Extracted main text — title through conclusion, appendix excluded. This is what our citation measures are computed over, published so the extraction can be checked by eye.
92,806 characters · 23 sections · 0 citation commands
Causal mechanism and mediation analysis for macroeconomics dynamics: a bridge of Granger and Sims causality
\pagenumbering{arabic} \setcounter{section}{0} \setcounter{page}{1}
\onehalfspacing
Understanding the dynamic causal effects of external interventions and their transmission mechanisms is a central objective in modern macroeconomics. Since the seminal contribution of \nocite*{sims1980macroeconomics}, impulse response functions (IRFs) have become a standard tool for analyzing dynamic causality. A large body of literature has focused on IRFs identification.\footnote{See, e.g., \nocite*{sims1980macroeconomics}, \nocite*{blanchard1989dynamic}, \nocite*{gali1999technology}, \nocite*{romer2004new}, \nocite*{uhlig2005effects}, \nocite*{mertens2013dynamic}, \nocite*{stock2018identification}, \nocite*{nakamura2018high}, and \nocite*{ramey2018government}} While IRFs offer an aggregate view of shock propagation, they do not disentangle the contributions of distinct transmission mechanisms. In particular, they overlook the role of mediator variables, commonly referred to as causal channels, that transmit the effects of shocks over time.
Although empirical studies have examined specific channels in macroeconomic causal transmission,\footnote{See, e.g., \nocite*{ramey2011identifying}, \nocite*{mian2013household}, \nocite*{bekaert2013risk}, \nocite*{jurado2015measuring}, \nocite*{jorda2015leveraged}, \nocite*{gali2015monetary}, \nocite*{ramey2018government}, \nocite*{jarocinski2020deconstructing}, \nocite*{greenwald2020credit}, and \nocite*{bauer2023reassessment}} a formal econometric framework for decomposing the dynamic causal effects of structural shocks remains underdeveloped. This limitation is particularly salient given that mediator variables can exert dynamic influences across different horizons, shaping both the interpretation of macroeconomic dynamics and the policy implications. To address this gap, this paper develops a structured econometric framework that quantifies the contribution of mediator variables to the dynamic causal effects of an exogenous shock, thereby offering a more nuanced understanding of the underlying transmission mechanisms.
This paper introduces a novel econometric framework, impulse response decomposition, which integrates mediation analysis into a time series setting (\citename{baron1986moderator} \citeyear*{baron1986moderator}, \citename{mackinnon2012introduction} \citeyear*{mackinnon2012introduction}, \citename{vanderweele2015explanation} \citeyear*{vanderweele2015explanation}, \citename{pearl2022direct} \citeyear*{pearl2022direct}) . This approach systematically decomposes impulse response functions into distinct components, providing a more precise characterization of macroeconomic transmission mechanisms. Specifically, we isolate the contribution of mediator variables to the total impulse response, assessing their dynamic roles in the propagation of shock. The decomposition framework formally links Sims and Granger causality, that is, Sims’ impulse responses capture the total effect of an exogenous intervention, while Granger (non)-causality from the mediator to the outcome variable reflects the (non)-existence of the mediator’s attribution to the total effect. By clarifying these causal relationships, our methodology improves the interpretability of impulse responses while offering a structured econometric tool to identify macroeconomic transmission channels.
First, we extend Sims’ impulse response analysis by developing a decomposition framework that quantifies the dynamic role of mediator variables at different stages of shock transmission. Our approach emphasizes a key but often overlooked insight: shocks influence outcomes only through the responses of endogenous variables within the dynamic system. By isolating the response of the outcome variable from each mediator, we attribute specific portions of the overall impulse response to individual channels, clarifying their distinct contributions. Importantly, the framework captures the time-varying nature of mediation effects in dynamic model, recognizing that different mediators can become relevant on different horizons, some exerting influence in the immediate aftermath of a shock, others emerging later. This dynamic decomposition provides a structured and granular view of evolving causal mechanisms.
Second, we justify our impulse response decomposition by embedding it within the framework of causal mediation analysis. We begin by defining total and mediation effects nonparametrically using the potential outcomes framework and demonstrate that, under a parametric VAR model for the data-generating process, the impulse response decomposition with respect to mediators coincides with the average mediation effect. We further establish that multi-horizon Granger non-causality (\citename{dufour1998short}, \citeyear*{dufour1998short}) is a sufficient condition for zero average mediation effect at that horizon. This result offers a novel interpretation of Granger causality: beyond reflecting the prevailing predictive content, it captures the mediation effect.
Third, we introduce a novel dynamic mediation index to quantify the evolving role of a specific mediator in the transmission of structural shocks over time. The index is constructed via a cosine-based inner product applied to the weighted components of the impulse response decomposition, capturing whether a given mediator amplifies or attenuates the effect of a shock at each horizon. By plotting these time-varying weights, we offer both a visual and quantitative depiction of the mediator’s shifting influence, shedding light on how transmission channels develop across the lifespan of impulse response functions. This methodology enables a more nuanced understanding of structural shock propagation and uncovers previously unobserved dimensions of the mechanisms underlying macroeconomic shock propagation.
Finally, we employ our framework to examine the role of investment sentiment, often associated with the sentiment channel, in the transmission of monetary policy shocks to aggregate output. Specifically, we decompose the impulse response function to isolate the mediation effect of investment sentiment, proxied by the excess bond premium (\citename{gilchrist2012credit}, \citeyear*{gilchrist2012credit}), and assess its dynamic contribution over time. We contrast these findings with the expected default risk channel and find that the component of investment sentiment induced by monetary policy shocks explains approximately 60% of the output response within the first 12 months. In contrast, the expected default risk contributes negligibly throughout the transmission process.
It is important to emphasize, particularly in light of the Lucas Critique, that our framework does not evaluate hypothetical policy interventions or simulate counterfactual regimes. Instead, our impulse-response decomposition provides an ex post attribution of how mediator variables contribute to the dynamic response of the outcome within the observed policy environment. Consequently, this analysis provides a descriptive framework for characterizing the prevailing data-generating process. This orientation distinguishes our approach from conventional counterfactual analyses in macroeconomics and mitigates concerns about the instability of structural relationships across alternative policy regimes.
Literature—Our motivation stems from the long-standing challenge of uncovering the causal mechanisms through which structural shocks propagate in macroeconomic systems. \nocite*{bernanke1995inside} described monetary-policy transmission as a “black box,” emphasizing the need to identify the roles of certain intermediate variables. Since then, a large empirical literature has examined specific channels, including credit frictions, balance-sheet effects, and uncertainty, see, e.g., \nocite*{gilchrist2012credit}, \nocite*{jurado2015measuring}, \nocite*{baker2016measuring}, and \nocite*{mian2013household}. More recent work highlights investor sentiment in monetary transmission \nocite*{bauer2023risk}, \nocite*{jarocinski2020deconstructing}. While informative, the literature lacks a unified framework for dynamic causal analysis.
