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Linear Regression with Common Unobserved Heterogeneity
\maketitle
\begin{abstract}
We study inference on common slope parameters in linear panel models with two-way
unobserved heterogeneity. In linear regression, bias arises from unobserved
heterogeneity that is shared between the residual and the covariates. Our common unobserved
heterogeneity (CUH) estimator targets this common variation directly:
it searches over functionals of the residualised outcome for variation that also
resides in the covariates, and seeks to further project out only bias-inducing variation.
This makes the estimator less sensitive to weak
outcome factors while avoiding the complete removal of exogenous variation in covariates
that is unrelated to the confounding heterogeneity. We develop a
weak-factor theory based on a singular-value strength index and establish
$\sqrt{NT}$-asymptotic normality of the cross-fitted CUH estimator. Under
homoskedastic errors, its asymptotic variance is no larger than that of a
common correlated effects factor estimator. Simulations and empirical
applications illustrate the resulting bias--variance trade-off and the gains from selectively removing common variation.
\end{abstract}
\section{Introduction}\label{sec:intro}
This paper studies estimation of common slope parameters in linear panel
regressions with two-way unobserved heterogeneity. We observe $(y_{it},x_{it})$
for $i=1,\dots,N$ and $t=1,\dots,T$, with $x_{it}\in\mathbb{R}^{d}$ and $d$ fixed,
generated as
\begin{equation}\label{eq:y}
y_{it}=x_{it}'\beta_0+g(\alpha_i,\gamma_t)+\varepsilon_{it},
\end{equation}
where $\alpha_i\in\Omega_\alpha\subset\mathbb{R}^{d_\alpha}$ and
$\gamma_t\in\Omega_\gamma\subset\mathbb{R}^{d_\gamma}$ are unobserved individual and
time effects, $g$ is unknown and $\varepsilon_{it}$ is an idiosyncratic shock.
The covariates are exogenous with respect to $\varepsilon_{it}$ but correlated
with $(\alpha_i,\gamma_t)$. They obey a \emph{two-block} structure that
separates heterogeneity common to both equations from heterogeneity present
only in $x_{it}$,
\begin{equation}\label{eq:x}
x_{it}=\underbrace{g_x(\alpha_i,\gamma_t)}_{\text{common (endogenous)}}
+\underbrace{\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)}_{\text{$x$-only (exogenous)}}
+\eta_{it},
\qquad
\mathbb{E}[\eta_{it}\mid\alpha_i,\gamma_t,\widetilde\alpha_i,\widetilde\gamma_t]=0,
\end{equation}
where $(\widetilde\alpha_i,\widetilde\gamma_t)$ are latent effects absent from \eqref{eq:y}.
Equation~\eqref{eq:x} is a construction, not an assumption. Set
$g_x(\alpha_i,\gamma_t):=\mathbb{E}[x_{it}\mid \alpha_i,\gamma_t]$,
$\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t):=\mathbb{E}[x_{it}\mid\alpha_i,\gamma_t,\widetilde\alpha_i,\widetilde\gamma_t]-g_x(\alpha_i,\gamma_t)$
and $\eta_{it}=x_{it} - \mathbb{E}[x_{it}\mid\alpha_i,\gamma_t,\widetilde\alpha_i,\widetilde\gamma_t]$.
Then $\mathbb{E}[\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)\mid \alpha_i,\gamma_t]=0$, and this
normalisation assigns to $g_x$ the \emph{entire} dependence of $x_{it}$ on
$(\alpha_i,\gamma_t)$. The correlation of $g_x$ with $g(\alpha_i,\gamma_t)$ is
the source of endogeneity, while $\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)$ and $\eta_{it}$ are
exogenous variation that identifies $\beta_0$ and should be preserved. Our estimator is therefore designed to remove from $x_{it}$ only the block $g_x$ \emph{common} to $y_{it}$ and $x_{it}$, together with the heterogeneity in the residualised
outcome.
Three leading approaches to interactive unobserved heterogeneity frame our
contribution.
\begin{example}[Factor model]\label{ex:factor}
The factor model identifies $g(\alpha_i,\gamma_t)$ from
$y_{it}-x_{it}'\beta_0=g(\alpha_i,\gamma_t)+\varepsilon_{it}$.
Assume $g(\alpha_i,\gamma_t)=\varphi(\alpha_i)'\psi(\gamma_t)$ for
$\varphi(\alpha_i),\psi(\gamma_t)\in\mathbb{R}^{R_0}$ with $R_0<\infty$ and the
strong-factor conditions
\begin{align}\label{eqn:strongfactor}
\frac1N\varphi(\alpha)'\varphi(\alpha)>0,
&&
\frac1T\psi(\gamma)'\psi(\gamma)>0 .
\end{align}
Then, under regularity conditions, factor-model estimates of $\beta$ are
$\sqrt{NT}$-consistent and asymptotically normal. When
\eqref{eqn:strongfactor} fails they are in general slower than $\sqrt{NT}$.
The approach residualises only through $y_{it}-x_{it}'\beta$, so the $x$-only
block $\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)$ of \eqref{eq:x} is never projected
out, however strong or low-rank, and keeps identifying $\beta_0$.
\end{example}
We call the next model the common correlated effects factor (CCEF) model to
distinguish it from the CCE model of \citet{Pesaran2006}, whose conditions are
tied to projection on cross-sectional means. CCEF lets the interactive effects
enter $x_{it}$ by assumption and admits generic factor-model estimators. {Given a preliminary
$\widetilde\beta$, estimate factor spaces jointly from $Y-X\widetilde\beta$ and the
covariates $X_j$. The CCEF estimator is then OLS after residualising both $Y$
and $X_j$ on these spaces.}
\begin{example}[Common correlated effects factor model (CCEF)]\label{ex:cce}
The CCEF model takes \eqref{eq:y}--\eqref{eq:x} and identifies functionals of
$(\alpha_i,\gamma_t)$ from functionals of $y_{it}$ and $x_{it}$ jointly. Assume
$g(\alpha_i,\gamma_t)=\varphi(\alpha_i)'\psi(\gamma_t)$ and
$g_x(\alpha_i,\gamma_t)=\varphi_x(\alpha_i)'\psi_x(\gamma_t)$ with
$\varphi,\psi,\varphi_x,\psi_x\in\mathbb{R}^{R_0}$, $R_0<\infty$. If either
\eqref{eqn:strongfactor} or
\begin{align}\label{eqn:CCEXstrongfactor}
\frac1N\varphi_x(\alpha)'\varphi_x(\alpha)>0,
&&
\frac1T\psi_x(\gamma)'\psi_x(\gamma)>0,
&&
\varphi(\alpha)\in{\rm span}(\varphi_x(\alpha)),\
\psi(\gamma)\in{\rm span}(\psi_x(\gamma)),
\end{align}
holds, CCEF estimates of $\beta$ are $\sqrt{NT}$-consistent and asymptotically
normal. Requiring only one of the two is weaker than
\eqref{eqn:strongfactor} alone. The price is higher variance, and in some cases
complete loss of identification. A functional
of $x_{it}$ cannot distinguish $g_x$ from $\widetilde g_x$, so
with $\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)=\widetilde\varphi_x(\widetilde\alpha_i)'
\widetilde\psi_x(\widetilde\gamma_t)$ the projected span is that of the \emph{combined}
loadings $(\varphi_x,\widetilde\varphi_x)$ and $(\psi_x,\widetilde\psi_x)$, and the exogenous
$x$-only variation is projected out with the common block.
\end{example}
\begin{example}[Bias-aware confidence intervals]\label{ex:ba}
The bias-aware approach allows preliminary estimates of $g(\alpha_i,\gamma_t)$
to be biased and widens the confidence interval for $\beta$ accordingly. For an
estimate $\widehat\beta^{(\rm ba)}$ and an upper bound $\bar B$ on its absolute bias,
the interval is
\begin{align*}
\big\{\widehat\beta^{(\rm ba)}\pm[\bar B+z_{1-\alpha/2}\cdot se(\widehat\beta^{(\rm ba)})]\big\},
\end{align*}
where $z_{1-\alpha/2}$ is the $1-\alpha/2$ Gaussian quantile. Any $\bar B>0$
costs power. The bound must cover the worst-case alignment of the unremoved
heterogeneity with the regressors. The method cannot tell the endogenous $g_x$
from the exogenous $\widetilde g_x$ in the common structure of $x_{it}$, so a strong
$x$-only block inflates the worst case and hence the width, although it
carries no bias.
\end{example}
Our common unobserved heterogeneity (CUH) estimator takes three steps. Start from a preliminary estimate
$\widetilde\beta$, for example the interactive-effects estimator of
\citet{Bai2009}, and the residuals $y_{it}-x_{it}'\widetilde\beta$.
\begin{enumerate}[label=\arabic*., leftmargin=*]
\item Compute the leading $R$ principal components of the residuals, unit
loadings and time factors, as in a factor model.
\item Regress each regressor on every principal component of the residuals,
unit by unit on the time factors and period by period on the unit loadings.
On each side keep the $L$ coefficient vectors that explain the most
variation in that regressor.
\item Project the leading components of steps 1 and 2 from both $y_{it}$ and
$x_{it}$. With $\widetilde y_{it}$ and $\widetilde x_{it}$ the projected data, the estimator is,
\[
\widehat\beta=\Big(\sum_{i,t}\widetilde x_{it}\widetilde x_{it}'\Big)^{-1}
\sum_{i,t}\widetilde x_{it}\widetilde y_{it}.
\]
\end{enumerate}
Step 2 sets CUH apart. Its covariate projections are built from outcome
directions, so the part of $x_{it}$ that the outcome does not share survives
step 3 and identifies $\beta_0$. Without step 2 this is the factor model of
Example~\ref{ex:factor}. CCEF replaces step 2 by principal components of the
regressors, which removes exogenous variation as well (Proposition~\ref{prop:eff}).
Section~\ref{sec:cuh-estimator} states the steps formally, and
Algorithm~\ref{alg:cf} gives the cross-fitted version used for inference.
The factor model is efficient but can be asymptotically biased under weak
factors. The CCEF approach also uses common structure in the covariates
to recover the latent factor spaces, at a cost in variance, and bias-aware
intervals keep coverage at a cost in power. CUH combines features of the first
two approaches, exploiting directions that are weak in the outcome but strong in
the covariates. Under homoskedastic errors its asymptotic variance is no larger
than that of CCEF, and the weak-factor theory quantifies how weak the common
factors may be while retaining $\sqrt{NT}$ inference without widening the interval.
Two smoothing variants extend the projection estimator
(Section~\ref{sec:smoothing}). NPCUH replaces projection by an additive kernel
smoother over the estimated proxies, and BCUH applies the smoother after the
CUH projection. Appendix~\ref{app:npcuh} gives conditions under which NPCUH is
asymptotically normal.
Figure~\ref{fig:intro-apps} previews the empirical applications.
{It reports point estimates for two
outcomes from the fiscal application as the number of purged outcome directions
$R$ varies, and, for one outcome, as the number of purged covariate components
varies.} In Section~\ref{sec:app-overlap} we show that when the residualised covariates
across different estimators have high correlation, they unsurprisingly have similar estimates,
and large gaps arise only where the correlation is low.
\begin{figure}[!ht]
\centering
\includegraphics[width=\textwidth]{figures/fig_overlap_coef.pdf}
\caption{Fiscal application of Section~\ref{sec:apps}. (a) Factor, CCEF and
CUH across the number $R$ of purged outcome directions. (b) Same outcome at $R=5$,
across $L=R_X$ covariate components. (c) Factor, NPCUH, BCUH and the two-way grouped
estimator of \citet{beyhum2025inference} (BM) on total expenditure. Bars are
95\% intervals, shaded bands the factor (b) and BM (c) intervals.}
\label{fig:intro-apps}
\end{figure}
Figure~\ref{fig:headline} previews the main simulation finding over a design
manifold of factor strength and the relative size of $T$ and $N$. Coverage of
the factor benchmark collapses along the weak-factor boundary. The CUH class
restores it, and BCUH and NPCUH have at least near-nominal coverage at almost all combinations of
strength and panel length.
Section~\ref{sec:sims} gives the full design with bias and variance surfaces.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.96\textwidth]{figures/comparison_manifold_coverage_kappa.pdf}
\caption{Simulated coverage of nominal 95\% intervals at $N=200$, over $T=N^{t}$ and
factor strength $\min(N,T)^{-w}$. The grey factor benchmark fails along the
weak-factor boundary ($w\geq 0.2$). In the CUH, NPCUH and BCUH panels, blue
cells are closer to nominal than the benchmark beyond Monte Carlo noise, amber
within noise, red farther. Details in Section~\ref{sec:sim-manifold}.}
\label{fig:headline}
\end{figure}
\subsection*{Related literature}
CUH belongs to the literature on large-panel regression with interactive
fixed effects. Under strong factors, the least-squares estimator of
\citet{Bai2009} iterates between slope estimates and principal components of
the residualised outcome and is $\sqrt{NT}$-normal.
