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Local Asymptotic Power of Honest Confidence Intervals
\maketitle
\begin{abstract}
Confidence intervals that are conservative against an untestable bias, called bias-aware or honest, are now standard in DiD, IV, RD, and factor-model settings. This paper characterises the local power of the tests they induce. Power is governed by the rate of the bias bound relative to the sampling rate, giving three regimes: when the bound vanishes faster than the standard error, conservatism is asymptotically free; when the two are of the same order it costs a bounded, explicit amount; and when the bound dominates, the typical case at the parametric rate, the honest test has \emph{zero} local power, failing to reject local alternatives with probability approaching one. A minimax argument shows this loss is intrinsic to honesty itself, not a property of any particular construction. No honest procedure recovers it, and the standard bias-aware interval is rate-optimal. Broadly, any confidence interval whose width fails to shrink fast enough has no local power in the interior of the set it traces out, and at best one-sided power at the boundary. Partial identification is the limiting case of this argument. Simulations and two empirical applications illustrate the three regimes. The practical recommendation is to report the half-width of the power ``dead zone'' alongside bias-aware intervals.
\end{abstract}
\medskip
\noindent\textbf{Keywords:} honest inference; bias-aware confidence intervals; local asymptotic power; dead zone; partial identification.\\
\noindent\textbf{JEL classification:} C12, C14, C23.
\section{Introduction}
A growing body of applied work reports confidence intervals that are deliberately conservative to untestable biases, where an estimate is widened by a worst-case bound on its bias so that coverage holds uniformly over a class of data-generating processes the data cannot rule out. These \emph{honest} or \emph{bias-aware} intervals are now standard for difference-in-differences under possible violations of parallel trends \citep{rambachan2023more}, instrumental variables under possible exclusion failures \citep{conley2012plausibly}, regression discontinuity \citep{armstrong2018optimal,armstrong2020simple}, and panels with potentially weak factors \citep{armstrong2022robust}.
This paper asks what such intervals cost in power, and shows the cost can be total.
Indeed, at the usual parametric rate, an honest interval may reject no local alternative with probability approaching one. The loss is not a flaw of any one procedure; a minimax argument shows it is intrinsic to honesty against an untestable bias.
The same argument cuts the other way: the standard bias-aware interval is rate-optimal among honest procedures, and the directional critical value of \citet{armstrong2018optimal} attains the sharp honest power frontier. The results therefore price the honesty requirement itself, rather than fault the constructions that meet it.
Taking a local-alternative approach makes the size--power trade-off of conservative tests precise, and the duality between the length of a confidence interval and the power of its dual test is itself classical \citep{pratt1961length}.
The organising idea is that this power loss is not, in essence, about conservatism or bias-awareness, but about \emph{interval width that does not shrink}. A level-$\alpha$ confidence interval whose width fails to vanish fast enough relative to the alternatives traces out a set of positive volume, and against any alternative in the interior of that set the dual test is powerless.
Only at the boundary can one-sided power survive. Partial identification is the limiting case, in which the width converges to a fixed identified set that does not shrink at all \citep[cf.][]{imbens2004confidence,stoye2009more}.
Honest inference under an untestable bias is the same phenomenon at a slower rate, the bound playing the role of the identified-set width. Seen this way, the three regimes developed below are one statement: local power degenerates exactly when the suitably rescaled interval width has positive limiting volume. Conservatism against an untestable bias is the leading economic instance, and the one developed in detail.
Assume the analyst possesses an estimator $\hat\beta$, and upper bound $\sup_\theta |\hat{B}|$, such that,
\begin{align*}
\hat\beta - \beta = \hat{B} + O_p(n^{-\epsilon}),
\end{align*}
where $\sup_\theta |\hat{B}|$ is a function of untestable user-defined parameters $\theta$ and may be much larger than the true bias, $\hat{B}$.
The term $O_p(n^{-\epsilon})$ represents error from noise, e.g. in a parametric model $\epsilon = 1/2$.
Confidence intervals built from such a worst-case bias bound are the \emph{bias-aware} intervals of \citet{armstrong2018optimal,armstrong2020simple} and \citet{imbens2019optimized}. That the bound is governed by $\theta$, which cannot be learned from the data and so must be supplied by the analyst, is fundamental rather than a deficiency of any particular estimator \citep{low1997nonparametric,bahadur1956nonexistence}.
For $\hat{B} = o_p\left(n^{-\epsilon}\right)$, there is negligible asymptotic bias.
For $\hat{B} \gtrsim c\cdot n^{-\epsilon}$, non-negligible asymptotic bias persists.
Note, the analyst only has $\sup_\theta |\hat{B}|$, not $\hat B$, so cannot use this for debiasing.
Also note, the bound $\sup_\theta |\hat{B}|$ may be very conservative - indeed this framework admits estimators with negligible true asymptotic bias, $\hat B$, but non-negligible bounds $\sup_\theta |\hat{B}|$.
The object that organises everything is the test's \emph{dead zone}: the set of true effects
so close to the hypothesised value that an honest test cannot tell them apart, rejecting them no
more often than its nominal size. A dead zone is the unavoidable price of the bound. Because the
worst-case $\sup_\theta|\hat B|$ lets the bias move the centre of $\hat\beta$ by as much as
$\pm\sup_\theta|\hat B|$, a true effect and the null produce the same data whenever they lie
within twice that bound of each other, exactly so in the leading applied settings, such that no honest test can separate them
(Figure~\ref{fig:deadzone}). The half-width of this zone, twice the bias bound, is the paper's headline disclosure.
A two-number example makes the accounting concrete.
Let the bound be $\sup_\theta|\hat B|=1$, suppose the realised bias also equals one, and ignore sampling noise, so $\hat\beta=\beta_0+1$.
The honest interval $\hat\beta\pm1=[\beta_0,\beta_0+2]$ covers the truth exactly at its edge and retains every value out to twice the bound above the truth: its far half defends not the world that generated the data but the observationally identical twin with bias $-1$, in which the truth would be $\beta_0+2$ (Figure~\ref{fig:deadzone}(b)).
Here the width is well spent, with bias genuinely at the bound, the full allowance is needed for coverage.
The genuine cost appears when the bound is conservative, sharpest at $\hat B=0$: the instrument in fact valid, the trend truly parallel.
Then $\hat\beta=\beta_0$, yet the interval $[\beta_0-1,\beta_0+1]$ still retains every value within one bound of the truth, spending the full width $2\sup_\theta|\hat B|$ where an oracle who knew $\hat B=0$ would spend only the sampling width $2z_{1-\alpha/2}\,se(\hat\beta)$.
Because the bias is untestable, the saving can never be identified or harvested: the analyst pays for the worst admissible world in every world, and the dead zone is that payment expressed in power rather than length.
Whether the zone matters asymptotically turns on how fast the bound shrinks relative to the local
alternatives. If the bound shrinks faster, the dead zone closes and conservatism is
asymptotically free; if the two shrink at the same rate, it persists at a fixed, explicit width
and the power loss is bounded (Theorem~\ref{thm:nondeg}, Proposition~\ref{prop:sharp}); and if
the bound shrinks more slowly, typical in the parametric case, the dead zone
engulfs every local alternative and local power is \emph{zero}
(Theorems~\ref{thm:LocalPower}--\ref{thm:LocalCI2}). A minimax argument shows the zone is
intrinsic to honesty itself, not to any one interval (Theorem~\ref{thm:lowerbound}). The same
picture recurs across the four settings: the factor-model, difference-in-differences, and
instrumental-variables cases are developed now, and regression discontinuity, the matched-rate
case, from Section~\ref{sect:primitives} onward.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.96\textwidth]{deadzone_concept.pdf}
\caption{The power dead zone. \textbf{(a)} The sharp honest power frontier
$\pi_\alpha(\xi)=\Phi\big((|\xi|-2\bar b)_+/\sigma-z_{1-\alpha}\big)$ of
Proposition~\ref{prop:sharp}: worst-case power (over the untestable bias) stays at the nominal
level $\alpha$ across the dead zone
$|\xi|\le 2\bar b$ (twice the bias bound) and rises only outside it; under any single bias the
undetected window has half-width $\bar b$, and the dead zone is the union of these windows. \textbf{(b)} Observational
equivalence. The estimator's centre can lie anywhere within $\pm\bar b$ of the true value, so the
null's centre range and the alternative's overlap whenever $\Delta<2\bar b$; the laws are
indistinguishable and the test is powerless. Detection requires $\Delta\ge 2\bar b$.}
\label{fig:deadzone}
\end{figure}
\begin{example}[Conservative to weak factor models]\label{rem:WeakFactors}
Consider the panel data linear regression model with fixed-effects setting studied in \cite{armstrong2022robust}.
Take data generated as
\begin{align*}
y_{it} = x_{it}'\beta + \Gamma_{it} + \varepsilon_{it}, &&
i\in \{1,\dots, N\}, t\in \{1,\dots, T\}.
\end{align*}
The term $\Gamma_{it}$ may be correlated with $x_{it}$.
For a given estimate $\hat\beta_{ba}$, an upper bound on absolute bias, $\bar B:= \sup_\theta|\hat B|$,
is defined by $\theta = R_{weak}$, i.e. the number of factors that are specified as weak.
The proposed confidence intervals in \cite{armstrong2022robust}, which take the fixed-length bias-aware form of \citet{donoho1994statistical}, are widened to,
\begin{align*}
\{\hat\beta_{ba} \pm [\bar B + z_{1-\alpha/2}\cdot se(\hat\beta_{ba})]\}
\end{align*}
where $z_{1-\alpha/2}$ is the $1-\alpha/2$ quantile of the Gaussian distribution.
When weak factors are truly present, $\{\bar{B}, \hat B\} = O_p(\max\{N,T\}/n)$, where $n = NT$.
However, when strong factors are truly present, $\bar{B} = O_p(\max\{N,T\}/n)$, whilst it can be that $\hat B = o_p(1/\sqrt{n})$.\footnote{Refer to \cite{Bai2009} Theorems 2 and 4 for parametric rate convergence of $\hat\beta$ under strong factors. }
\end{example}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.45\linewidth]{WeakPrateNonRejectionRegionsN=200T=24.pdf}
\includegraphics[width=0.45\linewidth]{StrongPrateNonRejectionRegionsN=200T=24.pdf}
\caption{
Weak vs. Strong factor model.
Rejections for $n^{-1/2}$ local alternative.
Left to Right: Weak vs. Strong factor.
Blue dashed line: Factor model;
Red solid line: Bias-aware.
Factor model over-rejects the truth with weak factors, nominally sized with strong factors; tight rejection regions in both.
}
\label{fig:IntroSimulation}
\end{figure}
Rejection rates for Example~\ref{rem:WeakFactors}, the debiased estimator in \cite{armstrong2022robust}, in Figure~\ref{fig:IntroSimulation} are taken from simulations in Section~\ref{sect:simFactor}.
The CIs conservative to weak factors have absolute upper bounds on bias of order $\bar{B} = O_p(\max\{N,T\}/n)$, where $n = NT$.
Take $T = c\cdot N^\tau$ for $c\in (0,1)$ and $\tau \in [0,1]$, i.e. $N = \max\{N,T\}$, such that $\sqrt{n}$ multiplied by bias is of order $\sqrt{n}\bar{B} = O_p(N^{(1 - \tau)/2})$.
Then, for $\tau = 1$, i.e. $N$ and $T$ proportional to each other, scaled sup bias is $\sqrt{n}\bar B = O_p(1)$, i.e. may be bounded by a constant but is not growing asymptotically.
This still leads to finite sample power loss, but with CIs converging still at the parametric rate.
However, when $\tau \neq 1$, scaled asymptotic sup bias is asymptotically divergent, and the test based on bias-aware confidence intervals retains no local asymptotic power when local power is defined at the $\sqrt{n}$ rate.
This power loss is without gain when true bias is asymptotically negligible, e.g. under a strong true factor model.
The same structure underlies two settings common in applied work.
\begin{example}[Conservative to violations of parallel trends]\label{ex:DiD}
Consider the honest difference-in-differences setting of \cite{rambachan2023more}.
An event study delivers pre- and post-treatment coefficients, with the post-period coefficient $\hat\beta_{post}$ targeting the treatment effect $\beta$,
\begin{align*}
\hat\beta_{post} - \beta = \hat B + O_p(1/\sqrt{n}),
\end{align*}
where $\hat B$ is the post-period violation of parallel trends.
The analyst supplies $\theta = M$, a bound on the second differences of the differential trend (the class $\Delta^{SD}(M)$), giving the worst-case bias $\bar B := \sup_\theta |\hat B|$ of exact order $M$, constant in $n$.
The honest interval widens to $\{\hat\beta_{post} \pm [\bar B + z_{1-\alpha/2}\cdot se(\hat\beta_{post})]\}$.
When parallel trends are violated up to the bound, $\{\bar B, \hat B\}$ are of exact order $M$.
However, when parallel trends hold, $\bar B$ is of exact order $M$ whilst $\hat B = o_p(1/\sqrt{n})$ ($\hat B = 0$ here).
\end{example}
\begin{example}[Conservative to invalid instruments]\label{ex:PlausExog}
Consider the plausibly-exogenous IV setting of \cite{conley2012plausibly}, with structural outcome $y_i = \beta x_i + \gamma z_i + \varepsilon_i$, instrument $z_i$, and first stage $x_i = \pi z_i + v_i$.
The IV estimator that imposes exogeneity satisfies
\begin{align*}
\hat\beta_{IV} - \beta = \hat B + O_p(1/\sqrt{n}), \qquad \hat B \xrightarrow{p} \gamma/\pi,
\end{align*}
the asymptotic bias from a true exclusion violation $\gamma$.
