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1. The probability density function of a random variable X is given in the following table:
Recall that: Xf ( x ) P( X x ) = =
x Xf ( x ) P( X x ) = =
1 0.1
3 0.8
4 0.1
Compute E(X), Var(X), and s.d.(X) (i.e. the standard deviation of X).
2. The new management at a bakery claims that workers are now more productive than they
were under old management, which is why wages have ‘generally increased’. Let Wb be
the wage under the old management and let Wa be the wage after the change. The
difference is a b DW W = − .
i. Suppose that the expected wage under the old management 15 b E(W ) = $ per hour
and the expected wage under the old management 20 a E(W ) = $ per hour. Let D be
the difference a b DW W = − . Compute the expected difference (rounded to 2 decimal
places).
ii. Suppose that the standard deviation of Wb is 2$ and the standard deviation of Wa is 4$.
Furthermore, suppose that the covariance between the two wages is 1. Compute the
standard deviation of the difference (rounded to 2 decimal places).
Answers
i. P(Y=0) = P(X=0,Y=0) + P(X=1,Y=0) = 0.3+ 0.3 =0.6.
ii. P(Y=1) = P(X=0,Y=1) + P(X=1,Y=1) = 0.2+ 0.2 =0.4.
iii. P(X=1|Y=0) = P(X=1,Y=0) / P(Y=0) = 0.3 / 0.6 =0.5.
iv. E(X) = 0×0.5 + 1×0.5 =0.5
since P(X=1)= P(X=1,Y=0) + P(X=1,Y=1) = 0.3+ 0.2 =0.5.
and P(X=0) = P(X=0,Y=0) + P(X=0,Y=1) = 0.3 + 0.2 =0.5
v. E(X|Y=0) = 0×P(X=0|Y=0) + 1×P(X=1|Y=0) = 0×0.3/0.6 + 0.5 =0.5.
3
4. The equation 𝑦𝑦𝑖𝑖 = 𝛽𝛽1 + 𝛽𝛽2𝑥𝑥𝑖𝑖 + 𝑒𝑒𝑖𝑖 has been estimated by OLS, where 𝑒𝑒𝑖𝑖 is the error term,
using a sample size of 62. You are provided with the following output from the
regression.
Fitted equation: 𝑦𝑦� = 1.7 − 3.2𝑥𝑥
Std. Error: (0.2) (0.41)
i. What is the test statistic for a left-tail hypothesis test that the coefficient of 𝑥𝑥𝑖𝑖 is less
than minus 3? (3 decimal places)
ii. What is the critical value for the test at a significance level of 5%? (3 decimal places)
Answer
i. t = (-3.2 + 3)/0.41 = -0.488
ii. The critical value is 𝑡𝑡𝑛𝑛−𝐾𝐾,𝛼𝛼 = 𝑡𝑡60,0.05 = −1.671
5. What is the value of the goodness of fit measures R2 and adjusted R2 for an OLS
regression, given the following information?
SST = 0.125523
SSR = 0.035273
Number of regression coefficients estimated = 4
Number of observations = 30
6. What is the critical value from the F(3,20) distribution (where 3 is the numerator d.f. and 20 is the
denominator d.f.) if you are performing a test at a 5% level of significance? (2 decimal places)
Answer the answer is 3.098 which is 3.10 at two decimal places.
4
7. What is the value of the F-statistic for the test for overall significance of the following model,
which has been estimated by OLS, given the following information? (2 decimal places)
Regression model: 𝑦𝑦𝑖𝑖 = 𝛽𝛽1 + 𝛽𝛽2𝑤𝑤𝑖𝑖 + 𝛽𝛽3𝑥𝑥𝑖𝑖 + 𝛽𝛽4𝑧𝑧𝑖𝑖 + 𝑒𝑒𝑖𝑖
SSR = 546.2
SSE = 713.1
Number of Observations = 128
Answer
F=𝑆𝑆𝑆𝑆𝑆𝑆/(𝐾𝐾−1)
𝑆𝑆𝑆𝑆𝑆𝑆/(𝑁𝑁−𝐾𝐾) = 713.1/3
546.2/(128−4) = 53.96
8. What is the unbiased estimator of 𝜎𝜎�2 in the general multiple regression case with k
explanatory variables including the constant?
A SST/(n – k)
B SSR/(n – k)
C SSR/(n – k)
D SSE/(n – k)
Answer B
5
9. Which one of the following statements is not a correct assumption of the simple linear
regression model?
i. The error variable is not random and is correlated with the dependent variable.
ii. The dependent variable is linearly dependent on the explanatory variable and the
error term.
iii. The sample observations on the independent variable are not all the same value.
iv. The error variable has a zero mean and constant standard deviation.
