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PDF Submissiоn Only Midterm Exаm 2 Pаrt 2: Dаta Analysis (Gradescоpe) (10-minute submissiоn window) Canvas file upload here Part I: ARIMA-GARCH Modelling 1a. (3 Pts.) Evaluate the stationarity properties of the Market Index Returns, GDP Growth and Unemployment Rate time series. Support your analysis with appropriate plots (e.g., time series plots, ACF/PACF) and statistical tests (e.g., Augmented Dickey-Fuller or KPSS) as needed. 1b (7 Pts.) Using the **Market Index Return** series, divide the data into training and testing sets, with the period from Q1 1999 to Q4 2023 as the training set and the last four quarters (Q1 2024 and Q4 2024) as the testing set. Using the training set, fit an ARIMA model of order (6,1,6). Then obtain the residuals from the fitted ARIMA model and examine their properties by plotting the ACF and PACF of both the residuals and the squared residuals, and by conducting appropriate diagnostic tests. Finally, evaluate whether the residuals exhibit evidence of heteroscedasticity, and provide written interpretation of the results, clearly explaining what the plots and test outcomes imply about the adequacy of the model. 1c (7 Pts.) Estimate a ARIMA(8,1,5)-GARCH(1,1) model for the Market Index Return (MIR). After fitting the model, evaluate whether it has adequately captured both the serial correlation and volatility clustering. Plot the ACF of the standardized residuals and the ACF of the squared standardized residuals to assess remaining structure, and check whether the conditional variance process is stationary based on the estimated GARCH parameters. Provide written interpretations of your plots and test results, clearly explaining what they indicate about the adequacy of your model. 1d (6 Pts.) Apply the selected model from (1c) to obtain one-lag rolling forecasts for the testing period. Visualize the predictions versus the observed data and calculate the Mean Absolute Percentage Error (MAPE) and Prediction Mean (PM) for each time series. Discuss the accuracy of the predictions. 1e (7 Pts.) Using the final order for your model from question 1b for the Market Index Return data, estimate a APARCH model. Write the model equation and evaluate whether it is necessary to control for asymmetry in the model. Support your conclusion by comparing the News Impact curve of the APARCH model with that of the GARCH model from question 1c. Note: If your model uses differenced data, you will need to convert the forecasts back to the original time series data. Part II: Multivariate Modeling 2a (8 Pts.) Fit an unrestricted VAR(p) model using the Market Index Return, GDP and Unemployment rate. Select the optimal lag order using the **BIC** information criterion, with a maximum order of p = 7. Evaluate the stability of the estimated VAR model. Assess the model fit, and support your comments with relevant plots and statistical tests (e.g., residual diagnostics, ACF/PACF of residuals). *Hint:* You can analyze the roots of the characteristic polynomial to check for stability. 2b (6 Pts.) For each time series in the VAR model from question 2a, apply the Wald test to identify any lead and lag relationships between the two time series, using a significance level of $alpha =0.05$. Comment on any potential relationships. Additionally, are there any contemporaneous relationships between the two time series? 2c (8 Pts.). Fit a VARX(p) model for p up to an order of 8 using the training data, where mir is the endogenous variable and ur and gdp are the exogenous variables. Use the AIC as the order selection criterion. Display the model summary of the selected VARX model. What is the selected order? Part III: Forecast 3a (8 Pts.) Using the VAR models fitted in questions 2a and 2c, obtain 4-quarter ahead predictions for the Market Index Return. Visualize the predictions versus the observed data and calculate the Mean Absolute Percentage Error (MAPE) and Prediction Mean (PM) accuracy measures. Comment on the accuracy of the predictions. Which model—ARIMA-GARCH, VAR, or Restricted VAR provides better predictions?