Econometrics Tutor Columbia University New York, NY

Some of the cleanest causal evidence in economics comes from a simple move: compare the change in a group a policy hit to the change in a group it missed. That is difference-in-differences — it turns a law that lands in one place but not another into something close to a natural experiment. It is one of the first quasi-experimental designs Columbia students meet, and this page builds it from the four cell means up.

1 · Two differences, and why one alone misleads

You want the causal effect of a policy on an outcome Y. You have two groups — a treated group that got the policy and a control group that did not — each observed before and after.

Two obvious comparisons both fail. Compare the treated group before and after, and you credit the policy with whatever else changed over that period — the business cycle, a national trend. Compare the treated and control groups after the policy, and you credit it with whatever made the two groups differ in the first place — different regions, industries, histories.

Difference-in-differences takes both differences and subtracts one from the other. The common time trend cancels. The fixed gap between the groups cancels. What survives is the treated group’s extra movement — the part no trend and no fixed difference can explain.

2 · The 2×2 table

Line the four group-by-period means up. Take the change over time within each group, then difference those changes:

The order does not matter — difference across groups first instead, and you land on the same number. The control group’s change is doing one job here: it estimates the trend the treated group would have followed on its own. Subtract it, and δ̂ is the treated group’s departure from that trend.

3 · The parallel-trends assumption

Everything rests on one condition. Absent the policy, the treated and control groups would have moved in parallel. The control’s change is a valid stand-in for the treated group’s missing counterfactual only if the two trends would have tracked each other.

This assumption is about a world you never observe, so you cannot test it directly — but you can threaten it. Anything that bends one group’s underlying path differently around the policy date — a regional shock, a group-specific trend, selection into treatment — breaks parallel trends and biases δ̂. Levels may differ freely; it is the slopes that must match.

4 · The regression formulation

The 2×2 table has a regression twin that scales to real data and delivers a standard error. Estimate:

Read the coefficients off the cells. β₀ is the control group before, β₁ the fixed treated-minus-control gap, β₂ the common before-to-after trend. The interaction coefficient δ is the difference-in-differences — identical to the double difference from the table. With more groups and periods this generalises, and δ can also be recovered as a two-way fixed-effects estimate. One caveat: observations within a group are correlated, so standard errors are usually clustered at the group level. (Regression discontinuity instead identifies its effect from a jump at a cutoff — a different design.)

5 · Diagnosing pre-trends

You cannot test parallel trends after the policy, but you can look before it. With several pre-periods, estimate the treated-minus-control gap in each, normalised to the last period before treatment. Under parallel trends these pre-treatment coefficients should sit near zero — the groups tracked each other before anything happened. A coefficient that drifts from zero before the policy is a warning: the groups were already diverging, and δ̂ will absorb that as if it were the effect. This lead-and-lag plot is the event study, and examiners treat clean pre-period zeros as what earns a DiD its credibility.

Worked example — a minimum-wage rise in one state

One state raises its minimum wage. A neighbouring state does not. You measure average employment per fast-food restaurant, before and after.

Step 1 — The four cell means. Treated state: 20 before, 21 after. Control state: 22 before, 26 after.

Step 2 — The first naive read. Treated before-to-after: 21 − 20 = +1. Alone, this says the higher minimum wage raised employment. But both states’ economies grew over the period.

Step 3 — The second naive read. Treated minus control, after only: 21 − 26 = −5. Alone, this says the policy cost five jobs per restaurant. But the treated state already sat 2 below the control before anything changed.

Step 4 — The double difference. Treated change (+1) minus control change (+4) = −3. Equivalently, the after-gap (−5) minus the before-gap (−2) = −3. Both routes agree.

Step 5 — The counterfactual. Apply the control’s +4 trend to the treated group’s start: 20 + 4 = 24 — where the treated state would have landed under parallel trends. It reached 21. The gap, 21 − 24 = −3, is δ̂: the policy lowered employment by three per restaurant.

Step 6 — A pre-trend that breaks it. Add one earlier period: treated 18, control 19. Between then and “before”, the treated state gained 2 while the control gained 3 — a placebo DiD of 2 − 3 = −1. The groups were not parallel beforehand; the treated state was already losing ground at 1 per period.

