Name:
Introduction to Econometrics - Midterm exam
Suggested Solution
by Hieu Nguyen
NOTES:
• There are 11 pages for this exam, make sure you check all 11 pages when doing;
• Calculation questions should be explained carefully rather than a number only to get the full grades;
• Make sure your handwriting is readable. Otherwise, you cannot be graded;
GOOD LUCK!
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Multiple Choice Questions (30 pts = 10 * 3 pts)
1. What is the meaning of the term ”heteroscedasticity”?
a. The variance of the errors is not constant
b. The variance of the dependent variable is not constant
c. The errors are not linearly independent of one another
d. The errors have non-zero mean
2. Data on one or variables collected at a given point of time
a. Time series data
b. Cross-section data
c. Pooled data
d. Panel data
3. The coefficient of determination (R2
) shows how many %...
a. Variation in the dependent variable Y is explained by the variation in independent variable
X
b. Variation in the independent variable Y is explained by the variation in dependent variable X
c. Variation in the dependent variable Y explains the variation in independent variable X
d. Variation in the independent variable Y explains the variation in dependent variable X
4. Rejecting a true hypothesis results in which type of error
a. Type I error
b. Type II error
c. Structural error
d. Hypothesis error
5. Which of the following statements is true of hypothesis testing?
a. The t test can be used to test 3 coefficient restrictions.
b. A test of single restriction is also referred to as a joint hypotheses test.
c. A restricted model will always have fewer parameters than its unrestricted model.
d. OLS estimates maximize the sum of squared residuals.
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6. Which of the following statements is true?
a. If the calculated value of F-test is higher than the F-critical value, we reject the alternative hypothesis
and accept the null hypothesis
b. The value of F-test is always nonnegative because SSRr is never smaller than SSRur
c. Degrees of freedom of a restricted model is always less than the degrees of freedom of an unrestricted
model
d. The F statistic is more flexible than the t statistic to test a hypothesis with a single restriction
7. The hypothesis testing with H1 : βj ̸= 0, where βj is a regression coefficient associated with an explanatory
variable, represents a one-sided alternative hypothesis.
a. true
b. false
8. In the following equation, GDP refers to gross domestic product (in million USD), bankcredit refers to the
amount of loans a bank provides to its customers (in million USD), and FDI refers to foreign direct investment.
log(GDP) = 2.65 + 0.527 log(bankcredit) + 0.222 FDI
Which of the following statements is then true?
a. If GDP increases by 1%, bank credit will increase by 0.527%, given the level of FDI remaining constant.
b. If bank credit increases by 1%, GDP will increase by 0.527%, given the level of FDI
remaining constant.
c. If GDP increases by 1%, bank credit increases by 0.527 million USD, given the level of FDI remaining
constant.
d. If bank credit increases by 1%, GDP will increase by 0.527 million USD, given the level of FDI remaining
constant
9. Which of the following correctly identifies an advantage of using adjusted R2
over R2
?
a. Adjusted R2
corrects the bias in R2
b. Adjusted R2
is easier to calculate than R2
c. Adjusted R2
has the penalty of adding new (irrelevant) independent variable(s) while R2
doesn’t have any
d. The adjusted R2
can be calculated for models having logarithmic functions while R2
cannot be calculated
for such models
10. The term u in an econometric model below is usually referred to as the
y = β0 + β1x1 + β2x2 + β3x3 + u
a. error term
b. parameter
c. hypothesis
d. dependent variable
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Theoretical Question (20 pts):
Choose ONE out of 2 following questions to answer:
1. Discuss in detail 4 importance specification criteria to decide whether a variable belongs to the regression
equation.
2. Discuss in detail 2 types of error in doing hypothesis testing. Provide 1 example for each type.
Suggested solution:
1. When determining whether a variable should be included in a regression equation, there are four important
specification criteria to consider:
• Theory: A variable’s inclusion should be supported by theoretical justification. This means that the
variable’s role in the equation should be unambiguous and theoretically sound. If economic theory
or domain-specific knowledge strongly suggests that a variable influences the dependent variable, it
should be included. For instance, when modeling consumer demand, variables like price and income are
theoretically justified as determinants.
