EMET3007/8012 Assignment 2 Instructions: This assignment is worth either 20% or 25% of the final grade, and is worth a total of 75 points. All working must be shown for all questions. For questions which ask you to write a program, youmust pro- vide the code you used. If you have found code and then modified it, then the original source must be cited. The assignment is due by 5pm Friday 1st of October (Friday of Week 8), using Turnitin on Wattle. Late submissions will only be accepted with prior written approval. Good luck. Question 1: [10 marks] In this exercise we will consider four differ- ent specifications for forecasting monthly Australian total retail sales. The dataset (available onWattle) AUSRetail2021.csv contains three columns; the first column contains the date; the second contains the sales figures for that month, and the third contains Australian GDP for that month.1 The data runs from January 1992 to January 2021. Let Mit be a dummy variable that denotes the month of the year. Let Dit be a dummy variable which denotes the quarter of the year. The four specifications we consider are S1 : yt = a0 + a1t+ a4D4t + et S2 : yt = a1t+ 4 Â i=1 aiDit + et S3 : yt = a0 + a1t+ b12M12,t + et S4 : yt = a1t+ 12 Â i=1 biMit + et where Eet = 0 for all t. 1The GDP data is extremely unreliable. Monthly GDP data is hard to come by, so this is quarterly data which I have made monthly by assuming smooth transition between quarters. This data is fine for this assignment, but cannot be used for any other purpose. 1 a) For each specification, describe this specification in words. b) For each specification, estimate the values of the parameters, and compute the MSE, AIC, and BIC. If you make any changes to the csv file, please describe the changes you make. As always, you must include your code. c) For each specification, compute the MSFE for the 1-step and 3-step ahead forecasts, with the out-of-sample forecasting exercise begin- ning at T0 = 60. d) For each specification, plot the out-of-sample forecasts and comment on the results. Question 2: [10 marks]Now add to Question 1 the additional assump- tion that et ⇠ N (0, s2). One estimator2 for s2 is sˆ2 = 1 T k T Â t=1 (yt yˆt)2 where yˆt is the estimated value of yt in the model and k is the number of regressors in the specification. a) For each specification (S1, . . . , S4), compute sˆ2. b) For each specification, make a 95% probability forecast for the sales in April 2021. c) For each specification, compute the probability that the retail sales in April 2021 will be greater than $31bn. According to the FRED series AUSSARTMDSMEI, what was the actual retail sales value for that month. d) Do you think the assumption that et is iid is a reasonable assumption for this data series. 2As in Assignment 1, this may not be the optimal estimator for s2 in a forecasting setting. In your final project, please use the MLE-estimator for s2. 2 Question 3: [10 marks] Here we investigate whether adding GDP as a predictor can improve our forecasts. Consider the following modified specifications: S01 : yt = a0 + a1t+ a4D4t + gxth + et S02 : yt = a1t+ 4 Â i=1 aiDit + gxth + et S03 : yt = a0 + a1t+ b12M12,t + gxth + et S04 : yt = a1t+ 12 Â i=1 biMit + gxth + et where Eet = 0 for all t, and xth is GDP at time t h. For each specifi- cation, compute the MSFE for the 1-step ahead, and the 3-step ahead fore- casts, with the out-of-sample forecasting exercise beginning at T0 = 60. For each specification, plot the out-of-sample forecasts and comment on the results. Question 4: [15marks]Herewe investigatewhetherHolt-Winters smooth- ing can improve our forecasts. Use aHolt-Winters smoothingmethodwith seasonality, to produce 1-step ahead and 3-step ahead forecasts and com- pute the MSFE for these forecasts. You should use smoothing parame- ters a = b = g = 0.4 and start the out-of-sample forecasting exercise at T0 = 50. Plot these out-of-sample forecasts and comment on the results. Additionally, estimate the values for a, b, and g which minimise the MSFE. Find the MSFE for these parameter vales and compare it to the baseline a = b = g = 0.4. Question 5: [5 marks] Questions 1, 3 and 4 each provided alternative models for forecasting Australian Retail Sales. Compare the efficacy of these forecasts. Your comparison should include discussions of MSFE, but must also make qualitative observations (typically based on your graphs). 3 Question 6: [10 marks] Develop another model, either based on mate- rial from class or otherwise, to forecast Australian Retail Sales. Your new model must perform better (have a lower MSFE or MAFE) than all models from Questions 1, 3, and 4. As part of your response to this question you must provide: a) a brief written explanation of what your model is doing, b) a brief statement on why you think your new model will perform better, c) any relevant equations ormathematics/statistics to describe themodel, d) the code to run the model, and e) the MSFE and/or MAFE error found by your model, and a brief dis- cussion of how this compares to previous cases. Question 7: [15 marks] Consider the AR(2) process with drift yt = µ+ r1yt1 + r2tt2 + et where the errors follow an AR(1) process et = fet1 + ut, u ⇠ N (0, s2 I) for t = 1, . . . , T and e0 = 0. Suppose f is known. Find (analytically) the maximum likelihood estimators for µ, r1, r2, and s2. [Hint: First write y and e in vector/matrix form. You may wish to use different looking forms for each. Find the distribution of e and y. Then apply some appropriate calculus.] 4