Question 1: Application of regression analysis to asset pricing models
You will be provided with an individual dataset, entitled “MANG6482_Question_1_XXXXXX.dta” (where XXXXXX is your university username), which contains data on the time indicator (date), returns on an individual asset (ri), a risk-free asset (rf), the market portfolio (rm), as well as the four Fama and French pricing factors (smb, hml, rmw, and cmw). The description of these factors is available in Fama and French (1993) and in Fama and French (2015). Where relevant, use the significance level of 5%. Required:
a) Depict the individual asset's risk premium against the market risk premium on a scatter diagram. Comment on the diagram.
b) Estimate the CAPM by means of the Ordinary Least Squares estimation procedure. Discuss the explanatory power and the goodness of fit of the model. Interpret the coefficients.
c) The constant term of the CAPM can be thought of as an estimate of abnormal returns (“ALPHA”). Is ALPHA positive or negative? Perform appropriate tests, which would allow you to conclude that ALPHA is either positive or negative, or insignificant. Carefully outline the null and alternative hypotheses for each test.
d) Is the systematic risk (“BETA”) of investment in the individual asset higher or lower than the market systematic risk? Perform appropriate tests, which would allow you to conclude that BETA is either higher or lower (or is not significantly different) than the market systematic risk. Carefully outline the null and alternative hypotheses for each test. Explain how the test statistic is constructed and how it is distributed.
e) Estimate the Fama and French three-factor model. Perform a test, which would allow you to conclude if this model outperforms the CAPM. Discuss the explanatory power and the goodness of fit of the model.
f) In the Fama and French three-factor model, perform appropriate tests, which would allow you to conclude if the model satisfies the CLRM assumptions. Interpret the outcome of those tests.
g) Estimate the Fama and French five-factor model. Perform a test, which would allow you to conclude if this model outperforms the Fama and French three-factor model. Discuss the explanatory power and the goodness of fit of the model.
Question 2: ARMA Model Selection
a) Illustrate the time variation on the variable y. Analyse the time series plot.
b) Proceed to test of there is a unit root in the variable y. Carefully outline the test equation, as well as the null and alternative hypotheses for this test.
c) Now estimate the autocorrelation and partial autocorrelation functions for y. Discuss the estimation output.
d) Estimate the AR(p) model for p=1,2,3,4; and the ARMA(p,q) model for p,q=1,2. Summarise the results in a table with one column for each model. For each model, provide the Akaike and Schwarz Bayesian Information Criteria, as well as the Ljung-Box test for the first 10 autocorrelations. Comment on the results.
e) Based on the estimation output in d), select the optimal AR(p) / ARMA(p,q) model. Carefully justify your choice.
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