Question 1 (Linear regression and dummy variables; 9 pts)
Margarita would like to explain the level of risk (𝑉𝑉𝑖𝑖 ) (in percentages) of a company i by several
independent variables:
- 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 : The amount of compensation (in dollars) of the CEO of the company
- 𝐼𝐼𝑉𝑉𝑖𝑖 : The implied volatility of the company in percentages (this is the expected risk of the
future of the company, based on option prices)
- 𝐷𝐷𝑠𝑠𝑗𝑗,𝑖𝑖 , A sector dummy if company i belongs to sector j (j = 1, 2 or 3).
- 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑖𝑖 : A dummy variable that equals 1 if the CEO is a Male, 0 otherwise
Margarita has a cross-sectional dataset of 1000 observations. She writes down the following linear
regression models:
(EQ 1.1) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
(EQ 1.2) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
(EQ 1.3) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽2 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽4 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽5 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
a) What is the economic interpretation of 𝛽𝛽2 in (EQ 1.2)? (1 pt)
An increase of 1% POINT in IV increases the risk of a company by 𝛽𝛽2 percentage POINTS, corrected
for the effect of CEO and sectors. (1 pt)
b) Margarita asks you to interpret (economically) the coefficients corresponding with the first
sector dummy (𝐷𝐷𝑠𝑠1,𝑖𝑖 ) in (EQ 1.2) AND (EQ 1.3). What is your answer? (2 pt)
EQ1.2: 𝛽𝛽3 the additional risk (in percentage points) of a company that belongs to sector 1 on top
of a company that belongs to sector 2, corrected for IV and CEO. (1 pt)
EQ1:3: we can NOT interpret this coefficient as there if perfect multicollinearity! (1 pt)
Margarita is unfortunately not able to create the dummy (𝐷𝐷𝑠𝑠3,𝑖𝑖 ). Hence, she can not estimate the
parameters of (EQ 1.2). She asks you whether it is possible to create a linear regression model that
explains the risk of a company by CEO, IV and the sector dummies, but it should NOT contain the
dummy (𝐷𝐷𝑠𝑠3,𝑖𝑖 ).
c) Write down this linear regression model. (1 pt)
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝑢𝑢𝑖𝑖 (1 pt)
After thinking for a while, Margarita writes down a fourth linear regression model:
(EQ 1.4) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽5 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽6 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
Page 3 of 11
, d) What is the economic interpretation of 𝛽𝛽5 in (EQ 1.4)? (1 pt)
The extra effect of CEO on V of a company that belongs to sector 2 will be 𝛽𝛽5 percentage POINTS
higher than the effect of CEO on V of a company that belongs to sector 1, corrected for IV and
sectors (1 pt)
Anna, a friend of Margarita, helps her to create the third sector dummy. Margarita is now able to
estimate the parameters of the linear regression models of (EQ 1.1) and (EQ 1.4). The residual sum
of squared (RSS) associated with these models are equal to 0.40 and 0.30, respectively.
Margarita hypothesizes that the sector dummies are completely unnecessary when explaining the
risk of a company.
e) Perform an appropriate test to investigate this hypothesis. Write down H0, HA, the test
statistic and your conclusion? (3 pt)
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽5 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽6 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
H0: 𝛽𝛽3 = 𝛽𝛽4 = ⋯ = 𝛽𝛽6 = 0
HA: at least one 𝛽𝛽𝑖𝑖 (i = 3,…6) is nonzero. (1 pt)
We have to perform the F-test with the following procedure.
1) Estimate the model under the null hypothesis
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
And store the SSR as SSR0 (1 pt)
2) Estimate the model of EQ(1.4), and store the SSR as SSR1.
3) The F-stat is now given by
(0.40 − 0.30)/4
= 82.75
0.30/(1000 − 7)
Which follows an F(4,993) distribution. (1/2 pt)
This value exceeds the critical value (check your book, its around 2.38 using a 5 % significance
level). Hence we reject the null hypothesis. That is, it: sector dummies are useful when explaining
the risk of a company! (1/2 pt)
Finally, Margarita is worried about heteroscedasticity when interpreting the significance of the
estimated parameters. Anna suggests to use Weighted Least Squares (WLS).
