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Spectral Truncation in Synthetic Control
Mojtaba Eslami
Intelligence
Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
This paper investigates Spectral Synthetic Control (SC), an estimator that matches treated units using the leading temporal singular vectors of donor panels rather than raw pre-treatment trajectories. The authors prove that Spectral SC reduces to raw-path SC at full rank and demonstrate that exact spectral balance is underdetermined when the number of donors exceeds the retained dimensions plus one. Through simulation across eleven regimes, they find that truncated Spectral SC generally yields higher RMSE than tuned raw-path SC. However, a robustness check reveals that removing unit and time fixed effects before spectral decomposition significantly reduces this performance gap, suggesting that basis-estimation noise and fixed-effects contamination are key drivers of Spectral SC's failure in raw data settings.
Entities (8)
Relation Signals (6)
Abadie et al. → authored → Synthetic Control
confidence 99% · Synthetic control (SC) (Abadie et al., 2003, 2010)
Spectral SC → isvariantof → Synthetic Control
confidence 95% · We study Spectral SC, which instead matches the treated unit in coordinates defined by the leading temporal singular vectors of the donor panel
Hybrid Estimator → nests → Spectral SC
confidence 90% · a hybrid estimator that places separately tunable weight on retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints.
Hybrid Estimator → nests → Synthetic Control
confidence 90% · nesting raw-path SC and truncated Spectral SC as endpoints.
Fixed Effects → causes → Basis-Estimation Noise
confidence 88% · basis-estimation noise from unremoved fixed effects
Weight Underdetermination → affects → Spectral SC
confidence 85% · exact spectral balance with K retained dimensions and N0 donors is underdetermined whenever N0>K+1
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Abstract
Abstract:Synthetic control (SC) matches a treated unit's pre-treatment trajectory to a weighted combination of donor units. We study Spectral SC, which instead matches the treated unit in coordinates defined by the leading temporal singular vectors of the donor panel, and a hybrid estimator that places separately tunable weight on retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints. We prove that the family reduces exactly to raw-path SC at full rank, that exact balance on $K$ retained dimensions with $N_0$ donors is underdetermined whenever $N_0>K+1$, with an affine solution set of dimension $N_0-K-1$, and that spectral imbalance maps to treatment-effect bias through a finite-sample best-linear-predictor decomposition. We evaluate the estimators across eleven data-generating regimes, using $400$ replications per regime and donor-only placebo validation to select regularization and the mixing weight. Truncated Spectral SC has significantly higher RMSE than tuned raw-path SC in every regime, with paired differences equal to $4$ to $11$ Monte Carlo standard errors. The hybrid estimator selects raw-path matching in most replications and is statistically indistinguishable from tuned SC in most regimes. The result is highly sensitive to preprocessing. With raw inputs, the performance gap is large; after removing unit and time fixed effects before spectral decomposition, as suggested by the assumptions behind our bound, the gap nearly disappears and placebo validation begins to favor truncation. We interpret these findings diagnostically rather than as evidence that Spectral SC should replace raw-path SC. Basis-estimation noise, balancing underdetermination, and fixed-effects contamination determine when spectral matching can help.
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- Source: https://arxiv.org/abs/2607.25074v1
- Canonical: https://arxiv.org/abs/2607.25074v1
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Spectral Truncation in Synthetic Control: Weight Underdetermination and Basis-Estimation Noise Mojtaba Eslami University of Calgary, mojtaba.eslami@alumni.ucalgary.ca Abstract Synthetic control (SC) matches a treated unit’s raw pre-treatment trajectory against a weighted combination of donor units. We study Spectral SC, which instead matches in the coordinates of the donor panel’s leading temporal singular vectors, and a hybrid estimator that assigns independently tunable weight to retained and discarded directions, nesting raw-path SC and truncated Spectral SC as endpoints. We prove this family reduces exactly to raw-path SC at full rank, prove that exact spectral balance with K retained dimensions and N0N_0 donors is underdetermined whenever N0>K+1N_0>K+1 (an affine solution set of dimension N0−K−1N_0-K-1), and give a finite-sample bound relating spectral balancing quality to treatment-effect bias via a best-linear-predictor decomposition of loadings in spectral scores. We evaluate the family – with regularization and mixing weight, but not rank, selected by donor-only placebo validation – across eleven data-generating regimes chosen to favor truncation (400400 replications each, with Monte Carlo standard errors and paired replication-level comparisons). Truncated Spectral SC has significantly higher RMSE than tuned raw-path SC in every regime (paired differences 44–1111 standard errors from zero); the hybrid estimator selects raw-path matching