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Quasi-Bayesian Local Projection Instrumental-Variables Method: Application to Renewable Energy and Electricity Prices
\paragraph*{Keywords:}
local projections; instrumental variables; quasi-Bayesian inference; renewable energy; electricity price
\paragraph*{JEL Classification:}
C11; C26; C32; Q41; Q42
Estimating dynamic causal effects is essential in applied econometrics. Local projections assess impulse responses by employing regressions specific to each horizon Jorda2005.\footnote{See Jorda2023,Jorda2025,Inoue2026 for a review.} When the treatment variable is endogenous, dynamic causal effects can be identified by combining local projections with instrumental variables, which results in LP--IV estimators Jorda2015,Ramey2016,Stock2018,Ramey2018. Since LP--IV does not require defining a complete dynamic system, it provides a flexible framework for causal impulse-response analysis across macroeconomics, finance, and applied microeconomics. A key limitation of LP--IV is that estimates based on finite samples can vary greatly across different horizons. Since the impulse response is estimated independently at each horizon, sampling noise can create irregular response patterns, particularly at medium- and long-term horizons. This issue is especially pronounced in high-frequency contexts with volatile outcomes and low signal-to-noise ratios. In these cases, unrestricted LP--IV estimates might be hard to interpret, even when economic adjustment is presumed to be smooth. This paper introduces a quasi-Bayesian LP--IV approach that regularizes impulse responses over different horizons. It considers the impulse response as a function of the horizon and applies a roughness-penalty prior to reduce differences between neighboring horizons. The quasi-posterior is derived from the generalized method of moments (GMM) objective function based on the IV moment conditions, following the moment-based quasi-Bayesian framework of Chernozhukov2003. The estimator integrates IV identification with structured regularization, without requiring a full likelihood specification.
The proposed approach has two main features. First, it offers a quasi-Bayesian formulation of LP--IV. Unlike existing Bayesian and quasi-Bayesian methods for local projections, such as Tanaka2020 and Ferreira2025, which are designed for setups where impulse responses are identified through narrative restrictions or vector autoregressions Ramey2016, these frameworks do not directly include IV moment restrictions. By deriving the quasi-posterior from the GMM objective function, this method integrates LP--IV moment conditions directly into Bayesian-like inference. Second, assuming certain regularity conditions, the quasi-posterior mean is asymptotically equivalent to the standard GMM estimator. Additionally, the roughness-penalty prior improves finite-sample estimates by sharing information across horizons. This prior is a proper Gaussian Markov random field with a hierarchical scale parameter, enabling data-driven regularization. Unlike frequentist smoothing techniques for LPs Barnichon2018,Barnichon2019,El-Shagi2019, this approach jointly estimates the smoothing and model parameters.
Joint inference plays a crucial role in impulse-response analysis. An impulse response represents a trajectory showing how an outcome changes over time after an intervention, rather than just a set of isolated scalar parameters. Therefore, uncertainty about this trajectory requires simultaneous inference across all points, not just individual, pointwise estimates. As highlighted by Jorda2023,Jorda2025, relying solely on pointwise intervals can be misleading because response estimates are correlated across different horizons. Consequently, inference about the overall shape of the response should incorporate cross-horizon dependence.
The proposed quasi-Bayesian framework also facilitates joint inference by estimating the entire sequence of LP--IV coefficients as a combined parameter vector. In our empirical application, we present pointwise confidence intervals and simultaneous confidence bands derived from a sandwich covariance matrix for the stacked moment conditions. Therefore, the baseline uncertainty measures are frequentist, while the quasi-Bayesian approach helps stabilize the estimation of the impulse-response path. This approach differs from Ferreira2025, which assesses uncertainty on an equation-by-equation basis. Unlike such individual equation methods, the stacked moment-based formulation used here simplifies simultaneous inference for LP--IV response paths.
A simulation study indicates that the quasi-Bayesian estimator using a flat prior closely matches the classical GMM estimator, reinforcing the connection between the quasi-posterior and traditional LP--IV estimation. When employing the roughness-penalty prior, the estimator applies moderate shrinkage and generally lowers RMSE in the baseline and block-diagonal weighting configurations, particularly at medium and longer horizons.
The application analyzes how renewable generation affects Danish wholesale electricity prices over time. Using daily data from Western Denmark (DK1) and Eastern Denmark (DK2), it models wind and solar generation based on weather-driven renewable potential derived from ERA5 reanalysis data. Findings reveal that wind generation significantly lowers prices in the short term in both zones, aligning with the merit-order effect described in electricity market studies. The effect of wind is more pronounced in DK1 and diminishes over the following days. Solar effects are smaller and less precisely measured in daily data. Numerous robustness tests, including alternative LP models, lag lengths, seasonal controls, prior hyperparameters, lagged renewable states, and placebo exercises, consistently support these core findings.
The paper makes contributions to two key areas. First, it advances the econometric field of impulse-response estimation by introducing a moment-based quasi-Bayesian approach for regularized LP--IV estimation. Unlike methods that rely on post-estimation smoothing or impose parametric restrictions on the impulse response, this new approach integrates smoothness directly into the estimation process while maintaining IV-based identification. Second, it adds to the empirical research on renewable energy and electricity prices by analyzing how weather-driven fluctuations in renewable supply influence dynamic price responses. The findings indicate that renewable integration affects both immediate prices and short-term price movements, which has important implications for balancing markets, storage incentives, and risk management.
The remainder of the paper is structured as follows. Section 2 discusses the economic background and empirical motivation. Section 3 outlines the quasi-Bayesian LP--IV framework. Section 4 details the simulation study. Section 5 covers the empirical application to renewable generation and electricity prices. Finally, Section 6 offers concluding remarks.
Electricity markets are ideal for analyzing how supply shocks change over time. The rise of variable renewable energy sources, like wind and solar, has caused significant weather-related fluctuations in supply, affecting price volatility, investment in generation, and system reliability. For policymakers, understanding the effect of renewable generation on prices is key for shaping energy and climate strategies, such as renewable subsidies, carbon pricing, and capacity mechanisms. Market participants also benefit from knowing how renewable supply shocks spread over time, aiding in operational planning, risk management, and investment decisions. Due to high-frequency data and clearly defined market structures, electricity markets are particularly suited for studying how external supply changes lead to dynamic price responses.
Extensive research shows that renewable energy reduces wholesale electricity prices via the merit-order effect Sensfuss2008,Wuerzburg2013,Clo2015. Wind and solar power have low marginal costs and can displace more expensive thermal generation, thereby lowering prices. While this immediate effect is well known, less is understood about how renewable supply shocks influence prices over later periods. This dynamic aspect is important for shaping storage incentives, reserve requirements, and cross-border trade. Persistence might stem from thermal units' ramping and startup constraints Borenstein2002, hydroelectric and storage options enable intertemporal substitution; expectations about short-term supply influence bidding strategies; and interconnections transmit shocks across regions and trading periods Green2011. More generally, renewable intermittency poses dynamic adjustment challenges beyond simple supply-demand interactions Joskow2011.
