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Matrix Completion Methods for Causal Panel Data Models
Open accessAbstract
In this article, we study methods for estimating causal effects in settings with panel data, where some units are exposed to a treatment during some periods and the goal is estimating counterfactual (untreated) outcomes for the treated unit/period combinations. We propose a class of matrix completion estimators that uses the observed elements of the matrix of control outcomes corresponding to untreated unit/periods to impute the “missing” elements of the control outcome matrix, corresponding to treated units/periods. This leads to a matrix that well-approximates the original (incomplete) matrix, but has lower complexity according to the nuclear norm for matrices. We generalize results from the matrix completion literature by allowing the patterns of missing data to have a time series dependency structure that is common in social science applications. We present novel insights concerning the connections between the matrix completion literature, the literature on interactive fixed effects models and the literatures on program evaluation under unconfoundedness and synthetic control methods. We show that all these estimators can be viewed as focusing on the same objective function. They differ solely in the way they deal with identification, in some cases solely through regularization (our proposed nuclear norm matrix completion estimator) and in other cases primarily through imposing hard restrictions (the unconfoundedness and synthetic control approaches). The proposed method outperforms unconfoundedness-based or synthetic control estimators in simulations based on real data.
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TL;DR
The paper proposes a matrix-completion estimator with nuclear norm minimization (MC-NNM) for causal panel data, imputing untreated potential outcomes for treated unit-periods. It shows that unconfoundedness, synthetic control, difference-in-differences, and matrix completion can be viewed as minimizing the same objective function but differing in restrictions or regularization, and finds MC-NNM performs well across thin, fat, and square matrices in simulations.
Key findings
- MC-NNM uses nuclear norm regularization plus unregularized unit and time fixed effects; the authors argue this is computationally attractive and avoids the fixed-rank computational difficulties of interactive fixed effects methods.
- The paper proves that horizontal regression (unconfoundedness), vertical regression (synthetic control), elastic net vertical regression, difference-in-differences, and nuclear norm matrix completion all minimize the same objective \(Q(Y; R, A, B, \gamma, \delta)\), differing only in hard restrictions or regularization (Theorem 1).
- In the California smoking data (\(N=38\), \(T=31\)), MC-NNM performs well across \(T_0/T\) ratios; in staggered adoption, VT-EN performs poorly and MC-NNM is the superior approach.
- In stock market data (daily returns for 2453 stocks over 3082 days; subsamples with \(N \times T = 4900\), from \((490,10)\) to \((10,490)\), 25% missing in a block design), MC-NNM adapts to both thin and fat regimes, does as well as the best estimator in both, and better than both in approximately square settings; it approximates data with rank 4 to 12.
Methods
- Panel data setting with \(N\) units over \(T\) periods, binary treatment \(W_{it}\), and potential outcomes \(Y_{it}(0)\), \(Y_{it}(1)\); the target is average treatment effect on the treated, imputing missing \(Y(0)\) entries for treated unit-periods.
- Estimator: \(\min_{L,\Gamma,\Delta} \frac{1}{|O|} \|P_O(Y - L - \Gamma 1_T^\top - 1_N \Delta^\top)\|_F^2 + \lambda \|L\|_*\), where \(O\) indexes observed control outcomes and \(\|\cdot\|_*\) is the nuclear norm; solved via soft-impute with cross-validation for \(\lambda\).
- Simulations use real data matrices where no unit is treated, artificially designate some units/periods as treated, and compare RMSE of DID, VT-EN, HR-EN, SC-ADH, and MC-NNM.
Limitations
- The paper states that studying the asymptotic distribution of \(L^* - L\) to construct confidence intervals is beyond scope and is future research; it suggests resampling methods instead.
- The paper states MC-NNM does not take into account the time series nature of observations and may have autocorrelated errors; it proposes an autoregressive modification but does not implement it in the main simulations.
- The paper states MC-NNM puts equal weight on all observed elements of \(Y-L\), and proposes propensity-score weighting as an extension; the theoretical bound depends on \(p_c = \min_i \pi_T^{(i)}\), which can be relaxed only under restrictions on \(L^*\).
