Appendix C — HonestDiD mathematical details

Formal statements of the two Rambachan-Roth-style bound families reported in §4.6, plus implementation notes.

C.1 Setting

Let $\delta_t$ denote the event-study coefficient at relative time $t$, with $\delta_{-1} = 0$ by convention (the reference period). Under no bias from pre-treatment trends, $\delta_t = 0$ for all $t < 0$ and $\delta_t = \tau_t$ for $t \ge 0$, where $\tau_t$ is the dynamic ATT at lag $t$. The pooled post-period ATT is

$$\tau = \frac{1}{T_\text{post}} \sum_{t=0}^{T_\text{post}-1} \tau_t$$

In practice, $\hat\delta_t$ for $t < 0$ is not identically zero; the pre-period event-study coefficients $\{\hat\delta_t\}_{t<0}$ carry the empirical signature of the parallel-trends violation. The Rambachan & Roth (2023) framework expresses the bias in the pooled post-period ATT as a function of the (unobservable) counterfactual trend $\delta_t^{CF}$ at post-period lags, and bounds the bias under a restriction on how much $\delta_t^{CF}$ can deviate from the extrapolated pre- trend.

The identified set for $\tau$ under restriction $R$ is

$$\mathcal{I}(R) = \left\{ \hat\tau - B(R) : B(R) \text{ consistent with } R \right\}$$

where $B(R)$ is the bias under $R$ and $\hat\tau$ is the observed pooled post-period coefficient.

C.2 Relative-magnitudes (RM-$\bar M$)

Restriction. Define $D_\text{pre} = \max_{t<0} |\hat\delta_t|$. The RM-$\bar M$ restriction asserts

$$|\delta_t^{CF}| \le \bar M \cdot D_\text{pre} \quad \forall\, t \ge 0$$

— the counterfactual post-period deviation is bounded by a multiple of the largest observed pre-period deviation. At $\bar M = 0$, the restriction is "parallel trends hold" (vanilla DiD); at $\bar M = 1$, "post-period deviations are no larger than the biggest pre-period deviation"; as $\bar M \to \infty$, the restriction imposes nothing and the identified set is the real line.

Bound. Under RM-$\bar M$, the bias is

$$B(\bar M) = \frac{1}{T_\text{post}} \sum_{t=0}^{T_\text{post}-1} \delta_t^{CF} \in [-\bar M D_\text{pre}, \bar M D_\text{pre}]$$

so the identified set is

$$\mathcal{I}(\bar M) = [\hat\tau - \bar M D_\text{pre},\, \hat\tau + \bar M D_\text{pre}]$$

This is a symmetric interval centered on the observed pooled ATT with half-width $\bar M D_\text{pre}$.

Breakdown point. The smallest $\bar M$ such that $\mathcal{I}(\bar M)$ includes zero:

$$\bar M^* = \frac{|\hat\tau|}{D_\text{pre}}$$

In our panel (per-unit event-study spec), $\hat\tau = -2.03$, $D_\text{pre} = 19.17$, so $\bar M^* = 0.106$. Reported as "breakdown at $\bar M = 0.5$" in the paper because we only sweep the grid $\{0, 0.5, 1.0, 1.5, 2.0\}$.

C.3 Linear-trend extrapolation (LT-$\bar M$)

The RM family is coarse: it bounds deviation in levels, not deviation from a clear trend signal. When the pre-period coefficients visibly trend (as ours do, with a fitted slope of $+0.48$ complaints per month), the relevant "counterfactual trend" is the continuation of that linear trend, not the absolute level.

Setup. Fit an OLS line to the pre-period coefficients:

$$\hat\delta_t = \hat a + \hat b \cdot t + \hat e_t \quad \text{for } t < 0$$

The extrapolated counterfactual trend at post-period lag $t \ge 0$ is $\hat a + \hat b \cdot t$. Define the maximum absolute pre- period residual as

$$R_\text{pre} = \max_{t < 0} |\hat e_t|$$

Restriction. The LT-$\bar M$ restriction asserts

$$|\delta_t^{CF} - (\hat a + \hat b t)| \le \bar M R_\text{pre} \quad \forall\, t \ge 0$$

— the counterfactual deviation from the extrapolated linear trend is bounded by a multiple of the largest pre-period residual. At $\bar M = 0$, we assume the linear extrapolation holds exactly; at $\bar M = 1$, we allow deviations up to the largest residual.