From a policy standpoint, the focus is not just on the immediate price effect of renewable generation but also on the entire adjustment process after a supply shock. Let $\gamma_{\left(h\right)}$ denote the causal effect of a one-unit increase in renewable generation at time $t$ on the electricity price $h$ periods ahead. The sequence $\left\{ \gamma_{\left(h\right)}\right\} _{h=0}^{H}$ characterizes how prices respond over time and helps distinguish transitory fluctuations from more persistent effects that may influence investment incentives and system reliability. This response path is also important for market participants in their price forecasting, risk management, as well as production and trading decisions.
A significant empirical challenge is that observed renewable generation might be correlated with other factors influencing electricity prices, such as demand patterns, seasonal fluctuations, and broader weather conditions. To accurately evaluate policies, it is essential to isolate exogenous variations in renewable supply. Recent studies have tackled this by employing meteorological variables as instruments for renewable generation. Wind speed and solar irradiance directly affect renewable output and, when controlling for demand, weather, and seasonal factors, offer plausibly exogenous supply variation Staffell2016,Pfenninger2016. Research using weather-based instruments demonstrates economically significant effects of renewable generation on electricity prices and related market outcomes Hirth2013.
Empirical analysis in this context faces several statistical challenges. First, renewable supply shocks have effects that change over time, necessitating estimates at multiple horizons. Second, electricity prices show significant high-frequency volatility, which can lead to imprecise horizon-specific estimates Weron2014. Third, although prices may spike sharply initially, economic adjustments following the shock are generally expected to develop smoothly across nearby horizons. These characteristics suggest a method that captures the entire dynamic response while smoothing out sampling noise, aligning with realistic economic adjustment patterns.
The objective is to estimate the dynamic causal effect of a treatment $r_{t}$ on an outcome $y_{t}$. In the empirical application, $r_{t}$ denotes renewable generation and $y_{t}$ denotes the wholesale electricity price. The parameter of interest is the impulse-response sequence $\left\{ \gamma_{\left(h\right)}\right\} _{h=0}^{H}$ , where $\gamma_{\left(h\right)}$ measures the effect of a treatment shock at time $t$ on the outcome $h$ periods ahead. Estimation encounters two main challenges: the treatment may be endogenous, and horizon-specific estimates of $\gamma_{\left(h\right)}$ can be noisy, especially at longer horizons. This section presents a framework that tackles these challenges by integrating IV identification with regularized quasi-Bayesian inference.
We determine the impulse response by conducting a series of horizon-specific regressions. For clarity, this section discusses a single treatment variable $r_{t}$. However, the framework can handle multiple treatment variables, as demonstrated in the empirical application. Throughout the paper, we focus on just-identified models where the number of instruments matches the number of regressors. This restriction simplifies both estimation and computation and enables the initial IV estimator to be expressed in a closed form. We consider two LP--IV specifications. The first is the level specification:
\[ y_{t+h}=\gamma_{\left(h\right)}r_{t}+\eta_{\left(h\right)}+\sum_{l=1}^{L}\mu_{\left(h\right),l}y_{t-l}+\boldsymbol{\delta}_{\left(h\right)}{}^{\top}\tilde{\boldsymbol{x}}_{t}+e_{\left(h\right),t}, \] where $\tilde{\boldsymbol{x}}_{t}$ is a vector of control variables and $e_{\left(h\right),t}$ is an error term. The coefficient $\gamma_{\left(h\right)}$ is the target causal effect at horizon $h$, while $\eta_{\left(h\right)}$, $\left\{ \mu_{\left(h\right),l}\right\} _{l=1}^{L}$, and $\boldsymbol{\delta}_{\left(h\right)}$ are nuisance parameters.
The second is the long-differenced (LD) specification, which replaces the dependent variable $y_{t+h}$ with $y_{t+h}-y_{t-1}$ and uses lagged first differences of the outcome as controls Stock2018: \[ y_{t+h}-y_{t-1}=\gamma_{\left(h\right)}r_{t}+\eta_{\left(h\right)}+\sum_{l=1}^{L}\mu_{\left(h\right),l}\Delta y_{t-l}+\boldsymbol{\delta}_{\left(h\right)}{}^{\top}\tilde{\boldsymbol{x}}_{t}+e_{\left(h\right),t}, \] where $\Delta y_{t-l}=y_{t-l}-y_{t-l-1}$. Under no anticipation, $y_{t-1}$ is unaffected by the treatment shock at $t$, so the LD coefficient $\gamma_{\left(h\right)}$ can be interpreted as the impulse response of $y_{t+h}$ relative to the pre-shock baseline. As the LD specification can reduce bias and improve coverage in finite samples Piger2025, we use the LD specification as the baseline in the analysis below.
We define the $J$-dimensional vectors of regressors and corresponding coefficients as \[ \boldsymbol{x}_{t}=\left(r_{t},1,\Delta y_{t-1},\cdots,\Delta y_{t-L},\tilde{\boldsymbol{x}}_{t}^{\top}\right)^{\top}, \] and \[ \boldsymbol{\theta}_{\left(h\right)}=\left(\gamma_{\left(h\right)},\eta_{\left(h\right)},\mu_{\left(h\right),1},...,\mu_{\left(h\right),L},\boldsymbol{\delta}_{\left(h\right)}{}^{\top}\right)^{\top}. \] The horizon-$h$ regression can then be written compactly as \[ y_{t+h}-y_{t-1}=\boldsymbol{\theta}_{\left(h\right)}^{\top}\boldsymbol{x}_{t}+e_{\left(h\right),t}. \]
Inference is based on the IV moment conditions. For each horizon, the structural error is assumed to be orthogonal to a $J$-dimensional instrument vector $\boldsymbol{z}_{t}$. Stacking the horizon-specific coefficient vectors gives \[ \boldsymbol{\theta}=\left(\boldsymbol{\theta}_{\left(0\right)}^{\top},...,\boldsymbol{\theta}_{\left(H\right)}^{\top}\right)^{\top}. \] We define the stacked moment function: \[ \boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)=\left(\boldsymbol{m}_{\left(0\right),t}\left(\boldsymbol{\theta}\right)^{\top},...,\boldsymbol{m}_{\left(H\right),t}\left(\boldsymbol{\theta}\right)^{\top}\right)^{\top}. \] Here, each $\boldsymbol{m}_{\left(h\right),t}\left(\boldsymbol{\theta}\right)$ is the horizon-$h$ IV moment contribution. Specifically, we define \[ \boldsymbol{m}_{\left(h\right),t}\left(\boldsymbol{\theta}\right)=\left(y_{t+h}-y_{t-1}-\boldsymbol{\theta}_{\left(h\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{z}_{t}. \] Because each horizon-specific regression generates $J$ orthogonality conditions, the stacked moment vector has dimension $K=J\left(H+1\right)$. Therefore, the population moment condition is \[ \mathbb{E}\left[\boldsymbol{m}_{t}\left(\boldsymbol{\theta}_{0}\right)\right]=\boldsymbol{0}_{K}, \] where $\boldsymbol{0}_{K}$ denotes the $K$-dimensional vector of zeros and $\boldsymbol{\theta}_{0}$ is the true parameter vector.