The analyst supplies $\theta = \bar\gamma$, a bound on $|\gamma|$, giving the worst-case bias $\bar B := \sup_\theta |\hat B|$ of exact order $\bar\gamma/\pi$, constant in $n$.
The union-of-intervals procedure of \cite{conley2012plausibly} widens to $\{\hat\beta_{IV} \pm [\bar B + z_{1-\alpha/2}\cdot se(\hat\beta_{IV})]\}$.
When the instrument is invalid up to the bound, then $\{\bar B, \hat B\}$ are of exact order $\bar\gamma/\pi$.
However, when the instrument is valid ($\gamma = 0$), $\bar B$ is of exact order $\bar\gamma/\pi$ whilst $\hat B = o_p(1/\sqrt{n})$ ($\hat B = 0$ here).
\end{example}
\subsection*{Related Literature\footnote{The literature on honest and bias-aware inference is vast. The references below are selective, and further suggestions are welcome.}}
The confidence intervals studied here belong to the literature on \emph{honest} and \emph{bias-aware} inference, in which an interval is widened by a worst-case bound on the estimator's bias so that coverage holds uniformly over a class of data-generating processes.
The feature central to this paper is that the asymptotic width of such intervals is governed by a user-specified bound that indexes this class and is, in general, not consistently estimable from the data.
In nonparametric regression this bound is a smoothness or magnitude constant. \citet{donoho1994statistical} characterises fixed-length intervals for linear functionals through the modulus of continuity over a convex parameter space, and \citet{li1989honest} shows that honest intervals must account for the bias that smoothing induces.
\citet{armstrong2018optimal} and \citet{armstrong2020simple} develop the modern bias-aware construction, replacing the usual critical value with one that incorporates the worst-case bias.
Applications and extensions include regression discontinuity with a discrete running variable \citep{kolesar2018inference}, finite-sample minimax regression discontinuity under a bound on the second derivative \citep{imbens2019optimized}, and inference under a bound on the magnitude of control coefficients \citep{armstrong2020biasaware,noack2024bias}.
The bias-aware intervals of \citet{armstrong2022robust} studied in Example~\ref{rem:WeakFactors} are of exactly this form, with the role of the user-specified bound played by the assumed number of weak factors.
That the governing bound cannot be learned from the data is fundamental rather than incidental.
\citet{low1997nonparametric} and \citet{cailow2004adaptation} establish that, within standard smoothness classes, one cannot adapt the length of an honest interval to the unknown regularity of the underlying function while maintaining uniform coverage.
The length is pinned down by the worst case, echoing the classical impossibility result of \citet{bahadur1956nonexistence}.
An early form of this floor appears in the one-dimensional subproblem at the heart of our limit experiment: in the bounded normal mean, the minimax half-length of a fixed-length $1-\alpha$ interval, affine or nonlinear, equals the bias bound itself whenever the bound does not exceed $z_{1-\alpha}$ noise units \citep[eq.~(9)]{donoho1994statistical}.
This is a length statement at the scale of one bias budget; its power counterpart, at twice that scale and uniform over all honest procedures, is the subject of Section~\ref{sect:main}.
Conservatism against an untestable bias therefore carries an unavoidable cost in length, and hence, as emphasised here, in power.
A parallel strand applies the same logic to inference that is robust to model misspecification, where the analyst specifies how far the model may depart from a baseline.
\citet{conley2012plausibly} relax the instrument exclusion restriction by a user-chosen amount and widen intervals accordingly; \citet{armstrong2021sensitivity} conduct sensitivity analysis in approximate moment-condition models under a bound on local misspecification; and \citet{andrews2017measuring} and \citet{bonhomme2022minimizing} study the sensitivity of estimates to misspecified moments. \citet{rambachan2023more} construct honest intervals for difference-in-differences that remain valid under bounded violations of parallel trends, and explicitly characterise the power of the resulting tests as a function of the assumed bound, a design motivated in part by the low power of conventional pre-trend tests \citep{roth2022pretest}.
The trade-off between interval length and the power of the associated test is classical.
\citet{pratt1961length} relates the expected length of a confidence interval to the power of the dual family of tests.
Not all of these procedures face the asymptotic power properties studied herein, and several make the trade-off explicit \citep[e.g.][]{li1989honest,low1997nonparametric,rambachan2023more,noack2024bias}.
The contribution of this paper is to make the trade-off precise in the bias-aware setting, where the relevant length is not a fixed multiple of the standard error but a user-specified bias bound whose asymptotic order may exceed the estimator's usual convergence rate.
The weak-factor example is convenient precisely because the power trade-off is a direct function of how conservative the analyst chooses to be, namely the number of factors specified as weak.
\section{Main Result}\label{sect:main}
This section begins with a simple and instructional result regarding power of the dual test under bias-aware intervals.
This establishes the breadth of the paper's contribution.
Subsection~\ref{sect:minimaxpowerLossHonest} establishes the power loss is an inherent feature of the honest requirement rather than the particular procedure proposed by bias-aware intervals.
That is, the power loss is unavoidable once honesty under undetectable bias is deemed strictly necessary.
Assume that $\bar B := \sup_\theta |\hat B|$ is bounded. Notation $\Theta_p $ implies an exact asymptotic bound: $A_n = \Theta_p(B_n) \iff A_n = O_p(B_n)$ and $A_n^{-1} = O_p(B_n^{-1})$.
\subsection{Power Loss of Bias-Aware Intervals}\label{sect:powerLossBACI}
\begin{assumption}\label{ass:AnalystEst}
Assume the analyst possesses $\hat{\beta}$, and $\bar B:= \sup_\theta |\hat B|$ such that,
\begin{align*}
\hat\beta - \beta = \hat{B} + O_p(n^{-\epsilon}).
\end{align*}
Moreover, assume the analyst possesses $se(\hat\beta) = \Theta_p(n^{-\epsilon})$.
\end{assumption}
\begin{assumption}\label{ass:BiasBound}
For sample size $n$, and $\bar B := \sup_\theta |\hat B|$, there is:
$\bar B = \Theta (n^{-\delta})$
for $\delta \in [0, \infty)$.
\end{assumption}
Assumption~\ref{ass:BiasBound} upper bounds the supremum over asymptotic bias to be constant, and admits settings with non-negligible asymptotic bias if $\delta < \epsilon$.
Local alternative hypotheses are defined per \cite{newey1994large} as follows:
\begin{definition}\label{def:LocalAlt}
For $\rho \in [0, \epsilon]$ the null and local alternative, with set $\Xi$ is:
\begin{align*}
H_0: \beta = \beta_0
&&
\textrm{vs. }
&&
H_{\rho}: \beta = \beta_0 + \xi\cdot n^{-\rho},
&&
\xi\in\Xi
.
\end{align*}
\end{definition}
Throughout the power analysis $\beta_0$ is the null value and the data are generated under it,
$\beta=\beta_0$: the null is correctly specified. It is then asked how well the dual test excludes local
perturbations of this truth, the local alternatives $\beta_0+\xi\,n^{-\rho}$, rejecting such a
value when it falls outside confidence interval $\mathcal B_n$. The local power is
$\Pr_{\beta_0}\{\,\beta_0+\xi\,n^{-\rho}\notin\mathcal B_n\,\}$, and zero local power means
$\mathcal B_n$ covers these perturbations with probability approaching one. Since the dead zone has
the same width about any value, the applications of Section~\ref{sect:applications} report it about
the null of no effect simply by convention.
Fixed alternative hypotheses are admitted in Definition~\ref{def:LocalAlt} by setting $\rho = 0$.
\begin{theorem}\label{thm:LocalCI}
Under Assumptions~\ref{ass:AnalystEst} and \ref{ass:BiasBound}, and $\rho \in [0, \epsilon]$ with $\delta<\rho$:
\begin{align*}
n^{\rho}\cdot(\bar B + z_{1-\alpha/2}\cdot se(\hat{\beta})) = \Theta_p(n^{\rho-\delta}) + O_p(1) \to \infty.
\end{align*}
Moreover, for $\rho \in [0, \epsilon]$ and $\delta\geq \rho$,
\begin{align*}
n^{\rho}\cdot(\bar B + z_{1-\alpha/2}\cdot se(\hat{\beta})) = O_p(1) .
\end{align*}
\end{theorem}
\begin{proof}
Theorem~\ref{thm:LocalCI} is immediate under the stated assumptions.
\end{proof}
Theorem~\ref{thm:LocalCI} states an immediate implication of scaling up confidence intervals at a rate faster than the worst-case bias shrinks: the intervals diverge.
Its hypothesis is satisfied by a range of honest procedures in current use \citep{armstrong2022robust,rambachan2023more,conley2012plausibly}, so the power loss is inherited wholesale rather than by any one construction.
Section~\ref{sect:minimaxpowerLossHonest} shows that the loss it induces is intrinsic to honesty under user defined biases.
The results of this subsection share one mechanism. After rescaling,
the local power function of any interval is the non-coverage function of its limiting rescaled
form. This is the organising claim of the introduction: the limit of the rescaled
width determines local power.
\begin{lemma}[Width--power duality]\label{lem:containment}
Let $\mathcal B_n=[\,l_n,u_n\,]$ be any interval and suppose that, with data generated at
the truth $\beta = \beta_0$, the rescaled endpoints converge jointly in distribution,
\begin{align*}
\big(\,n^{\rho}(l_n-\beta_0),\ n^{\rho}(u_n-\beta_0)\,\big)\ \xrightarrow{d}\ (L,U),
\end{align*}
with values in the extended reals. Then for every $\xi$ that is an atom of neither endpoint,
$\Pr\{L=\xi\}=\Pr\{U=\xi\}=0$,
\begin{align*}
\lim_{n\to\infty}\Pr\left\{A_n\notin\mathcal B_n\right\}\ =\ \Pr\left\{\xi\notin[L,U]\right\},
\qquad A_n=\beta_0+\xi\,n^{-\rho}.
\end{align*}
In particular every $\xi$ with $\Pr\{L\le\xi\le U\}\ge1-\alpha$ lies in the dead zone of
$\mathcal B_n$.
\end{lemma}
\begin{proof}
Write $L_n=n^{\rho}(l_n-\beta_0)$ and $U_n=n^{\rho}(u_n-\beta_0)$. Since $l_n\le u_n$, the
exclusion event is the disjoint union
$\{A_n\notin\mathcal B_n\}=\{L_n>\xi\}\cup\{U_n<\xi\}$. Convergence in distribution of the
endpoints means their distribution functions converge at every continuity point of the limits,
and $\Pr\{L=\xi\}=\Pr\{U=\xi\}=0$ makes $\xi$ such a point for both; hence
$\Pr\{L_n>\xi\}\to\Pr\{L>\xi\}$ and $\Pr\{U_n<\xi\}\to\Pr\{U<\xi\}$. The limit events can occur
together only if $U<\xi<L$, which $L\le U$ rules out (equality included), so adding the two
limits gives $\Pr\{\xi\notin[L,U]\}$. Only the marginal laws of the
endpoints enter.
\hfill
\end{proof}
Theorems~\ref{thm:LocalPower}--\ref{thm:nondeg} below are instances, differing only in the
limit $(L,U)$ of the bias-aware interval: $(-\infty,+\infty)$ in the divergent regime
$\delta<\rho$ (zero power); the constants $\big((c-1)\bar b,(c+1)\bar b\big)$ at the matched
rate with negligible noise (a step-function power); and Gaussian endpoints when the noise
survives. Proposition~\ref{prop:partialID} is the same statement with the identified set as
the limit.
Theorem~\ref{thm:LocalPower} states that whenever $\hat B = o_p(\bar B)$, local alternatives that are asymptotically small with respect to the bias bound, $\bar B$, are not rejected with probability approaching one (wpa1).
Whilst the set of local alternatives, $\Xi_n$, can grow with sample size, hence be unbounded in the limit, the rate this can grow is potentially very slow. Thus, it is easier to interpret this as a compact set.
\begin{theorem}\label{thm:LocalPower}
Make Assumptions~\ref{ass:AnalystEst} and \ref{ass:BiasBound}, and use Definition~\ref{def:LocalAlt}, with $\beta = \beta_0$, where the alternative set may grow with $n$: $\xi\in\Xi_n\subset\mathbb{R}$ with $\sup|\Xi_n| := \sup_{\xi\in\Xi_n}|\xi| = o(n^{\rho - \delta})$.
Let $A_n = \beta_0 + \xi\cdot n^{-\rho}$, and $\mathcal{B}_n = \left[\hat\beta \pm (\bar B + z_{1-\alpha/2}\cdot se(\hat{\beta}))\right]$ with $\rho\in[0,\epsilon)$ and $\delta < \rho$.
Then, when $\hat B = o_p(\bar B)$, for any $\xi\in \Xi_n$,
\begin{align*}
\lim_{n\to\infty}Pr\left\{A_n\in \mathcal{B}_n\right\} = 1.
\end{align*}
\end{theorem}
\begin{proof}
By definition, $A_n \in \mathcal{B}_n$ if and only if
\begin{align*}
\hat\beta - \left(\bar B + z_{1-\alpha/2}\cdot se(\hat{\beta})\right)
\leq A_n \leq
\hat\beta + \left(\bar B + z_{1-\alpha/2}\cdot se(\hat{\beta})\right).