Our framework is based on impulse response literature and complements dynamic causal analysis. Although IRFs are the workhorse tool for summarizing the total dynamic effects of shocks \nocite*{blanchard1989dynamic}, \nocite*{kilian2011reliable}, \nocite*{nakamura2018high}, they do not identify transmission channels in themselves. Counterfactual analyses, such as \nocite*{bernanke1997systematic}, \nocite*{sims2006does}, \nocite*{kilian2011does}, and \nocite*{chen2023direct}, simulate transmission restrictions but are often scenario specific. Our method provides a systematic attribution of dynamic responses to mediators.
Methodologically, we extend the causal mediation analysis, originating in psychology and epidemiology, to a macroeconomic time series context.\footnote{See, e.g., \nocite*{baron1986moderator}, \nocite*{robins1992identifiability}, \nocite*{robins2003semantics}, \nocite*{avin2005identifiability}, \nocite*{imai2010identification}, \nocite*{vanderweele2015explanation}, and \nocite*{pearl2022direct}.} Although mediation tools are widely used in microeconometrics, applications to dynamic systems remain limited. Our approach embeds mediation within a VAR environment and delivers a dynamic causal decomposition.
Finally, we connect Sims causality, which measures total dynamic effects via impulse responses \nocite*{sims1980macroeconomics}, with multi-horizon Granger causality, which captures predictive content across horizons \nocite*{granger1969investigating}, \nocite*{dufour1998short}. Although their relationship has been explored theoretically \nocite*{chamberlain1982general}, \nocite*{dufour1993relationship}, \nocite*{white2010granger}, \nocite*{kuersteiner2010granger}, it has rarely been studied through a mediation lens. We show that Granger non-causality of a mediator for the outcome implies a zero average mediation effect, establishing a direct structural link between predictive relevance and causal transmission.
Outline--Section (ref) reviews Sims’ impulse response functions and Granger causality. Section (ref) outlines the motivation for our study, highlighting the limitations of Sims’ impulse response analysis in identifying the transmission mechanisms of macroeconomic shocks. Section (ref) introduces our impulse response decomposition methodology. Section (ref) provides a theoretical foundation for the decomposition from the perspective of causal mediation analysis and establishes a connection between Sims and Granger causality. Section (ref) develops a dynamic mediation index to assess the evolving importance of different causal channels over time. Section (ref) applies our methodology to study monetary policy transmission through the sentiment channel. Finally, Section (ref) concludes.
In this section, we review the concepts of Granger and Sims causality.
We consider a \( K \)-dimensional macroeconomic time series process \( W_t\) governed by a vector autoregressive (VAR) model\footnote{For simplicity, the process \( W_t \) is assumed to be demeaned and detrended.}:
where \( \Phi(L) := I - \sum_{h=1}^p \Phi_h L^h \), the lag order \( p \) may be infinite, and the innovation process \( u_t \) is a \( K \times 1 \) white noise vector with mean zero and full-rank covariance matrix, \( u_t \sim (\mathbf{0}, \Sigma_u) \). For the purposes of this paper, we focus on the case \( K = 3 \), where \( W_t = (X_t, Y_t, M_t)' \) consists of three variables. Readers may interpret \( X \) as the variable receiving an exogenous intervention, \( Y \) as the outcome of interest, and \( M \) as a potential mediating variable.
Granger causality assesses whether past values of a variable $X$ improve the forecast of another variable $Y$ at one period ahead, beyond the predictive content of $Y$'s own past. This concept, originally introduced by \nocite*{wiener1956theory} and \nocite*{granger1969investigating}, provides a widely used framework for evaluating predictive relationships in time series. A more general formulation, multi-horizon Granger causality, has been thoroughly studied by \nocite*{dufour1998short}, recognizing that causal effects may emerge indirectly over time.\footnote{Also see \nocite*{lutkepohl1993testing}, \nocite*{dufour2006short}, \nocite*{dufour2010short}.}
We formally present the definition of Granger non-causality at horizon $h \geq 1$ as follows:
The concept of multi-horizon Granger causality highlights an important insight: a variable $X$ may Granger cause another variable $Y$ at longer horizons through intermediate channels, such as a mediator $M$, even if no causal relationship is detected at the one-period horizon. In this paper, we further explore this insight by investigating its implications in the context of dynamic treatment effects.
The VAR model in (ref) provides a parametric framework for evaluating causal relationships among variables. To capture causality that varies across horizons, we adopt the multi-horizon linear projection (MHLP) approach (see \nocite*{dufour1998short}), which extends the traditional VAR framework by generating a sequence of conditional expectations for $W_{t+h}$ based on the information available at time $t$. The MHLP representation is specified as follows:
where $u_t^{(h)} = (u_{X,t}^{(h)}, u_{Y,t}^{(h)}, u_{M,t}^{(h)})'$ denotes the projection error of $h$ steps ahead and $\{ \Phi_j^{(h)} \}$ are the projection coefficients specific to the horizon. These coefficients, termed generalized impulse responses (gir) by \nocite*{dufour1998short}, evolve recursively according to:
Note that \nocite*{dufour1998short} introduced the gir coefficients, which collect all coefficients in this MHLP. They demonstrate that these coefficients are necessary to test multi-horizon Granger causality. In particular, Theorem 3.1 in \nocite*{dufour1998short} establishes that the condition $X \overset{h}{\not\rightarrow} Y \mid M$ holds if and only if the generalized impulse responses $\Phi_{YX,k}^{(h)}=0$ for all $k \geq 1$. In this paper, we show that these coefficients also provide a convenient tool to assess the mediation effect of a specific variable within a shock transmission; more details are given in Section (ref).
Our gir coefficients differ fundamentally from the generalized impulse responses of \nocite*{pesaran1998generalized}. The gir coefficients in this paper comprise the entire set of MHLP coefficients. By contrast, Pesaran and Shin’s gir is conceptually closer to Sims’ impulse response: it is obtained by post-multiplying the first MHLP coefficient by a vector that accounts for the contemporaneous effect of the shock. As a result, their gir measures the causal effect of a structural shock, possibly in a nonlinear setting, but does not capture the mediation effects.
