Sharp Bounds on Continuous Treatment Effects under Unobserved Confounding

Junzhe Zhang

math.STOct 23, 2026

Sharp Bounds on Continuous Treatment Effects under Unobserved Confounding

Authors: Junzhe Zhang

Organizations: Syracuse University

Abstract

We study partial identification of conditional treatment effects with a continuous treatment under unobserved confounding. We target the conditional average potential outcome (CAPO) E[Y_x|z] and the conditional average treatment effect (CATE) between two treatment levels. We assume a bounded outcome and Lipschitz continuity of the response functions and of the propensity scores given the latent confounder, and show that these conditions imply the smooth treatment response and smooth treatment selection conditions of Kim et al. (2018). The combined bound of Kim et al. uses only two reference points on the identified diagonal and is generally loose. Bounding through every diagonal reference point instead gives explicit bounds whose average is the sharp identified set of the CAPO at every treatment, inside or outside the observational support. Capping the difference between the upper and lower bounds gives sharp causal bounds on the CATE. These bounds are informative without monotonicity or overlap assumptions.

Figures & tables

Panel of Figure 1 — Figure 1: Two confounded systems that induce the same observational law $P(X,Y)$ ( blue ; regression $\mathbb{E}[Y\mid x]$ in dark blue ) yet different interventional laws $P_{x}\left(Y\right)$ ( orange ), illustrating non-identifiability: (a) $\mathcal{M}$ with $Y\leftarrow-X^{2}+U^{2}$ has a causal effect $\mathbb{E}_{x}\left[Y\right]=1-x^{2}$ ( dark orange curve); (b) $\mathcal{M}^{\prime}$ with $Y\leftarrow X^{2}+\tfrac{1}{2}U^{2}$ has a causal effect $\mathbb{E}_{x}\left[Y\right]=x^{2}+\tfrac{1}{2}$ ( dark orange curve).

Figure 2: Half-width of the bound on the ETT over the $(x,x^{\prime})$ plane for Exs. 2 and 3 . The identified diagonal $x=x^{\prime}$ is dashed, and arrows point from each query to the diagonal point bounding it. (a) STR bound ( (STR) ); (b) STS bound ( (STS) ); (c) sharp bound over all diagonal reference points (Thm. 10 ); (d) enlarging a finite reference set $\mathbb{I}$ shrinks the averaged width toward the sharp bound. Panels (a)–(c) saturate at $\tfrac{1}{2}\operatorname{diam}(\mathbb{Y})=10$ .

Panel of Figure 3 — Figure 3: The L-shaped path of Lem. 5 (blue) from $(\bm{x},\bm{x}^{\prime})$ through $(\bm{s},\bm{x}^{\prime})$ to the diagonal point $(\bm{s},\bm{s})$ , at cost $\rho(\bm{s},\bm{x}^{\prime})$ (Eq. 14 ). A zig-zag path (grey) costs at least as much; query $\bm{x}$ may lie outside the support.

$x^{\prime}$	$s=0$	$s=1$	$s=2$	two-point	sharp
$0$	$2$	$1$	$2$	$2$	$1$
$1$	$3$	$0$	$1$	$0$	$0$
$2$	$4$	$1$	$0$	$0$	$0$

Table 1: The counterexample of Prop. 9 at the query

x=2

: terms

\mu(s)+|2-s|+|x^{\prime}-s|

, the two-point bound of Kim et al. (2018) (columns

s\in\{2,x^{\prime}\}

), and the sharp bound. Averaging over

P

gives

2/3

and

1/3

Setting	cov.	width	cov.	width	cov.	width	cov.	width	vs. Kim (%)
	Ours		Kim et al.		CMSM oracle		Khan et al.		Reduction
Discrete SCMs	1.00	0.11181	1.00	0.13832	1.00	0.04810	0.14	0.00000	19.17
Continuous SCMs	1.00	0.10627	1.00	0.13071	1.00	0.04841	0.04	0.00000	18.70
IST	1.00	0.03726	1.00	0.03777	1.00	0.07558	0.04	0.00000	1.34

Table 2: Query-averaged coverage and interval width normalized by the outcome range. Reduction compares pooled mean widths with Kim. CMSM uses a per-model coverage oracle, so its coverage holds by construction.

Figure 4: Example bounds and true means for discrete, smooth V-shaped continuous, and controlled IST models. Wine and gray shading denote ours and oracle CMSM; dashed and dotted boundaries denote Kim and CMSM. Discrete intervals use bars. Panels have separate outcome scales; example selection is specified in App. C .

Appendix figures & tables9 assets

Supplementary material from the paper’s appendix.