\end{align*}
Substituting $A_n = \beta_0 + \xi\cdot n^{-\rho}$ and $\hat\beta - \beta_0 = \hat B + O_p(n^{-\epsilon})$ from
Assumption~\ref{ass:AnalystEst}, subtracting $\beta_0$, and multiplying through by $n^{\rho}$, this is
equivalent to $L_n \leq \xi \leq U_n$, where
\begin{align}
\label{eqn:containment condition}
L_n &= n^{\rho}(\hat B - \bar B) - z_{1-\alpha/2}\cdot n^{\rho} se(\hat{\beta}) + O_p(n^{\rho - \epsilon}),
\nonumber
\\
U_n &= n^{\rho}(\hat B + \bar B) + z_{1-\alpha/2}\cdot n^{\rho} se(\hat{\beta}) + O_p(n^{\rho - \epsilon}).
\end{align}
From $\delta < \rho < \epsilon$, these terms are, by Assumptions~\ref{ass:AnalystEst}
and \ref{ass:BiasBound},
\begin{align*}
n^{\rho}\bar B = \Theta(n^{\rho - \delta}) \to \infty,
\qquad
n^{\rho}\hat B = o_p\!\left(n^{\rho}\bar B\right),
\qquad
n^{\rho} se(\hat{\beta}) = \Theta_p(n^{\rho - \epsilon}) = o_p(1),
\end{align*}
since $\hat B = o_p(\bar B)$. Hence
$n^{\rho}(\hat B \pm \bar B) = \pm\, n^{\rho}\bar B\,(1 + o_p(1)) = \pm\,\Theta_p(n^{\rho - \delta})$, so that
\begin{align*}
U_n = \Theta_p(n^{\rho - \delta}) \to +\infty,
\qquad
L_n = -\Theta_p(n^{\rho - \delta}) \to -\infty.
\end{align*}
Every $\xi \in \Xi_n$ satisfies $|\xi| \leq \sup|\Xi_n| = o(n^{\rho - \delta})$, while $U_n$ and $-L_n$
are of exact order $n^{\rho - \delta}$. Therefore the interval $[L_n, U_n]$ contains
$[-\sup|\Xi_n|, \sup|\Xi_n|] \supseteq \{\xi\}$ with probability approaching one, and event
\eqref{eqn:containment condition} holds wpa1. Consequently,
\begin{align*}
\lim_{n\to\infty} \Pr\left\{ A_n \in \mathcal{B}_n \right\} = 1.
\end{align*}
\hfill
\end{proof}
Theorem~\ref{thm:LocalPower} shows zero local power under a possibly slack bound, $\hat B = o_p(\bar B)$.
In the sequel, Theorem~\ref{thm:LocalCI2} shows that this power loss persists even when the sign of the bias is correct and the bound is tight, provided the alternatives shrink faster than the bias bound, i.e. $\delta<\rho$.
Knowing the sign of bias is of course outside the capabilities of bias-aware confidence intervals, since then a simple debias could be performed. It is nonetheless a useful exercise to study power properties had these been pinned down in the confidence interval widenings.
\begin{theorem}\label{thm:LocalCI2}
Make the assumptions of Theorem~\ref{thm:LocalPower}, with the condition $\hat B = o_p(\bar B)$ replaced by,
\begin{align*}
\hat B = c\cdot\bar B + o_p(n^{-\epsilon}), \qquad c\in[-1,1].
\end{align*}
Assume $\delta<\rho$ and $\Xi$ compact. Then, for every fixed $\xi\in\Xi$ with $\xi\neq 0$,
\begin{align*}
\lim_{n\to\infty}\Pr\left\{A_n\in\mathcal{B}_n\right\}
=
\begin{cases}
1, & c\in(-1,1),
\\
\mathbbm{1}\{\operatorname{sign}(\xi)=c\}, & c\in\{-1,1\}.
\end{cases}
\end{align*}
\end{theorem}
\begin{proof}
By the proof of Theorem~\ref{thm:LocalPower}, $A_n\in\mathcal{B}_n$ if and only if $\xi\in[L_n,U_n]$, with $L_n,U_n$ as in \eqref{eqn:containment condition}. Substituting $\hat B = c\bar B + o_p(n^{-\epsilon})$, and using $n^{\rho}\bar B=\Theta(n^{\rho-\delta})\to\infty$ and $n^{\rho}se(\hat\beta)=\Theta_p(n^{\rho-\epsilon})=o_p(1)$ (since $\rho<\epsilon$),
\begin{align*}
L_n = (c-1)\,n^{\rho}\bar B + o_p(n^{\rho}\bar B),
\qquad
U_n = (c+1)\,n^{\rho}\bar B + o_p(n^{\rho}\bar B).
\end{align*}
If $c\in(-1,1)$ then $c-1<0<c+1$, so $L_n\to-\infty$ and $U_n\to+\infty$, and $[L_n,U_n]$ contains any fixed $\xi$ wpa1. If $c=1$ then $U_n\to+\infty$ while $(c-1)n^{\rho}\bar B=0$, leaving $L_n=-z_{1-\alpha/2}\,n^{\rho}se(\hat\beta)+O_p(n^{\rho-\epsilon})=o_p(1)$; hence $[L_n,U_n]$ covers a fixed $\xi$ wpa1 if and only if $\xi>0$. Symmetrically, if $c=-1$ then $[L_n,U_n]$ covers $\xi$ wpa1 if and only if $\xi<0$. In both boundary cases coverage occurs wpa1 exactly when $\operatorname{sign}(\xi)=c$.
\hfill
\end{proof}
Theorem~\ref{thm:LocalCI2} isolates the role of the bound's tightness.
When $\delta<\rho$ the scaled bound $n^{\rho}\bar B$ diverges, so even a bound that exactly equals the bias in the limit with known sign cannot restore two-sided power.
The test is asymptotically powerless against local alternatives on the same side as the bias, and detects only opposite-signed alternatives.
The best that pinning down bias can do is provide one-sided power.
For any $c\in(-1,1)$ all local power is lost, exactly as in Theorem~\ref{thm:LocalPower}.
The matched-rate case $\delta=\rho$, where the alternative shrinks at precisely the rate of the bias bound, is the knife edge between the divergent regime above and the standard regime $\delta>\rho$. Theorem~\ref{thm:nondeg} pins down the resulting non-degenerate power loss.
\begin{theorem}\label{thm:nondeg}
Make Assumptions~\ref{ass:AnalystEst} and \ref{ass:BiasBound}, and use Definition~\ref{def:LocalAlt} with $\beta=\beta_0$. Take $\delta=\rho\in[0,\epsilon]$, bias $\hat B = c\cdot\bar B + o_p(n^{-\epsilon})$ with $c\in[-1,1]$, and suppose $n^{\delta}\bar B\xrightarrow{p}\bar b\in(0,\infty)$. Fix $\xi\in\Xi$.
\begin{enumerate}[(i)]
\item If $\rho<\epsilon$, then $n^{\rho}se(\hat\beta)=o_p(1)$ and, for every $\xi\notin\{(c-1)\bar b,\,(c+1)\bar b\}$, the local power is degenerate,
\begin{align*}
\lim_{n\to\infty}\Pr\left\{A_n\notin\mathcal{B}_n\right\}
=
\mathbbm{1}\big\{\,\xi\notin[\,(c-1)\bar b,\;(c+1)\bar b\,]\,\big\}.
\end{align*}
\item If $\rho=\epsilon$, assuming ${n}^\epsilon\,(\hat\beta-\beta_0-\hat B)\xrightarrow{d}\mathcal{N}(0,\sigma^2)$ and ${n}^\epsilon\,se(\hat\beta)\xrightarrow{p}\sigma$; local power is:
\begin{align*}
\lim_{n\to\infty}\Pr\left\{A_n\notin\mathcal{B}_n\right\}
=
\Pr\big\{\,|\,G+c\bar b-\xi\,| > \bar b + z_{1-\alpha/2}\,\sigma\,\big\},
\qquad
G\sim\mathcal{N}(0,\sigma^2).
\end{align*}
\end{enumerate}
\end{theorem}
\begin{proof}
By the proof of Theorem~\ref{thm:LocalPower}, $A_n\in\mathcal{B}_n$ if and only if $\xi\in[L_n,U_n]$, where $[L_n,U_n]$ has centre $n^{\rho}\hat B + O_p(n^{\rho-\epsilon})$ and half-width $n^{\rho}\bar B + z_{1-\alpha/2}\,n^{\rho}se(\hat\beta)$. With $\delta=\rho$, $n^{\rho}\bar B = n^{\delta}\bar B\xrightarrow{p}\bar b$ and $n^{\rho}\hat B = c\,n^{\delta}\bar B + o_p(n^{\rho-\epsilon})\xrightarrow{p}c\bar b$.
(i) For $\rho<\epsilon$, $n^{\rho}se(\hat\beta)=\Theta_p(n^{\rho-\epsilon})=o_p(1)$ and the $O_p(n^{\rho-\epsilon})$ term vanishes, so $L_n\xrightarrow{p}(c-1)\bar b$ and $U_n\xrightarrow{p}(c+1)\bar b$. For fixed $\xi$ away from the endpoints, $\Pr\{\xi\in[L_n,U_n]\}\to\mathbbm{1}\{\xi\in[(c-1)\bar b,(c+1)\bar b]\}$, which gives the stated rejection probability.
(ii) For $\rho=\epsilon$, ${n}^\epsilon(\hat\beta-A_n)={n}^\epsilon\hat B-\xi+{n}^\epsilon(\hat\beta-\beta_0-\hat B)\xrightarrow{d}c\bar b-\xi+G$, while the half-width ${n}^\epsilon\bar B + z_{1-\alpha/2}{n}^\epsilon se(\hat\beta)\xrightarrow{p}\bar b+z_{1-\alpha/2}\sigma$. Acceptance is the event $|{n}^\epsilon(\hat\beta-A_n)|\le$ half-width, and the continuous mapping theorem yields the stated limit.
\hfill
\end{proof}
Theorem~\ref{thm:nondeg} displays the cost of conservatism.
When $\rho=\delta$ the bound neither washes out the alternative ($\delta<\rho$) nor becomes negligible ($\delta>\rho$), and contributes a constant $2\bar b$ to width.
When $\rho<\epsilon$ the sampling error is of smaller order, so the limiting power is a step function.
Alternatives in $[(c-1)\bar b,(c+1)\bar b]$ are never rejected, while every alternative outside is rejected wpa1.
This window has half-width $\bar b$ and depends on the realised bias $c$, which is untestable;
its union over $c\in[-1,1]$ is $[-2\bar b,2\bar b]$, the worst-case dead zone of
Section~\ref{sect:minimaxpowerLossHonest}.
Conservatism thus leads to several scenarios.
Widths exactly twice the limiting bound, against which the test has no power, and setting $\bar b=0$ recovers the oracle that rejects every $\xi\neq0$. At the rate $\rho=\epsilon$ the sampling error survives, and the power curve is smooth.
It is the usual Gaussian power function, with the rejection threshold inflated from $z_{1-\alpha/2}\sigma$ to $\bar b+z_{1-\alpha/2}\sigma$ and recentered by the bias $c\bar b$. The added $\bar b$ is the exact, finite power cost of bias-aware conservatism.
\begin{remark}[The cost in sample-size currency]\label{rem:pitman}
The dead zone can be restated in the currency of Pitman relative efficiency
\citep[Ch.~14]{vandervaart1998asymptotic}: the number of observations needed to reach a
prescribed level and power. Fix a level $\alpha$, a target power $\gamma$, an effect
$\Delta>0$ and a fixed bias bound $\bar B$ (the $\delta=\rho=0$ case), with
$se(\hat\beta)=\sigma n^{-\epsilon}$ and $z_\gamma=\Phi^{-1}(\gamma)$. Setting the one-sided
honest worst-case power $\Phi\big(n^{\epsilon}(\Delta-2\bar B)/\sigma-z_{1-\alpha}\big)$ equal
to $\gamma$ gives the minimal sample sizes
\begin{align*}
n_{\mathrm{honest}}\approx\Big[\frac{(z_{1-\alpha}+z_{\gamma})\,\sigma}{\Delta-2\bar B}\Big]^{1/\epsilon},
\qquad
n_{\mathrm{oracle}}\approx\Big[\frac{(z_{1-\alpha}+z_{\gamma})\,\sigma}{\Delta}\Big]^{1/\epsilon},
\qquad
\frac{n_{\mathrm{honest}}}{n_{\mathrm{oracle}}}\;\longrightarrow\;
\Big(\frac{\Delta}{\Delta-2\bar B}\Big)^{1/\epsilon},
\end{align*}
the square of $\Delta/(\Delta-2\bar B)$ at the parametric rate $\epsilon=1/2$. The ratio
diverges as $\Delta\downarrow2\bar B$, and for $\Delta\le2\bar B$ no sample size delivers
power $\gamma>\alpha$: the Pitman efficiency of honest inference relative to the oracle is
zero inside the dead zone. Unlike the classical comparison
\citep[Thm.~14.19]{vandervaart1998asymptotic}, where the sample-size ratio is a single number
independent of $(\alpha,\gamma)$ and the alternative, here it depends on $\Delta$, because
honesty translates the power curve rather than rescaling its slope, and that dependence is
the content. The burden of proof grows without bound as the effect approaches the width of
the dead zone.
\end{remark}
\subsection{A minimax lower bound for honest inference}\label{sect:minimaxpowerLossHonest}
The results above concern a single procedure, i.e. the bias-aware interval
$\mathcal{B}_n$ and the test it induces.
It is shown here that every test that
is honest faces the same three-regime power behaviour, so the loss
documented in Section~\ref{sect:powerLossBACI} is a property of the inference problem rather than of only the
bias-aware confidence interval widening.
