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arxiv: 2605.14943 · v2 · pith:DTATQLFYnew · submitted 2026-05-14 · 📊 stat.ME

Piece-wise linear isotonic regression

Timo Kuosmanen , Juan F. Monge , Jos\'e L. Ruiz , Xun Zhou This is my paper

Pith reviewed 2026-05-20 20:59 UTC · model grok-4.3

classification 📊 stat.ME
keywords isotonic regressionpiece-wise linear smoothingmonotonic estimationmarginal effectsconditional convexitybilevel optimizationnon-convex monotonicity
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The pith

A bilevel-optimized piece-wise linear smoother recovers usable marginal estimates from isotonic regression even when the true relationship is non-convex.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Isotonic regression flexibly estimates monotonic functions without global curvature assumptions, yet its step-function output prevents calculation of meaningful marginal quantities such as elasticities or shadow prices. The paper introduces a smoothing procedure that fits a continuous, monotonic piece-wise linear function to the isotonic steps by solving a bilevel optimization problem that incorporates conditional convexity. Monte Carlo experiments indicate that the smoothed estimates improve accuracy relative to unsmoothed isotonic regression in both convex and non-convex regimes for univariate and multivariate data. An empirical illustration applies the method to agglomeration economies across Finnish municipalities.

Core claim

Fitting a continuous monotonic piece-wise linear function to initial isotonic regression predictions via bilevel optimization recovers meaningful marginal estimates in non-convex settings. The procedure rests on conditional convexity to enforce local convexity while preserving global monotonicity, yielding a smoothed function whose derivatives supply the desired marginal properties.

What carries the argument

Bilevel optimization that fits a continuous monotonic piece-wise linear function to isotonic predictions subject to conditional convexity constraints.

If this is right

  • Marginal effects such as elasticities and shadow prices become directly computable from the smoothed function.
  • The approach improves finite-sample accuracy for both convex and non-convex data-generating processes in simulations.
  • The method extends to multivariate isotonic settings without requiring global convexity.
  • It supplies a practical route to apply isotonic regression in economic contexts that need marginal information.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The smoothed functions could serve as inputs to downstream optimization models that require differentiable monotonic constraints.
  • Similar bilevel smoothing might be tested on other step-function estimators to extract marginal properties.
  • Computational scaling of the bilevel problem in high dimensions remains an open practical question.

Load-bearing premise

The bilevel smoothing step produces reliable marginal estimates without distorting the underlying monotonic signal.

What would settle it

Generate data from a known non-convex monotonic function, apply the method, and test whether the recovered marginal effects match the true derivatives more closely than those from standard isotonic or convex alternatives.

Figures

Figures reproduced from arXiv: 2605.14943 by Jos\'e L. Ruiz, Juan F. Monge, Timo Kuosmanen, Xun Zhou.

Figure 1
Figure 1. Figure 1: Illustration of the logistic functions estimated by INLS, CC-INLS (vertices), [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the Cobb-Douglas functions estimated by INLS, CC-INLS [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of out-of-sample performance across all scenarios. [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Illustration of the estimated CC-INLS regression functions in the test set. [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

Isotonic regression provides a flexible, tuning-free approach to estimating monotonic functions without imposing global curvature constraints, yet the estimated regression function is inherently a step function. This paper addresses a key limitation of such estimators: their inability to provide meaningful marginal properties, such as shadow prices or elasticities. We propose a novel piece-wise linear smoothing framework that recovers meaningful marginal estimates even in non-convex settings. Building on the concept of conditional convexity originally developed in deterministic frontier analysis, we formulate the smoothing process as a bilevel optimization problem that fits a continuous, monotonic, piece-wise linear function to the initial isotonic regression predictions. Monte Carlo simulations demonstrate that the proposed approach can significantly improve estimation accuracy in both convex and non-convex settings for univariate and multivariate data. We apply this approach to analyze agglomeration economies in Finnish municipalities, illustrating its practical value.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims that standard isotonic regression yields step-function estimates of monotonic relationships but cannot directly supply interpretable marginal quantities such as shadow prices or elasticities. It proposes a piecewise-linear smoothing procedure cast as a bilevel optimization problem that exploits the notion of conditional convexity (imported from deterministic frontier analysis) to produce a continuous, monotonic piecewise-linear approximant to the isotonic fit. Monte Carlo experiments are reported to show accuracy gains in both convex and non-convex univariate and multivariate settings, and the method is illustrated on an agglomeration-economies application using Finnish municipal data.

