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Impulse response functions (IRFs) measure the total dynamic causal effect of macroeconomic shocks on variables of interest, such as inflation and unemployment. However, the mechanisms -- or transmission channels -- through which these shocks influence the variables of interest are unexplored in IRF analysis. We propose a formal framework for quantitatively analysing transmission channels of structural shocks that is applicable to a large family of dynamic models. Our framework allows to dynamically decompose the effects of structural shocks into a set of disjoint dynamic partial effects, where the choice of decomposition -- the structural shock's transmission channel -- is determined by the research question.
Our first contribution is the development of a formal framework that allows the study of precisely defined transmission channels. While there is an extensive literature in macroeconomics and time series econometrics on methodology for studying the dynamic effects of interventions of shocks RameyMacroeconomicShocks2016,kilianStructuralVectorAutoregressive2017,nakamuraIdentification2018, the literature on formal analysis of transmission channels is relatively sparse. In macroeconomics, transmission channels are typically loosely defined as a collection of economic mechanisms that indirectly affect key macroeconomic outcomes through intermediate variables. For policymakers, however, it is imperative to quantitatively understand how policy effects economic outcomes, requiring precisely defined transmission channels. Yet, a precise definition and a coherent quantitative framework for analysing transmission channels is missing. Instead, many studies analyse transmission channels qualitatively. We fill this gap by providing a formal framework, including a precise definition, that allows for a quantitative analysis of transmission channels.
Our second contribution is to formulate transmission channel analysis (TCA henceforth) in terms of impulse response analysis. We formulate TCA as a decomposition of total effects, obtained through impulse response analysis, and establish that the calculation of impulse responses between different variables in the system is sufficient for the analysis of all transmission channels. Importantly, we also prove that only the structural identification of the initial shock of interest driving the impulse responses is required, and reduced-form impulse responses otherwise suffice. Consequently, TCA can be performed under the same conditions as traditional impulse response analysis, only requiring a structural identification scheme for a single shock plus a specified dynamic model. We prove this equivalence for a large class of dynamic models, including structural vector autoregressive (SVAR) and linearised dynamic stochastic general equilibrium (DSGE) models.
Our third contribution is of a more technical nature. In order to establish the results discussed above, we develop a graphical approach to study transmission channels. Specifically, we show how to connect the impulse-response-space representation of the (structural) model's equilibrium dynamics to a Directed Acyclic Graph (DAG). This involves a re-parameterisation of the dynamic equilibrium representation, which allows us to uniquely define a transmission channel as a collection of paths along the graph, connecting causal effects from one variable (resp. shock) to another variable and over time. To complement the graphical framework, we also develop an alternative representation of transmission channels in terms of potential outcomes. We prove that both representations are equivalent, implying that transmission channels can be defined either as paths through a graph or as a specific potential outcome. Apart from this specific result that is used throughout our theoretical analysis, the graphical and potential outcomes frameworks are of interest in themselves for the analysis of dynamic causal effects beyond our specific transmission questions.
Finally, we contribute novel insights into the functioning of monetary and fiscal policy by applying TCA to three distinct, well-established empirical macroeconomic models. First, we investigate the differences in monetary policy transmission when identified through either the shock series proposed by romerNewMeasureMonetary2004 or gertlerMonetaryPolicySurprises2015. We find that the former appears to mostly capture instantaneously implemented changes in the policy instrument, while the latter rather picks up other dimensions of monetary policy, such as forward guidance effects mckayWhatCanTime2023. Our findings thus provide quantitative support for the qualitative claims made by gertlerMonetaryPolicySurprises2015. Second, we shed light on anticipation effects of fiscal policy. Using the military spending news series of rameyGovernmentSpendingMultipliers2018, we specify transmission channels to distinguish between anticipation effects and implementation effects. This is achieved by defining an anticipation channel as the effect of the news shock not driven by the response of government military spending. Our findings agree with the conclusions of Ramey2011, who qualitatively assesses the importance of anticipation effects from the shape and timing of impulse responses to various macroeconomic variables. Third, we study transmission channels in the DSGE model of smetsShocksFrictionsUS2007. Here we decompose the total effect of monetary policy on inflation into a wage channel and a demand channel, respectively capuring the effect through wages and aggregate demand.