Let $P_{\beta,\theta}^{(n)}$ denote the data-generating process with target coefficient
$\beta$ and untestable nuisance $\theta\in\mathcal{C}_n$, and let
$\mathcal{P}_n = \{P_{\beta,\theta}^{(n)}: \beta\in\mathbb{R},\, \theta\in\mathcal{C}_n\}$ be
the class over which coverage is required. A test is a measurable
$\phi_n\in\{0,1\}$, rejecting $H_0:\beta=\beta_0$ when $\phi_n=1$.
\begin{definition}\label{def:honest}
A test sequence $\{\phi_n\}$ is \emph{asymptotically honest at level $\alpha$} over
$\{\mathcal{P}_n\}$ if
\begin{align*}
\limsup_{n\to\infty}\ \sup_{\theta\in\mathcal{C}_n}\ \Pr_{P_{\beta_0,\theta}^{(n)}}\left\{\phi_n = 1\right\}\ \leq\ \alpha.
\end{align*}
\end{definition}
Definition~\ref{def:honest} is the hypothesis test counterpart of uniform coverage.
Recall that an interval $\mathcal{B}_n$ is honest over $\{\mathcal{P}_n\}$
at level $1-\alpha$ if it covers each process's own parameter value uniformly,
\begin{align}\label{eqn:standardHonest}
\liminf_{n\to\infty}\ \inf_{\beta\in\mathbb{R},\ \theta\in\mathcal{C}_n}\ \Pr_{P_{\beta,\theta}^{(n)}}\left\{\beta\in\mathcal{B}_n\right\}\ \geq\ 1-\alpha.
\end{align}
Pairing $\mathcal{B}_n$ with its dual tests $\phi_n(b):=\mathbbm{1}\{b\notin\mathcal{B}_n\}$,
covering the true value is non-rejection of the true value.
For every process,
$\Pr_{P_{\beta,\theta}}\{\beta\in\mathcal{B}_n\}=1-\Pr_{P_{\beta,\theta}}\{\phi_n(\beta)=1\}$.
Taking the infimum and using $\inf(1-x)=1-\sup x$ together with
$\liminf_n(1-s_n)=1-\limsup_n s_n$, condition \eqref{eqn:standardHonest} is equivalent to uniform
asymptotic size control of the dual tests over the whole class,
\begin{align}\label{eqn:dualHonest}
\limsup_{n\to\infty}\ \sup_{\beta\in\mathbb{R},\ \theta\in\mathcal{C}_n}\ \Pr_{P_{\beta,\theta}^{(n)}}\left\{\phi_n(\beta)=1\right\}\ \leq\ \alpha.
\end{align}
Retaining only the term $\beta=\beta_0$ in \eqref{eqn:dualHonest}, which can only lower the
supremum, yields Definition~\ref{def:honest} for the dual test
$\phi_n(\beta_0)=\mathbbm{1}\{\beta_0\notin\mathcal{B}_n\}$, and equally the size control of the
dual test $\phi_n(b)=\mathbbm{1}\{b\notin\mathcal{B}_n\}$ at any value $b$. Thus honest coverage
\eqref{eqn:standardHonest} implies Definition~\ref{def:honest}.
The proof of Theorem~\ref{thm:lowerbound} invokes this size control at the tested perturbation
$A_n$, coverage of $A_n$ under the confounding law whose parameter is $A_n$, which any honest
interval provides, so the bound applies to every honest interval.
The primitive driving the bound is that the class can confound a shift in
$\beta$ with a change in the nuisance.
This is the information-theoretic dual of the
worst-case bias $\bar B$, where $\bar B$ measures the largest shift in $\hat\beta$ the
class can induce.
The condition below measures the largest shift in $\beta$ the class can \emph{hide in the likelihood}.
It is a genuine assumption on the information in the class, and is not implied by Assumption~\ref{ass:AnalystEst}, which restricts only the bias of $\hat\beta$.
Define the total variation distance,
\begin{align*}
TV(P, Q) : = \sup_{A\in\mathcal{F}}|P(A) - Q(A)|.
\end{align*}
\begin{assumption}[Least-favorable confounding]\label{ass:confounding}
There is a function $\tau:[0,1]\to[0,1-\alpha)$, continuous and nondecreasing with
$\tau(0)=0$, such that for every $n$ and every $\Delta\in[0,2\bar B]$ there exist
$\theta_0,\theta_1\in\mathcal{C}_n$ with $P_{\beta_0,\theta_0}^{(n)},
P_{\beta_0+\Delta,\theta_1}^{(n)}\in\mathcal{P}_n$ and
\begin{align*}
TV\left(\,P_{\beta_0,\theta_0}^{(n)},P_{\beta_0+\Delta,\theta_1}^{(n)}\right)\ \leq\ \tau\!\left(\Delta/2\bar B\right) + o(1),
\end{align*}
where the $o(1)$ term is uniform in $\Delta$.
\end{assumption}
Assumption~\ref{ass:confounding} says that a coefficient shift of up to \emph{twice} the
worst-case bias can be absorbed by the class, with detectability governed by the fraction
$\Delta/2\bar B$ of that reach used. The factor of two is Figure~\ref{fig:deadzone}(b): the
adversary spends the bias in both laws, pushing the observable centre of the true law up by
$\bar B$ and that of the confounding law down by $\bar B$, so the two laws approach each other
from both sides. The observable centres are $\beta_0+m_0$ and $\beta_0+\Delta+m_1$ with
$|m_0|,|m_1|\le\bar B$, and they coincide only if
\begin{align*}
\Delta\ =\ m_0-m_1\ \le\ 2\bar B,
\end{align*}
with equality at $m_0=+\bar B$, $m_1=-\bar B$.
Beyond $2\bar B$ the centres cannot meet, and the
separation between the laws is forced positive.
A shift that is a vanishing fraction of $2\bar B$ is
asymptotically undetectable ($\tau(0)=0$), while a shift equal to the full reach remains
only borderline detectable, $\tau(1)<1-\alpha$, so that even a full-reach shift leaves
the power of any honest test bounded away from one. The restriction to $[0,1-\alpha)$ is
costless in the regular case: Section~\ref{sect:primitives} shows that when the confounding
is measured through the modulus, the Neyman--Pearson analogue of this bound has
$\tau(u)=\Phi(s(u)-z_{1-\alpha})-\alpha<1-\alpha$ because $\Phi<1$. The calibration in terms of the ratio
$\Delta/2\bar B$, rather than $\Delta n^{\epsilon}$, is exactly the statement that the
worst-case bias corresponds to $O(1)$ units of statistical separation, i.e. the
modulus-of-continuity relation $\bar B \asymp \omega(n^{-1/2})$ of
\citet{donoho1994statistical}.
The function $\tau$ bundles the shape of the modulus with
the two-point affinity and need not be specified beyond its qualitative properties for
the rate conclusions.
\begin{theorem}[Honest power is bounded]\label{thm:lowerbound}
Make Assumptions~\ref{ass:BiasBound}-\ref{ass:confounding}; use
$n^{\delta}\bar B\to\bar b\in(0,\infty)$; use Definition~\ref{def:LocalAlt}. Let
$\mathcal{B}_n$ be any honest interval \eqref{eqn:standardHonest}, with the data generated at
the truth $\beta_0$, and let its dual test exclude the local alternative
$A_n=\beta_0+\xi\,n^{-\rho}$ when $A_n\notin\mathcal{B}_n$. Fix $\xi\neq 0$.
\begin{enumerate}[(i)]
\item If $\delta<\rho$, there is a nuisance $\theta_0\in\mathcal{C}_n$ such that, under the
true law $P_{\beta_0,\theta_0}^{(n)}$,
\begin{align*}
\limsup_{n\to\infty}\ \Pr\left\{A_n\notin\mathcal{B}_n\right\}\ \leq\ \alpha.
\end{align*}
No honest interval excludes $n^{-\rho}$-perturbations of the truth with more than its size.
\item If $\delta=\rho$ and $|\xi|< 2\bar b$, the same construction yields $\theta_0$ with
\begin{align*}
\limsup_{n\to\infty}\ \Pr\left\{A_n\notin\mathcal{B}_n\right\}\ \leq\ \alpha + \tau\!\left(|\xi|/2\bar b\right)\ <\ 1.
\end{align*}
The power to exclude a perturbation of the truth is non-degenerately bounded away from one
throughout the window $|\xi|<2\bar b$.
\end{enumerate}
\end{theorem}
\begin{proof}
Fix $\xi$ and set $\Delta_n=|\xi|\,n^{-\rho}$; the argument is symmetric in the sign
of $\xi$. For any two probability measures $P_0,P_1$ and any test $\phi$,
\begin{align*}
\mathrm{E}_{P_1}\phi - \mathrm{E}_{P_0}\phi
= \int \phi\,(dP_1 - dP_0)
\leq \int_{dP_1>dP_0}(dP_1-dP_0)
= TV(P_1, P_0),
\end{align*}
since $0\leq\phi\leq1$. In both cases below $\Delta_n\leq2\bar B$ for $n$ large, so
Assumption~\ref{ass:confounding} supplies $\theta_0,\theta_1$ with
$P_{0,n}:=P_{\beta_0,\theta_0}^{(n)}$ the true law ($\beta=\beta_0$) and
$P_{1,n}:=P_{\beta_0+\Delta_n,\theta_1}^{(n)}\in\mathcal{P}_n$ a confounding law whose own
parameter is $A_n=\beta_0+\Delta_n$. Apply the displayed inequality to the dual test
$\phi=\mathbbm{1}\{A_n\notin\mathcal{B}_n\}$ with $P_1=P_{0,n}$ and $P_0=P_{1,n}$. Honesty of
$\mathcal{B}_n$ at $A_n$, the parameter of $P_{1,n}$, controls the size of this test,
$\mathrm{E}_{P_{1,n}}\phi\leq\alpha+o(1)$, so
\begin{align*}
\mathrm{E}_{P_{0,n}}\{A_n\notin\mathcal{B}_n\}
\ \leq\ \alpha + TV\left(P_{0,n}, P_{1,n}\right) + o(1)
\ \leq\ \alpha + \tau\!\left(\Delta_n/2\bar B\right) + o(1).
\end{align*}
Under $P_{0,n}$ the true value is $\beta_0$. For (i), $\delta<\rho$ gives
$\Delta_n/2\bar B=\Theta(n^{\delta-\rho})\to0$, and continuity with $\tau(0)=0$ yields the bound
$\alpha$. For (ii), $\delta=\rho$ gives $\Delta_n/2\bar B\to|\xi|/2\bar b$, and continuity yields
$\alpha+\tau(|\xi|/2\bar b)$, which is below one because $\tau(|\xi|/2\bar b)\le\tau(1)<1-\alpha$.
Taking $\theta_0$ as above completes the proof.
\hfill
\end{proof}
In the two leading location-type settings the confounding is not merely bounded but
\emph{exact}: the class contains observationally identical laws whose targets differ by up to
twice the bias bound, so Assumption~\ref{ass:confounding} holds with $\tau\equiv0$ and the
dead zone follows with no regularity conditions at all.
\begin{lemma}[Exact confounding]\label{lem:exactconf}
Assumption~\ref{ass:confounding} holds with $\tau\equiv0$ on $[0,1)$, the total variation
distance being exactly zero at every $n$, in:
\begin{enumerate}[(i)]
\item \emph{Plausibly-exogenous IV} (Example~\ref{ex:PlausExog}), with $\bar B=\bar\gamma/\pi$:
for $\Delta\in[0,2\bar\gamma/\pi]$ the laws $(\beta_0,\ \gamma=\bar\gamma,\ \varepsilon)$ and
$(\beta_0+\Delta,\ \gamma=\bar\gamma-\Delta\pi,\ \varepsilon'=\varepsilon-\Delta v)$ induce the
same joint law of $(y,x,z)$ and $|\bar\gamma-\Delta\pi|\le\bar\gamma$ exactly when $\Delta\le2\bar\gamma/\pi$. The redefinition of the structural error is admissible because
the class leaves the dependence between the structural and first-stage errors unrestricted;
\item \emph{Honest DiD over $\Delta^{SD}(M)$} (Example~\ref{ex:DiD}), with $\bar B=M$: the
post-period violation $\delta_1$ consistent with flat pre-periods ranges over $[-M,M]$, so the
identified set for $\beta$ has diameter $2M$. For $\Delta\in[0,2M]$ the laws with trend paths
$(0,\dots,0,\ \delta_1=M)$ at target $\beta_0$ and $(0,\dots,0,\ \delta_1=M-\Delta)$ at target
$\beta_0+\Delta$ have identical event-study means, both paths lie in $\Delta^{SD}(M)$ exactly
when $\Delta\le2M$, and the pre-period coefficients are zero under both. (Over the full class
the raw post coefficient's bias is unbounded since linear trends have zero second differences,
which is why the honest procedure extrapolates; the extrapolation estimator's worst-case bias
over the whole class is exactly $M$, half the identified-set diameter.)
\end{enumerate}
\end{lemma}
\begin{proof}
Immediate from the displayed constructions: in each case the two laws induce the same
distribution of the data, so $TV=0$, while the targets differ by $\Delta$.
\hfill
\end{proof}
\begin{corollary}[Flat dead zone]\label{cor:flatdz}
In the settings of Lemma~\ref{lem:exactconf}, Theorem~\ref{thm:lowerbound}(ii) with
$\tau\equiv0$ gives, for every honest interval and every $|\xi|<2\bar b$,
$\limsup_n\Pr\{A_n\notin\mathcal{B}_n\}\le\alpha$: the minimax dead zone has half-width exactly
$2\bar b$ (the bias-aware interval itself excludes every $|\xi|>2\bar b$ wpa1 by
Theorem~\ref{thm:nondeg}), and the conclusion requires no limit experiment and no regularity
beyond the class itself.
