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

Technical Summary: Quasi-Bayesian Local Projection Instrumental-Variables Method

Problem Statement
Estimating dynamic causal effects using Local Projections (LP) is a standard approach in applied econometrics. When the treatment variable is endogenous, researchers combine LP with Instrumental Variables (IV) to form LP–IV estimators. However, a significant limitation of standard LP–IV is that estimates at different horizons are computed independently. In finite samples, particularly with high-frequency data and low signal-to-noise ratios, this independence leads to sampling noise that creates irregular, difficult-to-interpret impulse response functions (IRFs). Furthermore, standard inference often relies on pointwise confidence intervals, which fail to account for the cross-horizon correlation inherent in a response trajectory, potentially leading to misleading conclusions about the overall shape of the dynamic effect.

Methodology
The paper proposes a Quasi-Bayesian LP–IV framework that regularizes impulse responses across horizons while maintaining the identification properties of IV estimation. The methodology integrates three core components:

1. Moment-Based Quasi-Posterior: Following Chernozhukov & Hong (2003), the approach constructs a quasi-posterior distribution using the Generalized Method of Moments (GMM) objective function derived from IV moment conditions. This avoids the need for a full likelihood specification, thereby preserving the robustness of IV identification against likelihood misspecification.

   • The quasi-posterior is defined as:
     $$\pi(\theta) \propto \exp\left\{ -\frac{T}{2} \bar{m}(\theta)^\top W \bar{m}(\theta) \right\} p(\theta)$$
     where $\bar{m}(\theta)$ represents the sample moment conditions, $W$ is a weighting matrix, and $p(\theta)$ is a prior.

2. Roughness-Penalty Prior: To address the noise in horizon-specific estimates, the paper introduces a proper Gaussian Markov random field prior with a roughness penalty.

   • For a coefficient path $\theta_j$ across horizons, the prior is $\theta_j | \tau_j \sim N(0, \tau_j^2 Q^{-1})$.
   • The precision matrix $Q = D^\top D + \frac{8}{\rho^2}I$ penalizes adjacent differences (via the first-order difference matrix $D$) and the overall level (via the ridge term).
   • The scale parameter $\tau_j$ follows a hierarchical half-Cauchy prior, allowing the data to determine the degree of shrinkage. This enables the estimator to borrow information across neighboring horizons, smoothing the IRF path without imposing rigid parametric restrictions.

3. Inference and Computation:

   • Computation: The quasi-posterior is sampled using a Gibbs sampler. The conditional distribution of coefficients given the scale parameters is Gaussian, leveraging the sparse structure of the precision matrix for efficiency. Scale parameters are updated via a scale mixture of inverse-gamma distributions.
   • Inference: While the point estimates are derived from the quasi-posterior mean, uncertainty quantification remains frequentist. The paper constructs pointwise confidence intervals and simultaneous confidence bands using a sandwich covariance matrix based on the stacked moment conditions. This ensures that the uncertainty measures reflect the sampling variability of the moment-based estimator rather than posterior uncertainty about the smoothing parameters.

Key Contributions

1. Methodological Integration: The paper introduces the first moment-based quasi-Bayesian formulation specifically for LP–IV. Unlike existing Bayesian LP methods (e.g., Tanaka, 2020; Ferreira et al., 2025) that rely on narrative restrictions or VARs, this approach directly incorporates IV moment restrictions into the quasi-posterior.
2. Regularization without Likelihood: It offers a way to smooth impulse responses across horizons using a roughness-penalty prior without requiring a full structural likelihood, thus maintaining the flexibility and identification advantages of the LP–IV framework.
3. Joint Inference: By estimating the entire sequence of coefficients as a combined parameter vector, the framework facilitates joint inference (simultaneous bands) that accounts for cross-horizon dependence, addressing a known limitation of pointwise inference.
4. Asymptotic Equivalence: The paper establishes that under standard regularity conditions, the quasi-posterior mean is asymptotically equivalent to the standard GMM estimator, ensuring that the regularization does not compromise the large-sample properties of the IV estimator.

Empirical Results
The method is applied to daily data from the Danish wholesale electricity market (DK1 and DK2) to estimate the dynamic causal effect of wind and solar generation on electricity prices.

• Instrumentation: Weather-driven renewable potential (derived from ERA5 reanalysis data) is used as an instrument for observed renewable generation to address endogeneity.
• Findings:
  • Wind: A one-standard-deviation increase in wind generation leads to a significant, immediate decrease in wholesale electricity prices (approx. 8 EUR/MWh in DK1 and 4 EUR/MWh in DK2). This effect diminishes over 3–4 days, consistent with the merit-order effect.
  • Solar: The effect of solar generation is smaller and less precisely estimated in daily data, with confidence intervals often including zero.
  • Methodological Performance: The quasi-Bayesian estimator with the roughness-penalty prior produces smoother, more interpretable IRF paths compared to the unrestricted GMM estimator. Simulation studies confirm that this regularization reduces the Root Mean Squared Error (RMSE), particularly at medium and longer horizons, by mitigating sampling noise.

Significance and Claims
The paper claims to advance the field of impulse-response estimation by providing a tool that balances the flexibility of non-parametric LP with the stability of regularization.

• Econometric Significance: It demonstrates that quasi-Bayesian regularization is effective for parameters indexed by a horizon (paths) rather than scalar coefficients. It improves finite-sample performance (lower RMSE) and facilitates joint inference without sacrificing the identification guarantees of IV methods.
• Policy Significance: The empirical application highlights that renewable energy integration affects not only immediate prices but also short-term price dynamics. The findings suggest that the merit-order effect is most relevant at short horizons, with implications for balancing markets, storage incentives, and risk management strategies in electricity systems with high renewable penetration.

The authors maintain a modest stance, noting that the method is a regularization tool for point estimation while the identification relies on the IV moment conditions. They acknowledge that the instruments function as "weather-state" instruments rather than pure "weather-innovation" instruments, and the results are interpreted under a conditional exclusion restriction. Future research directions suggested include extending the framework to overidentified systems and developing procedures for weak instruments.