Appendix

Setting	SCMs	Queries	Ours width	Kim width	Reduction (%)	Coverage (both)
Canonical	1	7	0.1119	0.1391	19.57	1.00/1.00
Random	500	3544	0.1118	0.1383	19.17	1.00/1.00

Appendix

Table 3: Stronger discrete variant with a V-shaped response. Widths are divided by the attained outcome range,

\operatorname{diam}(\mathcal{Y})=2

. Ours and Kim use identical exact structural constants; all random draws are retained.

Figure 5: Discrete V-shaped SCMs with matched structural constants and exact outcome-range clipping. Left: the canonical V-shaped response, its confounded observational regression, and both intervals. Middle: mean-width reductions for all $500$ random systems. Right: the reduction across the assignment-tilt sweep, including settings where the methods tie.

Figure 6: Continuous V-shaped and smooth V-shaped response families. Top: canonical bounds at identical exact structural constants. Bottom: mean-width reductions against Kim across all $250$ random draws per family, including every draw without rejection.

Age	Ours width	Kim width	Reduction (%)	Ours cov.	Kim cov.
16–67	0.01972	0.01981	0.47	1.00	1.00
68–77	0.04844	0.04909	1.32	1.00	1.00
78–99	0.04362	0.04440	1.76	1.00	1.00

Appendix

Table 4: Controlled IST-derived binary SCM: mean widths and coverage at identical oracle STR/STS constants. Each age group has 25 treatment queries. This model has a new, explicitly constructed target.

Figure 7: Controlled IST-derived binary SCM, with identical oracle STR/STS constants for both methods. Top: bounds, constructed interventional targets, and observational means in each age group. Bottom: the reduction in mean width as global references are added to the per-pair endpoints.

Setting	Family	Ours	Kim	Ours cov.	Kim cov.	Reduction (%)
Canonical	Polynomial	0.3480	0.3639	1.00	1.00	4.37
Canonical	Logistic	0.3243	0.3282	1.00	1.00	1.18
Canonical	Phase	0.2136	0.2279	1.00	1.00	6.29
Canonical	Curved linear	0.1316	0.1362	1.00	1.00	3.36
Random	Polynomial	0.3471	0.3616	1.00	1.00	4.01
Random	Logistic	0.3235	0.3271	1.00	1.00	1.08

Appendix

Table 5: Designed smooth continuous SCMs. Widths are normalized by each system’s outcome range. Random rows retain all 125 draws per family; reduction compares pooled mean widths. Canonical settings and neighborhoods were chosen after exploratory checks.

Figure 8: Designed smooth continuous SCMs with identical structural constants and outcome ranges for both methods. Each panel shows our full-reference bounds (dark red), Kim’s bounds (purple dashed), the true mean (orange), and the observational regression (blue). Top: polynomial and logistic; bottom: phase and curved linear. These are fixed canonical examples chosen after exploratory checks; Table 5 also reports all random draws in the stated neighborhoods.

References / method	Lower	Upper	Width	Coverage
Kim et al.	-1.000	0.667	1.667	1.00
Ours: endpoints	-1.000	0.667	1.667	1.00
Ours: endpoints + $\{0\}$	-1.000	0.667	1.667	1.00
Ours: endpoints + $\{0,1\}$	-1.000	0.333	1.333	1.00
Ours: all references	-1.000	0.333	1.333	1.00

Appendix

Table 6: Controlled reference ablation: fixed observational law,

a=b=1

, and known outcome range

[-2,2]

. Endpoints are

\{x,x^{\prime}\}

per stratum. Bounds and widths are at

x=2

in outcome units; coverage is over all three queries. Adding

s=1

reduces width by

20\%

without losing coverage.

Figure 9: Controlled reference ablation with identical data and constants. Left: Kim’s intervals and our full-reference intervals at all three queries, with the true means marked. Right: the interval at $x=2$ as the extra global reference set $R$ grows. The interior reference $s=1$ lowers the upper endpoint to the attained sharp value $1/3$ ; adding the remaining endpoint does not tighten it further.

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Minimax rates for pointwise CATE estimation: the role of design smoothness

Yuxuan Hou, Zhenghan Li

Qiuzhen College, Tsinghua University · School of Statistics, University of Minnesota

Conditional Average Treatment Effect Estimation Minimax Estimation

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We study the role of design smoothness in the minimax estimation of conditional average treatment effects at a fixed covariate point. For design densities bounded above and away from zero, without further regularity, we derive upper and lower risk bounds with an arbitrarily small gap in the convergence-rate exponent. Under uniform design, we obtain matching upper and lower bounds at interior points in the regime of low nuisance smoothness. These results show that rough covariate designs can lead to slower rates than uniform designs, even under the same smoothness assumptions on the treatment effect and nuisance functions, highlighting the role of the covariate distribution in pointwise causal estimation.

Abstract

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