A related stream of literature uses impulse response analysis for studying counterfactual questions mckayWhatCanTime2023, simsDOESMONETARYPOLICY2006, kilianDoesFedRespond2011, caravelloEvaluatingPolicyCounterfactuals2024. While transmission channel and counterfactual analysis superficially appear to be related, it is important to highlight their fundamental difference. Investigating counterfactual questions requires the researcher to specify a different model, describing a different equilibrium, where specific behavioural changes result in the counterfactual scenario. TCA, however, focuses on decomposing impulse responses within a given equilibrium rather then understanding the difference between responses across different equilibria. Therefore, TCA evaluates the relative importance of specific variables in transmitting the total effect of a particular identified economic shock, whereas counterfactual analysis investigates total causal effects that this shock would have, had the policy response been different. As such, TCA complements counterfactual analysis; indeed, TCA can be used within a counterfactual study to investigate the importance of specific transmission channels across different equilibria.
TCA also shares similarities with mediation analysis imaiIdentificationInferenceSensitivity2010, chanEfficientNonparametricEstimation2016, danielCausalMediationAnalysis2015, pearlCausalMediationFormula2012, hayesIntroductionMediationModeration2018 as it investigates causal effects going through intermediate variables. In contrast to TCA, mediation analysis focuses on causal models that do not exhibit any feedback mechanisms, limiting its usefulness for studying transmission channels in dynamic general equilibrium macroeconomics.
The use of DAGs for transmission channel analysis is related to the graphical causal analysis literature in computer science pearlCausalityModelsReasoning2009. Contrary to that literature, we do not require that the causal system - the set of behavioural equations - can be cast in a DAG. Instead, we show that for a large class of linear macroeconomic models, a DAG representation of the equilibrium equations, rather than of the behavioural equations, always exists. This DAG representation, obtained using a QL-decomposition of the contemporaneous matrix, is related to the orthogonal reduced-form parameterisation of ariasInferenceBasedStructural2018. Importantly, this also covers non-recursive models which exhibit contemporaneous feedback effects.
Our work also relates to the recent literature which uses potential outcomes in macroeconomics to gain more insights into total causal effects. asheshrambachanWhenCommonTime2021, kolesarDynamicCausalEffects2024 investigate the non-parametric meaning of common macroeconomic causal estimands. cloyneStateDependentLocalProjections2023 use potential outcomes to investigate state dependent causal effects. Lastly, angristSemiparametricEstimatesMonetary2018 estimate monetary policy effects semi-parametrically. Contrary to the aforementioned, our focus is on the decomposition of total effects rather than on total effects themselves.
Finally, our work is related to other decomposition methods used in the literature, such as subspace Granger causality, forecast error variance decompositions (FEVD) and historical decompositions. Contrary to TCA, subspace Granger causality focuses on predictability and not on the effect of shocks dufourShortRunLong1998. The aim of FEVD and historical decompositions also differs from the aim of TCA. FEVD is used to decompose the forecast error variance into individual contributions of each structural shock. Historical decompositions provide information about which structural shocks drove specific historical developments. Neither, therefore, focuses on explaining the transmission of the shock through the economy; their focus is essentially still on total effects.
The paper is organised as follows. Section (ref) illustrates the main ideas behind TCA using a simple three-variable example. The general TCA framework is developed in Section (ref). Section (ref) contains our main theoretical results, while Section (ref) applies TCA empirically to three different macroeconomic models. Section (ref) concludes, while supplementary results are contained in the appendix.
We present the main idea and intuition behind TCA using a textbook version of the three-equation New Keynesian model galiMonetaryPolicyInflation2015 including the output gap $\mathrm{x}_t$, inflation $\mathrm{\pi}_t$ and nominal interest rates $\mathrm{i}_t$. The model is given by
and consists of an IS equation and a Phillips curve (PC) equation, together summarising the equilibrium in the goods market, as well as a (Taylor-type) rule specifying the central bank's interest rate policy (MR). The coefficients of the IS and PC equations, $\alpha_1$ and $\alpha_2$, depend on deep structural parameters $\bm \theta$ that specify the behaviour of firms and consumers in the economy. The coefficients of the interest rate rule, $\alpha_3$ and $\alpha_4$, have a behavioural interpretation since they explicitly specify the policy of the central bank. The set of deep structural parameters in the model is thus $\bm \vartheta = \{\bm \theta, \alpha_3, \alpha_4\}$. The demand, supply and interest rate (structural) shocks are respectively given by $\varepsilon^d_t$, $\varepsilon^s_t$ and $\varepsilon^i_t$.