- Evident from the text: the simulations use artificial missingness in real datasets rather than actual treatment assignments, so they assess imputation accuracy under constructed treatment patterns, not real causal effects.
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A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We find that the nuclear norm matrix completion estimator does well in a range of cases, including when T is small relative to N , when T is large relative to N , and when T and N are comparable.”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
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In the paper
“the unconfoundedness approach does not work very well if T N”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The proposed method outperforms unconfoundednessbased or synthetic control estimators in simulations based on real data.”
§ Matrix Completion Methods for Causal Panel Data Models * > Abstract
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In contrast, the unconfoundedness and synthetic control approaches break down in some of these settings in the expected pattern (the unconfoundedness approach does not work very well if T N , and the synthetic control approach does not work very well if N T ).”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“the synthetic control approach does not work very well if N T”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesThe paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Although formally the fixed effects can be subsumed in the matrix L (Γ1 T and 1 N ∆ are both rank one matrices), in practice, including these fixed effects separately and not regularizing them greatly improves the quality of the imputations.”
§ Matrix Completion Methods for Causal Panel Data Models * > 4 The Matrix Completion with Nuclear Norm Minimization Estimator
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- The authors state this as an assumption: The complete outcome matrix is the sum of a low-rank matrix L* and mean-zero noise ε.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Lemma 3. If the estimator L satisfies L -L * 2 F ≥ 4L 2 max θ/p c for a positive number θ, then there is a constant C ≥ 0.001 such that, when Cθ > T , we have P π p c 2 L -L * 2 F > (i,t)∈O A it , L -L * 2 + 8L 2 max T √ N ≤ 2 exp - Cθ T .”
§ Matrix Completion Methods for Causal Panel Data Models * > A.2 Proof of Theorem 2
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In the paper
“find that generally it outperforms the synthetic control estimators proposed Abadie et al. (2010) and the elastic net estimators proposed by Doudchenko and Imbens (2016).”
§ Matrix Completion Methods for Causal Panel Data Models * > 9 Conclusions
What the sources say
- Not checked yetcites Abadie et al. (2010)
- Not checked yetcites Doudchenko and Imbens (2016)
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesA result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“As T 0 /T grows closer to one, VT-EN, HR-EN, SC-ADH and MC-NNM methods all do well.”
§ Matrix Completion Methods for Causal Panel Data Models * > 7.1 The Abadie-Diamond-Hainmueller California Smoking Data
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIn the paper
“Lemma 1 (Adapted from Negahban and Wainwright (2011)).Then for all λ ≥ 3 E op /|O|, (i,t)∈O A it , L * -L 2 |O| ≤ 10λ √ R L * -L F .”
§ Matrix Completion Methods for Causal Panel Data Models * > A.2 Proof of Theorem 2
What the sources say
- Not checked yetcites Negahban and Wainwright (2011)
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- It may state two separate claims as one. 18
- Probably not a substantial claim. 18
- Borderline: one of the tests for a substantial claim is unclear. 1
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“For small values of T 0 /T , SC-ADH and HR-EN perform poorly while VT-EN is superior.”
§ Matrix Completion Methods for Causal Panel Data Models * > 7.1 The Abadie-Diamond-Hainmueller California Smoking Data
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The most interesting finding is that the proposed MC-NNM method adapts well to both regimes and does as well as the best estimator in both settings, and better than both in the approximately square setting.”
§ Matrix Completion Methods for Causal Panel Data Models * > 7.2 Stock Market Data
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In the simultaneous adoption case, DID generally does poorly, suggesting that the data are rich enough to support more complex models.”
§ Matrix Completion Methods for Causal Panel Data Models * > 7.1 The Abadie-Diamond-Hainmueller California Smoking Data
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The staggered adoption results are similar with some notable differences; VT-EN performs poorly (similar to DID) and MC-NNM is the superior approach.”