Rather than specifying a full likelihood for the observed variables, we use a GMM-based quasi-posterior to obtain a regularized moment-based estimator. Following Chernozhukov2003, we define \[ \pi\left(\boldsymbol{\theta}\right)=\frac{\exp\left\{ -\frac{T}{2}\bar{\boldsymbol{m}}\left(\boldsymbol{\theta}\right)^{\top}\boldsymbol{W}\bar{\boldsymbol{m}}\left(\boldsymbol{\theta}\right)\right\} p\left(\boldsymbol{\theta}\right)}{\int\exp\left\{ -\frac{T}{2}\bar{\boldsymbol{m}}\left(\boldsymbol{\theta}\right)^{\top}\boldsymbol{W}\bar{\boldsymbol{m}}\left(\boldsymbol{\theta}\right)\right\} p\left(\boldsymbol{\theta}\right)d\boldsymbol{\theta}}, \] where \[ \bar{\boldsymbol{m}}\left(\boldsymbol{\theta}\right)=\frac{1}{T}\sum_{t=1}^{T}\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right), \] $\boldsymbol{W}$ is a positive definite weighting matrix, and $p\left(\boldsymbol{\theta}\right)$ is a prior. Throughout, $T$ denotes the number of usable projection-origin observations after trimming initial lags and terminal leads. The quasi-posterior uses the moment restrictions directly and does not require a fully specified likelihood for data generation.
It is helpful to differentiate between the flat-prior and regularized cases. Under the flat prior $p\left(\boldsymbol{\theta}\right)\propto1$, the quasi-posterior is centered at the minimizer of the quadratic GMM criterion. In the linear exactly identified LP--IV model considered here, this estimator coincides with the conventional IV/GMM estimator. The flat-prior quasi-posterior, therefore, provides a direct moment-based representation of classical GMM inference. Under the roughness-penalty prior introduced in Section 3.3, the quasi-posterior mean is generally not equal to the conventional GMM estimator in finite samples. The prior regularizes the stacked coefficient vector by shrinking horizon-indexed coefficient paths toward smoother trajectories. Hence the resulting estimator trades off fit to the IV moment conditions against smoothness across horizons. This finite-sample regularization serves as the mechanism enabling the proposed method to reduce sampling variability compared to unrestricted LP--IV.
The quasi-likelihood is formulated from the quadratic GMM objective function linked to IV moment conditions Hansen1982. The resulting quasi-posterior merges insights from these moments with prior knowledge of the parameter vector. Since it relies on moment restrictions rather than a comprehensive likelihood, this approach does not necessitate specifying the data's joint distribution and avoids issues stemming from likelihood misspecification. Alternatively, the quasi-posterior can be interpreted as a Gibbs or generalized Bayesian posterior derived from the quadratic GMM loss Jiang2008,Bissiri2016, connecting it to Bayesian inference based on estimating equations and moment restrictions Kim2002,Yin2009,Liao2011,Li2016.
We assume that the marginal prior satisfies the standard local prior-negligibility condition \[ \log p\left(\boldsymbol{\theta}_{0}+\frac{c}{\sqrt{T}}\right)-\log p\left(\boldsymbol{\theta}_{0}\right)=o\left(1\right) \] uniformly over compact sets of $c$. Therefore, the quasi-posterior mean $\hat{\boldsymbol{\theta}}$ is first-order equivalent to the conventional GMM estimator Chernozhukov2003,Hong2021: \[ \sqrt{T}\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}_{0}\right)\rightarrow\mathcal{N}\left(\boldsymbol{0},\boldsymbol{V}\right), \] \[ \boldsymbol{V}=\left(\boldsymbol{G}^{\top}\boldsymbol{W}\boldsymbol{G}\right)^{-1}\boldsymbol{G}^{\top}\boldsymbol{W}\boldsymbol{\Sigma}_{0}\boldsymbol{W}\boldsymbol{G}\left(\boldsymbol{G}^{\top}\boldsymbol{W}\boldsymbol{G}\right)^{-1}. \] Here $\boldsymbol{G}=\mathbb{E}\left[\partial\boldsymbol{m}_{t}\left(\boldsymbol{\theta}_{0}\right)/\partial\boldsymbol{\theta}^{\top}\right]$ is the Jacobian of the population moment condition $\boldsymbol{m}_{t}\left(\boldsymbol{\theta}_{0}\right)$, and $\boldsymbol{\Sigma}_{0}$ is the covariance matrix of $\boldsymbol{m}_{t}\left(\boldsymbol{\theta}_{0}\right)$. We use the contemporaneous covariance $\boldsymbol{\Sigma}_{0}=\mathrm{Var}\left[\boldsymbol{m}_{t}\left(\boldsymbol{\theta}_{0}\right)\right]$, rather than a HAR long-run covariance matrix, as the baseline. The motivation is close to MontielOlea2021, who show that lag-augmented local projections do not require correction for serial correlation in the multi-step residuals: although the projection residuals are serially correlated, the relevant regression scores are serially uncorrelated under their assumptions, so heteroskedasticity-robust standard errors suffice. In the present LP--IV formulation, the analogous objects are the IV moment scores $\boldsymbol{m}_{\left(h\right),t}\left(\boldsymbol{\theta}\right)=\boldsymbol{z}_{t}e_{\left(h\right),t}\left(\boldsymbol{\theta}\right)$. Accordingly, the baseline asymptotic variance uses the covariance of these stacked moment scores rather than a HAR long-run covariance estimator. This choice is an identifying and inferential convention for the baseline specification; it is distinct from the cross-horizon covariance captured by stacking $\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)$.
In the LP--IV model, $\boldsymbol{G}$ is estimated by its sample analogue, \[ \hat{\boldsymbol{G}}=\boldsymbol{I}_{H+1}\otimes\left(-\frac{1}{T}\boldsymbol{Z}^{\top}\boldsymbol{X}\right), \] where \[ \boldsymbol{X}=\left(\boldsymbol{x}_{1},...,\boldsymbol{x}_{T}\right)^{\top},\quad\boldsymbol{Z}=\left(\boldsymbol{z}_{1},...,\boldsymbol{z}_{T}\right)^{\top}, \] $\boldsymbol{I}_{H+1}$ is the $\left(H+1\right)\times\left(H+1\right)$ identity matrix, and $\otimes$ denotes the Kronecker product. This first-order equivalence does not imply equality in finite samples: the quasi-posterior mean under an informative prior may differ from the GMM estimator.
The quasi-Bayesian approach offers two main benefits. First, it provides a straightforward way to include structured regularization through hierarchical priors. Second, since it estimates the entire sequence of horizon-specific coefficients together, it naturally supports joint uncertainty quantification for the full IRF. In the basic implementation described below, uncertainty is calibrated using sandwich-based frequentist bands centered on the quasi-posterior mean, while the hierarchical prior helps regularize the point estimate by determining the level of cross-horizon smoothing based on the data.