For ease of exposition, we assume the structural shocks to be white noise and mutually uncorrelated. The equilibrium in (ref) exhibits the static representation
A core task of macroeconomists is to assess dynamic causal effects of structural shocks on relevant, endogenously determined economic variables. Generally the focus is on total effects, i.e.\ total changes in equilibrium quantities triggered by an economically interpretable exogenous event. Since model (ref) is static, total effects of structural shocks die out after a single period, and impact effects of the three structural shocks are summarised by the columns of the impulse response matrix
where $\eta(\bm \vartheta) = -\alpha_1(\bm \theta) \alpha_2(\bm \theta) \alpha_4-\alpha_1(\bm \theta) \alpha_3+1$.
We focus on decomposing the total effect of a demand shock on interest rates. The total effect (TE), given by
can be decomposed into two transmission channels. The first channel, labelled the indirect transmission channel, measures how much of the interest rate response to a demand shock can be explained by the response of inflation to a demand shock ceteris paribus, i.e.\ holding all other endogenous and exogenous variables constant. The second channel, labelled the direct transmission channel, is the remainder of the total effect that is not explained by the indirect channel. It measures how much of the interest rate response to a demand shock cannot be explained ceteris paribus by the response of inflation to the demand shock. \footnote{In the following, for brevity, we suppress explicitly mentioning the ceteris paribus condition, but it should be kept in mind that this is intended when we discuss effects.}
Throughout, we take the deep structural parameterisation $\bm \vartheta$ as given. This is contrary to counterfactual analysis where changes in total effects $\bm \varPhi(\bm \vartheta)$ are analysed under changes in deep structural parameters $\bm \vartheta$, hence under different equilibria. By taking $\bm \vartheta$ and thus the dynamic equilibrium representation as given, TCA focuses on explaining the effects within a specific equilibrium rather then understanding effects across different equilibria. TCA is therefore, by definition, not subject to the Lucas Critique. To keep notation simple, we subsequently suppress the dependence of $\alpha_1$ and $\alpha_2$ on $\bm \theta$. Furthermore, we first discuss TCA under simplified equilibrium dynamics ($\alpha_1(\bm \theta)=0$) in Section (ref) before turning to the more general equilibrium dynamics ($\alpha_1(\bm \theta)\neq 0$) in Section (ref).
Consider model (ref) with $\alpha_1=0$ which then implies that $\bm A$ is lower-triangular; we label this the recursive (R) model. Equilibrium dynamics of the output gap are fully determined by the demand shock and the demand shock's total effect on interest rates is given by $\alpha_2\alpha_4 + \alpha_3$.
Here there exist two transmission channels that jointly explain the total effect. First, the direct transmission channel consists of the effect of the demand shock on the output gap which is directly carried forward to interest rates. Second, the indirect transmission channel consists of the effect of the demand shock on the output gap, which is carried forward to inflation and in turn to interest rates.
TCA quantifies the effect that goes through each transmission channel, the transmission effect. To simplify this task, we cast the analysis of transmission channels in a graphical framework by connecting the model to an associated Directed Acyclic Graph (DAG), visualised in Figure (ref) panel (R). The three key ingredients of the graph are the nodes, edges, and path coefficients. The nodes represent either the variables in the model (solid black circles) or the shocks (shaded blue squares). The edges are directed and represent the effects. Edges originating from shocks represent causal effects; their path coefficient represents the direct causal effect size of the structural shock on the destination variable. We label edges originating from variables as carrying effects, as they carry the causal effect forward; their path coefficients quantify the direct effect of a unit increase in the origin variable (irrespective of what drove this increase) on the destination variable.
We decompose the total effect of the demand shock $\varepsilon_t^d$ on interest rates (TE, $\alpha_2\alpha_4 + \alpha_3$) into direct effects (DE) and indirect effects (IE) that go through the direct and indirect channels respectively. The total effect ($\alpha_2\alpha_4 + \alpha_3$) can be read of Figure (ref) panel (TE) by multiplying the path coefficients of the edges along each path from $\varepsilon_t^d$ to $\mathrm{i}_t$ and then adding up the results across all paths. The direct channel is visualised in Figure (ref) panel (DE), the indirect channel in panel (IE). Transmission effects can be read from the graph in a similar way as the total effect. The transmission effects of the direct and indirect channel are given by
The transmission effects of the disjoint direct and indirect channels thus add up to the total effect; thereby decomposing the total effect. Throughout only information about $\varepsilon^d$ is required; other structural shocks do not have to be identified.
