You are tasked with estimating the causal impact of a carbon-tax that took effect in Province X in 2023 Q1, using Province Y (which never adopted the tax) as a control. Quarterly panel data on industrial CO₂ emissions from 2018 Q1 to 2025 Q4 are modeled with the two-way fixed-effects difference-in-differences specification
where Treat_i = 1 for Province X and 0 for Province Y, Post_t = 1 for all quarters t ≥ 2023 Q1 and 0 otherwise, γ_i are province fixed effects, and δ_t are quarter fixed effects.
Assuming (i) parallel pre-tax trends, (ii) no anticipation of the tax, and (iii) no time-varying province-specific confounders, which interpretation of β₃ is consistent with this model?
The baseline difference in average log-emissions between Provinces X and Y prior to 2023 Q1.
The average treatment effect on both provinces once province fixed effects are included, regardless of which province actually received the tax.
The average treatment effect on the treated (Province X) after accounting for province fixed effects and quarter fixed effects that remove time-invariant provincial factors and shocks common to both provinces.
The effect of unobserved time-varying confounders that differ across provinces but are unrelated to the carbon-tax policy.
The interaction coefficient β₃ is the difference-in-differences estimator. Because province fixed effects absorb all time-invariant differences between X and Y and quarter fixed effects absorb shocks common to both provinces, the remaining variation in the Treat × Post term isolates the change in log-emissions experienced only by the taxed province after the policy, relative to what would have happened in the absence of the tax. Under the parallel-trends and no-anticipation assumptions, this quantity equals the average treatment effect on the treated (ATT).
Choice "Average treatment effect on the treated, net of province-specific constants and common quarter shocks" is therefore correct.
The baseline level gap between provinces is captured by β₁, not by β₃, so the second option is wrong.
Time-varying confounders that differ by province would violate the identifying assumptions; β₃ does not estimate their effect, making the third option incorrect.
β₃ does not represent an average effect on all provinces; the control province never receives the treatment, so the last option is incorrect.
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What is the role of province fixed effects (γ_i) and quarter fixed effects (δ_t) in this model?
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What does the parallel pre-tax trends assumption ensure in difference-in-differences analysis?
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Why is β₃ interpreted as the average treatment effect on the treated (ATT) in this context?