outright in a majority of replications throughout and is statistically indistinguishable from tuned SC in most regimes. A robustness check shows this gap is highly preprocessing-dependent: under raw input the gap is large, but removing unit and time fixed effects before the spectral decomposition – consistent with the assumption underlying our bound – nearly eliminates it, and the placebo procedure itself then switches to preferring truncation. We report this as a diagnostic result, identifying specific mechanisms (basis-estimation noise, balancing underdetermination, and fixed-effects contamination of the spectral basis) governing when spectral matching can help, rather than establishing that the version tested here should replace raw-path SC. Keywords: synthetic control, panel data, low-rank factor models, spectral methods, causal inference 1 Introduction Synthetic control (SC) (Abadie et al., 2003, 2010) estimates a treated unit’s counterfactual path as a weighted average of untreated donor units, with weights chosen so the weighted donor pre-treatment trajectory matches the treated unit’s own pre-treatment trajectory. This matching is performed on the raw, T0T_0-dimensional pre-treatment path. This paper studies an alternative: matching in the coordinates of the panel’s leading temporal singular vectors rather than in raw time coordinates. The motivating hypothesis is that, if the untreated outcome process follows a low-rank interactive fixed-effects model, explicitly matching in the directions estimated to carry the panel’s systematic variation might target the underlying factor loadings more directly than matching the full raw trajectory. This is a hypothesis, not a settled conclusion: raw-path matching may already estimate factor-relevant weights efficiently, since repeated time observations carry information even when the underlying process is low rank, and projecting onto an estimated subspace first introduces its own estimation error while discarding whatever implicit regularization the full trajectory provided. We test the hypothesis with a simulation study designed specifically to find regimes in which it holds, and we characterize, formally, two specific mechanisms – weight underdetermination and basis-estimation noise from unremoved fixed effects – that can make it fail. Scope and claims. Low-rank and spectral structure in panel causal inference is already well established. Matrix completion (Athey et al., 2021), generalized synthetic control (Xu, 2017), robust synthetic control (Amjad et al., 2018), and augmented synthetic control (Ben-Michael et al., 2021) all use low-rank or factor-model structure to correct or regularize SC-type estimators, and synthetic difference-in-differences (SDID) (Arkhangelsky et al., 2021) is analyzed under a latent factor model. Synthetic Principal Component Design (Lu et al., 2022) uses a spectral optimization for the experimental-design problem of which units to treat; a functional extension of generalized synthetic control uses functional principal component scores for sparse, irregular panels (Shao et al., 2026); and Harmonic Synthetic Control (Liu and Xu, 2026) introduces a continuously tunable spectral allocation for a related but distinct purpose (Section 8.1). We do not claim that using spectral or low-rank structure in panel causal inference is new. What we study, narrowly, is a simplex-constrained score-balancing estimator and a hybrid generalization, with two new formal results (weight underdetermination and a score-to-loading bound) and an honest empirical account of when the resulting truncation helps – including a preprocessing-dependent reversal that we think is the paper’s most useful empirical finding. Roadmap. Section 2 sets up notation. Section 3 defines the spectral and hybrid estimators and proves the full-rank equivalence and weight-underdetermination results. Section 4 gives a formal representability condition and the bias bound. Section 5 addresses how fixed effects are removed before the spectral decomposition and reports a robustness check that reverses the paper’s main conclusion under one preprocessing choice. Section 6 describes hyperparameter selection, including which parameters are tuned and which are fixed by design, and a placebo-sample-size sensitivity check. Section 7 reports the eleven-regime study with paired statistics. Section 8 discusses related work and open questions. Appendix A gives reproducibility details. 2 Setup Let i=1,…,Ni=1,…,N index units and t=1,…,Tt=1,…,T index periods, with T0T_0 pre-treatment and T1=T−T0T_1=T-T_0 post-treatment periods. Unit 11 is treated starting at T0+1T_0+1; units i=2,…,Ni=2,…,N (the donor pool C, ||=N0|C|=N_0) are never treated. Yit=Yit(0)+τitWitY_it=Y_it(0)+ _itW_it, with Wit∈0,1W_it∈\0,1\ the treatment indicator, and Yit(0)=αi+δt+ℓi⊤ft+εit,Y_it(0)= _i+ _t+ _i f_t+ _it, (1) with αi,δt _i, _t unit and time fixed effects, ℓi∈ℝR _i ^R factor loadings, ft∈ℝRf_t ^R common factors, R≪min(N,T)R (N,T), εit _it idiosyncratic noise – the standard generating process used to motivate SC-type estimators (Abadie et al., 2010; Xu, 2017; Athey et al., 2021; Arkhangelsky et al., 2021). Stacking donor pre-treatment outcomes gives Y0,pre∈ℝN0×T0Y_0,pre ^N_0× T_0, approximately low rank under (1). Let ω∈ΔN0=ωi≥0,∑iωi=1ω∈ ^N_0=\ _i≥ 0, _i _i=1\. Raw-path SC solves ω^=argminω∈ΔN0‖y1,pre−∑iωiyi,pre‖22 ω= *arg\,min_ω∈ ^N_0\|y_1,pre- _i _iy_i,pre\|_2^2, with Y^1t(0)=∑iω^iYit Y_1t(0)= _i ω_iY_it for t>T0t>T_0. 