Consider the general three-equation New Keynesian model (ref) with $\alpha_1\neq 0$. The contemporaneous matrix $\bm A$ is no-longer lower-triangular; the model is now non-recursive (NR). Equilibrium dynamics of the output gap are no-longer fully determined by the demand shock, as visualised in Figure (ref) panel (NR) by two incoming edges into $\mathrm{x}_t$. This complicates the equilibrium dynamics and TCA cannot be performed directly on this graph since the feedback loop (or cycle) between the output gap and inflation in Figure (ref) panel (NR) implies that no logically consistent definition of a transmission channel exists.
While simply restricting $\alpha_1$ to zero would break the cycle in the graph, it would also change the equilibrium since it would require an explicit change in the deep structural parameters $\bm \vartheta$. Restricting $\alpha_1=0$ would thus correspond to a counterfactual analysis, comparing the total dynamic effect across two equilibria; one equilibrium under $\alpha_1\neq 0$ and one under $\alpha_1=0$. TCA, in contrast, takes $\bm \vartheta$ and thus the dynamic equilibrium representation as given. To decompose the equilibrium dynamics leading to the total effects, TCA requires model (ref) to be re-parameterised into an equivalent equilibrium representation that lends itself to logically consistent definitions of transmission channels - a parameterisation that does not involve a feedback loop (cycle) and which can be represented by a DAG.
Since our focus is on the transmission of shocks in an equilibrium representation, any equivalent equilibrium representation can be used that permits the quantification of the transmission channels of interest. The re-parameterisation step is therefore crucially determined by the research question. To quantify the effect corresponding to the direct and indirect transmission channel, we seek a representation of the form
where equilibrium interest rates depend on inflation and the output gap; here $\bm L$ is a lower-triangular matrix and $\bm Q$ is a rotation matrix. Model (ref) describes the same dynamic equilibrium as model (ref) and is suitable to perform TCA for the research question at hand, but the interpretation of individual equations may be different and the re-parameterised relationships among variables may no longer have a behavioural interpretation. For example, the re-parameterised formulation of the “inflation equation” and the “interest rate equation” may now include the demand shock; and the re-parameterised relationship between the interest rate and inflation does not characterise the central bank's policy reaction anymore.
Representation (ref) can be obtained using a QL-decomposition of the matrix ${\bm A}$ which specifies the contemporaneous relationships among the endogenous variables. Multiplying both sides of (ref) by $\bm Q$ and noting that $\bm Q\bm Q'=\bm I$, we obtain
where $\bm A=\bm Q\bm L$ is unique,\footnote{This uniqueness is only up to sign, but we require the diagonal of $\bm L$ to be positive, as standard in software applications.} assuming that $\bm A$ is non-singular.
Model (ref) can now be represented as a DAG, see Figure (ref) panel (NR') with path coefficients omitted for clarity. Unlike to the DAG of Section (ref), edges now exist from all shocks to all variables. Focusing on the total effect of the demand shock on interest rates in Figure (ref) (TE), edges exist from the demand shock to the output gap, as before, but also to inflation and interest rates; their interpretation remains the same as in Section (ref). There are two paths from the demand shock to the output gap not going through inflation that form the direct transmission channel in Figure (ref) (DE) and two paths going through inflation that together form the indirect channel in Figure (ref) (IE). The corresponding transmission effects can be computed as before and are respectively given by
To compute these effects, only the structural shock $\varepsilon^d$ needs to be identified. We will show in Section (ref) that, in general, only one shock -- the initial shock of interest -- needs to be identified.
Finally, note that the transmission channels and effects obtained through model (ref) depend on the ordering of the variables in the QL-decomposition. Each ordering defines a set of possible transmission channels, with the chosen ordering being entirely determined by the research question. Given its crucial role for TCA, we encode this variable ordering in a permutation matrix $\bm T$ which we label the transmission matrix. Importantly, the ordering encoded in the transmission matrix is not needed to identify the structural shock of interest and its meaning is therefore fundamentally different from the ordering for the Cholesky decomposition in recursively identified structural models.
