2021 Fall 2 MSF 1F
Linear Econometrics for Finance: Homework 2
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Each question carries the same weight of 4 points.
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1) Answer the following questions by conducting appropriate analyses to “data-stocks-FF-monthly.dta” saved during Class 5. Be sure to use monthly observations only up to July 2021.
(a) For this question, consider the following three stocks only: HP, Verizon and Wells Fargo.
Which is the best model among CAPM, Fama-French 3 factor model, and Fama-French 5 factor model? While it is difficult to determine a definite answer to this question, we can check which model is more appropriate for an individual security on the statistical basis.
For example, we can conduct a joint hypothesis test to decide whether Fama-French 5 factor model reduces to Fama-French 3 factor model. In other words, we decide whether to reject the null hypothesis that the coefficients of RMW5 and CMA5 are simultaneously zero in the following regression model.
Similarly, we can test whether Fama-French 3 factor model reduces to CAPM.
For each of three stocks, by conducting appropriate F-tests, make conclusions about CAPM vs. Fama-French 3 factor model vs. Fama-French 5 factor model.
(b) For this question, consider Pfizer only.
First, run the Fama-French 5-factor model shown below.
Then conduct both a t-test and an F-test for the following null hypothesis:
What are your conclusions based on the tests? What is the relationship between the obtained t-statistic and F-statistic?
(c) For this question, consider JP Morgan and Walmart only.
For each of these two stocks, predict the August 2021 expected excess return on the basis of the coefficients estimated from CAPM, Fama-French 3-factor model, and Fama-French 5-factor model, respectively. As for the values of mrp, smb3, hml3, smb5, hml5, rmw5, and cma5, use the actual figures included in data. Which of the three models predicts the excess return that is closed to the actual figure?
2) Textbook Chapter 5 Problem 5.8 (a) & (c) only.
There were 79 countries who competed in the 1996 Olympics and won at least one medal. For each of these countries, let MEDALS be the total number of medals won, POPM be population in millions, and GDPB be GDP in billions of 1995 dollars. Using these data we estimate the regression model MEDALS = β1 + β2POPM + β3GDPB + e to obtain
a) Given assumptions MR1–MR6 hold, interpret the coefficient estimates for β2 and β3.
c) Using a 1% significance level, test the hypothesis that there is no relationship between the number of medals won and GDP against the alternative that there is a positive relationship. What happens if you change the significance level to 5%?
3) Textbook Chapter 5 Problem 5.22 (b) & (d) only.
Using the data in the file toody5, estimate the model
where Yt = wheat yield in tons per hectare in the Toodyay Shire of Western Australia in year t; TRENDt is a trend variable designed to capture technological change, with observations 0, 0.1, 0.2, …, 4.7; 0 is for the year 1950, 0.1 is for the year 1951, and so on up to 4.7 for the year 1997; RAINt is total rainfall in decimeters (dm) from May to October (the growing season) in year t (1 decimeter = 4 inches).
b) For 1974, when TREND = 2.4 and RAIN = 4.576, use a 5% significance level to test the null hypothesis that extra rainfall will not increase expected yield against the alternative that it will increase expected yield.
d) There is concern that climate change is leading to a decline in rainfall over time. To test this hypothesis, estimate the equation RAIN = α1 + α2TREND + e. Test, at a 5% significance level, the null hypothesis that mean rainfall is not declining over time against the alternative hypothesis that it is declining.
4) Textbook Chapter 6 Problem 6.2 (b) & (d) only.
Consider the following model that relates the percentage of a household’s budget spent on alcohol WALC to total expenditure TOTEXP, age of the household head AGE, and the number of children in the household NK.
Using 1200 observations from a London survey, this equation was estimated with and without the AGE variables included, giving the following results:
b) Use an F-test and a 5% significance level to test whether NK should be included in the first equation.[Hint: F = t2]
d) After estimating the following equation, we find SSE = 46086.
Relative to the original equation with all variables included, for what null hypothesis is this equation the restricted model? Test this null hypothesis at a 5% significance level.
5) Textbook Chapter 6 Problem 6.8 (a) & (b) only.
Consider the wage equation
where the explanatory variables are years of education (EDUC) and years of experience (EXPER). Estimation results for this equation, and for modified versions of it obtained by dropping some of the variables, are displayed in Table 6.10. These results are from 200 observations in the file cps5_small.
a) What restriction on the coefficients of Eqn (A) gives Eqn (B)? Use an F-test to test this restriction. Show how the same result can be obtained using a t-test.
b) What restrictions on the coefficients of Eqn (A) give Eqn (C)? Use an F-test to test these restrictions. What question would you be trying to answer by performing this test?
6) Textbook Chapter 6 Problem 6.20 (a) & (f) only.
In Example 6.18, using 900 observations from the data file br5, we identified three potentially influential observations in the estimation of the model
Those observations were numbers 150, 411 and 540.
a) Estimate the model with (i) all observations, (ii) observation 150 excluded, (iii) observation 411 excluded, (iv) observation 540 excluded, and (v) observations 150, 411, and 540 excluded. Report the results and comment on their sensitivity to the omission of the observations.
f) Using the estimates obtained when observations 150, 411, and 540 are excluded, find the
out-of-sample forecast errors for observations 150, 411, and 540.
7) Textbook Chapter 6 Problem 6.28 (a) & (b) for US only.
Using time-series data on five different countries, Atkinson and Leigh18 investigate the impact of the marginal tax rate paid by high-income earners on the level of inequality. A subset of their data can be found in the file inequality.
a) Using data on Australia, estimate the equation SHARE = β1 + β2TAX + e where SHARE is the percentage income share of the top 1% of incomes, and TAX is the median marginal tax rate (as a percentage) paid on wages by the top 1% of income earners. Interpret your estimate for β2. Would you interpret this as a causal relationship?
b) It is generally recognized that inequalitywas high prior to the great depression, then declined during the depression and WorldWar II, increasing again toward the end of the sample period. To capture this effect, estimate the following model with a quadratic trend
where YEAR is defined as 1 = 1921, 2 = 1922,…, 80 = 2000. Interpret the estimate for α2. Has adding the trend changed the effect of the marginal tax rate? Can the change in this estimate, or lack of it, be explained by the correlations between TAX and YEAR and TAX and YEAR2?
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