3 Spectral and Hybrid Balancing 3.1 Spectral representation and the baseline estimator Let Y0,pre=UΣV⊤Y_0,pre=U V be the SVD of the donor pre-treatment matrix, estimated from donor pre-treatment cells only. Fix K and keep the leading components, VK∈ℝT0×KV_K ^T_0× K. Project every pre-treatment path onto VKV_K: s1=VK⊤y1,pres_1=V_K y_1,pre, si=VK⊤yi,pres_i=V_K y_i,pre for i∈i , and solve ω^=argminω∈ΔN0‖s1−∑i=1N0ωisi‖22+λω‖ω‖22,Y^1t(0)=∑i=1N0ω^iYit(t>T0). ω= *arg\,min_ω∈ ^N_0 \|s_1- _i=1^N_0 _is_i \|_2^2+ _ω\|ω\|_2^2, Y_1t(0)= _i=1^N_0 ω_iY_it\ \ (t>T_0). (2) Proposition 1 (Equivalence to raw-path SC at full rank). Suppose T0≤N0T_0≤ N_0 and K=T0K=T_0, so VK=V∈ℝT0×T0V_K=V ^T_0× T_0 is square and orthogonal. Then for every ω∈ΔN0ω∈ ^N_0, ‖VK⊤(y1,pre−Y0,pre⊤ω)‖22=‖y1,pre−Y0,pre⊤ω‖22\|V_K (y_1,pre-Y_0,pre ω)\|_2^2=\|y_1,pre-Y_0,pre ω\|_2^2, so (2) with K=T0K=T_0 has exactly the same objective, feasible set, and minimizer as raw-path SC with the same ridge penalty. Proof. Let x=y1,pre−Y0,pre⊤ωx=y_1,pre-Y_0,pre ω. Since V is orthogonal, ‖V⊤x‖22=x⊤VV⊤x=x⊤x=‖x‖22\|V x\|_2^2=x V x=x x=\|x\|_2^2, for every ω; the objectives coincide pointwise on ΔN0 ^N_0. ∎ This shows the projection preserves the raw Euclidean pre-treatment balancing discrepancy at K=T0K=T_0; it does not, by itself, establish equivalence to any estimator built from more than this discrepancy (see the remark on SDID in Section 3.3). Any difference between (2) and raw-path SC for K<T0K<T_0 comes specifically from truncating modes. 3.2 Weight underdetermination The title’s second mechanism is a precise linear-algebra fact about (2), not merely a qualitative observation. Consider the equality-constrained version of the spectral balancing problem obtained by dropping the ridge penalty and the sign constraints on ω (i.e. exact balance on the affine hull of the simplex): with donor score matrix SK=[s2⋯sN0]∈ℝK×N0S_K=[s_2\ ·s\ s_N_0] ^K× N_0, exact spectral balance requires SKω=s1,⊤ω=1,S_Kω=s_1, 1 ω=1, (3) a system of K+1K+1 linear equations in N0N_0 unknowns. Proposition 2 (Underdetermination of low-rank balancing). Let A=[SK⊤]∈ℝ(K+1)×N0A= bmatrixS_K\\ 1 bmatrix ^(K+1)× N_0. If N0>K+1N_0>K+1, system (3) is consistent (i.e. has at least one solution), and A has full row rank K+1K+1, then the solution set ω∈ℝN0:Aω=[s1;1]\ω ^N_0:Aω=[s_1;1]\ is a nonempty affine subspace of dimension N0−K−1N_0-K-1. Proof. By the rank-nullity theorem, dimnull(A)=N0−rank(A)=N0−(K+1) (A)=N_0-rank(A)=N_0-(K+1). If ω0 _0 is one solution, the full solution set is ω0+null(A) _0+null(A), an affine translate of null(A)null(A), hence of the same dimension. ∎ Whenever N0>K+1N_0>K+1 – generically true, since K is chosen small by design while N0N_0 is the size of the available donor pool – exact spectral balance does not pin down a unique donor weight vector: an entire (N0−K−1)(N_0-K-1)-dimensional family of weight vectors achieves identical pre-treatment spectral balance while generally implying different post-treatment counterfactuals, since nothing in (3) constrains behavior outside range(VK)range(V_K). The ridge penalty λω‖ω‖22 _ω\|ω\|_2^2 (together with the simplex sign constraints, which cut down but do not generally eliminate this indeterminacy) is what selects a particular point from this set; as N0−K−1N_0-K-1 grows, that selection carries correspondingly more of the estimator’s effective information content, since the data no longer pin it down. This is the formal counterpart of the empirical finding in Section 7: with N0=30N_0=30 and K=2K=2, the affine solution set of (3) has dimension up to 2727, and different regularization strengths can select donor weight vectors with materially different post-treatment behavior even though all of them balance the retained two-dimensional score equally well. Raw-path SC, with K=T0K=T_0, faces the analogous system with T0+1T_0+1 equations rather than K+1K+1; whenever T0T_0 is not much smaller than N0N_0 this system is far less underdetermined, and in the T0≫N0T_0 N_0 regime of Section 7 it can be overdetermined, which is one reason raw-path matching is implicitly well regularized in that regime without any explicit penalty. 3.3 A hybrid estimator A less abrupt alternative to truncation assigns discarded directions a strictly positive but shrunk weight η∈[0,1]η∈[0,1], using ΠK=VKVK⊤ _K=V_KV_K : ω^η=argminω∈ΔN0(y1,pre−Y0,pre⊤ω)⊤MK,η(y1,pre−Y0,pre⊤ω)+λω∥ω∥22,MK,η=ΠK+η(IT0−ΠK). ω_η= *arg\,min_ω∈ ^N_0(y_1,pre-Y_0,pre ω) M_K,η(y_1,pre-Y_0,pre ω)+ _ω\|ω\|_2^2, M_K,η= _K+η(I_T_0- _K). (4) By Proposition 1’s argument, η=1η=1 recovers raw-path SC exactly for any K; η=0η=0 recovers truncated Spectral SC (2) exactly. Values η∈(0,1)η∈(0,1) interpolate, and by Proposition 2, moving η above 0 directly reduces the dimension of directions left unconstrained by the balancing loss – η>0η>0 constrains all T0T_0 directions, just with unequal weight, rather than leaving T0−KT_0-K of them entirely unconstrained. This turns truncation into a continuous parameter tunable by the same placebo procedure used for the other estimators (Section 6). Remark 1 (On recovering SDID). At full rank with identity mode metrics, (2)–(4) preserve the raw Euclidean balancing discrepancy (Proposition 1). Recovering the complete SDID estimator (Arkhangelsky et al., 2021) additionally requires its precise unit-weight, time-weight, intercept, regularization, and