Figure (ref) gives all possible permutations of the transmission matrix in the example of this section. With $\bm A^*=\bm A\bm T'$ and $\bm y_t^*=\bm T\bm y_t$, the QL-decomposition of $\bm A^*$ results in six equivalent equilibrium representations, whose DAG representations are visualised in panels (a) to (f). The choice of transmission matrix determines the possible paths through the DAG and therefore transmission channels up for analysis. To answer our research question we need an equilibrium representation in which the interest rate depends on inflation and the output gap. This holds for the transmission matrices $\bm T$ in panels (a) and (c), directly ruling out the choice of $\bm T$ in the other panels. By ordering the output gap before inflation as in panel (a), we also enforce that inflation is dependent on the output gap, making it the only natural and logical choice to quantify our two transmission effects of interest.
We now formalise the intuition developed in the previous section. Section (ref) introduces our general dynamic model and its impulse-response representation which we use to compute total effects. Section (ref) then introduces the associated DAG which we use to decompose the total effect into effects through transmission channels, and formally defines transmission channels and transmission effects.
We assume $\bm y_t$ to be a $K$-dimensional stationary stochastic process with a structural Vector Autoregressive Moving Average (VARMA) representation given by
where $\{\bm A_i\}_{i=1}^{\ell}$ and $\{\bm \varPsi_j\}_{j=1}^{q}$ are $K\times K$ coefficient matrices which are statistically identified using any common scheme such as the echelon form (e.g., poskitt1992), $\bm A_0$ is a contemporaneous coefficient matrix assumed to be (partially) identified using some economic identification scheme, and $\bm \varepsilon_t$ is a $K\times 1$ vector of white noise.
The structural VARMA model (ref) encompasses a wide range of popular dynamic models used in macroeconomic research. The SVAR is recovered by setting $\bm \varPsi_j = \bf 0$ (for $j=1, \ldots, q$), while many linearised DSGE models can be represented as structural VARMAs with cross-equation restrictions ravennaVectorAutoregressionsReduced2007, fernandez-villaverdeABCsDsUnderstanding2007. We make two standard assumptions on $\bm \varepsilon_t$ and $\bm A_0$.
Assumption (ref) assures that impulse responses with respect to the white noise disturbances $\bm \varepsilon_t$ have a causal interpretation. Assumption (ref) assures that these causally interpretable impulse responses can be obtained from reduced-form IRFs. Section (ref) uses this last implication to show how transmission effects can be obtained from a single identified shock. Hence, $\bm A_0$ may be only partially identified.
While all equivalent equilibrium representations can be used to analyse total effects, decomposing these effects into effects through transmission channels requires a precise definition of which variables depend on which other variables in equilibrium. TCA thus involves choosing the ordering of variables using the transmission matrix $\bm T$.
Having a fixed and completely specified transmission matrix assures a fixed equilibrium representation. However, Section (ref) shows that, under certain conditions when two distinct transmission matrices lead to the same transmission effect, the transmission matrix only needs to be partially specified, so only a partial variable ordering is required in such case.
Under Assumptions (ref)-(ref) and a fixed $\bm T$, the QL-decomposition of $\bm A_0^*=\bm A_0\bm T'$ is unique, such that (ref) can be written in unique lower-triangular form
where $\bm L$ is a $K\times K$ lower-triangular matrix, $\bm y^*_t=\bm T\bm y_t$, $\bm A^*_i=\bm A_i\bm T'$ for $i=1,\dots,\ell$ and $\bm Q$ is a $K\times K$ rotation matrix.
Throughout the paper, we focus on transmission channels of the total effect of shock $\varepsilon_{i, t}$ to $y^*_{j, t+h}$ over the fixed finite impulse-response horizon $h$. Then the lower-triangular form (ref) can be re-written in systems form, capturing all dynamics of the shock $\varepsilon_{i, t}$ up to horizon $h$. Defining $\bm x=(\bm y_t^{*'}, \dots, \bm y_{t+h}^{*'})'$ and $\bm \varepsilon=(\bm \varepsilon_t', \dots, \bm \varepsilon_{t+h}')'$, the systems form (see Appendix (ref) for the derivation) is given by
where $\bm B$ is lower-triangular with zeros on the diagonal, $\bm \varOmega$ is lower-block-triangular, and both are respectively given by\footnote{For expositional ease, we henceforth omit dimension subscripts, letting $\bm I = \bm I_K$ and $\bm O = \bm O_K$.}
with $\bm D=\text{diag}(\bm L)^{-1}$, where $\text{diag}(\bm X)$ is a diagonal matrix of the diagonal of $\bm X$, $\bm A^*_i = \bm O$ for $i>\ell$ and $\bm \varPsi_j = \bm O$ for $j > q$.
