# Does the unconditional variance implied by a GARCH equal the sample variance?

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?

In this context, unconditional variance refers to the stationary variance level predicted by your GARCH model. This quantity need not coincide with the sample variance of the data on which the latter model has been calibrated.

That being said, in an effort to reduce the complexity of the GARCH parameters' estimation process (nasty non-linear optimisation problem), it is frequent amongst practitioners to impose that the unconditional variance (model-bound) matches sample variance (data-bound) perfectly. This technique, which effectively reduces the GARCH parameter space (i.e. constrains the intercept to become a function of the other GARCH parameters and sample variance) is known as variance targeting.

Although I do not know the particulars of the MATLAB function you are using, I guess "presample variance" simply refers to the first value of conditional variance $h_1$ (hence the adjective pre-sample). The knowledge of $h_1$ indeed allows all future conditional variances $(h_2,\dots,h_N)$ to be inferred since you already know the realised series of $(r_1, \dots, r_N)$ where $N=2000$ and $r_N$ the most recent past log-return in your case. Another possibility would be to consider $h_1$ as yet another parameter of your GARCH model and to determine it through maximum likelihood estimation.

• Thank you! Is any of the two superior to the other? Do you maybe have a reference which discusses approaches for selecting initial values? – Kondo May 17 '16 at 21:04
• @Kondo Not in general. See this paper google.be/url?sa=t&source=web&rct=j&url=http://… which discusses at least 6 different ways of picking $h_1$. Note that you'll face a similar problem when you'll want to simulate from your (now calibrated) GARCH process: you'll need to fix both the initial $y_0$ and $r_0$ (and once again you'll have different possible ways of choosing $r_0$). – Quantuple May 18 '16 at 6:16