Building Econometric Models

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Ch9 slides

News Impact Curves

The news impact curve plots the next period volatility (ht) that would arise from various positive and negative values of ut-1, given an estimated model.
News Impact Curves for S&P 500 Returns using Coefficients from GARCH and GJR Model Estimates:

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
 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.

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 t2 will give us models and forecasts for yt as well.
Variance forecasts are additive over time.

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,t2),
What is needed is to generate forecasts of T+12 T, T+22 T, ..., T+s2 T where T 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
T+12 = 0 + 1uT2 +T2 , T+22 = 0 + 1uT+12 +T+12 , T+32 = 0 + 1 uT+2+T+22

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