Quasi-Bayesian Local Projection Instrumental-Variables Method: Application to Renewable Energy and Electricity Prices

Masahiro Tanaka

arxiv: 2605.15966 · v1 · pith:THEJTLRNnew · submitted 2026-05-15 · 💰 econ.EM · stat.ME

Quasi-Bayesian Local Projection Instrumental-Variables Method: Application to Renewable Energy and Electricity Prices

Masahiro Tanaka This is my paper

Pith reviewed 2026-05-19 18:07 UTC · model grok-4.3

classification 💰 econ.EM stat.ME

keywords local projectioninstrumental variablesquasi-Bayesianimpulse responsesroughness penaltyGMM estimationrenewable energyelectricity prices

0 comments

The pith

A roughness-penalty prior smooths LP-IV impulse responses without changing their first-order asymptotics.

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

The paper introduces a quasi-Bayesian method for estimating local projection instrumental variables models. It forms a quasi-posterior from the GMM objective and adds a roughness penalty to encourage smooth impulse responses across time horizons. This keeps the estimator consistent with standard LP-IV in large samples but reduces estimation error in smaller samples. The approach also supports joint inference across horizons using simultaneous bands. An application shows its use in analyzing how renewable energy affects electricity prices.

Core claim

This paper develops a quasi-Bayesian approach to local projection instrumental-variables estimation by constructing a moment-based quasi-posterior from the GMM objective and applying a roughness-penalty prior to smooth impulse responses over horizons. The procedure preserves the first-order asymptotic properties of traditional LP-IV estimators while improving finite-sample performance and enabling joint inference.

What carries the argument

The roughness-penalty prior on the impulse response coefficients, which penalizes non-smooth variations across horizons to stabilize estimates.

If this is right

Simulations show lower root mean squared error than standard GMM, especially at medium and longer horizons.
The method supports simultaneous confidence bands for inference across multiple horizons.
It can be applied to empirical questions such as the effects of renewable energy on electricity prices in real markets.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Similar regularization might help in other instrumental variable settings with sequential parameters.
Extensions could incorporate different penalty forms tailored to specific economic theories.
Testing on larger datasets or alternative instruments would further validate the finite-sample gains.

Load-bearing premise

The roughness-penalty prior can be added without affecting the first-order asymptotic distribution of the LP-IV estimator.

What would settle it

If a Monte Carlo simulation with known data-generating process shows no reduction in root mean squared error at longer horizons compared to standard GMM, the claimed finite-sample improvement would be falsified.

Figures

Figures reproduced from arXiv: 2605.15966 by Masahiro Tanaka.

**Figure 1.** Figure 1: Estimated IRF (a) DK1, wind h 0 1 2 3 4 5 6 7 -15 -10 -5 0 5 10 Estimated Pointwise 90% interval Simultaneous 90% band (b) DK1, solar h 0 1 2 3 4 5 6 7 -2 -1 0 1 2 3 4 (c) DK2, wind h 0 1 2 3 4 5 6 7 -8 -6 -4 -2 0 2 4 6 (d) DK2, solar h 0 1 2 3 4 5 6 7 -2 -1 0 1 2 3 4 24 [PITH_FULL_IMAGE:figures/full_fig_p024_1.png] view at source ↗

read the original abstract

This paper introduces a quasi-Bayesian approach for local projection instrumental-variables (LP-IV) estimation. It builds a moment-based quasi-posterior using the generalized method of moments (GMM) objective and applies a roughness-penalty prior to smooth impulse responses over different horizons. The approach maintains the key first-order features of traditional LP-IV methods, while enhancing stability in finite samples and allowing for joint inference through simultaneous bands. Simulations indicate that this regularization decreases root mean squared error compared to standard GMM, especially at medium and longer horizons. An application to Danish electricity markets highlights the method's practical usefulness.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper adds a roughness penalty to a GMM-based quasi-posterior for LP-IV and reports lower RMSE in simulations, but the key asymptotic claim needs an explicit rate argument on the penalty term.

