Session 9A-Overview to Dynamic Time Series Analysis

A gjr model we obtain the following results 32 news

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Unformatted text preview: ted model. News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR Model Estimates: 33 GARCH-in Mean • We expect a risk to be compensated by a higher return. So why not let the return of a security be partly determined by its risk? • Engle, Lilien and Robins (1987) suggested the ARCH-M specification. A GARCH-M model would be • δ can be interpreted as a sort of risk premium. • It is possible to combine all or some of these models together to get more complex “hybrid” models - e.g. an ARMA-EGARCH(1,1)-M model. 34 What Use Are GARCH-type Models? • GARCH can model the volatility clustering effect since the conditional variance is autoregressive. Such models can be used to forecast volatility. • We could show that Var (yt ⏐ yt-1, yt-2, ...) = Var (ut ⏐ ut-1, ut-2, ...) • So modelling σt2 will give us models and forecasts for yt as well. • Variance forecasts are additive over time. 35 Forecasting Variances using GARCH Models • Producing conditional variance forecasts from GARCH models uses a very similar approach to producing forecasts from ARMA models. • It is again an exercise in iterating with the conditional expectations operator. • Consider the following GARCH(1,1) model: , ut ∼ N(0,σt2), • What is needed is to generate are forecasts of σT+12 ⏐ΩT, σT+22 ⏐ΩT, ..., σT+s2 ⏐ΩT where ΩT denotes all information available up to and including observation T. • Adding one to each of the time subscripts of the above conditional variance equation, and then two, and then three would yield the following equations σT+12 = α0 + α1 +βσT2 , σT+22 = α0 + α1 +βσT+12 , σT+32 = α0 + α1 +βσT+22 36 Forecasting Variances using GARCH Models (Cont’d) • • • • Let be the one step ahead forecast for σ2 made at time T. This is easy to calculate since, at time T, the values of all the terms on the RHS are known. would be obtained by taking the conditional expectation of the first equation at the bottom of slide 36: Given, how is , the 2-step ahead forecast for σ2 made at time T, calculated? Taking the conditional expectation of the second equation at the bottom of slide 36: = α0 + α1E( ⏐ ΩT) +β where E( ⏐ ΩT) is the expectation, made at time T, of , which is the squared disturbance term. 37 Forecasting Variances using GARCH Models (Cont’d) • • • • We can write E(uT+12 | Ωt) = σT+12 But σT+12 is not known at time T, so it is replaced with the forecast for it, , so that the 2-step ahead forecast is given by = α0 + α1 +β = α0 + (α1+β) By similar arguments, the 3-step ahead forecast will be given by = ET(α0 + α1 + βσT+22) = α0 + (α1+β) = α0 + (α1+β)[ α0 + (α1+β) = α0 + α0(α1+β) + (α1+β)2 Any s-step ahead forecast (s ≥ 2) would be produced by 38 What Use Are Volatility Forecasts? 1. Option pricing C = f(S, X, σ2, T, rf) 2. Conditional betas 3. Dynamic hedge ratios The Hedge Ratio - the size of the futures position to the size of the un...
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