\citet{MoonWeidner2015,MoonWeidner2017} develop its asymptotic expansions and
its robustness to overstating the number of factors, and
\citet{moon2018nuclear} target the low-rank component by nuclear-norm
regularisation. The common correlated effects approach of \citet{Pesaran2006}
instead proxies the factor space by cross-sectional averages of outcomes
\emph{and} covariates, with extensions to non-stationary factors
\citep{KapetaniosPesaranYamagata2011}, spatial correlation
\citep{PesaranTosetti2011} and nonlinear models \citep{chen2023common}.
\citet{westerlund2015cross} compare the averages-based and principal-components
routes, and \citet{BaiNg2002,Onatski2005,ahn2013eigenvalue} select the number
of factors.
When strong-factor conditions fail, factor-based slope estimates are generally biased. \citet{onatski2012}
shows that principal-component estimates are inconsistent once factor eigenvalues fail to
dominate the upper edge of the noise spectrum, and characterises the bias
through the Marchenko--Pastur law. \citet{bai2023approximate} establish
consistency and asymptotic normality of principal components when loadings have
strength $N^{\alpha}$ with $\alpha\in(0,1]$. \citet{uematsuyamagata2022}
estimate weak-factor structures with heterogeneous, consistently estimable
strength exponents, and \citet{ChudikPesaranTosetti2011} classify weak,
semi-strong and strong cross-section dependence in the CCE setting.
\citet{armstrong2026robust} develop the bias-aware intervals of
Example~\ref{ex:ba} for slope inference under weak factors. We
measure strength by the singular values of $g(\alpha_i,\gamma_t)$, an
index symmetric in loadings and factors (Remark~\ref{rem:alpha}).
We derive separate detection, selection and inference thresholds and their
dependence on the relative dimension $N\asymp T^\kappa$. Detection is necessary
to estimate the common directions. Below the detection threshold
the bias-aware construction is the natural fallback.
A third strand treats the heterogeneity nonparametrically.
\citet{freeman2023linear} approximate a smooth $g(\alpha_i,\gamma_t)$ by a
growing-rank factor structure, under which the least-squares slope estimator is
slower than $\sqrt{NT}$. \citet{fernandez2021low} study low-rank approximations
of nonseparable panel models,
\citet{chen2021nonlinear,zeleneev2020identification} nonlinear factor
structures in panel and network data, and \citet{freeman2026multidimensional}
multidimensional effects. Proxies for latent effects can be built by
discretisation and clustering \citep{bonhomme2021discretizing,beyhum2025inference},
from latent similarities \citep{deaner2025inferring}, or from leading
eigenfunctions. \citet{freeman2026nonparametric} develop
the identification theory for these eigenfunction proxies, and
\citet{freeman2026inference} combine them with Neyman-orthogonal moments and
two-way nonparametric regression to restore $\sqrt{NT}$-normality. Our
smooth-function regime (Section~\ref{sec:smooth}) uses the same spectral-decay
calibration, and Appendix~\ref{app:npcuh} adapts cross-fitted additive
smoothing to the CUH proxies.
The inference theory draws on orthogonal moments and bias corrections.
The moment condition is the partially linear score of
\citet{robinson1988root}.
Incidental-parameter bias in panels and its analytical and jackknife
corrections are studied by
\citet{HahnKuersteiner2002,HahnNewey2004,ArellanoHahn2007,FernandezValWeidner2013},
and the two-way sample split of \citet{freeman2023linear} is closest to our
construction. Cross-fitting centres the stochastic errors, so inference is
valid under primitive conditions without a separate bias correction.
Section~\ref{sec:cuh-estimator} defines the estimators.
Sections~\ref{sec:weak} and~\ref{sec:inference} develop the weak-factor and
inference theory, and Sections~\ref{sec:sims} and~\ref{sec:apps} report
simulations and empirical applications. The appendices contain proofs,
implementation details and the theory for NPCUH.
\section{The CUH estimator}\label{sec:cuh-estimator}
{This section defines the CUH estimator and the objects used in the weak-factor theory.}
Throughout, the data follow \eqref{eq:y}--\eqref{eq:x}.
{
Under exogeneity, $\mathbb{E}[x_{it}\varepsilon_{it}]=0$, the population
least-squares coefficient is
\begin{align*}
\mathbb{E}[x_{it} x_{it}']^{-1} \mathbb{E}[x_{it} y_{it}]
=
\beta +
\mathbb{E}[x_{it} x_{it}']^{-1} \mathbb{E}[x_{it} g(\alpha_i, \gamma_t)],
\end{align*}
so the bias comes from $\mathbb{E}[x_{it} g(\alpha_i,\gamma_t)]$.
Factor models and CCE identify $g(\alpha_i,\gamma_t)$ and partial it out by a
Frisch--Waugh--Lovell reduction. This is sufficient but not necessary. What
must be controlled is only the \emph{common} variation that makes
$\mathbb{E}[x_{it} g(\alpha_i,\gamma_t)]$ non-zero.
Projecting the covariates on the outcome directions strengthens the estimates
of factors that bias the slope but are hard to detect against the noise in
$y_{it}$. In this sense the covariate projections act as an additional sieve
that selects which outcome components induce bias.
CUH therefore needs far less factor strength than the strong-factor
conditions require.
}
\subsection{Notation and asymptotic regime}\label{sec:notation}\label{sec:regime}
Stack the data into $N\times T$ matrices $Y=(y_{it})$, $X_j=(x_{it,j})$ for
$j=1,\dots,d$, $G=(g(\alpha_i,\gamma_t))$ and $E=(\varepsilon_{it})$. Every
singular value decomposition (SVD) uses the statistical normalisation, so a
generic $N\times T$ matrix $M=\sum_r s_r u_r v_r'$ has $u_r'u_r/N=1$ and
$v_r'v_r/T=1$. This matches the population expansion
\begin{align*}
g(a,c)=\sum_{r\ge1}\sigma_r u_r(a)v_r(c), \quad \mathbb{E}[u_r(\alpha)u_s(\alpha)]
=\mathbb{E}[v_r(\gamma)v_s(\gamma)]=\mathbf 1\{r=s\},
\end{align*}
which, evaluated at the sample, gives
\begin{equation}\label{eq:Gsvd}
G=\sum_{r\ge1}\sigma_r\,u_r^{(g)}\,{v_r^{(g)}}',
\qquad
u_r^{(g)}=(u_r(\alpha_1),\dots,u_r(\alpha_N))',\quad
v_r^{(g)}=(v_r(\gamma_1),\dots,v_r(\gamma_T))',
\end{equation}
so the strength of common direction $r$ is carried entirely by $\sigma_r$. We
write $\left\lVert\cdot\right\rVert_{\rm op}$ and $\left\lVert\cdot\right\rVert_F$ for the operator and
Frobenius norms, $a\wedge b=\min\{a,b\}$, $a\vee b=\max\{a,b\}$ and
$\left\langle A,B\right\rangle_F=\operatorname{tr}(A'B)$. For a frame $U=(u_1,\ldots,u_K)$ and
$\mathcal I\subseteq\{1,\ldots,K\}$, let $U_{\mathcal I}=\{u_r:r\in\mathcal I\}$.
We work in the regime
\begin{equation}\label{eq:regime}
N\asymp T^{\kappa},\qquad \kappa\in(0,\infty),
\end{equation}
where $\kappa=1$ is the square case and $N$ grows faster than $T$ when
$\kappa>1$. Then $N\wedge T\asymp T^{\min\{1,\kappa\}}$,
$N\vee T\asymp T^{\max\{1,\kappa\}}$ and $NT\asymp T^{1+\kappa}$. Most results
are stated in $N\wedge T$, $N\vee T$ and $NT$, and so hold for every $\kappa>0$.
\subsection{The CUH construction}\label{sec:cuh-alg}
Given a preliminary estimator $\widetilde\beta$, the residualised outcome matrix is
\begin{equation}\label{eq:W}
W:=Y-\sum_{j=1}^d X_j\widetilde\beta_j
=X(\beta_0-\widetilde\beta)+G+E,
\end{equation}
where $X(\beta_0-\widetilde\beta):=\sum_jX_j(\beta_{0,j}-\widetilde\beta_j)$. Let
$W=\widehat U\widehat\Sigma\widehat V'$ be its SVD, whose singular vectors are the outcome
directions, and let $R$ be the number of detectable directions
(Assumption~\ref{ass:strength}). Write $\mathcal I_R=\{1,\ldots,R\}$,
$\widehat U_R=\widehat U_{\mathcal I_R}$ and $\widehat V_R=\widehat V_{\mathcal I_R}$. For
$j\le d$ and $r\le N\wedge T$, the covariate projections and their projection
energies are
\begin{align*}
\widehat\lambda_{j,r}(\alpha_i)&:=\frac1T\sum_t x_{it,j}\widehat v_{tr},
&
\widehat f_{j,r}(\gamma_t)&:=\frac1N\sum_i x_{it,j}\widehat u_{ir},\\
\widehat\sigma_{\alpha,j,r}^2&:=\frac1N\sum_i\widehat\lambda_{j,r}^2(\alpha_i),
&
\widehat\sigma_{\gamma,j,r}^2&:=\frac1T\sum_t\widehat f_{j,r}^2(\gamma_t).
\end{align*}
For each regressor, rank the outcome directions separately by the two
energies, let $\mathcal J_j^\alpha$ and $\mathcal J_j^\gamma$ index the
leading $L_j$ in each ranking, and set
\[
\widehat\Lambda_j
:=\frac1T X_j\widehat V_{\mathcal J_j^\alpha},
\qquad
\widehat F_j
:=\frac1N X_j'\widehat U_{\mathcal J_j^\gamma}.
\]
The stacked projections $\widehat\Lambda=(\widehat\Lambda_1,\ldots,\widehat\Lambda_d)$ and
$\widehat F=(\widehat F_1,\ldots,\widehat F_d)$, with $\widehat U_R$ and $\widehat V_R$, determine the
CUH transforms \eqref{eq:transforms}.
Section~\ref{sec:weak} shows that the outcome directions, covariate
projections and energy rankings are consistent for the detectable candidates
$r\in\mathcal I_R$ under the detectability condition of
Assumption~\ref{ass:strength}. The implementation
screens all estimated outcome directions, which can help near the detection
boundary, while the inference results are established for ranking restricted
to $\mathcal I_R$. {The smoothing variants of Section~\ref{sec:smoothing} accommodate the undetectable tail by smoothing rather than projecting.}
Our main proposal is the cross-fitted CUH estimator. Cross-fitting builds the
transforms on training subsamples disjoint from the block they residualise,
which limits their dependence on the evaluation errors.
Algorithm~\ref{alg:cf} in Appendix~\ref{app:crossfit} states the procedure.
\subsection{Transforms and scores}\label{sec:transforms}
Let $P_C:=C(C'C)^{+}C'$ project onto the column span of $C$, with
$(\cdot)^{+}$ the Moore--Penrose inverse, and let $M_C:=I-P_C$ be the residual
maker. The CUH transforms are
\begin{equation}\label{eq:transforms}
\widehat A=M_{\widehat\Lambda }M_{\widehat U_R},\qquad
\widehat B=M_{\widehat V_R}M_{\widehat F },
\end{equation}
each a product of two residual makers that need not commute. The oracle
transforms $A=M_{\Lambda}M_{U_R}$ and $B=M_{V_R}M_{F}$ use the population
analogues $U_R,V_R,\Lambda,F$.
For any $N\times T$ matrix $H$, the feasible and oracle operators are
\[
\widehat{\mathcal R}[H]
:=\widehat AH\widehat B,
\qquad
\mathcal R^{0}[H]
:=AHB .
\]
The CUH estimator is $\widehat\beta=\widehat Q^{-1}\widehat S$, where, for
$j,\ell=1,\dots,d$,
\begin{equation}
[\widehat Q]_{j\ell}
=\frac1{NT}\left\langle \widehat{\mathcal R}[X_j],\widehat{\mathcal R}[X_\ell]\right\rangle_F,
\qquad
[\widehat S]_j
=\frac1{NT}\left\langle \widehat{\mathcal R}[X_j],\widehat{\mathcal R}[Y]\right\rangle_F. \label{eq:gram_score}
\end{equation}
Substituting $Y=\sum_{q=1}^d X_q\beta_{0,q}+G+E$ gives
\begin{equation}
\widehat S
=\widehat Q\beta_0 +\frac1{NT}
\left(
\left\langle
\widehat{\mathcal R}[X_j],
\widehat{\mathcal R}[G]
\right\rangle_F
\right)_{j=1}^d +\frac1{NT}
\left(
\left\langle
\widehat{\mathcal R}[X_j],
\widehat{\mathcal R}[E]
\right\rangle_F
\right)_{j=1}^d .
