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The Augmented Synthetic Control Method

Eli Ben-Michael, Avi Feller, Jesse Rothstein

arXiv:1811.04170v3stat.MEecon.EM

TL;DR

SCM is intended for cases with excellent pre-treatment fit, but that fit can be infeasible when no suitable synthetic control can be constructed. Augmented SCM adjusts for poor fit using an outcome model, and simulations show large gains over alternative estimators, especially under model misspecification, in bias and root mean squared error.

  • Problem

    SCM is intended for cases with excellent pre-treatment fit, but such fit can be infeasible when a synthetic control cannot be constructed.

    Source: §1 · p. 2

  • Method

    Augmented SCM adjusts SCM for poor pre-treatment fit using an outcome model, with Ridge ASCM using ridge regression.

    Source: §1 · p. 2, §3.1 · p. 7

  • Results

    ASCM shows large gains over alternative estimators, especially under model misspecification, in both bias and root mean squared error.

    Source: §1 · p. 3

  • Takeaways & Limitations

    ASCM offers a middle ground for settings where a conventional synthetic control cannot be constructed.

    Source: §1 · p. 2

    Under the higher-noise factor model, gains from augmentation relative to SCM are more limited.

    Source: §7.1 · p. 22

Abstract

from arXiv · show

The synthetic control method (SCM) is a popular approach for estimating the impact of a treatment on a single unit in panel data settings. The "synthetic control" is a weighted average of control units that balances the treated unit's pre-treatment outcomes as closely as possible. A critical feature of the original proposal is to use SCM only when the fit on pre-treatment outcomes is excellent. We propose Augmented SCM as an extension of SCM to settings where such pre-treatment fit is infeasible. Analogous to bias correction for inexact matching, Augmented SCM uses an outcome model to estimate the bias due to imperfect pre-treatment fit and then de-biases the original SCM estimate. Our main proposal, which uses ridge regression as the outcome model, directly controls pre-treatment fit while minimizing extrapolation from the convex hull. This estimator can also be expressed as a solution to a modified synthetic controls problem that allows negative weights on some donor units. We bound the estimation error of this approach under different data generating processes, including a linear factor model, and show how regularization helps to avoid over-fitting to noise. We demonstrate gains from Augmented SCM with extensive simulation studies and apply this framework to estimate the impact of the 2012 Kansas tax cuts on economic growth. We implement the proposed method in the new augsynth R package.

1 Introduction

Augmented SCM extends synthetic control to cases where excellent pre-treatment fit is infeasible, using an outcome model to estimate and correct bias from imperfect fit. The paper develops Ridge ASCM, analyzes its error and regularization, extends it to auxiliary covariates, and evaluates it in simulations and a Kansas tax-cut application.

1 Introduction

  • ASCM addresses settings where SCM cannot achieve excellent pre-treatment fit by using an outcome model to estimate and remove the bias caused by imperfect fit.When pre-treatment fit is good, the estimated bias is small and SCM and ASCM estimates are similar; otherwise they diverge, with ASCM relying more on extrapolation.Source: §1 · p. 2
  • Ridge ASCM chooses weights to target SCM’s imbalance metric while penalizing distance from SCM weights, allowing negative weights to improve fit while controlling extrapolation.Unlike regression alone, its regularization parameter directly parameterizes the level of extrapolation.Source: §1 · p. 2
  • Under linear and latent factor models, improved pre-treatment fit reduces bias, while Ridge ASCM’s penalty manages the bias–variance trade-off and the risk of over-fitting.The authors propose cross-validation for selecting the hyperparameter.Source: §1 · pp. 2–3
  • ASCM incorporates auxiliary covariates alongside lagged outcomes or residualizes outcomes against them before Ridge ASCM; the latter controls estimation error under a linear factor model.The paper also implements the methodology in the augsynth R package.Source: §1 · p. 3
  • ASCM shows large gains over alternative estimators, especially under model misspecification, in bias and root mean squared error; its Kansas application finds a substantial negative effect of the 2012 tax cuts on economic output.The method is evaluated using calibrated simulation studies and the Kansas application.Source: §1 · p. 3

2 Overview of the Synthetic Control Method

SCM imputes the treated unit’s missing untreated outcome using control-unit outcomes weighted to balance pre-treatment outcomes. Simplex-constrained weights can prevent adequate fit, while the proposed modified SCM enables its use in many settings where standard SCM is infeasible.

