From the Moodle site, download EViews workfile ar2.wfl. The file contains a time series {y} from 1980m01 to 2000M10, a total of 250 observations. 1. Present the correlogram of the time series up to 10 lags. 2. What is the Q statistic for testing Ho : corr(yt,yt-1) = corr(99-2) = 0 against the alternative Ha : corr(ye,y-1) or corr(ye, yt-2) is not zero? Do you reject the null at the 5% significance level? 3. Based on the correlogram, a researcher decides to fit the AR (2) model to {y}. Justify his choice. 4. Use the observations from 1980M01(t = 1) to 1999M12 (t = 240) to estimate the parameters of an AR(2) model. Write down the fitted model including the variance of the shocks (round to three decimal points). 5. Use EViews to obtain the out-of-sample one-step ahead forecast for observations from 2000M1 to 2000M10 using the estimation results in question 4. Present the forecast graph. 6. From the EViews output in question 5, what is the one-step ahead forecast for y250 (the obser- vation on 2000M10)? 7. What is the information set for the forecast of y250 in question 6? 8. A researcher used the estimation results in question 4 and Dynamic forecast in EViews to forecast {yt} from 2000M1 to 2000M10. The forecast graph and the forecasted value of y from 2000m08 (t=248) to 2000m 10(t=250) is presented in Figure A and Table A. 2.5 20 15 1.0 Question B (20 points) We have two time series, {a} and {y}. Suppose that both follow an AR(1) process with 0.5-1 + y = 0.94-1 + where ... N(0,4) (i.e. the variance of eis 4) and ! WN(0,1). 11. (4 points) Derive and compute the unconditional mean of {r} and {y}. 12. (4 points) Derive and compute the unconditional variance of {} and {y} (round to two decimal places) 4 13. (4 points) Suppose that I = -2, compute the one-step ahead forecast, E(+1) and E(+1). o POE DDD Short Answers Question A (12 points) Suppose that the log price of an asset, pe, follows a random walk with drift, pe = 0.01 + pe-1 +, where E-1() = 0. The value of pe and the continuously compounded return rt in Jan, Feb, and Mar, 2016 are given in the following table. date 02/2020 01/2020 1 t 2 log Price pt Monthly return 03/2020 06/2020 3 6 P3 = 4.03 E3(P6) =? T3 = 1% E3(r6) =? P1 = 4.04 Ti = 4% P2 = 4.02 12 = -2% 8. (4 points) Suppose that we are at Mar 2020 (t = 3), what is your prediction of pe for June 2020? Justify your answer (show your steps using law of iterated expectation). 9. (4 points) Suppose that we are at Mar 2016 (t = 3), what is your prediction of the monthly return in June, E3(re)? Justify your answer (show your steps using law of iterated expectation). 10. Let r.(3) denote the quarterly return from month t - 2 to t. Suppose that we are at Mar 2020 (t = 3), what is your prediction of the quarterly return from Apr, May, and June, Es (re(3))? Justify your answer. 3. (4 points) Suppose that you have a pet octopus (octopuses are known to be very smart) and you want to use it to help you bet on the 2020 Australian Open. What is the probability of the octopus predicting the outcome of 7 consecutive matches correctly? (a) 1/2 (b)1/22 (c) 1/26 (d)1/27 i.id N(0,0%), 4. (4 points) Suppose that {yt} is generated from y = 0.2 +0.9* 4-1 + et, with e which of the following statements is not correct? (a) The unconditional mean of ye does not change with t. (b) The conditional mean of yt changes over time. (c) {y} exhibits autocorrelation. (d) {yt} is nonstationary. 5. (4 points) Given a time series {y}, we run the following regression y = co + C1yt-1 + 0241-2 +et and obtains OLS estimates as follow: @o = 1.6, i = 0.63, and 2 = 0.14. which of the following statements is not correct? (a) The autocorrelation of y: with its first lag is 0.63. (b) The autocorrelation of y with its second lag is 0.14. (c) The partial autocorrelation of yt with its first lag is 0.63. (d) The partial autocorrelation of ye with its second lag is 0.14. From the Moodle site, download EViews workfile ar2.wfl. The file contains a time series {y} from 1980m01 to 2000M10, a total of 250 observations. 