• t-test: The significance of a variable’s estimated coefficient can be tested using a t-test. The t-test assesses
whether the coefficient is statistically different from zero, indicating that the variable has a meaningful
impact on the dependent variable. Moreover, it should be significant in the expected direction (positive
or negative) as suggested by theory. For example, in a demand model, a positive coefficient on income
is expected, as higher income should increase demand.
• R2
: Another criterion is whether the overall fit of the model improves with the inclusion of the variable.
This can be evaluated using the R2
statistic, which measures the proportion of variance explained by the
model. If adding a variable significantly increases R2
, it suggests that the variable contributes valuable
information to the model. However, the adjusted R2
should also be considered to account for the addition
of multiple variables.
• Bias: The inclusion of a variable can affect the coefficients of other variables in the model. If adding
a variable causes significant changes in other coefficients, this suggests that the variable controls for an
important factor and reduces omitted variable bias. For instance, in a model predicting wages, including
education as a variable might change the coefficients of other demographic factors, indicating its relevance
in explaining wage variation.
2. 2 types of error in doing hypothesis testing:
• Type I Error (False Positive) occurs when we reject the null hypothesis when it is actually true. This
is also known as a ”false positive” result. The probability of committing a Type I error is denoted
by α, which is the significance level of the test (commonly set at 5% or 1%). Example: Suppose a
pharmaceutical company tests a new drug to determine if it is more effective than the current standard
treatment. The null hypothesis (H0) is that the new drug has no effect (i.e., it is no more effective than
the standard treatment). A Type I error would occur if the test results lead the company to conclude
that the new drug is effective when, in reality, it is not. This could result in the company promoting an
ineffective drug, potentially causing harm to patients and wasting resources.
• Type II Error (False Negative) occurs when we fail to reject the null hypothesis when it is actually
false. This is also known as a ”false negative” result. The probability of committing a Type II error is
denoted by β. Example: Consider a criminal trial where the null hypothesis (H0) is that the defendant
is innocent. A Type II error would occur if the jury fails to reject the null hypothesis (and thus finds the
defendant not guilty) when the defendant is actually guilty. This would allow a guilty person to go free,
which could have serious consequences for society and undermine trust in the justice system.
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Practice Exercises (50 pts):
1. From 40 randomly selected large US cities in 1988, the researchers aim to identify what factors can affect the
demand for urban transportation by bus. Potential explanatory variables include:
• bustravel ... demand for urban transportation by bus (in thousands of person);
• fare ... bus fare (in USD,);
• gasprice ... price of a gallon of gasoline (in USD);
• income ... average annual income per capita (in USD);
• pop ...population of the city (in thousands);
• density ... population density (persons/sq. mile);
• landarea ... land area of the city (sq. miles).
Answer the following questions:
a. (2 pts) Construct a regression model in which demand for urban transportation by bus (in thousands
of person) is estimated based on bus fare (in USD), average annual income per capita (in USD), price
of a gallon of gasoline (in USD), and population density (persons/sq. mile). Name this model as model 1.
Suggested solution:
bustravel = β0 + β1 · fare + β2 · income + β3 · gasprice + β4 · density + ε
b. (2 pts) Based on the regression output of model 1 shown below, write down the regression equation.
You are required to write numeric intercept, coefficients, and standard errors in parentheses under the
corresponding coefficients
Suggested solution:
bustravel = −734.115 + 416.430 · fare − 0.144735 · income + 2140.86 · gasprice + 0.407760 · density
6096.67 1019.54 0.150612 6200.56 0.0781852
c. (4 pts) From (b), interpret the estimated coefficients of fare and gasprice
Suggested solution:
• The coefficient of fare is estimated to be 416.43, which suggests that for each 1 USD increase in bus
fare, the demand for urban transportation by bus (bustravel) increases by 416.43 thousand persons,
holding other variables constant. This result is counterintuitive, as we would typically expect higher
fares to decrease demand (signal for the problem in the regression model!)
