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Find a time series data set which seems to show conditional volatility. (There must be a WEBSITE LINK for the data set, you can’t make the data up)!!! The time series plot should show periods of relatively high volatility interspersed with periods of relative calm. It is often easier to identify conditional volatility if the sampling interval is short (e.g., daily data) than when the sampling interval is longer. Leave out the last data point (let’s call this the n +1’st observation) for the ARMAARCH modeling. You will use it at the end of the project for checking the performance of the forecast intervals. Using first n observations of your data set, carry out the same type of analysis as you did in Homework 8. Below is the analysis requirements: 1) Plot the logs of data. Based on this plot, and the ACF and PACF of the logs and differenced logs, does the series appear to be stationary? Can you identify an ARIMA(p , d , q ) model from these plots? 2) Using AICC , select an ARIMA(p , 1 , q ) (without constant) with 0 ≤ p ≤ 2, 0 ≤ q ≤ 2. Write the complete form of the fitted model. Save the residuals and fitted values for the model you selected, using Storage → Residuals, Fits. The residuals will be stored in RESI1 and the fitted values will be stored in FITS1. (Note that FITS1 starts with one missing value, while at time t it represents f t −1 , 1, the one-step forecast for the log exchange rate at time t made from time t −1). Also, get Minitab to compute the (ARIMA) one step ahead forecast and 95% forecast interval. 3) Plot the residuals, as well as ACF and PACF of both the residuals and the squared residuals. Use these plots to argue that the residuals, although approximately uncorrelated, are not independent; instead, they show evidence of conditional heteroscedasticity. - 2 - We next need to save the residuals (RESI1) to an external file, to read into R. (To do this, use File → New → Worksheet in Minitab, then select the column of residuals in the old worksheet and copy and paste into the new worksheet. Save the new worksheet using File → Save Worksheet As. Save as type: Text (ANSI). I will use RES.TXT for the output file name. If you will be using R from apps.stern, save the file in your H: directory. If you will be using a version of R that is installed on your computer, save the file in a local directory (see the document on using R for detailed instructions). Next, minimize the Minitab window, and open R. To read the data set into R use the command >scan() or >file.choose() as described in the usingR document. Download and load the tseries package that will be used to fit ARCH models, as described in the usingR document. You can now fit your arch models, as described in class and in the handout on Estimation and Automatic Selection of ARCH models. ) 4) Using R on the residuals from the ARIMA model, find the log likelihood values and AICC values for ARCH (q ) models where q ranges from 0 to 10. You will need to calculate the log likelihood for the ARCH (0) model by hand. Next, consider a GARCH (1,1) model. If the residuals from Minitab are stored in an R data set x , then the R command is >model=garch(x,c(1,1)). Evaluate AICC for the GARCH (1,1) model, using q =2 in the formula for AICC . If the GARCH (1,1) is preferred by AICC , use it as your selected model. Comment on the statistical significance of the parameter values of your selected model, as given by the summary(model) command. Write the complete form of the ARCH or GARCH model you have selected. Hand in the R output for the selected model, that is, the results of both summary(model) and logLik(model), but only for the one model that was selected by AICC . Also evaluate the unconditional (marginal) variance of the shocks in this model. 5) Using the Minitab output from problem 2, and the R output from your selected model in problem 4, construct a 95% one step ahead forecast interval for the log exchange rate, based on your ARIMA-ARCH model. (If you decided to use a GARCH (1,1) model, you will need to first get the conditional variances from R. See Problem 6.) Compare this to the interval based on the ARIMA only model from problem 2. Also compute the 5th percentile of the conditional distribution of the next period’s log exchange rate. 6) Plot the conditional variances, ht, for your fitted ARCH model from problem 4. (See instructions below). Use this plot to locate bursts of high volatility. Do these highly volatile periods agree with those found from examination of the time series plot of the log exchange rates themselves? To save the conditional variances and read them into Minitab, proceed as follows. First, re-fit your selected ARCH model and store it in the variable "model". Compute and store the conditional variances with the command >ht=model$fit[,1]ˆ2. Next, write the ht dataset to a file named htfile.txt using the write() or file.choose() command as described in the usingR document. Minimize R and re-enter Minitab. Read htfile.txt into a new Minitab worksheet, using File → Open. The resulting column (ht) should have a length of 1259. So, for example, the last value is h1259, the conditional variance for time 1259 (which can be computed based on information available at time 1258). Copy and paste this column into the original Minitab worksheet. 7) Make a time series plot which simultaneously shows the log exchange rates, together with the ARIMA-ARCH one-step-ahead 95% forecast intervals based on information available the previous day. (See instructions below). Using the plot, together with the numerical values in your Minitab worksheet, comment on the accuracy and practical usefulness of the forecast intervals. Keep in mind that the performance may be somewhat better here than in an actual forecasting context, since the ARIMA-ARCH parameters are estimated from the entire data set, not just the observations up to the time at which the forecast is to be constructed. To compute the forecast intervals in Minitab, proceed as follows. First, get "low" and "high", the lower and upper endpoints of the 95% forecast intervals, using Calc → Calculator → Store result in variable: low , Expression : FITS1-1.96*sqrt(ht) → OK, and similarly for high. For a given t , the interval between low and high represents a one-step-ahead 95% forecast interval for the log exchange rate at time t based on information which was available at time t −1. To plot the intervals along with the log exchange rates in Minitab, use Graph → Time Series Plot → Multiple. In Data View, click the box for "Connect Line" but un-check the box for "Symbols". 8) Compute the residuals from your ARIMA-ARCH model, that is, et = εt /√⎯ ⎯ht . If the ARIMA-ARCH model is adequate, these residuals should be normally distributed with mean zero and variance 1. To compute these residuals in Minitab, use Calc → Calculator → Store result in variable: archres, Expression: RESI1/sqrt(ht). Make a normal probability plot of archres, using Stat → Basic Statistics → Normality Test. Does the model seem to have adequately described the leptokurtosis ("long-tailedness") in the data? 9) From the formula for the prediction intervals, it follows that the 95% prediction interval constructed yesterday fails to cover today’s log exchange rate whenever today’s residual exceeds 1.96 in absolute value. Use Calculator to count up how many failures there were, using sum(abs(archres)>1.96). What percentage of the time did the intervals fail? Of course, there may be a few modifications to the steps you took in Homework 8, depending on what you see here. For example, in part 2, you may or may not decide to use d = 1. Also in part 2, if you are in fact using d = 1, you may or may not decide to fit a constant term in the ARIMA model. (You can make this decision on the basis of the time series plot alone, if you wish.) Finally, check whether either or both of the one-step-ahead forecast intervals calculated in part 5 actually contained the n +1’st observation. Based on this, does the ARMA only interval seem too wide, too narrow, or just about right? Then answer the same question for the ARMA-ARCH interval. Please try to keep the length of the project to ten pages or less. The graphs cannot exceed 5 pages.