# RUGARCH package - t value when using normal distribution

I am new to the RUGARCH package in R however I am finding it strange how I am using the normal distribution specification, yet the fitted parameters are showing a t-value.

Code:

spec <- CreateSpec(variance.spec = list(model = c("sGARCH")), distribution.spec = list(distribution = c("norm")));

fit.ml<-FitML(spec, y,ctr=list(do.se=TRUE));

summary(fit.ml)


Part of the result:

*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(1,0,1)
Distribution    : norm

Optimal Parameters
------------------------------------
Estimate  Std. Error  t value Pr(>|t|)
mu     -0.001481    0.004234 -0.34965 0.726602
ar1    -0.492941    0.453775 -1.08631 0.277342
ma1     0.507445    0.449122  1.12986 0.258535
omega   0.000819    0.000293  2.79095 0.005255
alpha1  0.064641    0.009511  6.79614 0.000000
beta1   0.923276    0.012148 76.00059 0.000000

• I have added the results. I know this is not an R forum, however I have seen a lot of questions related to specific R statistical packages. – Anna Feb 26 '18 at 11:27
• @NickCox Can you maybe direct me to a paper or a website which helps me understand why the parameters of a GARCH process follow a t distribution? – Anna Feb 26 '18 at 13:05

\begin{aligned} y_t &= \mu_t + u_t, \\ \mu_t &= \varphi_1 y_{t-1} + \theta_1 u_{t-1} + u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \\ \varepsilon_t &\sim i.i.N(0,1). \\ \end{aligned} The last equation gives the normality assumption for the standardized innovations (shocks, errors). This shows up in the R output in the line Distribution : norm.
The point estimates of the model parameters are given in the column Estimate. Each estimate has an asymptotic normal distribution. You may want to test whether the actual effect of a given variable is nonzero in population., e.g. whether $\alpha_1\neq0$ (it is nonzero in sample, but that might be due to randomnes). You formulate a null hypothesis $H_0\colon \ \alpha_1=0$ and test it with a $t$-test. The column t value contains these test statistics for each variable in the model. It is typical to report these statistics even if you have not requested them.
So you see that the normality assumption for the standardized errors has nothing to do with the $t$-test results reported in the column t value.
• Yes, the variance is estimated rather than known, so the distribution of the test statstic is $t$ distribution. – Richard Hardy Feb 26 '18 at 16:58