Econ 423 Assignment 3
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Econ 423
Assignment 3
Due on Nov 30
1. Derive the unconditional variance of a GARCH(2,1) process. And derive the log-likelihood function for GARCH(2,1). What is the positivity condition and stationarity condition respectively? [20]
2. Monte Carlo simulation. [30]
(1) Set the values for μ , α 1 and β1 and assume that e follows a standard normal distribution.
(2) Based on (1), simulate 2000 data from a standard GARCH(1,1) pro- cess.
(3) Apply ARCH(1), GARCH(1,1) and GARCH(2,2) to fit the data. Re- port the estimates for each model along with the AIC and BIC measures. Give some comments.
(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. [50]
Use your empirical data set (pick any two out of your 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).
(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 func- tions from page 877 to evaluate your forecasts. (hint: you could use Tt(2) as the benchmark to evaluate your forecasts). In your empirical data, is GARCH(1,1) always the best?
(5) Apply a bivariate-VAR(1) on these two time series. Any Granger Causality effect between these two time series? And explain.
*(6) Construct back-testing Value-at-Risk measures across your sample period for each of your sample data. (Optional Question) [5]
2022-12-01