Significance. A reliable method for extracting marginal estimates from isotonic regressions without imposing global convexity would be useful in applied work that requires monotonicity together with local slopes. The Monte Carlo evidence and empirical illustration, if they survive closer scrutiny of the simulation design and the behavior of the bilevel solver under noise, would constitute a concrete contribution to the isotonic-regression literature.

major comments (3)
  1. [§3] §3 (Bilevel formulation): The central claim that the segment slopes of the fitted piecewise-linear function constitute reliable marginal estimates rests on the unproven assertion that conditional convexity prevents distortion of the underlying monotonic signal when the input is a noisy isotonic step function. No consistency or unbiasedness argument is supplied for these slopes under standard regression noise assumptions, particularly in flat regions or near jumps.
  2. [Monte Carlo section] Monte Carlo section: The abstract states that the approach 'significantly improve[s] estimation accuracy' in non-convex settings, yet the reported results contain no information on the number of replications, the precise data-generating processes (including how non-convexity is operationalized), the exact loss used to measure accuracy, or whether standard errors accompany the reported gains. Without these details the Monte Carlo evidence cannot be evaluated.
  3. [Empirical application] Empirical application: The agglomeration-economies illustration does not report a direct comparison of the marginal estimates obtained from the proposed smoother against those from convex regression or from local-polynomial methods applied to the same data; such a comparison is needed to assess whether the bilevel step adds value beyond the initial isotonic fit.
minor comments (2)
  1. [§3] The notation distinguishing the isotonic step function from the subsequent piecewise-linear approximant is introduced only informally; an explicit equation defining the bilevel objective and the monotonicity constraints would improve readability.
  2. [Figures] Figure captions should state the number of Monte Carlo replications and the precise performance metric plotted.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment point by point below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (Bilevel formulation): The central claim that the segment slopes of the fitted piecewise-linear function constitute reliable marginal estimates rests on the unproven assertion that conditional convexity prevents distortion of the underlying monotonic signal when the input is a noisy isotonic step function. No consistency or unbiasedness argument is supplied for these slopes under standard regression noise assumptions, particularly in flat regions or near jumps.

    Authors: We acknowledge that the manuscript does not supply a formal consistency or unbiasedness proof for the recovered slopes. The bilevel formulation is motivated by the property that conditional convexity, as imported from frontier analysis, enforces a local convexity structure that preserves the monotonic ordering from the isotonic fit while producing a continuous piecewise-linear approximant. Monte Carlo results provide supporting evidence across convex and non-convex designs. We agree that explicit discussion of behavior near jumps and in flat regions is warranted and will revise §3 to clarify the role of conditional convexity, add a limitations paragraph on the absence of asymptotic guarantees, and outline conditions under which the slopes are expected to remain reliable. revision: partial

  2. Referee: [Monte Carlo section] Monte Carlo section: The abstract states that the approach 'significantly improve[s] estimation accuracy' in non-convex settings, yet the reported results contain no information on the number of replications, the precise data-generating processes (including how non-convexity is operationalized), the exact loss used to measure accuracy, or whether standard errors accompany the reported gains. Without these details the Monte Carlo evidence cannot be evaluated.

    Authors: The referee is correct that these implementation details were omitted from the reported Monte Carlo section. We will revise the section to specify the number of replications, fully describe the data-generating processes (including explicit constructions for the non-convex cases), state the loss functions employed to quantify accuracy gains, and report standard errors or variability measures around the tabulated improvements. These additions will make the simulation evidence fully reproducible and evaluable. revision: yes

  3. Referee: [Empirical application] Empirical application: The agglomeration-economies illustration does not report a direct comparison of the marginal estimates obtained from the proposed smoother against those from convex regression or from local-polynomial methods applied to the same data; such a comparison is needed to assess whether the bilevel step adds value beyond the initial isotonic fit.

