1. Daily returns of Amazon (amzn) stock from January 2, 2009 to December 31, 2014.
Percentage log returns used in the analysis.
(a) Is the expected value of rt zero? Why? Are there any serial correlations in rt? Why?
(b) Fit a Gaussian ARMA-GARCH model to the rt series. Obtain the normal QQ-plot of the standardized residuals, and write down the tted model. Is the model adequate? Why?
(c) Build an ARMA-GARCH model with Student-t innovations for the rt series. Per-form model checking and write down the tted model.
(d) Obtain 1-step to 5-step ahead mean and volatility forecasts using the ftted ARMA-GARCH model with Student-t innovations.
2. Consider again the daily log returns, in percentages, of Amazon stock in Problem 1.
(a) Let at = rt - r, where r is the sample mean of rt. Fit an IGARCH(1,1) model with a constant term in the volatility equation to the at series. Write down the ftted model.
(b) Let t be the tted volatility of the IGARCH(1,1) model. De ne the standardized residuals as t = (rt - r)=t. Is there any serial correlation in the standardized residuals? Why?
(c) Is there any serial correlation in the squares of the standardized residuals? Why?
(d) Based on the model checking, is the IGARCH model adequate? Obtain 1-step to 4-step ahead volatility forecasts for the rt series (forecast origin is the last data point).
3. Monthly returns of McDonald's stock from August 1966 to December 2014.
(a) Is the expected MCD log return zero? Why? Is there any serial correlation in the log returns? Why? Is there any ARCH effect in the log returns? Why?
(b) Build a GARCH model with Gaussian innovations for the log return series. Per-form model checking and write down the tted model.
(c) Fit an IGARCH(1,1) model for the MCD log returns. Write down the ftted model.
(d) Fit a GARCH model with skew-Student-t innovations to the log return series. Perform model checking and write down the tted model.
(e) Based on the tted model, is the monthly log returns of MCD stock skewed? Why?
(f) Fit a GARCM-M model to the monthly log returns. Write down the model? Is the risk premium statistically signi cant? Why?
(g) Fit a TGARCH(1,1) model to the monthly log returns. Write down the fitted model. Is the leverage effect statistically signi cant? Why?
4. Monthly returns of the value-weighted index, including dividends from 1966 to 20014.
(a) Find an adequate model for the monthly log return series. Perform model checking to justify your model.
(b) Obtain 1-step to 5-step ahead predictions of the log return and its volatility at the forecast origin December 2014.
(c) Fit a GJR model (may use the APARCH command) to the monthly log return series. Write down the model. Is the leverage e ect statistically signi cant? Why?
5. Consider the log return of daily exchange rate between UK Pounds and U.S. Dollars from January 4, 2006 to March 20, 2015.
(a) Build a GARCH model (including mean equation) for the log return series. Per-form model checking. Write down the fitted model.
(b) Let rt be the daily log return. For numeric stability, consider the percentage log return, i.e. xt = 100rt. Fit a volatility model with leverage effect to xt. Write down the fitted model. Is the leverage effect statistically significant? Why?