\label{eq:score-decomp}
\end{equation}
The $\widehat{\mathcal R}[G]$ term is the heterogeneity left after
projection, and the $\widehat{\mathcal R}[E]$ term carries the score of the
central limit theorem.
\subsection{Smoothing variants, NPCUH and BCUH}\label{sec:smoothing}
The transforms \eqref{eq:transforms} remove the span of the estimated
directions, so they need those directions estimated accurately. Two variants
smooth over the same directions instead. A smoother removes any function of
the proxies, so it needs only injectivity of the proxy collection, not
accurate estimates of the population directions. Collect the outcome
directions and the selected covariate projections of
Section~\ref{sec:cuh-alg} into the unit-side and time-side proxy vectors
\[
\widehat\pi_i^{(\alpha)}
:=\big(\{\widehat u_{ir}\}_{r\in\mathcal I_R},\,
\{\widehat\lambda_{j,r}(\alpha_i)\}
_{j\le d,\,r\in\mathcal J_j^\alpha}\big),
\qquad
\widehat\pi_t^{(\gamma)}
:=\big(\{\widehat v_{tr}\}_{r\in\mathcal I_R},\,
\{\widehat f_{j,r}(\gamma_t)\}
_{j\le d,\,r\in\mathcal J_j^\gamma}\big),
\]
each with $Q_\pi:=R+\sum_jL_j$ coordinates. Write
$\widehat\pi_{i,q}^{(\alpha)}$ and $\widehat\pi_{t,q}^{(\gamma)}$ for the $q$th
coordinates and $K_h(u):=h^{-1}K(u/h)$ for a univariate kernel $K$. For each
coordinate $q$ the row-normalised kernel smoothers are
\begin{equation}\label{eq:np-smoother}
\big[S^{(\alpha)}_{q,h_\alpha}\big]_{ij}
:=\frac{K_{h_\alpha}\big(\widehat\pi^{(\alpha)}_{j,q}-\widehat\pi^{(\alpha)}_{i,q}\big)}
{\sum_{\ell\le N}K_{h_\alpha}\big(\widehat\pi^{(\alpha)}_{\ell,q}-\widehat\pi^{(\alpha)}_{i,q}\big)},
\qquad
\big[S^{(\gamma)}_{q,h_\gamma}\big]_{ts}
:=\frac{K_{h_\gamma}\big(\widehat\pi^{(\gamma)}_{s,q}-\widehat\pi^{(\gamma)}_{t,q}\big)}
{\sum_{\ell\le T}K_{h_\gamma}\big(\widehat\pi^{(\gamma)}_{\ell,q}-\widehat\pi^{(\gamma)}_{t,q}\big)}.
\end{equation}
The nonparametric CUH estimator (NPCUH) residualises with these
smoothers,\footnote{Appendix~\ref{app:npcuh} analyses the backfitted form of
this smoother. The product form here is usually numerically very close and is
cheaper to compute.}
\begin{equation}\label{eq:np-sweep}
\widehat{\mathcal M}[Z]:=(I-\widehat W^\alpha)Z(I-\widehat W^\gamma)',
\quad
I-\widehat W^\alpha:=\prod_{q\le Q_\pi}
(I-S^{(\alpha)}_{q,h_\alpha}),
\quad
I-\widehat W^\gamma:=\prod_{q\le Q_\pi}
(I-S^{(\gamma)}_{q,h_\gamma}).
\end{equation}
The NPCUH estimator is then constructed as
$\widehat\beta=\widehat Q^{-1}\widehat S$, replacing
$\widehat{\mathcal R}$ with $\widehat{\mathcal M}$ in \eqref{eq:gram_score}. In words, NPCUH compares observations in terms of their estimated latent heterogeneity and removes the shared systematic variation.
The blended estimator (BCUH) smooths the CUH projection residual, using
$\widehat{\mathcal M}\circ\widehat{\mathcal R}$ in place of $\widehat{\mathcal R}$. Both
maps are linear, so \eqref{eq:score-decomp} holds with either in place of
$\widehat{\mathcal R}$.
Sections~\ref{sec:sims} and~\ref{sec:apps} report the cross-fitted versions of
both, built as in Algorithm~\ref{alg:cf} with the smoothers in place.
\section{Weak-factor theory}\label{sec:weak}
This section bounds the estimated singular subspaces of $W$ in
\eqref{eq:W} using a single factor-strength index and a
singular-vector perturbation bound. Write
$\Delta_\beta:=X(\beta_0-\widetilde\beta)$, so that
$W=G+E+\Delta_\beta$. We maintain the standard noise condition
$\left\lVertE\right\rVert_{\rm op}=O_p(\sqrt{N\vee T})$ (\citealp{Bai2009,MoonWeidner2015}), which implies
\[
s_{NT}:=\frac{\left\lVertE\right\rVert_{\rm op}}{\sqrt{NT}}
=O_p((N\wedge T)^{-1/2}).
\]
The noise edge $s_{NT}$ bounds how weak a factor can be and remain detectable.
Appendix~\ref{app:weak-proofs} provides sufficient conditions
for the noise bound under independence or weak dependence.
\subsection{Factor strength and detectability}
\begin{assumption}[Singular-value strength and detectability]\label{ass:strength}
Let $\sigma_1(G)\ge\sigma_2(G)\ge\ldots$ be the singular values of $G$
under the statistical normalisation \eqref{eq:Gsvd}, and write
$\sigma_{R}:=\sigma_{R}(G)$ for the weakest candidate strength.
(i) (Detectability) $c_{NT}:=\sigma_{R}$ satisfies
\begin{equation}\label{eq:SNR}
c_{NT}\sqrt{N\wedge T}\to\infty,
\end{equation}
equivalently $c_{NT}/s_{NT}\to\infty$ with $s_{NT}$ read as its
rate $(N\wedge T)^{-1/2}$ from Lemma~\ref{lem:edge}.
(ii) (Spectral gap)
$\delta_{NT}:=\sigma_{R}(G)-\sigma_{R+1}(G)$ satisfies
\begin{align}
\label{eq:spectral_gap}
\delta_{NT}\asymp c_{NT} .
\end{align}
(iii) (Distinct candidate strengths) There is $c>0$ such that
$|\sigma_r(G)^2-\sigma_s(G)^2|\ge c\,\sigma_r(G)\,\sigma_s(G)$ for all
$r\neq s\le R$.
\end{assumption}
Condition \eqref{eq:SNR} is the detection threshold of the spiked-covariance
literature. A population singular direction appears in the sample spectrum of
$W$ if and only if its singular value exceeds the upper edge of the noise
spectrum \citep{onatski2012,bai2023approximate}.
Part (ii) is the spectral gap that the perturbation argument needs in the
spiked (finite-rank) case. When the spectrum decays the gap vanishes, and
Section~\ref{sec:smooth} treats that case. Part (iii) makes the candidate
strengths asymptotically distinct, so that results hold for individual singular
directions (Proposition~\ref{prop:xi}, Corollary~\ref{cor:select}). At an exact
tie only the span of the tied block is estimable, but the transforms are
invariant to rotations within it, so for slope estimation part (iii) is a
simplification rather than a restriction.
\begin{remark}[Calibrating the singular-value strength]\label{rem:alpha}
Assumption~\ref{ass:strength} restricts the singular values $\sigma_r$ and is
agnostic to whether weakness originates in the loadings, the factors, or both.
Write $G=\Lambda^0F^{0\prime}$, let $\Lambda^0_r\in\mathbb{R}^N$ and $F^0_r\in\mathbb{R}^T$,
$r=1,\ldots,N\wedge T$, be the columns of the true loading and factor matrices,
and let $s_r(G)$ be the ordinary singular values of $G$. Three calibrations of
$c_{NT}=\sigma_{R}$ follow.
\begin{enumerate}[(a)]
\item One-sided weak loadings \citep{bai2023approximate},
$\Lambda^{0\prime}_r\Lambda^0_r/N^{\alpha_r}\to c_r\in(0,\infty)$ with
$\alpha_r\in(0,1]$ and strong factors, so $s_r(G)^2\asymp N^{\alpha_r}T$ and
$c_{NT}\asymp N^{(\alpha_{R}-1)/2}\asymp T^{\kappa(\alpha_{R}-1)/2}$.
\item Symmetric weak loadings and factors,
$\Lambda^{0\prime}_r\Lambda^0_r/N^{\alpha_r}\to c_r\in(0,\infty)$ \emph{and}
$F^{0\prime}_rF^0_r/T^{\tau_r}\to d_r\in(0,\infty)$ with $\alpha_r,\tau_r\in(0,1]$,
so $s_r(G)^2\asymp N^{\alpha_r}T^{\tau_r}$ and
$c_{NT}\asymp N^{(\alpha_{R}-1)/2}T^{(\tau_{R}-1)/2}$. Here $N$ and $T$
enter symmetrically, and case~(a) is $\tau_r\equiv1$.
\item Smooth function or decaying spectrum \citep{freeman2023linear,
freeman2026inference,freeman2026nonparametric}, where the rank is not fixed
and $\sigma_r$ decays in $r$ (Section~\ref{sec:smooth}).
\end{enumerate}
Under (b), detectability \eqref{eq:SNR} reads
\begin{equation}\label{eq:detect-rho}
\kappa(1-\alpha_{R})+(1-\tau_{R})<\min\{1,\kappa\},
\end{equation}
and under (a), the calibration used in the rest of the paper, it reads
\begin{equation}
\alpha_{R}>\max\{0,1-1/\kappa\}. \label{eq:detectability}
\end{equation}
\end{remark}
\subsection{Estimation of the singular subspaces}
Define the population covariate projections
$\lambda_{j,r}(\alpha):=\int_{\Omega_\gamma}g_{x,j}(\alpha,c)v_r(c)\pi_\gamma(c)\,dc$
and symmetrically $f_{j,r}(\gamma)$, with feasible counterparts
$\widehat\lambda_{j,r},\widehat f_{j,r}$ from Section~\ref{sec:cuh-alg}. The same
symbols denote the stacked vectors, such as
$\lambda_{j,r}=(\lambda_{j,r}(\alpha_i))_{i\le N}$ and
$f_{j,r}=(f_{j,r}(\gamma_t))_{t\le T}$. For vectors indexed by units or
periods, write $\|a\|_N^2=N^{-1}\sum_i a_i^2$ and
$\|b\|_T^2=T^{-1}\sum_t b_t^2$.
\begin{assumption}[Regularity]\label{ass:reg}
(i) $\mathbb{E}\eta_{it}^4<\infty$, $\mathbb{E}\varepsilon_{it}^4<\infty$,
$g_{x,j}\in L^4(\pi_\alpha\otimes\pi_\gamma)$,
$u_r\in L^4(\pi_\alpha)$, $v_r\in L^4(\pi_\gamma)$ for every fixed
$j\le d$, $r\in\mathcal I_{R}$, and
$\left\lVertg\right\rVert_{L^2(\pi_\alpha\otimes\pi_\gamma)}<\infty$.
(ii) The eigenfunctions are orthonormal,
$\int u_r u_{r'}\pi_\alpha=\mathbbm 1\{r=r'\}$ and
$\int v_r v_{r'}\pi_\gamma=\mathbbm 1\{r=r'\}$.
(iii) For every fixed $r,r'\in\mathcal I_{R}$ and $j\le d$, the
evaluated eigenfunctions and regressors satisfy
$\big|N^{-1}\sum_i u_r(\alpha_i)u_{r'}(\alpha_i)-\mathbbm 1\{r=r'\}\big|
=O_p(N^{-1/2})$,
$\big|T^{-1}\sum_t v_r(\gamma_t)v_{r'}(\gamma_t)-\mathbbm 1\{r=r'\}\big|
=O_p(T^{-1/2})$ and $\|X_j\|_F/\sqrt{NT}=O_p(1)$.
(iv) For every fixed $j\le d$, $r\in\mathcal I_{R}$, the oracle covariate
projections and their energies obey
\begin{align*}
\|T^{-1}X_jv_r-\lambda_{j,r}\|_N&=O_p(T^{-1/2}),&
\|N^{-1}X_j'u_r-f_{j,r}\|_T&=O_p(N^{-1/2}),\\
\big|\|\lambda_{j,r}\|_N^2-\mathbb{E}[\lambda_{j,r}^2(\alpha)]\big|&=O_p(N^{-1/2}),&
\big|\|f_{j,r}\|_T^2-\mathbb{E}[f_{j,r}^2(\gamma)]\big|&=O_p(T^{-1/2}).
