Targeted Local Projections

Aleksei Nemtyrev; Otilia Boldea

REVIEW 3 major objections 4 minor 1 cited by

Targeted Local Projections

T0 review · 3 major / 4 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper proposes targeted local projections, a shrinkage estimator that combines LP and SVAR impulse responses horizon-by-horizon with a closed-form MSE-optimal weight, and a double bootstrap for coverage.

desk verdict A plausible shrinkage estimator undermined by a load-bearing proof error: the risk criterion double-counts variance and the weight doesn't converge to a constant. read the letter →

arxiv 2603.00248 v2 pith:WRBYIBKH submitted 2026-02-27 econ.EM

Aleksei Nemtyrev , Otilia Boldea This is my paper · ORCID

classification econ.EM MSC 62P2062F4062J07

keywords targetedlocalprojectionsstructuralVARimpulseresponsefunctionsmeansquarederrorshrinkagedoublebootstrapmisspecification

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a frequentist shrinkage estimator—targeted local projections (TLP)—that replaces the plain local projection impulse response at each horizon with a linear combination of the LP estimate and the corresponding structural VAR estimate. The combination weight is chosen to minimize an estimated mean-squared error, balancing the LP's small bias against the VAR's lower variance, and it has a closed form. The paper argues this reduces variance at long horizons while introducing only modest bias, and that a mean-symmetric double bootstrap restores near-nominal coverage in small samples despite the VAR's asymptotic bias. If correct, the method gives macroeconomists a computationally cheap way to obtain tighter impulse-response confidence intervals than LP alone.

What carries the argument

The key object is the shrinkage weight v(λ) = X'X/(X'X + λ), which after projecting out controls makes TLP a convex combination of the two estimators. The optimal λ (hence v) is obtained by minimizing a feasible risk criterion that estimates squared bias by the squared gap between the LP and VAR estimates plus twice the estimated variance of the combination. The closed-form weight in equation (21) and the mean-symmetric double bootstrap that centers on the bootstrap mean and uses symmetric intervals are what carry the argument.

What would settle it

Simulate the local-to-VAR DGP with bias of order T^{-1/2}, compute T(βhat_LP − βhat_VAR)^2 across many samples for large T, and check whether its distribution collapses to a point (the squared bias) or remains spread out; a spread-out limit means the weight is random and the claimed asymptotic normal approximation for TLP fails.

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Extended reading notes

Core claim

The central claim is that the TLP estimator, defined as the penalized LP solution after projecting out controls, equals v(λ) times the LP estimate plus (1−v(λ)) times the VAR estimate, with v determined by the data so as to minimize an asymptotic MSE criterion. Under local misspecification (VAR bias of order T^{-1/2}), the paper derives a unique closed-form weight, shows the estimator is asymptotically normal with bias smaller than the VAR's and variance smaller than the LP's, and provides a double-bootstrap inference procedure that maintains near-nominal coverage in simulations.

Load-bearing premise

The whole optimal-weight construction assumes that T(βhat_LP − βhat_VAR)^2 consistently estimates the squared asymptotic bias of the VAR, which requires the gap that appears in the numerator to converge to a constant rather than a random variable.

Editorial extensions

If this is right

Applied researchers can obtain materially narrower confidence intervals for long-horizon impulse responses without sacrificing nominal coverage, provided enough lags are used in the target VAR.
The closed-form weight makes the estimator as easy to compute as a ridge regression, with no numerical optimization per horizon.
The double-bootstrap procedure offers a general template for inference when an estimator is a linear combination of a robust and a biased-but-efficient estimator.
The paper's theory clarifies when it is optimal to trust the VAR: essentially when the two estimators are close relative to their variance difference.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same targeting idea could be applied to smooth local projections or to any pair of estimators with complementary bias-variance profiles, replacing the VAR target with any lower-variance estimator.
Because the weight depends on a squared gap that is only a noisy estimate of the bias in finite samples, the double bootstrap may be doing more work than the asymptotic theory suggests; a finite-sample study of the weight distribution would clarify this.
The horizon-specific weights could be exploited as a diagnostic: a sudden shift of weight toward the VAR at long horizons is a tell that LP variance is dominating, and the method's confidence intervals at those horizons reflect that trade-off.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Short version: the TLP estimator is a plausible addition to the local-projections toolkit, and the simulation work is serious. But the paper's central theoretical result—the closed-form weight that minimizes MSE—does not hold, because the risk criterion in Eq. (20) is not a consistent estimator of the MSE in Eq. (18), and the proof in Appendix B confuses second moments with squared expectations.

