CBS TWFE decomposition weights

The question: What is the effect on an industry of a 2001 tariff of l, compared to an industry that had no tariff at all?

Level weight: $w^{\text{lev}}(l) = \dfrac{(l - E[D])\cdot f_D(l)}{\text{Var}(D)}$

Six ingredients

$E[D]$

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$\text{Var}(D)$

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$P(D=0)$

—

$d_L$

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$d_U$

—

$f_D(l)$

—

Plug in the numbers at the chosen l

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Per-unit weight $W_i = \dfrac{(l_i - E[D]) \cdot \tfrac{1}{n}}{\text{Var}(D)}$ — same shape as $w^{\text{lev}}(l)$, with $\tfrac{1}{n}$ as the empirical density in place of $f_D(l)$. Sums to zero across the 155 industries.
Simulated first differences: $\Delta Y_i = 0.10 + 2\, l_i - 1.5\, l_i^{2} + \varepsilon_i, \;\; \varepsilon_i \sim N(0, 0.10^{2})$.
β^twfe = Σ_i W_i · ΔY_i = — — identical to the OLS slope.

Drag the l slider: industries with $l_i$ within ±bw of your chosen l get highlighted in orange on both scatters below.

The question: What is the effect on an industry, expressed per percentage point of 2001 tariff?

Coming soon

$w^{s}(l) = \dfrac{l \cdot (l - E[D]) \cdot f_D(l)}{\text{Var}(D)}$

Integrates to 1 but still has negative weights below the mean.

The question: For an industry already at tariff l, what is the effect of a small additional bump in tariff?

Coming soon

$w^{\text{acrt}}(l) = \dfrac{(E[D \mid D \ge l] - E[D])\cdot P(D \ge l)}{\text{Var}(D)}$

Non-negative and integrates to 1 — but not the dose density.

The question: Comparing a lower-tariff industry (l) to a higher-tariff industry (h), what is the outcome slope per percentage point of tariff gap?

Coming soon

$w^{2\times 2}(l, h) = \dfrac{(h - l)^{2} \cdot f_D(h) \cdot f_D(l)}{\text{Var}(D)}, \quad h > l$

Heatmap over the triangular region.