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Quasi-Bayesian Local Projections: Simultaneous Inference and Extension to the Instrumental Variable Method
\title{Quasi-Bayesian Local Projections: Simultaneous Inference and Extension
to the Instrumental Variable Method}
\author{Masahiro Tanaka\thanks{Faculty of Economics, Fukuoka University, Fukuoka, Japan. Address:
8-19-1, Nanakuma, Jonan, Fukuoka, Japan 814-0180. E-mail: [email removed].}}
\maketitle
\begin{abstract}
Local projections (LPs) are widely used for impulse response analysis,
but Bayesian methods face challenges due to the absence of a likelihood
function. Existing approaches rely on pseudo-likelihoods, which often
result in poorly calibrated posteriors. We propose a quasi-Bayesian
method based on the Laplace-type estimator, where a quasi-likelihood
is constructed using a generalized method of moments criterion. This
approach avoids strict distributional assumptions, ensures well-calibrated
inferences, and supports simultaneous credible bands. Additionally,
it can be naturally extended to the instrumental variable method.
We validate our approach through Monte Carlo simulations.\bigskip{}
\end{abstract}
\paragraph*{Keywords:}
local projections; Bayesian inference; Laplace-type estimator; instrumental
variable method; generalized method of moments
\section{Introduction}
Local projections (LPs) \citep{Jorda2005} relate an outcome variable
across horizons to exogenous variables and are primarily used for
impulse response analysis. For reviews, see \citet{Jorda2023} and
\citet{Jorda2025}.
Although frequentist inference for LPs is well studied, Bayesian approaches
remain limited \citep{Tanaka2020a,Ferreiraforthcoming}. A key challenge
is that LPs do not define a likelihood, forcing existing studies to
rely on pseudo-likelihoods. \citet{Tanaka2020a} treats LPs as seemingly
unrelated regressions with multivariate normal errors, estimated using
Gibbs sampling. While \citet{Tanaka2020b} demonstrates consistency
under a specific data-generating process, the posterior is misspecified
and credible intervals can have poor coverage. \citet{Ferreiraforthcoming}
propose a quasi-Bayesian method using \citeauthor{Mueller2013}'s
(2013) sandwich estimator. However, their approach relies on asymptotic
arguments and equation-by-equation variance estimates, which prevents
proper belief updating and constructing simultaneous credible bands
for impulse response functions (IRFs).
This study introduces a quasi-Bayesian framework using the Laplace-type
estimator (LTE) \citep{Kim2002,Chernozhukov2003}. The LTE constructs
a quasi-likelihood based on the generalized method of moments \citep{Hansen1982},
avoiding strong distributional assumptions and extending naturally
to LPs with instrumental variables (LP-IV) \citep{Ramey2018,Stock2018}.\footnote{\citet{Goh2022} construct the quasi-likelihood for an IV regression
in a similar manner.}
The approach offers three advantages. First, the LTE-based quasi-posterior
is ``well calibrated'' with credible intervals aligning with asymptotic
theory. Second, it enables simultaneous credible bands \citep{MontielOlea2019}.
Third, it accommodates IV estimation, providing what appears to be
the first feasible Bayesian method for LP-IV.\footnote{\citet{Huber2024} extend \citeauthor{Tanaka2020a}'s (2020a) pseudo-likelihood
approach, inheriting its misspecification and ignoring the correlations
between first- and second-stage errors.}
The remainder of the paper is structured as follows. Section 2 presents
the LTE framework and posterior analysis and extends the framework
to IV estimation; Section 3 conducts a simulation study; and Section
4 concludes.
\section{Quasi-Bayesian Inference of LPs}
\subsection{LPs and the pseudo-posterior}
LPs estimate the relationship between a response variable observed
at different horizons, $y_{t},y_{t+1},...,y_{t+H}$, and $J$ regressors,
$\boldsymbol{x}_{t}$. Specifically, the model is given by
\[
y_{t+h}=\boldsymbol{\theta}_{\left(h\right)}^{\top}\boldsymbol{x}_{t}+u_{\left(h\right),t+h},\quad h=0,1,...,H;t=1,...,T,
\]
where $\boldsymbol{x}_{t}$ includes an exogenous shock, an intercept,
lags of $y_{t}$, and controls. The first coefficient, $\theta_{1,\left(h\right)}$,
represents the response to the structural shock and its sequence,
$\theta_{1,\left(0\right)},...,\theta_{1,\left(H\right)}$, forms
the IRF. An alternative long-differenced (LD) specification replaces
$y_{t+h}$ with $y_{t+h}-y_{t-1}$, which can reduce bias and improve
coverage \citep{Piger2025}.
