1. The document discusses using regression analysis to estimate the effects of alcohol consumption on college GPA while controlling for relevant variables. It considers whether to include attendance and whether attendance could be used as an instrumental variable to address potential endogeneity. 2. The document also discusses using panel data from US states from 2000-2015 to investigate the effect of minimum wages on teenage employment. It compares models with and without state and time fixed effects and how this impacts the coefficient of interest. 3. Finally, the document discusses unit root testing of UK money supply data and variables using augmented Dickey-Fuller tests to inform forecasting of money growth rates. It considers a Granger causality test to evaluate whether lags

1. Section A: Answer only 2 of the 3 questions
1. Suppose we want to estimate the effects of alcohol consumption (alcohol) on college
grade point average (colGPA). In addition to collecting information on grade point
average and alcohol usage, we also obtain attendance information (say, percentage of
lectures attended, called attend). A standardized test score (say, SAT) and high school
GPA (hsGPA) are also available.
(i) Should SAT and hsGPA be included as explanatory variables? Explain.
A We would want to include SAT and hsGPA as controls, as these measure
student abilities and motivation. Drinking behavior in college could be
correlated with one’s performance in high school and on standardized tests.
Other factors, such as family background, would also be good controls.
(ii) Should we include attend along with alcohol as explanatory variables in a
multiple regression model? (Think about how you would interpret alcohol.)
A The answer is not entire obvious, but one must properly interpret the coefficient
on alcohol in either case. If we include attend, then we are measuring the effect
of alcohol consumption on college GPA, holding attendance fixed. Because
attendance is likely to be an important mechanism through which drinking affects
performance, we probably do not want to hold it fixed in the analysis. If we do
include attend, then we interpret the estimate of alcohol
 as being those effects on
colGPA that are not due to attending class. (For example, we could be measuring
the effects that drinking alcohol has on study time.) To get a total effect of alcohol
consumption, we would leave attend out.
(iii) We suspect that poor college performance might lead to increased alcohol
consumption. What effect would this have on our estimates?
A This would be an example of simultaneous causality and would cause bias.
(iv) Should we use attend as an instrumental variable to allow for this?
Explain and specify the model and estimation method that you think would best
estimate the effect that we are interested in.
A attend should be correlated with alcohol (relevant), but might also be
endogenous. Nevertheless, IV estimation with attend as the IV for alcohol and
not as a control variable is likely to yield the best estimate.
2. You want to investigate the effect of the minimum wage on teenage employment. You
get access to panel data from the United States that includes the following variables for
each of the 50 states over the period 2000-2015: the employment to population ratio of
teenagers (E), the nominal minimum wage (M), and the average wage (W).
Your baseline regression is as follows: Eit = β0 + β1(Mit/Wit) + uit
i. Briefly discuss the advantage of using panel data in this situation rather than
pure cross sections or time series.
A There are likely to be omitted variables in the above regression. One way to
deal with some of these is to introduce state and time effects. State effects will
capture the influence of omitted variables that are state specific and do not
vary over time, while time effects capture those of country wide variables that
are common to all states at a point in time. Furthermore, there are more
observations when using panel data, resulting in more variation.

2. ii. Write down the expression(s) for the regression that includes state and time
fixed effects.
A It can be written in two ways.
(1) “n-1, t-1" regression:
Eit = β0+ β1(Mit/Wit)+ γ2D2i +:::+ γ50D50i + δ2B2t +:::+ δ16B16t +uit, where
the D variables are state dummies and the B variables are time dummies.
That is, D2 = 1 for state 2, and 0 otherwise. Etc.
(2) Fixed-effects: Eit = β1(Mit/Wit) + αi + λt + uit, where αi captures state-fixed-
effects, a different constant for every state and λt captures time-fixed effects, a
different constant for every year. Note this regression does not include β0.
Table 2 reports the OLS estimates of the model, including only time fixed effects
results (column 1) and including both time and state fixed effects (column 2).
iii. Comment on the above results in column (1). Is the coefficient of interest
statistically significant? Imagine that the time fixed-effects are not statistically
different from zero, how would you interpret the coefficient of interest?
A There is negative relationship between minimum wages and the
employment to population ratio. Also, 20 percent of employment to population
of teenagers variation is explained by the above regression. Apriori, the effect
of the minimum wage on employment can vary across time and states. If the
time fixed effects are not significant, it means that the effect is constant over
time. Thus the coefficient is the sample mean for all states.
iv) Compare the results in columns (1) and (2). Why would the inclusion of state
fixed effects change the coefficients in this way? What does this result tell you
about testing the hypothesis that all of the state fixed effects can be restricted
to have the same coefficient? How would you test such a hypothesis?
A The parameter of interest was highly significant in column (1) , while in
column (2) it has changed signs and is statistically insignificant. The
explanatory power of the equation has increased substantially. The results
suggest that omitted variables, which are now captured by state fixed effects,
were correlated with the regressors and caused omitted variable bias. The
influence of the state effects is large. These are bound to be statistically
significant and the hypothesis to restrict these coefficients to zero is bound to
fail. Since these are linear hypothesis that are supposed to hold
simultaneously, an F-test is appropriate here.