    Authors: We agree that a direct comparison would help quantify the incremental contribution of the bilevel smoother. In the revised manuscript we will augment the empirical section with marginal estimates obtained from convex regression and from local-polynomial regression applied to the same Finnish municipal data, allowing readers to evaluate whether the piecewise-linear isotonic smoother yields substantively different or more interpretable slopes than these alternatives. revision: yes

Circularity Check

0 steps flagged

No circularity: bilevel smoothing is an independent optimization step, not a redefinition of inputs.

full rationale

The paper introduces isotonic regression as the first stage and then formulates a separate bilevel optimization that fits a continuous monotonic piecewise-linear function to those step-function predictions. No equation reduces the claimed marginal estimates (slopes) back to the isotonic values by algebraic identity or by renaming a fitted parameter as a prediction. The conditional-convexity concept is imported from prior deterministic frontier literature rather than defined circularly within this work, and no self-citation chain is shown to be load-bearing for the central claim. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of isotonic regression and on the imported notion of conditional convexity; no new free parameters are explicitly introduced in the abstract, and no new physical or mathematical entities are postulated.

axioms (2)
  • standard math Isotonic regression produces a valid monotonic step-function estimator
    Invoked in the opening sentence as the starting point for the smoothing step.
  • domain assumption Conditional convexity from deterministic frontier analysis can be used to guide the bilevel smoothing while preserving monotonicity
    Explicitly referenced as the conceptual foundation for formulating the smoothing as a bilevel problem.

pith-pipeline@v0.9.0 · 5673 in / 1497 out tokens · 28050 ms · 2026-05-20T20:59:34.546708+00:00 · methodology

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Works this paper leans on

300 extracted references · 300 canonical work pages

  1. [1]

    European Journal of Operational Research , author =

    A nonparametric least-squares model in network data envelopment analysis , volume =. European Journal of Operational Research , author =. 2026 , keywords =. doi:10.1016/j.ejor.2025.09.045 , abstract =

    work page doi:10.1016/j.ejor.2025.09.045 2026

  2. [2]

    European Journal of Operational Research , author =

    Sparse convex quantile regression:. European Journal of Operational Research , author =. 2026 , keywords =. doi:10.1016/j.ejor.2026.03.001 , abstract =

    work page doi:10.1016/j.ejor.2026.03.001 2026

  3. [3]

    European Journal of Operational Research , author =

    Optimal resource allocation:. European Journal of Operational Research , author =. 2025 , keywords =. doi:10.1016/j.ejor.2025.01.003 , abstract =

    work page doi:10.1016/j.ejor.2025.01.003 2025

  4. [4]

    Education for Chemical Engineers , author =

    Utilising interactive applications as educational tools in higher education:. Education for Chemical Engineers , author =. 2024 , keywords =. doi:10.1016/j.ece.2023.10.001 , abstract =

    work page doi:10.1016/j.ece.2023.10.001 2024

  5. [5]

    Lu, Qingzhao and Chen, Tao and Duan, Huicun and Zhou, Xun and Lin, Zhiwei (CJ) , month = may, year =. How. Journal of Travel Research , publisher =. doi:10.1177/00472875261441570 , abstract =

    work page doi:10.1177/00472875261441570

  6. [6]

    Energy Science , author =

    On the role of net metering and time-of-use pricing in distributed renewable energy adoption:. Energy Science , author =. 2026 , pages =. doi:10.1142/S2972379526300014 , abstract =

    work page doi:10.1142/s2972379526300014 2026

  7. [7]

    The Journal of Technology Transfer , author =

    Enhancing green innovation through university–industry collaboration and artificial intelligence: insights from regional innovation systems in. The Journal of Technology Transfer , author =. 2025 , keywords =. doi:10.1007/s10961-025-10232-8 , abstract =

    work page doi:10.1007/s10961-025-10232-8 2025

  8. [8]

