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Spectral Truncation in Synthetic Control: Weight Underdetermination and Basis-Estimation Noise
Keywords: synthetic control, panel data, low-rank factor models, spectral methods, causal inference
Synthetic control (SC) abadie2003,abadie2010 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, $T_0$-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.
\paragraph{Scope and claims.} Low-rank and spectral structure in panel causal inference is already well established. Matrix completion athey2021, generalized synthetic control xu2017, robust synthetic control amjad2018, and augmented synthetic control benmichael2021 all use low-rank or factor-model structure to correct or regularize SC-type estimators, and synthetic difference-in-differences (SDID) arkhangelsky2021 is analyzed under a latent factor model. Synthetic Principal Component Design lu2022 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 shao2026; and Harmonic Synthetic Control liu2026 introduces a continuously tunable spectral allocation for a related but distinct purpose (Section (ref)). 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.
\paragraph{Roadmap.} Section (ref) sets up notation. Section (ref) defines the spectral and hybrid estimators and proves the full-rank equivalence and weight-underdetermination results. Section (ref) gives a formal representability condition and the bias bound. Section (ref) 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 (ref) describes hyperparameter selection, including which parameters are tuned and which are fixed by design, and a placebo-sample-size sensitivity check. Section (ref) reports the eleven-regime study with paired statistics. Section (ref) discusses related work and open questions. Appendix (ref) gives reproducibility details.
Let $i=1,\dots,N$ index units and $t=1,\dots,T$ index periods, with $T_0$ pre-treatment and $T_1=T-T_0$ post-treatment periods. Unit $1$ is treated starting at $T_0+1$; units $i=2,\dots,N$ (the donor pool $\mathcal{C}$, $|\mathcal{C}|=N_0$) are never treated. $Y_{it}=Y_{it}(0)+\tau_{it}W_{it}$, with $W_{it}\in\{0,1\}$ the treatment indicator, and
with $\alpha_i,\delta_t$ unit and time fixed effects, $\ell_i\in\mathbb{R}^R$ factor loadings, $f_t\in\mathbb{R}^R$ common factors, $R\ll\min(N,T)$, $\varepsilon_{it}$ idiosyncratic noise -- the standard generating process used to motivate SC-type estimators abadie2010,xu2017,athey2021,arkhangelsky2021.
Stacking donor pre-treatment outcomes gives $Y_{0,\text{pre}}\in\mathbb{R}^{N_0\times T_0}$, approximately low rank under (ref). Let $\omega\in\Delta^{N_0}=\{\omega_i\ge0,\sum_i\omega_i=1\}$. Raw-path SC solves $\hat\omega=\operatorname*{arg\,min}_{\omega\in\Delta^{N_0}}\|y_{1,\text{pre}}-\sum_i\omega_i y_{i,\text{pre}}\|_2^2$, with $\hat Y_{1t}(0) = \sum_i\hat\omega_i Y_{it}$ for $t>T_0$.
Let $Y_{0,\text{pre}}=U\Sigma V^\top$ be the SVD of the donor pre-treatment matrix, estimated from donor pre-treatment cells only. Fix $K$ and keep the leading components, $V_K\in\mathbb{R}^{T_0\times K}$. Project every pre-treatment path onto $V_K$: $s_1=V_K^\top y_{1,\text{pre}}$, $s_i=V_K^\top y_{i,\text{pre}}$ for $i\in\mathcal{C}$, and solve
This shows the projection preserves the raw Euclidean pre-treatment balancing discrepancy at $K=T_0$; it does not, by itself, establish equivalence to any estimator built from more than this discrepancy (see the remark on SDID in Section (ref)). Any difference between (ref) and raw-path SC for $K<T_0$ comes specifically from truncating modes.
The title's second mechanism is a precise linear-algebra fact about (ref), 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 $\omega$ (i.e.\ exact balance on the affine hull of the simplex): with donor score matrix $S_K=[s_2\ \cdots\ s_{N_0}]\in\mathbb{R}^{K\times N_0}$, exact spectral balance requires
a system of $K+1$ linear equations in $N_0$ unknowns.
Whenever $N_0>K+1$ -- generically true, since $K$ is chosen small by design while $N_0$ is the size of the available donor pool -- exact spectral balance does not pin down a unique donor weight vector: an entire $(N_0-K-1)$-dimensional family of weight vectors achieves identical pre-treatment spectral balance while generally implying different post-treatment counterfactuals, since nothing in (ref) constrains behavior outside $\mathrm{range}(V_K)$. The ridge penalty $\lambda_\omega\|\omega\|_2^2$ (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 $N_0-K-1$ 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 (ref): with $N_0=30$ and $K=2$, the affine solution set of (ref) has dimension up to $27$, 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 $K=T_0$, faces the analogous system with $T_0+1$ equations rather than $K+1$; whenever $T_0$ is not much smaller than $N_0$ this system is far less underdetermined, and in the $T_0\gg N_0$ regime of Section (ref) it can be overdetermined, which is one reason raw-path matching is implicitly well regularized in that regime without any explicit penalty.