\end{corollary}
The dead zone in these settings is thus a statement about observational equivalence, not about
asymptotic approximation: within $2\bar b$ of the truth there are laws the data literally cannot
tell apart. Equivalently, $2\bar B$ is the diameter of the local identified set.
The bound in part (ii) of Theorem~\ref{thm:lowerbound} is sharp under regularity, and the sharp
value is an explicit Gaussian power function with a power ``dead zone''. Consider the
noise-surviving matched rate $\delta=\rho=\epsilon$ and localise $\beta=\beta_0+h\,n^{-\epsilon}$
where $h\in\mathbb{R}$ is the local coefficient of Definition~\ref{def:LocalAlt} treated as a
free coordinate that indexes the experiment.
The perturbation of interest is the point
$h=\xi$.
\begin{assumption}[Gaussian least-favourable limit]\label{ass:LE}
There is a nuisance path $g\mapsto\theta_n(g)\in\mathcal{C}_n$ and a scale
$\sigma\in(0,\infty)$ such that the localised experiments
$\{P^{(n)}_{\beta_0+h\,n^{-\epsilon},\theta_n(g)}\}$ converge, in the sense of
\citet{lecam1973convergence} \citep[see also][Chs.~9 and~15]{vandervaart1998asymptotic}, to the bounded-normal-mean experiment of observing
$X\sim N(h+m,\sigma^2)$, the nuisance contributing a free location shift
$m\in[-\bar b,\bar b]$ with $\bar b=\lim_n n^{\delta}\bar B$; and $\hat\beta$ is
asymptotically sufficient, $n^{\epsilon}(\hat\beta-\beta_0)\xrightarrow{d}X$.
\end{assumption}
Assumption~\ref{ass:LE} is the exact-confounding form of
Assumption~\ref{ass:confounding}.
In the limit a shift in $\beta$ of up to $\bar B$ is
matched by a nuisance shift, so the data cannot separate the two, and honesty reduces to
size control over the contaminated null mean,
$\sup_{|\nu|\le\bar b}\Pr_{N(\nu,\sigma^2)}\{\phi=1\}\le\alpha$.
In the location settings of Lemma~\ref{lem:exactconf} it
holds exactly: the confounding laws coincide, so the estimator carries all the usable
information and the sufficiency clause is innocuous. In genuinely nonparametric settings such
as regression discontinuity it does \emph{not} hold for the full data: the confounding there is
approximate rather than exact and the separation $s(u)$ of Section~\ref{sect:primitives} is
strictly positive, so the estimator is not asymptotically sufficient, and a directional
honest test tuned to a specific perturbation can retain some power inside the window. In those
settings Proposition~\ref{prop:sharp} describes the experiment generated by the estimator,
hence every procedure in the bias-aware (affine) class, while the frontier over all honest
tests is the smooth $\Phi(s(\xi)-z_{1-\alpha})$ of Proposition~\ref{prop:confprim}.
\begin{proposition}[Sharp honest power]\label{prop:sharp}
Under Assumption~\ref{ass:LE}, let $s(\xi)=(|\xi|-2\bar b)_+/\sigma$. With data at truth
$\beta_0$, every honest interval excludes the perturbation $\beta_0+\xi n^{-\epsilon}$ with
worst-case power (over the nuisance) no larger than
$\pi_\alpha(\xi):=\Phi\!\big(s(\xi)-z_{1-\alpha}\big)$. In particular
$\pi_\alpha(\xi)=\alpha$ for every $|\xi|\le 2\bar b$: no honest interval excludes a
perturbation within a window of half-width $2\bar b$ about the truth.
\end{proposition}
\begin{proof}
By \citet{lecam1973convergence} it suffices to argue in the limit $X\sim N(\nu,\sigma^2)$,
$\nu=h+m$, $|m|\le\bar b$ (the truth $\beta_0$ sits at $h=0$ by the centring of the localisation, a change of origin only); take $\xi>0$. If $\xi\le 2\bar b$, the nuisance $m'=-\bar b$
gives alternative mean $\nu_1=\xi-\bar b\in[-\bar b,\bar b]$, an admissible null mean, so
honesty forces power $\le\alpha$. If $\xi>2\bar b$, the closest null and alternative
means $\nu_0=\bar b$ and $\nu_1=\xi-\bar b$ are separated by $s(\xi)\sigma$, and the
Neyman--Pearson lemma bounds the power by $\Phi(s(\xi)-z_{1-\alpha})$, attained by the
honest test $\mathbbm{1}\{X>\bar b+z_{1-\alpha}\sigma\}$. The two-sided case differs by a
negligible tail \citep[Le~Cam's two-point method;][]{tsybakov2009}.
\hfill
\end{proof}
In the same limit the bias-aware interval excludes the perturbation $\xi$ with worst-case power
$\Phi(s(\xi)-z_{1-\alpha/2})$.
This reproduces the dead zone
$|\xi|\le 2\bar b$ exactly and matches the frontier $\pi_\alpha$ up to the fixed
critical-value gap $z_{1-\alpha/2}-z_{1-\alpha}$, which the directional value
$\mathrm{cv}_\alpha$ of \citet{armstrong2018optimal} removes (there
$\mathrm{cv}_\alpha(t)-t\to z_{1-\alpha}$).
\begin{definition}[Power dead zone]\label{def:deadzone}
With the data at the truth $\beta_0$ and the bias untestable, the \emph{dead zone} of an honest
interval $\mathcal{B}_n$ is the set of local perturbations of the truth whose exclusion cannot be
guaranteed beyond size: with $A_n=\beta_0+\xi\,n^{-\rho}$,
\begin{align*}
\mathcal{D}\;=\;\Big\{\,\xi:\ \limsup_{n\to\infty}\ \inf_{\theta\in\mathcal C_n}\
\Pr_{P^{(n)}_{\beta_0,\theta}}\{A_n\notin\mathcal{B}_n\}\le\alpha\,\Big\}.
\end{align*}
Inside $\mathcal{D}$ there is an admissible bias under which the perturbation is excluded no more
often than the size of the dual test, so its exclusion cannot be promised to carry evidence.
\end{definition}
The worst case over the nuisance is the operative reading precisely because the bias is
untestable. Under any \emph{single} law with bias $c\bar b$ the undetected window is
$[(c-1)\bar b,(c+1)\bar b]$ by Theorem~\ref{thm:nondeg}(i), which has half-width $\bar b$ about the
bias-shifted centre, which is what the shaded regions of the simulations realise. The dead zone
is the union of these windows over the admissible biases,
\begin{align*}
\bigcup_{c\in[-1,1]}\big[(c-1)\bar b,\ (c+1)\bar b\big]\;=\;[-2\bar b,\ 2\bar b],
\end{align*}
the least-favourable bias being chosen per perturbation ($m_0=\operatorname{sign}(\xi)\,\bar b$):
a perturbation such as $\xi=\tfrac32\bar b$ is excluded wpa1 under a zero-bias law, yet never
excluded under any bias $c\ge\tfrac12$; since the analyst cannot know $c$, only the union can be
promised. By Proposition~\ref{prop:sharp}, at the matched rate the dead zone of every honest
interval contains the window $|\xi|\le 2\bar b$, and for the bias-aware interval with
$\rho<\epsilon$ it is exactly $[-2\bar b,2\bar b]$. In the units
of the estimand this is a half-width $2\bar B$ (as $\bar b=\lim_n n^{\delta}\bar B$), the
diameter of the local identified set.
The scope follows Lemma~\ref{lem:exactconf}: in the exactly confounded settings the statement
binds every honest procedure with no further conditions, while in the approximately confounded
settings it binds the bias-aware class, with the all-tests frontier given by
Proposition~\ref{prop:confprim}. Reporting this single
number alongside the interval is the disclosure recommended in Section~\ref{sect:discussion}.
Figure~\ref{fig:unionzone} makes the per-law/union distinction visible in a single plot.
Sweeping the realised bias over its admissible range produces the family of sliding windows,
whose lower envelope, the empirical counterpart of the $\inf_\theta$ in
Definition~\ref{def:deadzone}, is flat on exactly $\pm2\bar b$. In contrast, the \emph{average}
over a uniformly drawn bias first reaches one at the zone's edge: a randomly drawn
bias locates $2\bar b$, but only the worst case is flat, because the dead zone is an infimum,
not a mean.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.75\textwidth]{plausexogUnionZone.pdf}
\caption{The union dead zone realised by simulation: plausibly exogenous IV as in
Section~\ref{sect:simulations}, with $\bar\gamma=0.3$, $\pi=1$ (so $\bar b=0.3$) and $n=1000$.
Thin grey: rejection rates under
$\gamma=c\bar\gamma$, $c\in[-1,1]$ with per-law half-width $\bar b$,
sliding with the realised bias. Black: pointwise minimums over $c$, the empirical
counterpart of Definition~\ref{def:deadzone}, flat on exactly $\pm2\bar b$
and rising outside, as in Proposition~\ref{prop:sharp}. Red dashed: average over
biases which first reaches one
at the zone edge $\pm2\bar b$. Shaded region is where envelope is at most $\alpha$, spanning $[-0.6,0.6]=\pm2\bar b$.}
\label{fig:unionzone}
\end{figure}
This is the dead zone of Figure~\ref{fig:deadzone}, now made sharp.
The frontier $\pi_\alpha$
is flat at $\alpha$ throughout $|\xi|\le 2\bar b$, and its half-width is $2\bar b$ rather than
$\bar b$ precisely because the least-favourable bias shifts the centre by $\bar b$ under the null
and again under the alternative (Figure~\ref{fig:deadzone}(b)).
\begin{corollary}[Rate-optimality of the bias-aware interval]\label{cor:rateoptimal}
Under the assumptions of Theorem~\ref{thm:lowerbound}, the bias-aware interval
$\mathcal{B}_n$, whose dual test excludes $A_n$ when $A_n\notin\mathcal{B}_n$, attains the
transition rate of the bound. By Theorems~\ref{thm:LocalPower}-\ref{thm:nondeg} its power to
exclude perturbations of the truth is asymptotically zero for $\delta<\rho$ and non-degenerate
for $\delta=\rho$; for $\delta>\rho$ with $\rho<\epsilon$, the rescaled half-width vanishes
(Theorem~\ref{thm:LocalCI}) while $|\hat B|\le\bar B$ pins the rescaled centre at the truth,
so Lemma~\ref{lem:containment} gives exclusion of every fixed $\xi\neq0$ wpa1. Hence honest
local power transitions at $\rho=\delta$,
and no honest interval improves on this rate.
\end{corollary}
Corollary~\ref{cor:rateoptimal} establishes that the bias-aware interval is not
needlessly wasteful.
The rate at which power degrades is the fastest any honest
procedure can achieve.
The comparison is sharpest at the matched rate $\delta=\rho$, where
Proposition~\ref{prop:sharp} gives the achievable frontier and
Theorem~\ref{thm:nondeg} the interval's actual power. By Proposition~\ref{prop:sharp}
the gap between them is exactly the critical-value gap $z_{1-\alpha/2}-z_{1-\alpha}$, the
slack a directional critical value recovers without violating coverage.
This is the asymptotic counterpart of the finite-sample finding of \citet{armstrong2022robust} that
the bias-aware critical value cannot be reduced by more than roughly a factor of two
without sacrificing coverage.
In the weak-factor setting of Example~\ref{rem:WeakFactors},
Assumption~\ref{ass:confounding} is the formal content of weak-factor
non-detectability.
A factor at the boundary of detection is statistically
invisible at the available sample size \citep{onatski2012}, so a coefficient shift
aligned with its loading is absorbed into the nuisance, realising a difference in $\beta$ up to
$\bar B$ with vanishing detectability and $\tau$ given by the Gaussian divergence.
Therefore the lower bound applies straightforwardly to Example~\ref{rem:WeakFactors}.
Proposition~\ref{prop:sharp} is the local-power counterpart of the length
non-adaptivity of \citet{low1997nonparametric} and \citet{cailow2004adaptation}.
They show that an honest interval cannot be shorter than the worst-case modulus of
continuity, whereas here it is shown that its dual test cannot have local power inside the resulting
modulus-width window. The two-point modulus construction of
\citet{donoho1994statistical} is common to both, and the constant that lower-bounds
their length is the $\bar b$ that bounds our power.
By the length-power duality of
\citet{pratt1961length}, an interval that cannot beat the worst-case modulus induces a
test that cannot detect within it.
The mechanism is in fact more general than honest inference, and is most transparent
through partial identification. Say $\beta$ is partially identified with identified set
$\mathrm{ID}(P)$, the values of $\beta$ consistent with the data law $P$ over the
maintained class, and recall the dual test excludes a value when that value leaves
the interval.
\begin{proposition}[Partial identification forces local-power degeneracy]\label{prop:partialID}
Let $\mathrm{ID}(P)$ be a compact interval of positive length and let $\mathcal{B}_n$ be any
honest interval, with the data generated at the truth $\beta_0$ (law $P$). For perturbations
$A_n=\beta_0+\xi\,n^{-\rho}$ with $\rho>0$:
\begin{enumerate}[(i)]
\item if $\beta_0\in\operatorname{int}\mathrm{ID}(P)$, then for every fixed $\xi$,
$A_n\in\mathrm{ID}(P)$ for $n$ large and $\limsup_n\Pr_P\{A_n\notin\mathcal{B}_n\}\le\alpha$:
the interval cannot exclude perturbations of the truth, and there is no local power;
\item if $\beta_0$ is an endpoint of $\mathrm{ID}(P)$, the same holds for every $\xi$
pointing into $\mathrm{ID}(P)$, so there is at best one-sided local power.