For computational stability, we fix the weighting matrix $\boldsymbol{W}$ throughout the simulation. This follows Chernozhukov2003 and avoids recomputing the covariance matrix of the moment conditions at each draw. In the reported inference, uncertainty is calibrated using the asymptotic sandwich covariance described below. We set $\boldsymbol{W}$ equal to the inverse of the empirical covariance matrix of the moment functions evaluated at an initial consistent estimator. In the exactly identified specification considered here, this initial estimator is \[ \boldsymbol{\theta}^{*}=\textrm{vec}\left(\left(\boldsymbol{Z}^{\top}\boldsymbol{X}\right)^{-1}\boldsymbol{Z}^{\top}\boldsymbol{Y}\right), \] where \[ \boldsymbol{Y}=\left(
\right), \] and $\textrm{vec}\left(\cdot\right)$ denotes the column-wise vectorization.
When $K$ is large, estimating and inverting the entire covariance matrix of the moment functions can become numerically unstable. In these situations, we opt for a block-diagonal weighting matrix that spans different horizons, \[ \boldsymbol{W}=\textrm{blkdiag}\left(\boldsymbol{W}_{\left(0\right)},...,\boldsymbol{W}_{\left(H\right)}\right), \] where $\boldsymbol{W}_{\left(h\right)}$ is the inverse of the empirical covariance matrix of $\boldsymbol{m}_{\left(h\right),t}\left(\boldsymbol{\theta}\right)$. This approximation maintains the within-horizon weighting to prevent unstable inversion of the entire stacked covariance matrix. The block-diagonal restriction applies solely to the weighting matrix for estimation. Meanwhile, the sandwich covariance matrix used for inference is derived from the complete stacked moment vector, preserving the cross-horizon dependence.
In impulse-response analysis, the coefficients across different horizons form a dynamic adjustment path rather than just a set of unrelated scalar parameters. In many cases, nearby horizons are expected to contain similar information because economic adjustments are driven by persistent mechanisms such as production constraints, storage, expectations, or institutional frictions. Therefore, smoothness across horizons can serve as a scientifically justified regularization technique: it helps reduce sampling noise while still allowing the data to shape the response's level, slope, and form.
For each covariate $j$, define the horizon-indexed coefficient path \[ \boldsymbol{\theta}_{j}=\left(\theta_{\left(0\right),j},...,\theta_{\left(H\right),j}\right)^{\top}. \] We assign the conditional Gaussian prior \[ \boldsymbol{\theta}_{j}|\tau_{j}\sim\mathcal{N}\left(\boldsymbol{0}_{H+1},\;\tau_{j}^{2}\boldsymbol{Q}^{-1}\right), \] where $\tau_{j}>0$ controls the overall scale of the path. The matrix $\boldsymbol{Q}$ is a baseline smoothing precision matrix over the projection horizon and is specified as \[ \boldsymbol{Q}=\boldsymbol{D}^{\top}\boldsymbol{D}+\frac{8}{\rho^{2}}\boldsymbol{I}_{H+1}, \] where $\boldsymbol{D}$ is the first-order difference matrix of dimension $H\times\left(H+1\right)$ and $\rho>0$ is a correlation-range parameter. The prior penalizes both adjacent differences and the overall level of the coefficient path. The term $\boldsymbol{D}^{\top}\boldsymbol{D}$ regularizes the shape of the coefficient path by penalizing adjacent differences, while leaving constant paths unpenalized. The ridge term makes the prior proper by penalizing this constant component that would otherwise remain unregularized, thus directly shrinking the overall path level toward zero. In the implementation, the prior is applied to each horizon-indexed coefficient path, including both nuisance coefficients and the IRF coefficients, although the latter are the main objects of interest.
This prior can be viewed as a proper Gaussian Markov random field over the projection horizon Lindgren2011. It is also closely connected to the discrete approximation of a Mat�rn-type Gaussian field. In the notation of a continuous horizon, the related field corresponds to the stochastic differential operator \[ \left(\frac{8}{\rho^{2}}-\frac{\partial^{2}}{\partial h^{2}}\right)f\left(h\right)=\mathcal{W}\left(h\right), \] where $\mathcal{W}\left(h\right)$ denotes Gaussian white noise. The precision matrix is a finite-difference analog of the continuous precision operator $8/\rho^{2}-\partial^{2}/\partial h^{2}$, up to boundary and grid-spacing constants. The correlation-range parameter $\rho$ has a quantitative interpretation. For an interior point of a long horizon grid, the prior correlation between two coefficients separated by $\left|h-h^{\prime}\right|$ horizons is approximately \[ \mathrm{Corr}\left(\theta_{\left(h\right),j},\theta_{\left(h^{\prime}\right),j}\right)\approx\exp\left(-\frac{\sqrt{8}}{\rho}\left|h-h^{\prime}\right|\right). \] Thus, larger values of $\rho$ imply slower decay of prior dependence and smoother coefficient paths. Quantitatively, the prior correlation falls to approximately 0.5 after $0.25\rho$ horizons, to $1/e\approx0.37$ after $0.35\rho$ horizons, and to 0.1 after $0.8\rho$ horizons.
The limiting case $\rho\rightarrow\infty$ corresponds to the intrinsic first-order random-walk prior Lang2004, which penalizes roughness; however, it is improper. For any finite $\rho$, the ridge component makes $\boldsymbol{Q}$ positive definite and defines a valid prior distribution for the full coefficient path.
To enable data-driven assessment of shrinkage strength, we assign a hierarchical prior to the scale parameter, \[ \tau_{j}\sim\mathcal{C}^{+}\left(0,\kappa\right), \] where $\mathcal{C}^{+}\left(0,\kappa\right)$ denotes a half-Cauchy distribution with scale parameter $\kappa$. This heavy-tailed prior enables large coefficient paths when supported by data, while pushing weakly supported paths toward zero and promoting smoother variation across horizons Polson2012.
The hierarchical prior regularizes the IRF across different horizons, with the smoothing degree estimated together with the coefficients. It offers a systematic method for the regularized estimation of horizon-indexed parameters identified through moments. The prior functions solely as a regularization tool, while the dynamic causal effects are identified by the IV moment conditions.
A related class of priors was introduced by Tanaka2020. That prior is improper, corresponding to the limiting case $\rho\rightarrow\infty$, whereas the present formulation is proper for any finite $\rho$. Unlike frequentist smoothing approaches to LPs Barnichon2018,Barnichon2019,El-Shagi2019, the quasi-Bayesian framework treats the prior scale parameters as unknown, rather than selecting the amount of smoothing outside the joint estimation problem.
