FINAL RM 708/BUS 386 DUE ON MAY 24, 2016 Instructions: Read the questions carefully. For some questions, you will have to refer to the relevant chapter in the Tsay textbook. For your answers, create a word document that includes your explanations along with the relevant Rstudio output. 1. (5 pts) The data set q msft.txt contains the quarterly earnings per share of Microsoft from the second quarter of 1986 to the third quarter of 2013. The earnings per share is in the third column. (i) Build a time series model for the log earnings series. Perform model checking and write down the fitted model. [For simplicity, you may ignore the outliers, if (ii) Fit the following model to the log earnings series: tmp<-arima(xt,order=c(0,1,1),seasonal=list(order=c(0,0,1),period=4)) where xt denotes the log earnings series. Write down the fitted model. fitting? Why? (iii) Compare the two time series models. Which model is preferred in terms of (iv) Use the backtest procedure to compare the two models via 1-step ahead fore- casts. You may use t = 81 as the starting forecast origin. Which model is 2. (5 pts) The data set d msft.txt contains daily simple returns of Microsoft (MSFT) stock from January 3, 2001 to December 31, 2013. Transform the simple returns (in the third column) to log returns. (i) Is the expected log return zero? Why? Are there any serial correlations in the (ii) If there are serial correlations in the log return, build a mean equation for the preferred? Why? log returns? Why? log returns. Write down the fitted model. Is there ARCH effect in the log return (iii) Fit a Gaussian ARMA-GARCH model to the log return series. Obtain the nor- mal QQ-plot of the standardized residuals, and write down the fitted model. Is (iv) Build an ARMA-GARCH model with Student-t innovations for the log return the model adequate? Why? garchFit( arma(p,q)+garch(m,n),data=data,trace=F) series. Perform model checking and write down the fitted model. 1 (v) Obtain 1-step to 5-step ahead mean and volatility forecasts using the fitted ARMA-GARCH model with Student-t innovations. 3. (5 pts) Using the quantmod package download the adjusted closing prices of two stocks, namely Apple (AAPL) and Procter & Gamble (PG). The sample period is from January 3, 2002 to May 30, 2014. The data can be downloaded from Yahoo. Use the adjusted closing prices to compute the daily log returns. For example, dataset<-getSymbols("AAPL",from="2002-01-03",to="2014-05-30") tmp<-diff(log(zoo(GE✩GE.Adjusted))) #install zoo package first Assume that the tail probability of interest is 1% and that you hold a portfolio which consists of both stocks valued at $1 million each. Consider the Apple stock only. (i) Calculate the VaR of your position for the next trading day using the RiskMetrics method on May 30, 2014. You must estimate the corresponding IGARCH(1,1) model. What is the associated expected shortfall? Also, what is the VaR for the next 10 trading days? (ii) Build a GARCH(1,1) model for the log return series with Gaussian innovations. What is the VaR based on the fitted model for the next trading day? What is the corresponding expected shortfall? (iii) Build a GARCH(1,1) model with Student-t innovations for the log return series. What is the VaR for the next trading day based on the fitted model? What is the corresponding expected shortfall? 4. (5 pts) For a log return series rt , let at = rt−μt . Then,the GARCH(m, s) specification at = σtt 2 t = α0 + σ Xm 2 t−i + αia i=1 i=1 (αi + βi) < 1. Show that the kurtosis of Xs 2 βjσ t−j j=1 where α0 > 0, αi ≥ 0, βj ≥ 0, and Pmax(m,s) is greater than 3 for m = s = 1. 5. (5 pts) The data set csco.csv contains observations on simple returns for Cisco Systems (CSCO) stock. Consider the daily log returns of CSCO stock (calculate it from simpe returns). You will use out-of-sample forecasts to compute the annualized h-period volatility, i.e., volatility term structure for CSCO stock from December 29, 2008 to December 31, 2010. (i) Estimate a GARCH(1,1) model using the entire sample. You can confirm your results from Table 5.1 on page 245. 2 (ii) Use the fitted model to calculate 1-step to 40-step ahead volatility forecasts for t from December 29, 2008 to December 31, 2010, i.e., σ (iii) Obtain conditional variance of the h-period log returns, i.e., σ for h = 1, 5, 10, 15, 20, 25, 30, 35, 40 and for t from December 29, 2008 to December 31, 2010. (iv) Calculate the annualized h-period volatility as σt,h,a = for h = 1, 5, 10, 15, 20, 25, 30, 35, 40. (v) Plot annualized h-period volatilities that you obtained in the previous part. You can comfirm your results from Figure 5.3 on page 248. 2 (l) for l = 1, . . . , 40. t q 252 σt,h h