log likelihood function for ar(1)-garch(1)

Question

I am trying to find log likelihood function for ar(1)-garch(1) model

I tried below links for solution:

Maximum likelihood in the GJR-GARCH(1,1) model

How to derive the conditional likelihood for a AR-GARCH model?

I understood that in this case we have to put value of mu=0zero +0one*rt-1 and value of variance as sigma^2

After that I don't know how to proceed?

Could anyone help me out?

Thank You

Answer 1

Suppose $r_t=\phi_0+\phi_1r_{t-1}+a_t$

$a_t=\epsilon_t\sigma_t$

$\sigma_t^2=\alpha_0+\alpha_1a_{t-1}^2+\beta_1\sigma_{t-1}^2$

where {$\epsilon_t$} is Gaussian white noise series with mean 0 and variance 1. Let's $\theta = (\phi_0,\phi_1,\alpha_0,\alpha_1,\beta_1)$. We have

$$r_t\mid r_{t-1},\sigma_{t-1},a_{t-1},\theta \sim \mathcal{N}(\phi_0+\phi_1r_{t-1}, \alpha_0+\alpha_1a_{t-1}^2+\beta_1\sigma_{t-1}^2)\,.$$ Then $$\mathcal{L} = p(r_1,r_2,\dots,r_T\mid\theta) = p(r_1\mid\theta)p(r_2\mid r_1,\sigma_1,a_1,\theta)p(r_3\mid r_2,\sigma_2,a_2,\theta)\dots p(r_T\mid r_{T-1},\sigma_{T-1},a_{T-1},\theta)$$

You can then compute the log likelihood recursively by supposing $r_1\sim\mathcal{N}(\frac{\phi_0}{1-\phi_1},\frac{\alpha_0}{1-\alpha_1-\beta_1})$.

Those mean and variance are obtained as follows :

Suppose the mean of $r_t$ is constant : $\mu = \mathbb{E}[r_t]$ then $$\mu = \mathbb{E}[r_t] = \phi_0 + \phi_1\mathbb{E}[r_{t-1}] + \mathbb{E}[a_t] = \phi_0+\phi_1\mu\,.$$ So $\mu = \frac{\phi_0}{1-\phi_1}$.

The same analyze for $\sigma_t^2$. First remark that $\mathbb{E}[\sigma_t^2] = \mathbb{E}[a_t^2]$ then $$\mathbb{E}[\sigma_t^2] = \alpha_0 + \alpha_1\mathbb{E}[\sigma_{t-1}^2] + \beta_1\mathbb{E}[\sigma_{t-1}^2]\,.$$ So $\mathbb{E}[\sigma_t^2] = \frac{\alpha_0}{1-\alpha_1-\beta_1}$.

It optimization can be done numerically. Suppose $T=2$, then $$log\mathcal{L} = \log p(r_1\mid\theta) + \log p(r_2\mid r_1,\sigma_1,a_1,\theta)\,.$$

As $r_1\sim\mathcal{N}(\frac{\phi_0}{1-\phi_1},\frac{\alpha_0}{1-\alpha_1-\beta_1})$ and $r_2\sim\mathcal{N}(\phi_0+\phi_1\,r_1,\alpha_0+\alpha_1\,a_1^2+\beta_1\sigma_1^2)$, for each $\theta$ the loglikelihood can be computed and maximized.