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1. Download the same dataset of SPY return from 1/3/2000 that we have been working with again to get the most recent data. You want to wait for the return data for the last day before Due Date (why?). But don't wait to the last day to work on this problem set... Start working early, and leave the last few day's return entry empty (or some random numbers as placeholders). After the market closes on Due Date-1, update the live return data to your excel, which should automatically update the prediction result if you have designed it well. Calculated daily log return and daily log return squared from 1/4/2000 to Due Date-1. That is our full sample to work with. 2. On each day in the full sample, suppose you are in the morning and don't know the day's return yet, predict the day's return? using - 1-month moving average (use past 21 observations) - 1-year moving average (use past 252 observations) Hints: - Some methods do not apply to some earlier days in the full sample. Please just leave them blank. It should not matter for later procedures. I prefer to have the prediction target and the corresponding predicted value on the same row. But always remember which variable is in what information set. For example, the prediction target and the predicted value are known on what dates respectively? 3. Do the same task with the Risk Metrics model. Repeat the task twice with = 0.94 (the recommended Risk Metrics parameter) and = 0.96 (a supposedly less preferred parameter) respectively. Hints: for the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction of as a start. Write the general recursive formula from the second day of the sample onward. 4. Now we are about a month 5. Estimate the GARCH model in the full sample. Hints: - Make a sensible guess of the GARCH parameters (w,a, B) as the initial guess. - For the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction o; as a start (some suggests using the sample variance as an estimation of the unconditional variance). Write the general recursive formula from the second row of the sample onward. - Calculate the log likelihood with respect to the initial guess parameters as placeholders. - Use solver to maximize the log likelihood by changing over the choices of parameters. - The excel solver is rather sensitive to the initial guess parameters. It only finds the local optimum, which depends on initial guess. To overcome, try a few different initial guesses, and record the local optimum with the greatest log likelihood target as the global optimum. - To automatize this process, choose "multistart" at Solver options GRG Non-linear Multistart. Check both "Use Multistart" and "Require Bounds on Variables". For this you need to set bounds for all three parameters. I simply restrict the three to be all within (0,1). Then solver takes a bit longer (about 15 secs) to automatically try a series of random initial guesses within the bounds. 6. Calculate the conditional expectation under the five models above one day into the future as your prediction of the Due Date. Report the five predictions. We are now about a month into the high volatility period from the Coronavirus. Between the two MA methods, which one tends to bias the measure of risk upward or downward? How about the two Risk Metrics predictions? 7. Now we need to compare the five methods. Full sample GARCH has an unfair advantage when compared with the other four directly (you will be asked why). Therefore, we are going to compare the five methods on an even ground by competing OOS. Let the first decade 1/4/2000 - 12/31/2009 be the training sample (aka in-sample), let the next portion, 1/4/2010 - the last day before Due Date be the validation sample (aka out-of-sample). Pretend now we are on 1/1/2010 morning, estimate a GARCH model in the training sample. Fix the parameters estimated IS. Use that set of parameters to predict returns? for 1/1/2010. Then pretend it's 1/2/2010 morning, use the same parameters estimated from the same IS (2000's data), but use the newest 1/1/2010 return to predict for 1/2/2010. Repeat this day by day for all 2010's onward. These are our OOS predictions. Now we have six series of best guesses in the validation sample: 4 non-estimation based methods, GARCH full sample, and GARCH OOS. Report the prediction accuracy of the six methods respectively. Here, measure prediction accuracy in terms of mean squared error, i.e. the average of prediction errors squared in the validation sample. Which sample should be used for the comparison? 8. Which method is the best? Do the six numbers make sense to you (Are they on the same scale, does the scale make sense? Does the ordering make sense? Is RM indeed better than MA? Is the RM with the recommended indeed better than the other RM. Is GARCH full sample better than GARCH OOS? Is that expected?) When arguing for the superiority of GARCH, can you cite the small MSE in the validation sample of the GARCH estimated with the full sample? Briefly explain why not, and why the GARCH estimated IS but validated OOS is a more fair comparison with MA and RM. 9. Draw a time-series line chart of the target series (return?) and the GARCH OOS predicted series to illustrate the validity of the prediction. This is also a good visual inspection for the other methods in the previous steps. Try playing with the Excel options (e.g., axis, lables, gridlines, etc.) to make the visualization clear. 