The impulse response function (IRF), introduced in the seminal work of \nocite*{sims1980macroeconomics}, is a foundational tool in macroeconomics to quantify the dynamic treatment effects of structural shocks. To characterize the underlying structural dynamics, \nocite*{sims1980macroeconomics} suggest orthogonalizing VAR innovations using identifying restrictions, such as Cholesky decomposition, and refer to the resulting model as a structural vector autoregressive (SVAR) process, given by
for \( t \in \mathbb{Z} \), where \( \Theta(L) = \Theta_0 - \sum_{h=1}^p \Theta_h L^h \), \( \Theta_0 \) is a full-rank matrix with ones on the main diagonal, \( \Theta_0^{-1}D\Theta_0^{'-1} = \Sigma_u \), and \( {\Greekmath 0122}_t =\Theta_0 u_t \) is a \( 3 \times 1 \) vector of serially uncorrelated structural disturbances with \( {\Greekmath 0122}_t \sim (\mathbf{0},D) \) and $D = \relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{diag}({\Greekmath 011B}_i^2)_{i=1,2,3}$. It is well known that $\Theta_0$ is typically not identified from $\Sigma_u$ alone. Rather than identifying the entire $\Theta_0$, we impose identification only for the shock of interest ${\Greekmath 0122}_{X,t}$.
Assumption (ref) deliberately abstracts from the way ${\Greekmath 0122}_{X,t}$ is identified. It accommodates the recursive/Cholesky orderings identification method with $X_t$ ordered first. Then, the impulse response function is defined in terms of latent shock, e.g., see \citename{hamilton1994time} (\citeyear*{hamilton1994time}, Equation (11.4.4)).
For a similar definition in terms of variables with contemporaneous and past controls, see \citename{chamberlain1982general} (\citeyear*{chamberlain1982general}, Definition 2) and \citename{wolf2021same} (\citeyear*{wolf2021same}, Equation (2)).
The impulse response function admits the equivalence between the VAR-based estimand and the local projection estimand; see \nocite*{wolf2021same}. It can be expressed as
where \( \Psi_h \) is the coefficient matrix in the lag polynomial \( \Psi(L) :=I - \sum_{h=1}^\infty \Psi_h L^h= \Phi(L)^{-1} \), under the stationarity assumption. It should be noted that \( \Psi_h \) is shown to be identical to the first coefficient matrix \( \Phi_1^{(h)} \) in the multi-horizon linear projection of \nocite*[Equation (3.6)]{dufour1998short}.
This section explores the limitations of impulse response functions to capture the full dynamics of causal transmission and emphasizes the necessity of examining causal mechanisms in macroeconomics.
Causal relationships in macroeconomics are frequently articulated through specific transmission channels. A textbook account of monetary policy shocks, for instance, typically describes a sequential mechanism in which higher interest rates raise the cost of capital, thereby reducing investment and consumption, and ultimately lowering output and employment. However, this narrative oversimplifies the complexities involved in monetary policy transmission. As highlighted in \nocite*{bernanke1995inside}, the credit channel plays a pivotal role: monetary policy affects the supply of credit through its influence on bank lending behavior and the strength of balance sheets. A contractionary policy may reduce the willingness of banks to lend, thus tightening credit conditions, especially for financially constrained firms and households.
More recent research has emphasized the sentiment channel as a key mechanism that shapes the macroeconomic effects of monetary policy. In particular, investor sentiment and perceptions of financial conditions, often proxied by the excess bond premium (EBP), can substantially influence how the economy responds to policy shocks (\citename{gilchrist2012credit}, \citeyear*{gilchrist2012credit}). A contractionary shock can increase risk aversion, thus raising the EBP and tightening credit conditions. This, in turn, magnifies the declines in investment and consumption, strengthening the contractionary impact on aggregate output.
These perspectives highlight the importance of recognizing that the effects of a structural shock on economic outcomes are mediated by key intermediate variables within a dynamic system. Failure to adequately account for such mediators can result in an incomplete or misleading interpretation of the transmission mechanism. Identifying these causal pathways requires adhering to foundational principles of causal inference in economics, where the objective is to isolate the effect of a specific variable. This involves systematically removing the influence of a particular mediator from impulse responses to assess its marginal contribution.
In the empirical literature that examines the role of monetary policy in the transmission of oil price shocks, \citename*{bernanke1997systematic} (\citeyear*{bernanke1997systematic}, hereafter BGW) propose a thought experiment in which the Federal Reserve holds the Federal Funds Rate (FFR) constant during an oil price shock. By deriving impulse responses of output under this fixed interest rate policy, they find that aggregate output would have declined more sharply in the absence of a monetary policy response. In contrast, \citename{kilian2011does} (\citeyear*{kilian2011does}, hereafter KL) offers an alternative identification strategy that nullifies the contemporaneous and lagged SVAR coefficients of the FFR with respect to oil prices, effectively insulating interest rate responses from oil shocks while allowing reactions to other macroeconomic variables.
These studies provide empirical illustrations of how to isolate and evaluate the contribution of specific mediator variables within the causal transmission of structural shocks, albeit through distinct approaches. Both methods compare impulse responses under restricted and unrestricted scenarios, interpreting the difference as the causal effect attributable to the mediator. However, KL critiques BGW’s method as unrealistic, given the implausibility of a central bank fully suppressing its policy reaction to oil shocks. In contrast, KL's own approach has a limitation. By their construction, a variable with zero SVAR coefficients in terms of the output variable of interest is treated as having no participation in shock propagation, thus failing to capture potential dynamic contributions at longer horizons.
Our perspective highlights the inherently dynamic nature of causal attribution. A mediator may not appear causally relevant on one horizon, but can exert influence on others. This notion aligns with the philosophy of multi-horizon Granger causality, which recognizes that causal relationships may emerge over extended time frames.
This subsection presents a numerical example that illustrates the distinction between Sims and Granger causality. Specifically, we construct a setting in which the Sims impulse response function remains identically zero across multiple horizons, while Granger causality is nonetheless present. The purpose of this example is to emphasize that a zero dynamic treatment effect, as captured by the Sims impulse response, does not preclude the existence of causal relationships in the Granger sense.
Although many economists have traditionally viewed Granger causality as capturing only statistical association, lacking structural or causal interpretation\footnote{See the discussion in \nocite*{granger1980testing}.}, we argue later in this article that Granger causality can, in fact, reveal causal mediation effects in the context of macroeconomic shock transmission. This interpretation highlights the existence of dynamic causal mechanisms, specifically mediation pathways, that are detectable through Granger causality, but may be obscured in Sims impulse response analysis.
To illustrate this point, consider the trivariate VAR(6) model specified below, where the vector $W_t$ evolves according to:
where $u_t$ is a white noise process with an identity covariance matrix. This specification eliminates contemporaneous effects by construction, focusing purely on dynamic interactions across lags.
Table (ref) reports the horizon-specific coefficients $\Phi_{XY,k}^{(h)}$, where $\Phi_{XY,1}^{(h)}$ represents the impulse response à la Sims, and the remaining terms correspond to generalized impulse response coefficients defined in (ref) that capture Granger causality from $Y$ to $X$ at the horizon $h$.
