Statistics and Algorithms / GARCH Models / GARCH Models (Univariate) /

UV GARCH Mean Model

While the need for a proper choice of the mean model was discussed earlier, the examples so far have used only an intercept. There is one model for the mean that is even simpler than this, where the dependent variable is already thought to be a mean zero process. To estimate the model under that assumption, use the option NOMEAN.

For a less trivial model, you can include the option REGRESSORS and add a supplementary card with the explanatory variables for the mean. Include CONSTANT if it’s one of the explanatory variables. For instance, the following will do a GARCH(1,1) with an AR(1) process for the mean:

garch(p=1,q=1,regressors) / dlogdm

# constant dlogdm{1}

If you want an ARMA model for the mean, use lags of %MVGAVGE in your list of regressors for any moving average terms. For instance, the same model as above, but with an ARMA(1,1) is:

garch(p=1,q=1,regressors) / dlogdm

# constant dlogdm{1} %mvgavge{1}

With the ARMA model, the pre-sample values for the moving average process are set to zero. The residuals are generated recursively conditioned on that. Note that an ARMA model is usually a poor choice—if the autocorrelation in the data is fairly weak (which is usually the case), the AR and MA coefficients will not be well-determined by the model, and you are likely to see (unreasonably) large coefficients for both with similarly large standard errors.

You can also use the option EQUATION to indicate both the dependent variable and the mean model. For instance,

linreg(define=ar1) dlogdm

# constant dlogdm{1}

garch(p=1,q=1,equation=ar1)

estimates an AR(1) model first by least squares, then with GARCH errors.

Output

With a more complex mean model, a blank line is inserted in the output to separate the mean model from the variance model parameters.

GARCH Model - Estimation by BFGS

Convergence in 20 Iterations. Final criterion was 0.0000075 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1865

Log Likelihood -2063.0191

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.022623191 0.016669924 -1.35713 0.17474110

2. DLOGDM{1} -0.075892967 0.022990732 -3.30102 0.00096333

3. C 0.015791350 0.004861763 3.24807 0.00116190

4. A 0.110511663 0.016053181 6.88410 0.00000000

5. B 0.868670147 0.017885895 48.56733 0.00000000

GARCH Model - Estimation by BFGS

Convergence in 27 Iterations. Final criterion was 0.0000004 <= 0.0000100

Dependent Variable DLOGDM

Usable Observations 1865

Log Likelihood -2062.4516

Variable Coeff Std Error T-Stat Signif

************************************************************************************

1. Constant -0.027037475 0.018680487 -1.44736 0.14779484

2. DLOGDM{1} -0.271873474 0.161346655 -1.68503 0.09198336

3. Mvg Avge{1} 0.194283914 0.163218781 1.19033 0.23391743

4. C 0.015810577 0.004697818 3.36551 0.00076401

5. A 0.110603223 0.013764273 8.03553 0.00000000

6. B 0.868534818 0.017203796 50.48507 0.00000000