ETC3460 final S 1 2020 PDF

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Semester One 2020 Exam - Alternative Assessment Task

UNIT CODE:

ETC3460-BEX3460-ETC5346

UNIT TITLE:

Financial Econometrics

ASSESSMENT DURATION:

3 hours 30 minutes (includes reading, downloading, and uploading time)

 This is an individual assessment task.  All responses must be included in ONE separate document. Clearly state your ID number, surname and given name at the top of this document.  Clearly state the question numbers corresponding to your answers.  Students are required to answer ALL questions.  This assessment accounts for 60% of the total in the unit and has a hurdle requirement of 40% to pass the unit.  Upon completion of this assessment task, please upload this document to Moodle using the assignment submission link.  You will sign the Student Statement shown on the next page automatically when using the assignment submission link.  Your submission must occur within 3 hours 30 minutes of the official commencement of this assessment task (Australian Eastern Daylight Time).

Please read the next page carefully (Student Statement) before commencing the assessment task.

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Date: Privacy Statement The information on this form is collected for the primary purpose of assessing your assessment and ensuring the academic integrity requirements of the University are met. Other purposes of collection include recording your plagiarism and collusion declaration, attending to the course and administrative matters and statistical analyses. If you choose not to complete all the questions on this form it may not be possible for Monash University to assess your assessment task. You have a right to access personal information that Monash University holds about you, subject to any exceptions in relevant legislation. If you wish to seek access to your personal information or inquire about the handling of your personal information, please contact the University Privacy Officer: [email protected] Page 2 of 3

MA MARKS RKS ALLO ALLOCATED CATED TO Q QUE UE UESTIO STIO STIONS NS WI WITHIN THIN THIS A ASSESS SSESS SSESSMEN MEN MENTT TA TASK SK Que Question stion 1 2 3 4 5 6 7 8 9 All Allocat ocat ocated ed M Marks arks 15 15 15 15 Office Use O Only nly Mar Markk re receive ceive ceived d Sec Second ond m markin arkin arkingg

TOTA TOTALL 60

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INSTRUCTIONS TO STUDENTS  When testing a hypothesis, to obtain full marks you need to specify the null and the alternative hypotheses, the test statistic and its distribution under the null, and then perform the test and state your conclusion.  If a question does not specify the level of signi…cance of a hypothesis test explicitly, use 5%.  Statistical tables are provided after Question 4. Question 1 (15 marks) This question uses monthly returns on the Fama-French manufacturing portfolio, denoted by t. Consider the following Fama-French three factor model + 1 ( mt  ft ) + 2 SMBt + 3 HMLt + t (1) t  ft = where ( mt  ft) is the market factor, ft is the risk free interest rate, SMB is the size factor, and HML is the value factor. r

r

r

r





r

r

r





v ;

r

1. The expression on the left hand side of (1), ( t  ft ), is referred to as excess return on the manufacturing portfolio at time . a. Provide a de…nition of the term ‘excess return’ on a …nancial asset. (1pt) b. Explain how this quantity relates to parameter in (1). (1pt) r

r

t



2. When estimating (1) over the period January 1990 to January 2020, we obtain the results shown in Figure 1. a. Provide the …tted regression equation. Neatly report the coe¢cient estimate of the size factor and test for its statistical signi…cance at 1% signi…cance level. Based on your conclusion interpret this coe¢cient estimate. (3pts) b. How would you test for joint signi…cance of the size and value factors? Provide all required steps. If you found that the size and value factors were jointly not statistically signi…cant what conclusion would you draw in terms of (1)? (3pts) 3. Use the residuals obtained from regression (1), ^t. a. Explain how you would check whether the E¢cient Market Hypothesis (EMH) holds and why. Provide the required steps. (2pts) b. Assume that the 2 of the relevant auxiliary regression amounts to 0.56. What conclusion would you draw when comparing your results with the appropriate distribution with 5 degrees of freedom? (1pt) v

R

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4. Given the results in Q1.3 we re-run (1) and results are shown in Figure 2.

a. Explain what change was made to the regression speci…cations and why. (1pt) b. For what other reason would you make the same change? Explain how you would go about testing this property using the residuals in (1). [Hint: we have discussed 3 such tests. Either one of these is acceptable]. (2pts) c. Do the results in Figure 2 change your conclusions in Q1.2.a on the coe¢cient estimate of the size factor? (1pt)

Figure 1

Figure 2

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Question 2

(15 marks)

Assume a time series

fytgtT=1.