    The role of user perceptions of intelligence, anthropomorphism, and self-extension on continuance of use of personal intelligent agents , volume =

    Moussawi, Sara and Koufaris, Marios and Benbunan-Fich, Raquel , month = may, year =. The role of user perceptions of intelligence, anthropomorphism, and self-extension on continuance of use of personal intelligent agents , volume =. European Journal of Information Systems , publisher =. doi:10.1080/0960085X.2021.2018365 , abstract =

    work page doi:10.1080/0960085x.2021.2018365 2021

  9. [9]

    Destination

    Xia, Haiyang and Muskat, Birgit and Karl, Marion and Li, Qian and Li, Gang , month = apr, year =. Destination. Journal of Travel Research , publisher =. doi:10.1177/00472875251322512 , abstract =

    work page doi:10.1177/00472875251322512

  10. [10]

    European Journal of Operational Research , author =

    Estimating most productive scale size using data envelopment analysis , volume =. European Journal of Operational Research , author =. 1984 , pages =. doi:10.1016/0377-2217(84)90006-7 , abstract =

    work page doi:10.1016/0377-2217(84)90006-7 1984

  11. [11]

    Ji, Yong-bae and Lee, Choonjoo , month = jul, year =. Data. The Stata Journal , publisher =. doi:10.1177/1536867X1001000207 , abstract =

    work page doi:10.1177/1536867x1001000207

  12. [12]

    Research Policy , author =

    Innovation under dual policies:. Research Policy , author =. 2026 , keywords =. doi:10.1016/j.respol.2025.105378 , abstract =

    work page doi:10.1016/j.respol.2025.105378 2026

  13. [13]

    Kuosmanen, Natalia and Kuosmanen, Timo and Zhou, Xun , month = feb, year =. Do

    work page

  14. [14]

    Technological Forecasting and Social Change , author =

    Overtourism and sustainability:. Technological Forecasting and Social Change , author =. 2023 , keywords =. doi:10.1016/j.techfore.2022.122285 , abstract =

    work page doi:10.1016/j.techfore.2022.122285 2023

  15. [15]

    Annals of Tourism Research , author =

    Conceptualising overtourism:. Annals of Tourism Research , author =. 2020 , keywords =. doi:10.1016/j.annals.2020.103025 , abstract =

    work page doi:10.1016/j.annals.2020.103025 2020

  16. [16]

    The Journal of Wildlife Management , author =

    Causality and wildlife management , volume =. The Journal of Wildlife Management , author =. 2023 , note =. doi:10.1002/jwmg.22412 , abstract =

    work page doi:10.1002/jwmg.22412 2023

  17. [17]

    Ecosystem Services , author =

    Digging for quality and quantity:. Ecosystem Services , author =. 2026 , keywords =. doi:10.1016/j.ecoser.2026.101822 , abstract =

    work page doi:10.1016/j.ecoser.2026.101822 2026

  18. [18]

    and Lynas, Mark and Ellis, Erle C

    Quagraine, Kwesi A. and Lynas, Mark and Ellis, Erle C. , month = jan, year =. As we breach 1.5 °. Nature , publisher =. doi:10.1038/d41586-026-00246-z , abstract =

    work page doi:10.1038/d41586-026-00246-z

  19. [19]

    Multi-regional

    Ouyang, Qianhong and Meng, Jing and Wang, Yafei and Zheng, Heran , month = dec, year =. Multi-regional. Scientific Data , publisher =. doi:10.1038/s41597-025-06377-8 , abstract =

    work page doi:10.1038/s41597-025-06377-8

  20. [20]

    Ecological Economics , author =

    From fee to tax:. Ecological Economics , author =. 2026 , keywords =. doi:10.1016/j.ecolecon.2026.108917 , abstract =

    work page doi:10.1016/j.ecolecon.2026.108917 2026

  21. [21]

    Journal of Environmental Management , author =

    How do uncertainties drive the risk spillover across. Journal of Environmental Management , author =. 2026 , pages =. doi:10.1016/j.jenvman.2026.128623 , abstract =

    work page doi:10.1016/j.jenvman.2026.128623 2026

  22. [22]