2.1 Notation and setup

  • The theoretical setup considers one treated unit among N units observed over T periods, with one post-treatment observation for exposition and potential outcomes defined under SUTVA.The authors note that results extend to more post-treatment periods.Source: §2.1 · p. 4

2.2 Synthetic Control Method

  • SCM imputes the treated unit’s missing untreated outcome as a weighted average of control outcomes, choosing weights to balance pre-treatment outcomes.The overview considers simplex-constrained weights, which are non-negative and sum to one.Source: §2.2 · p. 6
  • When the treated unit’s lagged outcomes lie inside the controls’ convex hull, SCM achieves perfect pre-treatment fit and has attractive properties, including a bias bound.The cited bias bound is established by Abadie et al. (2010).Source: §2.2 · p. 7
  • The curse of dimensionality can make near-perfect fit infeasible with simplex-constrained weights, and the authors’ modified SCM enables use in many such settings.Abadie et al. (2015) recommend against SCM when pre-treatment fit is poor or the pre-treatment period is short.Source: §2.2 · p. 7

3 Augmented SCM

Augmented SCM corrects imperfect pre-treatment fit by using an outcome model to estimate and remove bias from imbalance. The choice of outcome model determines how that adjustment works, with ridge regression yielding a penalized synthetic control estimator.

3.1 Overview

  • Augmented SCM adjusts the SCM estimate for imbalance in an outcome model’s predictions, treating that imbalance as estimated bias in the post-treatment control outcome.The paper frames this as analogous to bias correction for inexact matching.Source: §3.1 · p. 7

3.2 Choice of estimator

  • Ridge-regularized linear augmentation is itself a penalized synthetic control estimator, enabling direct analysis of regularization’s role.The paper identifies ridge regression as linear in both pre-treatment outcomes and comparison units.Source: §3.2 · p. 8
  • In a linear outcome model, imbalance in pre-treatment periods with larger absolute regression coefficients produces a larger adjustment.The coefficients allow the adjustment to emphasize periods that are empirically more predictive of the post-treatment outcome.Source: §3.2 · p. 8
  • When the outcome model is a linear combination of comparison units, augmentation adjusts SCM weights and can produce negative overall weights.Examples include k-nearest-neighbor matching and kernel weighting.Source: §3.2 · pp. 8–9

4 Ridge ASCM improves pre-treatment fit while penalizing extrapolation

Ridge ASCM improves pre-treatment fit by allowing controlled extrapolation beyond the convex hull. It uses SCM weights when balance is exact and negative weights when exact fit is unavailable.

4 Ridge ASCM improves pre-treatment fit while penalizing extrapolation

  • Outside the controls’ convex hull, Ridge ASCM improves pre-treatment fit over SCM by allowing negative weights, while λridge controls extrapolation from the SCM weights.It directly penalizes distance from sparse, non-negative SCM weights and resorts to negative weights only when the treated unit is outside the convex hull.Source: §4 · p. 10

4.1 Ridge ASCM as a penalized SCM estimator

  • When SCM exactly balances lagged outcomes, Ridge ASCM and SCM weights are identical; otherwise, Ridge ASCM uses negative weights to extrapolate, with the extent depending on imbalance and λridge.When SCM fit is good or λridge is large, the adjustment is small and Ridge ASCM weights remain close to SCM weights.Source: §4.1 · p. 11

4.2 Ridge ASCM improves pre-treatment fit relative to SCM alone

  • Ridge ASCM’s pre-treatment imbalance is strictly smaller than SCM’s except in corner cases, including when the SCM residual is orthogonal to control lagged outcomes.The section states that Ridge ASCM’s pre-treatment fit is at least as good as SCM’s, with equality possible in those corner cases.Source: §4.2 · pp. 12–13

5 Error under restrictions on the data generating processes

The section analyzes Ridge ASCM’s estimation error under linear and latent-factor data-generating processes, including approximately linear outcome models with Lipschitz deviations. It examines how pre-treatment fit, pre-period length, and regularization relate to error, and describes leave-one-period-out selection of the regularization parameter.