10. Finally, give the one number best prediction of return squared of the Due Date. Between GARCH estimated in the full sample, and GARCH estimated IS but validated OOS, which one should we use? 1. Download the same dataset of SPY return from 1/3/2000 that we have been working with again to get the most recent data. You want to wait for the return data for the last day before Due Date (why?). But don't wait to the last day to work on this problem set... Start working early, and leave the last few day's return entry empty (or some random numbers as placeholders). After the market closes on Due Date-1, update the live return data to your excel, which should automatically update the prediction result if you have designed it well. Calculated daily log return and daily log return squared from 1/4/2000 to Due Date-1. That is our full sample to work with. 2. On each day in the full sample, suppose you are in the morning and don't know the day's return yet, predict the day's return? using - 1-month moving average (use past 21 observations) - 1-year moving average (use past 252 observations) Hints: - Some methods do not apply to some earlier days in the full sample. Please just leave them blank. It should not matter for later procedures. I prefer to have the prediction target and the corresponding predicted value on the same row. But always remember which variable is in what information set. For example, the prediction target and the predicted value are known on what dates respectively? 3. Do the same task with the Risk Metrics model. Repeat the task twice with = 0.94 (the recommended Risk Metrics parameter) and = 0.96 (a supposedly less preferred parameter) respectively. Hints: for the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction of as a start. Write the general recursive formula from the second day of the sample onward. 4. Now we are about a month 5. Estimate the GARCH model in the full sample. Hints: - Make a sensible guess of the GARCH parameters (w,a, B) as the initial guess. - For the first day of the sample, since there is no history yet, put an arbitrary sensible number as the prediction o; as a start (some suggests using the sample variance as an estimation of the unconditional variance). Write the general recursive formula from the second row of the sample onward. - Calculate the log likelihood with respect to the initial guess parameters as placeholders. - Use solver to maximize the log likelihood by changing over the choices of parameters. - The excel solver is rather sensitive to the initial guess parameters. It only finds the local optimum, which depends on initial guess. To overcome, try a few different initial guesses, and record the local optimum with the greatest log likelihood target as the global optimum. - To automatize this process, choose "multistart" at Solver options GRG Non-linear Multistart. Check both "Use Multistart" and "Require Bounds on Variables". For this you need to set bounds for all three parameters. I simply restrict the three to be all within (0,1). Then solver takes a bit longer (about 15 secs) to automatically try a series of random initial guesses within the bounds. 6. Calculate the conditional expectation under the five models above one day into the future as your prediction of the Due Date. Report the five predictions. We are now about a month into the high volatility period from the Coronavirus. Between the two MA methods, which one tends to bias the measure of risk upward or downward? How about the two Risk Metrics predictions? 7. Now we need to compare the five methods. Full sample GARCH has an unfair advantage when compared with the other four directly (you will be asked why). Therefore, we are going to compare the five methods on an even ground by competing OOS. Let the first decade 1/4/2000 - 12/31/2009 be the training sample (aka in-sample), let the next portion, 1/4/2010 - the last day before Due Date be the validation sample (aka out-of-sample). Pretend now we are on 1/1/2010 morning, estimate a GARCH model in the training sample. Fix the parameters estimated IS. Use that set of parameters to predict returns? for 1/1/2010. Then pretend it's 1/2/2010 morning, use the same parameters estimated from the same IS (2000's data), but use the newest 1/1/2010 return to predict for 1/2/2010. Repeat this day by day for all 2010's onward. These are our OOS predictions. Now we have six series of best guesses in the validation sample: 4 non-estimation based methods, GARCH full sample, and GARCH OOS. Report the prediction accuracy of the six methods respectively. Here, measure prediction accuracy in terms of mean squared error, i.e. the average of prediction errors squared in the validation sample. Which sample should be used for the comparison? 8. Which method is the best? Do the six numbers make sense to you (Are they on the same scale, does the scale make sense? Does the ordering make sense? Is RM indeed better than MA? Is the RM with the recommended indeed better than the other RM. Is GARCH full sample better than GARCH OOS? Is that expected?) When arguing for the superiority of GARCH, can you cite the small MSE in the validation sample of the GARCH estimated with the full sample? Briefly explain why not, and why the GARCH estimated IS but validated OOS is a more fair comparison with MA and RM. 9. Draw a time-series line chart of the target series (return?) and the GARCH OOS predicted series to illustrate the validity of the prediction. This is also a good visual inspection for the other methods in the previous steps. Try playing with the Excel options (e.g., axis, lables, gridlines, etc.) to make the visualization clear. 10. Finally, give the one number best prediction of return squared of the Due Date. Between GARCH estimated in the full sample, and GARCH estimated IS but validated OOS, which one should we use

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