This example highlights the limitations of relying solely on Sims IRFs to characterize dynamic causal mechanisms. Although a zero IRF suggests the absence of a dynamic treatment effect of a structural shock on the outcome variable at a given horizon, it does not rule out the presence of causal mediation processes, as captured by Granger causality. This distinction is particularly relevant for empirical researchers analyzing dynamic models in which the identification and quantification of intermediate variables, often called transmission channels, is of central interest. Exclusive reliance on impulse responses may obscure the fact that dynamic causal effects are often mediated through other variables over periods.
This section develops a novel method, impulse response decomposition, which expresses impulse responses as the contributions of endogenous variables over time. The underlying dynamic structure is further clarified using a directed acyclic graph (DAG).
Sims’ definition of impulse responses in (ref) is based on the response of the system to a unit-size exogenous innovation. This subsection introduces an alternative variable-based definition that reinterprets impulse responses as the dynamic evolution of variables, thereby enabling their decomposition across time. This shift in perspective serves a critical purpose: by formulating impulse responses in terms of variables rather than innovations, one can trace how each variable of the system contributes inter-temporally to the impulse response.
We initiate the decomposition framework by reinterpreting the impulse response as the causal effect of an exogenous intervention on the vector of observables $W_t$ at the initial period. In particular, we focus on the contemporaneous impact of the intervention, which is characterized by an impulse response of magnitude ${\Greekmath 0112}_0$ applied directly to the system through $W_t$.
This perspective motivates an alternative definition of the impulse response: as the dynamic reaction of the system to an exogenous perturbation in the variable $W_t$ of size ${\Greekmath 0112}_0$. Accordingly, the impulse response function can be equivalently defined as:
Accordingly, the impulse response admits the following closed-form representation,
for all $h>0$. Note that the projection matrix $\Phi_1^{(h)}$, which defined in (ref) is the first gir coefficient at horizon $h$, coincides with the Sims' impulse response operator $\Psi_h$. Although the standard definition in (ref) expresses impulse responses with respect to latent shocks, the formulation in (ref) reframes them as arising from direct interventions in the observables. This equivalence highlights a key insight: a latent shock can be viewed as a sudden shift in the observables of the system, with its dynamic propagation governed by the sequence of the gir coefficient matrix $\{ \Phi_1^{(h)} \}_{h > 0}$.
Building on this formulation in terms of observables, we next examine the contribution of a specific variable $M$ in the initial period to the total impulse response at horizon $h$. Given that ${\Greekmath 0112}_h = \Phi_1^{(h)} {\Greekmath 0112}_0$, the marginal effect of $M$ can be isolated by keeping all other variables fixed. This approach ensures that the response of $W_{t+h}$ captures the fluctuations of $M$ alone, thereby avoiding the confounding effects of other contemporaneous variables. In contrast, failing to condition the remaining variables would risk misattributing mediated or joint effects to $M$, resulting in a distorted characterization of its dynamic impact.
Formally, the contribution of mediator $M$ at horizon zero is defined as:
where the reference value of the mediator variable is set to zero, and the intervened value is given by ${\Greekmath 0112}_{M,0}$. This normalization is adopted for notational simplicity and can be readily extended to cases where the reference value is any fixed constant, as long as the difference between the intervened and reference values remains ${\Greekmath 0112}_{M,0}$. In this setting, the contribution of the mediator $M$ within the VAR framework admits the following closed-form representation:
The term, ${\Greekmath 0112}_{h}^{(M_0)}$, is named as impulse response decomposition to the mediator $M$ at horizon zero.
Consequently, the impulse response can be decomposed into variable-specific contributions at horizon zero: $ {\Greekmath 0112}_h = {\Greekmath 0112}_{h}^{(Y_0)} + {\Greekmath 0112}_{h}^{(X_0)} + {\Greekmath 0112}_{h}^{(M_0)},$ where ${\Greekmath 0112}_{h}^{(Y_0)}$ and ${\Greekmath 0112}_{h}^{(X_0)}$ denote the contributions of the remaining variables, obtained analogously.
This decomposition broadens the conventional interpretation of impulse responses by showing that they can be understood not only as reactions to structural shocks, but also as the cumulative effects of exogenous shifts in economic observables. Although the horizon-zero decomposition isolates the contribution of each variable at the initial period, the framework naturally extends to accommodate subsequent periods. This extension preserves the dynamic nature of impulse responses and underscores how the effects of an intervention propagate through the system’s dynamic structure over time.
To further examine the dynamics of propagation, we extend the decomposition framework beyond horizon zero by incorporating variables at period one. This extension is essential because the passage of time following an exogenous intervention activates an expanding set of endogenous responses that collectively shape the impulse response. As a result, the attribution of the overall response to individual variables can evolve across horizons, reflecting their time-varying roles within the transmission mechanism.
To extend the analysis beyond the initial period, we redefine the impulse response on horizon $h$ in terms of the observables at periods $t$ and $t+1$, conditional on the information available throughout the period $t+1$. By projecting $W_{t+h}$ onto this expanded information set, we account not only for the contemporaneous effect of the intervention, but also for its immediate endogenous propagation within the system. This yields an alternative representation of the impulse response that captures the combined influence of adjustments occurring at both the initial and subsequent periods:
This expression admits the following closed-form representation:
for all $h >1$, where $\Phi_1^{(h-1)}$ and $\Phi_2^{(h-1)}$ are the first two gir coefficients at horizon $(h-1)$ presented in (ref), originally proposed by \nocite*{dufour1998short}.
This formulation highlights that the dynamic causal effect of an exogenous intervention in period $t$ can equivalently be represented as the joint impact of adjustments in observables in both periods $t$ and $t+1$. The projection coefficients $\Phi_1^{(h-1)}$ and $\Phi_2^{(h-1)}$ quantify how shocks to observables at these two periods map to the outcome variables. In particular, $\Phi_1^{(h-1)}$ captures the propagation of the shift ${\Greekmath 0112}_1$ in period $t+1$, while $\Phi_2^{(h-1)}$ reflects the continued influence of the ${\Greekmath 0112}_0$ impulse of the initial period. This decomposition emphasizes that a structural shock may be interpreted as a sequence of shifts in observables, whose influence unfolds through successive projection matrices. This generalizes the conventional impulse response operator $\Psi_h$ to account for the sequential movements in variables over time.
Building on this extended formulation, we next examine the contribution of variable $M$ at periods $t$ and $t+1$ to Sims' impulse response at horizon $h > 1$. Given the decomposition in (ref), the marginal effect of $M$ can be studied by isolating the endogenous response of $M$ to the initial intervention captured by ${\Greekmath 0112}_{M,1}$ and ${\Greekmath 0112}_{M,0}$ on horizons one and zero, respectively, while keeping the responses of all other variables constant. This approach ensures that the projected response of $W_{t+h}$ reflects only the dynamic influence of $M$, free from the confounding effects of other contemporaneous or lagged variables.