Bayesian LP studies \citep{Tanaka2020a,Ferreiraforthcoming} typically
stack LPs into a system of seemingly unrelated regressions with multivariate
normal errors:
\[
\boldsymbol{y}_{t}=\boldsymbol{\Theta}^{\top}\boldsymbol{x}_{t}+\boldsymbol{u}_{t},\quad\boldsymbol{u}_{t}\sim\mathcal{N}\left(\boldsymbol{0}_{H+1},\boldsymbol{\Sigma}\right),
\]
with $\boldsymbol{\Theta}=\left(\boldsymbol{\theta}_{\left(0\right)},...,\boldsymbol{\theta}_{\left(H\right)}\right)$.
This leads to a pseudo-likelihood in standard form. While the point
estimates of $\boldsymbol{\Theta}$ may be consistent, posterior uncertainty
is misspecified; therefore, credible intervals lack proper coverage.
\citet{Ferreiraforthcoming} address this by applying \citeauthor{Mueller2013}'s
(2013) sandwich estimator equation by equation. However, their approach
faces two key issues: (i) the posterior does not represent a proper
belief update, as the contribution of priors to posteriors is not
appropriately handled, leading to invalid point and interval estimates;
and (ii) they cannot generate simultaneous credible bands because
uncertainty is quantified equation by equation. Adjusting the pseudo-likelihood
with a learning rate, as in generalized/Gibbs posteriors \citep{Martin2022,Wu2023},
is a potential remedy but existing selection methods are unsuitable
for this.
\subsection{The quasi-posterior}
We propose inferring LPs using the LTE \citep{Kim2002,Chernozhukov2003},
which is based on moment conditions, $\mathbb{E}\left[\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)\right]=\boldsymbol{0}$
with $\boldsymbol{\theta}=\textrm{vec}\left(\boldsymbol{\Theta}\right)$.
For LPs, the moment function is defined as
\[
\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)=\left(\begin{array}{c}
\left(y_{t}-\boldsymbol{\theta}_{\left(0\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{x}_{t}\\
\left(y_{t+1}-\boldsymbol{\theta}_{\left(1\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{x}_{t}\\
\vdots\\
\left(y_{t+H}-\boldsymbol{\theta}_{\left(H\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{x}_{t}
\end{array}\right).
\]
The sample mean is denoted by $\bar{\boldsymbol{m}}_{T}\left(\boldsymbol{\theta}\right)=T^{-1}\sum_{t=1}^{T}\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)$.
Under standard conditions \citep{Hansen1982}, $\sqrt{T}\bar{\boldsymbol{m}}_{T}\left(\boldsymbol{\theta}^{\text{true}}\right)$
is asymptotically normal:
\[
\sqrt{T}\bar{\boldsymbol{m}}_{T}\left(\boldsymbol{\theta}^{\text{true}}\right)\overset{d}{\rightarrow}\mathcal{N}\left(\boldsymbol{0},\;\boldsymbol{V}\left(\boldsymbol{\theta}^{\text{true}}\right)\right),\quad\text{as }T\rightarrow\infty,
\]
with covariance $\boldsymbol{V}\left(\boldsymbol{\theta}^{\text{true}}\right)=\mathbb{E}\left[\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)^{\top}\right]$.
A consistent estimator $\hat{\boldsymbol{V}}$ is obtained using the
ordinary least squares (OLS) estimator $\hat{\boldsymbol{\theta}}^{\text{OLS}}$.
The quasi-likelihood is then
\[
\tilde{\mathcal{L}}\left(\mathcal{D}|\boldsymbol{\theta}\right)\propto\exp\left\{ -\frac{T}{2}\bar{\boldsymbol{m}}_{T}\left(\boldsymbol{\theta}\right)^{\top}\hat{\boldsymbol{W}}\bar{\boldsymbol{m}}_{T}\left(\boldsymbol{\theta}\right)\right\} ,\quad\hat{\boldsymbol{W}}=\hat{\boldsymbol{V}}\left(\hat{\boldsymbol{\theta}}^{\text{OLS}}\right)^{-1}.
