ c_0 c_3 = 0$?
Would we use the log maximum likelihood?
Can this be estimated consistently?
For GARCH coefficients you get (under appropriate regularity conditions) consistent (quasi-) maximum likelihood estimates with their variance matrix and $p$-values from standard software such as the package "rugarch" in R. Then it seems hypothesis testing is straightforward.
On the other hand, a hypothesis such as $H_0 \ \colon c_0 c_3=0$ considers the boundary of the parameter space (GARCH coefficients are restricted to be nonnegative), so matters become complicated. There are several papers and a monograph, all of them by Christian Francq and Jean-Michel Zakoian, that dwell precisely on these issues, if you are interested. The most relevant paper appears to be Francq & Zakoian (2009). It is quite technical, but still quite readable. (I would digest it and put a summary here if I had more time, but for now I am putting just the reference to serve as a starting point.)
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