Building Econometric Models

Autoregressive Volatility Models

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Ch9 slides

A simple example of a stochastic volatility model is the autoregressive volatility specification
This model is simple to understand and simple to estimate, because it requires that we have an observable measure of volatility which is then simply used as any other variable in an autoregressive model
The standard Box-Jenkins-type procedures for estimating autoregressive (or ARMA) models can then be applied to this series
For example, if the quantity of interest is a daily volatility estimate, we could use squared daily returns, which trivially involves taking a column of observed returns and squaring each observation
The model estimated for volatility, t2, is then

The term ‘stochastic volatility’ is usually associated with a different formulation to the autoregressive volatility model, a possible example of which would be
where ηt is another N(0,1) random variable that is independent of ut
The volatility is latent rather than observed, and so is modelled indirectly
Stochastic volatility models are superior in theory compared with GARCH-type models, but the former are much more complex to estimate.

A limitation of univariate volatility models is that the fitted conditional variance of each series is entirely independent of all others
This is potentially an important limitation for two reasons:

If there are ‘volatility spillovers’ between markets or assets, the univariate model will be mis-specified
It is often the case that the covariances between series are of interest too
The calculation of hedge ratios, portfolio value at risk estimates, CAPM betas, and so on, all require covariances as inputs

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