\]
With the prior $p\left(\boldsymbol{\theta}\right)$, posterior draws
are generated from $\pi\left(\boldsymbol{\theta}|\mathcal{D}\right)\propto\tilde{\mathcal{L}}\left(\mathcal{D}|\boldsymbol{\theta}\right)p\left(\boldsymbol{\theta}\right)$.
Fixing $\hat{\boldsymbol{W}}$ avoids repeated matrix inversions,
unlike adaptive weighting schemes \citep{Yin2009,Frazier2024}, which
are computationally demanding and unstable. The posterior mean $\hat{\boldsymbol{\theta}}$
has an asymptotic normal distribution:
\[
\sqrt{T}\left(\hat{\boldsymbol{\theta}}-\boldsymbol{\theta}^{\text{true}}\right)\overset{d}{\rightarrow}\mathcal{N}\left(\boldsymbol{0},\boldsymbol{\Omega}^{*}\right),\quad\text{as }T\rightarrow\infty,
\]
\[
\boldsymbol{\Omega}^{*}=\left(\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{G}\right)^{-1}\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{V}\left(\boldsymbol{\theta}^{\text{true}}\right)\hat{\boldsymbol{W}}\boldsymbol{G}\left(\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{G}\right)^{-1},
\]
\[
\boldsymbol{G}=\boldsymbol{I}_{H+1}\otimes\left(-\frac{1}{T}\boldsymbol{X}^{\top}\boldsymbol{X}\right).
\]
Because of the consistency of the OLS estimator, the covariance of
$\hat{\boldsymbol{\theta}}$ is well approximated by $T^{-1}\left(\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{G}\right)^{-1}$;
therefore, the quasi-likelihood behaves like a proper likelihood when
$T$ is large. The covariance $\boldsymbol{\Omega}=T^{-1}\boldsymbol{\Omega}^{*}$
is estimated by replacing $\boldsymbol{V}\left(\boldsymbol{\theta}^{\text{true}}\right)$
with $\boldsymbol{V}\left(\hat{\boldsymbol{\theta}}\right)$.
Compared with \citet{Ferreiraforthcoming}, our uncertainty quantification
also relies on asymptotics but the quasi-posterior is comparatively
``well calibrated.'' Simulations show that unlike pseudo-posteriors,
our approach avoids discrepancies between Bayesian and frequentist
measures when priors are weak and the gap diminishes---even with
informative priors---as $T$ grows.
For posterior simulation, we reframe the quasi-likelihood as
\begin{eqnarray*}
\tilde{\mathcal{L}}\left(\mathcal{D}|\boldsymbol{\theta}\right) & \propto & \exp\left\{ -\frac{T}{2}\left[\frac{1}{T}\textrm{vec}\left(\boldsymbol{X}^{\top}\left(\boldsymbol{Y}-\boldsymbol{X}\boldsymbol{\Theta}\right)\right)\right]^{\top}\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{G}\left[\frac{1}{T}\textrm{vec}\left(\boldsymbol{X}^{\top}\left(\boldsymbol{Y}-\boldsymbol{X}\boldsymbol{\Theta}\right)\right)\right]\right\} \\
& \propto & \exp\left\{ -\frac{T}{2}\left(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}^{\text{OLS}}\right)^{\top}\boldsymbol{G}^{\top}\hat{\boldsymbol{W}}\boldsymbol{G}\left(\boldsymbol{\theta}-\hat{\boldsymbol{\theta}}^{\text{OLS}}\right)\right\} .
\end{eqnarray*}
This representation facilitates efficient posterior simulation (see
Appendix A.2).
\subsection{Simultaneous credible band}
In our framework, a simultaneous credible band can be computed using
the method proposed by \citet{MontielOlea2019}. A simultaneous $1-\alpha$
credible band for the coefficients of a structural shock, $\boldsymbol{\theta}_{1}=\left(\theta_{1,\left(0\right)},\theta_{1,\left(1\right)},...,\theta_{1,\left(H\right)}\right)$,
is given by the Cartesian product of intervals: $\hat{\mathcal{C}}=\prod_{h=0}^{H}\hat{\mathcal{C}}_{\left(h\right)}$,
where $\hat{\mathcal{C}}_{h}$ denotes an interval such that the true
value of $\boldsymbol{\theta}_{1}$, denoted by $\boldsymbol{\theta}_{1}^{\textrm{true}}$,
is asymptotically covered with a probability of at least $1-\alpha$:
\[
\liminf_{T\rightarrow\infty}P\left(\boldsymbol{\theta}_{1}^{\textrm{true}}\in\hat{\mathcal{C}}\right)=\liminf_{T\rightarrow\infty}P\left(\theta_{1,\left(h\right)}^{\textrm{true}}\in\hat{\mathcal{C}}_{\left(h\right)}\;\textrm{for }h=0,1,...,H\right)\geq1-\alpha.
