GARCH before and after a shock. How to test if volatilities are different?

Question

I have an intraday dataset with minute returns for a bond. At a specific point in time, say 10:30, there is an external shock (in my case an auction where that bond is traded). I want to know whether the volatility before the shock is the same as after the shock. If I understand it correctly, using standard deviation of returns before and after the shock is not a good idea, since these returns are not independent.

1. Then, would it be a good approach to estimate a GARCH model right before (e.g. between 8:30 and 10:30) and right after the shock (between 10:30 and 12:30 or even the end of the same day)?

2. If yes, then could you please help me with R implementation. I use the following code (package rugarch):

MyModSpecify=ugarchspec(mean.model = list(armaOrder=c(0,0)),
                            variance.model = list(model="sGARCH",garchOrder=c(1,1)),
                            distribution.model = "norm")

MyModFitting=ugarchfit(data=MyData$Return, spec=MyModSpecify, out.sample = 0)

Then running MyModFitting can print the results on the screen such that mu, omega, alpha1 and beta1 are estimated. But is there a way to save these in 4 different variables? For instance, to save the value of mu (which is simply printed on the screen) to variable MyMu.

3. Finally, suppose I run this for the period before the shock and then after the shock for different days. Then for each day I will have estimates of mu, omega, alpha1 and beta1 before the shock and after the shock. Then how can I test if volatility is different before the shock and after the shock?

Thank you!

Answer 1

Logarithmic returns may be dependent (as in GARCH), but they may still be uncorrelated. If that is the case, $\text{Var}(r_1+\dots+r_n)=\text{Var}(r_1)+\dots+\text{Var}(r_n)$. And if empirically that is not too far from reality, you could just estimate the variances as usual in the two periods and test whether they are equal; there are a few classical tests for that.

If you want to do GARCH, consider estimating the model for the entire time period (without splitting into two) with the following conditional variance equation: $$ \sigma_t^2 = \omega +\tilde\omega D + \alpha_1\varepsilon_{t-1}^2 + \tilde\alpha_1 D\varepsilon_{t-1}^2 + \beta_1\sigma_{t-1}^2 + \tilde\beta_1 D\sigma_{t-1}^2 $$ where $D$ is a dummy variable indicating the period during and after the event. Under $H_0$ of the event having no influence on the variance, $\tilde\omega=\tilde\alpha_1=\tilde\beta_1=0$. (You could simplify and skip some terms in the model; the simplest version would include only $\tilde\omega D$ but not the other two. Then it is just about the statistical significance of the estimate of $\tilde\omega D$.)

Implementation wise, look for external regressors in the variance equation. Including $\tilde\omega D$ is straightforward. Including $\tilde\alpha_1 D$ is probably easy if your conditional mean equation is just a constant. Including $\tilde\beta_1 D$ may be more challenging. For simplicity, some people use ARCHX instead of GARCH where they replace $\sigma_{t-1}^2$ from GARCH with estimated variance from a rolling window.