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Part I:  ARIMA and GARCH Modelling 1a. Plot the time series and the ACF for all variables and their differenced data (4 plots total for each time series) and comment on their stationarity properties and how it would affect the use of the VAR model.  1b. Divide the data in training and testing sets, using the period January 1980 to October 2022 to train and the last 4 observations for testing, i.e. January 2023 to October 2023.  Using the Interest Rate differenced data, apply the iterative BIC selection process to find the best, *non-trivial* ARIMA-GARCH model order using the ARIMA max orders given by pmax = 5, qmax = 5, d order of max =0 (you are already using differenced data), and max orders given by m=2, n=2 for the GARCH model. Explain how the selected model captures features of the time series. 1c. Evaluate the Box-Ljung test results and the ACF on the residuals and squared residuals from the model you chose in Question 1b. Comment on your results. 1d. Apply the selected model in (1b) and obtain the rolling forecasts for the 4 months of data for 2023. Visualize the predictions versus the observed data and derive the MAPE and PM for each time series. What can you say about the accuracy of the predictions over the two year period? Note: If your model uses the differenced data, you will have to get the predictions for the actual time series data from your forecast outcome. Part II: Multivariate Modeling  2a. Fit a VAR(p) model for p up to an order equal to 8 using the training data. Use the BIC as the order selection criterion. Display the model summary of the selected VAR model. What is the selected order? Is the fitted model stable? Comment on the model results in terms of the significance (at 95% of confidence) of the estimated coefficients and interpret it.  2b. Using the model in Question 2a, evaluate the model’s assumptions to ensure its validity, i.e. serial correlation, heteroscedasticity and normality of the residuals. 2c. For the equation that corresponds to Interest Rate, comment on any potential relationships with the other variables. Use a significance level of

Which of the following is a limitation of ARCH models (select all correct responses):  

When the p-value of the Ljung-Box test performed on the squared residuals is lower than 0.05, we reject the null hypothesis of uncorrelated variance with a 95% confidence level. 

A hypothesis test is used to assess the significance of individual coefficients in a regression model by comparing the estimated parameter to its hypothesized value. Which of the options below is the correct test? 

Assume we have a multivariate time series with n individual time series. If the AR(2) applied to each of the n time series in the multivariate time series is causal, we can infer that:  

The BIC criterion is preferred  in the order selection in both  ARMA-GARCH and VAR models when the objective is prediction since it selects less complex models (lesser overfitting).

In a stationary multivariate time series, the individual variances of each time series and cross-variances between different series must remain constant over time.

The ARCH model assumes positive and negative shocks have the same effect on volatility because it depends on the log of the previous shocks 

In the reading, “What educated citizens should know about statistics and probability”, Utts argues that several statistical concepts are commonly misunderstood by both the public and even researchers themselves. Choose one of the seven topics mentioned in the article that the author identifies as a source of confusion and describe it in your own words. Then, discuss a potential consequence of this misunderstanding (within a specific scientific discipline or society more broadly). How might scientists (and lifelong students of science like you) be able to reduce misunderstandings of the topic you chose? 

In the reading, “Striving for transparent and credible research: practical guidelines for behavioral ecologists”, Ihle and her co-authors make several concrete recommendations that they think will help improve the credibility of behavioral ecology research (and science more broadly). Describe one of these recommendations in your own words, including how and why they think it will improve the credibility of science. (Optional: you can also add your opinion here about whether or not this recommendation will be effective and why).