5 Error under restrictions on the data generating processes

  • Across both models, improved pre-treatment fit reduces bias, while the ridge penalty negotiates a bias-variance trade-off, making the choice of λridge important.Under latent factors, fitting noisy pre-treatment outcomes also creates over-fitting risk; the penalty helps control it.Source: §5 · p. 13

5.1 Error under linearity

  • With a linear outcome model, exactly balancing lagged outcomes eliminates systematic error; if coefficients are sparse, balancing only nonzero-coefficient lags suffices.For an AR(K) process, balancing the first K lags is sufficient.Source: §5.1 · p. 14
  • Ridge ASCM’s finite-sample error depends on lagged-outcome imbalance and the L2 norm of its weights; finite λ can improve fit but entails a bias-variance trade-off.When SCM achieves perfect pre-treatment fit, ASCM and SCM weights are equivalent and error comes only from weight variance and post-treatment noise.Source: §5.1 · p. 15

5.2 Error under a latent factor model

  • In high-noise settings, ASCM can increase error relative to SCM because balancing noisy outcomes risks over-fitting, though the extrapolation penalty can reduce that risk.SCM’s simplex constraint limits over-fitting in this case, although SCM itself is likely to perform poorly regardless.Source: §5.2 · p. 15
  • Under latent factors, estimation error additionally includes approximation error from balancing observed lags rather than latent loadings, and noisy lags can attract large weights because of idiosyncratic noise.Balancing observed pre-treatment outcomes does not necessarily balance the latent factor loadings.Source: §5.2 · pp. 16–17
  • When SCM achieves perfect pre-treatment fit under the latent-factor model, ASCM and SCM weights are equivalent and estimation error is due only to post-treatment noise.Source: §5.2 · p. 17
  • Without exact pre-treatment balance, a long pre-period may not control imbalance error, underscoring excellent pre-treatment fit’s importance for SCM estimates.Under suitable conditions on factor loadings, approximation error and thus bias converge to zero in probability as the pre-period grows.Source: §5.2 · p. 17
  • A small amount of Ridge ASCM extrapolation lowers Theorem 1’s error bound, but too much risks overfitting; sufficiently regularized ASCM has a lower bound than SCM even at high noise.The risk of overfitting is greater at higher noise, while augmentation benefits and the optimal regularization choice are larger when noise is less extreme.Source: §5.2 · p. 18
  • Simulations indicate augmentation’s pre-treatment-fit gains can outweigh increased approximation error across practical settings, including very high noise.Theorem 1 gives a worst-case bound; the cited simulations assess typical Ridge ASCM performance.Source: §5.2 · p. 18

5.3 Hyper-parameter selection

  • The proposed cross-validation approach selects λ using leave-one-period-out prediction MSE, minimizing that score or applying the one-standard-error rule.For outcome models where excluding only period t is inappropriate, the procedure can instead exclude all periods at or after t.Source: §5.3 · pp. 18–19

6 Auxiliary covariates

This section extends SCM and ASCM to auxiliary covariates, distinguishing joint incorporation from a small-K approach that exactly balances covariates and fits residualized outcomes.

6 Auxiliary covariates

  • Auxiliary covariates can enter both the SCM balance objective and the ASCM outcome model, with user-specified weights controlling their balance importance and coefficient regularization.Source: §6 · p. 19
  • For moderate-dimensional covariates, a default gives lagged outcomes and auxiliary covariates equal balance and regularization weights after standardizing them to the same standard deviation.Source: §6 · pp. 19–20
  • The resulting two-step procedure residualizes pre- and post-treatment outcomes on auxiliary covariates, then fits Ridge ASCM on the residualized outcomes.Source: §6 · p. 20
  • This approach penalizes extrapolation in the residualized lagged outcomes rather than the original outcomes, allowing potentially large extrapolation in auxiliary covariates while penalizing the outcome component orthogonal to them.Source: §6 · p. 21
  • When K is small relative to N0, exactly balancing auxiliary covariates and targeting residualized lagged outcomes can reduce pre-treatment-fit error with only small increases in approximation and post-treatment-noise error; with larger K, parallel incorporation is preferable.Source: §6 · p. 21

7 Simulations and empirical illustrations … Supplementary Materials for “The Augmented Synthetic Control Method”

This section reports simulations and a Kansas tax-cut illustration of Augmented SCM, including placebo estimates and the role of auxiliary covariates, and closes with discussion of the method’s contribution and directions for extension.

