Building Econometric Models

The Unconditional Variance under the GARCH Specification

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The unconditional variance of ut is given by
• when
• is termed “non-stationarity” in variance
• is termed intergrated GARCH
• For non-stationarity in variance, the conditional variance forecasts will not converge on their unconditional value as the horizon increases.

Estimation of ARCH / GARCH Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Since the model is no longer of the usual linear form, we cannot use OLS.
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• We use another technique known as maximum likelihood.
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• The method works by finding the most likely values of the parameters given the actual data.
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• More specifically, we form a log-likelihood function and maximise it.
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Estimation of ARCH / GARCH Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The steps involved in actually estimating an ARCH or GARCH model are as follows
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• Specify the appropriate equations for the mean and the variance - e.g. an AR(1)- GARCH(1,1) model:
• Specify the log-likelihood function to maximise:
• 3. The computer will maximise the function and give parameter values and their standard errors

Parameter Estimation using Maximum Likelihood

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

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• Consider the bivariate regression case with homoscedastic errors for simplicity:
• Assuming that ut  N(0,2), then yt  N( , 2) so that the probability density function for a normally distributed random variable with this mean and variance is given by
• (1)
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• Successive values of yt would trace out the familiar bell-shaped curve.
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• Assuming that ut are iid, then yt will also be iid.