A less abrupt alternative to truncation assigns discarded directions a strictly positive but shrunk weight $\eta\in[0,1]$, using $\Pi_K=V_KV_K^\top$:
By Proposition (ref)'s argument, $\eta=1$ recovers raw-path SC exactly for any $K$; $\eta=0$ recovers truncated Spectral SC (ref) exactly. Values $\eta\in(0,1)$ interpolate, and by Proposition (ref), moving $\eta$ above $0$ directly reduces the dimension of directions left unconstrained by the balancing loss -- $\eta>0$ constrains all $T_0$ directions, just with unequal weight, rather than leaving $T_0-K$ of them entirely unconstrained. This turns truncation into a continuous parameter tunable by the same placebo procedure used for the other estimators (Section (ref)).
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.
Smaller $(\delta_s,\delta_\ell)$ is a stronger, more favorable condition; $\delta_s=\delta_\ell=0$ recovers exact spectral balance with exact loading representability. Under Assumption (ref), the loading-mismatch term in Proposition (ref) below is bounded by $\delta_\ell\|f_t\|_2$ at the oracle $\omega^\star$; Proposition (ref) in Section (ref) relates the achieved estimator's $\hat\omega$ (which need not equal $\omega^\star$) to the observed spectral residual $\|s_1-\sum_i\hat\omega_is_i\|_2$, which is what an analyst can actually compute.
The converse fails whenever the residual has a nonzero component outside $\mathrm{range}(V_K)$, so Assumption (ref) at a given $\delta_s$ is different from, not proven weaker than, the corresponding full-path condition.
Proposition (ref) 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.
This inequality is close to algebraic once Definition (ref) 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 $\|A_K\|_{\mathrm{op}}$ times the achieved spectral residual -- exactly the quantity (ref) minimizes -- and (ii) that the BLP residual term $r_i$, which absorbs both omitted-factor error ($K<R$) and basis-estimation error, is invisible to the spectral objective and not controlled by it at all. When $r_i$ is large, (ref) is uninformative regardless of how well (ref) is solved -- consistent with the simulation evidence in Sections (ref)--(ref).
Lemma (ref) assumes unit and time fixed effects have already been removed before the SVD. The practical estimator in Section (ref), and the simulation study as originally run, applies the SVD to raw pre-treatment paths. This matters: if $\alpha_i$ varies substantially across donors, the leading singular directions of $Y_{0,\text{pre}}$ can be dominated by cross-unit level differences rather than by the interactive factor structure in (ref), in which case $V_K$ estimated from raw data is a poor estimate of the population factor subspace and the BLP residual $r_i$ in Proposition (ref) 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:
Table (ref) shows a reversal, not merely an attenuation. Under raw input, Spectral SC's RMSE exceeds tuned SC's by $0.148$ ($\mathrm{SE}\ 0.025$), about six standard errors -- consistent with the main study in Section (ref), which uses raw input throughout. Under unit-demeaning the gap falls to $0.025$ ($\mathrm{SE}\ 0.012$), about two standard errors. Under two-way demeaning the gap is $0.001$ ($\mathrm{SE}\ 0.004$), statistically indistinguishable from zero, and -- more strikingly -- the placebo-tuning procedure itself switches its preference: the mean selected $\eta$ falls from $0.96$ (raw) to $0.30$ (two-way demeaned), with a majority of replications ($57\%$) now selecting $\eta=0$, full truncation, rather than $\eta=1$. This is exactly the pattern Proposition (ref) predicts: removing fixed effects before the SVD shrinks the BLP residual $r_i$ by bringing $V_K$ 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 $\|A_K\|_{\mathrm{op}}$-scaled control on its own.
We do not rerun the full eleven-regime study of Section (ref) 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 (ref) as a robustness check on the baseline regime and flag explicitly that the negative results in Section (ref) are for raw-path input specifically, not for spectral matching under every reasonable preprocessing choice.
We tune $\lambda_\omega$, and, for the hybrid estimator, $\eta$, 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 $(\lambda_\omega,\eta)$, and record squared error on $j$'s actual post-treatment path, never using post-treatment or treated-unit data. We use the grids $\lambda_\omega\in\{10^{-4},10^{-3},10^{-2},10^{-1},1\}$ and $\eta\in\{0,0.15,0.35,0.5,0.65,0.85,1\}$.
