SMM248 Statistics for Finance 2020
Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit
MSc in Finance
MSc in Corporate Finance
SMM248
Statistics for Finance
2020
QUESTION 1 [35 marks]
a) A researcher has formulated the simple linear regression model:
yi = F1 + F2xi2 + F3xi3 + ei , i = 1, … , N
to explain the cross-sectional variation in the abnormal performance of N=500 funds (yi is the Jensen’s alpha), using as drivers two characteristics of the fund manager: xi2 is the years of experience of the fund manager, xi3 is a dummy variable equal to 1 if the fund manager has a post-graduate degree from an top international university, 0 otherwise. The researcher deploys a heteroskedasiticy test and finds that the errors are not homoskedastic. Discuss:
i) What is heteroscedasticity?
[3 marks]
ii) Discuss a test that the researcher may have used to detect this problem. Indicate clearly the null and alternative hypotheses and the probability distribution of the test statistic.
[3 marks]
iii) In the presence of this heteroscedasticity problem, should she estimate the model parameters and corresponding standard errors using OLS? or should (s)he instead consider another estimation method? Justify your answers.
[4 marks]
iv) Suppose that instead the researcher prefers to re-specify the model in order to eliminate the heteroscedasticity problem. Discuss potential model re-specifications that may result in homoscedastic errors.
[4 marks]
b) In the context of the simple regression model yt = F1 + F2xt + et , t = 1, … , T explain the Ordinary Least Squares (OLS) estimation method and the Maximum Likelihood (ML) estimation method as regards: How are the corresponding OLS and ML estimators derived? In which circumstances is it appropriate to employ OLS instead of ML, and vice versa? (briefly explain your answers).
[4 marks]
c) Explain the main differences between an ordinary linear regression model and a logit regression model. Hint: Focusing on a single-regressor case, discuss for each model the interpretation of the fitted values, the estimation method, and the marginal effect.
[9 marks]
d) Explain the concept of AutoCorrelation Function (ACF) in time-series regression analysis? Draw an example of ACF in a graph to explain what information it provides (label carefully the X axis and the Y axis).
[4 marks]
e) Consider the following time-series regression based on daily data yt = F1 + F2xt + et , t = 1, … , T
where yt are the returns of stock PIVECO plc and xt are the returns of the market portfolio. The researcher wants to test for the presence of autocorrelation up to 2 weeks. Discuss two different tests for this purpose. Outline in each case the null and alternative hypothesis of the test, the test statistic, and the corresponding probability distribution.
[4 marks]
QUESTION 2 [30 marks]
A researcher is interested in explaining the variation in abnormal performance across funds. For this purpose (s)he estimates the following regression model
ai = F1 + F2 Ln(FundSizei ) + F3 Ln(FundAgei ) + F4 Ln(Agei ) + F5 Malei + ui
using data on i = 1, … , N funds (N = 400) where ai is the fund performance measure which is
defined as the annualized CAPM one-factor alpha, FundS izei is total net assets under
management in millions of dollars, FundAgei is the number of years since the fund inception
date, Agei is the age of the fund manager, and Malei is a gender dummy equal to 1 if the fund
manager is a male, 0 otherwise. The researcher estimates a multiple linear regression model
using OLS and obtains the estimation output shown in EXHIBIT 1.
EXHIBIT 1
Dependent Variable: FUND PERFORMANCE
Method: Least Squares; Sample: 1 400 observations
Variable |
Coefficient |
Std. Error |
-Statistic Prob. |
C LN(FUNDSIZE) LN(FUNDAGE) LN(AGE) MALE |
0.275104 -0.032992 -0.045317 -0.169905 0.104161 |
0.031198 0.002924 0.027201 0.044995 0.055652 |
|
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic) |
0.921937 11.31757 13.52493 0.000000 |
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion |
0.051137 |
a) Do funds managed by younger managers tend to perform better? Conduct a (one-sided) test at the conservative 1% significance level to answer this question. Begin by formalizing clearly the null and alternative hypothesis of the test, indicate the name of the test statistic, and its probability distribution. Draw a graph to discuss the outcome of the test, and visualize the following elements in the graph: test statistic value, critical value and p-value.