\end{align*}
\end{assumption}
Parts (iii)--(iv) are sampling regularity conditions. They hold under
independent sampling with the displayed moments and under weakly dependent or
mixing designs with the same root-$N$ and root-$T$ laws of large numbers. The
product measure $\pi_\alpha\otimes\pi_\gamma$ reflects averaging over the
rectangular panel, and neither it nor parts (iii)--(iv) require the unit and
time effects to be independent. The first pair of bounds in part (iv)
follows, for example, from bounded average long-run variances of $x_{it,j}v_r(\gamma_t)$
over $t$ and of $x_{it,j}u_r(\alpha_i)$ over $i$. As $d$ and $R$ are fixed in the spiked
regime, all rates hold jointly over $j\le d$ and $r,r'\in\mathcal I_{R}$.
Let $\xi_u,\xi_v$ be the sign-aligned root-mean-square errors of the
estimated singular vectors relative to the evaluated population
eigenfunctions,
\[
\xi_u^2:=\max_{r\in \mathcal{I}_{R}}\,\min_{s\in\{\pm1\}}\,
\|u_r^{(g)}-s\,\widehat u_r\|_N^2,
\qquad
\xi_v^2:=\max_{r\in \mathcal{I}_{R}}\,\min_{s\in\{\pm1\}}\,
\|v_r^{(g)}-s\,\widehat v_r\|_T^2 .
\]
The minimum removes the sign indeterminacy of the SVD.
\begin{proposition}[Subspace singular-vector]\label{prop:xi}
Impose Assumptions~\ref{ass:strength}--\ref{ass:reg}(i)--(iii), the noise edge
$\|E\|_{\rm op} = O_p(\sqrt{N\vee T})$, and
$\|E+\Delta_\beta\|_{\rm op}/\sqrt{NT}=o_p(\delta_{NT})$. Then, for every
$\kappa>0$,
\begin{equation}\label{eq:xi}
\left\lVertP_{\widehat U_{R}}-P_{U_{R}}\right\rVert_{\rm op}^2,\ \left\lVertP_{\widehat V_{R}}-P_{V_{R}}\right\rVert_{\rm op}^2,\ \xi_u^2,\ \xi_v^2
\;=\;O_p\!\Big(\underbrace{\frac{1}{c_{NT}^2\,(N\wedge T)}}_{\text{noise}}\Big)
\;+\;O_p\!\Big(\underbrace{\frac{\left\lVertX\right\rVert_{\rm op}^2\,\left\lVert\beta_0-\widetilde\beta\right\rVert^2}{NT\,c_{NT}^2}}_{\text{first-step }\beta}\Big),
\end{equation}
where $U_{R}=(u_r(\alpha_i))_{i\le N,r\in \mathcal{I}_{R}}$ and
$V_{R}=(v_r(\gamma_t))_{t\le T,r\in \mathcal{I}_{R}}$
are the evaluated eigenfunction frames.
\end{proposition}
\begin{remark}[Sharp rate and admissible $\kappa$]\label{rem:xi-rho}
Bound \eqref{eq:xi} is dimension-symmetric and sharp for $\kappa\le1$. Under calibration (a) of Remark~\ref{rem:alpha},
$c_{NT}\asymp T^{\kappa(\alpha_{R}-1)/2}$, so the noise part is
$\xi_v^2\asymp T^{\kappa(1-\alpha_{R})-\min\{1,\kappa\}}$. It vanishes if and
only if $\kappa(1-\alpha_{R})<\min\{1,\kappa\}$, that is, if $\alpha_{R}>0$ for
$\kappa\le1$ and if $\kappa<1/(1-\alpha_{R})$ for $\kappa>1$, the upper bound
of \eqref{eq:detect-rho}. For $\kappa>1$ the symmetric Wedin bound is not
sharp. In an approximate factor model the one-sided rate
$\xi_v^2=O_p(N^{-\alpha_{R}}+(N^{1-\alpha_{R}}/T)^2)$ of
\citet{bai2023approximate} refines it with the same consistency window
$\kappa<1/(1-\alpha_{R})$. The mirror statement in the loading dimension
governs $\xi_u$.
\end{remark}
\begin{theorem}[Projection rates]\label{thm:proj}
Under Assumption~\ref{ass:reg}, with the estimated singular vectors
sign-aligned as in the definitions of $\xi_u,\xi_v$,
\[
\max_{j\le d,\,r\in\mathcal I_{R}}
\|\widehat\lambda_{j,r}-\lambda_{j,r}\|_N=O_p(\xi_v+T^{-1/2}),
\qquad
\max_{j\le d,\,r\in\mathcal I_{R}}
\|\widehat f_{j,r}-f_{j,r}\|_T=O_p(\xi_u+N^{-1/2}).
\]
\end{theorem}
With Proposition~\ref{prop:xi}, Theorem~\ref{thm:proj} gives consistent
covariate projections under the noise edge, detectability and the spectral
gap. The $T^{-1/2}$ and $N^{-1/2}$ terms are the oracle rates for known
population singular functions. Under strong factors $\xi_u$ and $\xi_v$ are
root-$N$ and root-$T$ consistent, and generally slower otherwise.
\subsection{Selection consistency}
Let $\sigma_{\alpha,j,r}^2:=\mathbb{E}[\lambda_{j,r}^2(\alpha)]$ and
$\sigma_{\gamma,j,r}^2:=\mathbb{E}[f_{j,r}^2(\gamma)]$ be the population counterparts
of the projection energies $\widehat\sigma_{\alpha,j,r}^2$ and
$\widehat\sigma_{\gamma,j,r}^2$ of Section~\ref{sec:cuh-alg}.
\begin{proposition}[Energy expansion]\label{prop:sigma}
Under Assumptions~\ref{ass:strength}--\ref{ass:reg},
uniformly over $r\in \mathcal{I}_{R}$,
\begin{align}
\widehat\sigma_{\alpha,j,r}^2
=&\sigma_{\alpha,j,r}^2
+O_p\big(\sigma_{\alpha,j,r}(\xi_v+T^{-1/2})\big)
+O_p\big(\xi_v^2+(N\wedge T)^{-1}\big), \label{eq:sigmahat_alpha} \\
\widehat\sigma_{\gamma,j,r}^2
=&\sigma_{\gamma,j,r}^2
+O_p\big(\sigma_{\gamma,j,r}(\xi_u+N^{-1/2})\big)
+O_p\big(\xi_u^2+(N\wedge T)^{-1}\big).\label{eq:sigmahat_gamma}
\end{align}
\end{proposition}
The estimated energies are therefore consistent whenever the covariate
projections are, and their error rates set the minimum $x$-relevance strength
that consistent ranking requires.
\begin{assumption}[Covariate-relevance strength]
\label{ass:Xstrength}
Among the candidates $r\in\mathcal I_R$, the selected directions have energies
$\sigma_{\alpha,j,r}^2\ge\epsilon_{NT}^2$ with
$\epsilon_{NT}^2\gg\xi_v^2+(N\wedge T)^{-1}$, while the non-selected directions
have $\sigma_{\alpha,j,r}^2=O_p((N\wedge T)^{-1})$, and symmetrically for the
time side.
\end{assumption}
\begin{corollary}[Thresholding consistency]\label{cor:select}
Make Assumptions~\ref{ass:strength}--\ref{ass:Xstrength}, and let the smallest
\emph{nonzero} eigenvalues of $\Lambda'\Lambda/N$ and $F'F/T$ be at least
$c\,\epsilon_{NT}^2$ for a fixed $c>0$. Then, with probability approaching one,
ranking by $\widehat\sigma_{\alpha,j,r}^2$ (resp.\ $\widehat\sigma_{\gamma,j,r}^2$) over
$r\in\mathcal I_{R}$ selects exactly the $x$-relevant common directions. Write
$\Lambda \in \mathbb{R}^{p_\alpha},F \in \mathbb{R}^{p_\gamma}$ for the
population components, and
\begin{equation}\label{eq:zeta}
\zeta_\Lambda:=(\xi_v+T^{-1/2})/\epsilon_{NT},\qquad
\zeta_F:=(\xi_u+N^{-1/2})/\epsilon_{NT} .
\end{equation}
Then the estimated components satisfy
\begin{equation}\label{eq:select-angle}
\left\lVertM_{\widehat\Lambda}P_\Lambda\right\rVert_{\rm op}
=\left\lVertP_\Lambda M_{\widehat\Lambda}\right\rVert_{\rm op}=O_p(\zeta_\Lambda),
\qquad
\left\lVertM_{\widehat F}P_F\right\rVert_{\rm op}
=\left\lVertP_FM_{\widehat F}\right\rVert_{\rm op}=O_p(\zeta_F),
\end{equation}
and for every $N\times T$ matrix $H$,
\begin{align}
\left\lVert(P_{\widehat\Lambda}-P_\Lambda)H\right\rVert_F
&\le\sqrt{p_\alpha}\,\left\lVertM_\Lambda H\right\rVert_{\rm op}
+O_p(\zeta_\Lambda)\left\lVertH\right\rVert_F, \label{eq:select-energy-L}\\
\left\lVertH(P_F-P_{\widehat F})\right\rVert_F
&\le\sqrt{p_\gamma}\,\left\lVertHM_F\right\rVert_{\rm op}
+O_p(\zeta_F)\left\lVertH\right\rVert_F. \label{eq:select-energy-R}
\end{align}
\end{corollary}
The ranking statistic $\widehat\sigma_{\alpha,j,r}^2=\left\lVert\widehat\lambda_{j,r}\right\rVert_N^2$
measures the strength of a common direction in the $x$ equation, not the
singular value $\sigma_r$ of the residualised outcome. Selection is therefore
governed by a second strength index $\alpha^x_r$, generally distinct from the
outcome strength $\alpha_r$ of Remark~\ref{rem:alpha}, so that $\epsilon_{NT}$
depends on $\alpha^x$ while $c_{NT}$ and $\delta_{NT}$ depend on $\alpha$.
\begin{remark}[Factor strength requirements]
\label{rem:strength-consequences}
Let $c_{NT}^2\asymp N^{\alpha_{R}-1}$ and $N\asymp T^\kappa$.
\begin{enumerate}[(i)]
\item By \eqref{eq:detectability}, detection requires
$\alpha_{R}>\max\{0,1-1/\kappa\}$.
\item By \eqref{eq:xi},
$\xi_u^2\vee\xi_v^2=O_p\big(T^{\kappa(1-\alpha_{R})-\min\{1,\kappa\}}\big)$
when the preliminary-estimator term is negligible.
\item Let $\epsilon_{NT}^2\asymp N^{\alpha_{R}^x-1}$ be the strength of the
weakest selected direction in $x$. The condition
$\epsilon_{NT}^2\gg\xi_v^2+(N\wedge T)^{-1}$ of Assumption~\ref{ass:Xstrength} is
implied by
$\alpha_{R}+\alpha_{R}^x>2-\min\{1,\kappa\}/\kappa$.
\item Under Assumption~\ref{ass:cf-exact-spike} of
Section~\ref{sec:inference}, $\xi_u+\xi_v+\zeta_\Lambda+\zeta_F$ is of order
$(c_{NT}\epsilon_{NT})^{-1}(N\wedge T)^{-1/2}$, and
$\sqrt{NT}\,(\xi_u+\xi_v+\zeta_\Lambda+\zeta_F)^3=o_p(1)$ holds when
\begin{equation}\label{eq:inference-frontier}
c_{NT}\,\epsilon_{NT}\gg\Big\{\frac{N\vee T}{(N\wedge T)^2}\Big\}^{1/6},
\qquad
\alpha_{R}+\alpha_{R}^x>\tau(\kappa):=
\begin{cases}
\frac43+\frac1{3\kappa}, & \kappa\le1,\\[2pt]
\frac73-\frac2{3\kappa}, & \kappa\ge1.
\end{cases}
\end{equation}
By \eqref{eq:zeta}, $\epsilon_{NT}$ must then exceed $\xi_v+T^{-1/2}$ and
$\xi_u+N^{-1/2}$ by a factor larger than $(NT)^{1/6}$, which is stronger than
Assumption~\ref{ass:Xstrength}. At $\kappa=1$, $\tau(1)=5/3$.
\end{enumerate}
\end{remark}
\subsection{The smooth-function regime}\label{sec:smooth}
The spiked calibrations (a)--(b) of Remark~\ref{rem:alpha} keep $R$ fixed.
When $g$ is a smooth nonparametric function, $G$ has infinitely many decaying
singular values and the candidate truncation grows, $R=R_{NT}\to\infty$.
Assume the spectral decay
\begin{equation}\label{eq:decay}
\sigma_r\asymp r^{-(\rho+\frac12)},
\qquad\text{so that the truncation error}\quad
\zeta_{R}^2:=\sum_{r>R}\sigma_r^2
\asymp R^{-2\rho},
\end{equation}
with smoothness index $\rho=p/\min\{d_\alpha,d_\gamma\}$ for
$g\in H^p(\Omega_\alpha,\Omega_\gamma)$. Applied to the weakest candidate,
$c_{NT}=\sigma_{R}\asymp R^{-(\rho+1/2)}$, detectability \eqref{eq:SNR}
becomes
\begin{equation}\label{eq:detect-R}
\sigma_{R}\sqrt{N\wedge T}\to\infty
\qquad\Longleftrightarrow\qquad
R=o\big((N\wedge T)^{1/(2\rho+1)}\big),
\end{equation}
the growing-$R$ counterpart of \eqref{eq:detect-rho}.
