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1. Derive the unconditional variance and kurtosis of a GARCH(2,1) process.
2. Monte Carlo simulation.
(1) Set the values for μ, α1 and β1 and assume that   follows a standard
normal distribution.
(2) Based on (1), simulate 2000 data from a standard GARCH(1,1) process.
(3) Apply ARCH(1), GARCH(1,1) and GARCH(2,2) to fit the data. Report
the estimates for each model along with the AIC and BIC measures.
(4) In-sample and Out-of-sample volatility forecasting.
(i) Break the whole sample into two sub-samples (half/half).
(ii) Use the first half sample (first 1000) as the observed data. Estimate
ARCH(1), GARCH(1,1) and GARCH(2,2).
(iii) Forecast the volatility series based on ARCH(1), GARCH(1,1) and
GARCH(2,2) estimates. Compare them with the true series (plot all the
graphs in the one figure using different colors).
3. Empirical application. (data provided )
Use your empirical data set (five time series) from your previous assignments:
(1) Using the whole sample data, estimate ARCH(1), ARCH(2), GARCH(1,1),
GARCH (1,2) and GARCH(2,2).
1
(2) Recommend the “best” model for this sample data. Justify your
choice.
(3) Plot the estimated conditional variances from each model and the
corresponding squared returns series of the data.
(4) Based on your estimates, forecast the next 60 days’ volatility (out of
your sample period). Please read section 3 of the paper “ A FORECAST
COMPARISON OF VOLATILITY MODELS: DOES ANYTHING BEAT
A GARCH(1,1)?” (which I posted on Learn). Pick at least two loss functions
from page 877 to evaluate your forecasts. (hint: you could use r2
t
as the benchmark to evaluate your forecasts). In your empirical data, is
GARCH(1,1) always the best?
*(5) Construct back-testing Value-at-Risk measures across your sample
period for your data. (Optional Question)