Its corresponding correlogram is displayed in Figure 3.

1. Provide a de…nition for stationarity of a time series. Based on the information in Figure 3 do you consider yt to be a stationary or non-stationary time series? Brie‡y explain why. (3pts) 2. Based on the information in Figure 3, a researcher decides to use the following model to …t yt :

yt = 0 + 1 yt1 + 2 yt2 + "t ; where "t

3. Let

 W N (0; "2).

Justify his choice. (2pts)

Ft1 denote all information available at time t  1. Derive the conditional mean of yt Ft1 and derive the unconditional mean of yt. What is the condition for this process to

given

have a constant unconditional mean? (3pts)

4. Derive the expression for the 3-step ahead point forecast based on this model. (4pts)

6

5. Derive the unconditional autocovariance function,  k = Cov (yt ; ytk ), for k = 0 and k = 0. Here we assume that the unconditional mean of yt derived in Q2.3 is equal to zero. (3pts)

Figure 3

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Question 3

(15 marks)

A researcher is interested in modelling the Nikkei index daily log returns over the period 5 January 2000 to 8 May 2020. He uses a constant mean equation and initially considers an ARCH(5) speci…cation 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). (1pt) 2. The estimation results from this ARCH(5) model are shown in Figure 4. Write the theoretical model that is implied by the regression output in Figure 4. Explain what conditions are needed on the slope parameters of this model and why. Given the information in Figure 4, are these conditions satis…ed in this case? (3pts) 3. Next, the researcher considers a GARCH speci…cation and re-estimates his model. The results are shown in Figure 5. a. What is his motivation for considering this new model for the variance equation? Derive the relationship between this new model and an ARCH model. [Hint: start with the GARCH equation and substitute recursively the lagged conditional variance component] (4pts) b. Using the information in Figures 4 and 5 evaluate which out of the GARCH and ARCH models you would opt for. Explain your reasoning and metrics that you used. (2pts) 4. Next, the researcher considers a GJR-GARCH speci…cation and re-estimates his model. The results are shown in Figure 6. a. What is his motivation for considering this new model for the variance equation? Write the theoretical model speci…cation that is implied by the regression output in Figure 6 and interpret the coe¢cient estimate of the threshold parameter. Is its sign reasonable? Brie‡y explain. (2pts) b. Using the information in Figures 5 and 6, evaluate which out of the GARCH and GJRGARCH models you would opt for. Explain your reasoning and metrics that you used. Provide all steps in your testing procedure to obtain full credit. (3pts)

Figure 4

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Figure 5

Figure 6

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Question 4

(15 marks)

Let Pt be a random walk process: where "t

 WN

 

Pt = Pt1 + "t ;

0;  2 .

1. Show that for an initial condition P0 = 1, we have that:

X t

Pt = P0 +

i

"i ;

=1

and compute E (Pt ). (2pts)

2. Show that  k = Cov (Pt ; Ptk ) = (t

 k) 2 . Brie‡y comment on this result. (3pts)

 

3. Assume the corresponding log-return is distributed as follows:

rt

N

r ; r2 :

a. Show that the simple net return has a log-normal distribution with: 2

E (Rt ) = er +0:5r (4pts)



 1:



b. Show that the simple net return has a log-normal distribution with: 2

2

V (Rt ) = e2r +r er

1

:

(6pts)

 

[Hint: Recall that rt = ln (1 + Rt ) and start with deriving ln E (1 + Rt ).]

[Hint: Use the fact that ln E (a) = E [ln (a)] + 0:5V [ln (a)].] [Hint: Recall that V (Rt ) = E R2t

—————

E (Rt )2 .]

END OF EXAMINATION

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STATISTICAL TABLES

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