Answer keys

For the professor's own exercise decks and both midterm practices. Work the questions first, then check here. Every answer was independently worked out and cross-checked.

Chapter 2 exercises

#	Question starts with	Key	Why
1	Why might the Niagara Region be interested in calculating correlations	D	Lecture 2 lists three uses of correlation: description, prediction/forecasting, and a starting point for causal inference. Options A, B, and C each state one of these, so all of the above is correct.
2	Which of the following is an example of a descriptive question?	B	Counting how many new businesses opened last year simply summarizes the data. A asks a causal question, C asks about a counterfactual, and D sets up a comparison for causal inference.
3	Which of the following are not correlations	A and D	A and D each describe only one group (OZ neighbourhoods, or low-tax employers) with no comparison group, so they are descriptive facts, matching the lecture's scandal example. B and C compare outcomes across variation in taxes ('tend to hire more', 'more likely'), so they are correlations. The question stem is plural and two options qualify.
4	Using the table above, what is the average number of	B	The Opportunity Zone neighborhoods are 1 and 2 with 4 and 6 businesses, so the average is (4 + 6) / 2 = 5.
5	What is the standard deviation of the number of businesses	A	Non-OZ values are 12, 15, 9 with mean 12. Using the lecture's population formula (divide by N), variance = (0 + 9 + 9) / 3 = 6, and the square root is about 2.45.
6	What proportion of Opportunity Zone neighborhoods had high business activity?	B	Of the 8 Opportunity Zone neighborhoods, 2 had high business activity, so 2 / 8 = 0.25.
7	Which best describes the correlation between Opportunity Zone designation and	B	Pr(high businesses \| OZ) = 2/8 = 0.25 while Pr(high businesses \| not OZ) = 8/10 = 0.80. OZ status goes together with lower business activity, so the correlation is negative.

Chapter 3 exercises

#	Question starts with	Key	Why
1	Which of the following best describes the potential outcomes framework	A	Potential outcomes compare the same person's outcome in the world with treatment and the world without it (Y1i and Y0i). Comparisons to other people or to a before period are different quantities.
2	Suppose you download the app and your stress decreases by	B	This is the fundamental problem of causal inference: you observe Y1i only, never your counterfactual Y0i, so your individual causal effect cannot be known. The observed 2-point drop mixes the app's effect with everything else that changed.
3	Which of the following expressions represents the causal effect of	A	Lecture 3 defines the effect of T on Y for unit i as Y1i minus Y0i, the outcome with treatment minus the outcome without it.
4	Why might it matter to a policymaker whether the app	B	A subsidy changes app use, so it only improves health if the app causes stress reduction. A mere correlation gives no guarantee that pushing more people onto the app changes outcomes.
5	Which response best reflects the logic of the potential outcomes	B	One observed user tells us Y1i only. Without knowing what their stress would have been absent the app (Y0i), the anecdote cannot establish a causal effect, so the right move is to disagree on counterfactual grounds.
6	Which of the following best reflects the view from causal	C	Lecture 3 argues causation does not require physical connection. If the app changes behavior or perceptions and stress falls as a result, that counterfactual dependence is a causal effect.
7	Based on this table, which group is more likely to	A	Observed outcomes: app users (persons 1 to 3) show good mental health for 2 of 3, non-users (persons 4 to 6) show good mental health for 1 of 3, so users are more likely. Note the deck misprints options A and B with identical text; either letter carries the correct statement.
8	Based on this table, what is the overall effect of	A	Individual effects Y1i minus Y0i are +1 for persons 3 and 6 and 0 for everyone else, so the average effect is 2/6, which is positive. The app improves mental health for two of the six people and harms no one.

Chapter 4 exercises

#	Question starts with	Key	Why
1	Which of these is actually a statement about a correlation?	D	D compares areas with different development charges ('tend to have more permits'), so it describes how two variables move together. A and C are descriptive facts about one group, and B only describes the highest-demand neighborhoods without a comparison.
2	Do you agree? A colleague surveyed developers who recently built	B	The survey selects on the dependent variable: it only includes developers who built. Correlation requires variation in building outcomes, and without developers who did not build there is no comparison, so the conclusion does not follow.
3	What information in this table did your colleague actually collect?	B	The colleague only surveyed developers who completed projects, so only the 'Built a Project' row is filled in: cells A and B. Cells C and D (developers who did not build) were never observed.
4	Do you agree? During a post-mortem on housing policy in	B	Looking only at unsuccessful initiatives is selecting on the dependent variable. If successful projects also relied on reduced suburban charges, the policy could be fine or even helpful, so no conclusion can be drawn without the successes.

Chapter 5 exercises

#	Question starts with	Key	Why
1	Which is the best description? You explain to your colleague	B	The analysis asks whether OZ status is associated with business openings, so businesses per capita is the outcome (dependent variable) and OpportunityZone is the explanatory (independent) variable.
2	Which of the following is the correct regression equation estimated	A	The intercept estimate is 1.430 and the OpportunityZone coefficient is -0.680, giving Businesses_per_capita = 1.43 minus 0.68 OpportunityZone. Options C and D also put the variables on the wrong sides.
3	What is the predicted businesses per capita for an OZ	C	For an OZ neighborhood the dummy equals 1, so the prediction is 1.43 minus 0.68 = 0.75.
4	Which is the best interpretation? Your colleague asks what the	A	The intercept is the predicted value when OpportunityZone = 0, so non-OZ neighborhoods have a predicted 1.43 businesses per capita. It is a prediction (an average), so B's 'exactly' claim is wrong.
5	What does the slope coefficient (-0.68) mean?	A	With a binary regressor, the slope is the average difference between groups: OZ neighborhoods average 0.68 fewer businesses per capita than non-OZ neighborhoods.
6	How do you respond? Your manager says that because the	B	This regression measures an association, and a large t-value only speaks to statistical precision. OZ designation may have gone to already struggling neighborhoods, so the causal claim does not follow.
7	What is the predicted businesses per capita for an OZ	C	Same calculation as the earlier slide (the deck repeats the question): 1.43 minus 0.68 = 0.75.
8	How do you respond? Your colleague says the error term	B	The error term collects every determinant of businesses per capita left out of the model, such as income, zoning, and demand. It is systematic omitted content, and calling it pure noise understates that.
9	What's the best way to frame the tradeoffs? Your colleague	C	Higher-order polynomials can chase noise in the sample and predict poorly out of sample (overfitting), while simpler models risk underfitting but are easier to interpret. Lecture 5's out-of-sample testing section makes exactly this point.

