How Well Are State-Dependent Local Projections Capturing Nonlinearities?
Pith reviewed 2026-05-15 22:10 UTC · model grok-4.3
The pith
A local projection specification augmented with squared shocks and state interactions best recovers true nonlinear impulse responses.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In quadratic VAR environments that mimic pruned second-order DSGE solutions, the specification augmenting linear local projections with a squared shock term and interactions with state proxies most closely matches the true population impulse responses, outperforming linear and basic state-dependent alternatives.
What carries the argument
The augmented local projection regression that adds a squared shock term and shock-state proxy interaction terms to capture higher-order and state-dependent nonlinearities.
If this is right
- State-dependent local projections using sign interactions capture asymmetric effects, but gains appear mainly for large shocks.
- Interactions with observable state proxies capture state dependence, yet performance depends on how well the proxies reflect the relevant states.
- The proposed augmented specification admits valid estimation and inference procedures for its nonlinear terms.
- In a monetary policy application, shocks show state dependence while higher-order effects differ across outcome variables.
Where Pith is reading between the lines
- Applied researchers could adopt the augmented specification to obtain more accurate impulse responses when nonlinearities are suspected.
- Policy evaluations might reveal stronger or weaker effects depending on the state or shock size once higher-order terms are included.
- The approach invites testing whether similar augmentations improve performance in other classes of nonlinear models beyond the quadratic laboratory.
Load-bearing premise
The quadratic VAR serves as a representative laboratory for the nonlinear environments encountered in practice, and the chosen state proxies are of sufficient quality to capture relevant state dependence.
What would settle it
Simulating data from a different nonlinear process, such as a threshold VAR or higher-order perturbation solution, where the proposed augmented specification no longer matches true impulse responses more closely than linear or basic state-dependent alternatives.
read the original abstract
We use quadratic vector autoregressions, motivated by pruned second-order perturbation solutions to DSGE models, as a laboratory to evaluate how well popular local projection (LP) specifications recover true impulse responses in nonlinear environments. We derive closed-form population impulse responses under each specification and compare them to truth. Linear LP fails to capture nonlinearities when the shock is symmetrically distributed. State-dependent LP specifications capture distinct aspects of nonlinearity: interacting the shock with its sign captures asymmetric effects, while interacting the shock with observable state proxies captures state dependence. However, their gains over linear LP are concentrated in tail shocks or states, and for the latter depend on proxy quality. Our proposed specification -- augmenting linear LP with a squared shock term and shock-state proxy interactions -- best approximates true responses. We also establish valid estimation and inference procedures for this specification. In a monetary policy application, we find state dependence, while higher-order effects differ across outcomes.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper evaluates how well linear and state-dependent local projection (LP) specifications recover true nonlinear impulse responses, using quadratic vector autoregressions (motivated by pruned second-order DSGE perturbations) as a laboratory. Closed-form population IRFs are derived for each LP specification and compared directly to the quadratic VAR truth. Linear LP fails to capture nonlinearities under symmetric shocks; sign-interaction and state-proxy LPs capture distinct aspects but with limited gains except in tails; the proposed augmentation (linear LP plus squared shock term and shock-state proxy interactions) performs best overall. Valid estimation/inference is established for the proposed spec, and a monetary policy application finds evidence of state dependence with varying higher-order effects.
Significance. If the central comparisons hold, the paper supplies concrete guidance on LP specification choices for nonlinear macro settings, with the analytical closed-form derivations providing a clean benchmark free of sampling error and the monetary policy exercise offering an out-of-sample illustration. The explicit ranking of specifications and the proposed augmentation represent a practical contribution to the growing literature on nonlinear local projections.
major comments (2)
- [Simulation Design / Laboratory] The laboratory relies exclusively on quadratic VARs as the DGP. Because the proposed LP augmentation (squared shock term plus state-proxy interactions) directly mirrors the quadratic structure, the reported superiority may be partly by construction and may not generalize to other common nonlinear forms such as threshold, regime-switching, or non-quadratic higher-order DGPs. No systematic robustness checks across alternative DGPs are provided.
- [State-Dependent LP Results] The claim that the proposed specification 'best approximates true responses' rests on the maintained quality of the chosen state proxies. While the paper notes that proxy quality matters, the quantitative sensitivity of the ranking to proxy measurement error or alternative proxies is not systematically quantified in the Monte Carlo or application sections.
minor comments (2)
- [Proposed Specification] Notation for the squared shock term and the precise definition of the state-proxy interaction should be made fully explicit in the main text (currently appears only in the appendix derivation).
- [Application] The monetary policy application would benefit from a brief table reporting the estimated coefficients on the squared shock and interaction terms to allow readers to gauge economic magnitude.
Simulated Author's Rebuttal
We thank the referee for the constructive and detailed comments. We address each major comment below and indicate the planned revisions.
read point-by-point responses
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Referee: [Simulation Design / Laboratory] The laboratory relies exclusively on quadratic VARs as the DGP. Because the proposed LP augmentation (squared shock term plus state-proxy interactions) directly mirrors the quadratic structure, the reported superiority may be partly by construction and may not generalize to other common nonlinear forms such as threshold, regime-switching, or non-quadratic higher-order DGPs. No systematic robustness checks across alternative DGPs are provided.
Authors: We agree that the quadratic VAR laboratory is a deliberate choice motivated by pruned second-order DSGE perturbations, which permits closed-form population IRFs and a clean analytical benchmark. While this setup aligns with the proposed augmentation, we acknowledge that performance in threshold, regime-switching, or other higher-order DGPs remains unexamined. In the revision we will add a new subsection discussing the scope and limitations of the quadratic laboratory and will include a limited robustness exercise with a simple threshold DGP to illustrate how the ranking of specifications changes outside the quadratic case. revision: partial
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Referee: [State-Dependent LP Results] The claim that the proposed specification 'best approximates true responses' rests on the maintained quality of the chosen state proxies. While the paper notes that proxy quality matters, the quantitative sensitivity of the ranking to proxy measurement error or alternative proxies is not systematically quantified in the Monte Carlo or application sections.
Authors: We thank the referee for highlighting this point. Although the manuscript already notes that proxy quality affects results, we agree that systematic quantification of sensitivity to measurement error and alternative proxies is missing. In the revised version we will expand the Monte Carlo section with experiments that introduce noisy state proxies and alternative proxy definitions, reporting how the relative performance of the specifications changes. We will also add corresponding sensitivity checks using different state variables in the monetary policy application. revision: yes
Circularity Check
No significant circularity; evaluation uses independent quadratic VAR laboratory
full rationale
The paper derives closed-form population impulse responses for each LP specification and compares them directly to the true responses generated from the quadratic VAR DGP. This constitutes a standard Monte Carlo-style evaluation against an externally specified truth rather than any parameter fitting or self-referential reduction. The proposed augmented LP (with squared shock and interactions) is tested for approximation quality, but its superiority is shown via explicit comparison to the independent benchmark, not by construction within the same equations. No load-bearing step reduces to a self-citation, fitted input renamed as prediction, or ansatz smuggled via prior work. The derivation chain remains self-contained against the simulated quadratic benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption Quadratic VAR approximates pruned second-order perturbation solutions to DSGE models
discussion (0)
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