Building Econometric Models

Likelihood Ratio Tests (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Example: We estimate a GARCH model and obtain a maximised LLF of 66.85. We are interested in testing whether  = 0 in the following equation.
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• yt =  + yt-1 + ut , ut  N(0, )
• = 0 + 1 + 
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• We estimate the model imposing the restriction and observe the maximised LLF falls to 64.54. Can we accept the restriction?
• LR = -2(64.54-66.85) = 4.62.
• The test follows a 2(1) = 3.84 at 5%, so reject the null.
• Denoting the maximised value of the LLF by unconstrained ML as L( )
• and the constrained optimum as . Then we can illustrate the 3 testing procedures in the following diagram:

Comparison of Testing Procedures under Maximum Likelihood: Diagramatic Representation

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

Hypothesis Testing under Maximum Likelihood

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The vertical distance forms the basis of the LR test.
• The Wald test is based on a comparison of the horizontal distance.
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• The LM test compares the slopes of the curve at A and B.
• We know at the unrestricted MLE, L( ), the slope of the curve is zero.
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• But is it “significantly steep” at ?
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• This formulation of the test is usually easiest to estimate.

An Example of the Application of GARCH Models - Day & Lewis (1992)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Purpose
• To consider the out of sample forecasting performance of GARCH and EGARCH Models for predicting stock index volatility.
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• Implied volatility is the markets expectation of the “average” level of volatility of an option:
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• Which is better, GARCH or implied volatility?
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• Data
• Weekly closing prices (Wednesday to Wednesday, and Friday to Friday) for the S&P100 Index option and the underlying 11 March 83 - 31 Dec. 89
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• Implied volatility is calculated using a non-linear iterative procedure.