• The coefficient of gasprice is 2140.86, indicating that for each 1 USD increase in the price of gasoline,
the demand for bus travel increases by 2140.86 thousand persons. This positive relationship aligns
with the expectation that higher gasoline prices may encourage more people to use bus transporta-
tion.
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d. (4 pts) From (b), construct 90% confidence interval for population coefficients of fare and 95% confidence
interval for gasprice
Suggested solution:
• The 90% confidence interval for the coefficient of fare can be calculated as:
CI90% = 416.43 ± (t0.1,35) × 1019.54
Using a t-table, we find that t0.1,35 ≈ 1.6895 ≈ 1.690. Thus:
CI90% = 416.43 ± (1.690 × 1019.54) = (−1306.59, 2139.45)
• The 95% confidence interval for the coefficient of gasprice is:
CI95% = 2140.86 ± (t0.05,35) × 6200.56
Using a t-table, we find that t0.05,35 ≈ 2.03. Thus:
CI95% = 2140.86 ± (2.03 × 6200.56) = (−10446.28, 14728.00)
e. (8 pts) From (b), test the null hypothesis at 10% level of significance that fare has no effect on bustravel
against the alternative that it has a negative effect. Note: You need to state null hypothesis, alternative
hypothesis, clearly calculate test value, critical value, decision of rejection or acceptance, and interpretation
of the decision.
Suggested solution:
• (H0): βfare = 0 and (H1): βfare < 0
• t-test statistic: Coefficient of fare
Standard error of fare = 416.43
1019.54 = 0.4084
• t-critical value at 10% significance level with 35 degrees of freedom (1-tail test): t0.10,35 ≈ 1.306
Since the absolute value of the calculated t-value (0.4084) is greater than the t-critical value (1.306),
we fail to reject the null hypothesis. This suggests that there is no significant evidence at the 10%
significance level to conclude that an increase in fare has a negative effect on bustravel.
f. (8 pts) From (b), I want to test the null hypothesis at 10% level of significance that the total effect of
fare and gasprice is 0 against the alternative that the total effect is different from 0. Specify the way
to proceed the t-test for this hypothesis testing. Note: You need to state null hypothesis, alternative hypothesis,
clearly calculate test value, critical value, decision of rejection or acceptance, and interpretation
of the decision.
Suggested solution:
• (H0): βfare + βgasprice = 0 and (H1): βfare + βgasprice ̸= 0
• Test statistic:
t =
(βfare + βgasprice) − 0
SE(βfare + βgasprice)
To find t, we need to know the SE(βfare + βgasprice)
• t-critical value at 10% significance level (two-tailed) with 35 degrees of freedom: t0.1,35 ≈ 1.690
If the absolute value of calculated t-value is greater than 1.690, we reject the null hypothesis. Otherwise,
we fail to reject it. This would indicate whether the combined effect of fare and gasprice is statistically
different from zero at the 10% significance level.
g. (8 pts) A person claims that income, gasprice, density each has no effect on bustravel. Conduct the
hypothesis testing for this claim at 5% level of significance.
Note: You may need information of model 2 as below:
Also, given that
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• Model 1 has R2
= 0.5375, SSR = 1933.175
• Model 2 has R2
= 0.2302, SSR = 2431.757
Suggested solution:
• Setting the hypotheses:
H0: βincome = βgasprice = βdensity = 0
H1: At least one of βincome, βgasprice, or βdensity is not equal to 0.
• Choosing the Test Statistic: Since we are testing multiple coefficients, we use the F-test for joint
significance. The F-test statistic is calculated as:
F =
(SSRrestricted − SSRunrestricted)/q
SSRunrestricted/(n − k − 1)
where:
– SSRrestricted is the sum of squared residuals for the restricted model (Model 2, where income,
gasprice, and density are excluded),
– SSRunrestricted is the sum of squared residuals for the unrestricted model (Model 1, which includes
all variables),
– q is the number of restrictions (3 in this case, for income, gasprice, and density),
– n is the number of observations (40),
– k is the number of parameters in the unrestricted model (4 in this case, for fare, income, gasprice,
and density).
