Building Econometric Models

Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Then the joint pdf for all the y’s can be expressed as a product of the individual density functions 
• (2)
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• Substituting into equation (2) for every yt from equation (1),
• (3)
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Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The typical situation we have is that the xt and yt are given and we want to estimate 1, 2, 2. If this is the case, then f() is known as the likelihood function, denoted LF(1, 2, 2), so we write
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• (4)
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• Maximum likelihood estimation involves choosing parameter values (1, 2,2) that maximise this function.
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• We want to differentiate (4) w.r.t. 1, 2,2, but (4) is a product containing T terms.
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Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Since , we can take logs of (4).
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• Then, using the various laws for transforming functions containing logarithms, we obtain the log-likelihood function, LLF:
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• which is equivalent to
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• (5)
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• Differentiating (5) w.r.t. 1, 2,2, we obtain
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• (6)

Parameter Estimation using Maximum Likelihood (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• (7)
• (8)
•  Setting (6)-(8) to zero to minimise the functions, and putting hats above the parameters to denote the maximum likelihood estimators,
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• From (6),
• (9)