Extra Practice Questions (S1-2025)
Question 1
A researcher is interested in modelling the Nikkei index daily log returns over the period 5 January 2000 to 7 May 2020. She uses a constant mean equation and initially considers an ARCH(5) specification for the variance equation.
1. Which stylised feature(s) of the Nikkei log returns does the researcher attempt to address by using an ARCH model? Give a brief explanation of the feature(s).
2. The estimation results from this ARCH(5) model are shown in Figure 1.1. Write the theoretical model that is implied by the regression output in Figure 1.1.
3. Next, the researcher considers a GARCH specification and re-estimates her model. The results are shown in Figure 1.2. Using Figures 1.1 and 1.2 evaluate which out of the GARCH and ARCH models you would opt for. Explain your reasoning and the statistics that you used.
4. Next, the researcher considers a GJR-GARCH specification and re-estimates her model. The results are shown in Figure 1.3.
a. What is her motivation for considering this new model for the variance equation? Write the theoretical model specification that is implied by regression output in Figure 1.3 and interpret the coefficient estimate of the threshold parameter. Is the sign of the estimate reasonable? Briefly explain.
b. Using the information in Figures 1.2 and 1.3, evaluate which out of the GARCH and GJR-GARCH models you would opt for. Explain your reasoning and the statistics that you used.
Figure 1.1
Figure 1.2
Figure 1.3
Question 2
1. Assume that y t is stationary and follows the model
y t = 0.5 + εt
εt = σtut; ut ~ i.i.d.N (0; 1)
σt(2) = 1.0 + 0.1εt(2)-1 + 0.8σt(2)-1
True or false? The unconditional variance of y t is 0.52. Explain your answer by providing the required derivations.
2. Let Ft-1 denote the information set available at time t- 1. Suppose rt is stationary and follows an MA(1) process:
rt = 0.1 + ut + 0.5ut-1
where ut ~ i.i.d. (0; 4) with V (ut jFt-1) = V (ut) . Provide numerical answers to the following questions where applicable.
(a) What is the conditional mean of rt?
(b) Using your answer above, what is the unconditional mean of rt?
(c) What is V (rt jFt-1)?
(d) Using your answers above, what is V (rt)?
(e) Compare your answers to parts (c) and (d). Are they as you would expect? Briefly explain.
(f) What is cov (rt; rt-1)?
Question 3
This question uses monthly returns on the Fama-French manufacturing portfolio, denoted by rt.
Consider the following Fama-French three factor model
rt - r ft = α + β 1(r mt - r ft) + β 2SMBt + β 3HMLt + vt ; (1)
where (r mt - r ft) is the market factor, rft is the risk free interest rate, SMB is the size factor, and HML is the value factor. When estimating (1) over the period January 1990 to January 2020, we obtain the results shown in Figure 3.1.
a. Provide the fitted regression equation. Neatly report the coefficient estimate of the size factor and test for its statistical significance at 1% significance level. Based on your conclusion interpret this coefficient estimate.
b. How would you test for joint significance of the size and value factors? Provide all required steps. If you found that the size and value factors were jointly not statistically significant what conclusion would you draw in terms of (1)?
Figure 3.1