NYU Econometrics Multivariate Linear Regression Worksheet

3. [20 Points] Let R be the expected return on a risky investment and Ry be the return on a risk-free investment. The fundamental idea of modern finance is that an investor needs a financial
incentive to take a risk. Hence, R must exceed Rf. According to the capital asset pricing model
(CAPM) the expected excess return on an asset is proportional to the expected excess return
on a portfolio of all available assets (the “market portfolio”) That is, the CAPM says that
R-R=ẞ (Rm-R) + u where Rm is the expected return on the market portfolio and ẞ is the
coefficient in the population regression of R – Rf on Rm – Rf.
In the following STATA output, variable freturn is the excess returns for two firms in
computer chip industry and mreturn is the excess returns for the market.
Linear regression
Number of obs =
F( 1, 382)=
Prob F
=
R-squared
=
Root MSE
384
104.52
0.0000
0.2175
.13447
freturn |
Coef.
Robust
Std. Err.
t
mreturn | 1.608313
cons | 0031122
1573139
0071605
P>|t|
10.22 0.000
0.43 0.664
[95% Conf. Interval]
1.299004
-.0109666
1.917623
0171911
2
(a) [7 Points] According to CAPM, the true intercept must be zero and the true slope must be
one. Using hypothesis testing at 10% significance level test if CAPM is correct according
to above results.
(b) [6 Points] What is the meaning of F test in this regression? What is it testing? and how is
that statistic related to t test statistic on the same output
(c) [7 Points] In a given year, the rate of return on 3-month Treasury bills is 2.1% and the rate
of return on a large diversified portfolio of stocks (the S&P 500) is 6.2%. For each company
listed below, use the estimated value of ẞ to estimate the stock’s expected rate of return.
Estimated ẞ Expected rate of return
Company
Kellogg (breakfast cereal)
-0.03
Amazon (online retailer)
2.65
Barnes and Noble (book retailer)
1.022. [30 points] Use Table 1 below and GPA4.dta data file to answer the following questions.
Table 1 presents the results of three regressions, one in each column. Estimate (replicate) the
indicated regressions and check the values your regression if they are similar to what is
reported in Table 1 below. For example, to check the values in column (1), estimate the
regression with colGPA as the dependent variable and hsGPA and skipped as the independent
variables, using the “robust” option, and see the estimated coefficients
(a) [5 points] Write the regression in column (1) in “equation form,” with the standard error
below the respective regression coefficient.
(b) [5 points] Explain in words what the coefficient on hsGPA means in regression (1),
holding skipped unchanged.
(c) [5 points] Using regression (1), test the hypothesis that the coefficient on skipped is zero,
against the alternative that it is nonzero, at the 5% significance level. In everyday words
(not statistical terms), what precisely is the hypothesis that you are testing?
(d) [5 points] Test the hypothesis that the coefficient on skipped is zero in regressions (1),
(2), and (3) at 1% using the p value, does your answer change depending on what other
variables are included in the regression?
(e) [5 points] Using regression (3), consider the coefficient on campus. Does the sign and
magnitude make sense? Explain.
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(f) [5 points] Using regression (3), consider the coefficient on bgfriend. Does the sign and
magnitude make sense? Explain In regression (3), is the coefficient on campus statistically
significant at the 1% significance level? Is the coefficient on bgfriend statistically
significant at the 1% significance level?
Table 1
College GPA Results
Dependent variable: colGPA
Regressor
hsgpa
(1)
(2)
(3)
.459
.455
.461
(.094)
(.092)
(.090)
skipped
-.077
-.065
-.071
(.025)
(.026)
(.026)
PC
.128
.137
(.060)
(.059)
bgfriend
.086
☐
(.054)
campus
-.124
(.079)
Intercept
1.579
1.527
1.490
(.325)
(.321)
(.317)
Regression summary statistics
R²
0.223
0.250
0.278
Regression RMSE
.331
.326
.322
n
141
141
1414. [25 Points] At a recent campus event, you observed that at one stand people’s weight was
forecasted, and were surprised by the accuracy (within a range). Thinking about how the
person could have predicted your weight fairly accurately (despite the fact that she did not
know about your “heavy bones”), you think about how this could have been accomplished.
You remember that medical charts for children contain 5%, 25%, 50%, 75% and 95% lines
for a weight/height relationship and decide to conduct an experiment with 200 of your peers.
You collect the data and calculate the following sums:
ΣΕΞ1 Υ = 35,000, ΣΕ ΧΑ
=
5,000, y² = 60,000, 1×2 = 1,250, 1 xiyi = 6,250,
where the height (X) is measured in inches and weight (Y) in pounds. (Small letters refer to
deviations from means as in x₁ = X₁ – X and y₁ = Y₁ – Y.)
(a) [4 Points] Find the OLS estimator values for the slope (B₁) and intercept (Bo).
(b) [4 Points] Calculate TSS, ESS, and SSR values. Using SSR, find the standard errors for
the error term assuming homoskedasticity-only.
(c) [4 Points] Calculate the homoskedasticity-only standard errors for the intercept (Bo) and,
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using the resulting t-statistic, perform a test on the null hypothesis that the intercept is not
significant.
(d) [4 Points] Calculate the homoskedasticity-only standard errors for the slope parameter (B₁)
and, using the resulting t-statistic, perform a test on the null hypothesis that there is no
relationship between height and weight in the population of college students.
(e) [3 Points] What is the alternative hypothesis in the above test, and what level of
significance did you choose?
(f) [3 Points] Statistics and econometrics textbooks often ask you to calculate critical values
based on some level of significance, say 1%, 5%, or 10%. What sort of criteria do you think
should play a role in determining which level of significance to choose?
(g) [3 Points] What do you think the relationship is between testing for the significance of the
slope and whether or not the regression R² is zero?