In the MATLAB default settings for GARCH estimation they say "presample conditional variance is the sample average of the squared disturbances of the offset-adjusted response data y". Am I right in interpreting this as the sample variance? (sorry my English is not so sophisticated for me to get that sentence)
(second not totally unrelated question) Let's say that I'm using 2000 daily log returns to estimate a GARCH(1,1), and obtain $\omega=0.0000026$, $\alpha_1=0.1381$ and $\beta_1=0.8587$. Therefore the unconditional variance is $\frac{w}{1-\alpha_1 - \beta_1}=0.0008$. Should this estimate theoretically be the same as the sample variance $\frac{1}{n}\sum_{i=1}^{2000}(r_t-\mu_t)^2$, or are unconditional variance and sample variance not the same thing?