Step 7 — Resolve and interpret. If that −1 trend persisted, the treated group’s counterfactual change is +4 − 1 = +3, so the honest counterfactual is 20 + 3 = 23 and the true effect is 21 − 23 = −2. The raw −3 splits cleanly: −1 was pre-existing divergence, −2 was the policy — exactly what a pre-trend check flags. Check the leads before you trust the estimate.

Can you compute the double difference and defend the parallel-trends assumption behind it, pre-trend check included? That combination is where difference-in-differences marks are won and lost. Rehearsing it on your own problem sets is exactly what a one-on-one econometrics tutor does with you. Book a trial session.

Practice

Q1. A city introduces a soda tax; a comparable city does not. Sales per store (cases): taxed city 500 before, 440 after; control city 480 before, 460 after. Compute the difference-in-differences estimate, and both naive differences.

Q2. A fitted DiD regression gives β₀ = 12, β₁ = 3, β₂ = −2, δ = −4. State (a) the estimated treatment effect, (b) the treated group’s mean before, (c) the predicted control mean after, (d) the predicted treated mean after.

Q3. Before a reform, the treated group’s outcome rose by 6 and the control’s by 4. Over the reform window, the treated rose by 10 and the control by 5. Find (a) the naive DiD, (b) the pre-trend (placebo) DiD, (c) the effect corrected for a persisting pre-trend.

Answers. Q1: DiD = (440 − 500) − (460 − 480) = −60 − (−20) = −40; naive before-after = −60; naive after cross-section = 440 − 460 = −20. Q2: (a) δ = −4; (b) β₀ + β₁ = 15; (c) β₀ + β₂ = 10; (d) β₀ + β₁ + β₂ + δ = 9. Q3: (a) 10 − 5 = +5; (b) 6 − 4 = +2, so parallel trends fails; (c) +5 − 2 = +3.

Key takeaways

• Difference the differences. Subtract the control’s change from the treated group’s; the common trend and the fixed group gap both cancel, leaving the treated group’s departure from trend.
• Parallel trends is the whole assumption. Levels may differ; slopes may not. The control’s change must be the trend the treated group would have followed on its own.
• The interaction coefficient is the estimate. In Y = β₀ + β₁Treat + β₂Post + δ(Treat×Post), the term δ equals the 2×2 double difference and comes with a (clustered) standard error.
• Check the leads. Pre-treatment event-study coefficients should be near zero. A nonzero pre-trend means part of δ̂ is pre-existing divergence, not the policy.

Why Columbia students choose our econometrics tutoring

• PhD econometricians, not generalists: sessions are led by tutors who have estimated these models in their own research, so the identification talk is first-hand.
• Aligned to your problem sets: tutors work from your course notation and your dataset — a 2×2 table by hand or a fixed-effects panel in Stata or R.
• The identifying assumption, drilled: the marks in a DiD question sit on parallel trends and the pre-trend check, rehearsed until they come out cleanly under exam pressure.

FAQ

Q: Is difference-in-differences the same as just comparing before and after?
A: No. A before-after comparison on the treated group alone credits the policy with every other change over the period. DiD subtracts the control group’s change to strip out that common trend.

Q: What is the parallel-trends assumption in plain terms?
A: That without the policy, the two groups would have moved in step. Their levels can differ; their trends cannot. If it holds, the control’s change is a valid stand-in for the treated group’s missing counterfactual.

Q: Can you test parallel trends?
A: Not directly — it concerns a counterfactual you never observe. You test it indirectly with an event study: estimate the group gap in several pre-treatment periods. Coefficients near zero support the assumption; a pre-trend undermines it.

Q: Where does the effect show up in the regression?
A: In the interaction term. Regress the outcome on a treated dummy, a post dummy, and their product; the product’s coefficient δ is the difference-in-differences estimate.

Q: Why are the standard errors clustered?
A: Observations within a group and over time are correlated, so treating them as independent understates the uncertainty. Clustering at the group level corrects the standard errors and leaves the estimate δ̂ untouched.

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