read the letter

The central point is that this paper takes the standard LP-IV estimator, builds a quasi-posterior from its GMM objective, and adds an explicit roughness penalty across horizons to reduce noise in the impulse responses. The simulations show lower root mean squared error than plain GMM, especially at medium and longer horizons, and the Danish electricity application illustrates how the method produces joint bands that look more stable in practice. That combination of quasi-Bayesian weighting and the penalty is not a routine extension of existing local-projection work, so the methodological step is real even if incremental. The construction stays close to the original moments, which keeps the first-order consistency claim plausible on paper. The main soft spot is the rate condition. The abstract states that the procedure preserves the key asymptotic features of ordinary LP-IV, but this only holds if the penalty strength shrinks fast enough that it does not affect the leading score and Hessian terms. The paper needs to show that the chosen tuning parameter satisfies something like lambda_n = o_p(sqrt(n)) uniformly over the horizons considered; without that step the equivalence is asserted rather than derived. The simulation design and exact form of the penalty are also not fully visible from the abstract, so it is hard to judge how sensitive the RMSE gains are to those choices. This is the kind of paper that applied macro and energy econometricians would find useful when they already run LP-IV on modest samples and want smoother responses plus joint inference. It is not a fundamental shift in what can be identified, but it is a practical regularization device. I would send it to referees if the rate condition is worked out cleanly in the proofs and the simulation details are transparent.

Referee Report

1 major / 2 minor

Summary. The paper introduces a quasi-Bayesian local projection instrumental-variables (LP-IV) estimator. It constructs a moment-based quasi-posterior from the GMM objective function and incorporates a roughness-penalty prior to smooth impulse responses over horizons. The authors claim that the procedure maintains the first-order asymptotic features of standard LP-IV methods, improves finite-sample stability (with simulations showing lower RMSE especially at medium and longer horizons), enables joint inference via simultaneous bands, and demonstrate its usefulness via an application to renewable energy shocks and Danish electricity prices.

Significance. If the first-order asymptotic equivalence holds and the reported finite-sample gains are robust, the method could offer a practical regularization tool for LP-IV applications in volatile settings such as energy markets. The joint-inference capability addresses a frequent practical limitation of horizon-by-horizon LP-IV. Credit is given for grounding the quasi-posterior directly in the GMM objective and for providing an empirical illustration in Danish electricity data.

major comments (1)

[Asymptotic analysis] The central claim that the quasi-Bayesian procedure 'maintains the key first-order features of traditional LP-IV methods' and remains consistent for the same parameters rests on the roughness-penalty prior not disturbing the leading term of the GMM objective. This holds only if the penalty strength λ_n satisfies λ_n = o_p(√n) (or a similar rate) so that its contribution to the score and Hessian vanishes in the limit. The abstract asserts this property, but the argument is least secure precisely where the paper must show that the chosen (possibly data-driven) penalty satisfies the required rate uniformly over horizons; without that step the first-order equivalence is an assumption rather than a derived result. (See the asymptotic analysis and the definition of the quasi-posterior.)

minor comments (2)

[Simulations] The simulation design, data-generating processes, and exact construction of the roughness penalty should be described with greater precision so that readers can replicate the reported RMSE reductions.
[Application] Clarify how the tuning parameter for the roughness penalty is selected in the empirical application and whether results are sensitive to its value.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive and detailed comments. We address the major comment below and indicate the changes we will make to strengthen the manuscript.

read point-by-point responses

Referee: [Asymptotic analysis] The central claim that the quasi-Bayesian procedure 'maintains the key first-order features of traditional LP-IV methods' and remains consistent for the same parameters rests on the roughness-penalty prior not disturbing the leading term of the GMM objective. This holds only if the penalty strength λ_n satisfies λ_n = o_p(√n) (or a similar rate) so that its contribution to the score and Hessian vanishes in the limit. The abstract asserts this property, but the argument is least secure precisely where the paper must show that the chosen (possibly data-driven) penalty satisfies the required rate uniformly over horizons; without that step the first-order equivalence is an assumption rather than a derived result. (See the asymptotic analysis and the definition of the quasi-posterior.)

Authors: We appreciate the referee pointing out the need for a more explicit rate condition on the penalty parameter. The current draft establishes first-order asymptotic equivalence under the maintained condition that λ_n = o_p(√n), but we agree that verifying this rate for our (data-driven) choice of λ_n, uniformly over horizons, would make the result fully derived rather than conditional. In the revised version we will add a supporting lemma that derives the required rate under standard regularity conditions on the LP-IV moments and the roughness penalty, thereby closing this gap. revision: yes

Circularity Check

0 steps flagged

Quasi-posterior from GMM objective plus explicit roughness penalty; no load-bearing reduction to inputs by construction

full rationale

The paper defines the quasi-posterior directly from the GMM moment objective and adds the roughness-penalty prior as an explicit additive regularization term. This construction preserves the first-order asymptotics of standard LP-IV under a rate condition on the penalty strength, but the procedure itself does not equate any derived quantity to its inputs by definition or by renaming a fitted value as a prediction. No self-citation chain or uniqueness theorem is invoked to force the central result, and the derivation remains independent of the target impulse responses. The approach is therefore self-contained against external benchmarks with only minor regularization, yielding a low circularity score.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on the standard GMM moment conditions for LP-IV together with the assumption that a roughness penalty can be added while preserving first-order asymptotics. No new entities are postulated.

