Signature
tau_adj = tau - delta_pre * s_post / s_pre
| Inputs | Definition | Unit |
|---|---|---|
tau | Two-by-two estimate across the policy (HE-FM-DID-001) | outcome unit |
delta_pre | Differential change between the groups over the pre-policy interval | outcome unit |
s_post | Time between the before and after periods used for tau | years |
s_pre | Time between the two pre-policy periods used for delta_pre, above zero | years |
tau_adj | Estimate after subtracting the extrapolated differential trend | outcome unit |
|---|
Function
Difference-in-differences estimate of a policy effect from treated and comparison groups
Maps outcomes observed before and after a policy in a group exposed to it and a group that is not to an estimate of the policy's average effect on the exposed: the change in the exposed group minus the change in the comparison group. Fixed differences between the groups and shocks common to both cancel. The estimate is causal only under parallel trends and no anticipation. The notation follows the Difference-in-Differences article.
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Implementations
Excel
Trend-adjusted estimate from named cells
With the unadjusted estimate in DiDEstimate, the placebo estimate in PlaceboDiD and the two interval lengths in PreYears and PostYears, the formula returns the adjusted estimate, held in AdjustedDiD.
=DiDEstimate-PlaceboDiD*PostYears/PreYears
Assumptions
Differential trend continues in a straight line
The difference in trends seen before the policy would have continued at the same rate per year. Rambachan and Roth bound how far the post-policy violation can deviate from such a linear extrapolation, which is preferable to a single corrected number.
Pre-policy periods free of anticipation
The placebo interval ends before the policy is announced, so delta_pre reflects trends and not early responses.
Worked examples
Admissions example with a two-year differential decline before the policy
The placebo estimate of minus 4.0 over 2021 to 2023, carried over the two years to 2025, explains a gap of 4.0, so the estimate of minus 3.0 becomes plus 1.0 admissions per 1,000, as in the article.
tau = -3; delta_pre = -4; s_pre = 2; s_post = 2; tau_adj = 1
Same pre-trend carried over a one-year post-policy interval
If the after period were one year after the before period, half of the pre-policy difference would be carried forward: minus 3.0 plus 2.0 gives minus 1.0 (computed here for illustration).
tau = -3; delta_pre = -4; s_pre = 2; s_post = 1; tau_adj = -1
No differential pre-trend leaves the estimate unchanged
With a placebo estimate of zero the adjusted estimate equals the unadjusted one.
tau = -3; delta_pre = 0; s_pre = 2; s_post = 2; tau_adj = -3
Common errors
Treating one trend-corrected number as the answer
In the article the correction turns a saving into a net cost, but it rests entirely on the linear extrapolation; reporting a bounded sensitivity analysis shows which conclusions survive.
Ignoring the pre-policy column
The 2023 to 2025 comparison alone gives minus 3.0 and cannot reveal the differential decline of 4.0 seen from 2021 to 2023.
Sources
Restrictions relative to a linear extrapolation of pre-treatment trend differences
Roth J, Sant'Anna PHC, Bilinski A, Poe J. What's trending in difference-in-differences? A synthesis of the recent econometrics literature. Journal of Econometrics. 2023;235(2):2218-2244 (read as arXiv 2201.01194 version 3). Section 4: researchers test for pre-trends, which often have low power; Rambachan and Roth bound post-treatment violations of parallel trends relative to pre-treatment violations, including restrictions on how far the violation can deviate from a linear extrapolation of the pre-treatment differences in trends, and construct confidence sets valid under them.
Canonical Identity
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