3. 3. You collect monthly data on the money supply (M2) for the United Kingdom from
1962:12002:12 to forecast future money supply behaviour. You define LM2 as the log
level of M2 and DLM2 as the month to month growth rate of M2.
For this exercise you may need the Large-Sample Critical Values of the Augmented
Dickey Fuller Statistic. You can find this on Table 14.5 at the end of this exercise.
i. In order to annualize monthly growth rates, how would you proceed?
A Using quarterly data, when analyzing inflation and unemployment in the
United States, the textbook converted log levels of variables into growth rates
by differencing the log levels, and then multiplying these by 400. To annualize
monthly growth rates, you would need to multiply them by 1,200. The
annualized growth rate of money would be1200∆ln(LM 2t).
ii. How would you go about testing for a stochastic trend in LM2 and DLM2? Be
specific about how to decide the number of lags to be included and whether or
not to include a deterministic trend in your test?
A The ADF statistic should be calculated to test for the presence of a unit root
in each of the series. Inspecting the graph of LM2 it makes sense to include a
time trend, while for DLM2 is not so clear. An information criterion (such as
BIC or AIC) should be used to determine the lag length. Additionally, students
may comment on the fact that given that money growth determines the
inflation rate in the long-run, your expectation would be to also find a unit root
for money growth.
iii. You decide to conduct an ADF unit root test for LM2, DLM2, and the change
in the growth rate ∆DLM2. This results in the following t-statistic on the
parameter of interest.
LM 2
with
trend
DLM 2
without
trend
DLM 2
with
trend
∆DLM 2
without
trend
-0.505 -4.100 -4.592 -8.897
Find the critical value at the 1%, 5%, and 10% level and decide which of the
coefficients are significant. What is the alternative hypothesis?
A In general: (i) in the intercept-only specification, the alternative is that Y is
stationary around a constant, no long term growth in the series. And (ii) in the
intercept & trend specification, the alternative is that Y is stationary around a
linear time trend, the series has long term growth. The series of LM2 contain
a time trend, and hence the critical values for an intercept and a time trend are
relevant (see table 14.5 below). These are (-3.96), (-3.41), and (-3.12) for the
three significance levels respectively. Hence you
cannot reject the null hypothesis of a unit root for LM 2. The growth rate of
money does not have a time trend for the entire sample period, so the intercept
only critical values should be used. These are (3.43), ( 2.86), and ( 2.57)
respectively. Hence you are able to reject the null hypothesis of a unit root for
money at the 1% significance level. And similarly for ∆DLM 2.
iv. In forecasting the money growth rate, you add lags of the monetary base
growth rate (DLMB) to see if you can improve on the forecasting performance
of a chosen AR(10) model in DLM2. You perform a Granger causality test on
the 9 lags of DLMB and find a F-statistic of 2.31. Discuss the implications.
A The critical value for the null hypothesis that monetary growth rates
do not Granger cause money supply growth rates is F9,∞ = 1.88 at the

