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Fоr this exаm, yоu will be explоring weekly аggregаted values of the DOW index from the beginning of 2010 through the end of 2020. Please use the provided starter script to begin the data analysis. Additionally you will find included a script containing a few helper functions that may prove useful. It is entirely optional to use them. Additionally, you may refer to any course notes or example code. Please read the open-book policy document to better understand what it allowed and not allowed during this exam. Analysis should be done on the 'train' data except for comparing ARIMA forecasts in question 2, when you will be using the 'test' data. MT Starter Code.R Midterm Helper Code.R Additional reminder to watch the time carefully and make sure you submit before it closes. Data: DJI.csv Question 1: Data Exploration (25 Points) Produce time series and ACF plots for the original series as well as the first and second differences. What assumptions (if any) does each level of differencing appear to violate, and which seems closest to stationary? Comment if any features stand out. Fit a third order parametric polynomial (cubed) and a splines model on the first difference of the data. Extract the residuals of each model and use graphical residual analysis to evaluate the goodness of fit. Also apply performance measurements (MAPE, Precision Measure, and MSE) to assess the performance of the prediction. Now fit a third order parametric polynomial, but also include a second order harmonic component. Assess the fit in the same way as in 1(B) above; does the quality of the fit seem to be improved? How about the predictions? What do the results seem to indicate? Question 2: ARIMA Modelling and Forecasting (45 Points) On each level of differencing, now build the PACF plots. When considered along with the plots from before, what (if anything) can you say about the potential order of the ARIMA model? Using the iterative AIC minimization process with a maximum order of (4,2,4), determine the non-trivial (at least one coefficient) orders associated with the lowest and second lowest AIC scores. Fit models with both orders and use Box tests and appropriate graphical analysis to assess the quality of the fits. Perform root analysis of both models (graphically or algebraically) and comment on the properties of the processes. Write out the algebraic function associated with the model that has the lowest AIC score, include a model summary for verification purposes. For the same model, forecast ahead 6 points and overlay the values with confidence intervals on top of the plot for the data of the same period. Calculate forecast accuracy metrics PM, MAE, and MAPE and comment on the prediction performance. Does ARIMA seem like a good method of modelling a financial index like the DOW? Why or why not? (To receive full credit, you must reference the results of your analysis as well as underlying theory). Please upload your full report below when complete. Do not click "submit" until you are done or it will end your test. Please upload in one of the following formats to receive full credit (-5 for incorrect format): PDF, Word, HTML