final doubly-weighted regression specification; matching one Euclidean discrepancy term is necessary but not sufficient for that equivalence, and we do not pursue a time-weighted analogue here. 4 Identification and Bias 4.1 A formal representability condition We use the following as a motivating condition rather than a claim we verify: it states what would need to be true for spectral balancing to be well targeted, in terms of explicit, interpretable tolerances. Assumption 1 (Approximate rank-K representability). Fix tolerances δs,δℓ≥0 _s, _ ≥ 0. There exists ω⋆∈ΔN0ω ∈ ^N_0 such that ‖ΠK(y1,pre−Y0,pre⊤ω⋆)‖2≤δs,‖ℓ1−∑iωi⋆ℓi‖2≤δℓ, \| _K (y_1,pre-Y_0,pre ω ) \|_2≤ _s, \| _1- _iω _i _i \|_2≤ _ , (5) where ΠK,ℓi _K,\ _i\ are the same population objects (not re-estimated) across the pre- and post-treatment windows. Smaller (δs,δℓ)( _s, _ ) is a stronger, more favorable condition; δs=δℓ=0 _s= _ =0 recovers exact spectral balance with exact loading representability. Under Assumption 1, the loading-mismatch term in Proposition 4 below is bounded by δℓ‖ft‖2 _ \|f_t\|_2 at the oracle ω⋆ω ; Proposition 5 in Section 4.3 relates the achieved estimator’s ω ω (which need not equal ω⋆ω ) to the observed spectral residual ‖s1−∑iω^isi‖2\|s_1- _i ω_is_i\|_2, which is what an analyst can actually compute. Proposition 3 (Full-path balance implies rank-K subspace balance). If ω∈ΔN0ω∈ ^N_0 satisfies y1,pre=∑iωiyi,prey_1,pre= _i _iy_i,pre exactly, it satisfies ΠKy1,pre=ΠK∑iωiyi,pre _Ky_1,pre= _K _i _iy_i,pre for every K,ΠK, _K. Proof. Apply ΠK _K to both sides. ∎ The converse fails whenever the residual has a nonzero component outside range(VK)range(V_K), so Assumption 1 at a given δs _s is different from, not proven weaker than, the corresponding full-path condition. 4.2 Exact bias decomposition Proposition 4 (Bias decomposition under the factor model). Under (1), for any ω∈ΔN0ω∈ ^N_0 and t>T0t>T_0, with Y^1t(0)=∑i∈ωiYit Y_1t(0)= _i _iY_it and τ^t:=Y1t−Y^1t(0) τ_t:=Y_1t- Y_1t(0), τ^t−τt=(α1−∑iωiαi)⏟level mismatch+(ℓ1−∑iωiℓi)⊤ft⏟loading mismatch+(ε1t−∑iωiεit)⏟idiosyncratic noise. τ_t- _t= ( _1- _i _i _i )_level mismatch+ ( _1- _i _i _i ) f_t_loading mismatch+ ( _1t- _i _i _it )_idiosyncratic noise. (6) Proof. Substitute (1) (with Y1tY_1t additionally containing τt _t) into τ^t=Y1t−∑iωiYit τ_t=Y_1t- _i _iY_it and use ∑iωi=1 _i _i=1 to cancel δt _t. ∎ 4.3 From spectral score balance to loading mismatch Proposition 4 isolates the loading-mismatch term as what spectral balancing targets. We now bound it in terms of the observable spectral residual, using a best-linear-predictor (BLP) construction so that the relevant linear map is defined, not merely assumed to exist. Definition 1 (BLP of loadings on spectral scores). Let AK∈ℝR×KA_K ^R× K be the least-squares (best linear predictor) coefficient obtained by regressing ℓii∈1∪\ _i\_i∈\1\ on sii∈1∪\s_i\_i∈\1\ : AK:=argminA∑i‖ℓi−Asi‖22A_K:= *arg\,min_A _i\| _i-As_i\|_2^2, and let ri:=ℓi−AKsir_i:= _i-A_Ks_i be the resulting residuals. AKA_K is well defined whenever the sis_i span ℝKR^K (generic whenever N0≥KN_0≥ K), with no distributional or correct-specification assumption required. Lemma 1 (AKA_K is exact when K=RK=R and VKV_K spans the true factor subspace). If unit and time fixed effects have been removed, so yi,pre=Fpreℓi+εi,prey_i,pre=F_pre _i+ _i,pre, and K=RK=R with A:=VK⊤FpreA:=V_K F_pre invertible, then the population BLP coincides with AK=A−1A_K=A^-1 and ri=−A−1VK⊤εi,prer_i=-A^-1V_K _i,pre. Proof. si=Aℓi+VK⊤εi,pres_i=A _i+V_K _i,pre, so ℓi=A−1si−A−1VK⊤εi,pre _i=A^-1s_i-A^-1V_K _i,pre exactly; since this linear relation holds exactly for every i, it is in particular the least-squares solution. ∎ Proposition 5 (Error decomposition and observable upper bound under approximate score representability). For AK,riA_K,r_i as in Definition 1 and any ω∈ΔN0ω∈ ^N_0, ‖ℓ1−∑iωiℓi‖2≤‖AK‖op‖s1−∑iωisi‖2+‖r1−∑iωiri‖2, \| _1- _i _i _i \|_2\;≤\;\|A_K\|_op\, \|s_1- _i _is_i \|_2\;+\; \|r_1- _i _ir_i \|_2, (7) and, at any t>T0t>T_0, |(ℓ1−∑iωiℓi)⊤ft|≤‖ft‖2[‖AK‖op‖s1−∑iωisi‖2+‖r1−∑iωiri‖2] |( _1- _i _i _i) f_t |≤\|f_t\|_2 [\|A_K\|_op\|s_1- _i _is_i\|_2+\|r_1- _i _ir_i\|_2 ]. Proof. ℓ1−∑iωiℓi=AK(s1−∑iωisi)+(r1−∑iωiri) _1- _i _i _i=A_K(s_1- _i _is_i)+(r_1- _i _ir_i) by construction and ∑iωi=1 _i _i=1; triangle inequality and submultiplicativity give the first bound, Cauchy–Schwarz the second. ∎ This inequality is close to algebraic once Definition 1 is in place, and we do not present it as a deep identification theorem; its value is interpretive. It makes precise (i) that spectral balancing controls loading mismatch only through ‖AK‖op\|A_K\|_op times the achieved spectral residual – exactly the quantity (2) minimizes – and (i) that the BLP residual term rir_i, which absorbs both omitted-factor error (K<RK<R) and basis-estimation error, is invisible to the spectral objective and not controlled by it at all. When rir_i is large, (7) is uninformative regardless of how well (2) is solved – consistent with the simulation evidence in Sections 5–7. 