We simulate from the quasi-posterior using a Gibbs sampler. The quasi-posterior kernel of $\boldsymbol{\theta}$ conditional on $\boldsymbol{\tau}$ can be written as \[ \pi\left(\boldsymbol{\theta}|\boldsymbol{\tau}\right)\propto\exp\left\{ -\frac{T}{2}\left(\boldsymbol{\theta}-\boldsymbol{\theta}^{*}\right)^{\top}\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{G}}\left(\boldsymbol{\theta}-\boldsymbol{\theta}^{*}\right)\right\} p\left(\boldsymbol{\theta}|\boldsymbol{\tau}\right). \] Combining the quadratic GMM loss and conditional Gaussian prior yields \[ \boldsymbol{\theta}|\boldsymbol{\tau}\sim\mathcal{N}\left(\boldsymbol{\Omega}\boldsymbol{\Upsilon}\boldsymbol{\theta}^{*},\;\boldsymbol{\Omega}\right), \] where \[ \boldsymbol{\Omega}=\left(\boldsymbol{\Upsilon}+\boldsymbol{\Pi}\right)^{-1},\quad\boldsymbol{\Upsilon}=T\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{G}},\quad\boldsymbol{\Pi}=\boldsymbol{Q}\otimes\textrm{diag}\left(\tau_{1}^{-2},...,\tau_{J}^{-2}\right). \] Sampling from this Gaussian distribution leverages the sparse structure of the prior precision matrix $\boldsymbol{Q}$. We implement the Gaussian Markov random field simulation algorithm of Rue2001, which improves computational efficiency.
The scale parameters are updated using the half-Cauchy representation as a scale mixture of inverse-gamma distributions Wand2011: \[ \tau_{j}^{2}|\nu_{j}\sim\mathcal{IG}\left(\frac{1}{2},\frac{1}{\nu_{j}}\right),\quad\nu_{j}\sim\mathcal{IG}\left(\frac{1}{2},\frac{1}{\kappa^{2}}\right), \] where $\nu_{j}$ is an auxiliary random variable and $\mathcal{IG}\left(a,b\right)$ denotes an inverse gamma distribution with shape parameter $a$ and rate parameter $b$. The corresponding full conditional distributions are derived as in Makalic2015:
The quasi-posterior mean is estimated by averaging the simulated draws of $\boldsymbol{\theta}$. While the MCMC algorithm samples both the coefficient vector and the smoothing-scale parameters, our primary uncertainty summaries are based on a frequentist approach rather than a Bayesian posterior. Specifically, we present sandwich standard errors calculated around the quasi-posterior mean. This approach distinguishes the regularization employed for point estimation from the calibration of sampling uncertainty. Consequently, the derived uncertainty measures, grounded in the moment conditions, preserve the conventional GMM interpretation, with the covariance estimated as \[ \hat{\boldsymbol{V}}=\frac{1}{T}\left(\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{G}}\right)^{-1}\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{\Sigma}}\boldsymbol{W}\hat{\boldsymbol{G}}\left(\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{G}}\right)^{-1}, \] where $\hat{\boldsymbol{\Sigma}}$ is the empirical covariance matrix of $\boldsymbol{m}_{t}\left(\hat{\boldsymbol{\theta}}\right).$ Pointwise confidence intervals are constructed from the corresponding diagonal elements of $\hat{\boldsymbol{V}}$. The sample analogue $\hat{\boldsymbol{\Sigma}}$ is computed as the empirical covariance matrix of $\boldsymbol{m}_{t}\left(\hat{\boldsymbol{\theta}}\right)$, consistent with the baseline covariance choice discussed in Section 3.2. Simultaneous confidence bands are constructed using the max-$t$ procedure of MontielOlea2019 (2019, Section 2.5), applied to the impulse-response coefficients. Since these intervals and bands are sandwich-based, they should be understood as frequentist confidence intervals and confidence bands centered on the regularized estimator, rather than as posterior credible intervals. Specifically, they do not directly account for posterior uncertainty concerning the smoothing-scale parameters $\tau_{j}$. In the baseline analysis, the hierarchical prior mainly functions to regularize the point estimate by determining the degree of cross-horizon shrinkage learned from the data.
To assess the finite-sample performance of the proposed LP--IV estimator under genuine endogeneity, we conduct a series of Monte Carlo experiments. See Appendix A.1 for the data-generating process. We consider two priors for the LP--IV coefficients. First, to make a direct comparison with frequentist methods, we use a flat prior, $p\left(\boldsymbol{\theta}\right)\propto1$. Second, to assess the practical value of regularization, we use the roughness-penalty prior introduced in Section 3.3. We refer to the former as QB--flat and the latter as QB--RP. We consider sample sizes $T\in\left\{ 200,500,1000\right\} $. The hyperparameter $\rho$ ensures that the prior correlation between consecutive parameters is approximately 0.5 and the prior correlation between the two endpoints is approximately zero. \footnote{$\exp\left(-\sqrt{8}/4\times1\right)\approx0.493,$ $\exp\left(-\sqrt{8}/4\times3\right)\approx0.120$, $\exp\left(-\sqrt{8}/4\times5\right)\approx0.029$, and $\exp\left(-\sqrt{8}/4\times7\right)\approx0.007$.} For each design, we generate 1,000 Monte Carlo samples. Each MCMC run uses 25,000 iterations, with the first 5,000 discarded as burn-in.
We compare the proposed quasi-Bayesian method with a traditional benchmark: the system-wide two-step GMM estimator. In the first stage, we use the identity weighting matrix, which is equivalent to the two-stage least-squares estimator, and in the second stage, we apply the weighting matrix evaluated at the first-stage estimate. Consistent with the covariance choice discussed in Section 3.2, the main simulation results use the empirical covariance matrix of the stacked moment scores rather than a HAR long-run covariance estimator. In supplementary simulations reported in Appendix A, we also examine alternative weighting and covariance estimators, including a block-diagonal covariance estimator and a HAR estimator with a Bartlett kernel Newey1987. For the HAR estimator, the bandwidth is $B=\left\lceil 1.3T^{1/2}\right\rceil $, following Lazarus2018. In the exactly identified case, the unregularized IV/GMM point estimate is invariant to the weighting matrix. By contrast, the roughness-penalty quasi-posterior mean can depend on the weighting matrix in finite samples because $\boldsymbol{W}$ affects the curvature of the GMM quasi-likelihood, $T\hat{\boldsymbol{G}}^{\top}\boldsymbol{W}\hat{\boldsymbol{G}}$, and therefore the finite-sample trade-off between fitting the IV moments and enforcing cross-horizon smoothness. Performance is evaluated by mean bias, RMSE, mean pointwise 90% interval length, pointwise 90% coverage, and simultaneous 90% coverage across horizons.
Tables 1 and 2 present Monte Carlo results for the LD specification, using the plain covariance estimator as the benchmark. Table 1 reports pointwise performance measures, while Table 2 reports simultaneous 90% coverage. The quasi-Bayesian estimator with a flat prior (QB--flat) closely matches the classical GMM estimator across sample sizes and horizons, due to the equivalence between the flat-prior quasi-posterior mean and the GMM estimate. Bias remains small and diminishes with larger samples (Panel (a) of Table 1), while RMSE decreases as T increases and tends to rise with the horizon because sampling uncertainty builds up (Panel (b) of Table 1). Pointwise interval lengths for GMM and QB--flat are also almost the same (Panel (c) of Table 1), and pointwise coverage is generally close to the nominal 90% level (Panel (d) of Table 1). Table 2 shows that simultaneous coverage is close to 90% for GMM and QB--flat. For QB--RP, coverage is conservative in smaller samples and approaches the nominal level as the sample size increases.