Formally, the contribution of mediator $M$ at horizon one is defined as:
The choice of reference value follows the same rationale as in the horizon-zero decomposition. Within the VAR framework, the contribution of the mediator $M$ at horizon one admits the following closed-form representation:
We refer to ${\Greekmath 0112}_{h}^{(M_1)}$ as the impulse response decomposition to the mediator $M$ at horizon one. Although we acknowledge the inherently dynamic nature of macroeconomic systems, where a change in one variable induces endogenous responses throughout the system, the exercise conducted here is purely a decomposition of the impulse response with respect to a specific mediator. It does not involve any variable manipulation or intervention in the structural sense.
Accordingly, the total impulse response at horizon $h$ admits the following variable-specific decomposition: $ {\Greekmath 0112}_h = {\Greekmath 0112}_{h}^{(Y_1)} + {\Greekmath 0112}_{h}^{(X_1)} + {\Greekmath 0112}_{h}^{(M_1)},$ where ${\Greekmath 0112}_{h}^{(Y_1)}$ and ${\Greekmath 0112}_{h}^{(X_1)}$ denote the contributions of the remaining variables at horizon one, defined analogously.
By explicitly modeling this two-period contribution, this framework offers a richer view of the transmission mechanism, revealing how the influence of each variable evolves as the system responds dynamically over time. In particular, the marginal effect of a given variable may shift as the system transitions across horizons. For example, a monetary policy shock can initially affect financial variables such as interest rates, which in turn influence real activity, such as consumption or investment, with a delay. As a result, the contribution of a mediator $M$ can differ substantially between horizon zero and one, depending on the timing of its activation within the propagation process.
We now extend the decomposition framework to an arbitrary intermediate horizon $n$, where $n < h$. This generalization yields a unified representation of dynamic responses by systematically tracing the evolution of endogenous variables and projecting their sequential effects onto the outcome at horizon $h$.
Under this generalization, the impulse response function admits the following representation:
for all $h > n$ and $n \geq 1$, where $\{\Phi_k^{(h-n)}\}_{k\geq 1}$ are a sequence of gir coefficients at horizon $(h-n)$ presented in (ref), originally proposed by \nocite*{dufour1998short}. This expression implies that the causal effect of a structural shock at period zero can be interpreted as the accumulated influence of successive changes in observables from periods $0$ through $n$, each projected onto horizon $h$ via its corresponding coefficient.
Accordingly, the impulse response at horizon $h$ can be decomposed into the sum of contributions from observables up to period $n$, as summarized in Table (ref). The table provides a structured overview of how the impulse response ${\Greekmath 0112}_h$ is decomposed across different intermediate horizons. This tabular representation highlights a key feature of the dynamic system: the time-specific contribution of each variable to the overall response at horizon $h$. Within this framework, impulse responses propagate sequentially through the system, with each variable potentially influencing future outcomes. A central insight from this decomposition is that the attribution of the response to individual observables evolves over time. This time-varying attribution underscores the importance of tracing period-specific contributions in order to fully characterize the underlying transmission mechanism.
Following the previous decomposition applied to mediator $M$, we extend the analysis to this general case. This yields the following expression for the contribution of $M$ to the impulse response at horizon $h$:
This formulation captures how the dynamic influence of $M$ evolves across periods and contributes to the total response. Consequently, the impulse response function admits the following variable- specific decomposition at horizon $n$: ${\Greekmath 0112}_h = {\Greekmath 0112}_h^{(M_n)} + {\Greekmath 0112}_h^{(Y_n)} + {\Greekmath 0112}_h^{(X_n)},$ where ${\Greekmath 0112}_h^{(Y_n)}$ and ${\Greekmath 0112}_h^{(X_n)}$ represent the contributions of the remaining variables, defined analogously.
The horizon-$n$ decomposition provides a flexible and informative framework for analyzing causal transmission in dynamic systems. By tracing how the influence of the mediator variable evolves across time, it sheds light on the sequential propagation of shocks through certain causal channels. This temporal mapping could enable researchers to identify dominant mediating variables at different stages of transmission and assess their relative importance across horizons. As a result, the framework supports more effective policy design by revealing when and through which variables interventions are most impactful.
To clarify the structure of our causal dynamics, we introduce a directed acyclic graph (DAG), which provides a transparent and intuitive representation of the dynamic relationships among treatment, mediators, outcomes, and post-treatment confounders.
In the dynamic setting considered, we investigate the causal effect of a structural shock on a future outcome, where this effect may be partially mediated through a sequence of intermediate variables. The DAG in Figure (ref) illustrates the assumed causal structure.
The DAG formalizes the causal channels that link treatment $X_t$ and outcome $Y_{t+h}$. Two intermediate blocks mediate this relationship: (i) a sequence of mediators $M_{t}, \dots, M_{t+n}$, and (ii) a set of post-treatment confounders $Y_{t}, \dots, Y_{t+n}, X_{t+1}, \dots, X_{t+n}$. This structure highlights the importance of accounting for such dependencies when identifying causal effects.
Within this framework, the total effect of $X_t$ on $Y_{t+h}$ can be decomposed into five distinct causal pathways:
Together, these pathways delineate a decomposition of the total treatment effect, measured by the impulse response function, into distinct causal channels, each representing a separate path through which the shock propagates over time.
This section rigorously develops the conceptual foundation for the impulse response decomposition within the framework of causal mediation analysis. We first derive a nonparametric definition of the average mediation effect using the potential outcomes framework. We then prove that the impulse response decomposition of the mediator is equivalent to the average mediation effect by constructing a linear structural equation model (SEM). Finally, we establish a formal connection between Sims causality and Granger causality through the lens of causal mediation analysis.
We depart from the parametric VAR framework to define the average total effect and the average mediation effect within a general nonparametric setting, drawing on insights from the causal inference literature. This broader formulation allows us to disentangle the mechanisms through which a treatment variable $X_t$ influences an outcome $Y_{t+h}$, conditional on the mediator path $M_{t:t+n}$.
Let $Y_{t+h}(x)$ and $M_{t:t+n}(x)$ denote the potential outcomes of the outcome and mediator variables, respectively, when the treatment $X_t$ is set to value $x$. In addition, let $Y_{t+h}\{x, M_{t:t+n}(x')\}$ denote the potential outcome at horizon $h$ under treatment $X_t = x$ and mediator path $M_{t:t+n} = M_{t:t+n}(x')$, where $x, x' \in \{0,1\}$ and $\mathbf{m} \in \mathbb{R}^{n+1}$.
The total effect is defined as:
The causal mediation effect is given by:
for $x \in \{0,1\}$. We adopt the binary treatment convention in alignment with the causal mediation literature, though our framework can be readily extended to accommodate continuous treatments.
The causal mediation effect captures the portion of the effect that operates through the mediator paths $M_{t:t+n}$. This conceptualization aligns with the standard definitions in the causal mediation literature, including \nocite*{robins1992identifiability}, \nocite*{robins2003semantics}, \nocite*{imai2010identification}, \nocite*{imai2013experimental}, and \nocite*{pearl2022direct}.
