Building Econometric Models

What Use Are Volatility Forecasts? (Cont’d)

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• What is the optimal value of the hedge ratio?
• Assuming that the objective of hedging is to minimise the variance of the hedged portfolio, the optimal hedge ratio will be given by
• where h = hedge ratio
• p = correlation coefficient between change in spot price (S) and change in futures price (F)
• S = standard deviation of S
• F = standard deviation of F
• What if the standard deviations and correlation are changing over time?
• Use

Testing Non-linear Restrictions or Testing Hypotheses about Non-linear Models

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Usual t- and F-tests are still valid in non-linear models, but they are not flexible enough.
• There are three hypothesis testing procedures based on maximum likelihood principles: Wald, Likelihood Ratio, Lagrange Multiplier.
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• Consider a single parameter,  to be estimated, Denote the MLE as and a restricted estimate as .

Likelihood Ratio Tests

• ‘Introductory Econometrics for Finance’ © Chris Brooks 2013

• Estimate under the null hypothesis and under the alternative.
• Then compare the maximised values of the LLF.
• So we estimate the unconstrained model and achieve a given maximised value of the LLF, denoted Lu
• Then estimate the model imposing the constraint(s) and get a new value of the LLF denoted Lr.
• Which will be bigger?
• Lr  Lu comparable to RRSS  URSS
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• The LR test statistic is given by
• LR = -2(Lr - Lu)  2(m)
• where m = number of restrictions