ECON2300: INTRODUCTORY ECONOMETRICS Research Project 1: CAPM
ECON2300: INTRODUCTORY ECONOMETRICS
Coordinator: Professor Alicia N. Rambaldi
Research Project 1
Due: 4 pm on 16 April
This project weighs 15% of your final overall mark. Total possible points 100.
Background1
A very famous model in finance is the Capital Asset Pricing Model (CAPM). This is an equilibrium model that assumes investors hold a portfolio that gives maximum expected return for a given level of risk (known as mean variance efficient portfolio). Under the assumptions that all investors hold the same beliefs about expected returns and (co)variances of individual assets, the aggregate of all individual portfolios, the market portfolio is also mean variance efficient. The key relationship is between two expected excess returns:
E(excess returns on risky portfolioj) = βjE(excess returns on market portfolio) (1)
where the term excess returns refers to returns in excess of a risk free rate and E() denotes this is
the ”expected value.”
“j” denotes portfolio j, and
βj is known as the ’beta’ of the portfolio.
The relationship indicates it is expected that the return of any risky portfolio (asset), in excess of the riskless rate, is proportional to its ‘beta’, which provides an indication of how sensitive the value of portfolio j is to general market movements.
The data file Hw1capm.xlsx contains 610 observations on a representative sample of portfolios for three industries (food, consumer durables, and construction). The dataset has additional variables.
• cnstrrf: excess return on valued weighted construction industry portfolio
• durblrf: excess return on valued weighted of consumer durables industry portfolio
• foodrf: excess return on valued weighted food industry portfolio
• Jan: =1 if portfolio excess returns are for the month of January
• rmrf: excess return on valued weighted market portfolio.
• hml: return on a value-stock portfolio minus the return on a growth-stock portfolio.
• smb return on a small-stock portfolio minus the return on a large-stock portfolio.
• umd: return on a high prior return portfolio.
Submission of your report
Your report must be single-spaced and in 12 Font size. You should give your answer to each of the following questions following a similar format of the solutions to the tutorial problem sets. When you are required to use R, you must show your R command and R outputs (screenshots or figures generated from R). You will lose 2 points whenever you fail to provide R commands and outputs. For each question, when you are asked to discuss or interpret, your answer has to be brief and compact. You will lose 2 points if your answer is needlessly wordy. You must upload your assignment via the course’s Blackboard provided link, in PDF format. (Do not submit a hard copy.)
1The data used in this project comes from Verbeek (2008). A Guide to Modern Econometrics. Third Edition.
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Research tasks
1. (20 points) Load data, libraries and conduct preliminary data analysis.
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(a) (4 points) You are given a dataset in Excel format. Load this dataset in R using a relevant package. Provide your R-commands to load the data. In particular, be clear about which R-package you install and use. (Hint: use the Internet. There are several different ways of doing it.)
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(b) (2.5 points) For this report you will need to compute summary statistics, plot data, estimate linear regression models, and perform hypothesis tests. List the packages that you have called using the “library” commands (Hint: Go over relevant tutorial solutions).
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(c) (13.5 points) Obtain summary statistics and histograms for the variables foodrf and rmrf and scatter diagram of those two variables. For the diagrams, give informative titles and variable names instead of just using the default titles and variable names. For example, you could use Food Industry Excess Return in place of foodrf. Discuss the data character- istics.
2. (13 points) CAPM without intercept
The conceptual model in equation (1) suggests a regression representation of the relationship
between the excess returns of the industry and market portfolios that might take the form: Industry’s j excess returnsi = βMarket portfolio excess returnsi + ui (2)
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(a) (4 points) Estimate the linear regression
foodrfi = βrmrfi + ui (3)where ui is the error and β is the unknown population coefficient. Assuming that E[u|rmrf] = 0, what is the estimated ‘beta’ of this portfolio? Provide an interpretation of the estimated ‘beta’ in a regression context.
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(b) (2.5 points) Produce a scatter plot with a plot of the fitted regression line from model (3) on the same graph.