    Journal of the Finnish Economic Association , author =

    Adaptation of. Journal of the Finnish Economic Association , author =. 2025 , keywords =. doi:10.33358/jfea.148331 , abstract =

    work page doi:10.33358/jfea.148331 2025

  23. [23]

    Bostian, Moriah and Lundgren, Tommy and Bostian, Moriah and Lundgren, Tommy , month = mar, year =. Valuing. Sustainability , publisher =. doi:10.3390/su14053035 , abstract =

    work page doi:10.3390/su14053035

  24. [24]

    Journal of Environmental Economics and Management , author =

    Using satellite-observed geospatial inundation data to identify the impacts of floods on firm-level performance:. Journal of Environmental Economics and Management , author =. 2026 , pages =. doi:10.1016/j.jeem.2025.103276 , abstract =

    work page doi:10.1016/j.jeem.2025.103276 2026

  25. [25]

    Journal of Environmental Management , author =

    Large language models for environmental modeling:. Journal of Environmental Management , author =. 2026 , pages =. doi:10.1016/j.jenvman.2025.128417 , abstract =

    work page doi:10.1016/j.jenvman.2025.128417 2026

  26. [26]

    Can the environment be an inferior good?

    Dupoux, Marion and Martinet, Vincent , month = apr, year =. Can the environment be an inferior good?

    work page

  27. [27]

    2024 , pages =

    Journal of Statistical Software , author =. 2024 , pages =. doi:10.18637/jss.v111.i06 , abstract =

    work page doi:10.18637/jss.v111.i06 2024

  28. [28]

    European Journal of Operational Research , author =

    Stochastic non-convex envelopment of data:. European Journal of Operational Research , author =. 2013 , pages =. doi:10.1016/j.ejor.2013.06.005 , abstract =

    work page doi:10.1016/j.ejor.2013.06.005 2013

  29. [29]

    and Bowen, William G

    Baumol, William J. and Bowen, William G. , month = jan, year =. Performing

    work page

  30. [30]

    Baumol, W. J. and Bowen, W. G. , year =. On the. The American Economic Review , publisher =

    work page

  31. [31]

    and de Ferranti, David and Malach, Monte and Pablos-Méndez, Ariel and Tabish, Hilary and Wu, Lilian Gomory , year =

    Baumol, William J. and de Ferranti, David and Malach, Monte and Pablos-Méndez, Ariel and Tabish, Hilary and Wu, Lilian Gomory , year =. The

    work page

  32. [32]

    The ones that got away?

    Buri, Riku and Heinonen, Miika and Pietola, Matias , month = jan, year =. The ones that got away?

    work page

  33. [33]

    Steven and Pakes, Ariel , year =

    Olley, G. Steven and Pakes, Ariel , year =. The. Econometrica , publisher =. doi:10.2307/2171831 , abstract =

    work page doi:10.2307/2171831

  34. [34]

    Estimating gross value added volumes and prices by institutional sector , copyright =

    Wieland, Elisabeth and Kavonius, Ilja Kristian , year =. Estimating gross value added volumes and prices by institutional sector , copyright =. doi:10.2866/533914 , abstract =

    work page doi:10.2866/533914

  35. [35]

    Journal of Productivity Analysis , author =

    Inter-industry and intra-industry switching as sources of productivity growth: structural change of. Journal of Productivity Analysis , author =. 2024 , pages =. doi:10.1007/s11123-023-00712-0 , abstract =

    work page doi:10.1007/s11123-023-00712-0 2024

  36. [36]

    Journal of Productivity Analysis , author =

    Modeling economies of scope in joint production:. Journal of Productivity Analysis , author =. 2025 , pages =. doi:10.1007/s11123-024-00739-x , abstract =

    work page doi:10.1007/s11123-024-00739-x 2025

  37. [37]

    , month = jun, year =

    Kuosmanen, Timo and Johnson, Andrew L. , month = jun, year =. Conditional. The Energy Journal , publisher =. doi:10.5547/01956574.42.S12.tkuo , abstract =

    work page doi:10.5547/01956574.42.s12.tkuo

  38. [38]

