Building Econometric Models

The Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• The “Base” Models
• For the conditional mean
• (1)
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• And for the variance (2)
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• or (3)
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• where
• RMt denotes the return on the market portfolio
• RFt denotes the risk-free rate
• ht denotes the conditional variance from the GARCH-type models while t2 denotes the implied variance from option prices.
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The Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Add in a lagged value of the implied volatility parameter to equations (2) and (3).
• (2) becomes
• (4)
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• and (3) becomes
• (5)
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• We are interested in testing H0 :  = 0 in (4) or (5).
• Also, we want to test H0 : 1 = 0 and 1 = 0 in (4),
• and H0 : 1 = 0 and 1 = 0 and  = 0 and  = 0 in (5).
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The Models (cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• If this second set of restrictions holds, then (4) & (5) collapse to
• (4’)
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• and (3) becomes
• (5’)
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• We can test all of these restrictions using a likelihood ratio test.
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In-sample Likelihood Ratio Test Results: GARCH Versus Implied Volatility

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

In-sample Likelihood Ratio Test Results: EGARCH Versus Implied Volatility

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

Conclusions for In-sample Model Comparisons & Out-of-Sample Procedure

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• IV has extra incremental power for modelling stock volatility beyond GARCH.
• But the models do not represent a true test of the predictive ability of IV.
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• So the authors conduct an out of sample forecasting test.
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• There are 729 data points. They use the first 410 to estimate the models, and then make a 1-step ahead forecast of the following week’s volatility.
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• Then they roll the sample forward one observation at a time, constructing a new one step ahead forecast at each step.
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