Matrix Completion Methods for Causal Panel Data Models

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.

“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.”

The unconfoundedness approach does not work very well if T is small relative to N.

“the unconfoundedness approach does not work very well if T N”

The proposed method outperforms unconfoundedness-based or synthetic control estimators in simulations based on real data.

“The proposed method outperforms unconfoundednessbased or synthetic control estimators in simulations based on real data.”

The unconfoundedness and synthetic control approaches break down in some of these settings in the expected pattern.

“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 ).”

The synthetic control approach does not work very well if N is small relative to T.

“the synthetic control approach does not work very well if N T”

Including fixed effects separately and not regularizing them greatly improves the quality of imputations compared to subsuming them in L.

“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.”

• AssumptionThe authors state this as an assumption: The complete outcome matrix is the sum of a low-rank matrix L* and mean-zero noise ε.

Lemma 3: If the estimator L satisfies ||L-L*||_F^2 ≥ 4L_max^2 θ/p_c for a positive number θ, then there is a constant C ≥ 0.001 such that, when Cθ > T, P_π( p_c/2 ||L-L*||_F^2 > sum_{(i,t)∈O} <A_it, L-L*>^2 + 8L_max^2 T/√N ) ≤ 2 exp(-Cθ/T).

“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 .”

The method generally outperforms the synthetic control estimators proposed by Abadie et al. (2010) and the elastic net estimators proposed by Doudchenko and Imbens (2016).

“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).”

• cites Abadie et al. (2010)Not checked yet

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• cites Doudchenko and Imbens (2016)Not checked yet

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As T0/T grows closer to one, VT-EN, HR-EN, SC-ADH and MC-NNM methods all do well.

“As T 0 /T grows closer to one, VT-EN, HR-EN, SC-ADH and MC-NNM methods all do well.”

Lemma 1: For all λ ≥ 3||E||_op/|O|, the sum over (i,t) in O of <A_it, L*-L>^2 / |O| is at most 10λ√R ||L*-L||_F.

“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 .”

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Lemma 5: Let Z_ρ ≡ sup_{M∈C(θ,ρ)} | ||M||_{L2(Π)}^2 - ||X_O(M)||_2^2 |, then there exist a constant C3 ≥ 0.0025 such that P_π( Z_ρ ≥ 1/(2ξ) ξρ + 2θ ) ≤ exp(-C3 ρξ/T).

“Lemma 5. Let Z ρ ≡ sup M∈C(θ,ρ) M 2 L 2 (Π) -X O (M) 2 2 , then there exist a constant C 3 ≥ 0.0025 such that P π Z ρ ≥ 1 2ξ ξρ + 2ϑ ≤ exp - C 3 ρξ T .”

Lemma 2: There exists a constant C1 such that ||E||_op ≤ C1 σ max(√(N log(N+T)), √(T log^{3/2}(N+T))) with probability greater than 1-(N+T)^{-2}.

“Lemma 2. There exist a constant C 1 such that E op ≤ C 1 σ max N log(N + T ), √ T log 3/2 (N + T ) , with probability greater than 1 -(N + T ) -2 .”

In the authors' simulations, it is often better to regularize estimators than to impose hard restrictions.

“We find the same in our simulations below.”

When N is much larger than T, the HR-EN estimator does poorly because it is trying to do the horizontal regression with too few observations relative to the number of regressors.

“When N T , the HR-EN estimator does poorly for the same reason: it is trying to do the horizontal regression with too few observations relative to the number of regressors.”

The method adapts well to different configurations of the data.

“In illustrations we show that the method adapts well to different configurations of the data”

In the stock market data, when T is much larger than N, the VT-EN estimator does poorly because it attempts to do the vertical regression with too few time periods to estimate that well.

“In the T N case the VT-EN estimator does poorly, not surprisingly because it attempts to do the vertical regression with too few time periods to estimate that well.”

Under the conditions of Theorem 2, ||L - L*||_F^2 ≤ 16C σ^2 R (N log(N+T) ∨ T log^3(N+T))/p_c^2 + 32 L_max^2 T/√N / p_c.

“∆ 2 F ≤ 16C σ 2 R N log(N + T ) ∨ T log 3 (N + T ) p 2 c + 32L 2 max T √ N p c”

If the only missing entry is in the last period for unit N, period T, then the vertical regression imputed value is linear in the observed outcomes of unit N in prior periods, with different weights from the horizontal regression.

“Similarly, if we estimate the vertical regression in (3.2), it is still the case that ŶNT is linear in Y N 1 , . . ., Y N,T -1 , just with different weights from the horizontal regression in (3.1).”

The vertical regression in (3.2) is a generalization of the original Abadie et al. (2010) synthetic control estimator, relaxing the restrictions that coefficients are nonnegative and that the intercept is zero.

“As shown in Doudchenko and Imbens (2016), this is generalization of the original Abadie et al. (2010) synthetic control estimator, relaxing two restriction: (i) that the coefficients are nonnegative and (ii) that the intercept in this regression is zero.”