To define these effects nonparametrically in a dynamic setting, we account for the presence of post-treatment confounders, denoted by $L_{t,n} = (Y_{t}, \dots, Y_{t+n}, X_{t+1}, \dots, X_{t+n})'$. The vector $L_{t,n}$ includes contemporaneous and future variables from horizon zero to $n$, excluding the treatment $X_t$ and the mediator path $M_{t:t+n}$. These variables are classified as post-treatment confounders because they are affected by the treatment and, through the dynamic structure of the VAR model, influence both the mediators and the outcomes. This categorization is consistent with the formal definition of post-treatment confounding. A nonparametric formulation of causal effects in the presence of such confounders allows us to express average potential outcomes conditional on both treatment and mediator trajectories. This approach follows the identification strategies developed in \nocite*{robins1986new}, \nocite*{robins2003semantics}, and \nocite*{vansteelandt2009estimating}.
We formalize the identification strategy through the following assumption:
By Assumption (ref) (i), the treatment is ignorable conditional on the pre-treatment covariates $\overline{W}_{t-1}$. Equivalently, structural innovation ${\Greekmath 0122}_{X,t}$ is independent of the potential outcomes given $\overline{W}_{t-1}$. This assumption can be broadened to include contemporaneous pre-treatment variables, e.g., variables ordered before $X_t$. Assumption (ref) (ii) further posits that the mediator is ignorable conditional on the treatment, the pre-treatment covariates, and post-treatment confounders. Under the parametric VAR specification in (ref), these conditional independence restrictions are maintained.
See the proof in Appendix (ref). It is important to emphasize that the validity of the nonparametric definition of the average mediation effect (AME) critically depends on the sequential ignorability of the mediator. This condition, also known as the stratified exchangeability assumption, requires that there be no unmeasured confounding in the mediator–outcome relationship, conditional on treatment and relevant covariates. Crucially, it also protects against collider bias, which may arise from inappropriate adjustment for post-treatment variables that lie on backdoor paths. The economic interpretation of our AME is to isolate the contribution of the mediator by varying its value while keeping post-treatment confounders fixed at the levels they would attain under treatment. In effect, our AME captures the portion of the treatment effect that is transmitted exclusively through the path from the mediator path to the outcome. Therefore, it can also be called as path-specific effect, as \nocite*{avin2005identifiability}. This structure facilitates a transparent decomposition of the total treatment effect, as previously illustrated through DAGs.
A closely related literature on causal mediation includes the framework developed by \nocite*{imai2010identification}, which is built upon the sequential ignorability of the mediator. A key distinction of our approach lies in its explicit incorporation of post-treatment confounders, motivated both by the critique in \nocite*{robins2003semantics} and by the nature of macroeconomic dynamics, where post-treatment variables often affect both the mediator and the outcome, also considered by \nocite*{avin2005identifiability}. Within this broader literature, our estimator moves beyond the framework of natural direct and indirect effects and instead accommodates settings where post-treatment confounding is present, thus aligning more closely with the path-specific effects proposed by \nocite*{avin2005identifiability}.
We now turn to a parametric framework for identifying the average mediation effects. Specifically, we derive a linear structural equation model (SEM) under the VAR model. This approach is consistent with traditional mediation analysis (e.g., \nocite*{baron1986moderator}; \nocite*{mackinnon2012introduction}; \nocite*{vanderweele2015explanation}) but is extended here to dynamic macroeconomic environments.
The total effect of the treatment variable $X_t$ on the outcome $Y_{t+h}$ is captured by the impulse response. Under the parametric VAR specification in (ref) and Assumption (ref), this yields the linear projection model, often termed the total-effect equation,
where ${\Greekmath 0112}_{Y,h}$ is the impulse response function, ${\Greekmath 0112}_{Y,h}=\partial Y_t/ \partial {\Greekmath 0122}_{X,t}$, $\overline{W}_{t-1}$ are pretreatment covariates and ${\Greekmath 0118}_{Y,t}^{(h)}$ is the projection error. Note that the parameter ${\Greekmath 0112}_{Y,h}$ has a causal interpretation of the treatment effect of the structural shock ${\Greekmath 0122}_{X,t}$ based on Assumption (ref) (i). This equation is often named as ‘local projection’ in macroeconomic literature (\citename{jorda2005estimation}, \citeyear*{jorda2005estimation},\citename{wolf2021same}, \citeyear*{wolf2021same}).
To investigate mediation effects, we define a decomposition horizon $n \in \{0, \dots, h-1\}$ and specify the corresponding dynamic mediator sequence $M_{t:t+n}$. This formulation highlights a key dynamic feature that increasing the decomposition horizon $n$ incorporates a longer history of the mediator into the analysis. Then, under the parametric VAR specification in (ref) and Assumption (ref), we regress the mediator sequence $M_{t:t+n}$ onto treatment and pre-treatment confounders, commonly referred to as the mediator equation,
The mediator equation characterizes the dynamic propagation of the shock to the mediator variables over time. which ${\Greekmath 0112}_{M,0}$ corresponds to the impulse response of the mediator to the structural shock.
Lastly, we regress the outcome variable on the treatment, the mediator sequence $M_{t:t+n}$, and the corresponding confounders; this specification is commonly referred to as the outcome equation,
where $\Phi_{Y\dot,i}^{(h-n)}$ is referred to as generalized impulse response coefficient (see \nocite*{dufour1998short}); here they map the endogenous responses of the treatment, mediator, and post-treatment confounders to the output variable.
Note that the VAR specification ensures that the outcome equation with mediators, given in (ref), does not suffer from endogeneity, as all relevant post-treatment confounders are explicitly taken into account. This feature provides a key practical advantage over frameworks that require the strong assumption of no post-treatment confounding, as highlighted by \nocite*{robins2003semantics}. By conditioning on all endogenous variables affected by the treatment and observed prior to the outcome, the model does not violate the sequential ignorability condition for the mediator, which is crucial for identifying mediation effects.
Equations (ref)–(ref) collectively define a dynamic structural equation model, whose validity is based on Assumption (ref) and the underlying VAR specification. This framework yields the following expression for the average mediation effect.
See the proof in Appendix (ref).
Theorem (ref) justifies our impulse response decomposition from the causal mediation analysis with the assumption of sequential ignorability. The VAR structure in our setting ensures that all relevant post-treatment variables are appropriately controlled, thereby satisfying the required conditional independence. This identification strategy is closely aligned with the framework of \nocite*{imai2010identification}, but allows confounders after treatment.
By embedding mediation analysis within a linear VAR framework, we obtain a tractable and interpretable model that supports rigorous causal decomposition. This setting is particularly well suited for macroeconomic applications, where policy interventions propagate through dynamic systems, and where full ignorability is rarely plausible without structural assumptions. The VAR framework not only ensures correct temporal ordering and control for feedback, but also enables estimation of mediation effects via direct projection, without the need for iterative simulation or integration steps typically required by G-computation.