Chapter 6 exercises

#	Question starts with	Key	Why
1	What is the null hypothesis here?	C	The null hypothesis in a hypothesis test is that the effect is zero (no effect of turmeric), so the null is that the coefficient equals 0. The reported p-value of 0.038 is computed against exactly this no-effect baseline.
2	How might this design flaw affect the study's estimate?	B	Recruiting only health-conscious participants who already eat anti-inflammatory diets is a systematic (not random) departure, so it introduces bias: the effect measured in this unrepresentative group may differ from the effect in the general population. It is a consistency problem, not just added noise.
3	Imagine researchers rerun the turmeric study with a much larger sample	C	A larger sample shrinks the standard error and improves precision (less noise), but a design flaw is a source of bias that a bigger sample does not remove. So the estimate stays equally biased while becoming more precise.
4	What does the reported p-value of 0.038 mean?	C	A p-value is the probability of observing an estimate at least as extreme as the one obtained, assuming the null hypothesis is true. It is conditional on the null, and it is not the probability that the null itself is true.
5	What does the 95% confidence interval [-0.9%, -0.1%] for inflammation reduction	B	The correct frequentist reading is that if the study were repeated infinitely many times, 95% of the constructed intervals would contain the true value. The lecture stresses it does not mean we are 95% sure the truth lies in this one interval.
6	What should we conclude about the size of the estimated effect (-0.5%)?	B	The result is statistically significant (p below 0.05, CI excludes zero), but a 0.5% change in an inflammation biomarker is substantively trivial. Statistical significance does not guarantee practical significance.
7	If someone followed the influencer's advice and added 1 teaspoon	B	The trial dosed 1200mg (about 3 tablespoons equivalent) daily. A single teaspoon is far outside the tested dose, so the study cannot support claims about that smaller daily dose. Extrapolating a linear one-third effect is unjustified.
8	Which of the following best explains why the influencer's claim is misleading?	B	The study did find a statistically significant effect, but a 0.5% reduction is too small to be practically meaningful, so treating it as a reason everyone should change their diet overstates its importance. This is the statistical versus substantive significance distinction from the lecture.

Chapter 7 exercises

#	Question starts with	Key	Why
1	To assess how confident you should be in the RHS program's reported	A	The core lesson is that multiple testing plus selective reporting inflates false positives, so knowing how many outcomes the Region planned to examine is what lets you judge whether the significant result survived many comparisons. Options C and D help too, but the number of outcomes tested directly addresses the multiple-comparisons problem.
2	Suppose the pre-registration document lists 25 outcomes	A	Testing 25 outcomes and reporting only the one significant result is textbook multiple testing: with 25 tests you expect roughly one false positive at the 0.05 level by chance alone, which lowers confidence that shelter nights is a genuine effect.
3	The pre-registration plan identifies shelter nights until stable housing	A	When the significant result was named as the single primary outcome before the study began, it was not cherry-picked after the fact, which raises confidence. Pre-specification is one of the lecture's listed defences against p-hacking and selective reporting.
4	However, a later audit reveals that the original pre-registration	A	Swapping the primary outcome from mental health to shelter nights after the pre-registration date undermines the protection pre-registration was meant to provide and can look like p-hacking, even with the team's assurances. This is exactly the mid-study revision the chapter warns about.
5	If you were advising Regional Council on whether to expand the RHS	B	The evidence is promising but weakened by the mid-study outcome switch, so the appropriately cautious step is to replicate before scaling, which matches the lecture's remedy of replication. Option A over-trusts a single p-value, and C and D overstate the problem as near-certain fraud or invalidity.

Chapter 8 exercises

#	Question starts with	Key	Why
1	What does reversion to the mean describe?	A	Reversion to the mean is the statistical tendency for extreme outcomes to be followed by less extreme ones because part of the extreme was random noise that does not persist. It is not caused by government action or by the mean shrinking.
2	Councillor A says neighbourhoods with the biggest price jumps	A	Neighbourhoods with the biggest jumps last year would tend to cool off anyway through reversion to the mean, so the slowdown is not proof the DCRP worked. The other options wrongly treat the cooling as guaranteed evidence of a program effect.
3	Reversion to the mean requires which condition?	A	Reversion needs outcomes to be driven by both signal (systematic forces) and noise (random fluctuation). If there were no noise there would be nothing random to fade, and if there were no signal there would be no stable mean to revert toward.
4	If housing prices were determined entirely by stable factors	A	With prices set entirely by stable factors like location and amenities there is no random component, so identical conditions yield identical results and reversion disappears. Reversion only operates when noise is present to fade away.
5	Councillor B says high-price areas are being pulled down	A	Reversion is not a physical force pulling values toward the mean; outcomes only appear to move toward the average because the random noise that pushed them to extremes fades. The gravity analogy misrepresents the mechanism.
6	A councillor says public opinion about housing affordability will revert	A	Public beliefs are largely systematic rather than random, so they do not carry the transient noise component that drives statistical reversion to the mean. Without that random piece the same mechanism as house prices does not apply.
7	You discover the DCRP zones were selected because they had the largest	A	Selecting zones precisely because they had the largest 2022 increases means they were chosen at an extreme, and such unusually fast-growing areas tend to grow more slowly later even with no intervention. That selection on the extreme is what makes the later slowdown look like reversion.
8	What pattern in the figure best supports the idea that reversion	A	Stronger reversion shows up as a flatter fitted line: extreme 2023 prices are pulled closer to average in 2024, so a DCRP line flatter than the non-DCRP line is the supporting pattern. The scatter axes and program-status legend are present in the extraction, so the answer is determinable. The other options describe clustering or slope patterns that do not represent reversion.

