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Attenuated Heterogeneity in Fixed-Effects Causal Forests, and a Cross-Fitted Correction
{\sloppy \noindentKeywords: best linear predictor; calibration; conditional average treatment effect; difference-in-differences; panel data; regression to the mean.}
\noindentJEL: C14; C21; C23.
Applied researchers increasingly use causal forests to estimate how treatment effects vary across units. In panel and difference-in-differences designs, the leading concern is that unit and time fixed effects contaminate the estimated heterogeneity: a forest applied to raw panel data may split on covariates correlated with the fixed effects and report level differences as effect differences. kattenberg2023 address this with causal forests with fixed effects (CFFE), which remove the fixed effects by a within-transformation computed locally at each tree node, so that splits respond to genuine treatment-effect heterogeneity rather than to between-node variation in the fixed effects.
This paper studies a separate property of a widely implemented class of such estimators: forests that form the CATE by averaging honest leaf-level treatment effects across trees, the causal-tree-averaging construction of athey2016, as distinct from the similarity-weight aggregation of the generalized random forest wager2018,athey2019grf. In the fixed-effects panel setting, such averaging systematically attenuates the heterogeneity it is meant to recover. Attenuation is not unique to this class: the similarity-weight GRF, estimated on the same within-transformed signal, is also biased toward the mean, as any regularized nonparametric estimator of a noisy surface will be. But the leaf-averaging construction attenuates materially and consistently more, and we quantify the gap directly against a GRF competitor across designs.
\paragraph{The attenuation.} Let $\tau(x)$ denote the true CATE and $\hat{\tau}_{\text{raw}}(x)$ the raw forest average. Across designs we find
so the estimated spread is compressed toward the mean by the factor $b$. This is a form of regression to the mean: honest leaf estimates on small, within-transformed nodes have limited treatment variation, and averaging noisy, shrunken leaf effects across trees compounds the shrinkage. The common device that recenters the predictions by an additive constant, so their mean equals a separately estimated, unbiased within-fixed-effects average treatment effect, corrects the level; being additive, it leaves the slope $b$ and the compressed heterogeneity unchanged.
\paragraph{The correction, and its relation to existing tools.} To undo the compression we use a best-linear-predictor calibration. In within-transformed space the outcome satisfies a moment condition linear in $\tau(x)$; regressing the residualized outcome on the residualized treatment interacted with a centered forest proxy recovers a slope that rescales the compressed heterogeneity, using no ground truth. This regression is not new. The slope is exactly the best-linear-predictor coefficient of cddf2025, and it is the quantity the grf package grf,atheywager2019 reports as differential.forest.prediction when testing whether a forest's heterogeneity is well calibrated. Our use of it differs on two points. Where those tools use the slope as a diagnostic, testing whether it equals one, we invert it to correct the predictions, shipping a de-attenuated CATE rather than a test statistic. And we require the proxy to be formed out-of-bag, for each unit averaging only the trees whose subsample excluded it, because an in-sample proxy calibrates the forest's noise to itself and manufactures spurious heterogeneity even when the true effect is homogeneous. The out-of-bag construction removes this covariation, so the correction is asymptotically inert under the null: its de-attenuation slope vanishes as the panel grows, with only a small residual dispersion at finite sample size.
\paragraph{Contributions and relation to the literature.} Our primary contribution is diagnostic. The best-linear-predictor slope and the observation that forest CATEs can be miscalibrated are established cddf2025,atheywager2019. What is not established is a characterization of the attenuation for the leaf-averaging class of causal forests athey2016 in the fixed-effects panel setting: how large the slope $b$ is, how it moves with the design, and how it compares to the similarity-weight alternative. We map $b$ across signal-to-noise, panel size, dimensionality, and forest hyperparameters, benchmark it against a generalized random forest on the same signal, and show it is systematically below one, and consistently further below one than the GRF's, while the standard level-recentering leaves it untouched (Section (ref)). Our secondary contribution is a correction suited to this setting. Rescaling attenuated CATE estimates is itself established: lengdimmery2024 fit a linear (Platt-style) rescaling and vanderlaan2023 an isotonic calibration map. Both calibrate against an external unbiased benchmark, a held-out randomized difference-in-means, and are developed for the randomized-experiment setting. Our correction requires no such benchmark. It is self-contained within the observational panel, using the within-fixed-effects average treatment effect as the level anchor and an out-of-bag forest proxy for the direction, under difference-in-differences identification. We are not aware of a panel or difference-in-differences causal-forest method, including the difference-in-difference causal forest of gavrilova2025, that de-attenuates its heterogeneity estimates. We show out-of-bag cross-fitting is what secures null-safety, provide simulation evidence (25--42% CATE mean-squared-error reduction), and ship the method in the causalfe Python package\ (Sections (ref)--(ref)). We do not claim the best-linear-predictor object, the miscalibration concept, or slope-rescaling of CATEs as new, and cite cddf2025, atheywager2019, lengdimmery2024, and vanderlaan2023 as their sources.