5 Preprocessing and the Choice of Basis Lemma 1 assumes unit and time fixed effects have already been removed before the SVD. The practical estimator in Section 3, and the simulation study as originally run, applies the SVD to raw pre-treatment paths. This matters: if αi _i varies substantially across donors, the leading singular directions of Y0,preY_0,pre can be dominated by cross-unit level differences rather than by the interactive factor structure in (1), in which case VKV_K estimated from raw data is a poor estimate of the population factor subspace and the BLP residual rir_i in Proposition 5 is correspondingly large through no fault of the truncation rule itself. We evaluate three preprocessing choices, applied identically to raw-path SC, Spectral SC, and the hybrid estimator so the comparison stays apples-to-apples: • Raw: balancing and basis estimation use Y0,pre,y1,preY_0,pre,y_1,pre directly (the estimator as defined in Section 3 and used in the main study). • Unit-demeaned: each unit’s own pre-treatment mean is subtracted before balancing and basis estimation; the counterfactual is reconstructed as ∑iω^iYit _i ω_iY_it plus an intercept correction y¯1,pre−∑iω^iy¯i,pre y_1,pre- _i ω_i y_i,pre, so the estimator remains a donor-weighted average with a level adjustment, not a forecast. • Two-way demeaned: unit means and donor-pool time means are both removed (double centering) before balancing and basis estimation, with the same intercept correction as above. Preprocessing SC RMSE Spectral RMSE Δ (Spectral−-SC) η¯ η Pr(η^=0) ( η=0) Raw 0.283 (0.019) 0.431 (0.025) 0.148 (0.025) 0.959 0.008 Unit-demeaned 0.213 (0.012) 0.238 (0.016) 0.025 (0.012) 0.878 0.032 Two-way demeaned 0.223 (0.013) 0.224 (0.013) 0.001 (0.004) 0.295 0.568 Table 1: Preprocessing robustness check, baseline regime, N0=30N_0=30, K=2K=2, 250250 replications per row. Standard errors in parentheses; Δ is the paired RMSE difference (same simulated panels). Table 1 shows a reversal, not merely an attenuation. Under raw input, Spectral SC’s RMSE exceeds tuned SC’s by 0.1480.148 (SE 0.025SE\ 0.025), about six standard errors – consistent with the main study in Section 7, which uses raw input throughout. Under unit-demeaning the gap falls to 0.0250.025 (SE 0.012SE\ 0.012), about two standard errors. Under two-way demeaning the gap is 0.0010.001 (SE 0.004SE\ 0.004), statistically indistinguishable from zero, and – more strikingly – the placebo-tuning procedure itself switches its preference: the mean selected η falls from 0.960.96 (raw) to 0.300.30 (two-way demeaned), with a majority of replications (57%57\%) now selecting η=0η=0, full truncation, rather than η=1η=1. This is exactly the pattern Proposition 5 predicts: removing fixed effects before the SVD shrinks the BLP residual rir_i by bringing VKV_K closer to the true factor subspace, and the placebo procedure – which never sees this proposition, only post-treatment squared error on held-out donors – detects the resulting improvement in ‖AK‖op\|A_K\|_op-scaled control on its own. We do not rerun the full eleven-regime study of Section 7 under two-way demeaning; doing so, and mapping out exactly which regimes benefit and by how much, is a natural next step this note does not complete. We report Table 1 as a robustness check on the baseline regime and flag explicitly that the negative results in Section 7 are for raw-path input specifically, not for spectral matching under every reasonable preprocessing choice. 6 Hyperparameter Selection We tune λω _ω, and, for the hybrid estimator, η, by donor-only leave-one-out placebo validation: for each donor j in a randomly drawn subset, we treat j as if treated, re-estimate the basis from the remaining donors, balance j’s pre-treatment path against them at each candidate (λω,η)( _ω,η), and record squared error on j’s actual post-treatment path, never using post-treatment or treated-unit data. We use the grids λω∈10−4,10−3,10−2,10−1,1 _ω∈\10^-4,10^-3,10^-2,10^-1,1\ and η∈0,0.15,0.35,0.5,0.65,0.85,1η∈\0,0.15,0.35,0.5,0.65,0.85,1\. K is fixed by design, not tuned. Throughout this paper, the rank K is a prespecified input to Algorithm 1, chosen per regime as part of the experimental design (e.g. set equal to, below, or above the data-generating process’s true factor rank R, to study misspecification directly, as in the weak-factor regimes of Section 7). We tune only (λω,η)( _ω,η) by placebo validation. We do not fold K into the same procedure because the placebo objective is comparable across candidate λω,η _ω,η values at fixed K (all evaluated in the same T0T_0-dimensional held-out squared-error units) but is not obviously comparable across K within a single, unified selection rule without additional structure; treating K as fixed avoids that complication and keeps the object of study – the consequence of a given truncation choice – separate from the object of the search. Algorithm 1 Spectral / hybrid SC with placebo-tuned (λω,η)( _ω,η) at fixed K 1:Donor pre-treatment matrix Y0,preY_0,pre; treated pre-path y1,prey_1,pre; full donor and treated outcomes; a fixed rank K; candidate grids for λω _ω and η; preprocessing mode. 2:Preprocess and estimate basis (donor data only): apply the chosen preprocessing (Section 5) to Y0,preY_0,pre; compute VKV_K from the processed donor matrix. 3:Placebo-tune (λω,η)( _ω,η) at the fixed K: as described above; select the pair minimizing average placebo post-treatment squared error. 4:Balance and predict: solve (4) for ω ω at the selected (λω,η)( _ω,η); form Y^1t(0) Y_1t(0) (with the intercept correction of Section 5 if preprocessing is not raw); τ^t=Y1t−Y^1t(0) τ_t=Y_1t- Y_1t(0). 