GARCH Unconditional Variance, Sample Variance, and Variance Targeting
Summary
The document distinguishes the unconditional variance implied by a fitted GARCH model from the sample variance of observed returns. The model-implied quantity is its stationary variance level, and it need not equal the variance calculated from the estimation sample. It describes variance targeting as a common estimation choice that constrains the unconditional variance to match the sample variance, reducing the number of free parameters by making the intercept depend on the other parameters and the sample variance.
The discussion also clarifies the role of a presample conditional variance: it can initialize the recursion that generates later conditional variances from observed returns. Alternatively, an initial variance may be estimated as another model parameter. The document cites a paper on GARCH variance initialization but gives no comparison of initialization procedures or empirical findings. Its main caveat is that the precise meaning of a software setting depends on the implementation; the explanation offered for MATLAB is presented as an interpretation, not verified documentation.
Key ideas
- A GARCH unconditional variance is the stationary variance implied by the model parameters.
- The model-implied unconditional variance need not equal the sample variance of the fitted data.
- Variance targeting constrains the model's unconditional variance to match the sample variance.
- A presample conditional variance initializes the sequence of conditional variances.
- An initial variance can instead be treated as an additional parameter for estimation.
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# Does the unconditional variance implied by a GARCH equal the sample variance?
# 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?
## Answer by Quantuple (score 7, accepted)
https://quant.stackexchange.com/a/26083
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.
## Answer by agritrader (score 1)
https://quant.stackexchange.com/a/61139
I believe the paper referenced by Quantuple in the comment 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.