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(c) (4 points) Test the hypothesis that β = 1 for the food industry portfolio. State the alternative hypothesis clearly.
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(d) (2.5 points) Given your conclusion in (c), interpret the estimated ‘beta’ for the food in- dustry portfolio.
3. (13 points) CAPM regression with intercept
The CAPM is often estimated with an intercept to study deviations from the hypothesis of efficient markets. The intercept of the CAPM model is known as “Jensen’s α”. If it is positive it might indicate abnormal returns of a portfolio over the theoretically expected return.
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(a) (4 points). Estimate the linear regression
foodrfi = β0 + β1rmrfi + ui (4)where ui is the error, β0 and β1 are the unknown population coefficients.
Assuming that E[u|rmrf] = 0, and using the estimates, what is the predicted change in foodrf for every additional one per cent of excess returns in the market portfolio? -
(b) (4 points) Test whether the intercept coefficient of the model is zero at the 1% significance level. Pay attention to the alternative hypothesis when performing the test.
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(c) (2.5 points) Predict the excess food industry portfolio return when the excess market return is 2 per cent.
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(d) (2.5 points) Interpret the R2.
4. (20 points) CAPM with additional regressors
(a) (4 points) It is often the case that empirical work considers the existence of the ’January effect’ in the CAPM. There has been some reported evidence that returns are higher in January than in any other month. To test this hypothesis we add a January dummy to the model. Estimate the regression:
foodrfi = β0 + β1rmrfi + β2Jan + ui (5) Discuss whether the results support the hypothesis for the food industry portfolio by formally
testing that the January effect is statistically significant at the 5% significance level.
(c) (2.5 points) Recent literature suggests that asset prices are well described by the so-called factor model, where excess returns are linearly explained by excess returns from a number of “factor portfolios” (in addition to the market portfolio). Discuss under what condition(s) the estimated coefficients in the previous questions would be biased due to the omission of the ‘factor portfolios’ .
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(d) (4 points) Estimate the linear regression model including the “factor portfolios”
foodrfi = β0 + β1rmrfi + β3smb + β4hml + β5umd + ui (6)
where ui is the error and β0, β1, ..., β5 are the unknown population coefficients. Compare the estimation results with those obtained from the one-factor (CAPM) in Question 3. Pay attention to the estimated partial slope coefficients and the R2.
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(e) (3 points) Which is the parameter of interest in the regressions you have estimated so far? Which variables can be considered ”controls” in model (6)?
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(f) (4 points) Perform an F-test for the hypothesis that the coefficients of the three additional factors included in model (6) are jointly equal to zero.
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(g) (2.5 points) Test the hypothesis that the “Jensen’s α” of model (6) is zero. Carefully consider what is the alternative hypothesis and reflect this when performing the test.
5. (20 points) Summarising the Results on the Food Industry Portfolio
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(a) (14 points) Present the results of the regressions of Excess Returns of the Food Industry on the Excess Returns of the Market Portolio and other variables used in the previous questions (i.e. regressions (3)-(6)) in a table format. Please use as a guide Textbook Table 7.1 (see slide 30 in Lecture 5), or Table 1 in the Solutions to Tutorial 5.
Briefly discuss the overall results. (Hint: Discuss the effect of adding control variables)
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(b) (6 points) Compute and present the 95% confidence intervals for the food industry ‘beta’ that are obtained from alternative models (3)-(6). Briefly conclude on the evidence for proportionality of the industry and market portfolios.
6. (14 points) An Additional Industry
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(a) (7 points) Choose one of the additional industry portfolios provided with the dataset (either the construction or consumer durable industries) and estimate the equivalent model to that in equation (6). Present the fitted regression in standard form (i.e.)
Yb = βˆ0+ βˆ1X1+ βˆ2X2 + ...+ βˆkXk (SE(βˆ )) (SE(βˆ )) (SE(βˆ )) (SE(βˆ ))
R2 =???
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(b) (7 points) Compare the ‘beta’ estimate for this portfolio to that obtained for the food industry and provide a brief conclusion in relation to the proportionality between the returns of the food industry and those of the market portfolio.
012k
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