7.1 Calibrated simulation studies

  • Ridge augmentation reduces SCM bias without conditioning on excellent pre-treatment fit; reductions exceed 75% under the baseline factor model and 90% under fixed effects, with smaller gains under AR(3) and higher noise.The simulations also find lower RMSE for ASCM than SCM in all settings.Source: §7.1 · p. 22
  • Ridge ASCM has slightly lower bias than ridge regression alone; when fixed effects are misspecified, combining SCM with fixed effects has much lower bias than either method alone.When fixed effects are correctly specified, de-meaned SCM has similar bias to the fixed-effects approach.Source: §7.1 · p. 22
  • Augmentation reduces bias across SCM-fit quartiles, but gains are more modest in the best-fit quartile; too-small λ can raise RMSE when pre-treatment fit is good.Bias is non-monotonic in λ for the best-fit quartile, and RMSE reaches a minimum at an intermediate λ.Source: §7.1 · p. 23
  • Across tested outcome models, augmentation almost always reduces SCM bias and nearly always beats pure outcome modeling; with good SCM fit, flexible-model gains are limited except for oracle gsynth.Figure 5 summarizes absolute bias for ridge, fixed effects, machine-learning and panel-data outcome models and their augmented versions.Source: §7.1 · pp. 23–24, Figure 5 · p. 24
  • Given its consistent performance and relative simplicity, the paper recommends penalized-regression augmentation—particularly ridge regression—as a default when SCM has poor pre-treatment fit.Ridge ASCM also lets practitioners diagnose the outcome model’s level of extrapolation.Source: §7.1 · p. 24

7.2 Illustration: 2012 Kansas tax cuts

  • SCM alone achieves fairly poor pre-treatment fit; its synthetic control exceeds the treatment unit by 2–4% in 2004–2005, comparable in scale to an estimated 3% average post-treatment decrease, and all four approaches estimate negative impacts.Augmenting SCM with ridge regression improves pre-treatment fit; adding auxiliary covariates and augmenting further improves pre-treatment fit and balance.Source: §7.2 · p. 25
  • Placebo effects were near zero for all three placebo treatment times in Q2 2009, 2010, and 2011.Figure 7 reports placebo estimates with 95% conformal confidence intervals.Source: §7.2 · p. 25, Figure 7 · p. 26

Appendix Figures F.6 and F.7 show the corresponding placebo estimates for SCM alone and Ridge

  • Adding auxiliary covariates to SCM and ridge optimization greatly improved balance; residualizing on them perfectly balanced covariates while Ridge ASCM retained very good lagged-outcome fit.Figure 8a displays covariate balance, and Figure 6 displays point estimates and confidence intervals for the tax-cut effect.Source: Figure 6 · p. 26, Figure 8 · p. 27, Appendix Figures F.6 and F.7 show the… · p. 27
  • Ridge ASCM weights remain similar to SCM weights with minimal extrapolation; only Louisiana receives a meaningful negative weight, and that property persists when covariates are included.The weights retain interpretability, with close weights for SCM’s positive-weight donors and relatively few non-negligible negative weights.Source: Figure 8 · p. 27, Appendix Figures F.6 and F.7 show the… · p. 27