Nevertheless, we acknowledge that the validity of the sequential ignorability condition depends on model specification. In practice, omitted variable bias remains a concern, especially when the VAR includes a limited set of macroeconomic variables. We argue that the VAR framework with a limited number of variables provides a reasonable approximation of the data-generating process, despite the risk of omitted variables. This approximation can be partially validated by comparing the estimated impulse response functions with those derived from a structural model, an alternative elegant specification that typically includes a similar or even smaller set of variables. More fundamentally, the issue of omitted variable bias can be mitigated by incorporating additional relevant variables. From an econometric perspective, the inclusion of a rich set of covariates is, in principle, feasible in certain high-dimensional settings under assumptions such as approximate sparsity or dense factor structures. In such cases, dimension reduction techniques, such as principal components or regularization methods, can be used in estimation. We leave this extension to future research.
In this subsection, we establish a conceptual bridge between Granger causality and mediation analysis by showing that Granger non-causality at multi-horizon is sufficient for a zero average mediation effect. This correspondence yields a novel interpretation of Granger non-causality as a sufficient condition for the absence of dynamic mediation from the mediator to the outcome variable.
We formalize this insight within a general structural equation model for mediation, defined at a decomposition time \( n \in \{0, 1, \dots, h-1\} \). Our impulse response decomposition separates the total effect into contributions from the treatment \( X \), the mediator \( M \), and dynamic feedback through \( Y \) itself. If the mediator does not Granger-cause the outcome at horizon \( h - n \), denoted \( M \overset{h-n}{\nrightarrow} Y \), then the generalized impulse response coefficients from \( M \) to \( Y \) at that horizon must vanish. Formally, following \nocite*{dufour1998short}, this non-causality condition is given by:
This condition immediately implies the absence of mediation via \( M \) at horizon \( h \), leading to the following proposition:
See the proof in Appendix (ref). This proposition establishes a fundamental connection between Granger non-causality and the absence of mediation effects in a dynamic time series framework. Specifically, it states that if the mediator $M$ does not Granger-cause the outcome variable $Y$, then the average mediation effect, $\mathbb{E}[ {\Greekmath 0120}_{ME,h,n}(x) ]$, is zero. This implies that in the absence of a Granger-causal relationship between $M$ and $Y$, any intervention that induces a change in $M$ will have no indirect effect on $Y$, thereby shutting down the mediation channel.
Statistically, a sufficient condition for the absence of mediation is the nullity of all relevant Granger-causal parameters, i.e., $\Phi_{YM,j}^{(h-n)} = 0$ for all $j \geq 1$. These zero coefficients eliminate any dynamic pathway through which mediation could occur. A weaker yet still sufficient condition, as formalized in the proposition, requires that $\Phi_{YM,j}^{(h-n)} = 0$ for $1 \leq j \leq n+1$. This follows directly from the parametric definition of the mediation effect, see (ref), $$ {\Greekmath 0112}_{Y,h}^{(M_n)} = \Phi_{YM,1}^{(h-n)} {\Greekmath 0112}_{M,n} + \cdots + \Phi_{YM,n}^{(h-n)} {\Greekmath 0112}_{M,1} + \Phi_{YM,n+1}^{(h-n)} {\Greekmath 0112}_{M,0}. $$ At decomposition horizon $n$, only the dynamic path $M_t, \ldots, M_{t+n}$, activated by the initial structural shock, contributes to the mediation effect. If all associated mapping coefficients $\Phi_{YM,j}^{(h-n)}$ equal to zero, the entire mediation mechanism is deactivated.
This result underscores that the role of a mediator can be horizon-dependent. It may be irrelevant at one stage of shock propagation, but pivotal at another. The proposed dynamic mediation framework explicitly accommodates this horizon-specific structure by allowing the importance of mediators to vary over time. Consequently, it is essential to evaluate generalized impulse response coefficients across multiple horizons to fully capture the temporal evolution of mediation effects.
From a practical point of view, the proposition offers a conceptual bridge between the Sims and Granger frameworks. Although Sims' impulse responses quantify the total effect of an exogenous shock, generalized impulse responses, grounded in Granger-causal relationships, enable a decomposition of that total effect into direct and mediated components. Using this decomposition, our framework facilitates dynamic mediation analysis in macroeconomic systems, providing a unified and robust approach to uncovering time-varying causal mechanisms.
In this section, we introduce a Dynamic Mediation Index for assessing causal channels, building on the novel concept of impulse response decomposition.
As established in the preceding discussion, the impulse response at horizon $h$ can be decomposed at any earlier time $n$, where $0 \leq n < h$. This decomposition is economically meaningful as it enables a structured evaluation of the contribution of mediator variable to the impulse response at a given horizon. Because the decomposition is indexed by both the response horizon $h$ and the time of decomposition $n$, it naturally forms a triangular array that captures how the mediator’s contribution evolves over time.
Although this two-dimensional structure provides rich information, it is often desirable to summarize the mediator’s influence using an index. Specifically, our objective is to quantify the attribution of the mediator, measured at a given decomposition time $n$, to the pattern of future impulse responses up to a terminal horizon $H$. This index offers a compact but informative characterization of the influence of the mediator over time while preserving the temporal structure of the decomposition.
To formalize this idea, we introduce the Dynamic Mediation Index, which measures the attribution of the mediator at horizon $n$ to the future trajectory of the impulse response up to horizon $H$. This index is constructed using the inner product:
for $0\leq n <H$, where \( {\mathbold{\Greekmath 0112}}_{Y,n,H}:=({\Greekmath 0112}_{Y,n+1},{\Greekmath 0112}_{Y,n+2},\cdots.{\Greekmath 0112}_{Y,H})' \) denotes the impulse response functions from horizon \( n+1 \) to the upper bound horizon \( H \), and \( {\mathbold{\Greekmath 0112}}_{Y,n,H}^{(M)}:=({\Greekmath 0112}_{Y,n+1}^{(M_n)},{\Greekmath 0112}_{Y,n+2}^{(M_n)},\cdots,{\Greekmath 0112}_{Y,H}^{(M_n)})' \) represents the contribution of mediator variable \( M \) to the impulse response over the same horizons, given the decomposition time \( n \). The symbol, \( \langle \cdot, \cdot \rangle \), denotes the inner product. Note that we define $\relax\protect\ifmmode\expandafter\text@\else\expandafter\mbox\fi{DMI}_{Y,H,H}^{(M)} = 0$, indicating that at the terminal horizon, the mediator contributes no further to the impulse response.
