Building Econometric Models

Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• From (7),
• (10)
•  
• From (8),

Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Rearranging,
• (11)
• How do these formulae compare with the OLS estimators?
• (9) & (10) are identical to OLS
• (11) is different. The OLS estimator was
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• Therefore the ML estimator of the variance of the disturbances is biased, although it is consistent.
•  But how does this help us in estimating heteroscedastic models?

Estimation of GARCH Models Using Maximum Likelihood

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Now we have yt =  + yt-1 + ut , ut  N(0, )
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• Unfortunately, the LLF for a model with time-varying variances cannot be maximised analytically, except in the simplest of cases. So a numerical procedure is used to maximise the log-likelihood function. A potential problem: local optima or multimodalities in the likelihood surface.
• The way we do the optimisation is:
• 1. Set up LLF.
• 2. Use regression to get initial guesses for the mean parameters.
• 3. Choose some initial guesses for the conditional variance parameters.
• 4. Specify a convergence criterion - either by criterion or by value.

Non-Normality and Maximum Likelihood

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Recall that the conditional normality assumption for ut is essential.
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• We can test for normality using the following representation
• ut = vtt vt  N(0,1)
•  
•  
• The sample counterpart is
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• Are the normal? Typically are still leptokurtic, although less so than the . Is this a problem? Not really, as we can use the ML with a robust variance/covariance estimator. ML with robust standard errors is called Quasi- Maximum Likelihood or QML.