We observe a panel $\{(Y_{it}, D_{it}, X_i)\}$ for units $i=1,\dots,N$ over periods $t=1,\dots,T$, with a binary, staggered-or-single-event treatment $D_{it}\in\{0,1\}$ and time-invariant covariates $X_i$. The outcome follows the two-way fixed-effects model
where $\alpha_i$ and $\gamma_t$ are unit and time fixed effects and $\tau(x)$ is the conditional average treatment effect at covariate value $x$, the estimand of interest. The effect enters (ref) without a unit-level unobserved gain, so under Assumption (ref) the conditional effect on the treated coincides with the population conditional effect; we therefore refer to $\tau(x)$ simply as the CATE throughout. We take identification of $\tau(\cdot)$ as given, under the standard conditions stated in Assumption (ref), and focus on its estimation. The attenuation we study and the correction we propose concern estimation only; they do not weaken or strengthen these conditions.
Assumption (ref) is the covariate-conditional two-way-fixed-effects analogue of the conditions in callaway2021difference and kattenberg2023; under it the within-transformed contrast identifies $\tau(x)$. The remaining conditions we need are regularity conditions on the data-generating process that govern the behavior of the forest itself, which we collect next.
Part (iv) is what makes the out-of-bag construction of Section (ref) a valid cross-fit: because units are the independent sampling blocks, a tree that excludes unit $i$ carries no information about $i$'s idiosyncratic noise. We invoke Assumption (ref) where it is used and flag the one place (Proposition (ref)) where a further condition on leaf size would be needed for a fully formal statement.
The estimator we study builds an ensemble of honest causal trees, each grown on a cluster-level subsample (units, not observations, are sampled so that all observations of a unit stay together). Within a tree, every node $\mathcal{N}$ is treated locally: the outcome and treatment are residualized by a within-transformation computed using only the observations in $\mathcal{N}$,
obtained by alternating unit/time demeaning (iterative projection, converging in a few passes). The local treatment effect is the within estimator $\hat\tau_{\mathcal{N}} = \sum_{\mathcal{N}} \tilde{D}_{it}\tilde{Y}_{it} \big/ \sum_{\mathcal{N}} \tilde{D}_{it}^2$, and candidate splits $S$ into children $\mathcal{L},\mathcal{R}$ are scored by the heterogeneity criterion $\Delta(S) = (n_{\mathcal{L}} n_{\mathcal{R}}/n^2)(\hat\tau_{\mathcal{L}} - \hat\tau_{\mathcal{R}})^2$. Honesty is enforced by splitting the tree's subsample into a structure half (used to choose splits) and an estimation half (used to compute the leaf effects), with the split taken at the unit level.
The feature that drives our results is the aggregation. Writing $\hat\tau_b(x)$ for the leaf effect that tree $b$ assigns to a point $x$, the forest prediction is the simple average
This is the causal-tree-averaging construction of athey2016. It differs from the generalized random forest of wager2018,athey2019grf, which does not average leaf effects but instead uses the forest to build similarity weights $\alpha_i(x)$ and solves a single locally weighted moment condition. The distinction matters quantitatively: as we show in Section (ref), both constructions compress the estimated heterogeneity, but averaging noisy, honestly-estimated leaf effects compresses it substantially more than similarity weighting does on the same signal.