8 Discussion

  • ASCM addresses settings where excellent pre-treatment SCM fit is infeasible by controlling fit while minimizing extrapolation; the paper shows error control under a linear factor model and proposes auxiliary-covariate extensions.Source: §8 · p. 28
  • Future work includes robust inference for imperfect fit, building on outlined conformal and jackknife+ approaches, evaluating broader settings, and sensitivity analysis for departures from the linear factor model.The discussion notes that more work is needed to understand performance in more general settings.Source: §8 · p. 28
  • The discussion says staggered adoption, when units take up treatment at different times, is more complicated, and reports exploring this extension in Ben-Michael et al. (2019).Some cited approaches simply average the units together (e.g., Kreif et al., 2016; Robbins et al., 2017).Source: §8 · p. 28
  • Further extensions could address multiple outcome series, hierarchical data with individual and aggregate outcomes, and discrete or count outcomes.Source: §8 · p. 28

A Inference · B Additional results · B.1 Specialization of Ridge ASCM results to SCM

The appendix evaluates conformal inference for SCM and ridge ASCM, then specializes Ridge ASCM results to SCM under linear and linear-factor models. Coverage depends on pre-treatment fit and approximation error, while exchangeability governs exact validity.

A Inference

  • Full conformal inference tests candidate treatment effects by adjusting the post-treatment outcome under the null, refitting weights, and inverting the test to form a confidence interval.The interval can be viewed as a conformal prediction set for the untreated post-treatment outcome, based on quantiles of pre-treatment residuals.Source: A Inference · p. 33
  • Under additive noise, conformal coverage is close to nominal when pre-treatment fit is good and approximation error is small; exact validity holds when residuals are exchangeable.In a linear factor model, exchangeability may hold when factor values are similar across periods, but wide pre/post differences or treatment adoption correlated with shocks can cause undercoverage.Source: A Inference · p. 34
  • Jackknife+ uses leave-one-period-out residuals and estimates, requiring refitting for each pre-treatment period; it has exact finite-sample coverage under exchangeable periods or residuals.Unlike full conformal inference, it does not enforce a sharp null when estimating weights; approximate validity without exchangeability is left for future work.Source: A Inference · p. 34
  • Ridge ASCM prediction intervals had coverage greater than or close to nominal, while SCM intervals sometimes undercover because of finite-sample bias from poor treatment fit.The simulation used 95% intervals for the first post-treatment counterfactual outcome and compared full conformal and jackknife+ methods across four settings.Source: A Inference · p. 34, Table A.1 · p. 35

B.1 Specialization of Ridge ASCM results to SCM

  • For the linear model with independent sub-Gaussian noise, Corollary A.1 specializes a Ridge ASCM result to SCM weights in the simplex that are independent of post-treatment outcomes.This is the SCM special case corresponding to the Ridge ASCM penalty tending to infinity.Source: §B.1 · p. 36
  • Corollary A.2 is stated under linear factor model (21) with independent sub-Gaussian noise of scale parameter σ, for simplex weights independent of post-treatment outcomes, and any δ > 0.Source: §B.1 · p. 36

B.2 Error under a partially linear model with Lipshitz deviations from linearity

This subsection bounds SCM and Ridge ASCM estimation error when the control outcome is partially linear with Lipschitz deviations from linearity. It identifies pre-treatment imbalance and distance from the treated unit as important to controlling error.

B.2 Error under a partially linear model with Lipshitz deviations from linearity

  • Ridge ASCM’s error bound separates contributions from imbalance in X, excess approximation error, and SCM approximation error.The theorem considers ridge ASCM weights with λ_ridge = N_0λ.Source: §B.2 · pp. 36–37
  • For simplex weights independent of post-treatment outcomes, the error bound has imbalance-in-X and approximation-error components.This is the SCM-weight corollary.Source: §B.2 · p. 37
  • Controlling error requires good pre-treatment fit and weighting controls near the treated unit; L/∥β∥₂ governs their relative importance.The model assumes an L-Lipschitz deviation from linearity and independent, ignorable sub-Gaussian noise.Source: §B.2 · pp. 36–37
  • Under the partially linear model, limiting lagged-outcome imbalance and the weighted sum of distances is sufficient to control interpolation error.The penalized SCM formulation of Abadie and L’Hour (2018) uses λ in the role of L/∥β∥₂.Source: §B.2 · p. 37

B.3 Error under a linear factor model with covariates

The section bounds error under a linear factor model with covariates and shows why targeting fit in residualized lagged outcomes can help while leaving overfitting to noise as a tuning challenge.