free parameters (1)

roughness penalty tuning parameter
The prior requires a smoothing parameter whose value must be chosen or estimated; its selection rule is not specified in the abstract.

axioms (1)

domain assumption GMM moment conditions can be used to construct a valid quasi-posterior
The paper explicitly builds the quasi-posterior from the GMM objective function.

pith-pipeline@v0.9.0 · 5621 in / 1390 out tokens · 56363 ms · 2026-05-19T18:07:07.789261+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We assign the conditional Gaussian prior θⱼ|τⱼ ∼ N(0, τⱼ² Q⁻¹) where Q = D^⊤D + 8/ρ² I (Section 3.3)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The quasi-posterior mean is asymptotically equivalent to the standard GMM estimator under local prior-negligibility (Section 3.2)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

44 extracted references · 44 canonical work pages

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[1] [1]

& Brownlees, C

Barnichon, R. & Brownlees, C. (2019). Impulse response estimation by smooth local projections. Review of Economics and Statistics , 101(3), 522--530

work page 2019

[2] [2]

& Matthes, C

Barnichon, R. & Matthes, C. (2018). Functional approximation of impulse responses. Journal of Monetary Economics , 99, 41--55

work page 2018

[3] [3]

G., Holmes, C

Bissiri, P. G., Holmes, C. C., & Walker, S. G. (2016). A general framework for updating belief distributions. Journal of the Royal Statistical Society Series B: Statistical Methodology , 78(5), 1103--1130

work page 2016

[4] [4]

Borenstein, S. (2002). The trouble with electricity markets: Understanding C alifornia's restructuring disaster. Journal of Economic Perspectives , 16(1), 191--211

work page 2002

[5] [5]

& Hong, H

Chernozhukov, V. & Hong, H. (2003). An MCMC Approach to Classical Estimation . Journal of Econometrics , 115(2), 293--346

work page 2003

[6] [6]

Cl \`o , S., Cataldi, A., & Zoppoli, P. (2015). The merit-order effect in the I talian power market: The impact of solar and wind generation on national wholesale electricity prices. Energy Policy , 77, 79--88

work page 2015

[7] [7]

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

work page 2019

[8] [8]

N., Miranda-Agrippino, S., & Ricco, G

Ferreira, L. N., Miranda-Agrippino, S., & Ricco, G. (2025). Bayesian local projections. Review of Economics and Statistics , 107(5), 1424--1438

work page 2025

[9] [9]

Goh, G. & Yu, J. (2022). Causal Inference with Some Invalid Instrumental Variables: A Quasi- B ayesian Approach . Oxford Bulletin of Economics and Statistics , 84(6), 1432--1451

work page 2022

[10] [10]

& Vasilakos, N

Green, R. & Vasilakos, N. (2011). The long-term impact of wind power on electricity prices and generating capacity. In 2011 IEEE Power and Energy Society General Meeting

work page 2011

[11] [11]

Hansen, L. P. (1982). Large Sample Properties of Generalized Method of Moments Estimators . Econometrica , 50(4), 1029--1054

work page 1982

[12] [12]

Hirth, L. (2013). The market value of variable renewables: The effect of solar wind power variability on their relative price. Energy Economics , 38, 218--236

work page 2013

[13] [13]

Hong, H., Li, H., & Li, J. (2021). BLP estimation using L aplace transformation and overlapping simulation draws. Journal of Econometrics , 222(1), 56--72

work page 2021

[14] [14]

Inoue, A., Jord \`a , \`O ., & Kuersteiner, G. M. (2026). Inference for local projections. Econometrics Journal , 29(1), 2--26

work page 2026

[15] [15]

& Tanner, M

Jiang, W. & Tanner, M. A. (2008). Gibbs posterior for variable selection in high-dimensional classification and data mining. Annals of Statistics , 36(5), 2207--2231

work page 2008

[16] [16]

Jord \`a , \`O . (2005). Estimation and inference of impulse responses by local projections. American Economic Review , 95(1), 161--182

work page 2005

[17] [17]

Jord \`a , \`O . (2023). Local Projections for Applied Economics . Annual Review of Economics , 15(1), 607--631

work page 2023

[18] [18]

Jord \`a , \`O ., Schularick, M., & Taylor, A. M. (2015). Betting the House . Journal of International Economics , 96, S2--S18

work page 2015

[19] [19]