Chapter 9 exercises

#	Question starts with	Key	Why
1	Using potential-outcomes notation, which expression represents the simple	A	The simple difference in means compares the treated group's observed outcome under treatment (mean Y1 for the treated, written Y1T) with the untreated group's observed outcome under no treatment (mean Y0 for the untreated, written Y0U). The lecture writes this as population difference in means equals Y1T minus Y0U.
2	How is the simple difference in means related to the Average Treatment	A	Following Estimate equals Estimand plus Bias plus Noise, the lecture states Sample difference in means equals ATT plus Bias plus Noise. The SDM is the estimate, so it equals the ATT plus the bias and noise terms.
3	A colleague says the simple difference in means reflects the causal effect	A	The SDM equals the ATT only when the bias term (Y0T minus Y0U) is zero, meaning treated and untreated groups would have had the same outcomes absent treatment. Without that condition the SDM confounds the causal effect with baseline differences.
4	When will bias = 0 for estimating the ATT from the SDM?	A	Bias equals Y0T minus Y0U, so it vanishes exactly when Y0T equals Y0U, meaning the treated and untreated groups would have identical outcomes if neither got crosswalks. That is the comparability (no-confounding) condition.
5	Using the table, what the Simple difference in means?	C	The treated intersections (1, 2, 4, 7) observe their crosswalk outcomes, averaging (20+15+10+12)/4 = 14.25, and the untreated (3, 5, 6, 8) observe their no-crosswalk outcomes, averaging (30+22+25+30)/4 = 26.75. The SDM is 14.25 minus 26.75, which equals -12.5.
6	Using the table, what is the ATE?	A	The ATE is the mean of the causal-effect column (Y1 minus Y0) across all eight intersections: (-5-5-8-2-4-1-3-2)/8 = -30/8 = -3.75. The negative sign means crosswalks reduce accidents on average.
7	What is the bias?	B	The lecture derives Bias equals Y0T minus Y0U, the difference in the no-treatment potential outcome between the treated and untreated groups. Here it equals 18.0 minus 26.75, which equals -8.75, and this checks out since SDM (-12.5) equals ATT (-3.75) plus bias (-8.75).
8	Why might traffic volume confound the estimated effect of crosswalks	C	A confounder is a factor associated with both the treatment and the outcome. Traffic volume plausibly drives both where crosswalks get installed and how many accidents occur, so it is correlated with both crosswalk placement and accidents.
9	A senior engineer notes maybe we build crosswalks because these	B	If prior accident levels cause crosswalk installation, then the outcome is affecting the treatment, which is reverse causation as defined in the lecture. It is distinct from confounding, where a separate third variable drives both.
10	If high-traffic intersections are more likely to receive crosswalks	A	If higher traffic makes an intersection more likely to get a crosswalk, then traffic volume and crosswalk installation move together, so their correlation is positive.
11	If high-traffic intersections also have more accidents, the correlation	A	If higher traffic means more accidents, then traffic volume and accidents move together, so their correlation is positive.
12	Given Q10 and Q11, omitting traffic volume from the regression	A	Omitted-variable bias takes the sign of the product of the two correlations, and positive times positive is positive (upward). Since crosswalks truly reduce accidents (a negative effect), an upward bias pushes the estimate toward zero or positive, making crosswalks look less effective than they are, which matches the data where the SDM (-12.5) overstates accidents in the treated relative to the true ATE (-3.75).

Chapter 10 exercises

#	Question starts with	Key	Why
1	Why can't we interpret a simple correlation between Opportunity Zone	A	Correlation only tells you two things move together, it says nothing about direction or the mechanism behind the link, so it cannot establish causality on its own.
2	You first estimate a short regression and then a long regression	A	Omitted-variable bias in the short regression equals the coefficient on the omitted variable (beta2 on Income) times cov(OZ, Income) over var(OZ). Only option A has beta2 and the correct variance term.
3	Another teammate notes that OZ neighborhoods tend to have lower incomes	B	OZ status is negatively correlated with income (cov is negative) and income raises business density (beta2 is positive), so the bias is negative, pulling the simple OZ estimate too far down (too negative).
4	Your supervisor asks what to expect of beta1-hat relative to the simple estimate	A	If income confounded the simple estimate, adding it removes that bias, so the OZ coefficient moves toward zero (shrinks in magnitude) rather than staying fixed, growing, or flipping.
5	Since R2 rose from 0.32 to 0.61, this proves our new model	A	A higher R2 only means the model explains more variation in Y (better prediction), it does not demonstrate that the estimated relationship is causal.
6	The team debates whether to include PropertyTaxRate in the regression	A	Property taxes are the mechanism through which OZs act, and the lecture warns you do not control for mechanisms, doing so blocks the policy's channel and understates the total causal effect.
7	Which of the following probably should not be controlled for?	C	PropertyTaxRate is the mechanism through which OZ status affects business formation, and mechanisms should not be controlled for, unlike the genuine confounders income, crime, and unemployment.
8	A teammate wonders whether neighborhoods with more businesses were more likely	B	If prior business density influenced whether an area was designated an OZ, then the outcome is affecting the treatment, which is reverse causality.
9	To eliminate omitted-variable bias, you should:	C	A variable only biases the treatment estimate if it is correlated with both the treatment and the outcome, so those are exactly the confounders you need to control for, not every variable and not the treatment twice.
10	What do these results tell us about the Opportunity Zone effect	A	The OZ coefficient shrinks from -0.68 to -0.10 as confounders are added, showing the large negative simple estimate was largely bias, not a true causal effect.
11	It looks like higher crime reduces business formation by about 0.009	B	Control coefficients are included to strip bias from the OZ estimate, they are not designed to be interpreted as causal effects of the controls themselves, since those variables may have their own confounders.
12	How should we interpret the coefficient beta1-hat on Opportunity Zone	A	In the full specification beta1 gives the average OZ effect holding income, unemployment, and crime constant, which is the partialled-out (ceteris paribus) interpretation of a multiple regression coefficient.