    Structural change decomposition of productivity without share weights , volume =

    Kuosmanen, Timo and Kuosmanen, Natalia , month = dec, year =. Structural change decomposition of productivity without share weights , volume =. Structural Change and Economic Dynamics , publisher =. doi:10.1016/j.strueco.2021.08.011 , abstract =

    work page doi:10.1016/j.strueco.2021.08.011 2021

  39. [39]

    Paradoxes of the public sector productivity measurement , copyright =

    Kuosmanen, Timo and Zhou, Xun , month = oct, year =. Paradoxes of the public sector productivity measurement , copyright =. doi:10.48550/arXiv.2509.14795 , abstract =

    work page doi:10.48550/arxiv.2509.14795

  40. [40]

    Secondary materials,

    Kuosmanen, Timo and Zhou, Xun , month = feb, year =. Secondary materials,. doi:10.48550/arXiv.2502.14636 , abstract =

    work page doi:10.48550/arxiv.2502.14636

  41. [41]

    Economica , author =

    An. Economica , author =. 2015 , note =. doi:10.1111/ecca.12159 , abstract =

    work page doi:10.1111/ecca.12159 2015

  42. [42]

    Erwin and Fox, Kevin J

    Diewert, W. Erwin and Fox, Kevin J. , editor =. Productivity. The. 2019 , pages =. doi:10.1007/978-3-030-23727-1_18 , abstract =

    work page doi:10.1007/978-3-030-23727-1_18 2019

  43. [43]

    Jorgenson, D. W. and Griliches, Z. , month = jul, year =. The. The Review of Economic Studies , publisher =. doi:10.2307/2296675 , abstract =

    work page doi:10.2307/2296675

  44. [44]

    Firm lifecycles and evolution of industry productivity , volume =

    Hyytinen, Ari and Maliranta, Mika , month = jun, year =. Firm lifecycles and evolution of industry productivity , volume =. Research Policy , publisher =. doi:10.1016/j.respol.2013.01.008 , abstract =

    work page doi:10.1016/j.respol.2013.01.008 2013

  45. [45]

    , month = nov, year =

    Hart, Oliver and Shleifer, Andrei and Vishny, Robert W. , month = nov, year =. The. The Quarterly Journal of Economics , publisher =. doi:10.1162/003355300555448 , abstract =

    work page doi:10.1162/003355300555448

  46. [46]

    Productivity

    Färe, Rolf and Grosskopf, Shawna and Norris, Mary and Zhang, Zhongyang , year =. Productivity. The American Economic Review , publisher =

    work page

  47. [47]

    Canadian Journal of Economics/Revue canadienne d'économique , author =

    Measuring real consumption and consumer price index bias under lockdown conditions , volume =. Canadian Journal of Economics/Revue canadienne d'économique , author =. 2022 , note =. doi:10.1111/caje.12545 , abstract =

    work page doi:10.1111/caje.12545 2022

  48. [48]

    Oxford University Press

    Diewert, W. Erwin , editor =. Productivity. The. 2018 , pages =. doi:10.1093/oxfordhb/9780190226718.013.7 , abstract =

    work page doi:10.1093/oxfordhb/9780190226718.013.7 2018

  49. [49]

    The Economic Journal , author =

    Multilateral. The Economic Journal , author =. 1982 , pages =. doi:10.2307/2232257 , number =

    work page doi:10.2307/2232257 1982

  50. [50]

    Reimagining government subsidy policies: facilitating echelon utilization and sustainable practices for retired battery systems , volume =

    Gu, Xiaoyu and Huang, Mengyi and Zhou, Li , month = oct, year =. Reimagining government subsidy policies: facilitating echelon utilization and sustainable practices for retired battery systems , volume =. Computers & Industrial Engineering , publisher =. doi:10.1016/j.cie.2025.111437 , abstract =

    work page doi:10.1016/j.cie.2025.111437 2025

  51. [51]

    , year =

    Solow, Robert M. , year =. Technical. The Review of Economics and Statistics , publisher =. doi:10.2307/1926047 , number =

    work page doi:10.2307/1926047

  52. [52]