The quasi-Bayesian estimator with the roughness-penalty prior (QB--RP) smooths coefficient paths across different horizons. This regularization introduces some bias compared to GMM and QB--flat, especially in small samples, but it often decreases RMSE at longer horizons. The most notable improvement occurs at $T=200$, while in larger samples, the benefits are mainly seen at later horizons. This improvement comes from the prior's ability to borrow information from nearby horizons and minimize spurious high-frequency variations in the impulse response estimate. Therefore, the quasi-Bayesian approach incorporates smoothness directly into the estimation process rather than applying it after estimation.
Tables A.1--A.5 in Appendix A report supplementary simulations comparing the level and LD specifications alongside alternative weighting and covariance estimators. The key patterns stay consistent across different scenarios. The LD specification generally shows slightly less bias in several cases, aligning with evidence that long-difference LP estimators can mitigate finite-sample bias when shocks contain noise Piger2025. The results from the baseline and block-diagonal covariance estimators are similar, with the block-diagonal option offering a more stable numerical approximation when estimating the full covariance matrix proves challenging. The HAR estimator tends to produce shorter confidence intervals in some cases and often results in lower coverage. Overall, QB--flat remains nearly equivalent to GMM across all specifications. The RMSE improvements with QB--RP are most notable with the baseline and block-diagonal estimators, especially at medium and longer horizons. However, when using the HAR estimator, QB--RP does not always improve RMSE and may perform worse in certain designs.
This section uses the framework to analyze daily renewable-generation and wholesale-price data. The goal is to estimate how external changes in renewable supply influence electricity prices in the following days.
The empirical analysis uses daily data for Denmark's two wholesale electricity bidding zones: Western Denmark (DK1) and Eastern Denmark (DK2). Market data are sourced from Open Power System Data. The boundaries of the bidding zones are defined using Eurostat GISCO country geometries and 2021 NUTS-2 regions: DK1 includes Syddanmark, Midtjylland, and Nordjylland, while DK2 covers Hovedstaden and Sj�lland. The dataset spans from January 1, 2015, to September 30, 2020.
Daily electricity prices are calculated as averages of hourly prices determined one day in advance. Wind and solar generation are measured as daily totals, aggregated from hourly data, while electricity demand is represented by the total daily load. Meteorological variables are derived from ERA5 hourly single-level reanalysis data available through the Copernicus Climate Data Store. Wind power potential is obtained through a nonlinear transformation of hourly wind speeds at 100 meters, calculated from zonal and meridional wind components. Solar potential is assessed by the daily accumulated surface downward solar radiation. Renewable potentials at the grid-cell level are aggregated to the bidding-zone level using technology-specific installed capacity weights. The daily wind and solar potential series are employed as instruments for observed wind and solar generation. Weather influences, including daily mean 2m temperature, total precipitation, and total cloud cover, are derived from the same ERA5 data to account for weather-driven variations in electricity demand.
Let $y_{t}$ denote the daily average wholesale electricity price, and $r_{t}^{wind}$ and $r_{t}^{solar}$ denote daily wind and solar generation, respectively. We estimate the LD specification
The coefficients $\gamma_{\left(h\right)}^{wind}$ and $\gamma_{\left(h\right)}^{solar}$ trace the dynamic price responses to wind and solar generation. Wind and solar generation are normalized to have a mean of zero and a variance of one, so these coefficients represent the average change in price in response to a one-standard-deviation increase in generation. To account for strong seasonal variation, we include a Fourier series in calendar time, $\varsigma_{t}\left(\boldsymbol{\alpha}_{\left(h\right)},\boldsymbol{\beta}_{\left(h\right)}\right)$. Let $d_{t}$ denote the day-of-year and $D\left(t\right)\in\left\{ 365,366\right\} $ be the number of days in the corresponding year. To handle leap years, define the normalized within-year position $s_{t}=\left(d_{t}-1\right)/D\left(t\right)$. The seasonal component is \[ \varsigma_{t}\left(\boldsymbol{\alpha}_{\left(h\right)},\boldsymbol{\beta}_{\left(h\right)}\right)=\sum_{n=1}^{N}\left[\alpha_{\left(h\right),n}\sin\left(2\pi ns_{t}\right)+\beta_{\left(h\right),n}\cos\left(2\pi ns_{t}\right)\right], \] where \[ \boldsymbol{\alpha}_{\left(h\right)}=\left(\alpha_{\left(h\right),1},...,\alpha_{\left(h\right),N}\right)^{\top},\quad\boldsymbol{\beta}_{\left(h\right)}=\left(\beta_{\left(h\right),1},...,\beta_{\left(h\right),N}\right)^{\top}. \] The baseline specification uses $N=4$.
The covariate vector $\breve{\boldsymbol{x}}_{t}$ includes variables capturing demand conditions and systematic market patterns. The continuous controls include total electricity load, daily mean 100m wind speed, daily mean 2m temperature, total precipitation, and total cloud cover; each is standardized to have a mean of zero and a variance of one. Additionally, we incorporate day-of-week indicators, excluding Monday, along with four calendar indicators: public holidays, Constitution Day, Christmas Eve, and New Year's Eve.\footnote{Public holidays include New Year's Day, Maundy Thursday, Good Friday, Easter Sunday, Easter Monday, Great Prayer Day, Ascension Day, Whit Sunday / Pentecost, Whit Monday, Christmas Day, and Second Christmas Day. }
The resulting regressor vector is
The corresponding coefficient vector $\boldsymbol{\theta}_{\left(h\right)}$ is defined as in Section 3.1. We set $H=L=7$ and use the roughness-penalty prior with $\rho=4$ and $\kappa=1$. The MCMC simulation uses 50,000 draws after discarding the initial 5,000 draws as burn-in.
Renewable generation might be endogenous due to potential correlation with unobserved factors affecting electricity prices, such as demand changes and weather patterns. To mitigate this issue, meteorological variables are employed as instruments for renewable generation. These instruments are derived from daily wind speed, adjusted to account for the nonlinear link between wind speed and turbine output, along with daily solar irradiance. Both variables serve as robust predictors of renewable generation because of the direct physical connection between weather conditions and energy yield Staffell2016,Pfenninger2016.