\end{enumerate}
\end{proposition}
\begin{proof}
If $\beta_0$ and $A_n$ both lie in $\mathrm{ID}(P)$ they are consistent with the same law
$P$, so under the truth $\beta_0$ the value $A_n$ is an equally admissible parameter; honest
coverage therefore forces the interval to cover it, $\Pr_P\{A_n\in\mathcal{B}_n\}\ge1-\alpha+o(1)$,
whence $\Pr_P\{A_n\notin\mathcal{B}_n\}\le\alpha+o(1)$. In (i), $A_n\to\beta_0\in
\operatorname{int}\mathrm{ID}(P)$ gives $A_n\in\mathrm{ID}(P)$ for $n$ large, for every
$\xi$; in (ii), the same holds for $\xi$ pointing inward, while perturbations leaving the
set are realised by distinct laws and may be excludable.
\hfill
\end{proof}
Proposition~\ref{prop:partialID} applies the paper's regimes under one principle.
Local power degenerates whenever a procedure's interval width fails to shrink fast enough
relative to the alternatives. Partial identification is the extreme since the width
converges to a set that does not shrink at all, and honest inference under a fixed
bound ($\delta=0$) is exactly this case.
The intermediate regimes $0<\delta\le\rho$ are
``partial identification at the local scale'', since after rescaling by $n^{\rho}$ the honest
half-width $n^{\rho}\bar B=\Theta(n^{\rho-\delta})$ does not vanish, so the rescaled
identified set has positive volume and the same degeneracy applies, by
Proposition~\ref{prop:partialID} with $\beta_0$ interior (zero power) or at the boundary
(one-sided power, recovering Theorem~\ref{thm:LocalCI2}). The power loss is thus a feature
of any procedure that endorses confidence intervals with slowly converging width, of
which conservative honest inference is one instance \citep[cf.][]{imbens2004confidence,stoye2009more}.
The bound in part~(i) is $\alpha$ rather than zero, and this is sharp.
Honesty caps the exclusion power \emph{in excess of size}, not the exclusion probability itself:
a dual test that exhausts its size at $A_n$ still excludes $A_n$, under the truth $\beta_0$, with
probability $\alpha+o(1)$.
A particular honest interval may exclude perturbations with vanishing
probability, but none can guarantee excluding one with more than $\alpha$, which is exactly the
local power that degenerates.
Taken together, Sections~\ref{sect:powerLossBACI}-\ref{sect:minimaxpowerLossHonest} show that the bounds on honest
local power coincide in rate and transition at $\delta=\rho$. The power loss of
conservative confidence intervals is not a deficiency of particular constructions, but intrinsic to the uniform coverage requirement over a class that the data cannot
discipline.
Against a class wide enough to contain the bias, no honest procedure recovers the local power that conservatism forgoes.
The only way to regain power is to relax the strict honesty requirement, for instance by accepting an undersized test that retains
power \citep[cf.][]{low1997nonparametric,rambachan2023more}.
\subsection{Primitive conditions and the modulus}\label{sect:primitives}
The Gaussian limits in Section~\ref{sect:minimaxpowerLossHonest} rest on standard primitives. The matched-rate normal
approximation of Theorem~\ref{thm:nondeg}(ii) and Proposition~\ref{prop:sharp} holds
whenever $\hat\beta$ is asymptotically linear:
$\hat\beta-\beta-\hat B=\sum_i w_{ni}u_i+o_p(n^{-\epsilon})$; and the array
$\{w_{ni}u_i\}$ satisfies the Lindeberg condition with
$n^{2\epsilon}\sum_i w_{ni}^2\,\mathrm{E}u_i^2\to\sigma^2$; the variance estimator is
consistent, $n^{\epsilon}se(\hat\beta)\xrightarrow{p}\sigma$; and the bias is stable,
$n^{\delta}\hat B\xrightarrow{p}c\bar b$, $n^{\delta}\bar B\to\bar b$. For
$\epsilon=1/2$ this is the usual Lyapunov central limit theorem.
For $\epsilon<1/2$ it is
the standard Lindeberg condition for kernel estimators with effective sample size
$n^{2\epsilon}$, and holds for the estimators of the designs in the simulations of Section~\ref{sect:simulations}.
The function $\tau$ of Assumption~\ref{ass:confounding} is not a free primitive but a
single scalar implied by the model.
It shows how statistically different two laws within the same class of DGPs must be under a shift in the parameter of interest.
\begin{definition}[Confounding separation]\label{def:confsep}
For $u\in[0,1]$ consider the admissible pairs that confound a target shift of $2u\bar B$, the
true law $P^{(n)}_{\beta_0,\theta_0}$ and a confounding law
$P^{(n)}_{\beta_0+2u\bar B,\theta_1}\in\mathcal P_n$. Let $s_n(u)$ be the separation of the
least-favourable such pair, in standard-error units of the local Gaussian experiment
(equivalently, the rescaled inverse of the \citet{donoho1994statistical} modulus of
continuity). The \emph{confounding separation} is $s(u)=\lim_n s_n(u)$, assumed continuous and
nondecreasing with $s(0)=0$.
\end{definition}
\begin{proposition}[Primitive condition for the bound]\label{prop:confprim}
Suppose the confounding separation $s(u)$ of Definition~\ref{def:confsep} exists. Then every
asymptotically honest level-$\alpha$ test has power at most $\alpha+\tau(u)$ at the confounded
alternative, with
\begin{align*}
\tau(u)=\Phi\big(s(u)-z_{1-\alpha}\big)-\alpha,
\end{align*}
which has $\tau(0)=0$ and, since $\Phi<1$, lies in $[0,1-\alpha)$. Definition~\ref{def:confsep}
thus delivers the conclusion of Theorem~\ref{thm:lowerbound} directly, the Neyman--Pearson
bound replacing the total-variation step in its proof with the sharper $\tau$, whose range
$[0,1-\alpha)$ comes for free.
Moreover, over convex classes and linear functionals the bound is \emph{attained}: the
likelihood-ratio test between the closest pair of the two convex slices is exactly honest over
the tested slice, by the separating-hyperplane property of the closest pair, and its worst-case
power is exactly $\Phi(s(u)-z_{1-\alpha})$, so
$\Phi\big(s(\xi)-z_{1-\alpha}\big)$ is the exact honest local-power frontier.
\end{proposition}
\begin{proof}
By Neyman--Pearson at the least-favourable pair, a level-$\alpha$ test at the null law has
power at most $\Phi(s_n(u)-z_{1-\alpha})$ at the alternative.
This replaces the
$\mathrm E_{P_1}\phi\le\alpha+TV$ step in the proof of Theorem~\ref{thm:lowerbound}, and the
limit follows by continuity.
Attainment over a convex class with a linear functional is by the same pair: test one-sidedly
along the difference of the closest pair, with the critical value set at the tested-slice end.
Because the slices are convex, the closest-pair difference separates them. Each slice's
projection onto that direction is extremised at its closest-pair point, so the test's size
over the whole tested slice is attained at that point and equals $\alpha$ exactly, while its
worst-case power over the other slice is attained at the opposite point and equals
$\Phi(s_n(u)-z_{1-\alpha})$. The limit again follows by continuity.
\hfill
\end{proof}
The four settings are instances, distinguished by the \emph{leading behaviour} of $s$ at the
origin. Where $s$ has a linear term, its coefficient $s_\star$ in $s(u)\approx s_\star u$ is the
honest analogue of the classical Pitman slope that summarises local power
\citep[Thm.~14.7]{vandervaart1998asymptotic}; in general the leading exponent matters as much as
the coefficient:
\begin{center}\small
\begin{tabular}{lll}
\hline
Setting & Leading behaviour of $s$ & Source \\
\hline
Plausibly-exog.\ IV & $s\equiv0$ on $[0,1)$ & exact confounding (Lemma~\ref{lem:exactconf}) \\
Honest DiD, $\Delta^{SD}(M)$ & $s\equiv0$ on $[0,1)$ & exact confounding (Lemma~\ref{lem:exactconf}) \\
Regression discontinuity & $s(u)\asymp u^{5/4}$ & inverse H\"older modulus at the noise scale \\
&& \citep{donoho1994statistical} \\
Weak factors & $s(0)=0$ only claimed & detection boundary \citep{onatski2012}; see below \\
\hline
\end{tabular}
\end{center}
The endpoints are the theory's two extremes. Plausibly-exogenous IV and honest DiD have
$s\equiv0$ on $[0,1)$: by Lemma~\ref{lem:exactconf} a target shift of up to $2\bar B$ is matched
\emph{exactly} within the class, which is the perfect-confounding form of
Assumption~\ref{ass:LE} and the flat-dead-zone, pure zero-power case. Regression discontinuity
has $s$ strictly increasing: the confounding is approximate, its cost from the H\"older
modulus, so the frontier $\Phi(s(\xi)-z_{1-\alpha})$ rises smoothly rather than staying flat.
There is no exactly flat dead zone over the full class, only the (bounded) resolution limit of
the bias-aware procedures in use.
The exponent is worth a remark: for $|f''|\le M$ the modulus is
$\omega(\varepsilon)\asymp\varepsilon^{4/5}$, and the calibration $\bar B\asymp\omega(n^{-1/2})$
gives $s(u)\asymp u^{5/4}$. Read literally as a derivative at the origin, then, $s'(0)=0$ in
\emph{every} row. A concave modulus $\omega\asymp\varepsilon^{r}$ with $r<1$ always yields the
leading power $1/r>1$, so the rows are distinguished by leading behaviour, not by a common
Taylor coefficient, and the RD frontier lifts off $\alpha$ only slowly near the origin, its
first-order term missing.
Where the linearisation $s(u)\approx s_\star u$ is used, $s_\star$ is
the slope at the operating scale, the modulus' slope at the noise level, a secant rather
than the origin derivative. The factor model is the one non-convex
instance, where the modulus machinery does not apply.
A factor at the boundary of detection
\citep{onatski2012} makes a loading-aligned shift undetectable, which gives the qualitative
content $s(0)=0$ that Theorem~\ref{thm:lowerbound}(i) needs but not a closed-form leading term,
which is not claimed here. So $\tau$, the dead
zone, and the interval length are three readings of the one object $s$.
In classical terms the rows deform the textbook power envelope
\citep[Thm.~15.4]{vandervaart1998asymptotic} in three distinct ways: exact confounding
preserves its slope and \emph{translates} it by $2\bar b$; regression discontinuity replaces
the parametric slope by the modulus' slope at the noise level; and the factor row flattens
slope and curvature alike at the origin.
\section{Simulations}\label{sect:simulations}
This section illustrates the power loss of Theorems~\ref{thm:LocalCI}--\ref{thm:lowerbound}
across five distinct honest-inference problems. Each design pairs a conservative,
honest procedure with a conventional competitor that ignores the
worst-case bias; the difference-in-differences design adds a third, intermediate
trend-adjusted procedure, described there. In every case the data are generated at a fixed true parameter
value, a level-$\alpha$ confidence interval is constructed once per Monte Carlo
replication, and the dual test is obtained by inversion, rejecting any value that falls
outside the interval. With the data generated at the true value $\beta_0$, sweeping the
local alternative $A_n=\beta_0+\xi\,n^{-\rho}$ and averaging the rejection indicator
$\mathbbm 1\{A_n\notin\mathcal B_n\}$ over replications traces the local-power curve as a
function of $\xi$. Throughout,
$\alpha=0.05$ and $z_{1-\alpha/2}$ denotes the standard normal quantile.
The five designs span the regimes of Section~\ref{sect:main}. The interactive
fixed-effects, difference-in-differences, and plausibly-exogenous designs place
the bound at a constant (or sample-size-inflated) order relative to the
$n^{-1/2}$ sampling rate, illustrating the zero-power collapse of
Theorem~\ref{thm:LocalCI}. The regression-discontinuity design is genuinely
nonparametric, with sampling error and worst-case bias of the same order,
illustrating the non-degenerate power loss of Theorem~\ref{thm:nondeg}. The
linear-functional design contrasts a parametric estimator with an honest one that
converges at a slower nonparametric rate, so the honest interval's $n^{1/2}$-rescaled
width diverges and its rejection probability is asymptotically flat in $\xi$: the dual
test cannot discriminate among $n^{-1/2}$ alternatives (Lemma~\ref{lem:containment}). In
the three bound-dominated designs the honest procedure attains size arbitrarily close to
zero; in the regression-discontinuity and linear-functional designs, where the worst-case
bias and the standard error are of the same order, size is conservative but bounded away
from zero. In every case the formal price is the documented loss of local
power.\footnote{Replication code for all five designs is available from the author; the
empirical applications of Section~\ref{sect:applications} are reproduced by the script
described there.}
\subsection{Interactive Fixed Effects: Weak and Strong Factors}
\label{sect:simFactor}
The leading example is the panel regression with interactive fixed effects of
\cite{armstrong2022robust}. With $n=NT$, data are generated as
\begin{equation*}
y_{it}=x_{it}\beta+c_n\,\lambda_i f_t+\varepsilon_{it},
\qquad
x_{it}=\lambda_i f_t+\eta_{it},
\qquad i\le N,\ t\le T,
\end{equation*}
with $\lambda_i,f_t\sim\mathrm{i.i.d.}\,N(0,1)$, the product $\lambda_i f_t$
normalised to unit variance, $\varepsilon_{it},\eta_{it}\sim\mathrm{i.i.d.}\,N(0,1)$,
$\beta=1$, and $T=\lfloor N^{3/5}\rfloor$. The scalar $c_n$ controls the
strength of the factor that contaminates the regressor: $c_n=n^{-1/3}$ yields a
\emph{weak} factor (Figure~\ref{fig:WeakFactors}) and $c_n=1$ a \emph{strong}
factor (Figure~\ref{fig:StrongFactors}). Local alternatives are taken at the
parametric rate, $A_n=\beta\pm\xi/\sqrt{n}$.
