How to analyze the volatility with GARCH?

Question

How could I use GARCH model to detect if the volatility is constant during all the series(time series)?

I can't do a visual check, I need to detect if the volatility is constant using R and GARCH function of tseries package

Answer 1

You can use the classic Engle test for ARCH effects. The test is implemented as follows (copy from Ch. Brooks, "Introductory Econometrics for Finance").

1. Estimate the model

   $$y_t=\beta_0+x_{1t}\beta_1+...+x_{kt}\beta_k + u_t, \quad t=1,...,T$$ using OLS and save the residuals $\hat{u}_i$. (If you do not have any explanatory variables, simply demean $y$.)

2. Square the residuals and estimate the following regression:

   $$\hat{u_t}^2=\gamma_0+\hat{u}_{t-1}^2\gamma_1+\hat{u}_{t-2}^2\gamma_2+...+\hat{u}_{t-q}^2\gamma_q+v_t,$$

   Obtain $R^2$ from this regression.

3. The test statistic, which is defined as $T\cdot R^2$, is distributed as $\chi^2(q)$.

4. The null hypothesis is $\gamma_i=0$, $i=1,\ldots,q,$ against the alternative $\exists j: \gamma_j\neq 0$.

This can be implemented in R as follows. Supposing we have $\hat{u}$, the p-value for the hypothesis is

> set.seed(13)
> u<-rnorm(100)
> 1-pchisq(summary(lm(X1~.,data=data.frame(embed(u^2,5))))$r.squared*100,4)
[1] 0.6867667

Naturally you will need to decide how many lags to include. Since usually GARCH models use small numbers of lags, some predefined number may entirely appropriate. I would do some MC simulations to determine which works best.