ETF3300 Quantitative Methods For Financial Markets

Individual Assignment 1

Due by Monday, 4pm, 12 September 2016 (Week 8)

When performing a hypothesis test, use α = 0.05.

This assignment must be submitted to your tutor at your tutorial, or to your tutor’s mailbox

at Level 5, Building H, Caulfield Campus.

This is an individual assignment.

Make sure the assignment cover sheet is on the top of your assignment.

This project will be marked out of 50 and this mark will be converted to a mark out of 10

for the purpose of establishing a final mark for you in this unit.

Your assignment can be hand written and/or typed.

Please note that this assignment is due by 4pm Monday 12 September 2016 instead of

Thursday 8 September 2016 as stated in the unit guide. You can submit your assignment

early if you wish.

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Question 1 [1+1+1+1+1+1+4+4+4+4+2+2 = 26 marks]

Go to the Yahoo Finance website. Pick your favourite U.S. stock and download its daily

data from 3 January 2012 to 31 December 2015. Perform your analysis on the daily adjusted

closing price. Please note that you cannot pick Apple as your favourite stock because we

have already considered this stock in our tutorials.

(a) Plot the price data of your stock.

(b) Compute daily simple returns of your stock. Express the simple returns as percentages

and plot the simple returns.

(c) Compute daily log returns of your stock. Express the log returns as percentages and

plot the log returns.

(d) Based on the time plots in parts (a) to (c), do any of them look like a stationary process?

Explain.

(e) Plot the simple returns and log returns on the same graph and compare the two returns.

(f) Download daily data for the S&P 500 index (ticker symbol ^GSPC) for the same time

period (ie. from 3 January 2012 to 31 December 2015). Compute daily log returns of

this index. Express the log returns as percentages and plot the log returns.

For the remaining parts, use the log returns in percentage terms.

(g) For the daily log returns of your stock and the daily log returns of the S&P index, report

the sample mean, sample standard deviation, skewness and kurtosis. Briefly comment.

(h) For your stock and the S&P index, test the hypotheses H 0 : log return is normally

distributed against H A : log return is not normally distributed. What do you conclude?

(i) The basic form of the efficient markets hypothesis is that all available information is

factored into the current price. An implication of the efficient markets hypothesis is

that the current price provides no information on the direction of the future price. This

suggests that log returns should be serially uncorrelated. Does the log return of your

stock violate this efficient markets hypothesis? Why?

(j) Do you think that the log return of your stock follows an MA(1) model? Explain why

or why not.

(k) Regardless of your answer in part (j), estimate an MA(1) model for the log return of

your stock. Write down the fitted model.

(l) Is the MA coefficient significant at the 5% level? Explain.

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Question 2 [3+3+4 = 10 marks]

Simulate T = 1500 observations from a zero mean AR(1) model

Y t = φ 1 Y t−1 + e t

with φ 1 = −0.85 (call it ARn085) and e t ∼ i.i.d.N(0,1). Assume that Y 1 = 0.

(a) Plot ARn085 and obtain its correlogram. Briefly comment on the correlogram.

(b) Using the simulated data, estimate the following models, display their EViews outputs

and write down ONLY the fitted AR(1) model:

(i) Y t = φ 1 Y t−1 + e t

(ii) Y t = φ 1 Y t−1 + φ 2 Y t−2 + e t

(iii) Y t = e t + θ 1 e t−1

(c) Based on your answers to part (b), which is the preferred model according to the SC

criterion? Are you surprised by the model selected by SC? Why or why not?

Question 3 [5+5 = 10 marks]

(a) Consider a stationary AR(2) model

Y t = φ 0 + φ 1 Y t−1 + φ 2 Y t−2 + e t

where e t ∼ i.i.d.(0,σ 2 ). Derive, step by step, the mean of Y t .

(b) Consider an MA(2) model

Y t = θ 0 + e t + θ 1 e t−1 + θ 2 e t−2

where e t ∼ i.i.d.(0,σ 2 ). Suppose θ 1 = 0 and so the "restricted" MA(2) model becomes

Y t = θ 0 + e t + θ 2 e t−2 .

Derive, step by step, the variance of Y t for this "restricted" MA(2) model.

Question 4 [4 marks]

The EViews workfile sim.wf1 contains a simulated series called y. Using the correlogram,

identify whether the series y is an AR(p) or an MA(q) model. If it is an AR(p) model, what

is the value for p and write down the fitted AR(p) model? If it is an MA(q) model, what is

the value for q and write down the fitted MA(q) model? State your reasonings.

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