\paragraph{Estimators (Figures~\ref{fig:WeakFactors}-\ref{fig:StrongFactors}).} The conventional \emph{factor} estimator is the iterated
least-squares estimator of \cite{Bai2009}, with
interval $\hat\beta\pm z_{1-\alpha/2}\,\mathrm{se}(\hat\beta)$. The
honest \emph{bias-aware} interval is that of \cite{armstrong2022robust}, $\{\hat\beta_{ba}\pm[\bar B+z_{1-\alpha/2}\,\mathrm{se}(\hat\beta_{ba})]\}$,
where the user supplies the number of factors treated as weak (here one) and
$\bar B$ is the implied worst-case bias bound. The analyst cannot detect whether
the factor is weak, and so must guard against it.
Under the weak factor the
factor estimator's interval undercovers and its dual test over-rejects the truth, while
under the strong factor the bias-aware
interval is conservative even though the true bias is negligible.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.32\linewidth, height = 5cm]{WeakPrateNonRejectionRegionsN=50T=10.pdf}
\includegraphics[width=0.32\linewidth, height = 5cm]{WeakPrateNonRejectionRegionsN=200T=24.pdf}
\includegraphics[width=0.32\linewidth, height = 5cm]{WeakPrateNonRejectionRegionsN=1200T=70.pdf}
\caption{Weak factors $n^{-1/2}$ local alternatives.
Dashed blue line: Factor model;
Solid red line: bias-aware intervals;
Grey area: dead-zone.
Left to Right: $(N,T) \in \{(50, 10),(200, 24),(1200, 70)\}$
}
\label{fig:WeakFactors}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.3\linewidth, height = 5cm]{StrongPrateNonRejectionRegionsN=50T=10.pdf}
\includegraphics[width=0.3\linewidth, height = 5cm]{StrongPrateNonRejectionRegionsN=200T=24.pdf}
\includegraphics[width=0.3\linewidth, height = 5cm]{StrongPrateNonRejectionRegionsN=1200T=70.pdf}
\caption{Strong factors $n^{-1/2}$ local alternatives.
Dashed blue line: Factor model;
Solid red line: bias-aware intervals;
Grey area: dead-zone.
Left to Right: $(N,T) \in \{(50, 10),(200, 24),(1200, 70)\}$
}
\label{fig:StrongFactors}
\end{figure}
\subsection{Honest Difference-in-Differences}
\label{subsec:did}
The second design is the honest difference-in-differences setting of \cite{rambachan2023more}. An event study with three pre-treatment periods and one
post-treatment period is summarised by its coefficient vector
$\hat\beta=(\hat\beta_{\mathrm{pre}}^\top,\hat\beta_{\mathrm{post}})^\top\in\mathbb{R}^4$,
generated directly as a Gaussian draw
\begin{equation*}
\hat\beta\sim N\!\left(\mu,\ \Sigma_0/N\right),
\qquad
\mu=\big(-3s,\ -2s,\ -s,\ \beta_{\mathrm{post}}+s+w\big),
\qquad \Sigma_0=I_4,
\end{equation*}
where $\beta_{\mathrm{post}}=0$ is the true post-period effect and the entries of
$\mu$ encode a linear differential trend $\delta_t=s\cdot t$ over event time
$t\in\{-3,-2,-1,1\}$, plus a second-difference kink $w$ at $t=1$, the reference
period $t=0$ being normalised out. The scalars
$s\ge 0$ and $w\ge0$ index the strength of the linear pre-trend and of the deviation
from linearity, and the conventional standard
error is of order $N^{-1/2}$. The target is the post-period effect,
$\theta=\ell^\top\hat\beta_{\mathrm{post}}$ with $\ell=1$, whose true value
remains $\beta_{\mathrm{post}}$ for every $(s,w)$. Local alternatives are taken at
$A_n=\beta_{\mathrm{post}}\pm\xi/\sqrt N$, and the sample sizes are
$N\in\{100,500,2500\}$.
Setting $s=w=0$ recovers exact parallel trends, in which the honest interval is
needlessly conservative. For $s>0$ with $w=0$ there is a linear pretrend, and the
pre-treatment coefficients trend linearly and the post-period coefficient is
contaminated by $\delta_1=s$. Because a linear trend has zero second differences,
it lies in $\Delta^{SD}(M)$ for \emph{every} $M\ge 0$ (here set to $M =0.2$). The honest interval
extrapolates the estimated pre-trend and continues to cover the true effect
$\beta_{\mathrm{post}}$, whereas the conventional event-study interval ignores the
pre-periods and centres on the biased post coefficient $\beta_{\mathrm{post}}+s$,
so its coverage of the truth collapses as $N$ grows.
Note the honest interval is not the event-study interval widened about a common centre, as in
the plausibly-exogenous design below: the extrapolation \emph{recentres} it at the truth, so the
two non-rejection windows separate, and with $s=M$ the event-study window collapses onto the
upper edge of the honest window. This is the upper endpoint of the local identified set $[-M,M]$.
The design therefore includes a third, intermediate procedure that realises the recentring
without the widening: the \emph{trend-adjusted} interval, the fixed-length interval over
$\Delta^{SD}(0)$, which extrapolates the estimated linear pre-trend with no allowance for
curvature. This is the parametric linear adjustment common in applied work
\citep{dobkin2018economic,bhuller2013broadband}, corresponding to $M=0$ in the class of
\citet{rambachan2023more}. Under an exactly linear trend it is correctly centred and nominally
sized and retains full local power.
It claims the power the honest interval forgoes by
asserting the untestable $w=0$, exactly as the event study asserts $\delta=0$, the same
wager, one derivative up. The kinked configuration ($s=0$, $w=M$) prices that wager. It is the
least-favourable path of Lemma~\ref{lem:exactconf}(ii), flat pre-periods with $\delta_1=M$, on
which the extrapolation of the flat pre-trend is zero, so the conventional and trend-adjusted
intervals are \emph{both} miscentred by $M$ and reject the truth with probability approaching
one, while the honest interval, operating at its designed worst case, spends exactly its size
$\alpha$ at the truth: its undetected window is the one-sided $[0,2M]$, the $c=1$ sliding
window of Theorem~\ref{thm:nondeg}(i). Figure~\ref{fig:did} reports
the strong pre-trend ($s=0.2$, $w=0$), the kinked violation ($s=0$, $w=0.2$), and no pre-trend
($s=w=0$).
\paragraph{Estimators (Figure~\ref{fig:did}).}
Terms $M$ and $\Delta^{SD}(M)$ in the following are defined in \cite{rambachan2023more}.
The conventional \emph{event-study} interval is the
standard confidence set that imposes parallel trends ($\delta=0$),
$\hat\theta\pm z_{1-\alpha/2}\,\mathrm{se}(\hat\theta)$. The \emph{trend-adjusted}
interval is the fixed-length interval over $\Delta^{SD}(0)$, computed with the same
machinery at $M=0$. The \emph{honest}
interval is the fixed-length confidence interval over the smoothness class
$\Delta^{SD}(M)$, which bounds the second differences of the differential trend
by a user-chosen $M$ (here $M=0.2$). The honest interval is centred at the
extrapolation-corrected estimator and widened by the worst-case post-period bias that a
violation within $\Delta^{SD}(M)$ can induce; as $N$ grows with $M$ fixed, this constant-order
term dominates the shrinking standard error.
\begin{figure}[!ht]
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{honestdid_strongPretrend.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{honestdid_kinkViolation.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{honestdid_noPretrend.pdf}
\caption{Honest DiD rejections for $n^{-1/2}$ local
alternatives. Left to right: $N\in\{100,500,2500\}$.
Blue dashed: event-study interval;
green dot-dash: trend-adjusted $\Delta^{SD}(0)$ interval;
red solid: honest $\Delta^{SD}(M)$ interval, $M=0.2$.
Shaded: the dead zone $\{$power $\le\alpha\}$.
Top to Bottom: strong pretrend; kinked violation; no pretrend.
With strong pretrends the event-study window rests at the honest region's right
edge: the honest estimator extrapolates the trend away and is centred at the truth, while the
event-study interval remains centred at its bias $s$, which here equals $M$; the
trend-adjusted interval is correctly centred and nominally sized, its full power bought by
asserting the untestable $w=0$. In the kinked row that wager fails: the event-study and
trend-adjusted windows coincide, both miscentred by $w=M$, while the honest interval, at its
designed worst case, spends exactly its size $\alpha$ at the truth.
}
\label{fig:did}
\end{figure}
\subsection{Honest Regression Discontinuity}
\label{subsec:rd}
The third design is the honest sharp regression-discontinuity setting of
\citet{armstrong2018optimal, armstrong2020simple}, with the Monte Carlo design of
their supplemental.
Running variable $x_i\sim\mathrm{Uniform}[-1,1]$,
$\varepsilon_i\sim\mathrm{i.i.d.}\,N(0,\sigma^2)$ with $\sigma^2=0.1295$, and
\begin{equation*}
y_i=\tau\,\mathbf{1}\{x_i\ge 0\}+ a\,f(x_i)+\varepsilon_i,
\end{equation*}
where $\tau=0$ is the true jump and $f(x)=C\,\operatorname{sign}(x)\,x^2$ ($C=1$)
is the least-favourable odd quadratic, saturating $|f''|=2C$ everywhere, scaled by
an amplitude $a\in[0,1]$. The sample sizes are $n\in\{500,2500,12500\}$ and, the
estimator being nonparametric, local alternatives are taken at the rate
$A_n=\tau\pm\xi\,n^{-2/5}$.
\paragraph{Estimators (Figure~\ref{fig:rd}).} Both intervals use the triangular-kernel
local-linear estimate $\hat\tau$ at a common bandwidth $h$ (H\"older class
$\{|f''|\le M\}$, $M=2C$). The
\emph{conventional} interval $\hat\tau\pm z_{1-\alpha/2}\,\mathrm{se}(\hat\tau)$
drops the smoothing bias, whereas the \emph{honest} interval replaces $z_{1-\alpha/2}$ by the bias-aware value
$\mathrm{cv}_\alpha(\overline{\mathrm{bias}}/\mathrm{se})$, with
$\overline{\mathrm{bias}}$ the worst-case bias over the class. At the optimal bandwidth the
bias-to-standard-error ratio $\overline{\mathrm{bias}}/\mathrm{se}$ is a bounded
constant ($\approx 0.5$), so the bias-aware critical value is only modestly above
$z_{1-\alpha/2}$ \citep[$2.18$ rather than $1.96$ at the $n^{-2/5}$ rate;][]{armstrong2020simple}.
RD is therefore the matched-rate case of Theorem~\ref{thm:nondeg} ($\delta=\rho=\epsilon=2/5$),
in which conservatism costs only a \emph{bounded} amount.
When $a=1$ the curvature
saturates the bound, so the conventional interval under-covers (coverage
$\approx 0.92$; honest needed), while when $a=0$ the design is linear, the
conventional interval is correct, and the honest interval is only
$\approx 11\%$ wider.
The much larger under-coverage documented by
\citet{armstrong2018optimal} arises with data-driven bandwidths that mis-estimate curvature; here only the honest/conventional margin at a common bandwidth is explored, for clarity.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.85\textwidth, height = 5cm]{honestRDNeeded.pdf}
\\
\centering
\includegraphics[width=0.85\textwidth, height = 5cm]{honestRDNotNeeded.pdf}
\caption{Honest RD rejection rates for $n^{-2/5}$ local
alternatives. Left to right: $n\in\{500, 2500, 12500\}$.
Top to bottom: $a=1$ curvature saturates, $a = 0$ linear.
Blue dashed:
conventional local-linear interval; red solid: honest H\"older interval.
Shaded: the realized dead zone $\{$power $\le\alpha\}$, the alternatives the honest test cannot
detect (where the red curve sits on/below $\alpha$); green dotted lines: $\pm2\bar b$, the
least-favourable dead-zone half-width of the bias-aware class (Proposition~\ref{prop:sharp};
in RD the confounding is approximate, so the full-class frontier is the smooth one of
Proposition~\ref{prop:confprim}). At
matched rate ($\delta=\rho=\epsilon$) both are \emph{bounded}, the bounded cost of
conservatism}
\label{fig:rd}
\end{figure}
\subsection{Estimation of a Linear Functional}
\label{subsec:donoho}
The fourth design is the stylised problem of \cite{donoho1994statistical}, and
unlike the others the honest procedure here converges at a \emph{slower rate}
than the conventional one. Design points $x_i\sim\mathrm{Uniform}[0,1]$,
$i=1,\dots,n$, generate
\begin{equation*}
y_i=f(x_i)+\varepsilon_i,\qquad
f(x)=\theta+a\,C\,\min(x,h),\qquad \varepsilon_i\sim\mathrm{i.i.d.}\,N(0,\sigma^2),
\end{equation*}
with $\sigma=1$ and target the boundary value $f(0)=\theta=0$. The term
$a\,C\min(x,h)$ is a Lipschitz wedge of amplitude $a\in[0,1]$ confined to the
window $[0,h]$, of width equal to the honest bandwidth $h\sim n^{-1/3}$ defined
below. Sample sizes are $n\in\{500,2500,12500\}$ and local alternatives are taken
at the \emph{parametric} rate $A_n=\theta\pm\xi/\sqrt n$.