& Taylor, A

Jord \`a , \`O . & Taylor, A. M. (2025). Local Projections . Journal of Economic Literature , 63(1), 59--110

work page 2025

[20] [20]

Joskow, P. L. (2011). Comparing the costs of intermittent and dispatchable electricity generating technologies. American Economic Review , 101(3), 238--241

work page 2011

[21] [21]

Kim, J. (2002). Limited Information Likelihood and B ayesian Analysis . Journal of Econometrics , 107(1-2), 175--193

work page 2002

[22] [22]

& Brezger, A

Lang, S. & Brezger, A. (2004). Bayesian P -splines. Journal of Computational and Graphical Statistics , 13(1), 183--212

work page 2004

[23] [23]

J., Stock, J

Lazarus, E., Lewis, D. J., Stock, J. H., & Watson, M. W. (2018). HAR inference: Recommendations for practice. Journal of Business and Economic Statistics , 36(4), 541--559

work page 2018

[24] [24]

& Jiang, W

Li, C. & Jiang, W. (2016). On oracle property and asymptotic validity of B ayesian generalized method of moments. Journal of Multivariate Analysis , 145, 132--147

work page 2016

[25] [25]

& Jiang, W

Liao, Y. & Jiang, W. (2011). Posterior consistency of nonparametric conditional moment restricted models. Annals of Statistics , (pp.\ 3003--3031)

work page 2011

[26] [26]

Lindgren, F., Rue, H., & Lindstr \"o m, J. (2011). An explicit link between G aussian fields and G aussian M arkov random fields: The stochastic partial differential equation approach. Journal of the Royal Statistical Society, Series B: Statistical Methodology , 73(4), 423--498

work page 2011

[27] [27]

& Schmidt, D

Makalic, E. & Schmidt, D. F. (2015). A simple sampler for the horseshoe estimator. IEEE Signal Processing Letters , 23(1), 179--182

work page 2015

[28] [28]

Montiel Olea, J. L. & Plagborg-M ller, M. (2019). Simultaneous Confidence Bands: Theory, Implementation, and an Application to SVAR s . Journal of Applied Econometrics , 34(1), 1--17

work page 2019

[29] [29]

Montiel Olea, J. L. & Plagborg-M ller, M. (2021). Local Projection Inference Is Simpler and More Robust Than You Think . Econometrica , 89(4), 1789--1823

work page 2021

[30] [30]

Newey, W. K. & West, K. D. (1987). A Simple, Positive Semi-definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix . Econometrica , 55(3), 703--08

work page 1987

[31] [31]

& Staffell, I

Pfenninger, S. & Staffell, I. (2016). Long-term patterns of E uropean PV output using 30 years of validated hourly reanalysis and satellite data. Energy , 114, 1251--1265

work page 2016

[32] [32]

& Stockwell, T

Piger, J. & Stockwell, T. (2025). Differences from Differencing: Should Local Projections with Observed Shocks Be Estimated in Levels or Differences? Available at SSRN: https://ssrn.com/abstract=4530799 or http://dx.doi.org/10.2139/ssrn.4530799

work page doi:10.2139/ssrn.4530799 2025

[33] [33]

Polson, N. G. & Scott, J. G. (2012). On the half- C auchy prior for a global scale parameter. Bayesian Analysis , 7(4), 887--902

work page 2012

[34] [34]

Ramey, V. A. (2016). Macroeconomic shocks and their propagation. In J. Taylor & H. Uhlig (Eds.), Handbook of Macroeconomics , volume 2A chapter 2, (pp.\ 71--162). Elsevier

work page 2016

[35] [35]

Ramey, V. A. & Zubairy, S. (2018). Government Spending Multipliers in Good Times and in Bad: Evidence from US Historical Data . Journal of Political Economy , 126(2), 850--901

work page 2018

[36] [36]

Rue, H. (2001). Fast sampling of G aussian M arkov random fields. Journal of the Royal Statistical Society, Series B: Statistical Methodology , 63(2), 325--338

work page 2001

[37] [37]

Sensfu , F., Ragwitz, M., & Genoese, M. (2008). The merit-order effect: A detailed analysis of the price effect of renewable electricity generation on spot market prices in G ermany. Energy Policy , 36(8), 3086--3094

work page 2008

[38] [38]

& Pfenninger, S

Staffell, I. & Pfenninger, S. (2016). Using bias-corrected reanalysis to simulate current and future wind power output. Energy , 114, 1224--1239

work page 2016

[39] [39]

Stock, J. H. & Watson, M. W. (2018). Identification and Estimation of Dynamic Causal Effects in Macroeconomics Using External Instruments . Economic Journal , 128(610), 917--948

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