Part I: ARIMA-GARCH Modelling (65 pts) Question: 1a.a (9 pts…

Pаrt I: ARIMA-GARCH Mоdelling (65 pts) Questiоn: 1а.а (9 pts) Evaluate the statiоnarity properties of the full time series corresponding to Financial Returns, Economic Activity Indicator, and Risk Conditions Index. Support your analysis with appropriate plots (including time series plots and ACF/PACF) and statistical tests (e.g., Augmented Dickey-Fuller or KPSS) as needed. Additionally, explore the correlation structure between the three time series using appropriate tools (e.g., contemporaneous correlation matrix, scatter plots, or lagged cross-correlation plots). Based on your findings, discuss the degree of comovement and interdependence across the three series. Conclude by providing an initial assessment of whether a multivariate framework such as a VAR model is appropriate for capturing the joint dynamics of these variables, or whether the series appear largely independent and better suited for separate univariate modeling approaches. Question: 1a.b (2 pts) Why is important to study the time series plots of multivariate time series? What features are important to evaluate?   Training vs Testing Data Using the Financial Returns (RF) series, divide the data into training and testing sets, leaving the last six observations (July 2025 to December 2025) as the testing set, and using the remaining data (January 2000 to June 2025) as the training set. For the questions below, you will estimate/fit and compare several models using the training data. You will then generate one-step-ahead rolling forecasts for each of the last six months as well as 6-step ahead forescasts. Specifically, for one-step-ahead rolling forecasts, each forecast origin, re-estimate the specified models using all data available up to that time point and produce a one-period-ahead prediction.   Question: 1b.a (7 pts) Using the training set (without the last six months), fit an ARIMA model of order (5,0,2). Then obtain the residuals and the standardized residuals from the fitted ARIMA model and examine their properties by plotting the ACF of the residuals and the squared (standardized) residuals, and by conducting appropriate diagnostic tests (e.g., Box-Ljung test). Evaluate whether the residuals exhibit evidence of heteroscedasticity, and provide a written interpretation of the results, clearly explaining what the plots and test outcomes imply about the adequacy of the model. Question: 1b.b (2 pts) Write the model equation for the model fit in Question: 1b.a.   Question: 1c.a (9 pts) Estimate an ARIMA(5,0,2)–GARCH(1,1) model for the Financial Returns (RF) series. Plot the ACF of the standardized residuals and the ACF of the squared standardized residuals to assess whether any remaining structure is present in the mean or variance. Evaluate whether the model has adequately captured both the serial correlation and volatility clustering present in the data using appropriate diagnostic tests, such as Box-Ljung tests applied to the standardized residuals and squared standardized residuals. Additionally, examine whether the conditional variance process is stationary by evaluating the estimated GARCH parameters (i.e., whether the sum of the ARCH and GARCH coefficients is less than one). Provide written interpretations of your plots and results, clearly explaining what they indicate about the adequacy of the model. Question: 1c.b (2 pts) How would the modeling the heteroskedasticity impact the estimate and the statistical inference on the first moment, the expectation? That is, what would be the difference between ARMA and ARMA-GARCH?   Question: 1d (11 pts) Apply the model from (1b) to obtain one-step-ahead rolling forecasts for the Financial Returns (RF) series over the testing period, consisting of the last six observations (July 2025 to December 2025).  Apply the model from (1b) to obtain six-step-ahead forecasts for the Financial Returns (RF) series over the testing period, consisting of the last six observations (July 2025 to December 2025).  Visualize the forecasts against the observed values, and calculate the Mean Absolute Percentage Error (MAPE) and the Prediction Mean (PM) for the testing period. Based on these results, discuss the accuracy of the forecasts and whether the model provides satisfactory predictive performance for the Financial Returns series, comparing the one-step-ahead rolling forecasts with the six-step-ahead forecasts.   Question: 1e (14 pts) Using the mean model specification in Question 1b for the Financial Returns (RF) series, fit an EGARCH(1,1) model using the training data. Write down the full model equations, including both the mean equation and the conditional variance equation, clearly defining all parameters. Plot the ACF of the standardized residuals and the ACF of the squared standardized residuals for both the standard GARCH model from Question 1c and the EGARCH model. Use Box-Ljung tests applied to the standardized residuals and squared standardized residuals to assess whether either model leaves remaining structure in the mean or variance. Evaluate whether it is appropriate to model the conditional variance using an EGARCH specification. Support your conclusion by comparing the EGARCH model with the standard GARCH model from Question 1c, focusing on whether the EGARCH model provides a better representation of volatility dynamics in the Financial Returns series.  In particular, assess whether there is evidence of asymmetry in volatility responses, that is, whether positive and negative shocks have different effects on future volatility. To support your discussion, plot and interpret the News Impact Curves (NICs) for both the GARCH and EGARCH models, and explain what these imply about the role of asymmetric shocks in the data. Comment on whether the EGARCH model appears to improve the overall adequacy of the volatility specification relative to the standard GARCH model.   Question: 1f (9 pts) Using the Economic Activity (EA) training data, estimate an ARMA(1,0,2) model. Using the fitted model, generate 6-step-ahead forecasts and 1-step-ahead  rolling forecasts for the testing period. Compare the forecast visually as well as using prediction accuracy measures.   Part II: Multivariate Modeling (20 pts) Question: 2a.a (9 pts) Using the training data (without the last six months), fit an unrestricted VAR(p) model using the Financial Returns (RF), Economic Activity (EA), and Risk Conditions (RC) series. Select the optimal lag order using the AIC information criterion, considering a maximum lag length of p = 7. Evaluate the stability of the estimated VAR model. Assess the overall fit of the model, and support your discussion with relevant plots and statistical tests, such as residual diagnostics and the ACF/PACF of the residuals. Hint: You can analyze the roots of the characteristic polynomial to assess whether the VAR model is stable. Question: 2a.b (3 pts) Would you expect the selected order to be higher or lower if the training data includes more observations? Explain your answer. Why do we prefer an unrestricted VAR model?   Question: 2b.a (6 pts) For each time series in the VAR model from question 2a, apply the Wald test to identify any lead–lag relationships among the Financial Returns (RF), Economic Activity (EA), and Risk Conditions (RC) series, using a significance level of