§ Matrix Completion Methods for Causal Panel Data Models * > 7.1 The Abadie-Diamond-Hainmueller California Smoking Data
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“The unconfoundedness literature (Rosenbaum and Rubin (1983); Rubin (2006); Imbens and Wooldridge (2009); Abadie and Cattaneo (2018)) focuses primarily on the single-treatedperiod block structure with a thin matrix (N T ), a substantial number of treated and control units, and imputes the missing potential outcomes in the last period using control units with similar lagged outcomes:”
§ Matrix Completion Methods for Causal Panel Data Models * > 3.3.1 Horizontal Regression and the Unconfoundedness Literature
What the sources say
- Not checked yetcites Rosenbaum and Rubin (1983)
- Not checked yetcites Rubin (2006)
- Not checked yetcites Imbens and Wooldridge (2009)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In contrast, the proposed method does not impose any restrictions but uses regularization to characterize the estimator.”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In particular in the case we focus on for part of this study, with only a single treated unit/period pair there would be no information in this value.”
§ Matrix Completion Methods for Causal Panel Data Models * > 2 Set Up
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The unconfoundedness approach assumes that patterns over time are stable across units, and the synthetic control approach assumes that patterns across units are stable over time.”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
What the sources say
- Not checked yetcites Abadie et al. (2010)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In setting with few values of Y(1) observed it is unlikely that the information in these values is important.”
§ Matrix Completion Methods for Causal Panel Data Models * > 2 Set Up
Strength:hedged· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
None of the sources cited for this claim have been checked yet.
In the paper
“The matrix completion literature (Candès and Recht (2009); Candès and Plan (2010); Mazumder et al. (2010)) focuses on imputing missing elements in a matrix assuming that: (i) the complete matrix is the sum of a low rank matrix plus noise and (ii), the missingness is completely at random (except Gamarnik and Misra (2016) that study a stylized rank one case).”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
What the sources say
- Not checked yetcites Candès and Recht (2009)
- Not checked yetcites Candès and Plan (2010)
- Not checked yetcites Mazumder et al. (2010)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The rank of the matrix is implicitly determined by the regularization through the addition of a penalty term to the objective function.”
§ Matrix Completion Methods for Causal Panel Data Models * > 1 Introduction
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“This is partly because compared to the settings studied in the matrix completion literature the fraction of observed values is relatively high, and so these fixed effects can be estimated accurately.”
§ Matrix Completion Methods for Causal Panel Data Models * > 4 The Matrix Completion with Nuclear Norm Minimization Estimator
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The complete outcome matrix is the sum of a low-rank matrix L* and mean-zero noise ε.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Because we focus on estimating the average effect for the treated, all the relevant values for Y it (1) are observed, and thus we only need to impute the missing entries in the Y(0) matrix for treated units with W it = 1.”
§ Matrix Completion Methods for Causal Panel Data Models * > 2 Set Up
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The stable-unit-treatment-value assumption holds and there are no dynamic effects: potential outcomes are indexed only by the contemporaneous treatment for that unit, not by past treatments or treatments for other units.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“We note that here the fixed effects are absorbed in L * since they are two rank 1 matrices and their addition does not affect our low-rank assumption on L * .”
§ Matrix Completion Methods for Causal Panel Data Models * > 4 The Matrix Completion with Nuclear Norm Minimization Estimator
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
- The authors state this as an assumption: The complete outcome matrix is the sum of a low-rank matrix L* and mean-zero noise ε.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The horizontal regression focuses on a pattern in the time path of the outcome Y it , specifically the relation between Y iT and the lagged Y it for t = 1, . . ., T -1, for the units for whom these values are observed, and assumes that this pattern is the same for units with missing outcomes.”
§ Matrix Completion Methods for Causal Panel Data Models * > 3.4 Fixed Effects and Factor Models
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“The vertical regression focuses on a pattern between units at times when we observe all outcomes, and assumes this pattern continues to hold for periods when some outcomes are missing.”
