1. Are the variables in your GARCH model relevant?
In rugarch you can estimate over 10000 different GARCH models. These models differ in terms of variables used. A good model has only those variables that are relevant. This is equivalent to saying the parameter multiplying the variable is not zero.
2. Example
Consider e.g. the AR(1) GJR GARCH model with skewed student t innovations. This model has eight parameters. For many time series, you don't need such a complex model and thus you can simplify the model by omitting the variables for which the parameter is zero.
Suppose, for example, that the AR(1) parameter is zero, then you should omit it and simply estimate a constant mean model. Similarly, if the gamma1 parameter is zero, then there is no garch in mean effect and you can use a standard GARCH model instead of the GJR GARCH model.
3. Challenge
Finding the zero parameters seems simple, but in fact it is not, because you never observe the true parameter values. You need to estimate them. Due to estimation error, your estimate is always going to be different from zero, even if the true parameter is zero. Let's take the example. Are the ar1 and gamma1 parameters zero? Their estimate is small, but is it small enough to conclude that the true value is zero?
4. Test of statistical significance
To answer this question you need to use a test for statistical significance. It solves the question of finding out whether the magnitude of the estimated coefficient is large enough to conclude that the true parameter is not zero.
It does this by comparing the estimated parameter to its standard error. This is the standard deviation of the parameter estimate.
5. t-statistic
The result is the t-statistic that equals the ratio between the estimate and its standard error. Both the standard error and the t-statistic are automatically computed when estimating a GARCH model.
6. Example for MSFT returns
As usual, you need to first specify and fit the model. Then you can print the table with parameter estimates, standard errors, and t-statistics using the fit dollar matcoef slot in your estimation output variable.
7. Parameter estimation table
This leads to a table with four columns, where the first column shows the estimated parameters. The second one shows the corresponding standard error. The third one shows the t-statistics. They equal the parameter estimate divided by the standard error.
8. Interpretation of t-statistic
The calculation of the t-statistic is thus simple. The key intuition for the interpretation is that its absolute value t-statistic can be interpreted as a distance measure. It measures how many standard errors the estimated parameter is separated from zero. The larger the distance, the more evidence the parameter is not zero.
As a rule of thumb, we decide by comparing the absolute value of the t-statics with 2. If it is larger than 2, then you decide that the estimated coefficient is sufficiently different from 0 to be statistically significant. And thus you conclude that the true parameter is not zero.
9. Conclusion for MSFT returns using t-statistics
In our example for the Microsoft returns, we find that all t-statistics are larger than 2. All parameter estimates are thus statistically significant.
You can reach the same conclusion using the pvalues shown in the last column.
10. Interpretation of p-value
Those p-values measures how likely it is that the parameter is zero. The lower its value, the more evidence there is that the parameter is not zero.
The rule of thumb to decide is then to compare the p-value with 5%. If it is less than 5% the estimated parameter is statistically significant and you can conclude that the true parameter is not zero.
11. Let's practice your skill to analyze statistical significance.
Time to put this into practice.