Because each leaf effect in (ref) is a within estimator on a small honest subsample, the average $\hat{\tau}_{\text{raw}}(x)$ is a biased estimate of the overall level. A standard remedy reports the average treatment effect separately, from the full-sample within-fixed-effects estimator
which is numerically the two-way fixed-effects regression coefficient and admits the usual cluster-robust (pair-clustered) standard error. Predictions are then recentered by the additive constant that equates their mean to (ref):
This is a common default reporting mode. It corrects the level: by construction $\overline{\hat{\tau}_{\text{rec}}} = \widehat{\mathrm{ATE}}$. Because it is an additive shift, it leaves the spread of the predictions unchanged, that is, their standard deviation, their slope on $\tau(x)$, and every centered moment. If the raw predictions attenuate the heterogeneity, recentering does not fix it. That is the gap the next two sections diagnose and close.
Define the attenuation slope $b$ as the coefficient in the population least-squares projection of the raw forest prediction on the truth,
so that $\hat{\tau}_{\text{raw}}(x)\approx a + b\,\tau(x)$ as in (ref). A slope $b=1$ means the forest tracks the heterogeneity one-for-one; $b<1$ means it is compressed toward the mean. Across every design we examine, $b$ is bounded well below one. Because recentering (ref) is additive it does not enter (ref): the recentered predictions have the same slope $b$ and the same standard deviation as the raw ones. Attenuation is therefore a property of the reported CATEs under the standard workflow, not an artifact removed by level correction.
The mechanism is regression to the mean, operating twice. First, each leaf effect is a within estimator on a small honest estimation sample in which the residualized treatment $\tilde{D}$ has limited variation; such estimates are noisy and, once assigned as a constant prediction over the leaf, shrink the effective contrast between high- and low-effect regions. Second, averaging across trees in (ref) combines many such noisy leaf assignments for the same point $x$; when a point is sometimes placed in a high-effect leaf and sometimes in a lower one, the average pulls its prediction toward the center of the effect distribution. Both forces push $b$ below one, and both intensify when the per-leaf signal is weak, whether from few units, low signal-to-noise, or many candidate covariates diluting the splits. This is the qualitative content of the comparative statics we document next. The GRF construction of Section (ref) does not assign and average constant leaf effects; it still shrinks a noisy surface toward the mean, so its slope is also below one, but it avoids the second, averaging-specific channel, which is why it attenuates less than the averaged-tree class. Table (ref) shows this gap directly. Two caveats keep the comparison honest. The GRF is fit on the outcome and treatment residualized on unit and period effects once over the full sample, whereas the averaged-tree forest residualizes locally within each node (ref); the two estimators therefore differ not only in aggregation but in where the fixed-effect transform is applied, so the measured gap bounds rather than exactly isolates the averaging channel. We hold the identifying signal fixed, feeding GRF the same within-transformed contrast, so the comparison is not confounded by different fixed-effect handling at the level of what is identified; it is the locality of the transform, not its target, that still differs.
Table (ref) reports the slope. For the averaged-tree forest it is everywhere below one, ranging from $0.31$ to $0.75$, and it moves in interpretable directions. Attenuation eases as the signal strengthens ($b$ rises from $0.38$ to $0.72$ as the signal-to-noise ratio grows) and as the panel grows ($b$ from $0.31$ at $100$ units to $0.75$ at $800$); it worsens as the covariate dimension rises and dilutes the splits ($b$ from $0.70$ at $p=2$ to about $0.51$ at $p=20$). It is nearly flat in tree depth. The generalized random forest, fit on the same within-transformed signal, tells the same qualitative story with one difference of degree: its slope is also below one in every design, so similarity weighting does not escape attenuation, but it is higher than the averaged-tree slope in every row, by roughly $0.10$ to $0.16$ in the small-panel, low-signal, and high-dimensional regimes where the averaging-specific channel bites hardest, and by less where the per-leaf signal is strong. The gaps are large relative to their Monte Carlo standard errors. No single number is universal, but the compression is pervasive, is governed by the per-leaf signal, and is systematically worse for leaf averaging than for similarity weighting, as the mechanism in the previous subsection predicts. The practical implication is that in the small or noisy panels common in difference-in-differences applications, the reported CATE distribution can understate the true spread by a factor of two or more, and by more for the averaged-tree forests that are our focus.
