Comparing volatility using GARCH

Question

I am investigating if the policy of inflation targeting lowers the volatility of inflation.

I have estimated an AR(2)-GARCH(1,1) model, where the conditional mean equations is

$$ Inflation_t = C + Inflation_{t-1} + Inflation_{t-2} + Error_t. $$

and $Error_t$ follows a GARCH(1,1) model. I am unsure of how exactly to conclude if the model estimated on the subsample after inflation targeting does indeed show a reduction in the volatility of inflation as compared with the model estimated on the subsample before inflation targeting.

Is it possible to reach a conclusion by comparing the significance of the GARCH(-1) coefficients before and after inflation targeting or comparing the significance of the RESID(-1)^2 coefficient before and after inflation targeting, or both, or some other way?

Answer 1

GARCH is more appropriate for forecasting volatility. Here, you can use $F$ test to compare the inflation volatility before and after the inflation targeting. It involves lesser assumptions and is more appropriate.

Let $X_1, \dotsc, X_n$ and $Y_1, \dotsc, Y_m$ be independent and identically distributed samples from two populations which each have a normal distribution. The expected values for the two populations can be different, and the hypothesis to be tested is that the variances are equal. Let

$$\overline{X} = \frac{1}{n}\sum_{i=1}^n X_i\text{ and }\overline{Y} = \frac{1}{m}\sum_{i=1}^m Y_i$$

be the sample means. Let

$$ S_X^2 = \frac{1}{n-1}\sum_{i=1}^n \left(X_i - \overline{X}\right)^2\text{ and }S_Y^2 = \frac{1}{m-1}\sum_{i=1}^m \left(Y_i - \overline{Y}\right)^2 $$

be the sample variances. Then the test statistic

$$ F = \frac{S_X^2}{S_Y^2} $$

has an $F$-distribution with $n − 1$ and $m − 1$ degrees of freedom if the null hypothesis of equality of variances is true. Otherwise it has a non-central $F$-distribution. The null hypothesis is rejected if $F$ is either too large or too small.

Answer 2

You seem to be interested in unconditional volatility since you ask whether the level of volatility has decreased after the regime change. Given the volatility equation of a GARCH(1,1) model

$$ \sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1\sigma_{t-1}^2, $$

the unconditional volatility is

$$ \sigma^2 = \frac{\omega}{1-\alpha_1-\beta_1}. $$

If you have estimated two models, you can compare the point estimates of $\sigma^2$ from before and after the regime change. However, you would need some assessment of accuracy of the point estimates to be able to see whether the difference is statistically significant. It may take some effort to get it, unless eViews (looks like eViews, isn't it?) can give you $\sigma^2$ and its standard error right away.

There is a pretty simple shorcut, albeit it comes with extra assumptions. You could estimate an AR(2)-GARCH(1,1) model for the whole sample (including both the pre-change and post-change periods) with a dummy variable in the volatility equation. That is,

$$ \sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1\sigma_{t-1}^2 + \gamma_1 dummy_t, $$

$dummy_t$ would have value 1 before the change and value 0 at and after the change. Its statistical significance would show whether the reduction in unconditional variance at and after the change has been statistically significant. The extra assumptions in this approach is that the AR coefficients as well as the GARCH coefficients remain unchanged througout the full sample. Alternatively, you could introduce more dummies and interactions to allow the AR and or GARCH coefficients to change at the point of the regime change.