• Calculating the F-Statistic
Given:
SSRrestricted = 2431.757, SSRunrestricted = 1933.175
q = 3, n = 40, k = 4
F =
(2431.757 − 1933.175)/3
1933.175/(40 − 4 − 1)
=
498.582/3
1933.175/35
=
166.194
55.2336
≈ 3.01
• Finding the Critical Value and Making a Decision
For an F-test with q = 3 and n − k − 1 = 35 degrees of freedom at the 5% significance level, we look
up the critical value in an F-distribution table. The critical value F0.05,3,35 ≈ 2.87.
• Conclusion: Since the calculated F-statistic (3.01) is greater than the critical value (2.87), we reject
the null hypothesis at the 5% significance level. This suggests that at least one of the variables
income, gasprice, or density has a statistically significant effect on bustravel.
h. (8 pts) Test the overall significance of a regression in model 1, given R2
= 0.5375, SSR = 1933.175.
Note: You need to state null hypothesis, alternative hypothesis, clearly calculate test value, critical value,
decision of rejection or acceptance, and interpretation of the decision.
Suggested solution:
• Setting the hypotheses:
(H0): βfare = βincome = βgasprice = βdensity = 0 (the model has no explanatory power)
(H1): At least one of the slope coefficients is not zero (the model has explanatory power)
• Test Statistic:
F =
R2
/q
(1 − R2)/(n − k − 1)
where:
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– R2
= 0.5375
– q = 4
– k = 4 (number of parameters in the model)
– n = 40 (number of observations)
F =
0.5375/4
(1 − 0.5375)/(40 − 4 − 1)
=
0.5375/4
0.4625/35
=
0.134375
0.0132143
≈ 10.17
• Critical Value:
For an F-test with degree of freedom 1 (df1) = q = 4 and degree of freedom 2 (df2) = n−k −1 = 35
degrees of freedom at the 5% significance level, the critical value F0.05,4,35 ≈ 2.61.
• Compare values and make decision: F-statistic (10.17) > F −cv(2.61), we reject the null hypothesis.
• Interpretation: The decision to reject the null hypothesis suggests that the regression model has
significant explanatory power at the 5% level. This means that at least one of the independent
variables (fare, income, gasprice, or density) contributes to explaining the variability in bustravel.
i. (6 pts) One student constructs a regression model (named model 3) in which demand for urban transportation
by bus (in thousands of person) is estimated based on bus fare (in USD), average annual
income per capita (in USD), price of a gallon of gasoline (in USD), population density (persons/sq.
mile), population of the city (in thousands), and land area of the city (sq. miles). You are asked to
answer:
• Discuss in detail what classical assumption is violated in model 3.
• What is the consequence of this violation?
• How to solve this violation problem?
Suggested solution:
• In Model 3, the classical assumption that is likely violated is the assumption of no multicollinearity
among the explanatory variables. Multicollinearity occurs when two or more independent variables
are highly correlated with each other. Given that the model includes variables such as population
density, population, and land area, which are often related to each other in urban settings,
multicollinearity is a probable issue.
• The presence of multicollinearity affects the precision of the estimated coefficients. Specifically, it
leads to:
– High standard errors for the affected coefficients, which reduces the statistical significance of
those variables.
– Unstable coefficient estimates that can vary greatly with small changes in the model or data,
making it difficult to interpret the true effect of each variable.
– Reduced reliability of the regression results, as high multicollinearity inflates the variance of the
estimated coefficients, making them less precise.
• There are several methods to address multicollinearity: (Notes: only one of these following solutions
is enough to get full credit.
– Remove one or more correlated variables: If two variables are highly correlated (e.g., population
and land area), consider removing one of them from the model to reduce multicollinearity.
– Combine correlated variables: Create a new variable that combines the information from the
correlated variables. For instance, instead of using both population and population density, one
might use only one of them or an index that reflects urban density.
Above is suggested solution, detailed grades for students will be based on students’ understanding
and explanation (with convinced arguments)
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