# I am trying to fit an GARCH(p,q) model to FX volatility. Should I be interested in the t-value of GARCH parameters?

So what I am trying to do is to model the volatility for different currencies by fitting a GARCH(p, q) model. I am selecting the values of (p, q) by iteratively going through p & q such that max(p, q) = 5. After going through the 24 different combinations of p & q (excluding where p = q = 0), I select the model which has the lowest Akaike Information Criterion. I am doing all this in R using the rugarch package.

For the "best" model selected for some currencies, I observed that the t-value is extremely low. I have observed t-value of omega in a GARCH(1, 1) model as low as 0.09. My question is that can I still use the model if the GARCH parameters are statistically insignificant? Or am I mistaken and the usual "statistical significance of model parameters" which I learned for linear regression doesn't apply for GARCH type models.

I am a newbie to this field but I do not mind trying to learn from the start so any pointers would be greatly appreciated.

The post became kind of lengthy, so if you're only interested in what you should do, just skip to the summary in the end. However, I think most of the confusion about these tests in general stems from a lack of knowledge about what one is actually doing, when applying the test.

First, let's take a look at what we are testing. As you said, you want to see, whether the GARCH parameters are "significant", according to the test, in order to choose a lag order.

Generally, for a parameter estimate $\hat\beta$, we want to test the null hypothesis $\mathcal{H}_0:\hat\beta=\beta_0$ against the alternative $\hat\beta\ne\beta_0$. As we want to see, whether our parameters are "significantly different from zero", we apply the test with $\beta_0=0$. Then, the t-statistic is given by \begin{align}\label{52} t_{\hat\beta}=\frac{\hat\beta-\beta_0}{\sigma_{\hat\beta}}=\frac{\hat\beta}{\sigma_{\hat\beta}}, \end{align} where $\sigma_{\hat\beta}$ denotes the standard error of our estimate $\hat\beta$, i.e. the standard deviation of its sampling distribution.

Now, one of the main assumptions of the test comes into play, namely that the estimator is normally distributed. As you use the rugarch package, estimation is done by MLE. Thus, one can note that the asymptotic distribution of $\hat\beta$ is $\mathcal{N}(0,\sigma_{\hat\beta}^2)$ under the null hypothesis and some regularity conditions. Hence, in this case the t-statistic is asymptotically standard normally distributed. Considering that the standard normal distribution is symmetric, we can approximate the p-value by \begin{align}\label{53} p=2\cdot(1-\Phi(|t_{\hat\beta}|)), \end{align} with $\Phi(\cdot)$ denoting the standard normal cumulative distribution function.

Numerically, to use this test, we firstly have to compute the standard error $\sigma_{\hat\beta}$. This can be (and usually is) done for every estimated parameter at once, by approximating the covariance matrix $\Sigma$ with the inverse observed information matrix ${\mathcal I}^{-1}$. The latter is given by the negative hessian of the likelihood $-\hat{\mathbf{H}}$, or the hessian of the negative likelihood, evaluated at the maximum likelihood estimate. Altogether, this means \begin{align}\label{54} \Sigma\approx\mathcal{I}^{-1}=(-\hat{\mathbf{H}})^{-1}, \end{align} which yields the standard errors for every parameter estimate, by calculating \begin{align}\label{55} \sqrt{\text{diag}\left((-\hat{\mathbf{H}})^{-1}\right)}. \end{align}

Now, let's look into some of the possible issues one might encounter:

• The estimator could be non-normally distributed. In your case, as you're using MLE, the asymptotic gaussian distribution could be inaccurate for small sample sizes.
• The inversion of the hessian may prove to be difficult and require high numerical precision, specifically for highly (in)significant parameters.
• It should be kept in mind that, inherently, statistical tests don't give you the probability that the null hypothesis is true, but the probability of observing the data, given that the null hypothesis is true.

In summary, what I wanted to say is: The test certainly has its issues and shouldn't be used as the only deciding factor. However, it can provide some insight into the impact that a parameter has on a model's estimates.
Thus, when deciding the lag order of models, one should always additionally consult other model selection measures, like e.g. information criteria (AIC/BIC). The t-values can be used as pointers, which parameters could be omitted for a potentially better fit.