    The Journal of Industrial Economics , author =

    Robustness of. The Journal of Industrial Economics , author =. 2007 , note =. doi:10.1111/j.1467-6451.2007.00322.x , abstract =

    work page doi:10.1111/j.1467-6451.2007.00322.x 2007

  53. [53]

    Journal of Economic Growth , author =

    Notes on. Journal of Economic Growth , author =. 1999 , pages =. doi:10.1023/A:1009828704275 , abstract =

    work page doi:10.1023/a:1009828704275 1999

  54. [54]

    Econometrica , author =

    Identification. Econometrica , author =. 2015 , note =. doi:10.3982/ECTA13408 , abstract =

    work page doi:10.3982/ecta13408 2015

  55. [55]

    Econometrica , author =

    Endogenous. Econometrica , author =. 2024 , pages =. doi:10.3982/ECTA20629 , abstract =

    work page doi:10.3982/ecta20629 2024

  56. [56]

    Journal of Econometrics , author =

    The unbalanced nested error component regression model , volume =. Journal of Econometrics , author =. 2001 , pages =. doi:10.1016/S0304-4076(00)00089-0 , abstract =

    work page doi:10.1016/s0304-4076(00)00089-0 2001

  57. [57]

    Journal of Productivity Analysis , author =

    Partial local and global returns to scale in efficiency analysis with different technologies , volume =. Journal of Productivity Analysis , author =. 2025 , pages =. doi:10.1007/s11123-025-00791-1 , abstract =

    work page doi:10.1007/s11123-025-00791-1 2025

  58. [58]

    Understanding and

    Ma, Shuang and Ge, Lin and Jia, He (Michael) and Wang, Yonggui , month = dec, year =. Understanding and. Information Systems Research , publisher =. doi:10.1287/isre.2023.0229 , abstract =

    work page doi:10.1287/isre.2023.0229 2023

  59. [59]

    , month = dec, year =

    McKinley, Galen A. , month = dec, year =. Revised estimates of. Nature , publisher =. doi:10.1038/d41586-025-03981-x , abstract =

    work page doi:10.1038/d41586-025-03981-x

  60. [60]

    and Li, Hongmei and van der Woude, Auke and Schwingshackl, Clemens and Pongratz, Julia and Regnier, Pierre and Andrew, Robbie M

    Friedlingstein, Pierre and Le Quéré, Corinne and O’Sullivan, Michael and Hauck, Judith and Landschützer, Peter and Luijkx, Ingrid T. and Li, Hongmei and van der Woude, Auke and Schwingshackl, Clemens and Pongratz, Julia and Regnier, Pierre and Andrew, Robbie M. and Bakker, Dorothee C. E. and Canadell, Josep G. and Ciais, Philippe and Gasser, Thomas and Jo...

    work page doi:10.1038/s41586-025-09802-5

  61. [61]

    Transportation Research Part D: Transport and Environment , author =

    Two-stage. Transportation Research Part D: Transport and Environment , author =. 2020 , pages =. doi:10.1016/j.trd.2020.102489 , abstract =

    work page doi:10.1016/j.trd.2020.102489 2020

  62. [62]

    The Review of Economic Studies , author =

    Revisiting. The Review of Economic Studies , author =. 2024 , pages =. doi:10.1093/restud/rdae007 , abstract =

    work page doi:10.1093/restud/rdae007 2024

  63. [63]

    The Quarterly Journal of Economics , author =

    Generative. The Quarterly Journal of Economics , author =. 2025 , pages =. doi:10.1093/qje/qjae044 , abstract =

    work page doi:10.1093/qje/qjae044 2025

  64. [64]

    and Krueger, Alan B

    Grossman, Gene M. and Krueger, Alan B. , month = may, year =. Economic growth and the environment , volume =. The Quarterly Journal of Economics , publisher =. doi:10.2307/2118443 , abstract =

    work page doi:10.2307/2118443

  65. [65]

