# Comparing volatility using GARCH

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?

• Your first GARCH model is awfully close to IGARCH(1,1), and the second violates nonnegativity restriction of the coefficients. Usually this is an indication that model is not well suited for data you are fitting. – mpiktas Nov 20 '15 at 11:28
• The updated model for the whole sample is problematic as there should be no negative coefficients in the GARCH equations. I wonder why eViews does not automatically restrict at least $\omega$, $\alpha_1$ and $\beta_1$ (as per my answer) to be positive. Strangely, the dummy has a negative coefficient, which implies the volatility of inflation has increased after the regime change (if you constructed the dummy as I suggested with ones in the beginning and zeros at and after the change). The second model looks better than the first: lower AIC, BIC, better Durbin-Watson, positive $R_{adj.}^2$. – Richard Hardy Nov 20 '15 at 19:10
• I don't see the ARCH-LM test. Also, if you have estimated a GARCH model, ARCH-LM is no longer suitable; you should use Li-Mak test instead that adjusts for the fact that a GARCH model has been fitted to the residuals. The test is to be applied on standardized residuals, i.e. residuals divided by their fitted standard deviations coming from the GARCH model. Meanwhile, ARCH-LM would work fine for model residuals if there only was a conditional mean model but not a GARCH model extra to it. (Few econometricians seem to be aware of this, so you might slip through with ARCH-LM regardless.) – Richard Hardy Nov 20 '15 at 19:15
• I'd say it's mostly fine. Perhaps your professor does not expect you to do all the complicated testing of model adequacy. You top-right model looks fine, you should just ensure that the coefficients in the GARCH model are all positive. In this particular model, just reverse the dummy: use zeros in place of ones and vice versa. You should then get a positive coefficient. You could add lagged GDP growth, unemployment and interest rates in the conditional mean model (lagged so as to prevent endogeneity), and that's it. – Richard Hardy Nov 20 '15 at 20:34
• Oh, but there is one thing: the finding seems to be the opposite of what you expected. That's tricky. Either the model is poor, or the theory is faulty. Perhaps your professor will not like that... – Richard Hardy Nov 20 '15 at 20:35

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.

• Could you elaborate on how the $F$-test should be designed? Do you mean comparing the variances of model errors in subsample 1 versus subsample 2? Would that be appropriate in case of GARCH-type model errors? I highly doubt that; compare that to testing equality of sample means when observations in the samples are not independent; a simple $t$-test will not work. – Richard Hardy Nov 20 '15 at 10:09
• Thanks! Now it's clear. This setting involves stricter rather than "lesser" assumptions (a fact) and is less appropriate (an opinion) for the problem at hand. IMHO, $i.i.d.$ + normality is far from appropriate when it comes to inflation. Fortunately, this can be tested, so @GARCHer who has his data can do that. – Richard Hardy Nov 20 '15 at 12:27
• I can confirm that with Jacque-Bera statistics of 35 and 62 respectively I have non-normality. – GARCHer Nov 20 '15 at 18:29
• @GARCHer you may used both. Even you required to make assumption about the distribution of residuals in GARCH model too. – Neeraj Nov 21 '15 at 6:18

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.

• I have now added screenshots of the regressions with dummy's. This work is only for undergraduate level so I don't plan to use such complicated econometrics. I may try to justify regressing inflation only on past inflation as above but fear I will be marked down as I should include these variables. I have also attached another screenshot. On the right is a test for arch effects after already running garch after correcting for non stationarity with infln = c + infln(-1) + infln (-2) + error and a dummy for inflation targeting in var eqn. Do these F values mean I have made an error? – GARCHer Nov 20 '15 at 16:23