The previous article showed time series using the autoregressive conditional heteroskedasticity (ARCH) model in STATA. The present article shows extensions of ARCH, i.e. GARCH model in STATA. Like the ARCH model, ARCH extensions like the Generalised ARCH (GARCH) model also need squared residuals as determinants of the equation’s variance.
Generalized ARCH (GARCH) model
In the GARCH model, the conditional variance of error term at time ‘t’ depends on the following:
- Squared error term in the previous time period (ARCH1)
- Its conditional variance in the previous time period.
Follow this command for using the GARCH model:
arch logRE_d1, arch(1) garch(1)
The interpretation of the syntax is as follows.
In the ARCH regression model, ‘logRE_d1’ is a dependent variable with no independent variables other than a constant. ‘arch(1)’ command adds a single lagged value of et to the modelled variance in STATA. ‘garch(1)’ command adds a single lag of the variance, ht, to the modelled variance.
Results of GARCH model
The figure below shows the results of the GARCH model. The estimate of a lagged value of error term is 0.491 whereas the coefficient estimate is 0.238. The p values are all significant and log-likelihood is also highest in all the iterations.
Like ARCH, generate variances for the GARCH model using the same command:
predict GTgarch, variance
Here ‘GTgarch’ is the name for the predicted series of variances. The results will not appear in the ‘Result’ window, but in the ‘data editor’ window of STATA. To examine the movement of GTgarch generates a time plot using this command:
STATA Command: tsline GTgarch
The figure below shows the results.
The above graph shows that there is a lot more volatility towards the entire span of the sample considering the lagged values of variances.
Threshold GARCH (TGARCH) model
The threshold GARCH model or TGARCH is another generalization of the GARCH model where positive and negative news are treated asymmetrically. This adds another option to the arch ‘Stock_RE_d1’ regression model in STATA. Use this command for the TGARCH model in STATA:
arch logRE_d1, arch(1) garch(1) tarch(1)
The figure below shows the results. The estimate of the lagged value of the error term is 0.08 and the coefficient on the lagged variance is 0.919. The p values are all significant and log-likelihood is also highest in all the iterations.
Further to generate variances for TGARCH use this command:
predict HTgarch, variance
Here ‘HTgarch’ is the name for the predicted series of variances. The results do not appear in the ‘Result’ window, but in the ‘data editor’ window of STATA. To examine the movement of the variance of HTgarch, generate a time plot using this command:
After analyzing ARIMA, VAR, VEC, ARCH and some extensions of the ARCH models, the next set of articles will focus upon some common diagnostics tests applied to time series analysis. These tests will pertain to normality, heteroskedasticity, autocorrelation, multicollinearity and stability.