• cites Doudchenko and Imbens (2016)Not checked yet

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• cites Abadie et al. (2010)Not checked yet

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The proposed method generalizes matrix completion results by allowing missing data patterns to have a time series dependency structure common in social science applications.

“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.”

The unconfoundedness approach assumes that patterns over time are stable across units.

“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.”

• cites Rosenbaum and Rubin (1983)Not checked yet

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• cites Imbens and Rubin (2015)Not checked yet

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All these estimators can be viewed as focusing on the same objective function.

“We show that all these estimators can be viewed as focusing on the same objective function.”

By focusing on only one of these patterns, cross-section or time series, the horizontal and vertical regression approaches ignore alternative patterns that may help in imputing missing values.

“However, by focusing on only one of these patterns, cross-section or time series, these approaches ignore alternative patterns that may help in imputing the missing values.”

Once regularization methods are used, unconfoundedness and synthetic control approaches are applicable in the same settings.

“We argue that once regularization methods are used, the two approaches, unconfoundedness and synthetic controls, are applicable in the same settings, leaving the researcher with a real choice in terms of methods.”

Given the common objective function, the unconfoundedness and synthetic control approaches impose different sets of restrictions on the factors in the matrix factorization.

“Given this common objective function, the unconfoundedness and synthetic control approaches impose different sets of restrictions on the factors in the matrix factorization.”

The proposed estimator includes fixed effects Γ and ∆, unlike the setting studied by Candès and Recht, Candès and Plan, and Mazumder et al.

“Compared to the setting studied by Candès and Recht (2009); Candès and Plan (2010); Mazumder et al. (2010), we include the fixed effects Γ and ∆.”

• cites Candès and Recht (2009)Not checked yet

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• cites Candès and Plan (2010)Not checked yet

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• cites Mazumder et al. (2010)Not checked yet

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• AssumptionThe 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 Abadie-Diamond-Hainmueller synthetic control method can be interpreted as regressing the outcomes for the treated unit prior to the treatment on the outcomes for the control units in the same periods.

“Imbens (2016)and Ferman and Pinto (2019) show how the Abadie-Diamond-Hainmueller synthetic control method can be interpreted as regressing the outcomes for the treated unit prior to the treatment on the outcomes for the control units in the same periods.”

• cites Imbens (2016)Not checked yet

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• cites Ferman and Pinto (2019)Not checked yet

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The proposed approach can be viewed as nesting both unconfoundedness and synthetic control approaches.

“In fact, we show that it can be viewed as nesting both.”

Negahban and Wainwright (2012) assume L* is not too spiky, formally that the ratio ||L*||max / ||L*||F should be of order 1/√(NT) up to logarithmic terms.

“An example of such restriction is given by Negahban and Wainwright (2012) where they assume L * is not too spiky.Formally, they assume the ratio L * max / L * F should be of order 1/ √ N T up to logarithmic terms.”

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The correlation structure of observations from each row deteriorates the type of upper bound that can be obtained for ||E||_op.

“In fact, the correlation structure deteriorates the type of upper bound that can be obtained for E op .”

One drawback of MC-NNM is that it does not take into account the time series nature of the observations.

“One drawback of MC-NNM is that it does not take into account the time series nature of the observations.”

A major advantage of using the nuclear norm is that the resulting estimator can be computed using fast convex optimization programs, such as the soft-impute algorithm by Mazumder et al. (2010).

“In contrast, a major advantage of using the nuclear norm is that the resulting estimator can be computed using fast convex optimization programs, e.g. the soft-impute algorithm by Mazumder et al. (2010) that will be described next.”

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Other commonly used Schatten norms would not work as well as the nuclear norm for this specific problem.

“Other commonly used Schatten norms would not work as well for this specific problem.”

The matrix completion, horizontal regression, vertical regression, synthetic control, elastic net, and difference-in-differences estimators can all be expressed as minimizing the same objective function Q(Y; R, A, B, γ, δ), differing in regularization and additional restrictions.

“They can all be charactized as focusing on the exact same objective function, but differing in the regularization and additional restrictions imposed on the parameters of the objective function.”

• Also stated in: § 5 The Relationship with Horizontal and Vertical Regressions · plain

  “The next result shows that horizontal regression, vertical regression, the Abadie-Diamond-Hainmueller synthetic control estimator, the difference-in-differences estimator, and the nuclear norm minimization matrix completion can all be expressed as minimizing Q(Y; R, A, B, γ, δ) under different restrictions on, or with different approaches to regularization of the unknown parameters (R, A, B, γ, δ)”

The rank pattern validates that there is a stronger correlation between daily returns of different stocks than between returns for different time periods of the same stock.

“This validates the fact that there is a stronger correlation between daily return of different stocks than between returns for different time periods of the same stock.”

In order to relax the constraint on p_c, one would need to restrict the family of matrices L*.

“In order to relax this constraint we would need to restrict the family of matrices L * .”

There is no direct way to test the horizontal and vertical methods against each other.