What's actually new: horizon-specific frequentist shrinkage of LP toward a VAR, with double-bootstrap inference. That specific combination is not in the cited literature. The positioning against BLP and SLP is fair, and the simulations cover relevant DGPs (VARMA with local misspecification, GARCH), with honest comparison across LP, SVAR, SLP, BLP, and TLP. The double-bootstrap procedure is described in enough detail to replicate.

The soft spot is load-bearing. Under the paper's own Theorem 2, √T(β̂_LP − β̂_VAR) has a non-degenerate normal limit, so T(β̂_LP − β̂_VAR)² converges to a random variable, not to the constant aBias_h². That means the weight in Eq. (21) converges to a random variable, not to the v in Eq. (24), and Theorem 5's asymptotic normality claim is unsupported. Appendix B makes the same problem worse: Eq. (46) writes T(E[β̂_LP])² as T(β*)² + asymptotic variance + o(1), which confuses E[β̂²] with (Eβ̂)²; Eq. (49) puts variance terms inside the squared-bias component; and Eq. (51) then adds variance twice. So the claim that (21) minimizes MSE is not established. This isn't a minor typo—it's the main theoretical contribution.

That said, the idea may be salvageable with a proper unbiased risk estimator (a SURE-style criterion that subtracts the variance terms correctly) and with inference that acknowledges the weight is random under local misspecification. The simulation evidence is not nothing; it suggests the method can work in practice even if the theory is currently wrong.

Who this is for: applied macroeconomists looking for a variance-reduction trick for LPs would get something from the simulations, but they shouldn't cite the theory as it stands. It deserves a serious referee—this is a relevant, well-motivated question with extensive simulations, so a desk reject would be too harsh. I'd send it out, expecting major revision or a conditional reject. A careful referee should check whether the weight-selection theory can be fixed.

Referee Report

3 major / 4 minor

Summary. The paper proposes Targeted Local Projections (TLP), a ridge-type shrinkage estimator that forms a horizon-specific linear combination of LP and VAR impulse-response estimates. The shrinkage weight is chosen by minimizing a feasible estimator of the asymptotic mean-squared error, yielding a claimed closed-form solution. The paper also proposes a Mean Symmetric Double Bootstrap (MSDB) for inference and reports simulations comparing TLP with LP, VAR, SLP, and BLP under locally misspecified VARMA DGPs.

Significance. The idea of shrinking LP toward a VAR target with data-driven, horizon-specific weights is practically appealing, and the simulation study is reasonably broad, including different sample sizes, lag lengths, misspecification strength, and GARCH errors. The double-bootstrap proposal is a useful practical ingredient. However, the central theoretical contribution is invalid: the feasible risk criterion is not a consistent estimator of the stated MSE, the closed-form weight does not minimize the population MSE, and the asymptotic-normality claim for TLP is unsupported. What remains is a heuristic estimator with simulation support, rather than the theoretically justified procedure claimed in the abstract and theorems.

major comments (3)

[Appendix B, Eqs. (46)-(51); Theorem 3, Eq. (20)] The proof of Theorem 3 confuses the second moment with the squared mean. Eq. (46) assigns Sigma_LP to T(E[beta_hat_LP])^2, but this variance term belongs to E[(beta_hat_LP)^2], not to the square of the expectation. Eq. (49) then puts variance terms inside the squared-bias component, and Eq. (51) adds the same variance block a second time through the factor 2. Consequently, the feasible criterion R_hat_h in Eq. (20) is not a consistent estimator of R_h in Eqs. (18)-(19), and Theorem 3 is not established.
[Theorem 2 vs. Theorem 3, Eqs. (20)-(21), (24)-(25)] Under Assumptions 1-2 with zeta=1/2, Theorem 2 implies sqrt(T)(beta_hat_LP - beta_hat_VAR) converges in distribution to a nondegenerate normal random variable Z. Hence T(beta_hat_LP - beta_hat_VAR)^2 converges in distribution to Z^2, not in probability to aBias_h^2. Therefore the 'bias' term in Eq. (20) and the weight in Eq. (21) converge to nondegenerate random variables, contradicting the definition of the constant v in Eq. (24). The claim in Theorem 5 that v(lambda_hat) ->_p v, and the resulting asymptotic normality of TLP, is unsupported.
[Section 2.3 and discussion after Theorem 4] Even when the VAR is correctly specified, beta_hat_LP - beta_hat_VAR is O_p(T^{-1/2}), so T times its square does not vanish. Eq. (22) then yields a random limit weight in (0,1), not zero. Thus the statements that TLP 'reduces to SVAR' under no misspecification (Section 2.3 and the discussion after Theorem 4) are contradicted by the authors' own formula. This is not a minor caveat but a direct consequence of the same missing variance correction that invalidates Theorem 3.