5:return τ τ, ω ω, selected (λω,η)( _ω,η). Sensitivity to placebo sample size. Hyperparameters above are tuned using 44 randomly drawn placebo donors per replication. Table 2 checks whether this is large enough by comparing 44, 1010, and all 2929 eligible donors on the baseline regime. Placebo donors N (replications) η¯ η Pr(η^=1) ( η=1) Δ (Spectral−-SC) RMSE 4 120 0.970 0.917 0.147 (0.038) 10 120 0.972 0.950 0.148 (0.038) 29 (all) 50 0.994 0.960 0.194 (0.056) Table 2: Placebo-sample-size sensitivity, baseline regime, raw preprocessing. N is reduced for the all-donor row for compute feasibility; standard errors in parentheses. The preference for raw-path matching does not depend on using a small, noisy placebo sample: it is, if anything, stronger with more placebo donors. This addresses the natural concern that four donors might be too few to reliably estimate the placebo objective and could itself explain the endpoint selection. 7 Simulation Study We compare DiD, raw-path SC, truncated Spectral SC, and the hybrid estimator, each (except DiD) with (λω,η)( _ω,η) tuned by the placebo procedure of Section 6 at a fixed, regime-specified K, across eleven raw-input regimes chosen to favor truncation, with 400400 replications per regime. The purpose of this comparison is not to identify the best available panel estimator. SDID, augmented synthetic control, generalized synthetic control, and matrix completion are established, independently benchmarked alternatives that we do not evaluate here. The purpose is narrower: to isolate the incremental effect of replacing raw-path SC’s matching metric with a truncated, or partially truncated, spectral metric, holding the donor-weighted-average architecture fixed. 7.1 Design The baseline regime uses (1) with N0=30N_0=30, T0=20T_0=20, T1=10T_1=10, αi,δt∼N(0,1) _i, _t N(0,1), ℓi∼N(0,IR) _i N(0,I_R), ftf_t a random walk (increment sd 0.30.3), εit∼N(0,0.32) _it N(0,0.3^2), constant post-treatment effect τt=2 _t=2, R=K=2R=K=2. Ten further regimes each change one feature: sparse oracle donors, clustered donor loadings, treated near a hull vertex, T0≫N0T_0 N_0 (N0=10,T0=150N_0=10,T_0=150), high-frequency noise, a weak third factor tested at K=2K=2 and K=3K=3, post-treatment factor rotation, and a treatment-correlated weak factor (elevated treated loading achieved via an in-hull convex combination concentrated on the donors with the highest loading on that factor, so representability holds by construction) tested at K=2K=2 and K=3K=3. Full generative detail for every regime is in Appendix A. Regime DiD SC (tuned) Spectral (tuned) Hybrid (tuned) Baseline 0.003 (0.013) 0.013 (0.013) 0.009 (0.021) 0.012 (0.013) Sparse oracle donors 0.019 (0.032) 0.011 (0.015) 0.016 (0.025) 0.014 (0.016) Clustered loadings −-0.060 (0.080) −-0.038 (0.024) −-0.050 (0.032) −-0.036 (0.024) Treated at hull edge −-0.085 (0.072) −-0.018 (0.027) −-0.030 (0.043) −-0.020 (0.027) T0≫N0T_0 N_0 0.063 (0.044) 0.023 (0.018) 0.034 (0.027) 0.022 (0.019) High-frequency noise 0.007 (0.017) 0.027 (0.021) 0.028 (0.026) 0.026 (0.021) Weak factor, K=2K=2 0.001 (0.014) −-0.014 (0.014) −-0.023 (0.021) −-0.012 (0.014) Weak factor, K=3K=3 0.001 (0.014) −-0.014 (0.014) −-0.022 (0.016) −-0.016 (0.014) Post-treatment rotation 0.003 (0.015) 0.023 (0.018) 0.020 (0.023) 0.022 (0.018) Confounded weak factor, K=2K=2 −-0.019 (0.079) 0.009 (0.068) −-0.013 (0.075) 0.015 (0.068) Confounded weak factor, K=3K=3 −-0.019 (0.079) 0.009 (0.068) 0.004 (0.072) 0.008 (0.069) Table 3: Bias (Monte Carlo standard error), τtrue=2 _true=2, 400400 replications per regime. Regime DiD SC (tuned) Spectral (tuned) Hybrid (tuned) Baseline 0.263 (0.014) 0.268 (0.016) 0.423 (0.019) 0.268 (0.015) Sparse oracle donors 0.646 (0.038) 0.304 (0.018) 0.509 (0.026) 0.314 (0.021) Clustered loadings 1.598 (0.099) 0.479 (0.028) 0.645 (0.036) 0.481 (0.026) Treated at hull edge 1.449 (0.090) 0.543 (0.038) 0.866 (0.048) 0.547 (0.038) T0≫N0T_0 N_0 0.884 (0.045) 0.369 (0.034) 0.548 (0.036) 0.384 (0.033) High-frequency noise 0.339 (0.013) 0.418 (0.017) 0.516 (0.022) 0.427 (0.017) Weak factor, K=2K=2 0.271 (0.014) 0.275 (0.016) 0.425 (0.032) 0.278 (0.016) Weak factor, K=3K=3 0.271 (0.014) 0.275 (0.016) 0.321 (0.021) 0.277 (0.016) Post-treatment rotation 0.297 (0.019) 0.351 (0.020) 0.455 (0.020) 0.353 (0.019) Confounded weak factor, K=2K=2 1.588 (0.068) 1.367 (0.071) 1.502 (0.069) 1.364 (0.065) Confounded weak factor, K=3K=3 1.588 (0.068) 1.367 (0.071) 1.440 (0.068) 1.383 (0.068) Table 4: RMSE (Monte Carlo standard error, 400400-fold bootstrap), 400400 replications per regime. Regime Δ RMSE(Spectral−-SC) SE Δ RMSE(Hybrid−-SC) SE Baseline 0.155 0.019 0.001 0.001 Sparse oracle donors 0.204 0.018 0.010 0.006 Clustered loadings 0.166 0.020 0.002 0.001 Treated at hull edge 0.323 0.031 0.004 0.002 T0≫N0T_0 N_0 0.180 0.021 0.015 0.005 High-frequency noise 0.098 0.020 0.009 0.006 Weak factor, K=2K=2 0.150 0.021 0.004 0.003 Weak factor, K=3K=3 0.046 0.010 0.003 0.002 Post-treatment rotation 0.103 0.019 0.002 0.003 Confounded weak factor, K=2K=2 0.135 0.029 −-0.003 0.017 Confounded weak factor, K=3K=3 0.073 0.018 0.016 0.005 Table 5: Paired RMSE differences (same simulated panels; 1,0001,000-fold paired bootstrap SE). Positive values favor SC. 7.2 What the simulation shows Truncated Spectral SC’s RMSE disadvantage is precisely estimated and large relative to its standard error in every regime. Table 5 reports paired differences on the same simulated panels rather than treating the two RMSE standard errors in Table 4 as independent, which is the more