“As a result there is also no direct way to test the two methods against each other.”

There is no unique solution to minimizing Q(Y; A, B) if we also minimize over the rank R.

“There is no unique solution to minimizing Q(Y; A, B) if we also minimize over the rank R.”

Under Assumptions 1 and 2, if rank(L*) = R, T ≥ C0 log(N+T), and λ is a constant multiple of σ√(N log(N+T) ∨ T log^3(N+T))/|O|, then with probability greater than 1 - 2(N+T)^-2, the RMSE of L is bounded by the expression in Theorem 2.

“Theorem 2. Suppose Assumptions 1 and 2 hold, rank of L * is R, T ≥ C 0 log(N + T ) for a constant C 0 , and the penalty parameter λ is a constant multiple of σ N log(N + T ) ∨ T log 3 (N + T ) |O| .”

• AssumptionThe authors state this as an assumption: L* is a deterministic matrix.
• AssumptionThe authors state this as an assumption: Conditional on L*, the adoption dates t_i are independent of each other and of ε.
• AssumptionThe authors state this as an assumption: The noise variables {ε_it} are independent of each other and are independent of {t_i}.
• AssumptionThe authors state this as an assumption: Lmax is a positive constant such that ||L*||max ≤ Lmax.

The rank norm is not computationally feasible for large N and T if the cardinality and complexity of the set M are substantial.

“The rank norm is not computationally feasible for large N and T if the cardinality and complexity of the set M are substantial.”

The RMSE of L converges to zero as N and T grow when L* is low-rank (R constant) and p_c (1/T ∨ 1/N) log^{3/2}(N+T).

“Interpretation of Theorem 2: In order to see when the RMSE of L converges to zero as N and T grow, we note that the right hand side of (6.2) converges to 0 when L * is low-rank (R is constant), and p c ( 1/T ∨ 1/N ) log 3/2 (N + T ).”

• AssumptionThe authors state this as an assumption: L* is a deterministic matrix.
• AssumptionThe authors state this as an assumption: Conditional on L*, the adoption dates t_i are independent of each other and of ε.
• AssumptionThe authors state this as an assumption: The noise variables {ε_it} are independent of each other and are independent of {t_i}.
• AssumptionThe authors state this as an assumption: Lmax is a positive constant such that ||L*||max ≤ Lmax.

In the case of N = T, in order to have consistent estimation, a factor √N (up to logarithmic factors) more entries per column should be observed than in the matrix completion literature.

“For example, as discussed above, in the case of N = T , in order to have a consistent estimation we need more data.Specifically, a factor √ N (up to logarithmic factors) more entries per column should be observed, than in the matrix completion literature.”

When T is the same order as N, a sufficient condition for the RMSE to converge to zero is that the lower bound for the average number of control units (N p_c) grows faster than a constant multiple of √N log^{3/2}(N).

“For example, when T is the same order as N , a sufficient condition for the latter is that the lower bound for the average number of control units (N p c ) grows faster than a constant multiple of √ N log 3/2 (N ).”

• AssumptionThe authors state this as an assumption: L* is a deterministic matrix.
• AssumptionThe authors state this as an assumption: Conditional on L*, the adoption dates t_i are independent of each other and of ε.
• AssumptionThe authors state this as an assumption: The noise variables {ε_it} are independent of each other and are independent of {t_i}.
• AssumptionThe authors state this as an assumption: Lmax is a positive constant such that ||L*||max ≤ Lmax.

The matrix completion, horizontal regression, vertical regression, synthetic control regression, elastic net version, and difference-in-differences estimators are very closely related.

“In this case the matrix completion, horizontal regression, vertical regression, synthetic control regression, the elastic net version, and difference-in-differences estimators are very closely related.”

Another limitation of MC-NNM is that it puts equal weight on all observed elements of the difference Y - L, ignoring the covariates.

“Another limitation of MC-NNM is that it puts equal weight on all observed elements of the difference Y -L (ignoring the covariates).”

The performance improvement of MC-NNM can be attributed to its use of additional observations, namely pre-treatment values of treatment units.

“The performance improvement of MC-NNM can be attributed to its use of additional observations (pre-treatment values of treatment units).”

The Fröbenius norm on the penalty term would not have been suitable for estimating L* in the case with missing entries because the solution for L_it for (i,t) in M is always zero.

“For example, the Fröbenius norm on the penalty term would not have been suitable for estimating L * in the case with missing entries because the solution for L it for (i, t) ∈ M is always zero (which follows directly from the representation of L F = (i,t)∈[N ]×[T ] L 2 it ).”

The impact of the additional correlation in the authors' observation model is that the estimation error deteriorates significantly compared to prior literature.

“The impact of this additional correlation is that the estimation error deteriorates significantly, compared to the ones in prior literature.”

Theorem 2 provides an upper bound for the root-mean-squared-error (RMSE) of the estimator L.

“The main result of this section is the next theorem (proved in §A.2) that provides an upper bound for L * -L F / √ N T , the root-mean-squared-error (RMSE) of the estimator L.”