1. Present the correlogram of the time series up to 10 lags. 2. What is the Q statistic for testing Ho : corr(yt,yt-1) = corr(99-2) = 0 against the alternative Ha : corr(ye,y-1) or corr(ye, yt-2) is not zero? Do you reject the null at the 5% significance level? 3. Based on the correlogram, a researcher decides to fit the AR (2) model to {y}. Justify his choice. 4. Use the observations from 1980M01(t = 1) to 1999M12 (t = 240) to estimate the parameters of an AR(2) model. Write down the fitted model including the variance of the shocks (round to three decimal points). 5. Use EViews to obtain the out-of-sample one-step ahead forecast for observations from 2000M1 to 2000M10 using the estimation results in question 4. Present the forecast graph. 6. From the EViews output in question 5, what is the one-step ahead forecast for y250 (the obser- vation on 2000M10)? 7. What is the information set for the forecast of y250 in question 6? 8. A researcher used the estimation results in question 4 and Dynamic forecast in EViews to forecast {yt} from 2000M1 to 2000M10. The forecast graph and the forecasted value of y from 2000m08 (t=248) to 2000m 10(t=250) is presented in Figure A and Table A. 2.5 20 15 1.0 Question B (20 points) We have two time series, {a} and {y}. Suppose that both follow an AR(1) process with 0.5-1 + y = 0.94-1 + where ... N(0,4) (i.e. the variance of eis 4) and ! WN(0,1). 11. (4 points) Derive and compute the unconditional mean of {r} and {y}. 12. (4 points) Derive and compute the unconditional variance of {} and {y} (round to two decimal places) 4 13. (4 points) Suppose that I = -2, compute the one-step ahead forecast, E(+1) and E(+1). o POE DDD Short Answers Question A (12 points) Suppose that the log price of an asset, pe, follows a random walk with drift, pe = 0.01 + pe-1 +, where E-1() = 0. The value of pe and the continuously compounded return rt in Jan, Feb, and Mar, 2016 are given in the following table. date 02/2020 01/2020 1 t 2 log Price pt Monthly return 03/2020 06/2020 3 6 P3 = 4.03 E3(P6) =? T3 = 1% E3(r6) =? P1 = 4.04 Ti = 4% P2 = 4.02 12 = -2% 8. (4 points) Suppose that we are at Mar 2020 (t = 3), what is your prediction of pe for June 2020? Justify your answer (show your steps using law of iterated expectation). 9. (4 points) Suppose that we are at Mar 2016 (t = 3), what is your prediction of the monthly return in June, E3(re)? Justify your answer (show your steps using law of iterated expectation). 10. Let r.(3) denote the quarterly return from month t - 2 to t. Suppose that we are at Mar 2020 (t = 3), what is your prediction of the quarterly return from Apr, May, and June, Es (re(3))? Justify your answer. 3. (4 points) Suppose that you have a pet octopus (octopuses are known to be very smart) and you want to use it to help you bet on the 2020 Australian Open. What is the probability of the octopus predicting the outcome of 7 consecutive matches correctly? (a) 1/2 (b)1/22 (c) 1/26 (d)1/27 i.id N(0,0%), 4. (4 points) Suppose that {yt} is generated from y = 0.2 +0.9* 4-1 + et, with e which of the following statements is not correct? (a) The unconditional mean of ye does not change with t. (b) The conditional mean of yt changes over time. (c) {y} exhibits autocorrelation. (d) {yt} is nonstationary. 5. (4 points) Given a time series {y}, we run the following regression y = co + C1yt-1 + 0241-2 +et and obtains OLS estimates as follow: @o = 1.6, i = 0.63, and 2 = 0.14. which of the following statements is not correct? (a) The autocorrelation of y: with its first lag is 0.63. (b) The autocorrelation of y with its second lag is 0.14. (c) The partial autocorrelation of yt with its first lag is 0.63. (d) The partial autocorrelation of ye with its second lag is 0.14