Forbidden Comparisons in Fixed-Effect Poisson Difference-in-Differences

For every unit $i=1,\ldots,N$ and period $t=1,\ldots,T$, $$E_{\mathcal P_N}\!\left[Y_{it}(0)\right] = b_{iN}\exp(\gamma_{t0}),$$ with the normalization $\gamma_{10}=0$.

For every unit $i=1,\ldots,N$ and period $t=1,\ldots,T$ with $D_{it}=1$, $$E_{\mathcal P_N}\!\left[Y_{it}(1)\right] = E_{\mathcal P_N}\!\left[Y_{it}(0)\right]\exp(\delta_{G_i t}).$$

The collapsed second-moment matrix $$\sum_{g\in\mathcal C}\sum_{t=1}^T q_{gt} r_{gt} r_{gt}'$$ is positive definite.

For every cohort-period cell $(g,t)$ with $D_{gt}=1$, $$\delta_{gt}>0.$$

The cohort set satisfies $$\{2,3,4,\infty\}\subseteq \mathcal C.$$

Let $T$ be positive and let $\mathcal C$ be a nonempty finite cohort set. Fix cohort shares $\pi_g\in(0,1)$, baseline levels $\bar b_g>0$, time effects $\gamma_t$, and a proportional-effect array $\delta$. Assume:

• (Rank.) The collapsed design satisfies Assumption A-5.
• (Cell.) The cohort $k$ belongs to $\mathcal C$, the period $s\in\{1,\ldots,T\}$, and the cohort-period cell is treated: $D_{ks}=1$.

Write $$\widetilde W_{ks}(\delta)$$ for the pseudo-true $q_{gt}\mu^\star_{gt}(\delta)$-weighted FWL residual of the treatment indicator in cell $(k,s)$, and define $$\mathcal E(\delta) = \sum_{g\in\mathcal C}\sum_{t=1}^T q_{gt}\,\mu^\star_{gt}(\delta)\, \widetilde W_{gt}(\delta)^2 .$$ Then $\mathcal E(\delta)>0$, and the one-coordinate path $$x\mapsto \beta^\star\!\bigl(\delta \text{ with } \delta_{ks}\text{ replaced by }x\bigr)$$ is differentiable at $x=\delta_{ks}$ with derivative $$\frac{ q_{ks}\,B_{ks}\exp(\delta_{ks})\,\widetilde W_{ks}(\delta) }{ \mathcal E(\delta) }.$$ Consequently, $$\frac{\partial \beta^\star(\delta)}{\partial \delta_{ks}}<0 \iff \widetilde W_{ks}(\delta)<0,\qquad \frac{\partial \beta^\star(\delta)}{\partial \delta_{ks}}=0 \iff \widetilde W_{ks}(\delta)=0,$$ and $$\frac{\partial \beta^\star(\delta)}{\partial \delta_{ks}}>0 \iff \widetilde W_{ks}(\delta)>0.$$

Fix $T$, a finite cohort set $\mathcal C$, sampling laws $\{\mathcal P_N\}_N$, potential outcomes $Y_{it}(d)$, positive unit baselines $b_{iN}$, cohort shares $\pi_g\in(0,1)$, positive limiting cohort baselines $\bar b_g$, untreated time components $\gamma_{t0}$, cohort-time log effects $\delta_{gt}$, and a scalar $\delta_0$. Suppose that

• (Untreated means.) Assumption A-2 holds.
• (Collapsed rank.) Assumption A-5 holds for $\mathcal C$ and $\{\pi_g\}_g$.
• (Common treated-cell effect.) For every $g\in\mathcal C$ and every period $t$, if $D_{gt}=1$, then $\delta_{gt}=\delta_0$.

Then $$\beta^\star(\delta)=\delta_0,$$ and, for every $g\in\mathcal C$ and every period $t$, $$\mu^\star_{gt}(\delta) = B_{gt}\exp(D_{gt}\delta_0).$$

Fix a panel length $T$, a finite cohort support $\mathcal C$, triangular-array sampling laws $\mathcal P_N$, potential outcomes $Y_{it}(d)$, deterministic cohort labels $G_i$, positive unit baselines $b_{iN}$, limiting shares $\pi_g\in(0,1)$, positive limiting cohort baselines $\bar b_g$, untreated time effects $\gamma_{t0}$, and log proportional effects $\delta_{gt}$. Suppose:

• (Horizon and support.) $T>0$, $T\ge 4$, the never-treated cohort belongs to $\mathcal C$, and every finite supported cohort has adoption date different from the first period.
• (Array support.) For every $N$ and every unit $i$, $G_i\in\mathcal C$.
• (Share limits.) Assumption A-1 holds for $(\mathcal C,G_i,\pi_g)$.
• (Untreated means.) Assumption A-2 holds for $(\mathcal P_N,Y_{it}(d),b_{iN},\gamma_{t0})$.
• (Baseline limits.) Assumption A-3 holds for $(\mathcal C,G_i,b_{iN},\bar b_g)$.
• (Proportional effects.) Assumption A-4 holds for $(\mathcal P_N,Y_{it}(d),G_i,\delta_{gt})$.
• (Rank.) Assumption A-5 holds for $(\mathcal C,\pi_g)$.
• (Frontier scope.) Assumption A-6 holds for $(T,\mathcal C)$.
• (Positive effects.) Assumption A-7 holds for $(\mathcal C,\delta_{gt})$.