4. 5% significance level, and 2.41 at the 1% significance level. Hence you
can reject the null hypothesis at the 5% level, but not at the 1% level.
Section B: Answer all questions
1. The Linear Regression Model is 𝑌𝑖 = 𝛽0 + 𝛽1𝑋𝑖 + 𝑢𝑖. The OLS estimator of the
intercept term is:
A) 𝛽0 = 𝑌
̅ − 𝛽
̂1𝑋
̅
B) 𝛽
̂0 =
1
𝑛
∑ 𝑌𝑖
𝑛
𝑖=1 − 𝛽
̂1
1
𝑛
∑ 𝑋𝑖
𝑛
𝑖=1
C) 𝛽
̂1 =
∑ (𝑋𝑖−𝑋
̅)
𝑛
𝑖=1 (𝑌𝑖 −𝑌
̅)
∑ (𝑋𝑖−𝑋
̅)
𝑛
𝑖=1
2
D) 𝛽
̂0 =
𝑆𝑋𝑌
𝑆2 𝑋
E) 𝛽
̂1 =
𝑆𝑋𝑌
𝑆2 𝑋
Answer: B
2. Which of the following statements are true:
A) The 𝑅2
and 𝑅
̅ tell you if you have chosen the most appropriate set of
regressors.
B) The 𝑅2
and 𝑅
̅ tell you whether the regressors are good at predicting the
values of the dependent variable.
C) The 𝑅2
and 𝑅
̅ tell you the regressors are a true cause of the dependent
variables.
D) Both statements (B) and (C).
E) None of the above.
Answer: B
3. Your regression model is 𝑌𝑖 = 𝛽0 + 𝛽1𝑊𝑖 + 𝛽2𝑋𝑖 + 𝛽3𝑍𝑖 + 𝑢𝑖. You run your
regression with a sample of data and wish to test the joint hypothesis 𝐻0: 𝛽1 =
0 𝑎𝑛𝑑 𝛽2 = 0 𝑎𝑛𝑑 𝛽3 = 0. To do this you can:
A) Look at the p-value associated with the F-statistic of the 𝑅2
in the
unrestricted regression.
B) You look at the size of the 𝑅2
, if it is bigger than 0.5 then you reject the null
hypothesis.
C) Rearrange the regressors so that the restriction becomes a restriction on a
single coefficient.
D) Calculate the t-statistics for 𝛽1,𝛽2 and 𝛽3 and compare each of them to the
relevant t-critical value.
E) None of the above.
Answer: A

5. 4. A Type 1 error is when:
A) The p-value is smaller than 0.05.
B) You reject the null hypothesis when it is actually true.
C) You reject the null hypothesis when it is actually false.
D) The law of large numbers does not hold.
E) Both statements (A) and (C) are correct.
Answer: B
5. Your run a regression and receive the following output: ln(𝑌)
̂ = 2.57 + 0.0034𝑋.
According to this regression, which of the following statements are true:
A) For each additional increase in X, Y increases by 0.0034.
B) For each additional increase in X, ln(Y) increases by 0.0034.
C) For each additional increase in X, Y increases by 0.34%.
D) For each additional increase in X, ln(Y) increases by 0.34%.
E) Both statements (B) and (C) are correct.
Answer: E
6. Omitted variable bias:
(A) is always there but is negligible in almost all economic examples.
(B) exists if the omitted variable is correlated with the included regressor and is a
determinant of the dependent variable.
(C) exists if the omitted variable is correlated with the included regressor but is
not a determinant of the dependent variable.
(D) exists if the omitted variable is not correlated with the included regressor and
is a determinant of the dependent variable.
(E) will always be present as long as the regression R2 < 1.
Answer: B
7. Under imperfect multicollinearity
(A) the OLS estimator cannot be computed.
(B) two or more of the regressors are highly correlated.
(C) the OLS estimator is biased even in samples if n > 100.
(D) the error terms are highly, but not perfectly, correlated.
(E) the OLS estimator has small standard errors
Answer: B
8. Stationarity means that the:
(A) time series are constant.
(B) time series has a unit root.
(C) probability distribution of the time series variable does not change over time.
(D) forecasts remain within 1.96 standard deviation outside the sample period.
(E) times series are a random walk.
Answer: C
9. Which of the following statements is true?
(A) A random walk process is stationary.
(B) The variance of a random walk process increases as a linear function of time.

6. (C) Adding a drift term to a random walk process makes it stationary.
(D) The variance of a random walk process with a drift decreases as an exponential
function of time.
(E) None of the above.
Answer: 10.B
10. In the model Yi = β0 + β1ln(Xi) + ui, what is the interpretation of the slope coefficient?
(A) a 1% change in X is associated with a β1% change in Y .
(B) a 1% change in X is associated with a change in Y of 0.01 β1 .
(C) a change in X by one unit is associated with a 100 β1% change in Y .
(D) a change in X by one unit is associated with a β1 change in Y .
(E) none of the above.
Answer: B.