How come the first conditional variance of an Inferred GARCH(1,1) model is not the fitted constant for that model?

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Mike A on 19 Nov 2016
Answered: Sebastian Kammer on 23 Nov 2016
\sigma _t^2 = 0.0001 + 0.75\sigma _{t - 1}^2 + 0.1\varepsilon _{t - 1}^2
Base off of this formula from the Matlab Documentation for "garch" if we infer this model the first conditional variance should be the constant. I have also noticed that is the case for ARIMA models as well can anyone help me with a good explanation.

Sebastian Kammer on 23 Nov 2016
Most MATLAB functions are available as M-files and can be opened in the MATLAB editor. This can be helpful in situations where you would like to find out how a certain function output is obtained.
In this case, you would be inferring the conditional variance using the "infer" function.
Typing
edit infer
at the command window will reveal the contents of the "infer" function.
If you look at line 287 and the comments above, that should shed some light on how the first conditional variance is calculated.
Basically the long-run unconditional variance of the input is calculated as the average squared residual and is then assigned to the first observation of the conditional variance (if there are no lags in the model).