(10 marks)
b) Which percentage of the variation in the abnormal performance across funds is this model able to explain? Indicate the name of the statistic that you are adopting to measure this explained variation.
(10 marks)
c) Conduct a statistical test to evaluate whether the regression is significant overall. Indicate the name of the test statistic that you are using for this purpose and the probability distribution that it follows. Write down clearly the null and alternative hypotheses of the test in two different ways: i) with reference to the model’s coefficients, and ii) with reference to the percentage of the variation in the abnormal performance across funds that the model is able to explain. Explain verbally each of those hypotheses. Draw a graph to explain the test outcome; label clearly in the graph the sample value of the test statistic, the 5% critical value of the test and thep-value. What shall the researcher we conclude?
(10 marks)
QUESTION 3 [35 marks]
Consider the following linear regression model to investigate the determinants of CEOs compensations across firms within a specific industry:
CE0Salary = F1 + F2 ln(stock) + F3profits + F4male + u
Where CE0Salary measured in thousands of Euros, stock is the stock market value of the company in millions of Euros, profits is the amount of profits in millions ofEuros realized by each company, and male is a gender dummy equal to 1 if the CEO is a male, and 0 otherwise. Using a random sample of N=25 CEOs, we estimate equation (1) by OLS and obtain the following results:
EXHIBIT 2
Dependent Variable: CEOsalary
Method: Least Squares; Sample: 1 25 observations
Variable |
Coefficient |
Std. Error t-Statistic Prob. |
C |
1.138569 |
0.031198 |
Ln(STOCK) |
0.145317 |
0.027201 |
PROFITS |
0.089905 |
0.044995 |
MALE |
0.094161 |
0.055652 |
R-squared Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F- |
0.178947
0.103892 11.31757 |
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion |
a) Interpret the estimate of the intercept F0 . In which units is this measure expressed?
[4 marks]
b) According to the model estimates, what is the differential salary between male CEOs and female CEOs on average? Can you argue that male CEOs tend to earn more than female CEOs any other thing equal? Conduct a test to answer this question, indicate clearly the name of the test statistic, the null and alternative hypotheses, and the probability distribution of the test statistic. Discuss the outcome of the test at the 1%, 5% and 10% significance levels.
[5 marks]
c) Ifthe market value ofthe firm’s stock increases by 10 million Euros, how does a CEO salary changes on average ceteris paribus? [5 marks]
d) Using the information from the table compute the adjusted-R2 and interpret the meaning of this measure in relation to the R2 . Compute the AIC and discuss what can this measure be used for.
[6 marks]
e) Suppose that years of experience, denoted exper, is a significant driver of CEO compensation and that adding this variable to the existing model results in an OLS estimate for the coefficient of exper equal to 0.21386 with standard error 0.10945. The estimated correlation between this variable and the first two dependent variables in the model (with significance p-value in square brackets) is corr(ln(stock), exper)=0.23[0. 124), corr(profits, exper)=0.46[0.002], and the variance of each of those regressors is var(ln(stock))=0.8945, var(profits)=0.2394, and var(exper)=0. 1345. Quantify the expected effect (if any) of omitting this variable in the coefficient estimates for the ln(stock) and profits variables.
[10 marks]
f) Re-formulate the regression model in order to accommodate two effects: (i) the effect of Exper on CEOsalary, and (ii) a differential marginal effect of on CEOsalary for males and females. Write down the new model, and the null/alternative hypotheses associated to the two statistical tests that you would conduct to assess (i) and (ii), respectively, in the context of this model. Accordingly, what is the expected marginal effect of experience on CEO salary for malels? and for females?
[5 marks]
2022-08-20