The wind instrument is constructed from 100m wind speed. For each grid cell $i$ and hour $s$, we define \[ WS_{i,s}=\sqrt{u_{i,s}^{2}+v_{i,s}^{2}}, \] where $u_{i,s}$ and $v_{i,s}$ are the 100m zonal and meridional wind components, respectively. Wind speed is transformed into wind-power potential using the turbine power curve \[ q_{i,s}^{wind}=
\] We set $\varrho^{in}=3$, $\varrho^{rated}=13$, and $\varrho^{out}=25$. The daily grid-cell wind potential is \[ q_{i,t}^{wind}=\frac{1}{24}\sum_{s\in t}q_{i,s}^{wind}. \] We then aggregate to the bidding-zone level using installed wind-capacity weights, \[ q_{t}^{wind}=\sum_{i}w_{i}^{wind}q_{i,t}^{wind}, \] \[ w_{i}^{wind}=\frac{c_{i}^{wind}}{\sum_{j}c_{j}^{wind}}. \] This procedure produces a wind-potential series for bidding zones that approximates the weather-driven part of total wind generation. Capacity weighting is intuitive, as it reflects how meteorological conditions relate to turbine locations. We create these weights using renewable power-plant data from Open Power System Data, which includes information on Danish renewable facilities and their installed capacities. The solar instrument is constructed from surface downward solar radiation. For each grid cell $i$ and hour $s$, let $SSR_{i,s}$ denote the hourly accumulated surface solar radiation from ERA5. The daily grid-cell solar potential is
\[ q_{i,t}^{solar}=\sum_{s\in t}SSR_{i,s}. \] We aggregate to the bidding-zone level using installed solar-capacity weights:
\[ q_{t}^{solar}=\sum_{i}w_{i}^{solar}q_{i,t}^{solar},\quad w_{i}^{solar}=\frac{c_{i}^{solar}}{\sum_{j}c_{j}^{solar}}. \]
The instrument vector is obtained by replacing the observed wind and solar generation in $\boldsymbol{x}_{t}$ with the corresponding weather-driven renewable potentials:
The horizon-$h$ moment condition is \[ \mathbb{E}\left[\left(y_{t+h}-y_{t-1}-\boldsymbol{\theta}_{\left(h\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{z}_{t}\right]=\boldsymbol{0}_{J}, \] which requires that, conditional on controls, weather-driven renewable potential affects electricity prices only through realized renewable generation.
The core assumption is that, once demand, weather, seasonal, and calendar factors are held constant, changes in renewable potential influence electricity prices solely via renewable generation. While weather can directly affect electricity demand, this is mitigated by flexibly controlling for variables like temperature, precipitation, cloud cover, seasonality, and calendar effects. With this exclusion restriction, any remaining variation in renewable potential serves as an exogenous source of renewable supply variation. This approach aligns with empirical research that treats renewable variability as quasi-experimental variation in electricity markets Hirth2013.
The instruments used in this study should be viewed as weather-state instruments rather than weather-innovation instruments. Weather-innovation instruments involve unexpected meteorological changes and are designed to be orthogonal to predetermined market variables by construction. Conversely, our instruments are based on actual wind and solar potential levels. These factors are strong physical predictors of renewable energy output but, due to weather persistence, may also be correlated with lagged renewable output, historical prices, or other prior market conditions. Thus, identification depends on a conditional weather-state exclusion restriction: once demand, weather controls, seasonality, calendar effects, and lagged outcomes are accounted for, the residual variation in renewable potential influences electricity prices solely through realized renewable generation. The placebo diagnostics in Appendix B serve as tests of this conditional restriction's plausibility, rather than as evidence that the instruments are purely weather innovations.
We conduct diagnostics to evaluate the IV design, with details available in Appendix B. To assess first-stage relevance, we regress renewable generation on renewable potential and baseline controls, reporting coefficient estimates, the partial $R^{2}$ for the excluded instruments, Wald-type first-stage relevance statistics, and the smallest singular value of the matrix of renewable-potential coefficients (see Table B.1). The lowest first-stage relevance statistic is 45.6, and the smallest singular value is 0.336, both indicating strong relevance. In placebo regressions, most partial $R^{2}$ values are small, except for a maximum of 0.301, showing that current renewable potential can predict lagged renewable generation in DK2.
Second, we perform placebo diagnostics for predetermined variables. Specifically, we regress lagged market variables---such as prices, total load, and renewable generation---on renewable potential and baseline controls. We then report the coefficient estimates, partial $R^{2}$, and Wald $p$-values (see Table B.2). The placebo partial $R^{2}$ values are typically small, but higher values and lower Wald $p$-values for lagged renewable generation suggest that the excluded instruments still contain predictable weather-state components. Some $p$-values remain small, indicating the instruments are more aligned with weather-state instruments than purely weather-innovation instruments. This is especially evident in DK2, where current potential predicts lagged renewable generation, likely reflecting serial correlation in wind and solar conditions. Consequently, Section 5.6 presents a specification including lagged renewable potential and lagged renewable generation. The baseline model excludes these lagged renewable-state controls because they can reduce interpretability and may absorb part of the persistent supply-shock variation that the IRF aims to capture.
These placebo results do not definitively disprove the empirical design, but they suggest that the instruments maintain predictable weather-state elements. Consequently, the estimates should be understood within the framework of the conditional weather-state exclusion restriction outlined earlier, rather than as outcomes driven by unforeseen weather changes. This distinction is particularly important for DK2, where recent renewable potential forecasts likely influence lagged renewable generation, plausibly indicating serial correlation in wind and solar conditions.
These diagnostics are not utilized to build the quasi-posterior. Instead, they evaluate whether the empirical design provides enough exogenous variation to identify the dynamic effects of wind and solar generation. They also check whether the excluded weather-potential instruments predict predetermined market conditions after accounting for the baseline controls. While the estimator has a quasi-Bayesian nature, its identifying assumptions impose restrictions on the data-generating process, which can be tested using traditional design-based diagnostics. Developing quasi-Bayesian tests for IV moment restrictions remains a topic for future research.
Finally, to identify pre-existing price movements, we perform a lead-placebo analysis. We choose the level specification because the goal of this exercise is to determine whether renewable-potential instruments forecast pre-existing price levels, not to measure the post-shock cumulative price response. We regress pre-treatment leads, $y_{t-8},...,y_{t-1}$, on the same regressors as in the baseline model. Significant responses at these leads would suggest pre-existing price movements linked to renewable potential; however, the absence of such responses supports, but does not prove, the exclusion restriction. As shown in Figure B.1, lead responses are generally near zero, and the uncertainty ranges do not show a clear pre-trend pattern.
Figure 1 reports the estimated IRFs for DK1 and DK2. The responses are measured in EUR/MWh and represent a one-standard-deviation increase in wind or solar generation. The key result is that wind generation significantly lowers wholesale electricity prices in the short run, while the effect of solar is smaller and less accurately estimated.
Wind generation has a negative estimated effect on both bidding zones. In DK1, a one-standard-deviation rise in wind output reduces the daily average electricity price by around 8 EUR/MWh initially. This effect diminishes over three to four days, approaching zero. In DK2, the negative effect is also observed but is smaller, about 4 EUR/MWh, and similarly decreases over time. These findings align with the merit-order principle: low-marginal-cost wind shifts the supply curve outward, displacing more expensive thermal generation and lowering wholesale prices. These results corroborate previous research on renewable energy and electricity prices Sensfuss2008,Wuerzburg2013,Clo2015,Hirth2013 and further show the dynamic adjustment process.