The index admits the following interpretation:
From an economic perspective, the index captures the extent to which the mediator’s contribution aligns with the pattern of impulse responses, preserving both shape and magnitude. This framework provides a systematic approach to evaluating the evolving role of the mediator in shock transmission, offering a more nuanced understanding of the mediator’s channel within the macroeconomic dynamics.
In this section, we implement the impulse response decomposition and explore the dynamic attributions of investor sentiment in shaping the macroeconomic effects of monetary policy shocks.
We perform our impulse response decomposition analysis using a monetary dataset consisting of 383 monthly observations from February 1988 to December 2019. The data set includes eight key U.S. macroeconomic indicators: Industrial Production (IP), Consumer Price Index (CPI), Excess Bond Premium (EBP), Expected Default Risk (EDR), Unemployment Rate (UNEMP), Personal Consumption Expenditures (PCE), the 2-Year Treasury Rate, and Wages. Monetary policy shocks are identified using the orthogonal high-frequency instrumental variable (HF-IV) proposed by \nocite*{bauer2023reassessment}.
The EBP and EDR series are proxy variables for investor sentiment and firm-level economic fundamentals, respectively, provided by \citename{gilchrist2012credit} (2012, hereafter GZ). GZ introduced the Gilchrist-Zakrajek credit spread as a market-based measure of corporate credit risk. The GZ spread combines two distinct components: expected default risk, estimated using firm-level fundamentals such as leverage and equity volatility; and the excess bond premium, which captures the compensation investors demand for bearing credit risk beyond expected losses and reflects market sentiment and risk pricing.
To evaluate impulse responses to a monetary policy shock, we estimate a Vector Autoregressive model with twelve lags, following the specification used by \nocite*{bauer2023reassessment} to ensure comparability of the estimated responses. Our primary objective is to assess the dynamic effects of monetary policy on aggregate output, proxied by industrial production. We then decompose the estimated impulse responses across horizons to examine the evolving contributions of each variable in the VAR model.
The decomposition is performed at four distinct periods: the initial period (horizon zero), and at 3, 6, and 12 months following the shock. At each period, we decompose the impulse responses of aggregate output, up to horizon 36, into the contributions of individual variables. This approach reveals the evolving structure of the transmission mechanism, highlighting how the relative importance of each channel in mediating the monetary policy shock varies over time.
Figure (ref) presents the decomposition of the impulse response function of industrial production to monetary policy shock. In each panel, the solid black line represents the total impulse responses, while the stacked bars illustrate the contributions of individual macroeconomic variables, as derived from our impulse response decomposition framework. Each colored bar isolates the contribution of a specific endogenous transmission channel, namely CPI, EBP, EDR, UNEMP, PCE, WAGE, and the 2-year interest rate, to the total response.
The initial response is primarily driven by personal consumption expenditures, consumer prices, and the excess bond premium, with consumption expenditures exhibiting the strongest contemporaneous contribution to the impulse response of the monetary policy shock. At the 3-month horizon, the contribution of the excess bond premium becomes more prominent, while the influence of personal consumption largely dissipates. This shift suggests that investment sentiment plays a dominant role in mediating the transmission of monetary policy shocks at this stage. By the 6-month horizon, the contribution of the excess bond premium decreases, while industrial production accounts for an increasing share of the response, indicating that real economic activity becomes more responsive as transmission progresses. At the 12-month horizon, the response is almost entirely attributed to industrial production, suggesting that the cumulative effects of policy tightening are fully reflected in output at this point.
This decomposition underscores the inherently dynamic and multifaceted nature of monetary policy transmission. It provides a more nuanced interpretation of the transmission mechanism than conventional impulse response analysis alone.
We employ a dynamic mediation index to trace the evolving contributions of the excess bond premium and expected default risk over time, thus evaluating how financial market conditions, particularly investor sentiment and perceived credit risk, shaped the transmission of monetary policy to real economic outcomes.
Our analysis sheds light on a key dimension of monetary policy: the amplification and attenuation of its causal effects through a sentiment channel, by which shifts in risk premia and investor confidence alter the transmission of policy. Although it is well established that financial markets serve as a key conduit for the transmission of policy shocks to the real economy, our framework advances this understanding by offering a dynamic, horizon-specific quantitative measure that uncovers when sentiment channels become most influential and how their strength evolves over the adjustment path.
In particular, our findings complement and extend the evidence in \nocite*{gilchrist2012credit} and \nocite*{benson2024understanding}, which show that the macroeconomic relevance of credit spreads arises primarily from their non-default component, the excess bond premium, which captures fluctuations in investor sentiment and the risk-bearing capacity of financial intermediaries. The dyanmic mediation index for the excess bond premium and expected default risk are presented in the Figure below.
In Figure (ref), the upper panel focuses on the excess bond premium. The top-left subplot displays the impulse response decomposition, showing the contribution of the excess bond premium at various points in time relative to the monetary policy shock. The upper right subplot reports the dynamic mediation index, which measures the extent to which the output response is transmitted through the excess bond premium, projected onto the direction of the total response. The index peaks within the first few months, indicating that intermediary-driven fluctuations in market sentiment constitute a dominant short-term mechanism in the transmission of monetary policy.
In contrast, the lower panel evaluates the expected default risk as an alternative mechanism. The decomposition reveals that its contribution to the total output response remains muted and stable over time. The corresponding mediation index remains near zero, indicating that changes in default probabilities, while relevant to bond pricing, play a minimal role in the macroeconomic propagation of monetary policy. This divergence in transmission strength underscores a key insight. What matters most for aggregate dynamics is not the probability of default per se, but the risk premium investors demand above and beyond expected losses, an interpretation that closely echoes the findings in \nocite*{gilchrist2012credit}.
Together, our results offer a new lens on the mechanism of transmission of monetary policy through the sentiment channel. For policymakers, this framework not only quantifies the role of financial market sentiment in amplifying the effects of monetary shocks but also clarifies when and to what extent this channel becomes more operative, offering practical insight into the structure of policy transmission.
This paper develops a novel econometric framework for decomposing impulse responses to study dynamic causal mechanisms. By integrating VAR models with the principles of causal mediation analysis, we provide a systematic approach to disentangling the role of mediator variables in the transmission of structural shocks. Building on a nonparametric definition of the mediation effect and a parametric VAR representation, we show that the impulse response decomposition to mediators corresponds exactly to the average mediation effect. This result establishes a formal connection between Granger causality and dynamic mediation.
We propose a Dynamic Mediation Index to capture the directional and time-varying contribution of mediators to the overall impulse response. Applying the framework to the transmission of US monetary policy shocks, we find that investor sentiment acts as a significant short-term amplifier, while expected default risk contributes minimally across all horizons. These findings underscore the importance of investor sentiment in shaping macroeconomic outcomes and highlight the usefulness of our methodology in evaluating the structure and effectiveness of policy transmission. More broadly, our approach offers a unified and flexible econometric tool for investigating causal channels and mechanisms in macroeconomic shock propagation.