MATLAB: How to get the historical GARCH Variance

forecastinggarchvariance

When one constructs forecasts with a GJR-GARCH model, the output matlab gives is the variance forecast (i.e. $\sigma^2_{t+1}$).

How can one extract the corresponding $\sigma^2_t$ (i.e. the historical variance) of the GJR-GARCH model?

For a standard GARCH model one could do this using the ugarchpred function (<http://nl.mathworks.com/help/finance/ugarchpred.html)>. The historical variance is the 'H' in the syntax window (see link). Unfortunately this function is not available for a GJR-GARCH and hence my question, how can I extract the historical variance?

Many thanks

Best Answer

Never mind, I found the answer to my question on the following link:

http://nl.mathworks.com/help/econ/infer-conditional-variances.html

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MATLAB: Prediction based on previous historical data

You don't have enough points to infer any kind of regression. I mean, if you look at:

data = [0.25        0.256
        0.24         0.24
       0.224         0.24
        0.26         0.27
        0.24         0.26];
scatter(data(:, 1), data(:, 2));

there's no pattern to your data.

If you want to do a linear regression anyway:

polystartend = polyfit(data(:,1), data(:, 2), 1); %linear regression
todaystart = 21;
todayend = polyval(polystartend, todaystart)

gives

todayend = 0.22688

MATLAB: Problems with GJR-GARCH/EGARCH estimation with fmincon (sqp)

Hi Duarte,

It is true that the log-likelihood surface of GJR and EGARCH are not a globally smooth function like y = x^2.

The situation is somewhat similar to y = abs(sin(x)) * (x<=2).

Is this function differentiable? Well, not globally. At x=0 and x=2, the derivatives are not well defined. However, it is differentiable almost everywhere. At the maximum (say x=pi/2), the function is differentiable. As long as the starting value is very close to that point, a gradient-based optimizer should converge to that point, because the function is concave in a neighborhood around the maximum.

Actually in most of the econometric models estimated by MLE, the likelihood function is complicated and there is no guarantee of convergence. My suggestion is to try many starting values and refine-tune the optimization options, so as to increase the chance of getting a good estimator.

In most cases, our functionalities of GJR-GARCH and EGARCH work well and are useful for volatility forecasting.

As for the default choice of algorithm, 'SQP', it was chosen because it offers a nice blend of accuracy and runtime performance. There have been instances in which other algorithms, such as 'Interior-Point', give better results, but in the vast majority of cases various algorithms provide very similar answers provided the model chosen is a good description of the data generating process.

If you want to use an algorithm other than 'SQP', then the "estimate" methods accepts an optional "Options" input which allows you to change it. Please see the following documentation for how to do so: http://www.mathworks.com/help/econ/cvm.estimate.html

Best Answer

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MATLAB: Prediction based on previous historical data

MATLAB: Problems with GJR-GARCH/EGARCH estimation with fmincon (sqp)

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