Statistical Modelling for Business
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1QBUS2810: Statistical Modelling for Business
Assignment Task #3
Submission Due Date: Sunday, 22nd November, 2020 (Week 12) before 11:59 pm (Sydney
time)
Instructions:
1. You are required to type up your entire assignment, including any equations. Copy and paste relevant
outputs into your text. If you are using Word, you should use the equation editor for any maths notation.
2. You should attach relevant analysis outputs (graphs, tables, etc.) while discussing your answer in the text.
3. Please answer all questions in the given order; i.e., 1a, 1b, etc. You do not need to re-write the assignment
questions again. Keep your answers clear, brief, and concise.
4. There is no requirement for font size and line spacing, but it must be legible and correctly oriented.
5. Please convert and submit your assignment in pdf, which must be uploaded to the Turnitin assignment
box on Canvas.
6. For hypothesis test question, use the p-value approach. Your answer should include the alternatives
(H0 and H1), decision, and conclusion.
7. Data used in this assignment are in the spreadsheet A3Dataset.xlsx.
8. You are encouraged to discuss the assignment with your classmates, tutors, and lecturer. However, you
MUST write up solutions on your own. Students caught cheating will automatically receive a mark of 0
and are subject to disciplinary action.
1. The capital asset pricing model (CAPM) is used in finance to determine a theoretically appropriate
required rate of return of an asset, where that asset is to be added to an already well-diversified
portfolio, given that asset’s non-diversifiable risk. Traditionally, applications of the CAPM use only
one variable to describe the returns of a portfolio or stock with the returns of the market as a whole:
rstock − rf = αstock + βstock(rm − rf ) + ut
In contrast, the Fama-French model uses three variables:
rstock − rf = αstock + βstock(rm − rf ) + β2SMB + β3HML + εt
rstock is the stock’s rate of return, rf is the risk-free return rate, and rm is the return of the
whole stock market. The parameter αstock is the stock’s ”alpha”. It measures how much the stock
outperforms its ”theoretical” predicted returns under the CAPM and βstock is the stock’s ”beta”,
which measures the stock’s exposure to the overall market. Different stocks will have different
parameters.
The Fama-French model contains two additional factors to explain stock returns. Small market cap-
italization Minus Big (SMB) measures the historic excess returns of small cap stocks over big caps.
High book-to-market ratio (BtM) Minus Low book-to-market ratio (HML) measures the historic
excess returns of value stocks (small BtM ratio) over growth stocks (High BtM ratio). These factors
are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Capital-
isation ranking) and available historical market data. Historical values are available on Kenneth
French’s web page for American stocks.
The variables used in this exercise are as follows:
rBHP = Monthly return on BHP stock as observed on the ASX.
rm = Monthly return on market index, here the All Ordinaries Index (AOI).
SMB = Small market capitalization Minus Big market capitalization factor.
HML = High book-to-market ratio Minus Low book-to-market ratio
You are to assume a risk-free rate of rf = 0.005 per month. Your task is to estimate the Fama-French
three factor model using the given data. and determine whether it is any better at explaining the
BHP stock returns compared to the market excess returns given by only the All Ordinaries Index.
2(a) Write down the five-number summaries plus mean, standard deviation, skewness, and kurtosis
coefficients of rBHP .
(b) Plot and comment the rBHP series over time.
(c) Generate two new variables rBHP − rf and rm− rf and estimate the one-factor CAPM model:
rstock − rf = β0 + β1(rm − rf ) + ut
Copy and paste the regression output into your answer sheet. Write down the fitted regression
equation.
(d) Comment on the sign of the estimated coefficient β1 and state whether this is what you expect.
(e) Test whether or not the excess market returns explain the excess returns of BHP shares at the
α = 0.05 level.
(f) Test whether or not the BHP’s ”beta” is greater than one at the α = 0.05 level.
(g) Estimate the Fama-French 3-Factor CAPM model:
rstock − rf = β0 + β1(rm − rf ) + β2SMB + β3HML + εt
Copy and paste the regression output into your answer sheet. Write down the fitted regression
equation.
(h) Set up the general linear hypothesis for testing whether or not the Fama-French 3-Factor CAPM
model explains the stock returns better than the one-factor CAPM model; i.e., determine L,
β, and c for H0: Lβ = c.
(i) Conduct a hypothesis test for part (h).
(j) A Financial Analyst believes that the effect of book-to-market values (HML) on stock returns is
twice as great as the effect of market capitalization (SMB). Formulate an appropriate hypothesis
test and use re-parametrisation to convert it to a simple t-test to test the assertion. Perform
the required regression and state your conclusion at the α = 0.05 level.
(k) Obtain the variance-covariance matrix for the estimators of parameters in a regression model in
part (g). Utilize the regression result in part (g) and the variance-covariance matrix to repeat
the hypothesis test in part (j) by means of a simple t-test.
2. The marketing manager of a company producing a new cereal aimed for children wants to examine
the effect of the shape of the box’s logo on the approval rating of the cereal. He combined 4 colours
and 2 shapes to produce a total of 8 designs. Each logo was presented to 2 different groups (a total
of 16 groups of children) and the approval rating for each was recorded and is shown below.
Color
Shape Red Green Blue Yellow
Circle 52, 44 67, 61 36, 44 45, 41
Square 34, 36 56, 58 36, 31 21, 25
(a) How many factors does this experiment have? Identify the factors and state how many levels
each factor has.
(b) If all combinations are compared, how many different treatments (cells) are there in the exper-
iment? What is the response variable?
(c) Consider the following regression model:
Y = β0 + β1C + β2R + β3G + β4B + β5CR + β6CG + β7CB + ε
where C = 1 if shape = circle; 0 otherwise. R = 1 if color = red; 0 otherwise. G = 1 if color
= green; 0 otherwise. B = 1 if color = blue; 0 otherwise.
Use the regression parameters to recover the cell means µij and fill in the following table:
3Colour