f) Which crucial ingredient(s) is/are needed such that WLS can be applied? (1 pt)
We need at least variable that is related to the variance of the disturbance term. (1 pt)
Page 4 of 11
,Question 2 (Binary models; 6 pts)
You investigate weekly S&P 500 index returns from 1978 – 2009. Based on these returns you
construct a binary variable that indicates if the market in is a bull state (rising prices of assets) or a
bear state (prices expected to decrease)
The data consist of the following variables
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐶𝐶𝑀𝑀𝑅𝑅𝑡𝑡 0/1 Indicator, 1 if the US market is in a bull state, 0 if in a bear state
Δ𝐵𝐵𝑈𝑈𝐶𝐶𝑀𝑀𝑈𝑈𝑡𝑡 Yearly unemployment growth in percentages
𝑌𝑌𝐵𝐵𝐷𝐷𝑌𝑌𝑈𝑈𝑅𝑅𝐶𝐶𝑀𝑀𝐷𝐷𝑡𝑡 Yieldspread, defined as the difference between the 1-year and 3-month
government bond yield in percentages
𝐷𝐷𝐼𝐼𝑉𝑉𝑌𝑌𝐼𝐼𝐶𝐶𝐵𝐵𝐷𝐷𝑡𝑡 Dividend Yield in percentages
𝑀𝑀𝑅𝑅𝑀𝑀𝑇𝑇𝐶𝐶𝑡𝑡 Nominal Exchange rate vis-à-vis the Euro
You would like to select an appropriate model to estimate the relationship between the BULLBEAR
indicator and the independent variables. Your friend suggests to estimate a linear probability model,
a Probit model or a Logit model. You estimate a Probit and Logit model using all above listed
variables and get the following output in Stata:
Table 1.1: Stata output of a Probit model
Table 1.2: Stata output of a Logit model
Page 5 of 11
, a) Suppose you had estimated a linear probability model using the same variables as above.
Suppose the estimated constant equals 1.34. Argue why or why NOT this value would make
sense? (1 pt)
This would mean that the probability of a bull market equals 1.34 when all x variables are equal to
zero. This is against the law of statistics. (1 pt)
b) Argue WHY or WHY NOT the -0.86 coefficient of yearly unemployment growth in Table 1.2 is
economically plausible (1 pt)?
An increase in unemployment will decrease the probability of bull markets (good times in the
market) (corrected for the other factors). This makes sense, as unemployment growth could go
together with bad times in the market. (1 pt)
Nikki states that the LOGIT model does – statistically - a much better job than the PROBIT model,
since the estimated coefficients of the LOGIT model are in absolute sense much higher than the
same coefficients in in the PROBIT model. For example, the yldspread coefficient almost doubles
from 0.17 to 0.31.
c) Is Nikki wright? Why (NOT)? EXPLAIN YOUR answer! (1 pt)
Nikki is not right, she should look at the Pseudo R2 or the log likelihood. These values are almost
the same. (1 pt)
Nikki types the following two commands in Stata after estimating the Logit model
1: dydx, (*)
2: dydx, (*) at means
She gets the following Stata output:
Table 1.3: Stata output after typing ``dydx, (*)’’
Page 6 of 11
Margarita would like to explain the level of risk (𝑉𝑉𝑖𝑖 ) (in percentages) of a company i by several
independent variables:
- 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 : The amount of compensation (in dollars) of the CEO of the company
- 𝐼𝐼𝑉𝑉𝑖𝑖 : The implied volatility of the company in percentages (this is the expected risk of the
future of the company, based on option prices)
- 𝐷𝐷𝑠𝑠𝑗𝑗,𝑖𝑖 , A sector dummy if company i belongs to sector j (j = 1, 2 or 3).
- 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑖𝑖 : A dummy variable that equals 1 if the CEO is a Male, 0 otherwise
Margarita has a cross-sectional dataset of 1000 observations. She writes down the following linear
regression models:
(EQ 1.1) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
(EQ 1.2) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
(EQ 1.3) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽2 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽4 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽5 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
a) What is the economic interpretation of 𝛽𝛽2 in (EQ 1.2)? (1 pt)
An increase of 1% POINT in IV increases the risk of a company by 𝛽𝛽2 percentage POINTS, corrected
for the effect of CEO and sectors. (1 pt)
b) Margarita asks you to interpret (economically) the coefficients corresponding with the first
sector dummy (𝐷𝐷𝑠𝑠1,𝑖𝑖 ) in (EQ 1.2) AND (EQ 1.3). What is your answer? (2 pt)
EQ1.2: 𝛽𝛽3 the additional risk (in percentage points) of a company that belongs to sector 1 on top
of a company that belongs to sector 2, corrected for IV and CEO. (1 pt)
EQ1:3: we can NOT interpret this coefficient as there if perfect multicollinearity! (1 pt)
Margarita is unfortunately not able to create the dummy (𝐷𝐷𝑠𝑠3,𝑖𝑖 ). Hence, she can not estimate the
parameters of (EQ 1.2). She asks you whether it is possible to create a linear regression model that
explains the risk of a company by CEO, IV and the sector dummies, but it should NOT contain the
dummy (𝐷𝐷𝑠𝑠3,𝑖𝑖 ).
c) Write down this linear regression model. (1 pt)
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠1,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝑢𝑢𝑖𝑖 (1 pt)
After thinking for a while, Margarita writes down a fourth linear regression model:
(EQ 1.4) 𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽5 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽6 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
Page 3 of 11
, d) What is the economic interpretation of 𝛽𝛽5 in (EQ 1.4)? (1 pt)
The extra effect of CEO on V of a company that belongs to sector 2 will be 𝛽𝛽5 percentage POINTS
higher than the effect of CEO on V of a company that belongs to sector 1, corrected for IV and
sectors (1 pt)
Anna, a friend of Margarita, helps her to create the third sector dummy. Margarita is now able to
estimate the parameters of the linear regression models of (EQ 1.1) and (EQ 1.4). The residual sum
of squared (RSS) associated with these models are equal to 0.40 and 0.30, respectively.