The relative gap now vanishes,
$\delta_{R}/\sigma_{R}=(\sigma_{R}-\sigma_{R+1})/\sigma_{R}\asymp R^{-1}\to0$,
so Lemma~\ref{lem:wedin}, which needs $\delta_{NT}\asymp c_{NT}$, no longer
gives a useful rate. Instead split $G=G_{R}+G_{>R}$ into its leading-$R$ part
and the tail, with $\left\lVertG_{>R}\right\rVert_F/\sqrt{NT}\asymp\zeta_{R}$, and carry the
tail as an additional perturbation. Increasing $R$ shrinks the tail but adds
weaker directions whose proxies are harder to estimate, so $R$ trades
truncation bias against accuracy within the limit
\eqref{eq:detect-R}.
CUH projection leaves the tail $G_{>R}$, which is nonzero for every finite $R$,
and its product with the common block of $X$ left after projection need
not be negligible. CUH and Theorem~\ref{thm:cf} are therefore confined to the
spiked regime. NPCUH (Section~\ref{sec:smoothing}) smooths instead of projecting, and
Appendix~\ref{app:npcuh} develops its theory for the decaying spectrum. Under a uniform immersion of the proxy collection NPCUH can
keep the number of outcome directions fixed, while an almost-everywhere
immersion requires growing $R$.
\section{Inference}\label{sec:inference}
\subsection{Cross-fitted CUH and inference}\label{sec:cf-lead}
Cross-fitting (Section~\ref{sec:cuh-alg}) is widely used in large panels
\citep[e.g.][]{freeman2023linear,beyhum2025inference,freeman2026inference}.
A temporal gap separates same-unit training and evaluation observations, which
allows weak temporal dependence.
\subsubsection{Sample split}\label{sec:sampleSplit}
Split the units into data-independent balanced halves
$\mathcal N_1,\mathcal N_2$, with $-a:=3-a$, and the periods into $K$
adjacent balanced blocks $\mathcal M_1,\ldots,\mathcal M_K$. Let
$\mathcal M_k^{\rm qc}$ collect the periods at distance greater than $q_T$
from $\mathcal M_k$. For $b=(a,k)$, $\mathcal I_b=\mathcal N_a\times\mathcal M_k$
is the evaluation block and $H^{(b)}$ the corresponding submatrix of $H$.
The preliminary estimator $\widetilde\beta_b$ uses
$\mathcal N_{-a}\times\mathcal M_k^{\rm qc}$. The unit-side transform uses
$\mathcal N_a\times\mathcal M_k^{\rm qc}$, with regressor and error arrays
$X_j^{A,b},E^{A,b}$, and the time-side transform uses
$\mathcal N_{-a}\times\mathcal M_k$, with arrays $X_j^{B,b},E^{B,b}$. Let
$\mathcal F_b$ be the sigma algebra generated by these training data. With
$U_{R,b},\Lambda_b$ and $V_{R,b},F_b$ the population frames restricted to
$\mathcal N_a$ and $\mathcal M_k$, the oracle block transforms and operator are
\[
A_b:=M_{\Lambda_b}M_{U_{R,b}},
\qquad
B_b:=M_{V_{R,b}}M_{F_b},
\qquad
\mathcal R_b^0[H^{(b)}]
:=A_bH^{(b)}B_b.
\]
Their feasible counterparts $\widehat A_b,\widehat B_b$ are built from the training
data, and $\widehat\beta^{\rm cf}$ aggregates the residualised Gram matrices and
moments over the evaluation blocks. As $K$ is fixed and $q_T=o(T)$, every
training rectangle has dimensions proportional to $N$ and $T$.
\subsubsection{Conditions and asymptotic normality}\label{sec:cf-conditions}
Let
$\mathcal C:=\sigma(\{\alpha_i,\gamma_t,\widetilde\alpha_i,
\widetilde\gamma_t\}_{i,t})$ and
$\mathcal C_0:=\sigma(\{\alpha_i,\gamma_t\}_{i,t})$, and write
$\beta_i(q\mid X,\mathcal C)$ for the beta-mixing coefficient at lag $q$ of
$\{\varepsilon_{it}:t\le T\}$ conditional on $(X,\mathcal C)$. Let
$\mathcal V_j:=X_j-\Gamma_{X,j}$, with $\Gamma_{X,j}$ the array of
$g_{x,j}(\alpha_i,\gamma_t)$, so that $\mathcal V_j$ has entries
$\widetilde g_{x,j}(\widetilde\alpha_i,\widetilde\gamma_t)+\eta_{it,j}$ by \eqref{eq:x}. Put
$\mathcal V_{j,b}:=\mathcal V_j^{(b)}$,
$\mathring{\mathcal V}_{j,b}:=
X_{j}^{(b)}-\mathbb{E}[X_{j}^{(b)}\mid\mathcal C_0,\mathcal F_b]$,
\begin{align*}
\Sigma_{E,b}:=&
\mathbb{E}[\operatorname{vec}(E^{(b)})\operatorname{vec}(E^{(b)})'
\mid X,\mathcal C],\\
\Sigma_{\mathring V,j,b}
:=&
\mathbb{E}\!\left[
\operatorname{vec}(\mathring{\mathcal V}_{j,b})
\operatorname{vec}(\mathring{\mathcal V}_{j,b})'
\,\middle|\,\mathcal C_0,\mathcal F_b
\right],
\end{align*}
and the oracle-residualised idiosyncratic-regressor Gram matrix
\[
Q_{\nu,NT}^0
:=
\frac1{NT}\sum_b
\mathbb{E}\!\left[
\left\{
\left\langle
\mathcal R_b^0[\mathcal V_{j,b}],
\mathcal R_b^0[\mathcal V_{k,b}]
\right\rangle_F
\right\}_{j,k=1}^d
\right].
\]
\begin{assumption}[Cross-fitting regularity]\label{ass:cf-reg}
For every block $b$ and $j\le d$:
\begin{enumerate}[(i)]
\item Conditional on $(X,\mathcal C)$, the unit processes
$\{\varepsilon_{it}:t\le T\}$ are independent across $i$, centred, and
uniformly geometrically beta mixing over time, with
$\max_{i\le N}\beta_i(q\mid X,\mathcal C)\le C_\beta e^{-c_\beta q}$ for
$q\ge1$ and fixed constants $C_\beta,c_\beta>0$. For some $\delta>0$,
$\max_{i,t}\mathbb{E}[|\varepsilon_{it}|^{4+\delta}\mid X,\mathcal C]=O_p(1)$.
\item $K$ is fixed, $\log N=o(q_T)$, $q_T=o(T)$, and
$\|\Sigma_{E,b}\|_{\rm op}
+\|\Sigma_{\mathring V,j,b}\|_{\rm op}=O_p(1)$.
\item $\liminf_{N,T\to\infty}
\lambda_{\min}(Q_{\nu,NT}^0)>0$, with $\lambda_{\min}$ the smallest
eigenvalue.
\end{enumerate}
\end{assumption}
The bound on $\Sigma_{E,b}$ in part (ii) follows from part (i) by
covariance inequalities for beta-mixing arrays. Part (iii) is no stronger than
the full-rank condition that CCEF places on the remaining idiosyncratic
regressor variation.
\begin{assumption}[Blockwise selected-frame strength]
\label{ass:cf-frame-strength}
For fixed $c>0$, with probability approaching one, every nonzero eigenvalue
of $\Lambda_b'\Lambda_b/N$ and of $F_b'F_b/T$ is at least $c\,\epsilon_{NT}^2$,
for every block $b$.
\end{assumption}
Assumption~\ref{ass:cf-frame-strength}, the block analogue of the eigenvalue
condition of Corollary~\ref{cor:select}, excludes selected columns that are
nearly but not exactly collinear. Lemma~\ref{lem:cf-transform} uses it to
show that the frame truncation of Algorithm~\ref{alg:cf} recovers the ranks of
$\Lambda_b$ and $F_b$.
We collect estimation errors of the outcome
directions and covariate projections together with the noise to signal ratio into an aggregated nuisance error rate:
\begin{equation}\label{eq:cf-varsigma}
\varsigma_{NT}:=\xi_u+\xi_v+\zeta_\Lambda+\zeta_F+\frac{1}{c_{NT}\sqrt{N\wedge T}},
\end{equation}
with $\zeta_\Lambda$ and $\zeta_F$ as in \eqref{eq:zeta}.
\begin{assumption}[Cross-fitted regularity]
\label{ass:cf-exact-spike}
\begin{enumerate}[(i)]
\item $G$ has rank $R_0$ and $R=R_0$. Uniformly over training samples, the
block analogues of Assumptions~\ref{ass:strength}--\ref{ass:Xstrength} hold
for $r\in\mathcal I_R$, and, for $s\in\{A,B\}$,
$\left\lVertE^{s,b}\right\rVert_{\rm op}=O_p(\sqrt{N\vee T})$ and
$\left\lVert\operatorname{Cov}\{\operatorname{vec}(E^{s,b})\mid X,\mathcal C\}\right\rVert_{\rm op}
=O_p(1)$.
\item The blockwise preliminary estimators obey
\begin{equation}\label{eq:cf-initial}
\max_b\left\lVert\widetilde\beta_b-\beta_0\right\rVert=O_p\big((N\wedge T)^{-1/2}\big).
\end{equation}
\item The population frames satisfy
\begin{equation}\label{eq:cf-commute}
\max_b\min\Big\{\left\lVertP_{U_{R,b}}P_{\Lambda_b}-P_{\Lambda_b}P_{U_{R,b}}\right\rVert_{\rm op},\
\left\lVertP_{V_{R,b}}P_{F_b}-P_{F_b}P_{V_{R,b}}\right\rVert_{\rm op}\Big\}
=O_p(\varsigma_{NT}).
\end{equation}
\item The spectral rates satisfy
\begin{equation}\label{eq:cf-exact-rate}
\sqrt{NT}\,\varsigma_{NT}^3=o_p(1).
\end{equation}
\end{enumerate}
\end{assumption}
Part (i) adds the noise edge of Proposition~\ref{prop:xi} on each training
rectangle and a covariance bound implied by Assumption~\ref{ass:cf-reg}(i).
Part (ii) holds for the least-squares estimator of \citet{Bai2009} on
$\mathcal N_{-a}\times\mathcal M_k^{\rm qc}$ with any preliminary rank
$R_{\rm pre}\ge R_0$, under the conditions of
\citet[Theorem~4.1]{MoonWeidner2015}, which require no strong factors. Part (iii) restricts the common block alone. It holds with both
commutators zero when $g_x$ is a multiple of $g$, and under the spanning
condition \eqref{eqn:CCEXstrongfactor} of Example~\ref{ex:cce}.
Section~\ref{sec:cf-transform-consistency} shows how to check it from the training frames. Both parts
(i) and (iv) restrict factor strengths. See Remark~\ref{rem:strength-consequences}(iv) for how these rates translate into factor strength.
\begin{theorem}[Asymptotic normality of cross-fitted CUH]\label{thm:cf}
Let $\widehat\beta^{\rm cf}$ be the estimator of Algorithm~\ref{alg:cf}, and
suppose Assumptions~\ref{ass:cf-reg}, \ref{ass:cf-frame-strength},
\ref{ass:cf-exact-spike} and~\ref{ass:cf-score} hold. Then
\[
\sqrt{NT}\,(\widehat\beta^{\rm cf}-\beta_0)
\Rightarrow\mathcal N\big(0,Q^{-1}\Omega Q^{-1}\big),
\]
where $Q$ and $\Omega$ are the limiting oracle Gram matrix and oracle score
variance of Assumption~\ref{ass:cf-score}.
\end{theorem}
The errors may be heteroskedastic and serially dependent, and
Proposition~\ref{prop:cf-csdep} relaxes their independence across units to
weak cross-sectional dependence of unknown ordering.
Remark~\ref{rem:strength-consequences}(iv) calibrates the factor strength that
Assumption~\ref{ass:cf-exact-spike}(iv) requires. The two-way robust standard
errors of Section~\ref{sec:cf-variance} are consistent by
Theorem~\ref{thm:cf-chs}.