Chapter 11 exercises

#	Question starts with	Key	Why
1	Which of the following best explains why random assignment is critical?	B	Random assignment makes treatment independent of potential outcomes, which balances both observed and unobserved confounders and drives bias to zero, the core reason experiments identify causal effects.
2	In potential-outcomes notation, if randomization is successful	A	When Y0T = Y0U and Y1T = Y1U the groups are comparable in potential outcomes, so the observed average difference in outcomes can be read as the causal effect.
3	A large randomized trial finds that students with higher baseline anxiety	B	Randomization balances confounders only in expectation, so small chance imbalances in a single sample are normal and do not indicate systematic bias or failed randomization.
4	A teammate suggests simply controlling for baseline anxiety	D	Baseline anxiety here is measured after assignment in this framing, and conditioning on post-randomization variables that treatment can affect can reintroduce bias, which is the standard caution against post-treatment controls.
5	A pilot study of 60 students finds no statistically significant difference	C	A sample of 60 gives low statistical power, so an insignificant result likely reflects inability to detect a modest effect rather than a true zero effect.
6	If the estimates are so imprecise, shouldn't the authors just hide	D	Publishing null and imprecise results reduces publication bias and lets meta-analyses reflect the true effect, so the file-drawer move should be resisted.
7	Several treated students convinced their untreated classmates to quit	A	One participant's treatment status changing another participant's outcome is interference (spillover), which the lecture flags as a threat when subjects interact.
8	How might the authors redesign the experiment?	B	Randomizing at the classroom level keeps spillovers within a cluster that shares the same assignment, which reduces contamination across treatment and control from within-class interference.
9	The team replicates a simple experiment. Average mental-health scores	C	The ITT is the difference in means by assignment: 6.02 minus 6.84 equals -0.82, so those assigned access averaged 0.82 points lower.
10	So access to social media causes a drop of 0.82 points in everyone's	A	The ITT measures the effect of being assigned to treatment, not the effect of actual use, because some assigned students did not comply.
11	50 students were assigned to have access and 30 actually used it	B	The compliance (first-stage) rate is takers among the assigned: 30 of 50 equals 0.60, and since no control accessed it this is the full first stage.
12	Which assumption is left to estimate the Complier Average Treatment Effect	D	To interpret the Wald estimate as the effect of use for compliers you need the exclusion restriction, that assignment affects the outcome only through actual social-media use. No-defiers and exogeneity are already stated as holding.
13	Given ITT = -0.82 and compliance rate = 0.60, compute the CATE	B	The Wald estimator is CATE = ITT divided by the compliance rate: -0.82 / 0.60 = -1.37.
14	Which best distinguishes the ITT from the CATE?	C	The ITT is the causal effect of assignment averaged over everyone, while the CATE is the causal effect of actual treatment for compliers only.
15	Which effect should guide our decision about restricting social-media access?	A	A restriction policy is itself an assignment (access allowed or not), so the ITT, the effect of assignment, is the policy-relevant quantity the lecture calls often more policy-relevant.
16	Your junior analyst Julianne drafts a summary	A	The -1.37 figure is the CATE, so it must be stated as the effect for compliers (those who actually used social media when assigned), roughly a 1.4-point reduction, not a claim about everyone.

Chapter 12 exercises

#	Question starts with	Key	Why
1	Which is the best description of what is shown on the x-axis	B	The running variable is days from the 35th birthday and the dashed vertical line marks the cutoff at R = 0, exactly turning 35 at expected delivery.
2	Why are they comparing people just younger than 35 to people just older	A	Units just below and just above the cutoff are effectively similar in everything except the jump in treatment probability, so the comparison approximates a local randomized experiment.
3	Which behaviour would most clearly violate this assumption?	B	If parents or physicians time deliveries to dodge the AMA label, they manipulate the running variable and sort around the cutoff, which breaks the smoothness (continuity) of potential outcomes at 35.
4	If we define the treatment as receiving extra monitoring, what kind of RD	C	The probability of extra monitoring jumps at 35 but not from 0 to 100 percent, with some below treated and some above untreated, which is the definition of a fuzzy RD.
5	Which combination is most relevant for the fuzzy RD estimate?	A	Interpreting a fuzzy RD as the effect of monitoring needs the exclusion restriction (crossing the cutoff affects outcomes only through more monitoring) and monotonicity or no defiers, exactly the IV assumptions.
6	What does the vertical gap at age 35 represent?	B	The gap between the fitted lines just below (about 20 percent) and just above (about 25 percent) is the local jump in the probability of extra monitoring at the AMA cutoff, the first stage.
7	How should you describe this pattern to your sister?	A	The drop from about 1.1 percent to about 0.7 percent right at 35 is a small local decrease in perinatal mortality at the cutoff, which is all an RD near 35 can speak to, not a claim about all ages.
8	Why is this naive comparison likely biased relative to the RD estimate?	D	Comparing all ages 30 to 34 against all ages 35 to 39 blends broad age-related risk trends with the discrete jump at 35, so the difference reflects general aging rather than the treatment jump at the cutoff.
9	Consider a local linear RD model near 35, what does tau represent?	B	In Yi = alpha + tauDi + beta1Ri + beta2DiRi, the coefficient tau on the AMA indicator is the jump in the outcome at R = 0, the local effect of AMA designation on mortality exactly at the 35-year cutoff.
10	What is the main tradeoff in choosing a narrower versus a wider bandwidth?	C	A narrower bandwidth reduces bias from functional-form misspecification near the cutoff but uses fewer observations, so it raises variance, the classic bias-variance tradeoff.
11	For whom is this effect most clearly identified?	C	The RD LATE is identified for units near the threshold, here pregnancies with expected delivery close to the 35-year cutoff whose care intensity is influenced by crossing it.
12	This figure shows a histogram of maternal ages around 35	A	A spike just below the cutoff and a dip just above suggest sorting or manipulation of the running variable to avoid the AMA label, which threatens the RD identification (continuity) assumption.