appropriate comparison here. The Spectral-vs-SC gap ranges from 0.0460.046 (SE 0.010SE\ 0.010, weak factor at K=3K=3) to 0.3230.323 (SE 0.031SE\ 0.031, hull edge) – four to eleven standard errors from zero in every regime, including the ones constructed to favor truncation. The hybrid estimator is statistically indistinguishable from tuned SC in most regimes. Nine of eleven paired Hybrid-vs-SC differences in Table 5 are under two standard errors from zero. The two exceptions – T0≫N0T_0 N_0 (0.0150.015, SE 0.0050.005) and the confounded weak factor at K=3K=3 (0.0160.016, SE 0.0050.005) – are real but an order of magnitude smaller than the corresponding Spectral-vs-SC gaps in the same regimes (0.1800.180 and 0.0730.073). Consistent with this, the placebo-tuned mixing weight selects η^=1 η=1 (raw-path matching) in a majority of replications in every regime (Table 6, unchanged from the mechanism described in Section 6), with Pr(η^=1) ( η=1) ranging from 0.670.67 to 0.910.91. Regime η¯ η Pr(η^=0) ( η=0) Pr(η^=1) ( η=1) Pr(0<η^<1) (0< η<1) Baseline 0.948 0.013 0.898 0.090 Sparse oracle donors 0.948 0.015 0.895 0.090 Clustered loadings 0.922 0.048 0.868 0.085 Treated at hull edge 0.945 0.025 0.910 0.065 T0≫N0T_0 N_0 0.788 0.123 0.705 0.173 High-frequency noise 0.821 0.060 0.673 0.268 Weak factor, K=2K=2 0.928 0.025 0.873 0.103 Weak factor, K=3K=3 0.854 0.088 0.805 0.108 Post-treatment rotation 0.940 0.020 0.898 0.083 Confounded weak factor, K=2K=2 0.832 0.098 0.760 0.143 Confounded weak factor, K=3K=3 0.788 0.150 0.730 0.120 Table 6: Empirical distribution of the hybrid estimator’s placebo-selected η η, 400400 replications per regime. The confounded weak-factor regime isolates the risk that unsupervised truncation discards a low-variance, causally relevant direction. With the relevant factor guaranteed in-hull by construction, Spectral SC’s RMSE falls from 1.5021.502 at K=2K=2 (excluding it) to 1.4401.440 at K=3K=3 (including it) – a real, partial recovery, and the paired Spectral-vs-SC gap shrinks correspondingly (from 0.1350.135 to 0.0730.073 SE 0.0180.018–0.0290.029) – while raw-path SC and the hybrid estimator, which retain at least partial weight on all directions regardless of K, are comparatively stable and remain the best performers at both values of K. This is exactly the pattern Proposition 5 predicts. We read the overall pattern as evidence for the two mechanisms formalized in Sections 3.2 and 5, not as a general indictment of the motivating hypothesis: under raw-input matching, low-dimensional balancing is underdetermined relative to the donor pool (Proposition 2) and the basis itself is contaminated by unremoved fixed effects, inflating the BLP residual in Proposition 5. Section 5 shows removing that contamination removes most of the gap on the one regime we checked; the eleven-regime study here does not. 8 Discussion 8.1 Related work Matrix completion (Athey et al., 2021), generalized synthetic control (Xu, 2017), robust synthetic control (Amjad et al., 2018), and augmented synthetic control (Ben-Michael et al., 2021) use low-rank or factor-model structure to correct or regularize SC-type estimators; SDID (Arkhangelsky et al., 2021) is analyzed under a latent factor model. Synthetic Principal Component Design (Lu et al., 2022) uses spectral ideas for the experimental-design problem of which units to treat, solved with a power-method-based combinatorial optimization – a different problem from the post-treatment score balancing studied here. A functional extension of generalized synthetic control (Shao et al., 2026) uses functional principal component scores for sparse, irregular panels within a Bayesian hierarchical model, addressing a data-sparsity problem this note does not consider. Harmonic Synthetic Control (Liu and Xu, 2026) is the closest prior work in spirit: it also introduces a continuously tunable spectral allocation governed by a single parameter. It addresses a different problem by a different mechanism, however. Harmonic Synthetic Control targets unit-specific stochastic trends in nonstationary data by splitting variation between donor matching and a residual time-series forecasting component, with its tuning parameter, selected by rolling-origin cross-validation, governing the allocation between these two branches; its spectral interpretation shows how this downweights low-frequency residual components in donor matching in favor of the forecasting branch. Our hybrid estimator introduces no forecasting branch: η modifies only the geometry of pre-treatment donor matching, by shrinking discrepancies outside a donor-estimated principal subspace, with the counterfactual always a donor-weighted average rather than a forecast. The two tuning parameters connect different pairs of objects – theirs interpolates between donor matching and extrapolation, ours between truncated and raw-path matching. Why spectral completion is a different estimator. A separate way to use the low-rank structure in (1) is to estimate the untreated panel directly via low-rank regression on donor and treated pre-treatment cells, then read off τ^t=Y1t−α^1−δ^t−L^1t τ_t=Y_1t- α_1- δ_t- L_1t for t>T0t>T_0. This is legitimate and correctly identified when the treatment effect is computed as a residual outside the fitting step, but it is a version of existing matrix-completion and interactive-fixed-effects estimators (Athey et al., 2021; Xu, 2017; Bai, 2009) rather than a variant of the donor-weighted balancing family studied here, and we do not evaluate it in Section 7. Fitting the treatment effect jointly with the low-rank component, rather than as a residual, requires an explicit restriction preventing the low-rank term from absorbing the treatment pattern itself, since otherwise the treatment parameter does not appear in a loss computed only on untreated cells. 