The correction inverts (ref) using a feasible moment restriction that requires no ground truth. In the within-transformed space of (ref), evaluated on the full sample, the model (ref) implies $\tilde{Y}_{it} = \tau(X_i)\,\tilde{D}_{it} + \text{error}$. Let $S(x) = \hat{\tau}_{\text{oob}}(x) - \overline{\hat{\tau}_{\text{oob}}}$ be a centered forest proxy for the heterogeneity (its out-of-bag construction is defined below). Anchoring the level at the unbiased $\widehat{\mathrm{ATE}}$ of (ref), we estimate a single slope $s$ from the moment regression
and form the corrected prediction
Equation (ref) is the best linear predictor of the residualized outcome in the direction of the proxy; its slope $s$ rescales the compressed heterogeneity back toward its true magnitude. The corrected prediction (ref) preserves the unbiased level, $\overline{\hat{\tau}_{\text{blp}}} = \widehat{\mathrm{ATE}}$, exactly as recentering does, so nothing is lost on the dimension the default already got right. Note that $s$ is not the naive inverse $1/b$: because the proxy $S$ is itself estimated with error, the MSE-optimal rescaling shrinks toward one, so $s$ is a partial de-attenuation that accounts for proxy noise, which is why it improves mean-squared error rather than overshooting.
The proxy $S$ must be cross-fitted, and the forest supplies a natural means at no extra cost. Each tree is grown on a subsample of units; for observation $i$ we form $\hat{\tau}_{\text{oob}}(X_i)$ by averaging only those trees whose subsample excluded unit $i$. This out-of-bag proxy is independent of the leaf estimates that see unit $i$, so the calibration in (ref) does not regress the forest's estimation noise on itself.
This requirement has teeth. If instead one uses the in-sample proxy $\hat{\tau}_{\text{raw}}(X_i)$, the slope $s$ picks up the spurious covariation between a point's prediction and its own contribution to the leaves that formed it. Under a homogeneous effect this manufactures heterogeneity out of noise, and the in-sample calibration reports a dispersed CATE distribution where none exists. The out-of-bag proxy removes that covariation, and the slope collapses toward zero when there is nothing to de-attenuate.
The slope in (ref) is a ratio,
and null-safety is a statement about where this ratio concentrates when there is nothing to de-attenuate. The delicate point is that under a homogeneous effect both the numerator and the denominator of (ref) are small: the proxy $S$ is then pure estimation noise, so the numerator has mean zero and the denominator, $\sum_{it}\tilde{D}_{it}^2 S(X_i)^2$, is itself shrinking with the proxy variance. The result below therefore controls the denominator explicitly, through a leaf-size condition, rather than treating $s$ as a numerator alone.
The leaf-size condition is the one substantive regularity requirement: it guarantees the proxy does not degenerate to a constant, which is what keeps the denominator of (ref) positive and the ratio well-defined. It is mild (honest forests with a fixed minimum leaf size satisfy it), and the null row of Table (ref) confirms the conclusion directly: with a homogeneous effect the out-of-bag correction leaves the CATE spread at the noise floor, manufacturing no heterogeneity.
Algorithm (ref) states the full estimate-then-correct procedure.
Table (ref) summarizes performance using the shipped implementation with out-of-bag calibration. Under heterogeneity the correction restores most of the lost spread (in the base design, from an attenuated standard deviation up toward the true $1.59$) and reduces CATE mean-squared error by $25$ to $42$ percent relative to the recentering default, with the largest gains in the small-panel and high-dimensional regimes where attenuation is worst; the Monte Carlo standard errors on the reduction (in parentheses) are small relative to the gains. The GRF column answers the natural “why not just use a generalized random forest” question on the quantity that matters: recentered to the same average effect, the GRF's CATE mean-squared error is above the corrected averaged-tree forest's in every heterogeneous design, so correcting the averaged-tree forest is preferable to switching estimators, not merely to leaving it uncorrected. The homogeneous row checks null-safety. With no true heterogeneity the corrected standard deviation is $0.15$, against a true value of zero: the correction does introduce a small residual dispersion at $n=300$, but it is an order of magnitude below the genuine heterogeneity in the other rows (true SD near $1.6$), and Proposition (ref) shows it vanishes as the panel grows. Its mean-squared error stays at the noise floor in absolute terms ($0.035$ against the recentering $0.020$); the large negative percentage in that row is the ratio of two values that are both close to zero, and its standard error is correspondingly wide, so it should be read as noise-floor movement rather than a meaningful degradation. The honest summary is that the correction is strongly active when there is heterogeneity to recover and close to quiet, though not exactly silent at finite $n$, when there is not.