minor comments (4)

[Eq. (18)] The notation 'argmin T E[...]' is imprecise; the criterion is minimized over lambda_h, so it should be min_{lambda_h} or argmin_{lambda_h} of the expectation.
[Theorem 5] The statement uses v(lambda_hat) without subscript h, and beta^* without subscript h, although all objects are horizon-specific. Please make the notation consistent.
[Section 5] The DGP labeled 'VARMA(1,100)' is referred to as having the 'highest degree of misspecification', but the truncation of the infinite MA polynomial at 100 lags is not explicitly defined in the text; please clarify.
[Section 4.1] The bootstrap validity discussion cites Brüggemann et al. (2016) for the correct specification case, but no formal theorem is given for the double bootstrap under the local misspecification used in the main text; a precise statement of the regularity conditions and convergence result would help.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: TLP's derivation rests on external asymptotics and standard risk estimation, not on self-citation or definitional equivalence.

full rationale

The derivation chain is self-contained conditional on external results. The local-to-VAR DGP, LP double robustness, VAR local bias, and joint CLT are taken from Olea et al. (2026) and Brüggemann et al. (2016); those authors have no overlap with Nemtyrev and Boldea, so the self-citation and imported-uniqueness patterns do not apply. The TLP weight is selected by minimizing a plug-in estimate of the TLP MSE, which is standard empirical risk minimization rather than a definitional equivalence, and the paper provides Monte Carlo comparisons against LP, VAR, SLP, and BLP. The closest issue is Theorem 3: the proof appears to conflate E[βhat^2] with (E[βhat])^2, and the gap term T(βhat_LP − βhat_VAR)^2 is not a consistent estimator of aBias_h^2 under the paper's own Theorem 2. That is a substantive correctness problem, but it is an estimation-consistency flaw, not circular reasoning: the feasible criterion is a (mistaken) plug-in estimator of the target MSE, not the target MSE defined in terms of the fitted value. The only unsupported assertion of the 'unreported simulations' kind (Section 2.4.2) concerns the choice of SLP's smoothing criterion and is not load-bearing for the TLP derivation. Therefore no circular step can be quoted, and the score is 0.

Assumptions & free parameters 6 free parameters · 6 assumptions · 0 invented entities

No new physical or structural entities are introduced. The load-bearing assumptions are the local-to-VAR framework, mixing conditions, and several external bootstrap-validity results from the cited literature. The shrinkage weight is a data-fitted parameter, and the simulation design adds several hand-chosen tuning constants.

free parameters (6)

shrinkage parameter lambda_h = data-dependent, per horizon
Central tuning parameter; chosen to minimize the estimated risk criterion (20). Because the criterion is inconsistently derived, this is effectively a fitted weight rather than a provably optimal one.
LP lag length p = 10 in simulations
Hand-chosen in the Monte Carlo; results may depend on it, though LP is lag-augmented.
VAR/target lag length q = 1 or 8 in simulations
Hand-chosen; Section 5.1 notes q=1 causes undercoverage, so q=8 is load-bearing for coverage claims.
First-level bootstrap replications B1 = 200
Simulation choice; affects bootstrap variance estimates.
Second-level bootstrap replications B2 = 100
Simulation choice.
Block length ℓ = T^{1/3}
Resampling block length, taken from Gonçalves and Kilian (2004); not varied.

assumptions (6)

domain assumption Assumption 1: local-to-VAR DGP with zeta>1/4, stationary solution, martingale difference shocks, Cholesky identification
Defines the entire theoretical environment; if zeta<=1/4 both estimators have larger biases and the paper's trade-off is void.
domain assumption Assumption 2: alpha-mixing of size -delta/(2+delta)
Needed for the CLT in Theorem 2 and for bootstrap validity.
domain assumption Olea et al. (2026) Propositions 3.1/3.2: LP is asymptotically unbiased, VAR has bias T^{-zeta} aBias_h
Theorem 1 is stated as a straightforward generalization; the paper does not rederive the central bias-variance trade-off.
domain assumption Moving-block bootstrap validity under conditional heteroskedasticity (Brüggemann et al. 2016)
Needed for the first-level resampling scheme used for all estimators.
domain assumption Double bootstrap restores coverage in the presence of asymptotic bias (Cavaliere et al. 2024)
Needed for the MSDB coverage claim.
standard math VAR is more efficient than LP in the correctly specified iid Gaussian case
Standard ML efficiency argument used to motivate shrinking toward VAR.