8.2 What remains open • The preprocessing reversal is checked on one regime. Table 1 shows two-way demeaning nearly eliminates the baseline regime’s gap and flips the placebo-selected η; whether this holds across all eleven regimes, and especially in the ones with the largest raw-input gaps (hull edge, sparse donors), is not yet known. • K-selection is not addressed. We fix K by design and tune only (λω,η)( _ω,η); a principled joint selection of K together with the other hyperparameters, and an understanding of why the placebo objective is not directly comparable across K, is left open. • Rotational identification. VKV_K is identified as a subspace, not as individually labeled components. • Inference after a data-dependent basis and hyperparameters. V^K,λ^ω,η V_K, λ_ω, η are estimated from the same sample used to form τ τ; naive standard errors treating them as fixed understate uncertainty. This remains the largest methodological gap in the paper. • Staggered adoption and multiple treated units are not addressed here. 9 Conclusion We characterized two specific mechanisms governing when spectral truncation can help or hurt synthetic control: an exact equivalence to raw-path SC at full rank, a proof that low-dimensional spectral balance is generically underdetermined by N0−K−1N_0-K-1 degrees of freedom, and a finite-sample bound tying spectral balancing quality to treatment-effect bias through a best-linear-predictor residual that basis-estimation noise and omitted factors inflate. Across eleven raw-input regimes chosen to favor truncation, truncated Spectral SC had significantly higher RMSE than tuned raw-path SC throughout, and a hybrid estimator free to select anywhere between truncated and raw-path matching selected raw-path matching in a majority of replications in every regime. A robustness check on the baseline regime shows this conclusion is specific to raw-input matching: removing fixed effects before the spectral decomposition – exactly the condition our bound assumes – nearly eliminates the gap and reverses the placebo procedure’s preference. We present the paper as a diagnostic result identifying when and why spectral truncation fails under raw-path matching, alongside concrete evidence for what changes when that condition is addressed, rather than as a method ready to replace raw-path SC. Appendix A Reproducibility Details Baseline parameters. N0=30N_0=30 donors, T0=20T_0=20 pre-treatment periods, T1=10T_1=10 post-treatment periods, true factor rank R=2R=2 (or 33 in the weak-factor and confounded-weak-factor regimes), K=2K=2 unless stated otherwise. αi,δt∼iidN(0,1) _i, _t iid N(0,1); ℓi∼iidN(0,IR) _i iid N(0,I_R) for non-cluster, non-sparse, non-edge, non-confounded regimes; ft=ft−1+ξtf_t=f_t-1+ _t, ξt∼N(0,0.32IR) _t N(0,0.3^2I_R), f0=0f_0=0; εit∼iidN(0,0.32) _it iid N(0,0.3^2); constant post-treatment effect τt=2 _t=2. Regime-specific generation. Sparse: treated loading ℓ1=∑iwiℓi _1= _iw_i _i with w∼Dirichlet(0.15⋅N0)w (0.15·1_N_0). Cluster: two component centers ∼N(0,1.62IR) N(0,1.6^2I_R); each donor loading is its assigned center plus N(0,0.32IR)N(0,0.3^2I_R) noise; treated loading is the first center plus independent N(0,0.32IR)N(0,0.3^2I_R) noise. Edge: treated loading equals a uniformly random donor’s loading plus N(0,0.052IR)N(0,0.05^2I_R) noise. High-frequency noise: an additional term 0.6×(−1)t×N(0,1)0.6×(-1)^t× N(0,1) is added to every donor and treated observation at every t. Weak factor: the third factor path is scaled by 0.250.25 at every t. Post-treatment rotation: for t>T0t>T_0, the first two factor coordinates are rotated by angle θ∼Uniform(0.3,0.9)θ (0.3,0.9) radians. Confounded weak factor: the third factor is scaled by 0.150.15 for t≤T0t≤ T_0 and left unscaled for t>T0t>T_0; donors are ranked by their third-factor loading, the top third form a pool, and the treated loading is ∑iwiℓi _iw_i _i with w supported only on that pool, w∼Dirichlet()w (1) on its support. T0≫N0T_0 N_0: N0=10N_0=10, T0=150T_0=150, T1=20T_1=20, otherwise as baseline. Optimization. All balancing problems are solved by projected gradient descent on the simplex (Euclidean projection via the standard O(nlogn)O(n n) algorithm), with step size 1/L1/L for Lipschitz constant L=2λmax(X⊤MX)+2λω+10−9L=2 _ (X MX)+2 _ω+10^-9; 6060 iterations for the final applied fit, 2525 iterations during placebo tuning (verified to match a 150150-iteration solve to within 0.1%0.1\% of objective value). Tie-breaking in placebo selection: grid search iterates λω _ω in the order listed, and (for the hybrid estimator) η nested within each λω _ω, in the order listed; the first (λω,η)( _ω,η) attaining the minimum average placebo squared error is retained. Randomization and replication. Replication m of a given regime uses NumPy generator default_rng(seed0 + m) with seed0=2000, controlling all randomness in that replication (panel generation and placebo donor subsampling) from a single stream. 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