§ Matrix Completion Methods for Causal Panel Data Models * > 3.4 Fixed Effects and Factor Models
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“They differ solely in the way they deal with identification, in some cases solely through regularization (our proposed nuclear norm matrix completion estimator) and in other cases primarily through imposing hard restrictions (the unconfoundedness and synthetic control approaches).”
§ Matrix Completion Methods for Causal Panel Data Models * > Abstract
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“The synthetic control literature (Abadie et al. (2010)) focuses primarily on the single-treatedunit block structure with a relatively fat (T N ) or approximately square matrix (T ≈ N ), and a substantial number of pre-treatment periods:”
§ Matrix Completion Methods for Causal Panel Data Models * > 3.3.2 Vertical Regression and the Synthetic Control Literature
What the sources say
- Not checked yetcites Abadie et al. (2010)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
In the paper
“For nuclear norm matrix completion representation a key insight is that (Lemma 6, Mazumder et al. ( 2010)) L * = min A,B:L=AB 1 2 A 2 F + A 2 F .”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
What the sources say
- Not checked yetcites Mazumder et al. (2010)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The proposed estimator, building on the interactive fixed effects and matrix completion literatures has attractive computational properties in settings with large N and T”
§ Matrix Completion Methods for Causal Panel Data Models * > 9 Conclusions
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Existing papers assume that entries (i, t) ∈ O are independent random variables whereas we allow for a time series dependency structure.”
§ Matrix Completion Methods for Causal Panel Data Models * > 6.2 Main Result
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A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Without some form of regularization they cannot be applied in the same setting, because the non-regularized versions require N > T or N < T respectively.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“In addition, if L is the solution to Equation (4.3) that has rank R, then one solution for A and B is given by A = SΣ 1/2 , B = RΣ 1/2 (5.2) where L = S N × RΣ R× RR T × R is singular value decomposition of L.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
What the sources say
- Not checked yetcites Mazumder et al. (2010)
- Not checked yetcites Hastie et al. (2015)
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The regularization in the elastic net version only affects the last row of this matrix, and replaces it with a regularized version of the regression coefficients.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
None of the sources cited for this claim have been checked yet.
In the paper
“Our estimator and its theoretical analysis are motivated by and generalize existing research on matrix-completion (Srebro et al., 2005;Mazumder et al., 2010;Candès and Recht, 2009;Candès and Tao, 2010;Keshavan et al., 2010a,b;Gross, 2011;Recht, 2011;Rohde et al., 2011;Negahban andWainwright, 2011, 2012;Koltchinskii et al., 2011;Klopp, 2014).”
§ Matrix Completion Methods for Causal Panel Data Models * > 6.2 Main Result
What the sources say
- Not checked yetcites Srebro et al., 2005
- Not checked yetcites Mazumder et al., 2010
- Not checked yetcites Candès and Recht, 2009
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The synthetic control estimator further restricts the values of the γ N and α i .”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedCausal:direct· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✗ Based: failsProbably not a substantial claim. The paper’s conclusions may not depend on it. The paper may give no reason why it holds.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Formally, the optimization problem withs the rank norm is NP-hard.”
§ Matrix Completion Methods for Causal Panel Data Models * > 4 The Matrix Completion with Nuclear Norm Minimization Estimator
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The nuclear norm estimator uses regularization to get around this by regularizing A and B.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“It is likely that the ε it are correlated over time.”
§ Matrix Completion Methods for Causal Panel Data Models * > 8.3 Autocorrelated Errors
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✗ Based: failsProbably not a substantial claim. The paper may give no reason why it holds.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Given a particular choice for regularization, however, one can use cross-validation methods to compare the two approaches.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The horizontal and vertical regressions are fundamentally different approaches, and they cannot easily be nested.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passesIt may state two separate claims as one.
In the paper
“Proposition 1 (Matrix Bernstein Inequality).Let Z 1 , . . ., Z N be independent matrices in R d 1 ×d 2 such that E[Z i ] = 0 and Z i op ≤ D almost surely for all i ∈ [N ].Let σ Z be such that σ 2 Z ≥ max N i=1 E[Z i Z i ] op , N i=1 E[Z i Z i op . Then, for any α ≥ 0 P N i=1 Z i op ≥ α ≤ (d 1 + d 2 ) exp -α 2 2σ 2 Z + (2Dα)/3 .”