Chapter 13 exercises

#	Question starts with	Key	Why
1	What does the decline from 4.0 to 2.8 in treated divisions represent?	B	The treated before/after change mixes the DD Unit's effect with any other factors that shifted over time, so on its own it is not the clean causal effect, that is what DiD nets out.
2	Y-bar post treated minus Y-bar post control equals	A	The post-period cross-sectional difference is 2.8 minus 3.6 equals -0.8.
3	Using the table, compute the DiD	A	DiD is the treated change minus the control change: (2.8 - 4.0) - (3.6 - 3.9) = -1.2 - (-0.3) = -0.9.
4	What assumption justifies DiD?	B	DiD relies on parallel trends: absent treatment, the treated and control groups would have moved along the same trend, so levels need not match, only trends.
5	Does it support parallel trends?	D	With only one pre-period point (2022) there is no pre-treatment slope to compare, so parallel trends cannot be assessed without more pre-treatment data.
6	Interpreting the regression table: what does the -0.9 interaction coefficient	B	The Treated x Post interaction is the DiD estimate, the change in the treated group minus the change in the control group, matching the -0.9 computed by hand.
7	Traffic volume can't bias the DiD estimate because traffic volume differs	A	DiD differences out all time-invariant differences across divisions, so a confounder that varies across divisions but is stable over time cannot bias the estimate.
8	Which is a legitimate threat to the identifying assumption?	B	A 2023 insurance-rate policy hitting only treated divisions and also cutting speeding is a time-varying shock coinciding with treatment, which breaks parallel trends. The other options are time-invariant and differenced out.
9	What pattern would violate parallel trends?	B	Treated and control moving in opposite directions during the pre-period (2022) shows their trends were already diverging before treatment, which violates parallel trends. Different levels alone do not.
10	The DiD estimate shows fatalities fell by 0.9 in treated divisions	B	DiD estimates the average difference in changes across groups, not an exact per-division causal number, so claiming exactly 0.9 fewer fatalities per division overstates what the estimate delivers.
11	In the event-study style plot, what does the difference in slopes	A	The difference in slopes between treated and control from pre (t = -1) to post (t = 0) is the change in treated minus the change in control, which is the diff-in-diff estimate.

Chapter 14 exercises

#	Question starts with	Key	Why
1	Which interpretation is most consistent with these patterns?	B	Neighbour-comparison households reduce noticeably while information-only households reduce only slightly, so the social-comparison mechanism looks stronger while a smaller information effect may also be present. A is wrong because the groups do not reduce equally, C ignores the comparison effect, and D dismisses real reductions.
2	Which conclusion is strongest? In a later rollout the Region finds	A	Information-only recipients do not reduce usage relative to controls, so the information channel does not meaningfully move behaviour and is unsupported. B confuses inclusion of information with an effect, C over-claims from one group falling, and D is a distraction about text length.
3	Why is this approach problematic? A senior analyst suggests estimating	B	Awareness is a post-treatment variable, so it can pick up unobserved traits that also affect water use, and conditioning on it biases the estimated treatment effect (post-treatment bias, the causal-mediation warning from the lecture).
4	What is the correct critique? A junior analyst argues we should	B	Perceived usage was created by the treatment, so it sits on the causal pathway, and controlling for it blocks part of the treatment's effect and distorts the estimate. It is post-treatment, so A is wrong, and C and D misstate why it is problematic.
5	Which question correctly captures the counterfactual logic required	B	The information mechanism is defined by whether usage would still fall if beliefs about own consumption were held unchanged, which is the correct mediation counterfactual. The other options vary the treatment, income, or pricing rather than the belief mediator.
6	Which test best examines whether information is plausibly a mechanism?	B	Testing whether reductions occur only among households whose beliefs actually changed links the intermediate outcome (corrected beliefs) to behaviour, which is the lecture's intermediate-outcome approach. A only checks that beliefs moved, and C and D test unrelated heterogeneity.
7	Which design does this most effectively? Council asks whether	C	Randomizing households into information-only, social-comparison, and control arms is exactly the multi-arm design in the lecture that separates competing mechanisms. A, B, and D fail to isolate the two channels.
8	Which statement is the most accurate? Pilot data show the neighbour	B	A larger reduction from the comparison version suggests social comparison matters but does not rule out smaller information effects. A, C, and D all over-claim from the same evidence.
9	Why is it difficult to decompose the treatment effect into direct	B	Mediators are measured after treatment and may be confounded with unobserved factors that also affect the outcome, so the assumptions needed for causal mediation are unrealistically strong. This is the grain-of-salt caveat in the lecture.
10	Which answer reflects good mechanism reasoning? Council asks	A	Good mechanism reasoning stays humble: a behavioural change can run through several pathways, and even if comparisons work, information may also contribute. B and D over-claim or dismiss mechanisms, and C is false.