Define $$h_{gt} = \pi_g\,\bar b_g\exp(\gamma_{t0})\exp(D_{gt}\delta_{gt}), \qquad R_g=\sum_{t=1}^T h_{gt}, \qquad C_t=\sum_{g\in\mathcal C} h_{gt},$$ $$M=\sum_{g\in\mathcal C}R_g, \qquad A=\sum_{g\in\mathcal C}\sum_{t=1}^T D_{gt}h_{gt}, \qquad \Phi = MA-\sum_{g\in\mathcal C}\sum_{t=1}^T D_{gt}R_gC_t .$$ Then $$\beta^\star(\delta)<0\iff \Phi<0,\qquad \beta^\star(\delta)=0\iff \Phi=0,\qquad 0<\beta^\star(\delta)\iff 0<\Phi .$$ Moreover, the primitive tuple induced by $(\pi_g,\bar b_g,\gamma_{t0},\delta_{gt})$ belongs to $\mathcal R_T$ if and only if $\Phi<0$. For every $x>1$ and $y>1$, in the four-period support $\{2,3,4,\infty\}$ with equal shares, unit limiting baselines, zero untreated time effects, multiplier $x$ on treated cells other than $(2,4)$, and multiplier $y$ at $(2,4)$, $$\Phi=\frac{5x^2+12x-2y-15}{16}.$$ Also, $$\frac{5(101/100)^2+12(101/100)-15}{2}=\frac{4441}{4000},$$ and the explicit witness $W_4$ has $$\Phi=-\frac{11559}{32000}.$$ Let $$m_{gt}=\bar b_g\exp(\gamma_{t0})\exp(D_{gt}\delta_{gt}), \qquad H=\{(g,t):g\in\mathcal C,\ D_{gt}=1\}.$$ For every $(g,t)\in H$, $$B^{obs}_{gt}=B_{gt},\qquad \tau_{gt}=\exp(\delta_{gt})-1,\qquad \omega_{gt}>0.$$ The counterfactual-share weights sum to one: $$\sum_{(g,t)\in H}\omega_{gt}=1.$$ The counterfactual-share PTT satisfies $$PTT = \frac{\sum_{(g,t)\in H} q_{gt}m_{gt}} {\sum_{(g,t)\in H} q_{gt}B^{obs}_{gt}} -1.$$ Finally, $$\Phi<0 \quad\Longrightarrow\quad \beta^\star(\delta)<0<PTT .$$

The four-cohort witness $W_4$ is constructed with cohort support $\mathcal C=\{2,3,4,\infty\}$, equal shares $\pi_g=1/4$, limiting baselines $\bar b_g=1$, untreated time component $\gamma_{t0}=0$, and proportional-effect log multipliers $$\delta_{gt}= \begin{cases} \log 4, & (g,t)=(2,4),\\ \log(101/100), & D_{gt}=1 \text{ and } (g,t)\neq(2,4),\\ 0, & D_{gt}=0 . \end{cases}$$ Then:

• (Region membership.) The primitive $W_4$ belongs to the sign-reversal region $\mathcal R_4$ of Definition P-5.
• (Positive effects.) Every treated supported cell has a strictly positive proportional-effect log multiplier: if $g\in\mathcal C$, $t\in\{1,2,3,4\}$, and $D_{gt}=1$, then $\delta_{gt}>0$.
• (Unique largest treated effect.) The treated cell $(2,4)$ has the strictly largest proportional-effect log multiplier: if $g\in\mathcal C$, $t\in\{1,2,3,4\}$, $D_{gt}=1$, and $(g,t)\neq(2,4)$, then $\delta_{gt}<\delta_{2,4}$.
• (Weighted FWL residuals.) At the no-effect vector, the weighted FWL residual $\widetilde W_{gt}$ of Definition P-4 equals $$\frac{1}{8} \begin{pmatrix} -3 & 3 & 1 & -1\\ -1 & -3 & 3 & 1\\ 1 & -1 & -3 & 3\\ 3 & 1 & -1 & -3 \end{pmatrix},$$ with rows $g=2,3,4,\infty$ and columns $t=1,2,3,4$. In particular, $\widetilde W_{2,4}=-1/8$.
• (Negative late-cell derivative at zero.) Holding all other effects at zero, $$\left.\frac{d}{dx}\, \beta^\star\!\left(\delta_{2,4}=x,\ \delta_{gt}=0\text{ for }(g,t)\neq(2,4)\right) \right|_{x=0} =-\frac{1}{10}.$$
• (Local negative derivative under positive effects.) There exists $\varepsilon>0$ such that, for every effect vector $\delta'$ satisfying $|\delta'_{gt}|<\varepsilon$ for all $g\in\mathcal C$ and $t\in\{1,2,3,4\}$, if $\delta'$ has strictly positive treated effects in the sense of Assumption A-7, then $$\left. \frac{d}{dx}\, \beta^\star\!\left(\delta' \text{ with } \delta'_{2,4}\text{ replaced by }x\right) \right|_{x=\delta'_{2,4}} <0 .$$