\]
Let $\hat{\boldsymbol{\Omega}}_{1}$ denote the estimated posterior
covariance matrix for $\boldsymbol{\theta}_{1}$, obtained by deleting
the rows and columns irrelevant to $\boldsymbol{\theta}_{1}$ from
the full matrix $\hat{\boldsymbol{\Omega}}$. Let $\hat{\varsigma}_{\left(h\right)}$
denote the point-wise standard error for $\theta_{1,\left(h\right)}$,
which is computed as $\hat{\varsigma}_{\left(h\right)}=T^{-\frac{1}{2}}\sqrt{\hat{\boldsymbol{\Omega}}_{1,\left(h,h\right)}}$,
where $\hat{\boldsymbol{\Omega}}_{1,\left(h,h\right)}$ denotes the
$\left(h+1\right)$th diagonal element of $\hat{\boldsymbol{\Omega}}_{1}$.
Given a critical value $c$, a credible band is defined as
\[
\hat{\mathcal{B}}\left(c\right)=\hat{B}\left(\hat{Q}_{1-\alpha}\right)=\bigtimes{}_{h=0}^{H}\left[\hat{\theta}_{1,\left(h\right)}-\hat{\varsigma}_{\left(h\right)}c,\;\hat{\theta}_{1,\left(h\right)}+\hat{\varsigma}_{\left(h\right)}c\right].
\]
Random vectors $\boldsymbol{e}_{1},...,\boldsymbol{e}_{N}$ are generated
from a multivariate normal distribution with mean zero and covariance
matrix $\hat{\boldsymbol{\Omega}}_{1}$,
\[
\boldsymbol{e}_{i}=\left(e_{i,\left(0\right)},e_{i,\left(1\right)},...,e_{i,\left(H\right)}\right)^{\top}\sim\mathcal{N}\left(\boldsymbol{0}_{H+1},\hat{\boldsymbol{\Omega}}_{1}\right),
\]
and then $c$ is chosen using the draws:
\[
c=q_{\xi}\left(\max_{h=0,1,...,H}\left|\hat{\Omega}_{1,\left(h,h\right)}^{-\frac{1}{2}}e_{i,\left(h\right)}\right|\right),
\]
where $q_{\xi}\left(\cdot\right)$ denotes the $\xi$-quantile function.
The pseudo-code of this procedure is shown in Algorithm A.1 in Appendix
A.3.
\subsection{Extension to the IV method}
Our framework can be naturally extended to the IV method \citep{Jorda2015,Ramey2016,Stock2018}.
Let $\boldsymbol{z}_{t}$ denote a vector consisting of an IV and
exogenous variables, which may overlap with the covariates $\boldsymbol{x}_{t}$.
As we assume that the dimensions of $\boldsymbol{z}_{t}$ and $\boldsymbol{x}_{t}$
are the same, the model is just-identified. An LP-IV is specified
as
\begin{eqnarray*}
x_{1,t} & = & g\left(\boldsymbol{z}_{t}\right)+u_{t}^{\prime},\\
y_{t+h} & = & \boldsymbol{\theta}_{\left(h\right)}^{\top}\boldsymbol{x}_{t}+u_{\left(h\right),t+h},\quad h=0,1,...,H,
\end{eqnarray*}
where $u_{t}^{\prime}$ and $u_{\left(h\right),t+h}$ are error terms.
We infer $\boldsymbol{\theta}$ using the following moment conditions:
\[
\boldsymbol{m}_{t}\left(\boldsymbol{\theta}\right)=\left(\begin{array}{c}
\left(y_{t}-\boldsymbol{\theta}_{\left(0\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{z}_{t}\\
\left(y_{t+1}-\boldsymbol{\theta}_{\left(1\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{z}_{t}\\
\vdots\\
\left(y_{t+H}-\boldsymbol{\theta}_{\left(H\right)}^{\top}\boldsymbol{x}_{t}\right)\boldsymbol{z}_{t}
\end{array}\right).
