GARCH(1,1) parameter estimation optimization method

Question

In estimating a GARCH(1,1) model, $$\sigma_{t+1}^2 = \omega+\alpha \epsilon_t^2+\beta\sigma_t^2$$ Usually the parameter tuple $(\omega,\alpha,\beta)$ is estimated by the quasi-maximal likelihood. However, it seems hard to find the optimal parameter estimation stably. Are there any references for explicitly dealing with the optimization issue?

Answer 1

Let's say we have a time series $\left\{\epsilon_t\right\}_{t=1}^T$ of daily log-returns and we want to estimate the model: \begin{align} \epsilon_t&=\sigma_tu_t ,\quad u_t \overset{iid}{\sim}{\cal N}(0,1)\\ \sigma_t^2&=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 \end{align} If I understood your idea right, then you want to estimate the regression: \begin{align} \epsilon_t^2=\alpha_0+(\alpha_1+\beta_1)\epsilon_{t-1}^2+\eta_t \end{align} However, I dont understand how you come up with this. In my opinion, you start with the conditional variance equation: \begin{equation} \sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2 \end{equation} Now add $w_t=\epsilon_t^2-\sigma_t^2$ on both sides. You obtain: \begin{align} &\sigma_t^2+w_t=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+w_t \\ \leftrightarrow &\sigma_t^2+\epsilon_t^2-\sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+w_t\\ \leftrightarrow &\epsilon_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1\sigma_{t-1}^2+ w_t \\ \end{align} Notice that $w_{t-1}=\epsilon_{t-1}^2-\sigma_{t-1}^2$ and therefore $\sigma_{t-1}^2=\epsilon_{t-1}^2-w_{t-1}$. You obtain: \begin{align} \epsilon_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2+\beta_1(\epsilon_{t-1}^2-w_{t-1})+ w_t \\ \leftrightarrow \epsilon_t^2=\alpha_0+(\alpha_1+\beta_1)\epsilon_{t-1}^2 -\beta_1w_{t-1} +w_t \end{align} If $E(\epsilon_t^4)<\infty $ $\left\{w_t \right\}$ has finite variance and is a weak white noise, so the GARCH(1,1)-model has an ARMA(1,1)-representation for the squared returns. If I remember correctly, OLS for ARMA(1,1) is inconsistent and ML for this model seems to be difficult too. Even if we assume that $u_t \overset{iid}{\sim}{\cal N}(0,1)$ what is the conditional distribution of $w_t$ ? I have no idea what the analytic form of the likelihood would be.

It seems like is is possible to estimate the model this way but to be honest, I have never seen this approach before and I have the feeling that the results will be terrible. Maybe try it and then compare the results with the standard approach via ML ?

Basically, my approach would be:

1. Choose initial value $\theta_0=(\alpha_0,\alpha_1,\beta_1)'$.
2. Choose inital values for $\epsilon_0^2$ and $\sigma_0^2$, $\epsilon_0^2=\sigma_0^2=\frac{1}{T}\sum_{t=1}^T\epsilon_t^2$ is a natural choice.
3. For the given parameter vector $\theta_i$ calculate $\sigma_t^2$.
4. Use the results to calculate the log conditional densities ${\cal l}_t(\theta_i)=-\frac{1}{2}\ln(\sigma_t^2)-\frac{1}{2}\frac{\epsilon_t^2}{\sigma_t^2}$
5. Use some optimization method, I think BHHH is often used. $$ \theta_{i+1}=\theta_i+\lambda\left[\sum_{t=1}^T\frac{\partial{\cal l}_t(\theta_i)}{\partial \theta_i}\frac{\partial{\cal l}_t(\theta_i)}{\partial \theta_i'}\right]^{-1}\sum_{t=1}^T\frac{\partial{\cal l}_t(\theta_i)}{\partial \theta_i} $$
6. Stop if $\vert\vert \theta_{i+1}-\theta_i\vert\vert<\epsilon$, for example $\epsilon=10e^{-4}$. If not return to 3.

However, I am not an expert in optimization, it would be nice to hear other opinions on that.

Answer 2

Would it make sense to do the following?

Let the estimation of $\sigma_t^2$ be $\epsilon_t^2$ and $$\epsilon_{t+1}^2 = \omega+(\alpha+\beta)\epsilon_t^2+u$$ where $u$ is the residual. We apply linear regression to the above AR model and obtain the estimate of $\omega$ and $\alpha+\beta$. We only need to estimate one variable, either $\alpha$ or $\beta$. We do so through maximizing the quasi-maximal likelihood. This time, it is a $1$ as opposed to $3$ dimensional problem, which would be much easier.