Margarita hypothesizes that the sector dummies are completely unnecessary when explaining the
risk of a company.
e) Perform an appropriate test to investigate this hypothesis. Write down H0, HA, the test
statistic and your conclusion? (3 pt)
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝛽𝛽3 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽4 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝛽𝛽5 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠2,𝑖𝑖 + 𝛽𝛽6 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 𝐷𝐷𝑠𝑠3,𝑖𝑖 + 𝑢𝑢𝑖𝑖
H0: 𝛽𝛽3 = 𝛽𝛽4 = ⋯ = 𝛽𝛽6 = 0
HA: at least one 𝛽𝛽𝑖𝑖 (i = 3,…6) is nonzero. (1 pt)
We have to perform the F-test with the following procedure.
1) Estimate the model under the null hypothesis
𝑉𝑉𝑖𝑖 = 𝛽𝛽0 + 𝛽𝛽1 𝐶𝐶𝐶𝐶𝑇𝑇𝑖𝑖 + 𝛽𝛽2 𝐼𝐼𝑉𝑉𝑖𝑖 + 𝑢𝑢𝑖𝑖
And store the SSR as SSR0 (1 pt)
2) Estimate the model of EQ(1.4), and store the SSR as SSR1.
3) The F-stat is now given by
(0.40 − 0.30)/4
= 82.75
0.30/(1000 − 7)
Which follows an F(4,993) distribution. (1/2 pt)
This value exceeds the critical value (check your book, its around 2.38 using a 5 % significance
level). Hence we reject the null hypothesis. That is, it: sector dummies are useful when explaining
the risk of a company! (1/2 pt)
Finally, Margarita is worried about heteroscedasticity when interpreting the significance of the
estimated parameters. Anna suggests to use Weighted Least Squares (WLS).
f) Which crucial ingredient(s) is/are needed such that WLS can be applied? (1 pt)
We need at least variable that is related to the variance of the disturbance term. (1 pt)
Page 4 of 11
,Question 2 (Binary models; 6 pts)
You investigate weekly S&P 500 index returns from 1978 – 2009. Based on these returns you
construct a binary variable that indicates if the market in is a bull state (rising prices of assets) or a
bear state (prices expected to decrease)
The data consist of the following variables
𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐵𝐶𝐶𝑀𝑀𝑅𝑅𝑡𝑡 0/1 Indicator, 1 if the US market is in a bull state, 0 if in a bear state
Δ𝐵𝐵𝑈𝑈𝐶𝐶𝑀𝑀𝑈𝑈𝑡𝑡 Yearly unemployment growth in percentages
𝑌𝑌𝐵𝐵𝐷𝐷𝑌𝑌𝑈𝑈𝑅𝑅𝐶𝐶𝑀𝑀𝐷𝐷𝑡𝑡 Yieldspread, defined as the difference between the 1-year and 3-month
government bond yield in percentages
𝐷𝐷𝐼𝐼𝑉𝑉𝑌𝑌𝐼𝐼𝐶𝐶𝐵𝐵𝐷𝐷𝑡𝑡 Dividend Yield in percentages
𝑀𝑀𝑅𝑅𝑀𝑀𝑇𝑇𝐶𝐶𝑡𝑡 Nominal Exchange rate vis-à-vis the Euro
You would like to select an appropriate model to estimate the relationship between the BULLBEAR
indicator and the independent variables. Your friend suggests to estimate a linear probability model,
a Probit model or a Logit model. You estimate a Probit and Logit model using all above listed
variables and get the following output in Stata:
Table 1.1: Stata output of a Probit model
Table 1.2: Stata output of a Logit model
Page 5 of 11
, a) Suppose you had estimated a linear probability model using the same variables as above.
Suppose the estimated constant equals 1.34. Argue why or why NOT this value would make
sense? (1 pt)
This would mean that the probability of a bull market equals 1.34 when all x variables are equal to
zero. This is against the law of statistics. (1 pt)
b) Argue WHY or WHY NOT the -0.86 coefficient of yearly unemployment growth in Table 1.2 is
economically plausible (1 pt)?
An increase in unemployment will decrease the probability of bull markets (good times in the
market) (corrected for the other factors). This makes sense, as unemployment growth could go
together with bad times in the market. (1 pt)
Nikki states that the LOGIT model does – statistically - a much better job than the PROBIT model,
since the estimated coefficients of the LOGIT model are in absolute sense much higher than the
same coefficients in in the PROBIT model. For example, the yldspread coefficient almost doubles
from 0.17 to 0.31.
c) Is Nikki wright? Why (NOT)? EXPLAIN YOUR answer! (1 pt)
Nikki is not right, she should look at the Pseudo R2 or the log likelihood. These values are almost
the same. (1 pt)
Nikki types the following two commands in Stata after estimating the Logit model
1: dydx, (*)
2: dydx, (*) at means
She gets the following Stata output:
Table 1.3: Stata output after typing ``dydx, (*)’’
Page 6 of 11