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Cite this review

Pith. "Pith review of Targeted Local Projections." pith.science (2026). https://pith.science/paper/WRBYIBKH

@misc{pith2026260300248,
  author       = {Pith},
  title        = {Pith review of: Targeted Local Projections},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WRBYIBKH}},
  note         = {Machine review of arXiv:2603.00248}
}

read the original abstract

Local projection (LP) and structural vector autoregression (SVAR) are commonly employed to estimate dynamic causal effects of macroeconomic policies at multiple horizons. With enough lags as controls, LP estimators have little bias but their variance can increase with the horizon due to accumulating additional shocks. Because they typically employ fewer lags or suffer from local misspecification, SVAR estimators typically incur higher bias, but their variance decreases with the horizon due to exponentiation. We propose to target the LP estimators towards their SVAR counterparts - constructed with fewer lags than LP at each horizon - to reduce their variance at the cost of incurring some bias. The resulting targeted LP estimator is a linear combination of the LP and SVAR estimators. We propose choosing this linear combination optimally to minimize the mean-squared error of the new estimator. Our simulations show that, under a locally misspecified SVAR model, targeting substantially reduces the LP variance at longer horizons while maintaining near-nominal coverage in small samples when a double bootstrap is employed.

Figures

Figures reproduced from arXiv: 2603.00248 by the authors.

**Figure 1.** Coverage for T = 200 (left) and T = 800 (right), DGP = VARMA(1,100). Red solid line: MSDB with q = 8. Purple solid line (alternative method): LP, VAR and TLP use double bootstrap centering around VAR estimates; SLP uses HAC with undersmoothing (Barnichon and Brownlees, 2019). Dashed lines for q = 1. Monte Carlo with 1000 replications, 200 × 100 bootstrap samples. 21 [PITH_FULL_IMAGE:figures/full_fig_p022_1.png] view at source ↗

**Figure 2.** Coverage, length, bias, standard deviation, and RMSE for [PITH_FULL_IMAGE:figures/full_fig_p025_2.png] view at source ↗

**Figure 3.** Coverage, length, bias, standard deviation, and RMSE for [PITH_FULL_IMAGE:figures/full_fig_p026_3.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: Coverage, length, bias, standard deviation, and RMSE for [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

**Figure 5.** Figure 5: Coverage, length, bias, standard deviation, and RMSE for [PITH_FULL_IMAGE:figures/full_fig_p033_5.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

Works this paper leans on

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Pith tools

Reviewed August 2, 2026 · model on record in the stance chip above.

[1] [1]

Barnichon, R. and C. Brownlees (2019). Impulse response estimation by smooth local projec- tions.Review of Economics and Statistics 101(3), 522–530. Brüggemann, R., C. Jentsch, and C. Trenkler (2016). Inference in VARs with conditional heteroskedasticity of unknown form.Journal of econometrics 191(1), 69–85

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Gonçalves, M

Cavaliere, G., S. Gonçalves, M. Ø. Nielsen, and E. Zanelli (2024). Bootstrap inference in the presence of bias.Journal of the American Statistical Association 119(548), 2908–2918

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2003

[4] [4]

Dufour, J.-M. and E. Renault (1998). Short run and long run causality in time series: Theory. Econometrica, 1099–1125

1998

[5] [5]

El-Shagi, M. (2019). A simple estimator for smooth local projections.Applied Economics Letters 26(10), 830–834

2019

[6] [6]

Ferreira, L. N., S. Miranda-Agrippino, and G. Ricco (2023). Bayesian local projections.Review of Economics and Statistics, 1–45. Gonçalves, S.andL.Kilian(2004). Bootstrappingautoregressionswithconditionalheteroskedas- ticity of unknown form.Journal of econometrics 123(1), 89–120. 27 González-Casasús, O. and F. Schorfheide (2025). Misspecification-robust s...

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[7] [7]

Hall, P. (1988). On symmetric bootstrap confidence intervals.Journal of the Royal Statistical Society Series B: Statistical Methodology 50(1), 35–45

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Hansen, B. E. (2016). Efficient shrinkage in parametric models.Journal of Econometrics 190(1), 115–132. Jordà, Ò. (2005). Estimation and inference of impulse responses by local projections.American Economic Review 95(1), 161–182. Jordà, Ò. (2023). Local projections for applied economics.Annual Review of Economics 15(1), 607–631. Jordà, Ò. and A. M. Taylor...

2016