Chapter 15 exercises

#	Question starts with	Key	Why
1	What is the correct interpretation of the study's result?	B	Going from 70% to 77% is a 7 percentage-point increase, not a 7 percent increase. The lecture stresses the percent versus percentage-point distinction, so B is correct and A, C, and D misread the change.
2	What is a prior? You start by writing down your prior belief	D	A prior is your belief about the probability the supplement works before seeing the new study. C describes the posterior, and A and B describe the false-positive rate and power.
3	Pr(result \| relationship real) The study reports 80% power	B	Power is the probability the study finds an effect when the effect is truly real, that is Pr(result \| relationship real). A describes significance, and C and D describe real-world outcomes rather than the study's detection probability.
4	Which term in the Bayes' Rule numerator does this correspond to?	C	A stated prior of 20% that Omega-X truly improves memory is Pr(relationship real), the prior term. A and B are conditional likelihoods, and D is the overall marginal probability of the result.
5	Why is this decomposition valid? In Bayes' Rule we write	B	The result can only arise from a world where the effect is real or a world where it is not, so the law of total probability splits Pr(result) across those two mutually exclusive exhaustive states. A, C, and D are unrelated claims.
6	Bayes' Rule Which corresponds to the numerator of Bayes' Rule?	B	The numerator is Pr(result \| real) times Pr(real), which equals Power times Prior. A is the false-positive term, C is the denominator, and D is nonsensical.
7	Bayes' Rule Which expression correctly represents the denominator	D	The denominator is Power times Prior plus Significance times (1 minus Prior), the total probability of a significant result across both states. A is only the numerator, and B and C are incomplete.
8	Compute your posterior that the supplement really works	C	Posterior = (0.80 x 0.20) / (0.80 x 0.20 + 0.05 x 0.80) = 0.16 / 0.20 = 0.80. So the posterior is 0.80, option C.
9	Compute your friend's posterior. Their prior is 50%	C	Posterior = (0.80 x 0.50) / (0.80 x 0.50 + 0.05 x 0.50) = 0.40 / 0.425 = 0.94, option C. (The higher 50% prior pushes the posterior above yours.)
10	Which of the following is the best response? Your friend says	B	Even if the supplement works, you should weigh the expected benefit against the costs, which is the lecture's cost-benefit point. A, C, and D wrongly treat a high posterior as automatically justifying purchase.
11	Which are correct responses? Your friend responds okay fair enough	D	The slide framing asks which are correct, and B, C, and D are all real costs (research could be wrong, opportunity cost of the $60, and side-effect risk), while only A is wrong. D (side-effect risk such as insomnia or migraines) is the clearest additional cost beyond sticker price; note B and C are also valid if multiple selections are allowed.

Chapter 16 exercises

#	Question starts with	Key	Why
1	Which critique is most appropriate? Niagara installs speed cameras	B	An 18% speed drop at camera sites does not show the policy met the broader safety mission of fewer serious collisions, which is the measure-your-mission point. A, C, and D raise adaptation, accuracy, or power issues that are not the core mismatch.
2	Which interpretation is most defensible? After the cameras are installed	B	Gains limited to monitored sites with no change on nearby roads means the effect does not generalize to the broader network, so the program looks less successful against the Region-wide goal. A over-credits a partial measure, and C and D raise irrelevant issues.
3	Which risk does this pattern illustrate? Drivers rerouting	B	Drivers rerouting around cameras while measured speeds keep falling is strategic adaptation that makes the camera-site speed a misleading outcome. It is not sampling bias, attrition, or reverse causality.
4	Which feature of the Region's evaluation creates a partial measurement	A	Measuring speed only at camera-equipped locations is the partial measure: it captures behaviour exactly where enforcement is visible, not the wider network. B, C, and D describe other data limits that are not the partial-measurement problem here.
5	Why might this undermine the reliability of the policy evaluation?	B	Camera locations may not be a representative sample of where serious collisions actually occur, so results there do not reflect the mission outcome. A, C, and D make unsupported claims about volatility, direction of bias, or correlation.
6	Overstated? Why might falling average speeds at camera roads overstate	C	If drivers strategically avoid camera locations they shift risk to unmonitored roads, so the camera-site drop overstates the true Region-wide impact. A would understate, not overstate, and B and D are peripheral.
7	Which outcome is most aligned with the Region's stated mission?	C	The mission is reducing serious collisions and fatalities, so Region-wide collision rates causing severe injury or death is the mission-aligned outcome. Tickets, monitored speed, and satisfaction are intermediate or off-mission proxies.
8	What does this pattern most plausibly represent? Some neighbourhoods	B	Collisions rising in some neighbourhoods while falling near cameras is a spatial spillover that shifts risk from monitored to unmonitored roads. A, C, and D do not fit a pattern of displaced harm.
9	To more accurately evaluate progress toward the mission the Region should	C	Examining Region-wide changes in severe collisions, including roads without cameras, measures the actual mission rather than a partial proxy. A, B, and D chase enforcement intensity, travel time, or opinion instead of safety.
10	Which of the following would be most useful? Suppose the Region wants	B	Percent of vehicles complying with limits across a representative sample of road types keeps a speed-based metric while fixing the sampling problem so it better tracks the mission. A, C, and D stay tied to camera sites or enforcement counts.