§ Matrix Completion Methods for Causal Panel Data Models * > A.4 Proof of Lemma 2
What the sources say
- Not checked yetcites Tropp (2012)
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“and allows for a relatively large number of factors.”
§ Matrix Completion Methods for Causal Panel Data Models * > 9 Conclusions
Load-bearingStrength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesBorderline: one of the tests for a substantial claim is unclear.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The restrictions for the horizontal regression on the one hand, and for the vertical regression, synthetic control and elastic net regression on the other hand, are quite different, and not directly comparable.”
§ Matrix Completion Methods for Causal Panel Data Models * > Comment 1.
Strength:plain· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
The authors state this as an assumption.
The stable-unit-treatment-value assumption holds and there are no dynamic effects: potential outcomes are indexed only by the contemporaneous treatment for that unit, not by past treatments or treatments for other units.
“Implicit in our set up is that we rule out dynamic effects and make the stable-unit-treatment-value assumption (Rubin (2006); Imbens and Rubin (2015)): the potential outcomes are indexed only by the contemporaneous treatment for that unit and not by past treatments or treatments for other units.”
The authors state this as an assumption.
The complete outcome matrix is the sum of a low-rank matrix L* and mean-zero noise ε.
“In the absence of covariates we model the N × T matrix of complete outcomes data matrix Y as Y = L * + ε, where E[ε|L * ] = 0 .”
The authors state this as an assumption.
The noise ε is independent of L*, and its elements are σ-sub-Gaussian and independent of each other.
“Assumption 1. ε is independent of L * , and the elements of ε are σ-sub-Gaussian and independent of each other.”
The authors state this as an assumption.
L* is a deterministic matrix.
“We also assume that L * is a deterministic matrix.”
The authors state this as an assumption.
Conditional on L*, the adoption dates t_i are independent of each other and of ε.
“Assumption 2. Conditional on L * , the adoption dates t i are independent of each other and of ε.”
The authors state this as an assumption.
The noise variables {ε_it} are independent of each other and are independent of {t_i}.
“We also assume that the noise variables {ε it } it∈[N ]×[T ] are independent of each other and are independent of {t i } N i=1 .”
The authors state this as an assumption.
Lmax is a positive constant such that ||L*||max ≤ Lmax.
“Let L max be a positive constant such that L * max ≤ L max (recall that L * max = max i,t |L * it |).”
The authors state this as an assumption.
The treatment effect is not constant but is such that the matrix Y(1) has a low-rank expectation.
“Assume the treatment effect is not constant but is such that the matrix Y(1) has a low-rank expectation.”
The authors state this as an assumption.
In §6.1, all entries (i, t) of Y for t ≤ t_i are observed.
“In §6.1 we assumed that all entries (i, t) of Y for t ≤ t i are observed.”
The authors state this as an assumption.
In the extension, each pre-treatment entry (i, t) can be missing independently with probability p, for p not too large.
“where each (i, t) ∈ ∪ N i=1 {(i, 1), (i, 2), . . ., (i, t i )} can be in O miss , independently, with probability p for p that is not too large.”
The paper relies on this without support.
The error terms ε_it are mean zero, σ-sub-Gaussian, and independent across i and t.
“random variables εit -E[ε it ] are centered, τ 2 -sub-Gaussian, and independent of each other (as i or t vary)”
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The cited source says what the claim attributes to it.
In the citing paper
“The complaints and use of force results are consistent across IFE and matrix completion (23) estimators; see SI Appendix.”
From Procedural justice training reduces police use of force and complaints against officers (2020) · cites this paper as [23]
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
Each contradiction between two passages listed here was confirmed by an independent check.
Rank is incorrectly defined as a limit of Schatten norms
MinorThe stated Schatten norm formula does not converge to rank as p approaches zero. For a matrix with two singular values equal to 1, it equals 2^(1/p), which diverges, while the rank is 2.
