5. Load the data in dailyibm.dat using the command ibm<-scan("dailyibm.dat", skip=1). This series is the daily closing price of IBM stock from Jan 1, 1980 to Oct 8, 1992. (a) Make a plot of the data and an ACF plot of the data. Does the time series appear to be stationary? Explain. Interpret the ACF plot in this situation. (b) Difference the data. Plot this differenced data, and make an ACF plot. What is your opinion of whether the series is stationary after differencing? (c) Another option for attempting to obtain stationary data when there is something similar to an exponential trend is to take the logarithm. Use the R command log() to take the logarithm of the data. Plot this transformed data. Does the transformed data appear stationary? Explain. (d) Perhaps some combination of differencing and the logarithmic transform will give us stationary data. Why would log(diff(ibm)) not be a very good idea? Try the opposite, difference the log transformed data difflogibm<-diff(log(ibm)). Except for a few extreme outliers, does this transformation succeed in creating stationary data? (e) Delete the extreme outliers using the following command: difflogibm<-difflogibm[difflogibm> -0.1] Plot this data and the ACF for this data. Sometimes with very long time series like this one, portions of the series exhibit different behavior than other portions. Break the series into two parts using the following commands: difflogibm1<-difflogibm[1:500] difflogibm2<-difflogibm[501:length(difflogibm)] Plot both of these and create ACF plots of each. Do you notice a difference between these two sections of the larger time series? (f) Assume the model for the data that we have called difflogibm2 is of the following form: dt = δ + wt 4 where wt , t = 1, ..., T is Gaussian white noise with variance σ 2 w. Is this reasonable from what you now know of this time series? How would you estimate δ and σw? What are the numerical values of these estimates?