# Initial value of the conditional variance in the GARCH process

From what I know, the GARCH(p,q) model is estimated via MLE and through an iterative process. Let's say if i wanted to recreate a GARCH(1,1) parameter estimation with excel solver (through maximizing the log-likelihood), how are my initial GARCH terms $\sigma_t^2$ set?

More specifically, given $\epsilon_t = v_t\sqrt{\sigma_t^2}$ where $$\sigma_t^2 = \alpha_0 +\alpha_1\epsilon^2_{t-1}+\alpha_2\sigma_{t-1}^2$$ how does the process of parameter estimation start since we do not know what $\sigma_{t-1}$ is?

One answer I've read from here shows that the program set the initial GARCH term to be the sample variance or its expected value. Is this how we approach it?

• It may depend on the particular software you want to mimic; but sample variance sounds like a good choice in general. Jan 13, 2015 at 20:17
• Let's say it is Matlab, with matlab, i fit the Garch model on the same data set with the estimate() function. Then I inferred the residuals through infer(estmdl, x) and found my fitted model through fitted = y-residuals; On a graph, the estimation was almost spot on. However when I transfer this over to the excel solver, I end up with an almost identical loglikelihood but very different results. I am taking the Loglikelihood function from here: ams.sunysb.edu/~yiyang/research/computational_finance/… Jan 14, 2015 at 4:10

I know of at least five ways of initializing the volatility process:

1) Set it equal to $\varepsilon_{t-1}^2$,

2) The sample variance,

3) Unconditional variance of the model ($\alpha_0/(1-\alpha_1 - \alpha_2)$),

4) Allow it it to be an parameter to be estimated,

5) Backcasting with an exponential filter.

The topic is discussed in further detail here

• The link from @Johan Stax Jakobsen is now dead. I believe the paper referenced is the following: Pelagatti, Matteo, and Francesco Lisi. "Variance initialisation in GARCH estimation." S. Co. 2009. Sixth Conference. Complex Data Modeling and Computationally Intensive Statistical Methods for Estimation and Prediction. Maggioli Editore, 2009. Feb 13, 2021 at 19:51