\paragraph{Estimators (Figure~\ref{fig:donoho}).} The conventional \emph{parametric} estimator assumes the
approximately linear model of \citet[\S9.1]{donoho1994statistical} holds exactly
($f$ affine).
It is the OLS intercept $\hat\theta_{ols}$ with interval
$\hat\theta_{ols}\pm z_{1-\alpha/2}\,\mathrm{se}(\hat\theta_{ols})$ and
$\mathrm{se}$ of order $(n^{-1/2})$. The \emph{honest} estimator only assumes $f$ lies
in the Lipschitz class $\{|f(x)-f(x')|\le C|x-x'|\}$ ($C=2$).
It is the one-sided
local average over $[0,h]$ at the MSE-optimal bandwidth $h=(\sigma^2/(C^2 n))^{1/3}$,
widened by the modulus-of-continuity bias $C\,h$, $\hat\theta_{loc}\pm(C\,h+z_{1-\alpha/2}\,\mathrm{se}(\hat\theta_{loc}))$,
with the slower nonparametric rate $\mathrm{se}$ of order $(n^{-1/3})$. When $a=0$ the
function is flat and honesty is not needed, and the parametric interval is valid and
sharper, the honest interval needlessly wide.
When $a>0$ the OLS line extrapolates the bulk level $a\,C\,h$ and undercovers the
boundary value $f(0)$, while the honest interval stays valid. Crucially the wedge is
\emph{untestable}: confined to $[0,h]$ it perturbs only $\sim nh=n^{2/3}$ observations
at magnitude $\sim n^{-1/3}$, so no specification test detects it. Indeed a $y\sim x+x^2$
test keeps size $0.05$ for every $n$, and the conservatism cannot be sidestepped by
first testing for non-linearity, unlike a \emph{global} quadratic whose curvature is
$\sqrt n$-estimable. In both cases the honest interval shrinks only at $n^{-1/3}$, so
against $n^{-1/2}$ alternatives its rejection probability is asymptotically flat in $\xi$
(Lemma~\ref{lem:containment}): no local discrimination, precisely the price of robustness.
\begin{figure}[!ht]
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{linearfunctionalStrongCurvepdf.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{linearfunctionalWeakCurvepdf.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{linearfunctionalNoCurvepdf.pdf}
\caption{Rejection rates for $n^{-1/2}$ local alternatives.
Left to right: $n\in\{500,2500,12500\}$. Blue dashed: OLS interval
($n^{-1/2}$); red solid: honest Lipschitz interval, $C=2$
($n^{-1/3}$). Localised wedge ($a>0$, honest needed): OLS rejects the truth;
flat design ($a=0$, honest not needed): OLS valid and sharper.
Shaded: the dead zone $\{$power $\le\alpha\}$; the honest interval shrinks only at
$n^{-1/3}$, so in $n^{-1/2}$ units the zone widens across panels.
Top to bottom: $a = 1$, $a = 0.25$, $a = 0$.
}
\label{fig:donoho}
\end{figure}
\subsection{Plausibly Exogenous Instrumental Variables}
\label{subsec:plausexog}
The final design is the plausibly-exogenous instrumental-variables setting of
\cite{conley2012plausibly}. With a single instrument $z_i$, endogenous
regressor $x_i$, and structural outcome $y_i$,
\begin{equation*}
z_i\sim N(0,1),\quad
x_i=\pi z_i+v_i,\quad
y_i=\beta x_i+\gamma z_i+\varepsilon_i,\quad
\varepsilon_i=\rho\, v_i+\sqrt{1-\rho^2}\,\nu_i,
\end{equation*}
where $v_i,\nu_i\sim\mathrm{i.i.d.}\,N(0,1)$, first-stage strength $\pi=1$,
endogeneity $\rho=0.6$, and structural coefficient $\beta=1$. The analyst relaxes the
exclusion restriction to $|\gamma|\le\bar\gamma$ with $\bar\gamma=0.30$. The knob is the
\emph{true} exclusion violation, $\gamma\in\{0.3,\ 0.05,\ 0\}$: a \emph{strongly endogenous},
\emph{weakly endogenous}, and \emph{exogenous} instrument, respectively (first-stage strength
is held at $\pi=1$ throughout). Under the strongly endogenous instrument
($\gamma=0.3$, the violation at the bound) the conventional IV estimator is miscentred by
$\gamma/\pi=0.3$ and honesty is needed; under the exogenous instrument ($\gamma=0$) the
exclusion restriction in fact holds, the instrumental-variables analogue of the strong-factor
design, and the honest interval is conservative despite negligible true bias. Sample sizes are
$n\in\{100,500,2500\}$ and local alternatives are at the parametric rate
$A_n=\beta\pm\xi/\sqrt n$.
\paragraph{Estimators (Figure~\ref{fig:plausexog}).} Conventional \emph{standard IV} imposes
$\gamma=0$, $\hat\beta_{IV}=(z^\top y)/(z^\top x)$, with interval
$\hat\beta_{IV}\pm z_{1-\alpha/2}\,\mathrm{se}(\hat\beta_{IV})$. The \emph{honest}
interval is the union of confidence intervals of \cite{conley2012plausibly}: for
each admissible $g\in[-\bar\gamma,\bar\gamma]$ one forms the IV interval for
$\beta$ from the netted outcome $y-g z$,
$\hat\beta(g)=z^\top(y-g z)/(z^\top x)$, and takes the union
$\big[\min_{|g|\le\bar\gamma}\mathrm{lb}(g),\ \max_{|g|\le\bar\gamma}\mathrm{ub}(g)\big]$.
The union widens the interval by an amount of order $\bar\gamma\,(z^\top z)/(z^\top x)\approx\bar\gamma/\pi$,
a constant that dominates the $O(n^{-1/2})$ sampling error as $n$ grows.
\begin{figure}[!ht]
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{plausexogWeakIV.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{plausexogSemiWeakIV.pdf}
\centering
\includegraphics[width = 0.85\textwidth, height = 5cm]{plausexogStrongIV.pdf}
\caption{Plausibly-exogenous IV rejection rates for $n^{-1/2}$ local
alternatives. Left to right: $n\in\{100,500,2500\}$. Blue dashed: standard IV
interval; red solid: honest union-of-CIs with $\bar\gamma=0.30$.
Shaded: the dead zone $\{$power $\le\alpha\}$; the $\bar\gamma$ term is constant in $n$,
so the zone does not shrink across panels.
Top to bottom: strongly endogenous instrument ($\gamma=0.3$, honest needed), weakly
endogenous ($\gamma=0.05$), exogenous ($\gamma=0$, honest not needed); $\pi=1$ throughout.
}
\label{fig:plausexog}
\end{figure}
\section{Empirical applications}\label{sect:applications}
The paper's practical recommendation is to report the power ``dead zone'' alongside
bias-aware intervals.
This is the set of effect sizes the chosen bound forgoes the ability to detect.
This section makes the recommendation concrete on two published studies, chosen so that the exercise is easily reproducible.\footnote{The script \texttt{Applications.R} reproduces both numbers and figures from
the replication data bundled in the \texttt{RDHonest} and \texttt{HonestDiD} packages; no data
need be downloaded.}
The two sit in the paper's two non-degenerate regimes: bounded and cheap at the matched rate,
decisive and non-vanishing when the bound dominates.
In both applications the relevant null is no effect, $\beta_0=0$, and the dead zone is the set of
true effects whose detection cannot be guaranteed.
That is, the zone symmetrically about $\beta_0=0$ that an honest test rejects no more often than its size.
The dead-zone half-width $2\bar B$ is a property of
the bound, not of the null: it is the same window around \emph{any} hypothesised value, so it is drawn at the economically natural $\beta_0=0$.
At the matched rate (as in RD) it is a fixed band of half-width $2\bar B$.
When the bound dominates (as in DiD) the band grows with the bound.
\subsection{Regression discontinuity: incumbency advantage (matched rate)}\label{subsec:app-rd}
Consider the \citet{lee2008randomized} regression-discontinuity estimate of the U.S. House
incumbency advantage, using the bias-aware procedure of \citet{armstrong2018optimal,
armstrong2020simple} with the data-driven smoothness constant of its default rule of thumb. The
point estimate is a $5.85$ percentage-point incumbency advantage. The conventional interval,
which drops the smoothing bias, is $(3.17,8.53)$; the honest interval, which inflates the
critical value from $z_{1-\alpha/2}=1.96$ to the bias-aware $\mathrm{cv}_\alpha=2.31$, is
$(2.69,9.01)$, only $18\%$ wider. This is the matched-rate case $\delta=\rho=\epsilon=2/5$ of
Theorem~\ref{thm:nondeg}. The worst-case bias and the standard error are of the same order
($\bar B/se\approx0.65$), so conservatism costs only a bounded amount. The worst-case bias is
$\bar B=0.89$, so the dead zone of the reported procedure has half-width
$2\bar B=1.78$ percentage points.
Here the estimate exceeds the dead-zone
half-width by a factor of more than three (Figure~\ref{fig:app_rd}), so the finding is robust to
the conservatism, and disclosing the dead zone confirms rather than overturns it.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{app_rd_deadzone.pdf}
\caption{Incumbency advantage \citep{lee2008randomized}: conventional (blue) and honest
bias-aware (red) confidence intervals, with the dead zone of the bias-aware procedure,
half-width $2\bar B=1.78$ percentage points, shaded about the null $\beta_0=0$ (the band has
the same width about any null). The estimate ($5.85$ pp) lies far outside the dead zone:
detectable despite the conservatism.}
\label{fig:app_rd}
\end{figure}
\subsection{Difference-in-differences: VAT pass-through to profits (bound dominates)}\label{subsec:app-did}
Consider the \citet{benzarti2019who} event study of a large French VAT cut, asking whether the
reduction was captured as firm profits, with the honest difference-in-differences procedure of
\citet{rambachan2023more}. Imposing exact parallel trends, the first post-reform year ($2009$)
shows a sharply significant profit effect of $0.196$ with conventional interval $(0.159,0.233)$
($t\approx10$).
This is the bound-dominated regime: under the relative-magnitudes restriction
$\Delta^{RM}(\bar M)$, which allows the post-period violation of parallel trends to be at most a
multiple $\bar M$ of the largest pre-period violation, the honest interval widens with $\bar M$
and does not shrink with the sample.
The lower limit reaches zero at a breakdown of
$\bar M\approx1.75$ (Figure~\ref{fig:app_did}).
\citet{rambachan2023more} analyse this application with the same machinery and report a
closely related sensitivity analysis of the profit effect; the incremental content here is
the dead-zone reading of that exercise: the breakdown $\bar M$ is the point at which the
estimate enters the dead zone of the no-effect test.
The conventionally significant effect becomes
undetectable once one concedes a post-period trend deviation roughly twice the largest one seen
before treatment.
Disclosure of the dead zone in this application makes explicit that the ability to reject a zero effect rests on the bound the analyst is prepared to defend, i.e. on the prescribed tolerance for pre-trend violations.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{app_did_bc_sensitivity.pdf}
\caption{VAT pass-through to profits \citep{benzarti2019who}: the honest interval for the
$2009$ profit effect as a function of the relative-magnitudes bound $\bar M$. Red bars exclude
zero (significant); grey bars cover zero, so the honest interval now covers $\beta_0=0$ and the
estimate has entered the dead zone of the no-effect test. The lower limit crosses zero at the
breakdown $\bar M\approx1.75$ (green dashed). The conventional interval is the $\bar M=0$ point.
The dead zone is set by $\bar M$, not by the sample size.}
\label{fig:app_did}
\end{figure}
The two applications bracket the message. In regression discontinuity the dead zone is a
bounded, modest object and the headline survives it; in difference-in-differences the dead zone
is governed entirely by an untestable bound, and can be wide enough that even a very large $t$-statistic is not, by itself, decisive. In both cases the dead-zone half-width is a single number, in the units of the estimand,
that the interval's endpoints alone do not convey.
\section{Conclusion and recommendations}\label{sect:discussion}
This paper establishes a local-alternative power trade-off intrinsic to conservative
inference. At the usual parametric rate, an honest interval may have no local power, and
the loss is shared by every honest procedure, not just bias-aware constructions. The paper is
closed with some practical discussion.
\paragraph{Disclosure of dead zone.} Consider comparing the bias bound
$\bar B$ with the standard error.
$\bar B=\sup_\theta|\hat B|$ is chosen by the analyst, so the comparison merely
re-expresses how conservative a bound is.
The power cost is not a hidden risk to
be detected but a deterministic consequence of a stated belief about unobservable
biases, i.e. a monotone function of that preference. What this analysis recommends is
\emph{disclosure}: report the dead-zone half-width $2\bar b$, in the units of the
estimand, alongside any bias-aware interval, since it states exactly which alternatives
the chosen bound forgoes the ability to detect. This is information the interval's endpoints
alone do not convey.
\paragraph{Honesty versus power.} When the bound dominates standard errors, which is the
typical parametric-rate case with zero honest local power, and the analyst
nonetheless needs power, the only escape is to relax strict honesty.
For instance, to defend a
\emph{tighter} bound on substantive grounds, or to report an undersized test that retains
local power \citep[cf.][]{low1997nonparametric,rambachan2023more}. Which is preferable is
a genuine modelling choice that the size--power frontier here makes explicit rather than
resolves.
Conservatism buys uniform coverage at a power cost that is, at the parametric
rate, total. Quantifying when the bound can be credibly tightened, and characterising the
optimal undersized test are natural next steps for research.
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