Answer: i
10. Eric Neumayer (2004) considers why there are more super-rich people in some countries
than in others. The data set contains information on the number of billionaires in 207
countries, there are 167 observations. The model to explain the number of billionaires is
1 2 3 4 billionaires Communist PoP GDPPWT u =+ + + + log( ) log( ) ββ β β
The estimated equation with standard errors in parentheses is
2
– 74.160 – .0726 2.6118 ( ) 4.432 ( )
(18.139) (0.059) (0.706) (1.469)
167, 0.107.
billionaires Communist log PoP log GDPPWT
n R
= + +
= =
The coefficient on Communist, -.0726 is in accordance with the conjecture that a higher
incidence of super wealth is associated with economic freedom. The fact that Communist
has an estimated coefficient different from zero could just be due to sampling error; to be
convinced of an effect, we need to conduct a t test.
Coefficient covariance matrix
const communist lnpop lngdppwt
329.019 0.0420106 -9.58951 -21.3875 const
0.00349785 -0.00489244 -0.0006144 communist
0.498937 0.222314 lnpop
2.15488 lngdppwt
i What is the value of the test statistic to test the hypothesis that at the 5% level (3
decimal places)
0 2 H : 0 β = against the alternative 1 2 H : 0 β < .
6
ii What is the 5% critical value for this test? (3 decimal places)
iii Do you reject the null hypothesis in favour of the alternative? (yes or no answer)
iv What is the value of the test statistic to test
02 3 H : 2 β β + = against the alternative 12 3 H : 2 β β + ≠ ? (3 decimal place)
Answer
i The t statistic on the coefficient of Communist is –0.0726/0.059 ≈ –1.231 . Under the
null it is an observation from a t distribution with n-k = 163 degrees of freedom
ii the 5% critical value for this test is t0.05,163 = -1.654
iii no
iv t=(b2+ b3-2)/s.e.(b2+ b3) ~ t163 under H0.
Note that V(b2 + b3) = V(b2) + V(b3) + 2×Cov(b2,b3).
Estimates for the terms on the right-hand side can be found in the Coefficient covariance
matrix:
0.0726 2.6118 2 0.7682
0.00349785 0.498937 2 ( 0.00489244)
t
−+ − = = + + ×−
1. The probability density function of a random variable X is given in the following table:
Recall that: Xf ( x ) P( X x ) = =
x Xf ( x ) P( X x ) = =
1 0.1
3 0.8
4 0.1
Compute E(X), Var(X), and s.d.(X) (i.e. the standard deviation of X).
2. The new management at a bakery claims that workers are now more productive than they
were under old management, which is why wages have ‘generally increased’. Let Wb be
the wage under the old management and let Wa be the wage after the change. The
difference is a b DW W = − .
i. Suppose that the expected wage under the old management 15 b E(W ) = $ per hour
and the expected wage under the old management 20 a E(W ) = $ per hour. Let D be
the difference a b DW W = − . Compute the expected difference (rounded to 2 decimal
places).
ii. Suppose that the standard deviation of Wb is 2$ and the standard deviation of Wa is 4$.
Furthermore, suppose that the covariance between the two wages is 1. Compute the
standard deviation of the difference (rounded to 2 decimal places).
Answers
i. P(Y=0) = P(X=0,Y=0) + P(X=1,Y=0) = 0.3+ 0.3 =0.6.
ii. P(Y=1) = P(X=0,Y=1) + P(X=1,Y=1) = 0.2+ 0.2 =0.4.
iii. P(X=1|Y=0) = P(X=1,Y=0) / P(Y=0) = 0.3 / 0.6 =0.5.
iv. E(X) = 0×0.5 + 1×0.5 =0.5
since P(X=1)= P(X=1,Y=0) + P(X=1,Y=1) = 0.3+ 0.2 =0.5.
and P(X=0) = P(X=0,Y=0) + P(X=0,Y=1) = 0.3 + 0.2 =0.5
v. E(X|Y=0) = 0×P(X=0|Y=0) + 1×P(X=1|Y=0) = 0×0.3/0.6 + 0.5 =0.5.
3
4. The equation 𝑦𝑦𝑖𝑖 = 𝛽𝛽1 + 𝛽𝛽2𝑥𝑥𝑖𝑖 + 𝑒𝑒𝑖𝑖 has been estimated by OLS, where 𝑒𝑒𝑖𝑖 is the error term,
using a sample size of 62. You are provided with the following output from the
regression.