Bound. Under LT-$\bar M$, the trend-adjusted ATT is

$$\tau_\text{adj}(\bar M) = \hat\tau - \overline{(\hat a + \hat b t)}_{t \ge 0}$$

where the second term is the mean of the extrapolated linear trend over the post-period. The identified set is

$$\mathcal{I}(\bar M) = [\tau_\text{adj} - \bar M R_\text{pre},\, \tau_\text{adj} + \bar M R_\text{pre}]$$

— a symmetric interval centered on the trend-adjusted ATT with half-width $\bar M R_\text{pre}$.

Breakdown point.

$$\bar M^* = \frac{|\tau_\text{adj}|}{R_\text{pre}}$$

In our panel, $\tau_\text{adj} = -22.07$, $R_\text{pre} = 7.97$, so $\bar M^* = 2.77$. Reported as "breakdown beyond $\bar M = 2.0$" because the sweep grid tops out at $2.0$.

C.4 Why we report LT alongside RM

When the pre-period is stationary around zero with bounded jitter, RM is the natural restriction family — there is no trend to extrapolate, just noise. When the pre-period visibly trends, RM is overly conservative because it bounds deviation in levels (which grow with time under a linear trend) rather than deviation from the trend's extrapolation. Our pre-period is the second case, so LT is the more informative restriction family.

The LT family is not the original Rambachan-Roth smoothness (SD) restriction — SD bounds the second difference of the counterfactual trend, which grows the identified-set half-width quadratically in the post-period horizon and quickly becomes uninformative for long post-windows. LT is a simpler cousin that keeps the half-width constant by construction and is more interpretable for a 19-month post-window.

C.5 Interpretation for a non-econometrician

The LT-$\bar M$ restriction says: "imagine the pre-period trend kept going — what post-period deviation from that smooth extrapolation can you tolerate before the estimate becomes zero?" At $\bar M = 2$ we're asking "can we tolerate a deviation twice as large as the biggest pre-period jitter?" The answer is yes — the identified set is still entirely negative.

This is the concrete operational content of "the HonestDiD bounds say the parallel-trends rejection is not fatal." The pre-period trend was doing something — but whatever the counterfactual post- period trend was, within a reasonable bound on deviation from its linear extrapolation, the treatment effect is robustly negative.

C.6 Caveats and limitations of this implementation

1. Single-number summary of the pooled post-period ATT. We report bounds on the pooled ATT rather than on individual post-period event-study coefficients. The original Rambachan- Roth construction is lag-by-lag; pooled bounds are a summary.
2. Linear extrapolation assumption. LT assumes the pre-period trend is well-approximated by OLS. In our panel the fit is good (R² ≈ 0.6 for the linear OLS on 23 pre-period points), but a non-linear pre-trend would require a more flexible fit.
3. No confidence-interval coverage. The bounds are identified sets, not confidence intervals. Combining statistical uncertainty (the event-study SEs) with identification uncertainty (the $\bar M$ sensitivity) requires the Rambachan-Roth conditional-coverage construction, which we have not implemented in the Python port. A follow-up revision will either add this or swap in the R HonestDiD package through an rpy2 bridge.
4. Bounds on pooled ATT, not on per-cohort ATT. The per-cohort decomposition in §4.4 is NOT subjected to the HonestDiD analysis directly; it uses the standard TWFE confidence intervals. An extension would apply HonestDiD to each cohort's own event study separately.

The implementation lives in notebooks/11_honestdid_sensitivity.py and consumes the event-study CSV from notebooks/04_diagnostics.py. Regeneration is deterministic (same CSV → same bounds) and takes under a second on the cached artifacts.