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Unformatted text preview: derlying exposure, i.e. the number of futures contracts to buy or sell per
unit of the spot good.
39 What Use Are Volatility Forecasts? (Cont’d) •
• 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)
σS = standard deviation of S
σF = standard deviation of F
• What if the standard deviations and correlation are changing over time?
40 Testing Non-linear Restrictions or
Testing Hypotheses about Non-linear Models
• 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. • Consider a single parameter, θ to be estimated, Denote the MLE as
and a restricted estimate as . 41 Likelihood Ratio Tests •
• 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
The LR test statistic is given by
LR = -2(Lr - Lu) ∼ χ2(m)
where m = number of restrictions 42 Likelihood Ratio Tests (cont’d)
• Example: We estimate a GARCH model and obtain a maximised LLF of
66.85. We are interested in testing whether β = 0 in the following equation.
yt = μ + φyt-1 + ut , ut ∼ N(0,
= α0 + α1
+β ) • 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 χ2(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:
43 Comparison of Testing Procedures under Maximum Likelihood:
Diagramatic Representation 44 Hypothesis Testing under Maximum Likelihood • The vertical distance forms the basis of the LR test. • The Wald test is based on a comparison of the horizontal distance. • 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. • But is it “significantly steep” at • This formulation of the test is usually easiest to estimate. ? 45 An Example of the Application of GARCH Models
- Day & Lewis (1992)
To consider the out of sample forecasting performance of GARCH and
EGARCH Models for predicting stock index volatility. • Implied volatility is the markets expectation of the “average” level of
volatility of an option: • Which is better...
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This note was uploaded on 09/20/2013 for the course FINA 5170 taught by Professor Janebargers during the Summer '13 term at Greenwich School of Management.
- Summer '13
- Financial Markets