Fitted equation: 𝑦𝑦� = 1.7 − 3.2𝑥𝑥
Std. Error: (0.2) (0.41)
i. What is the test statistic for a left-tail hypothesis test that the coefficient of 𝑥𝑥𝑖𝑖 is less
than minus 3? (3 decimal places)
ii. What is the critical value for the test at a significance level of 5%? (3 decimal places)
Answer
i. t = (-3.2 + 3)/0.41 = -0.488
ii. The critical value is 𝑡𝑡𝑛𝑛−𝐾𝐾,𝛼𝛼 = 𝑡𝑡60,0.05 = −1.671
5. What is the value of the goodness of fit measures R2 and adjusted R2 for an OLS
regression, given the following information?
SST = 0.125523
SSR = 0.035273
Number of regression coefficients estimated = 4
Number of observations = 30
6. What is the critical value from the F(3,20) distribution (where 3 is the numerator d.f. and 20 is the
denominator d.f.) if you are performing a test at a 5% level of significance? (2 decimal places)
Answer the answer is 3.098 which is 3.10 at two decimal places.
4
7. What is the value of the F-statistic for the test for overall significance of the following model,
which has been estimated by OLS, given the following information? (2 decimal places)
Regression model: 𝑦𝑦𝑖𝑖 = 𝛽𝛽1 + 𝛽𝛽2𝑤𝑤𝑖𝑖 + 𝛽𝛽3𝑥𝑥𝑖𝑖 + 𝛽𝛽4𝑧𝑧𝑖𝑖 + 𝑒𝑒𝑖𝑖
SSR = 546.2
SSE = 713.1
Number of Observations = 128
Answer
F=𝑆𝑆𝑆𝑆𝑆𝑆/(𝐾𝐾−1)
𝑆𝑆𝑆𝑆𝑆𝑆/(𝑁𝑁−𝐾𝐾) = 713.1/3
546.2/(128−4) = 53.96
8. What is the unbiased estimator of 𝜎𝜎�2 in the general multiple regression case with k
explanatory variables including the constant?
A SST/(n – k)
B SSR/(n – k)
C SSR/(n – k)
D SSE/(n – k)
Answer B
5
9. Which one of the following statements is not a correct assumption of the simple linear
regression model?
i. The error variable is not random and is correlated with the dependent variable.
ii. The dependent variable is linearly dependent on the explanatory variable and the
error term.
iii. The sample observations on the independent variable are not all the same value.
iv. The error variable has a zero mean and constant standard deviation.
Answer: i
10. Eric Neumayer (2004) considers why there are more super-rich people in some countries
than in others. The data set contains information on the number of billionaires in 207
countries, there are 167 observations. The model to explain the number of billionaires is
1 2 3 4 billionaires Communist PoP GDPPWT u =+ + + + log( ) log( ) ββ β β
The estimated equation with standard errors in parentheses is
2
– 74.160 – .0726 2.6118 ( ) 4.432 ( )
(18.139) (0.059) (0.706) (1.469)
167, 0.107.
billionaires Communist log PoP log GDPPWT
n R
= + +
= =
The coefficient on Communist, -.0726 is in accordance with the conjecture that a higher
incidence of super wealth is associated with economic freedom. The fact that Communist
has an estimated coefficient different from zero could just be due to sampling error; to be
convinced of an effect, we need to conduct a t test.
Coefficient covariance matrix
const communist lnpop lngdppwt
329.019 0.0420106 -9.58951 -21.3875 const
0.00349785 -0.00489244 -0.0006144 communist
0.498937 0.222314 lnpop
2.15488 lngdppwt
i What is the value of the test statistic to test the hypothesis that at the 5% level (3
decimal places)
0 2 H : 0 β = against the alternative 1 2 H : 0 β < .
6
ii What is the 5% critical value for this test? (3 decimal places)
iii Do you reject the null hypothesis in favour of the alternative? (yes or no answer)
iv What is the value of the test statistic to test
02 3 H : 2 β β + = against the alternative 12 3 H : 2 β β + ≠ ? (3 decimal place)
Answer
i The t statistic on the coefficient of Communist is –0.0726/0.059 ≈ –1.231 . Under the
null it is an observation from a t distribution with n-k = 163 degrees of freedom
ii the 5% critical value for this test is t0.05,163 = -1.654
iii no
iv t=(b2+ b3-2)/s.e.(b2+ b3) ~ t163 under H0.
Note that V(b2 + b3) = V(b2) + V(b3) + 2×Cov(b2,b3).
Estimates for the terms on the right-hand side can be found in the Coefficient covariance
matrix:
0.0726 2.6118 2 0.7682
0.00349785 0.498937 2 ( 0.00489244)
t
−+ − = = + + ×−