Reference no: EM132277261
EXERCISES
3.1. Derive multistep-ahead forecasts for a GARCH(1,2) model at the forecast origin h.
3.2. Derive multistep-ahead forecasts for a GARCH(2,1) model at the forecast origin h.
3.5. Consider the monthly simple returns of Intel stock from January 1973 to December 2008 in m- intc7308 . txt. Transform the returns into log returns. Build a GARCH model for the transformed series and compute 1-step- to 5-step-ahead volatility forecasts at the forecast origin December 2008.
3.6. The file m-mrk4608 . txt contains monthly simple returns of Merck stock from June 1946 to December 2008. The file has two columns denoting date and simple return. Transform the simple returns to log returns.
(a) Is there any evidence of serial correlations in the log returns? Use auto-correlations and 5% significance level to answer the question. If yes, remove the serial correlations.
(b) Is there any evidence of ARCH effects in the log returns? Use the residual series if there are serial correlations in part (a). Use Ljung-Box statistics for the squared returns (or residuals) with 6 and 12 lags of autocorrelations and 5% significance level to answer the question.
(c) Identify an ARCH model for the data and fit the identified model. Write down the fitted model.
3.7. The file m-3m4608 txt contains two columns. They are date and the monthly simple return for 3M stock. Transform the returns to log returns.
(a) Is there any evidence of ARCH effects in the log returns? Use Ljung-Box statistics with 6 and 12 lags of autocorrelations and 5% significance level to answer the question.
(b) Use the PACF of the squared returns to identify an ARCH model. What is the fitted model?
(c) There are 755 data points. Refit the model using the first 750 observations and use the fitted model to predict the volatilities for t from 751 to 755 (the forecast origin is 750).
3.8. The file m-gmsp5008 txt contains the dates and monthly simple returns of General Motors stock and the S&P 500 index from 1950 to 2008.
(a) Build a GARCH model with Gaussian innovations for the log returns of GM stock. Check the model and write down the fitted model.
(c) Build a GARCH model with Student-t distribution for the log returns of GM stock, including estimation of the degrees of freedom. Write down the fitted model. Let v be the degrees of freedom of the Student-t distribution. Test the hypothesis Ho : v = 6 versus Ha : v 6, using the 5% significance level.
(e) Obtain 1-step- to 6-step-ahead volatility forecasts for all the models obtained. Compare the forecasts.
3.10. Again, consider the returns in m-gmsp5 0 08 . txt.
(a) Build a Gaussian GARCH model for the monthly log returns of the S&P 500 index. Check the model carefully.
(b) Is there a summer effect on the volatility of the index return? Use the GARCH model built in part (a) to answer this question.
3.11. The file d-gmsp9 9 0 8 . txt contains the daily simple returns of GM stock and the S&P composite index from 1999 to 2008. It has three columns denoting date, GM return, and S&P return.
(a) Compute the daily log returns of GM stock. Is there any evidence of ARCH effects in the log returns? You may use 10 lags of the squared returns and 5% significance level to perform the test.
(b) Compute the PACF of the squared log returns (10 lags).
(c) Specify a GARCH model for the GM log return using a normal distri-bution for the innovations. Perform model checking and write down the fitted model.
3.12. Consider the daily simple returns of the S&P composite index in the file d-gmsp9908.txt.
(a) Is there any ARCH effect in the simple return series? Use 10 lags of the squared returns and 5% significance level to perform the test.
(b) Build an adequate GARCH model for the simple return series.
(c) Compute 1-step- to 4-step-ahead forecasts of the simple return and its volatility based on the fitted model.