\]
See \citet{Stock2018,Rambachan2025} for discussions on the point
identification of dynamic causal effects. The covariance of the moment
function $\boldsymbol{V}\left(\boldsymbol{\theta}^{\text{true}}\right)$
is estimated using the two-stage least squares estimator.
This approach differs from standard Bayesian IV methods \citep{Kleibergen2003,Conley2008,Lopes2014}
in two respects. First, it does not require assumptions about the
joint distribution of $u_{t}^{\prime}$ and $u_{\left(0\right),t},....,u_{\left(H\right),t+H}$.
Second, it avoids explicitly specifying or estimating the first-stage
equation.
In this study, we focus solely on the use of external instruments
\citep{Stock2012,Mertens2013}. LPs can also be applied with internal
instruments, obtained by estimating another model, such as structural
vector autoregressions \citep{Ramey2011,Plagborg-Moller2021}. However,
this two-step approach is not fully compatible with Bayesian analysis
because it ignores the uncertainty associated with the first-stage
estimation.
\section{Simulation study}
\subsection{LPs}
To assess finite-sample properties, we conducted a simulation comparing
four approaches. Pseudo-raw uses the pseudo-likelihood of \citet{Tanaka2020a}
with Gibbs sampling and computes credible intervals from the percentiles
of the draws. Pseudo-asymp, which corresponds to \citet{Ferreiraforthcoming},
applies \citeauthor{Mueller2013}'s (2013) sandwich estimator to the
same draws. LTE-raw uses draws from the LTE quasi-posterior with Gibbs
sampling, while LTE-asymp evaluates credible intervals with the asymptotic
covariance estimator. We employed the non-informative (NI) prior and
roughness-penalty (RP) prior \citep{Tanaka2020a}, with shrinkage
parameters inferred using standard half-Cauchy priors. Where necessary,
a scale-invariant Jeffreys prior is assigned to $\boldsymbol{\Sigma}$,
$p\left(\boldsymbol{\Sigma}\right)\propto\left|\boldsymbol{\Sigma}\right|^{-\left(H+2\right)/2}$.
Both the standard estimator and the heteroskedasticity- and autocorrelation-robust
(HAR) estimator \citep{Newey1987} were used to estimate the covariance.
We considered $T\in\left\{ 200,500,1000\right\} $, with $H=L=7$,
generating 1,000 datasets. A total of 50,000 draws were simulated
and the last 40,000 draws were used for the analysis. See Appendices
A.1 and A.2 for further details.
Tables A.1 and A.2 in the Appendix report the complete results for
the coverage of the point-wise 90\% credible interval (P-Coverage).
Three findings stand out. First, the HAR covariance estimator led
to under-coverage, while the standard estimator performed better,
consistent with \citet{MontielOlea2021}. Second, the LD and level
specifications yielded similar results. Third, Pseudo-raw coverage
was far from nominal, while Pseudo-asymp, LTE-raw, and LTE-asymp delivered
well-calibrated intervals.
Figure 1 displays the results for P-Coverage. We focus on the results
for the LD specification and standard covariance estimator. With the
NI prior, Pseudo-raw and Pseudo-asymp diverged substantially, while
LTE-raw and LTE-asymp were nearly identical and had close to nominal
coverage. With the RP prior, Pseudo-raw and Pseudo-asymp exhibited
significantly different coverage, implying that the contribution of
the prior was not properly accounted for in the posterior simulation.
By contrast, the discrepancies between LTE-raw and LTE-asymp were
smaller and diminished with larger samples.
\begin{figure}
\caption{Simulation results for LPs: Coverage probability of the point-wise
90\% credible interval}
\medskip{}
\centering{}\includegraphics[scale=0.5]{Figure_01}
\end{figure}
We also compare the point and interval estimates across the methods.
With the NI prior, the posterior means for Pseudo-raw and LTE-raw
were nearly identical (Figure A.2), whereas they diverged with the
RP prior, reflecting different prior contributions (Figure A.3). The
interval lengths for Pseudo-raw and Pseudo-asymp differed markedly
under both priors (Figures A.4 and A.5). By contrast, LTE-raw and
LTE-asymp produced nearly identical intervals with the NI prior (Figure
A.6), and their differences were smaller and diminished as sample
size grew under the RP prior (Figure A.7).