    Journal of Environmental Economics and Management , author =

    Heterogeneous responses to carbon pricing:. Journal of Environmental Economics and Management , author =. 2026 , pages =. doi:10.1016/j.jeem.2025.103266 , abstract =

    work page doi:10.1016/j.jeem.2025.103266 2026

  66. [66]

    Synthetic

    Agarwal, Anish and Shah, Devavrat and Shen, Dennis , month = dec, year =. Synthetic. Operations Research , publisher =. doi:10.1287/opre.2025.1590 , abstract =

    work page doi:10.1287/opre.2025.1590 2025

  67. [67]

    Energy Policy , author =

    Pensions as hidden green levers:. Energy Policy , author =. 2026 , pages =. doi:10.1016/j.enpol.2025.115022 , abstract =

    work page doi:10.1016/j.enpol.2025.115022 2026

  68. [68]

    Energy Policy , author =

    Decarbonizing residential space heating with heat pumps in the. Energy Policy , author =. 2026 , pages =. doi:10.1016/j.enpol.2025.114997 , abstract =

    work page doi:10.1016/j.enpol.2025.114997 2026

  69. [69]

    Energy Economics , author =

    Mitigating carbon leakage under the. Energy Economics , author =. 2026 , pages =. doi:10.1016/j.eneco.2025.109024 , abstract =

    work page doi:10.1016/j.eneco.2025.109024 2026

  70. [70]

    Journal of Environmental Economics and Management , author =

    Does central supervision mitigate border pollution?. Journal of Environmental Economics and Management , author =. 2026 , pages =. doi:10.1016/j.jeem.2025.103261 , abstract =

    work page doi:10.1016/j.jeem.2025.103261 2026

  71. [71]

    , month = dec, year =

    Bao, Chenzhang and Bardhan, Indranil R. , month = dec, year =. Performance of. Information Systems Research , publisher =. doi:10.1287/isre.2021.1080 , abstract =

    work page doi:10.1287/isre.2021.1080 2021

  72. [72]

    MIS Quarterly , author =

    Editor’s. MIS Quarterly , author =. 2022 , pages =. doi:10.25300/MISQ/2022/463E1 , abstract =

    work page doi:10.25300/misq/2022/463e1 2022

  73. [73]

    Production and Operations Management , author =

    On the causality and plausibility of treatment effects in operations management research , volume =. Production and Operations Management , author =. 2022 , pages =. doi:10.1111/poms.13863 , abstract =

    work page doi:10.1111/poms.13863 2022

  74. [74]

    Production and Operations Management , author =

    When. Production and Operations Management , author =. 2025 , pages =. doi:10.1177/10591478241310217 , abstract =

    work page doi:10.1177/10591478241310217 2025

  75. [75]

    MIS Quarterly , author =

    Causal. MIS Quarterly , author =. 2025 , pages =. doi:10.25300/MISQ/2024/18422 , abstract =

    work page doi:10.25300/misq/2024/18422 2025

  76. [76]

    MIS Quarterly , author =

    Reconfiguring for. MIS Quarterly , author =. 2021 , pages =. doi:10.25300/MISQ/2021/14997 , abstract =

    work page doi:10.25300/misq/2021/14997 2021

  77. [77]

    MIS Quarterly , author =

    Price,. MIS Quarterly , author =. 2024 , pages =. doi:10.25300/MISQ/2024/18466 , abstract =

    work page doi:10.25300/misq/2024/18466 2024

  78. [78]

    MIS Quarterly , author =

    The. MIS Quarterly , author =. 2023 , pages =. doi:10.25300/MISQ/2022/17265 , abstract =

    work page doi:10.25300/misq/2022/17265 2023

  79. [79]

    MIS Quarterly , author =

    Competing with the. MIS Quarterly , author =. 2022 , pages =. doi:10.25300/MISQ/2022/15666 , abstract =

    work page doi:10.25300/misq/2022/15666 2022

  80. [80]

    MIS Quarterly , author =

    Peer. MIS Quarterly , author =. 2024 , pages =. doi:10.25300/MISQ/2024/16308 , abstract =

    work page doi:10.25300/misq/2024/16308 2024

Showing first 80 references.