\subsection{Efficiency}\label{sec:eff}
CCEF (Example~\ref{ex:cce}), FK \citep{freeman2026inference} and CUH each
run OLS after a purge that residualises $Y$ and $X$ on an estimate of the
latent structure. Under homoskedastic errors each asymptotic variance is
proportional to the inverse second moment of the regressor variation
$z^e_{it}$ that the purge of estimator $e$ leaves,
\begin{equation}\label{eq:zscores}
z^{\rm CCEF}_{it}:=\eta_{it},\qquad
z^{\rm FK}_{it}:=\eta_{it}+\widetilde g_{x,it},\qquad
z^{\rm CUH}_{it}:=\eta_{it}+\widetilde g_{x,it}+w_{it},
\end{equation}
with $\widetilde g_{x,it}:=\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)$. CCEF removes the
common and $x$-only blocks, FK only the common block, and CUH additionally
keeps $w_{it}$, the part of $g_x(\alpha_i,\gamma_t)$ linearly orthogonal to
$g(\alpha_i,\gamma_t)$. Let $\Sigma_\eta$, $\widetilde G$ and $\mathcal W$ be the
limiting average second moments of $\eta_{it}$, $\widetilde g_{x,it}$ and $w_{it}$,
$\sigma^2$ the error variance, and $V_e$ the asymptotic variance of estimator
$e$. Assumption~\ref{ass:homo} in Appendix~\ref{sec:cf-efficiency} states
homoskedasticity and the standard expansions for CCEF and FK.
\begin{proposition}[Variance ordering]\label{prop:eff}
Under Assumptions~\ref{ass:cf-score} and~\ref{ass:homo},
\[
V_{\rm CUH}
\;\preceq\;
\sigma^2\big(\Sigma_\eta+\widetilde G+\mathcal W\big)^{-1}
\;\preceq\;
V_{\rm FK}
=\sigma^2\big(\Sigma_\eta+\widetilde G\big)^{-1}
\;\preceq\;
V_{\rm CCEF}
=\sigma^2\Sigma_\eta^{-1}.
\]
The equality \(V_{\rm FK}=V_{\rm CCEF}\) holds
if and only if \(\widetilde G=0\), i.e. the $x$-only block is negligible.
\end{proposition}
The CUH entry is an inequality because its residual makers need not commute,
so its variance is a full sandwich (Appendix~\ref{sec:cf-efficiency}). Under
heteroskedasticity the ordering can fail.
The factor estimator removes only the outcome directions. As
\[
A_bZB_b=M_{\Lambda_b}(M_{U_{R,b}}ZM_{V_{R,b}})M_{F_b}
\]
and residual makers are contractions, it retains weakly more regressor variation than CUH, but it
needs strong factors. Removing the covariate projections is the price CUH pays
for validity under weaker factors.
\begin{remark}[The post-purge floor]\label{rem:ppfloor}
Each estimator buys bias protection by removing regressor variation, and the
post-purge floor estimates the share that survives,
\[
\widehat{\mathcal F}
:=\frac{\lambda_{\min}\big(\widehat Q\big)}
{\lambda_{\min}\big(\tfrac1{NT}\textstyle\sum_{it}x_{it}x_{it}'\big)}
\qquad\Big(=\frac{\|\widehat AX\widehat B\|_F^2}{\|X\|_F^2}
\ \text{ for scalar } x_{it}\Big).
\]
A value near zero means all identifying variation has been purged, which the
CCE rank condition assumes away. Section~\ref{sec:apps}
reports floors for the empirical applications.
\end{remark}
{
\section{Monte Carlo evidence}\label{sec:sims}
\subsection{Design}\label{sec:sim-design}
The design follows \eqref{eq:y}--\eqref{eq:x}. The common effects
$\alpha_i,\gamma_t\in\mathbb{R}$ and the $x$-only effects
$\widetilde\alpha_i,\widetilde\gamma_t\in\mathbb{R}^3$ are i.i.d.\ standard Gaussian. The common
block $g_x(\alpha_i,\gamma_t)=\alpha_i\gamma_t$ has rank one and the $x$-only
block $\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)=\widetilde\alpha_i'\widetilde\gamma_t$ rank three,
each two-way demeaned and normalised to unit sample variance. The covariate
and outcome are
\begin{align*}
x_{it}&=g_x(\alpha_i,\gamma_t)
+\big\{\widetilde g_x(\widetilde\alpha_i,\widetilde\gamma_t)+\eta_{it}/\sqrt{NT}\big\}
\big/\mathrm{sd}\big(\widetilde g_x+\eta/\sqrt{NT}\big),\\
y_{it}&=x_{it}\beta_0+ c_{NT}\, g_x(\alpha_i,\gamma_t)+\varepsilon_{it},
\end{align*}
with $\beta_0=1$, $\varepsilon_{it},\eta_{it}$ i.i.d.\ standard Gaussian, and
common-factor strength $c_{NT}=(N\wedge T)^{-w}$ in $y_{it}$, so $w=0$ gives
strong factors and $w=1/2$ the detection boundary of \eqref{eq:SNR}. Both
$x_{it}$ and $y_{it}$ are two-way demeaned before estimation. Scaling
$\eta_{it}$ by $1/\sqrt{NT}$ makes $\Sigma_\eta$ of
Proposition~\ref{prop:eff} negligible, so almost all identifying variation
lies in the $x$-only block. CUH and the factor estimator keep it, while CCEF
projects it out.
The factor estimator is the least-squares estimator of \citet{Bai2009} with
$R=1$. CCEF projects out the outcome direction and the leading four singular
directions of $X$, the rank of $g_x+\widetilde g_x$. CUH follows Algorithm~\ref{alg:cf} with $R=L=1$,
$K=2$ time halves and a gap $q_T=2$. NPCUH and BCUH are cross-fitted
likewise, with the bandwidths of Remark~\ref{rem:npcuh-bandwidth}. The
bias-aware intervals of
\citet{armstrong2026robust} take the form of Example~\ref{ex:ba}, with their
own standard errors and a worst-case bias bound that allows the factor to be
weak. Table~\ref{tab:mc} adds BM, the cross-fitted two-way grouped estimator
of \citet{beyhum2025inference}, which projects out unit and time group means
formed by k-means on sample averages and is applied before two-way
demeaning. Apart from the bias-aware intervals, standard errors use the
two-way variance estimator of \citet{CHS_Restat_2024} (CHS hereafter) of
Section~\ref{sec:cf-variance}, with the degrees-of-freedom correction of
Remark~\ref{rem:cf-dfc}.
Theorem~\ref{thm:cf} gives valid inference for CUH inside the inference frontier of
Remark~\ref{rem:strength-consequences}(iv). With $T=N^t$ and $\alpha^x_R=1$,
the calibration of Remark~\ref{rem:alpha} gives $\kappa=1/t$ and
$\alpha_R=1-2wt$, so \eqref{eq:inference-frontier} reads
\[
w<\tfrac13-\tfrac1{6t},
\]
that is $w<0.03$, $0.10$, $0.14$ and $0.17$ at $t=0.55$, $0.7$, $0.85$ and
$1$.
\subsection{Design manifolds against the factor benchmark}
\label{sec:sim-manifold}
Figures~\ref{fig:headline}, \ref{fig:cmp-bias} and \ref{fig:cmp-sd} report
coverage, bias and standard deviation at $N=200$, with
$T=\mathrm{round}(N^{t})$ for $t\in\{0.55,0.7,0.85,1\}$ and
$w\in\{0,0.1,0.25,0.4,0.55,0.7,0.85\}$, and $400$ replications in each of
the $28$ cells. Blue marks performance better than the factor benchmark,
amber within noise and red worse.\footnote{Noise is an unpaired
two-standard-error band,
$1.96\{p_e(1-p_e)+p_f(1-p_f)\}^{1/2}/\sqrt{400}$ for coverage, with $p_e$ and
$p_f$ the coverages of the estimator and the benchmark,
$1.96({\rm sd}_e^2+{\rm sd}_f^2)^{1/2}/\sqrt{400}$ for bias, and the same
expression over $\sqrt{2\cdot399}$ for the standard deviation. The bands
ignore the positive covariance of the paired surfaces and so are
conservative. Dots give the verdict at each grid node, and facet shading
applies the same rule to the average of the four surrounding cells.}
Inside the inference frontier of Section~\ref{sec:sim-design}, CUH covers at
$0.935$ to $0.95$, and near the detection boundary its coverage dips to $0.77$ to
$0.84$ at $w=0.4$ for $T\ge41$. Beyond the boundary all three CUH variants
have smaller absolute bias than the factor benchmark
(Figure~\ref{fig:cmp-bias}) and similar or higher variance
(Figure~\ref{fig:cmp-sd}). BCUH is red nowhere on the coverage surface
(Figure~\ref{fig:headline}). NPCUH also covers near nominal at and beyond the
boundary, but over-covers in some strong-factor cells.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.96\textwidth]{figures/comparison_manifold_bias_kappa.pdf}
\caption{Bias surfaces over the design manifold, coloured against the
factor benchmark (grey) as in Section~\ref{sec:sim-manifold}. The
benchmark's bias ridge tracks the detection boundary. CUH-class panels are
red on a band around $w=0.25$ and blue beyond the boundary.}
\label{fig:cmp-bias}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.96\textwidth]{figures/comparison_manifold_sd_kappa.pdf}
\caption{Monte Carlo standard deviation surfaces, coloured as in
Figure~\ref{fig:cmp-bias}. NPCUH is red over most of the domain, and CUH and
BCUH are red beyond the detection boundary and along the shortest panel.}
\label{fig:cmp-sd}
\end{figure}
\subsection{Point comparison across sample sizes}\label{sec:sim-table}
Table~\ref{tab:mc} compares the estimators at three sample sizes. Under
strong factors CUH is almost as precise as the factor estimator, while CCEF
is uninformative by design. Under weak factors the factor estimator's
coverage falls from $0.07$ to $0.00$ as the sample grows, NPCUH and BCUH
cover at $0.93$ to $0.96$, CUH at $0.84$ to $0.87$, and the bias-aware
intervals at $1.00$ throughout. BM fails in both regimes. The common block is
two-way demeaned, so the sample averages on which BM forms its groups carry
no signal.
\begin{table}[!ht]
\centering
\caption{Bias, standard deviation and RMSE (all $\times100$) and coverage
of nominal $95\%$ intervals, with $T=\mathrm{round}(N^{0.7})$ and $2{,}000$
replications. The bias-aware row reports the bias, sd and RMSE of its point
estimate.}
\label{tab:mc}
\resizebox{\textwidth}{!}{
\begin{tabular}{l rrrr rrrr rrrr}
\hline\hline
& \multicolumn{4}{c}{$(N,T)=(100,25)$} & \multicolumn{4}{c}{$(200,41)$} & \multicolumn{4}{c}{$(400,66)$}\\
& bias & sd & rmse & cov & bias & sd & rmse & cov & bias & sd & rmse & cov\\ \hline
\multicolumn{13}{l}{\emph{Strong factors} ($w=0.00$)}\\
factor & 0.07 & 2.04 & 2.04 & 0.96 & 0.03 & 1.10 & 1.10 & 0.94 & 0.01 & 0.61 & 0.61 & 0.95\\
CCEF & 6.64 & 111.34 & 111.51 & 0.84 & 6.07 & 107.52 & 107.66 & 0.81 & 10.47 & 103.65 & 104.15 & 0.78\\
CUH & 0.22 & 2.12 & 2.13 & 0.96 & 0.07 & 1.12 & 1.12 & 0.93 & 0.02 & 0.62 & 0.62 & 0.95\\
NPCUH & 0.16 & 2.80 & 2.81 & 0.98 & 0.11 & 1.35 & 1.35 & 0.99 & 0.02 & 0.73 & 0.73 & 0.99\\
BCUH & 0.16 & 2.43 & 2.44 & 0.99 & 0.08 & 1.19 & 1.20 & 0.97 & 0.02 & 0.66 & 0.66 & 0.97\\
BiasAware & 0.06 & 4.72 & 4.72 & 1.00 & -0.15 & 3.25 & 3.25 & 1.00 & 0.06 & 2.18 & 2.18 & 1.00\\
BM & 44.97 & 9.55 & 45.97 & 0.00 & 45.16 & 8.91 & 46.03 & 0.00 & 45.24 & 8.53 & 46.04 & 0.00\\
\hline
\multicolumn{13}{l}{\emph{Weak factors} ($w=0.55$)}\\
factor & 7.77 & 2.54 & 8.17 & 0.07 & 6.11 & 1.30 & 6.24 & 0.02 & 4.85 & 0.62 & 4.89 & 0.00\\
CCEF & -0.70 & 94.69 & 94.66 & 0.78 & -1.07 & 88.67 & 88.66 & 0.69 & 2.98 & 83.61 & 83.64 & 0.61\\
CUH & 2.17 & 2.59 & 3.38 & 0.84 & 0.99 & 1.30 & 1.64 & 0.84 & 0.44 & 0.69 & 0.82 & 0.87\\
NPCUH & 2.31 & 3.27 & 4.00 & 0.94 & 1.17 & 1.55 & 1.94 & 0.95 & 0.53 & 0.80 & 0.96 & 0.96\\
BCUH & 1.60 & 2.96 & 3.36 & 0.94 & 0.63 & 1.43 & 1.56 & 0.93 & 0.23 & 0.73 & 0.77 & 0.94\\
BiasAware & 5.32 & 4.54 & 6.99 & 1.00 & 3.94 & 3.10 & 5.02 & 1.00 & 3.30 & 2.06 & 3.89 & 1.00\\
BM & 8.02 & 1.98 & 8.26 & 0.05 & 6.14 & 1.17 & 6.25 & 0.01 & 4.72 & 0.77 & 4.79 & 0.00\\
\hline
\hline
\end{tabular}
}
\end{table}
Table~\ref{tab:mc-re} uses a second design in which BM's sample averages
carry signal. The latent effects are i.i.d.\ uniform on $[0,1]$ without
demeaning, scaled to unit sample variance, and $x_{it}$ and $y_{it}$ each
carry an independent additive effect $a_i+b_t$, with $a_i,b_t$ i.i.d.\
$N(5,5^2)$. Two-way demeaning removes the means and the additive effects, so
for every estimator but BM only the latent distribution differs, and most of
their rows resemble Table~\ref{tab:mc}. BM now selects many more groups but
still fails, which suggests its groups track the additive effects rather than
the interactive confounder. Under weak factors it covers more often than the
factor estimator because its standard errors are larger.