Table 1 summarizes the results for the coverage of the simultaneous
90\% credible interval (S-Coverage). The results were similar for
LTE-raw and LTE-asymp. Although coverage tended to fall below 90\%,
it generally approached the nominal level with NI priors or larger
samples, consistent with the theory.
\begin{table}
\caption{Simulation results for LPs: Coverage probability of the simultaneous
90\% credible interval}
\medskip{}
\centering{}
\begin{tabular}{llccc}
\hline
\multirow{1}{*}{Prior} & \multirow{1}{*}{Method} & \multicolumn{3}{c}{Coverage probability}\tabularnewline
\hline
& & $T=200$ & $T=500$ & $T=1,000$\tabularnewline
\hline
\multirow{2}{*}{NI} & LTE-raw & 0.848 & 0.884 & 0.890\tabularnewline
& LTE-asymp & 0.849 & 0.881 & 0.891\tabularnewline
\hline
\multirow{2}{*}{RP} & LTE-raw & 0.769 & 0.849 & 0.883\tabularnewline
& LTE-asymp & 0.867 & 0.882 & 0.896\tabularnewline
\hline
\end{tabular}
\end{table}
\subsection{LP-IV}
We investigated the finite-sample properties of the proposed LP-IV
approach. The simulation setting was essentially identical to that
in Section 3.1. The vector $\boldsymbol{z}_{t}$ was the same as $\boldsymbol{x}_{t}$,
except that the structural shock (the first entry of $\boldsymbol{x}_{t}$)
was replaced with an IV.
Tables A.5-A.8 in the Appendix present the results. As in the case
without IVs, the LD and level specifications performed similarly,
with no clear dominance, while the standard covariance estimator outperformed
the HAR estimator, which tended to underestimate uncertainty. Accordingly,
we focus on the results for the LD specification and standard covariance
estimator.
Figure 2 shows the results for P-Coverage. Under the NI prior (first
row), LTE-raw and LTE-asymp exhibited nearly identical coverage, reaching
the nominal level. With the RP prior, their coverage differed but
converged as the sample size increased. Table 2 reports the S-Coverage
results. With the NI prior, LTE-raw and LTE-asymp again performed
almost identically. With the RP prior, their coverage diverged but
gradually converged as $T$ increased. Overall, these results suggest
that the proposed approach performs well for LP-IV.
\begin{figure}
\caption{Simulation results for LP-IV: Coverage probability of the point-wise
90\% credible interval}
\medskip{}
\centering{}\includegraphics[scale=0.5]{Figure_02}
\end{figure}
\begin{table}
\caption{Simulation results for LP-IV: Coverage probability of the simultaneous
90\% credible interval}
\medskip{}
\centering{}
\begin{tabular}{llccc}
\hline
\multirow{1}{*}{Prior} & \multirow{1}{*}{Method} & \multicolumn{3}{c}{Coverage probability}\tabularnewline
\hline
& & $T=200$ & $T=500$ & $T=1,000$\tabularnewline
\hline
\multirow{2}{*}{NI} & LTE-raw & 0.882 & 0.895 & 0.896\tabularnewline
& LTE-asymp & 0.882 & 0.896 & 0.895\tabularnewline
\hline
\multirow{2}{*}{RP} & LTE-raw & 0.833 & 0.882 & 0.867\tabularnewline
& LTE-asymp & 0.943 & 0.930 & 0.902\tabularnewline
\hline
\end{tabular}
\end{table}
\section{Conclusion}
In this study, we introduced a novel quasi-Bayesian approach for inferring
LPs. The proposed method offers three main advantages over existing
approaches. First, the quasi-posterior based on a generalized method
of moments criterion is ``well calibrated'': its credible intervals
closely match their asymptotic counterparts, ensuring a proper balance
between likelihood and prior in posterior estimation. Second, the
method enables the estimation of simultaneous credible bands. Third,
it naturally extends to IV estimation. While the frequentist literature
has made methodological and empirical contributions, this is the first
study to infer LP-IV within a Bayesian framework. These advances have
broad implications for applied macroeconomics and econometrics, where
LPs are increasingly used to study dynamic causal effects. We hope
this research will stimulate further methodological development and
empirical application.
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