1. Consider the daily returns of Amazon (amzn) stock from January 2, 2009 to December 31, 2014. The simple returns are available from CRSP and in the file d-amzn3dx0914. txt (the column with heading amzn). Transform the simple returns to log returns. Multiple the log returns by 100 to obtain the percentage returns. Let ~ be the percentage log returns. (a) Is the expected value of r, zero? Why? Are there any serial correlations in r,? Why? (b) Fit a Gaussian ARMA-GARCH model to the r, series. Obtain the normal QQ- plot of the standardized residuals, and write down the fitted model. Is the model adequate? Why? (c) Build an ARMA-GARCH model with Student-t innovations for the log return series. Perform model checking and write down the fitted model. (d) Obtain 1-step to 5-step ahead mean and volatility forecasts using the fitted ARMA-GARCH model with Student-t innovations.1. To transform the simple returns to log returns, we can use the formula: 7', : log(1 + 3,) where r, is the log return and 3,5 is the simple return. To obtain the percentage returns, we can multiply the log returns by 100. Let re, be the percentage log returns. (a) The expected value of T8, is not necessarily zero, because the log returns may have a non-zero mean. However, if the log returns are stationary and ergodic, then the sample mean of re, will converge to the true mean as the sample size increases. To test if the expected value of rat is zero, we can use a t-test or a z-test on the sample mean. The serial correlations in re, measure the dependence of the log returns on their past values. If the log returns are serially uncorrelated, then they are unpredictable and follow a random walk. To test for serial correlations in Tet, we can use the autocorrelation function (ACF) or the Ljung-Box test. (b) A Gaussian ARMA-GARCH model is a combination of an ARMA model for the mean equation and a GARCH model for the variance equation. The ARMA model captures the linear dependence of the log returns on their past values and past errors, while the GARCH model captures the volatility clustering and heteroskedasticity of the log returns. The Gaussian assumption implies that the log returns are normally distributed. To fit a Gaussian ARMA-GARCH model to the re, series, we can use the rugarch package in R. The ugarchspec function allows us to specify the orders of the ARMA and GARCH models, and the ugarchfit function allows us to estimate the parameters using maximum likelihood. The ugarchforecast function allows us to obtain the forecasts of the mean and volatility. The normal QQplot of the standardized residuals is a graphical tool to check the normality assumption of the log returns. The standardized residuals are obtained by dividing the raw residuals by the conditional standard deviation. If the log returns are normally distributed, then the standardized residuals should lie on a straight line. The qqnorm function in R can be used to plot the normal QQ-plot. The fitted model is adequate if it satisfies the following criteria: . The standardized residuals are serially uncorrelated, which can be checked by the ACF or the Ljung-Box test. . The standardized residuals are normally distributed, which can be checked by the normal QQ-plot or the Jarque-Bera test. . The standardized residuals have constant variance, which can be checked by the squared standardized residuals or the ARCH-LM test. An ARMA-GARCH model with Student-t innovations is similar to a Gaussian ARMA GARCH model, except that it relaxes the normality assumption and allows for heavier tails in the log returns distribution. The Studentt distribution has an extra parameter, the degrees of freedom, which controls the kurtosis of the distribution. The lower the degrees of freedom, the higher the kurtosis and the fatter the tails. To build an ARMA-GARCH model with Student-t innovations for the log return series, we can use the same rugarch package in R, but specify the distribution as std in the ugarchspec function. The rest of the steps are similar to the Gaussian ARMA-GARCH model. To perform model checking, we can use the same criteria as in part (b), but replace the normality tests with the Student-t tests. The qqplot function in R can be used to plot the Student-t QQ-plot, and the gofTest function in the fBasics package can be used to perform the Kolmogorov-Smirnov test for the Student-t distribution. (d) To obtain 1-step to 5-step ahead mean and volatility forecasts using the fitted ARMA- GARCH model with Student-t innovations, we can use the ugarchforecast function in the rugarch package, and specify the n.ahead argument as the desired number of steps. The function will return the point forecasts and the prediction intervals for the mean and volatility of the log returns