\paragraph{$K$ is fixed by design, not tuned.} Throughout this paper, the rank $K$ is a prespecified input to Algorithm (ref), 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 (ref)). We tune only $(\lambda_\omega,\eta)$ by placebo validation. We do not fold $K$ into the same procedure because the placebo objective is comparable across candidate $\lambda_\omega,\eta$ values at fixed $K$ (all evaluated in the same $T_0$-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.
\paragraph{Sensitivity to placebo sample size.} Hyperparameters above are tuned using $4$ randomly drawn placebo donors per replication. Table (ref) checks whether this is large enough by comparing $4$, $10$, and all $29$ eligible donors on the baseline regime.
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.
We compare DiD, raw-path SC, truncated Spectral SC, and the hybrid estimator, each (except DiD) with $(\lambda_\omega,\eta)$ tuned by the placebo procedure of Section (ref) at a fixed, regime-specified $K$, across eleven raw-input regimes chosen to favor truncation, with $400$ 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.
The baseline regime uses (ref) with $N_0=30$, $T_0=20$, $T_1=10$, $\alpha_i,\delta_t\sim N(0,1)$, $\ell_i\sim N(0,I_R)$, $f_t$ a random walk (increment sd $0.3$), $\varepsilon_{it}\sim N(0,0.3^2)$, constant post-treatment effect $\tau_t=2$, $R=K=2$. Ten further regimes each change one feature: sparse oracle donors, clustered donor loadings, treated near a hull vertex, $T_0\gg N_0$ ($N_0=10,T_0=150$), high-frequency noise, a weak third factor tested at $K=2$ and $K=3$, 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 $K=2$ and $K=3$. Full generative detail for every regime is in Appendix (ref).
Truncated Spectral SC's RMSE disadvantage is precisely estimated and large relative to its standard error in every regime. Table (ref) reports paired differences on the same simulated panels rather than treating the two RMSE standard errors in Table (ref) as independent, which is the more appropriate comparison here. The Spectral-vs-SC gap ranges from $0.046$ ($\mathrm{SE}\ 0.010$, weak factor at $K=3$) to $0.323$ ($\mathrm{SE}\ 0.031$, 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 (ref) are under two standard errors from zero. The two exceptions -- $T_0\gg N_0$ ($0.015$, SE $0.005$) and the confounded weak factor at $K=3$ ($0.016$, SE $0.005$) -- are real but an order of magnitude smaller than the corresponding Spectral-vs-SC gaps in the same regimes ($0.180$ and $0.073$). Consistent with this, the placebo-tuned mixing weight selects $\hat\eta=1$ (raw-path matching) in a majority of replications in every regime (Table (ref), unchanged from the mechanism described in Section (ref)), with $\Pr(\hat\eta=1)$ ranging from $0.67$ to $0.91$.
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 $1.502$ at $K=2$ (excluding it) to $1.440$ at $K=3$ (including it) -- a real, partial recovery, and the paired Spectral-vs-SC gap shrinks correspondingly (from $0.135$ to $0.073$ SE $0.018$--$0.029$) -- 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 (ref) predicts.
We read the overall pattern as evidence for the two mechanisms formalized in Sections (ref) and (ref), not as a general indictment of the motivating hypothesis: under raw-input matching, low-dimensional balancing is underdetermined relative to the donor pool (Proposition (ref)) and the basis itself is contaminated by unremoved fixed effects, inflating the BLP residual in Proposition (ref). Section (ref) shows removing that contamination removes most of the gap on the one regime we checked; the eleven-regime study here does not.
Matrix completion athey2021, generalized synthetic control xu2017, robust synthetic control amjad2018, and augmented synthetic control benmichael2021 use low-rank or factor-model structure to correct or regularize SC-type estimators; SDID arkhangelsky2021 is analyzed under a latent factor model. Synthetic Principal Component Design lu2022 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 shao2026 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 liu2026 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: $\eta$ 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.
\paragraph{Why spectral completion is a different estimator.} A separate way to use the low-rank structure in (ref) is to estimate the untreated panel directly via low-rank regression on donor and treated pre-treatment cells, then read off $\hat\tau_t=Y_{1t}-\hat\alpha_1-\hat\delta_t-\hat L_{1t}$ for $t>T_0$. 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 athey2021,xu2017,bai2009 rather than a variant of the donor-weighted balancing family studied here, and we do not evaluate it in Section (ref). 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.
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 $N_0-K-1$ 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.