The simulations above use fully synthetic designs. To show that the attenuation arises on the geometry of a real dataset, meaning its covariate distribution, panel dimensions, and staggered treatment timing, we run an empirical Monte Carlo on the county minimum-wage panel of callaway2021difference (the mpdta benchmark: 500 counties, 2003--2007, cohorts adopting in 2004, 2006, and 2007, with a large never-treated group). We use two real time-invariant covariates: standardized log county population, and each county's pre-treatment (2003) baseline log employment. We do not use this panel to estimate the true minimum-wage effect, which is unknown. Instead we take its real covariate values, unit and time structure, and treatment timing, and impose a known nonlinear heterogeneous effect, $\tau(x) = -0.08 + 0.10\,(\tilde{x}_1 + 0.5\,\tilde{x}_1^2 - 0.4\,\tilde{x}_1\tilde{x}_2)$, that varies with log population and its interaction with baseline employment. The nonlinearity is deliberate: because our correction is a single global linear rescaling, it cannot match this surface by construction, so any reduction in mean-squared error is a genuine gain rather than an artifact of imposing exactly the functional form the correction assumes. Because the truth is imposed, we can measure the attenuation directly on realistic data geometry, which a plain application cannot do because the true CATE is never observed. This is a design-geometry exercise on a standard public dataset, and it neither overlaps nor bears on any substantive application.
Table (ref) reports the result. The averaged-tree attenuation slope is $b=0.81$, milder than in the small, high-dimensional synthetic designs of Table (ref), as expected for a panel with a strong per-leaf signal and few covariates, but still visibly below one, so the recentered predictions understate the imposed spread (standard deviation $0.089$ against a true $0.106$). The GRF slope on the same residualized signal is $0.90$, again attenuated but less so, consistent with the synthetic evidence. The out-of-bag correction expands the averaged-tree spread toward the truth ($0.102$) and reduces CATE mean-squared error by about $17\%$, while recovering the average effect. It does so even though the imposed surface is nonlinear and the correction is a single linear rescaling, so the gain is not an artifact of matching the imposed functional form. The magnitude here is design-specific; what the exercise shows is that the attenuation and its correction behave on a familiar real-data geometry as the theory and synthetic evidence predict.
Causal forests that estimate heterogeneous effects by averaging honest leaf-level treatment effects, an attractive and widely implemented construction in fixed-effects panel settings, systematically compress the heterogeneity they are meant to recover. We characterized this attenuation, showing that the slope of the raw prediction on the truth is bounded well below one and moves predictably with signal-to-noise, panel size, and dimensionality, that a similarity-weight generalized random forest on the same signal attenuates in the same direction but consistently less, and that the additive recentering used to report an unbiased average treatment effect leaves the compression untouched. Because the underlying best-linear-predictor calibration slope is already available from the work of cddf2025 and the diagnostic tooling of atheywager2019, the remedy is a repurposing rather than a new estimator: we invert the slope into a correction, estimate it out-of-bag so that it is self-contained within the observational panel and asymptotically inert under a homogeneous effect, and ship it in the causalfe Python package. For applied work the recommendation is specific: when reporting the dispersion or tails of a causal-forest CATE distribution in a panel setting, recentering alone is not enough, and the reported spread should be read as a lower bound unless it has been calibrated.
Two limitations remain. The correction is a global linear rescaling; where attenuation varies across the covariate space, a vanderlaan2023-style monotone calibration may recover more, at the cost of the closed-form simplicity we exploit. And our evidence is based on simulation and empirical Monte Carlo; a formal characterization of the slope $b$ as a function of leaf size and subsample rate, in the spirit of the forest asymptotics of wager2018, is left to future work.
\ifanon\else
The author thanks colleagues for helpful comments. All errors are the author's own. \fi
The empirical illustration uses the publicly available mpdta county minimum-wage panel of callaway2021difference, distributed with the did R package on CRAN (https://cran.r-project.org/package=did). No new data were collected. All other results come from fully reproducible simulations; the code that generates every table and the empirical Monte Carlo, together with the software implementing the method, \ifanon is included as supplementary material and will be deposited in a public repository upon acceptance. \else is openly available at https://github.com/haytug/causalfe. \fi
The author received no specific funding for this work.
The author reports there are no competing interests to declare.
The author used a generative AI assistant to support drafting, code scaffolding, and literature search during the preparation of this manuscript; all methodological claims, simulation results, and their interpretation were verified by the author, who takes full responsibility for the content.