# Homework Assignment #4 | Good Grade Guarantee!

Summer 2019

MFE 6220 (Part 2)

Homework Assignment #4

Due at 11:59pm on Sunday, July 28th

Attention: Please write the answers on a separate sheet of paper. Then scan and upload to the Blackboard course website as a PDF document.

Question 1

A researcher wants to estimate a model with the dependent variable, Y, and three explanatory variables X2, X3, and X4. She suspects that time-varying heteroscedasticity will be an issue with her data, and decides to estimate a higher-order GARCH model. In particular, she estimates a GARCH(2,1) model.

a. Write a system of equations for the GARCH(2,1) model. Recall that the relevant system of equations will have three different equations. Clearly identify the conditional mean equation and the conditional variance equation.

b. Using formulas, show that the expected value of the error variable, , is zero and the variance of the error variable is time-dependent.

c. What is the stationarity condition for this GARCH(2,1) model? Remember to write the null hypothesis and the alternative hypothesis you would use to test the stationarity condition. Do we need to reject the null or not in order to achieve stationarity in the GARCH model?

Question 2

The researcherâ€™s colleague wants to estimate a different GARCH model using the same data variables. He believes that a higher-order GARCH model is not sufficient and that a GARCH-in-Mean or a TGARCH model would be a better choice.

a. The underlying reason why we would estimate a GARCH-in-Mean model is if a risk-return tradeoff exists in the data. Briefly explain in words what the risk-return tradeoff means.

b. Write a system of equations for the GARCH(1,2)-in-Mean model.

c. What is the stationarity condition for this GARCH(1,2)-in-Mean model? Remember to write the null hypothesis and the alternative hypothesis you would use to test this stationarity condition.

d. How can we formally test for the risk-return tradeoff? Using the relevant parameters from the model, write the null hypothesis and the alternative hypothesis needed to test the risk-return tradeoff. Do we need to reject the null or not in order to identify a significant risk-return tradeoff in the data?

e. The underlying reason why we would estimate a TGARCH model is if a leverage effect exists in the data. Briefly explain in words what the leverage effect means.

f. Write a system of equations for the TGARCH(1,1) model.

g. What is the stationarity condition for this TGARCH(1,1) model? Remember to write the null hypothesis and the alternative hypothesis you would use to test this stationarity condition.

h. How can we formally test for the leverage effect? Using the relevant parameters from the model, write the null hypothesis and the alternative hypothesis needed to test the leverage effect. Do we need to reject the null or not in order to identify a significant leverage effect in the data?

i. Out of the three models, which one is the best-fit model? Briefly explain in words, how you could determine the best-fit model.

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