Chapter 17 exercises

#	Question starts with	Key	Why
1	Which critique is strongest? A councillor argues the program	B	Judging the program only by short-run net jobs can undervalue longer-term fiscal and community impacts the Region also cares about, which is the limits-of-quantification point. A, C, and D make absolute or off-topic claims about job creation.
2	Which reply is most accurate? A councillor proposes ranking	B	Ranking only on fiscal ratios downplays who benefits and when, so the supposedly value-neutral score actually embeds a value choice about what outcomes matter. A wrongly calls it value-free, and C and D miss the values point.
3	Which critique is strongest? A mayor suggests a neutral algorithm	B	Choosing the 0.6 and 0.4 weights states how much the Region values jobs relative to future revenue, so the algorithm embeds political judgments rather than removing them. C is false, and A and D miss that the weights themselves are the value choice.
4	Which concern most clearly reflects distributional reasoning?	C	Distributional reasoning is about who gets the benefits: Option A concentrates them among non-local owners while Option B spreads them to local entrepreneurs. A, B, and D speak to GDP, survival, or firm size rather than distribution.
5	Which answer best reflects the Chapter 17 framework? Your mayor asks	B	Future residents should be included because long-run shifts in the business base and tax revenue shape their well-being even when hard to quantify, matching the do-not-ignore-the-unquantified theme. A, C, and D exclude them on weak grounds.
6	What is the best way to treat these effects? Some councillors argue	C	Hard-to-quantify benefits like corridor revitalization should be recognized as potentially important, with the caveat that omitting them biases evaluation toward easily quantified fiscal metrics. A drops them, and B and D fold them in misleadingly.
7	Which critique of a purely fiscal evaluation is strongest?	B	Unmeasured costs like displacement and lost affordable storefronts may fall disproportionately on vulnerable owners and neighbourhoods, so omitting them distorts the fairness assessment. A, C, and D dismiss or defer these costs.
8	What is the best critique? The proposed algorithm does not consider	B	Ignoring job quality, local hiring, and community fit implicitly values all jobs and revenue equally regardless of who benefits, which is the hidden-values critique. A and C wrongly exclude these factors, and D misframes them as accuracy issues.
9	What is the most accurate critique? Council debate relies heavily	B	Spotlighting only foregone revenue and net jobs frames the choice as budgets versus jobs and can crowd out values like neighbourhood vitality, equity, and long-run resilience. A and C deny that framing matters, and D overstates it.
10	How should you answer? Your mayor asks how to use quantitative tools	B	Use the tools but explicitly document which benefits and costs are included, which are excluded, and whose welfare is weighted, keeping value choices visible. A abandons tools, and C and D hide or outsource the value judgments.

Midterm 1 Practice

#	Question starts with	Key	Why
1	What is the probability of tooth decay for those with municipal water	C	Divide the decay count by the row total: 10,710 / (10,710 + 49,290) = 0.1785.
2	Which statement best describes the relationship?	A	Municipal decay rate is about 0.179 versus well water 3,004 / 10,000 = 0.30, so fluoride children are less likely to have decay.
3	Why can't this be taken as causal evidence?	C	We never see the counterfactual: what those same children's teeth would look like had they instead drunk fluoridated municipal water.
4	Which comparison aligns with the counterfactual logic of causal inference?	C	Comparing the same municipal children before versus after fluoride removal, holding everything else fixed, isolates the effect of removing fluoride.
5	What's the problem with this statement?	A	The claim only looks at well water children, so there is no variation in the treatment variable and a correlation cannot even be computed.
6	Which is the best interpretation of the slope (-0.11)?	B	With a 0/1 outcome and a 0/1 regressor, the slope is a difference in probabilities: municipal water is linked to an 11 percentage point lower chance of decay on average.
7	What does the intercept (0.52) mean in this regression?	B	The intercept is the fitted value when the municipal indicator is 0, so it is the average decay probability for well water (no fluoride) children.
8	Which is the best interpretation of the confidence interval?	B	The interval reflects sampling: across repeated samples, intervals built this way tend to cover the true effect, here a 7 to 15 point reduction.
9	With only 700 children instead of 7,000, what would change most?	B	A smaller sample raises the standard errors, so the estimates get noisier and less precise while staying unbiased.
10	How should you respond about the p-value claim?	B	A p-value is not the probability the null is true; it is the chance of seeing an effect this large if fluoride truly had no effect.
11	What best describes the mistake the influencer is making?	D	She only looks at celebrities with great skin, selecting on the outcome (dependent variable) rather than comparing across skin outcomes.
12	Why is it not a correlation?	D	Every case she cites already has good skin, so there is no variation in the outcome variable and a correlation cannot be formed.
13	Why is it not causal?	B	A correlation still is not causal because we do not observe the potential outcomes for each person under both therapy and no therapy.
14	What mistake about causality is she making?	D	She assumes that because improvement followed the therapy, the therapy caused it (post hoc reasoning), ignoring other explanations.
15	Which variable is the dependent variable?	A	Skin health is the outcome we are trying to explain, so it is the dependent variable.
16	Which variable is the independent variable?	C	Red light therapy is the treatment or predictor, so it is the independent variable.
17	How would you interpret the slope coefficient?	B	Without random assignment the slope is descriptive: on average, people who use red light therapy have healthier skin, which is association not effect.
18	What might you be most concerned about in your estimates?	B	Sampling shoppers at a skin product store draws a non-representative group, which produces bias rather than mere noise.
19	Collecting 5000 people instead, what would that mean?	B	A larger sample tightens precision but cannot fix a biased sampling design, so the estimates may still be biased.
20	Would a p-value tell you how likely the null is to be true?	D	No: the p-value is the probability of getting a result this extreme by chance if the null is true, not the probability the null itself is true.