“L 0 ≡ lim p↓0 L S p = i∈[N ] 1 σ i (L)>0”
“L S p ≡ i∈[N ] σ i (L) p 1/p , p ∈ (0, ∞)”
Conditioning data switch from rows to columns
MinorThe remark first identifies the conditioning data as the first T -1 rows of Y, then identifies them as the first T -1 columns. These are different parts of the matrix.
“conditional on the first T -1 rows of Y, the last column of the assignment W is independent of the last column of Y.”
“so the conditioning on the first T -1 columns of Y is identical to conditioning on L.”
Nuclear-norm factorization penalizes A twice and omits B
MinorThe claimed factorization identity has two terms involving A and none involving B, whereas Theorem 1 penalizes both factors. As written, rescaling A toward zero and B upward while keeping AB fixed makes the right-hand infimum zero even for a matrix with positive nuclear norm.
“L * = min A,B:L=AB 1 2 A 2 F + A 2 F .”
“Q(Y; R, A, B, γ, δ) + λ 2 A 2 F + λ 2 B 2 F”
Vertical prediction range includes a pre-treatment period
MinorTreatment is stated to begin in period T 0 + 1, but the vertical-regression description includes T 0 in its range of treated periods.
“for the treated unit in period t, for t = T 0 , . . ., T , the predicted outcome is”
“treated for a number of periods from period T 0 + 1 onwards”
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Self-citation
Flagged first, then by citations
4 of the 83 references couldn’t be matched to a known paper.
Research works printed but not linked: 4
- Alberto Abadie. Using synthetic controls: Feasibility, data requirements, and methodological aspects. Journal of Economic Literature, 2019.
- Joshua Angrist and Steve Pischke. Mostly Harmless Econometrics: An Empiricists' Com- panion. Princeton University Press, 2008.
- Guido W Imbens and Donald B Rubin. Causal Inference in Statistics, Social, and Biomedical Sciences. Cambridge University Press, 2015.
- Donald B Rubin. Matched sampling for causal effects. Cambridge University Press, 2006.
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Self-citationUsing Synthetic Controls: Feasibility, Data Requirements, and Methodological Aspects
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(2022) · arXiv (preprint) · doi:10.47004/wp.cem.2022.1122 · Cited by 607
Metadata
- Authors
- Athey, S., Bayati, M., Doudchenko, N., Imbens, G., & Khosravi, K.
- Affiliations
- Stanford Institute for Economic Policy Research (US)
- Stanford University (US)
- Google (United States) (US)
- Corresponding author
- Imbens, Guido
- Venue
- Journal of the American Statistical Association
- Publisher
- National Bureau of Economic Research
- Published
- 2018-10
- Volume
- 116
- Issue
- 536
- Pages
- 1716-1730
- ISSN
- 0162-1459 (ISSN-L)
- OpenAlex
- W2766301804
- Cited by
- 497
- References
- 83
- Citations by year
- 2026: 65 · 2025: 71 · 2024: 87 · 2023: 77 · 2022: 50 · 2021: 68 · 2020: 25 · 2019: 40 · 2018: 10
- FWCI
- 29.67
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Journal of the American Statistical Association (publishedVersion)
- National Bureau of Economic Research (publishedVersion)
- RePEc: Research Papers in Economics (submittedVersion)
- arXiv (Cornell University) (submittedVersion)
- SSRN Electronic Journal (submittedVersion)
- National Bureau of Economic Research (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (1554140)
- Office of Naval Research (N00014-17-1-2131), N00014-17-1, N00014)
- Division of Civil, Mechanical and Manufacturing Innovation
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Spatial and Panel Data Analysis
- Physical Sciences › Mathematics › Statistics and Probability › Statistical Methods and Inference
- Keywords
- matrix completion, potential outcomes, nuclear norm regularization, missing data patterns, synthetic control method, unconfoundedness, program evaluation, treatment effect estimation
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