The estimated response of solar generation is subdued. In DK1, the solar response stays near zero across different time horizons. In DK2, the point estimates are positive at shorter horizons and decrease over time, but the wide uncertainty intervals usually include zero. These findings should be viewed with caution. They might indicate that solar generation played a smaller role in Denmark during the period studied, reflect the high intraday concentration of solar output, or be affected by using daily average prices, which can mask within-day merit-order effects during daylight hours. Unlike wind generation, the daily data do not strongly suggest that solar generation causes a significant negative price response.
The comparison between DK1 and DK2 reveals spatial differences in how renewable generation affects prices. The wind effect is more pronounced in DK1 than in DK2, aligning with variations in wind penetration, interconnection, market structure, and local generation mixes. This variation is important for policy, as the price effects of expanding renewables can vary across bidding zones based on renewable capacity, transmission limitations, and the presence of flexible resources.
Overall, the findings indicate that integrating renewable energy mainly lowers wholesale prices through wind power, especially over short-term periods. Prices respond to weather-related wind supply shocks within a few days, though not instantaneously. This has important implications for balancing markets, storage investments, demand response, and managing short-term risks. Additionally, the results demonstrate the value of the proposed quasi-Bayesian LP--IV method: it generates smooth impulse responses that are easier to interpret economically, while also providing uncertainty estimates through both pointwise and simultaneous bands.
To evaluate robustness, we explore several alternative specifications. Figure C.1 shows estimates based on the level specification instead of the LD specification. Figures C.2 and C.3 modify the lag length and seasonal flexibility by setting $L=14$ and $N=8$, respectively. Figures C.4 and C.5 adjust the tuning parameter by setting $\rho=2$ and $\rho=8$. Figures C.6 and C.7 change the prior scale to $\kappa=0.1$ and $\kappa=10$. Guided by the weather-state nature of the instruments and the diagnostics for predetermined variables in Section 5.4, Figure C.8 extends the baseline specification by including lagged wind and solar generation, as well as lagged wind and solar potential from $t-1$ to $t-7$. Finally, Figure C.9 presents estimates under the flat prior.
The key findings remain consistent across different exercises. Wind responses consistently show a negative effect in both DK1 and DK2, gradually approaching zero. While the magnitude of responses varies depending on the specifications, the overall pattern remains the same: wind generation leads to a short-term decrease in wholesale electricity prices, with the most significant effect occurring at or shortly after impact. Solar responses follow the same overall trend as in the baseline but are smaller and less precisely measured than wind responses. Robustness checks do not indicate a consistent negative daily price response to solar generation.
The lag-augmented specification tackles the concern that baseline estimates might reflect serial correlation in weather or renewable generation. The estimates remain largely similar to the original, particularly for short-term wind responses. This suggests that the baseline results are primarily influenced by current weather-driven renewable supply fluctuations rather than ongoing renewable condition persistence. We do not adopt the lag-augmented model as our standard baseline because including lagged renewable state controls could absorb part of the dynamic adjustment that the IRF aims to capture.
The comparison between the roughness-penalty prior and the flat prior reveals that regularization primarily stabilizes the estimated response path. As anticipated, the flat prior results in less smooth estimates, yet the overall economic interpretation stays consistent. This aligns with the simulation findings in Section 4, where the roughness-penalty prior decreases sampling variability without altering the key implications of the IV estimates.
Overall, the robustness check reinforces the main results from Figure 1: wind generation consistently lowers prices in the short run in both Danish bidding zones, while solar effects are weaker and less precisely measured.
This paper presents a quasi-Bayesian LP--IV approach for estimating dynamic causal effects. The method addresses a common challenge in LP--IV applications: horizon-specific estimates can be flexible but often noisy, especially at longer horizons and with high-frequency data. To mitigate this, the approach treats the IRF as a series of related parameters and employs a roughness-penalty prior to smooth out variations across neighboring horizons. Since inference relies on moment conditions rather than a full likelihood, the framework preserves the interpretability of IV estimation while enabling structured regularization of the entire response path.
The method's theoretical and computational framework is straightforward. Building on Chernozhukov2003, the quasi-posterior is derived from the GMM objective function. Under typical regularity conditions, the quasi-posterior mean aligns asymptotically with the traditional GMM estimator, ensuring that the proposed estimator maintains the primary properties of standard LP--IV. Additionally, the roughness-penalty prior enhances finite-sample stability by sharing information across different horizons. The simulation of the quasi-posterior is carried out using a Gibbs sampler, which takes advantage of the conditional Gaussian structure of the quasi-posterior and the structured precision matrix created by the prior.
The simulation results indicate that the proposed estimator performs effectively in finite samples. Using the flat prior, the quasi-Bayesian estimator closely aligns with the classical GMM, reinforcing the connection between the quasi-posterior and traditional moment-based methods. When employing the roughness-penalty prior, the estimator applies moderate shrinkage, which often leads to a lower RMSE, especially at medium and longer horizons. This is because unrestricted LP--IV estimates tend to be more influenced by sampling noise at these horizons. Overall, these findings support applying cross-horizon regularization when the goal is to obtain a smooth impulse-response function.
The empirical study examines how renewable generation affects wholesale electricity prices in Denmark over time. By using weather-driven renewable potential as instruments for wind and solar output, the findings reveal that wind generation significantly lowers prices in both Danish bidding zones in the short term. The effect is more pronounced in Western Denmark and diminishes over subsequent days, suggesting that the merit-order effect is most relevant at short horizons. Conversely, solar effects are smaller and less precisely estimated with daily data. These results align with existing merit-order literature and advance it by analyzing the complete dynamic price response to renewable supply shocks. Robustness tests confirm that the main conclusions remain consistent across various LP models, lag structures, seasonal controls, prior hyperparameters, lag-augmented renewable controls, and placebo exercises.
The findings have implications for both econometric practice and energy-market policy. Methodologically, the study demonstrates that quasi-Bayesian regularization is effective when the target parameter is a path indexed by a horizon, rather than a single scalar coefficient. Substantively, the empirical results indicate that renewable energy integration influences not only current wholesale electricity prices but also the short-term price dynamics. This is significant for designing balancing markets, storage incentives, demand-response programs, and risk management strategies. As electricity systems increasingly include larger proportions of variable renewable energy, methods capable of reliably estimating dynamic effects amid endogeneity and high-frequency volatility will gain greater importance.
Several avenues remain for future research. One involves extending the framework to overidentified LP--IV systems and creating quasi-Bayesian procedures for testing overidentifying restrictions and assessing instrument validity within the same moment-based approach. Another potential is to develop procedures robust to weak instruments and methods for instrument selection in quasi-Bayesian LP--IV Goh2022. Additionally, enhancing scalability with respect to sample size, projection horizon, and the number of explanatory variables is crucial. These advancements would significantly expand the applicability of quasi-Bayesian LP--IV methods in applied econometrics.
\paragraph{Funding Statement}
This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
\paragraph{Conflict of Interest Statement}
The author declares no conflict of interest.
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