B.3 Error under a linear factor model with covariates

  • When the time-varying covariate function is linear, the approximation residuals are zero for every unit and time period.The model represents covariate effects through a time-varying function, and the residual is defined relative to the best linear approximation in the data.Source: §B.3 · pp. 38–39
  • Theorem A.2 bounds error through residualized-outcome imbalance, treated-unit approximation error, residual sum of squares, and SCM approximation error.The bound applies under the stated linear factor model with covariates and sub-Gaussian noise; e1max and RSSmax denote the maximal treated-unit residual and maximal residual sum of squares, respectively.Source: §B.3 · p. 38
  • Because auxiliary covariates enter additively and separably, controlling fit in residualized lagged outcomes partially controls error and directly targeting them is justified.The excess approximation error is small when the number of covariates is small relative to the number of controls and covariates are measured without noise.Source: §B.3 · p. 39
  • Ridge ASCM can improve balance in residualized lagged outcomes over SCM without increasing imbalance in covariates, which are orthogonal to those outcomes by construction.Overfitting to noise remains possible, so the tuning parameter still matters; the proposed cross-validation approach uses residualized rather than raw lagged outcomes.Source: §B.3 · p. 39

C Simulation data generating process · D Proofs · D.1 Proofs for Section 4

The section specifies calibrated simulation designs for evaluating Augmented SCM and gives proof arguments for the weighting representations and supporting lemmas. The simulations include linear-factor and fixed-effects designs, plus an AR(3) design.

C Simulation data generating process

  • The linear-factor simulation fits log GSP per capita with N = 50, T0 = 89, T = 105, and J = 3 using the Generalized Synthetic Control Method.The design is calibrated to the observed series.Source: C Simulation data generating process · p. 40
  • Simulations use estimated time effects and factors, include a variant with fourfold residual standard deviation, and impose a sharp null with ATT estimated at the final time point.The estimated unit-effect mean and standard deviation, factor-loading covariance, and residual standard deviation inform the data draws.Source: C Simulation data generating process · p. 40
  • The linear-factor design models selection using propensity scores, sets θ = 1/2 after rescaling factors and fixed effects to unit variance, and normalizes probabilities for one treated unit.Source: C Simulation data generating process · p. 40
  • Alternative designs specialize the factor model to unit and time fixed effects or generate outcomes from an AR(3) model, with the latter selecting units based on their last three outcomes.The fixed-effects design calibrates unit effects from gsynth and models selection proportional to fixed effects; the AR(3) design estimates ATT at T0 + 1.Source: C Simulation data generating process · pp. 40–41

D.1 Proofs for Section 4

  • The proofs derive weighting representations for the ridge and ASCM estimates from their optimization problems, using centered lagged outcomes and Lagrangian dual arguments.Source: §D.1 · pp. 42–43
  • The appendix proves Lemma 3 using orthogonality and states Lemma A.2 for ridge-augmented SCM weights, beginning its proof with a singular value decomposition and triangle inequality.Source: §D.1 · p. 43

D.2 Proofs for Sections 5, B.1, and B.2 · D.3 Proofs for Sections 6 and B.3

These appendices establish error bounds for ASCM under several assumptions and prove pre-treatment-fit bounds for the covariate-adjusted method. The results separate imbalance, approximation error, and noise, with the bounds depending on the assumed outcome structure.

D.2 Proofs for Sections 5, B.1, and B.2

  • Under the linear model, the unit-specific term is zero, so Corollary A.1 follows directly from Proposition A.1.The proof of Proposition 1 likewise applies Lemma 3 under the linear model.Source: §D.2 · p. 45
  • For mean-zero sub-Gaussian unit-specific terms, Lemma A.3 bounds approximation error, and Lemma A.4 gives a ridge-ASCM bound with an excess approximation-error term.The ridge result uses the specified hyperparameter λridge = λN0.Source: §D.2 · p. 45
  • Lemma A.5 shows that the linear factor model is a special case of the general model, enabling factor-model bounds and a result for ridge ASCM with time-dependent errors.The dependent-error result assumes the linear factor model and time-dependent errors.Source: §D.2 · pp. 47–48
  • Lemma A.6 states that if ξ_i = f(X_i) for an L-Lipshitz function f, the approximation error satisfies a bound for weights on the simplex.Source: §D.2 · p. 48