\begin{table}[!ht]
\centering
\caption{As Table~\ref{tab:mc}, in the second design of
Section~\ref{sec:sim-table}, with uniform latent effects that are not
demeaned and two-way additive effects in $x_{it}$ and $y_{it}$.}
\label{tab:mc-re}
\resizebox{\textwidth}{!}{
\begin{tabular}{l rrrr rrrr rrrr}
\hline\hline
& \multicolumn{4}{c}{$(N,T)=(100,25)$} & \multicolumn{4}{c}{$(200,41)$} & \multicolumn{4}{c}{$(400,66)$}\\
& bias & sd & rmse & cov & bias & sd & rmse & cov & bias & sd & rmse & cov\\ \hline
\multicolumn{13}{l}{\emph{Strong factors} ($w=0.00$)}\\
factor & 0.04 & 2.09 & 2.09 & 0.90 & -0.03 & 1.11 & 1.11 & 0.94 & -0.00 & 0.63 & 0.63 & 0.94\\
CCEF & 6.24 & 115.08 & 115.22 & 0.84 & 11.47 & 100.59 & 101.22 & 0.81 & 15.94 & 101.12 & 102.34 & 0.79\\
CUH & 0.17 & 2.13 & 2.14 & 0.93 & 0.02 & 1.12 & 1.12 & 0.96 & 0.01 & 0.64 & 0.64 & 0.94\\
NPCUH & 0.17 & 2.64 & 2.65 & 0.98 & 0.01 & 1.29 & 1.29 & 0.99 & 0.01 & 0.70 & 0.70 & 0.98\\
BCUH & 0.16 & 2.34 & 2.35 & 0.97 & 0.01 & 1.18 & 1.18 & 0.98 & 0.00 & 0.66 & 0.66 & 0.97\\
BiasAware & -0.05 & 4.94 & 4.94 & 1.00 & 0.05 & 3.15 & 3.15 & 1.00 & 0.09 & 2.17 & 2.17 & 1.00\\
BM & 42.12 & 9.95 & 43.28 & 0.07 & 41.93 & 7.72 & 42.64 & 0.01 & 42.01 & 6.26 & 42.47 & 0.00\\
\hline
\multicolumn{13}{l}{\emph{Weak factors} ($w=0.55$)}\\
factor & 7.79 & 2.52 & 8.18 & 0.06 & 6.18 & 1.21 & 6.30 & 0.01 & 4.84 & 0.66 & 4.89 & 0.00\\
CCEF & -2.41 & 99.11 & 99.11 & 0.78 & 1.35 & 81.19 & 81.18 & 0.68 & 5.04 & 82.00 & 82.13 & 0.63\\
CUH & 1.86 & 2.50 & 3.12 & 0.84 & 0.80 & 1.28 & 1.51 & 0.89 & 0.38 & 0.69 & 0.79 & 0.89\\
NPCUH & 2.03 & 2.94 & 3.58 & 0.95 & 0.93 & 1.51 & 1.77 & 0.95 & 0.47 & 0.78 & 0.91 & 0.95\\
BCUH & 1.34 & 2.77 & 3.07 & 0.94 & 0.46 & 1.39 & 1.46 & 0.95 & 0.19 & 0.73 & 0.76 & 0.94\\
BiasAware & 5.12 & 4.75 & 6.98 & 1.00 & 4.19 & 2.98 & 5.14 & 1.00 & 3.34 & 2.01 & 3.90 & 1.00\\
BM & 7.95 & 3.08 & 8.52 & 0.58 & 6.01 & 1.75 & 6.26 & 0.32 & 4.64 & 1.03 & 4.75 & 0.11\\
\hline
\hline
\end{tabular}
}
\end{table}
\subsection{The price of robustness in power}
Figure~\ref{fig:power} plots the rate at which each interval excludes
$\beta_0+\xi$, $\xi\in[-0.3,0.3]$, for data generated at $\beta_0$, with
$400$ replications per panel. The height at $\xi=0$ is size and elsewhere
power. The bias-aware intervals hold size in every panel but leave a wide
non-rejection region. CCEF neither holds size nor discriminates.
CUH matches the factor benchmark under strong factors. At $w=0.4$, near the
detection boundary, CUH rejects the truth with probability $0.22$ while BCUH
stays near nominal size. At $w=0.7$ the factor benchmark's non-rejection
region moves off the truth, while CUH and BCUH stay close to it, rejecting
the truth at rates near $0.10$ and $0.06$. NPCUH rejects the truth at or
below nominal size in all three panels, with a wider non-rejection region.
Near and beyond the detection boundary the smoothing variants are therefore
the safer choice.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.32\linewidth]{figures/comparison_power_w000.pdf}
\includegraphics[width=0.32\linewidth]{figures/comparison_power_w040.pdf}
\includegraphics[width=0.32\linewidth]{figures/comparison_power_w070.pdf}
\caption{Rejection rates at $\beta_0+\xi$, with data generated at $\beta_0$
and $(N,T)=(200,41)$, for $w=0$, $0.4$ and $0.7$. Blue dashed factor, purple
dot-dash CCEF, black solid CUH, orange long-dash NPCUH, green dotted BCUH,
red solid bias-aware, grey band the bias-aware non-rejection region, dotted
line $0.05$.}
\label{fig:power}
\end{figure}
}
{
\section{Empirical applications}\label{sec:apps}
We apply the estimators to the state fiscal panel of \citet{james2015}, also
used by \citet{beyhum2025inference}, and to the state cigarette demand panel
of \citet{baltagilevin1992} and \citet{baltagigriffinxiong2000}, the standard
testbed for interactive fixed effects estimators \citep{badaliebl2014}.
Standard errors, cross-fitting and bandwidths are as in
Section~\ref{sec:sims}, and the table notes give the remaining settings.
\subsection{State fiscal policy and resource revenue}\label{sec:app-fiscal}
\citet{james2015} asks how state governments absorb resource revenue. The
panel covers $N=50$ states over $T=51$ years (1958 to 2008). The regressor is
resource revenue and the five outcomes are non-resource revenue, income tax
revenue, total expenditure, education expenditure and savings, all as shares
of personal income.
\citet{beyhum2025inference} report their grouped estimator alongside two-way
fixed effects, the factor estimator with seven factors and CCE, and these
disagree by an order of magnitude on total expenditure. The post-purge floor
explains why.
After two-way demeaning, a single left singular vector accounts for
ninety-seven percent of the regressor variation, so any residualisation
against an estimated direction leaves only a few percent of it, the
near-zero case of Remark~\ref{rem:ppfloor}. Table~\ref{tab:fiscal} reports
each estimate with its post-purge floor. On four of the five outcomes every
purge-based estimator keeps at most eight percent of the regressor
variation, and NPCUH and BM keep under one percent. On education expenditure
the outcome directions are less aligned with resource revenue, so the
outcome-side purges keep more regressor variation, which the covariate-side
CCEF and BM purges remove.
On total expenditure BM reports a large positive effect, $0.53$ with
standard error $0.26$, while CUH, NPCUH and BCUH lie between $0.09$ and
$0.17$ with standard errors between $0.06$ and $0.09$. On savings the cross-fitted estimators lie between $0.38$ and
$0.61$, while the factor estimator is smaller and sensitive to $R$. At a
floor this low each estimator rests on a different thin slice of variation,
so choosing an estimator is choosing which covariate variation remains.
\begin{table}[!ht]
\centering\scriptsize
\caption{State fiscal policy and resource revenue, one outcome per column, $N=50$, $T=51$}
\label{tab:fiscal}
\setlength{\tabcolsep}{4pt}
\begin{tabular}{l ccccc}
\hline\hline
& non-resource rev. & income tax & total exp. & education exp. & savings\\
\hline
2WFE & $+0.006$ (0.020) & $+0.018$ (0.010) & $+0.397$ (0.031) & $+0.063$ (0.008) & $+0.609$ (0.040)\\
& [1.000] & [1.000] & [1.000] & [1.000] & [1.000]\\
factor $R=2$ & $-0.346$ (0.123) & $-0.063$ (0.048) & $-0.032$ (0.060) & $+0.004$ (0.019) & $+0.279$ (0.077)\\
& [0.061] & [0.053] & [0.078] & [0.217] & [0.030]\\
factor $R=3$ & $-0.335$ (0.133) & $-0.034$ (0.036) & $-0.011$ (0.058) & $+0.012$ (0.018) & $+0.165$ (0.071)\\
& [0.042] & [0.030] & [0.071] & [0.172] & [0.024]\\
CCEF $R=2$ & $-0.376$ (0.132) & $-0.081$ (0.054) & $+0.121$ (0.175) & $+0.060$ (0.043) & $+0.385$ (0.155)\\
& [0.023] & [0.020] & [0.022] & [0.021] & [0.016]\\
CUH $R=2$ & $-0.107$ (0.123) & $+0.072$ (0.059) & $+0.174$ (0.064) & $+0.031$ (0.013) & $+0.470$ (0.161)\\
& [0.031] & [0.070] & [0.058] & [0.090] & [0.020]\\
CUH $R=3$ & $-0.075$ (0.117) & $+0.048$ (0.021) & $+0.113$ (0.068) & $+0.033$ (0.020) & $+0.610$ (0.133)\\
& [0.031] & [0.030] & [0.035] & [0.114] & [0.020]\\
NPCUH $R=3$ & $-0.275$ (0.206) & $-0.033$ (0.042) & $+0.095$ (0.070) & $+0.020$ (0.038) & $+0.380$ (0.105)\\
& [0.002] & [0.002] & [0.003] & [0.011] & [0.003]\\
BCUH $R=3$ & $-0.074$ (0.130) & $+0.059$ (0.026) & $+0.085$ (0.085) & $+0.027$ (0.026) & $+0.610$ (0.180)\\
& [0.021] & [0.023] & [0.026] & [0.046] & [0.014]\\
BM (CHS s.e.) & $-0.478$ (0.161) & $-0.035$ (0.045) & $+0.527$ (0.264) & $-0.041$ (0.083) & $+0.587$ (0.221)\\
& [0.004] & [0.006] & [0.003] & [0.006] & [0.004]\\
BM, as published & $-0.478$ (0.143) & $-0.035$ (0.048) & $+0.527$ (0.284) & $-0.041$ (0.079) & $+0.587$ (0.228)\\
\hline\hline
\end{tabular}
\begin{minipage}{0.92\textwidth}\vspace{2pt}\footnotesize
Notes: variables are standardised for estimation, with coefficients rescaled as in \citet{beyhum2025inference}. Cells report the estimate (CHS s.e.) and, in brackets, the post-purge floor $\widehat{\mathcal F}$ of Remark~\ref{rem:ppfloor}. CCEF purges one covariate principal component and CUH uses one covariate projection. BM uses $2$ to $17$ unit groups and $1$ to $5$ time groups per quadrant, chosen by the authors' rule, and is reconstructed from its k-means labels so that its standard errors use the CHS variance. The last row reproduces the published estimates and Arellano standard errors of \citet[Table 3]{beyhum2025inference}, which the reconstruction matches to three decimals.