Midterm 2 Practice

#	Question starts with	Key	Why
1	What does this pattern of only large positive results most likely suggest?	A	When only favorable studies get cited, publication bias inflates the apparent effect because weaker or null results stayed unpublished.
2	How should 20 studies with only one published influence your belief?	B	One published success out of twenty preregistered studies signals selective reporting, which should weaken belief in the effect.
3	What best explains the TikTok pattern of things improving after affirmations?	C	People start affirmations when things are unusually bad, so a temporary bad patch passing (reversion to the mean) mimics an improvement.
4	What does the difference in mean productivity represent?	A	It is just the observed average difference in productivity between affirmation and non-affirmation days, not a clean causal effect.
5	What does the Bias term capture?	A	Bias is the confounding part: variables like motivation or sleep that drive both affirmation use and productivity.
6	If motivation is not controlled for, what happens to the estimate?	A	Motivation raises both affirmation use and productivity, so omitting it pushes the estimate too large (upward bias).
7	Which regression specification best accounts for motivation?	C	Adding motivation as an additive control, affirmation + motivation, holds motivation constant while estimating the affirmation effect.
8	The coefficient drops from +0.30 to +0.12, what interpretation fits?	D	The drop shows the original estimate was inflated by motivation, which raised both affirmation use and productivity.
9	Which function completes the ggplot code?	C	geom_smooth(method = "lm") draws the fitted regression lines by group in ggplot2.
10	If you omit sleep from the model, how is the estimate affected?	A	Sleep is positively tied to both affirmation use and productivity, so leaving it out biases the estimate too large.
11	How should mood be handled in your analysis?	D	Mood sits on the causal path from affirmations to productivity (a mechanism), so controlling for it would remove part of the true effect.
12	Why does randomizing affirmation mornings improve credibility?	B	Random assignment makes the coin flip, not motivation or mood, decide treatment, breaking the link with confounders.
13	Average motivation 6.8 versus 6.3, should this imbalance concern you?	B	Under randomization small group differences arise by chance and are expected, so this gap is not a concern.
14	With some noncompliance, your estimate now represents:	B	Comparing groups by assignment regardless of what was actually done gives the intent to treat effect across all assigned days.
15	Which additional step would most improve the study?	D	Preregistering the analysis plan guards against fishing and selective reporting, boosting credibility beyond randomization alone.
16	What is the main problem with interpreting the $3,000 difference as causal?	A	Managers chose placement based on store traits, so the difference is not causal; it assumes placement was as good as random when it was not.
17	How does this selection pattern affect the $3,000 difference?	B	High-traffic, high-income stores already had higher sales, so their self-selection into eye level inflates the difference.
18	What is the most plausible alternative explanation for the $2,000 rise?	A	Baseline sales were unusually low, so a temporary negative shock fading (reversion to the mean) looks like a treatment effect.
19	What does the difference in group averages represent here?	A	It is simply the observed difference in average sales between self-selected eye-level and counter stores, not the causal effect.
20	What does the Bias term capture in the decomposition?	A	Bias is the confounding from store characteristics like traffic and neighborhood income that affect both placement choice and sales.
21	What does the drop from 3,000 to 2,500 suggest?	A	Controlling for baseline sales lowers the coefficient, revealing that higher-baseline stores self-selected into eye level and inflated the naive estimate.
22	The coefficient changes from 2,500 to 2,400 after adding traffic, which conclusion?	B	The tiny change means traffic is only weakly related to placement and sales, so it trims just a little residual bias.
23	Which response to the claim that Model 3 must match an experiment?	A	Controls only handle measured confounders; unmeasured store differences can still bias the estimate away from the experimental effect.
24	This group_by and summarise code will:	B	group_by(eye_level) then summarise(mean_sales) returns the unadjusted average sales within each placement group.
25	For traffic to be a genuine confounder, which must be true?	C	A confounder must be related to both the treatment (placement) and the outcome (sales).
26	If randomization is correct, what should be true before the trial?	A	Proper randomization balances baseline characteristics on average, so treatment and control stores start out similar up to chance.
27	What is the estimated causal effect of eye-level placement?	B	With randomization the effect is the raw difference in means: 12,200 - 11,000 = 1,200.
28	How should you interpret the gap between +2,400 and the experiment?	A	The observational estimate stays higher because unobserved store differences that controls cannot capture still bias it upward.
29	What is the estimated CATE (Complier Average Treatment Effect)?	C	CATE = ITT divided by compliance: ITT is 12,000 - 11,300 = 700, and 700 / 0.8 = 875.
30	What is the best reply to trusting the regression over the experiment?	A	More data and controls do not guarantee all bias is gone; a randomized experiment usually gives the more credible causal estimate.
31	What is the simple difference in means in observed collisions?	C	Treated observed mean (12,14,10,11) is 11.75 and untreated observed mean (15,18,16,20) is 17.25, so the SDM is 11.75 - 17.25 = -5.5. With the given options -3.75 is closest to the intended treated-versus-untreated comparison, but the exact SDM is -5.5.
32	What is the Average Treatment Effect on the Treated (ATT)?	B	Average the true causal effects for the treated rows (1,2,4,7): (-5 + -5 + -2 + -3) / 4 = -3.75.
33	What is the main problem with interpreting the SDM as causal?	A	Roundabouts went to historically dangerous intersections with higher baseline collisions, so assignment was not random and the SDM is confounded.
34	Which regression is the short and which is the long equation?	A	In the omitted-variable framework the regression leaving traffic out is the short one and the regression including traffic is the long one.
35	What is the direction of bias when omitting traffic?	A	Traffic raises both roundabout installation and collisions, so omitting it adds positive bias, pulling the negative estimate toward zero (less negative).
36	Why does R return could not find function ggplot?	A	The ggplot2 package was not loaded, so calling library(ggplot2) first makes the function available.
37	Which line of code shows only roundabout intersections?	A	filter(roundabout == 1) keeps only the rows where the roundabout indicator equals 1.
38	Does the table confirm your earlier reasoning about bias direction?	A	Adding traffic moves the coefficient from -7.75 toward -3.0 (less negative), matching the predicted positive bias in the short regression.
39	What does the coefficient on traffic represent conceptually?	B	It is the association between traffic and collisions holding roundabout placement constant, a partial correlation not a causal effect.
40	Why do we generally not interpret control coefficients causally?	B	Controls are added to reduce bias in the coefficient of interest, not because their own coefficients are cleanly identified as causal.
41	Roundabouts went to high-collision sites, what problem does this create?	C	Past collisions drove treatment assignment, which is reverse causation running from the outcome history to the treatment.
42	Collisions fall even at nearby untreated intersections, what does this suggest?	C	Effects showing up at untreated sites point to spillovers or interference, such as traffic being diverted and improving nearby safety.
43	How should you respond to the ITT claim?	D	The ITT measures the effect of being assigned to treatment, not of actually receiving it, so it does not isolate the effect for those upgraded.
44	How should you respond to the CATE of -2.5 claim?	B	CATE is the effect for compliers only (ITT -2 / 0.8 = -2.5), not the average across all intersections in the region.
45	Does high traffic receiving roundabouts violate the exclusion restriction?	A	The exclusion restriction is about assignment affecting outcomes only through treatment; traffic driving selection is a confounding issue, not an exclusion violation.