D.3 Proofs for Sections 6 and B.3

  • Lemma 4 yields a bound on pre-treatment fit for the regression parameters used in the covariate-adjusted method.The supplied passage states the bound’s existence but omits its formula.Source: §D.3 · pp. 49–50
  • Theorem A.2’s proof separates projected outcomes from residuals; exact covariate balance removes one imbalance term, leaving noise and projection-related error to bound.The proof combines these components with Lemma 4 and a union bound.Source: §D.3 · pp. 50–52

E Connection to balancing weights and IPW

The section connects SCM’s balancing weights to inverse propensity score weighting through a Lagrangian dual. It generalizes this duality beyond squared L2 balance while cautioning that an IPW form does not always justify a propensity-score interpretation.

E Connection to balancing weights and IPW

  • SCM weights are approximate balancing weights, and the dual has a propensity-score-model form, though it is not always appropriate to interpret it as stochastic selection into treatment.Unlike traditional IPW, balancing weights directly minimize covariate imbalance; the dual link depends on the dispersion penalty.Source: E Connection to balancing weights and… · p. 53
  • The generalized formulation permits nonnegative-weight restrictions through the penalty domain and extends balance criteria beyond squared L2, suggesting SCM and ASCM extensions especially to L1 balance.Examples include re-centered L2 penalties for ridge regression and ridge ASCM, entropy, elastic net, and an L∞ constraint (Doudchenko and Imbens, 2017; Zubizarreta, 2015; Athey et al., 2018).Source: E Connection to balancing weights and… · pp. 53–54
  • The dispersion penalty determines the propensity-score link, while the balance criterion determines regularization; this recovers entropy balancing, Oaxaca-Blinder weights, and L∞ balance as special cases.These correspond to logistic link (Zhao and Percival, 2017), log-logistic link (Kline, 2011), and L1 regularization (Wang and Zubizarreta, 2018).Source: E Connection to balancing weights and… · pp. 53–54

F Additional figures

Appendix F presents additional figures on cross-validation, Kansas tax-cut estimates, placebo analyses, donor weights, and simulation results. The captions describe the plotted quantities and methods but do not report their numerical results.

F Additional figures

  • Figure F.1 plots cross-validation MSE and its one-standard-error estimate, highlighting the minimum and the largest λ within one standard error.The calculations follow Equation (25).Source: Figure F.1 · p. 55
  • Figures F.2–F.3 plot counterfactual log GSP per capita and estimated tax-cut effects using SCM, ridge ASCM, and ridge ASCM with covariates.They show point estimates with point-wise 95% conformal prediction intervals for the counterfactual and confidence intervals for the effects; observed log GSP per capita appears in black in Figure F.2.Source: Figure F.2 · p. 56, Figure F.3 · p. 56
  • Figure F.4 compares ATT estimates for de-meaned SCM, ridge regression, ridge ASCM, SCM with covariates, and two-way fixed-effects differences-in-differences.The figure shows estimates with ± two standard errors; the ridge ASCM λ minimizes cross-validated MSE, and the SCM-with-covariates method is attributed to Abadie et al. (2010).Source: Figure F.4 · p. 57
  • Figures F.5 and F.10–F.13 show averaged ridge coefficients, estimator RMSE and bias, and latent factors for calibrated simulation studies.The bias and RMSE figures include results conditioned on SCM fit in the top quintile; Figure F.13 presents latent factors for the calibrated simulations.Source: Figure F.5 · p. 57, Figure F.10 · p. 60, Figure F.11 · p. 61
  • Figures F.6–F.7 plot placebo estimates for SCM and ridge ASCM at treatment times in Q2 2009, 2010, and 2011, while Figures F.8–F.9 show donor-unit weights.The placebo plots show estimates with ± two standard errors; the weight figures cover lagged-outcome balancing, with Figure F.9 additionally residualizing auxiliary covariates.Source: Figure F.6 · p. 58, Figure F.7 · p. 58, Figure F.8 · p. 59

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