\end{minipage}
\end{table}
\subsection{Cigarette demand}\label{sec:app-cigar}
\begin{table}[!ht]
\centering\small
\caption{Cigarette demand elasticities, $N=46$, $T=30$}
\label{tab:cigar}
\begin{tabular}{l ll c}
\hline\hline
& log price & log income & $\widehat{\mathcal F}$\\
\hline
2WFE & $-1.02$ (0.22) [$-1.45$, $-0.60$] & $+0.52$ (0.18) [$+0.17$, $+0.87$] & 1.00\\
factor $R=4$ & $-0.38$ (0.04) [$-0.46$, $-0.31$] & $+0.36$ (0.11) [$+0.15$, $+0.57$] & 0.29\\
CCEF $R=1$ & $-0.40$ (0.05) [$-0.50$, $-0.30$] & $+0.37$ (0.15) [$+0.08$, $+0.66$] & 0.24\\
CUH $R=3$ & $-0.41$ (0.06) [$-0.53$, $-0.28$] & $+0.36$ (0.14) [$+0.09$, $+0.64$] & 0.20\\
NPCUH $R=2$ ($h=0.54$) & $-0.48$ (0.13) [$-0.74$, $-0.23$] & $+0.27$ (0.14) [$-0.00$, $+0.55$] & 0.04\\
BCUH $R=1$ & $-0.41$ (0.11) [$-0.63$, $-0.20$] & $+0.27$ (0.16) [$-0.04$, $+0.57$] & 0.21\\
BM & $-0.55$ (0.07) [$-0.69$, $-0.41$] & $+0.19$ (0.16) [$-0.13$, $+0.50$] & 0.13\\
\hline\hline
\end{tabular}
\begin{minipage}{0.92\textwidth}\vspace{2pt}\footnotesize
Notes: estimate (CHS s.e.) [95\% CI]. $\widehat{\mathcal F}$ is the post-purge floor of Remark~\ref{rem:ppfloor}. The small, insignificant neighbouring-price coefficient is omitted. Estimators that depend on $R$ are shown at the $R\le4$ giving the smallest standard error on log price, and their estimates move little across $R$. CCEF purges $R_X=3$ covariate principal components, matching the three covariate projections of CUH with $L=1$. BM uses groups $(9,3)$, $(12,3)$, $(15,5)$ and $(15,6)$ in the four quadrants, and its Arellano standard errors are $0.07$ and $0.13$.
\end{minipage}
\end{table}
}
The panel covers $N=46$ states over $T=30$ years (1963 to 1992). We
regress log per-capita sales on log real price, log real disposable income
and log real minimum price in neighbouring states, all two-way demeaned.
Every purge-based estimator other than BM puts the price elasticity between
$-0.38$ and $-0.48$ and the income elasticity between $0.27$ and $0.37$,
against $-1.02$ and $0.52$ for two-way fixed effects (Table~\ref{tab:cigar}).
The projection estimators keep post-purge floors of $0.20$ to $0.29$, and
Section~\ref{sec:app-overlap} shows that they draw on largely shared
variation. BM is the exception, at $-0.55$ on price and an insignificant
$0.19$ on income, with a floor of $0.13$.
{
\subsection{Shared identifying variation}\label{sec:app-overlap}
When two purge-based estimators disagree, their residualised regressors show
which identifying variation they share. For each $R$,
Table~\ref{tab:overlap} reports the post-purge floors of the factor
estimator, CCEF and CUH, the correlation $\rho$ of CUH's residualised
regressor with that of each of the other two, and the estimate gap
$|d|/{\rm se}$ in units of CUH's standard error. A large gap at high $\rho$
signals conflict on shared variation, and a large gap at low $\rho$ signals
that the two estimators purged different variation.
Figure~\ref{fig:apps-diag} plots both diagnostics.
On cigarette price $\rho$ lies between $0.61$ and $0.84$ and no gap exceeds
$1.7$ standard errors, so the agreement in Table~\ref{tab:cigar} is evidence
of shared identifying variation. On the fiscal outcomes $\rho$ is at most
$0.48$, and the two largest gaps, $12.0$ and $11.3$ standard errors on total
expenditure, come with correlations of $0.12$ and $0.13$, which is evidence
that the estimators disagree because they use different variation.
Figure~\ref{fig:intro-apps}(b) fixes $R=5$ and raises the number of CCEF and
CUH covariate components from one to nine. CUH's estimate stays near
$-0.03$, because the regressor is almost rank one and the frame truncation
of Algorithm~\ref{alg:cf} detects this and keeps the variation. CCEF purges
it indiscriminately, with estimates ranging from $-0.53$ to $0.25$ as its
post-purge floor falls from $0.008$ to $0.0005$.
Table~\ref{tab:overlap-bm} compares the smoothing variants with BM. The
correlations stay below $0.6$ throughout, so BM and the smoothing variants
keep different variation. The gaps stay within $1.4$ standard errors on cigarette
price but reach $6.2$ for NPCUH and $10.2$ for BCUH on total expenditure,
where BM stood apart in Table~\ref{tab:fiscal}.
\begin{table}[!ht]
\centering\footnotesize
\caption{Shared identifying variation and estimate divergence}
\label{tab:overlap}
\setlength{\tabcolsep}{5pt}
\begin{tabular}{l ccc c cc c cc}
\hline\hline
& \multicolumn{3}{c}{post-purge floor $\widehat{\mathcal F}$} &
& \multicolumn{2}{c}{$\rho$ with CUH} &
& \multicolumn{2}{c}{$|d|/{\rm se}$ against CUH}\\
\cline{2-4}\cline{6-7}\cline{9-10}
$R$ & factor & CCEF & CUH && factor & CCEF && factor & CCEF\\
\hline
\multicolumn{10}{l}{\emph{Cigarette, log price}}\\
$1$ & 0.74 & 0.35 & 0.47 && 0.78 & 0.78 && 1.7 & 1.3\\
$2$ & 0.50 & 0.33 & 0.35 && 0.80 & 0.84 && 1.1 & 0.5\\
$3$ & 0.40 & 0.33 & 0.27 && 0.77 & 0.77 && 0.3 & 0.4\\
$4$ & 0.35 & 0.29 & 0.20 && 0.68 & 0.70 && 0.2 & 0.2\\
$5$ & 0.30 & 0.26 & 0.15 && 0.64 & 0.65 && 0.1 & 0.2\\
$6$ & 0.24 & 0.21 & 0.13 && 0.61 & 0.62 && 0.4 & 0.5\\\hline
\multicolumn{10}{l}{\emph{Fiscal, non-resource revenue}}\\
$1$ & 0.197 & 0.033 & 0.043 && 0.38 & 0.44 && 0.9 & 1.8\\
$2$ & 0.061 & 0.023 & 0.031 && 0.33 & 0.34 && 2.0 & 2.2\\
$3$ & 0.042 & 0.014 & 0.031 && 0.32 & 0.38 && 2.2 & 1.3\\
$4$ & 0.052 & 0.009 & 0.023 && 0.29 & 0.30 && 0.6 & 7.1\\
$5$ & 0.045 & 0.008 & 0.022 && 0.37 & 0.22 && 0.9 & 7.0\\
$6$ & 0.032 & 0.007 & 0.008 && 0.22 & 0.27 && 0.1 & 3.8\\\hline
\multicolumn{10}{l}{\emph{Fiscal, total expenditure}}\\
$1$ & 0.156 & 0.024 & 0.056 && 0.48 & 0.37 && 1.1 & 5.0\\
$2$ & 0.078 & 0.022 & 0.058 && 0.40 & 0.32 && 3.2 & 0.8\\
$3$ & 0.071 & 0.013 & 0.035 && 0.29 & 0.34 && 1.8 & 2.8\\
$4$ & 0.040 & 0.014 & 0.042 && 0.30 & 0.21 && 1.4 & 0.3\\
$5$ & 0.038 & 0.009 & 0.020 && 0.26 & 0.13 && 0.7 & 2.9\\
$6$ & 0.008 & 0.005 & 0.013 && 0.12 & 0.13 && 12.0 & 11.3\\
\hline\hline
\end{tabular}
\begin{minipage}{0.92\textwidth}\vspace{2pt}\footnotesize
Notes: $\widehat{\mathcal F}$ is the share of the named regressor's variation
surviving each purge at $R$ outcome directions, $\rho$ the correlation of the
residualised regressors of CUH and of $e$, the factor estimator or CCEF, and
$|d|/{\rm se}=|\widehat\beta_{\rm CUH}-\widehat\beta_e|$ over the CHS standard error
of CUH. CUH uses $L=1$, and CCEF $R_X=3$ for cigarette (log price) and
$R_X=1$ for fiscal, with fiscal outcomes standardised as in
Table~\ref{tab:fiscal}.
\end{minipage}
\end{table}
\begin{table}[!ht]
\centering\footnotesize
\caption{Shared identifying variation against the grouped estimator}
\label{tab:overlap-bm}
\setlength{\tabcolsep}{3.4pt}
\begin{tabular}{l ccc c ccc}
\hline\hline
& \multicolumn{7}{c}{(a) Cigarette, log price}\\
\cline{2-8}
& \multicolumn{3}{c}{NPCUH} && \multicolumn{3}{c}{BCUH}\\
\cline{2-4}\cline{6-8}
$R$ & $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$ && $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$\\
\hline
$1$ & 0.11 & 0.46 & 0.5 && 0.25 & 0.53 & 1.3\\
$2$ & 0.07 & 0.56 & 0.6 && 0.20 & 0.58 & 1.4\\
$3$ & 0.05 & 0.52 & 0.8 && 0.16 & 0.56 & 1.0\\
$4$ & 0.04 & 0.46 & 1.0 && 0.12 & 0.51 & 0.7\\
$5$ & 0.02 & 0.42 & 1.2 && 0.09 & 0.43 & 0.5\\
$6$ & 0.02 & 0.37 & 1.2 && 0.07 & 0.39 & 0.4\\
\hline\hline
\end{tabular}
\vspace{10pt}
\begin{tabular}{l ccc c ccc c ccc c ccc}
\hline\hline
& \multicolumn{7}{c}{(b) Fiscal, income tax} &
& \multicolumn{7}{c}{(c) Fiscal, total expenditure}\\
\cline{2-8}\cline{10-16}
& \multicolumn{3}{c}{NPCUH} && \multicolumn{3}{c}{BCUH}
&& \multicolumn{3}{c}{NPCUH} && \multicolumn{3}{c}{BCUH}\\
\cline{2-4}\cline{6-8}\cline{10-12}\cline{14-16}
$R$ & $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$ && $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$ && $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$ && $\widehat{\mathcal F}$ & $\rho$ & $|d|/{\rm se}$\\
\hline
$1$ & 0.001 & 0.13 & 0.4 && 0.005 & 0.12 & 0.8 && 0.005 & 0.52 & 3.3 && 0.041 & 0.26 & 10.2\\
$2$ & 0.003 & 0.26 & 0.3 && 0.048 & 0.03 & 1.6 && 0.004 & 0.45 & 0.5 && 0.041 & 0.20 & 3.8\\
$3$ & 0.002 & 0.26 & 0.0 && 0.023 & 0.05 & 3.6 && 0.003 & 0.45 & 6.2 && 0.026 & 0.20 & 5.2\\
$4$ & 0.002 & 0.26 & 0.2 && 0.008 & 0.11 & 2.4 && 0.002 & 0.40 & 2.2 && 0.028 & 0.15 & 4.3\\
$5$ & 0.002 & 0.24 & 0.3 && 0.006 & 0.13 & 0.4 && 0.002 & 0.39 & 2.9 && 0.016 & 0.13 & 5.5\\
$6$ & 0.001 & 0.22 & 0.1 && 0.005 & 0.12 & 0.5 && 0.001 & 0.39 & 2.0 && 0.011 & 0.12 & 6.5\\
\hline\hline
\end{tabular}
\begin{minipage}{0.96\textwidth}\vspace{2pt}\footnotesize
Notes: as Table~\ref{tab:overlap}, with NPCUH or BCUH in place of CUH and BM
in place of $e$. BM does not vary with $R$. Its floor and
estimate (CHS s.e.) are $0.18$ and $-0.553$ $(0.071)$ for cigarette,
$0.006$ and $-0.035$ $(0.045)$ for income tax, and $0.003$ and $+0.527$
$(0.264)$ for total expenditure.
\end{minipage}
\end{table}
}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\textwidth]{figures/fig_overlap_diag.pdf}
\caption{Shared identifying variation across the number of purged outcome
directions $R$. Panel (a) compares CCEF with CUH on non-resource revenue,
panel (b) NPCUH and BCUH with BM on total expenditure.}
\label{fig:apps-diag}
\end{figure}
\section{Conclusion}
Bias in linear panels with two-way unobserved heterogeneity comes only from
the common block that the outcome shares with the covariates. CUH removes
it through the outcome directions and the covariate projections and keeps the
$x$-only variation. Under our conditions CUH is $\sqrt{NT}$-asymptotically
normal with weak factors, allowing $N$ and $T$ to grow at different rates,
and under homoskedastic errors its asymptotic variance is no larger than
those of FK and CCEF. In simulations it matches the factor estimator under strong factors,
and beyond the detection boundary, where the factor estimator's coverage
collapses, NPCUH and BCUH hold coverage near nominal at the cost of higher
variance. In the applications the post-purge floor and the shared-variation
diagnostics show when the estimators agree and why they diverge. We recommend
reporting this floor with any purge-based estimator.
\setlength{\bibsep}{2pt}
\bibliographystyle{chicago3}
\bibliography{refs}