Building the Conditional Variance Recursion in a GARCH(1,1) Model

Summary

The document describes a learner's attempt to estimate a GARCH(1,1) volatility model by hand after fitting an AR(1) model to a return or time series. The AR(1) residuals supply the lagged squared innovations used in the conditional variance equation, while the lagged conditional variance is itself generated recursively from the model rather than observed directly. This implies constructing a variance series from an initial variance value and iterating the recursion through the sample.

The author identifies maximum likelihood as the next step for estimating the model parameters and asks for guidance on the recursion. The document presents the question but contains no accepted solution, worked initialization, likelihood specification, or empirical results. Practical estimation therefore still requires choices about initialization and assumptions for the innovation distribution, as well as joint estimation or treatment of the mean equation.

Key ideas

• An AR(1) fit can provide residuals that serve as innovations for a GARCH variance model.
• The lagged conditional variance is produced recursively and is not directly observed.
• The recursion requires an initial variance value before it can be computed across the sample.
• Maximum likelihood is proposed for estimating the GARCH parameters, but the document does not supply a full procedure.

Tags

# Estimating an GARCH(1,1) model? Long hand method


# Estimating an GARCH(1,1) model? Long hand method












I am really trying to invest some time to estimate a GARCH(1,1) method, I know there is many statistical packages that will do this for me (Eviews, MATLAB, R), but I am trying to do this by hand, so that I can really understand the model.

Following the theory from Verbeek. 'A guide to modern econometrics',

I first estimate an AR(1) model,

$$Y_t= \alpha +\theta Y_{t-1} +\varepsilon_t $$

And then I save the residuals, $\varepsilon_t$,

I then want to model a GARCH(1,1) with these residuals.

$$\sigma_t^2=\omega+\alpha \varepsilon_{t-1}^2 +\beta\sigma_{t-1}^2$$

This is where I am struggling, I understand a few things, I understand how I can get the lagged $\varepsilon_{t-1}^2$, I already have it from the first equation, however my major question is how do I get a value for $\sigma_{t-1}^2$ this is what I am trying to estimate, so I am really struggling about how I derive this part. I suppose I need a series for $\sigma_{t-1}^2$, which in matrix terms is a long as the $\varepsilon_{t-1}^2$.

I am thinking I may need to use a for loop to get this $\sigma_{t-1}^2$, but any suggestions on how I might go about this would be helpful.

Second, when I do realise how to do the above step, I need to estimate the thing my maximum likelihood estimation. I think I read around this and try my best to solve it.

I would really like help on step one if anyone has got the time or direction to a good reference?

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.