The total effects can then be obtained from the impulse-response representation
Under our assumptions the impulse response matrix $\bm \varPhi$ always exists. The aim of TCA is to decompose these total effects into effects through transmission channels. However, before turning to the graphical representation used for this decomposition, we first revisit the examples of Section (ref) in our general framework.
To decompose the total effects given by $\bm \varPhi$ in (ref) into effects through transmission channels, we introduce the graph $\mathcal{G}(\bm B, \bm \varOmega)$ associated with system (ref). The graph $\mathcal{G}(\bm B, \bm \varOmega)$ is a DAG and has three ingredients: nodes, directed edges and path coefficients. A node exists for each shock $\varepsilon_{i,r}$ at time $r=t$ and for each variable $y_{i,r}$ at time points $r=t, \ldots, t+h$, where $h$ is the fixed horizon. Let $\bm x=(\bm y_t^{*'}, \ldots, \bm y_{t+h}^{*'})'$ and $\bm \varepsilon=(\bm \varepsilon_t', \ldots, \bm \varepsilon_{t+h}')'$ as in equation (ref). A directed edge $x_i \to x_j$ exists between variable $x_i$ and variable $x_j$ whenever $\bm B_{j,i}\neq 0$, where $\bm B_{j,i}$ denotes the $(j,i)$ element of $\bm B$. The path coefficient of this edge is given by $\omega_{x_i, x_j}=\bm B_{j,i}$ and is interpreted as the direct effect of a unit increase in $x_i$ (irrespective of the cause) on $x_j$. A directed edge $\varepsilon_i \to x_j$ from structural shock $\varepsilon_i$ to variable $x_j$ exists if $\bm \varOmega_{j,i}\neq 0$. The path coefficient of this edge is given by $\omega_{\varepsilon_i, x_j} = \bm \varOmega_{j,i}$ and is interpreted as the direct causal effect of a unit increase in $\varepsilon_i$ on $x_j$.
As each edge describes a direct effect, paths consisting of chained edges are interpreted as descriptions of causal flows. The collection $\mathcal{P}_{\varepsilon_i, x_j}$ of all paths connecting structural shock $\varepsilon_i$ to variable $x_j$ is a set of causal descriptions; so is any subset $P_{\varepsilon_i, x_j}$ of this collection. We are now ready to define transmission channels.
Through TCA we quantify transmission effects as the effects flowing through transmission channels. To this end, we define the effects along a specific path $p$ and the effects along a collection of paths $P_{\varepsilon_i, x_j}$, the latter leads to the definition of the transmission effect.
Having formally defined TCA in dynamic models for system (ref) and its associated graph $\mathcal{G}(\bm B, \bm \varOmega)$ (Definitions (ref) and (ref)), we provide some intuition for our three main results of Section (ref). For ease of exposition, consider a transmission channel $P_{\varepsilon_i, x_j}$ from $\varepsilon_i$ to $x_j$ through $x_k$ consisting of all paths starting at $\varepsilon_i$, going to $x_k$ and ending in $x_j$. First, note that the total effect of $\varepsilon_i$ on $x_j$ equals the total path-specific effect of $\mathcal{P}_{\varepsilon_i, x_j}$. This ensures the coherence of our framework.
Second, the transmission channels of the structural shock $\varepsilon_i$ only require $\varepsilon_i$ to be structurally identified. Edges leaving the structural shock $\varepsilon_i$ are causal effects of $\varepsilon_i$, thereby requiring structural identification. Edges leaving any intermediate variable $x_m$ on a path $p\in P_{\varepsilon_i, x_j}$, however, are effects of a unit increase in $x_m$, irrespective of the cause, and do not require structural identification. Only the single edge on any path $p\in P_{\varepsilon_i, x_j}$ originating from $\varepsilon_i$ requires structural identification.
Third, all transmission effects decomposing the total effect of $\varepsilon_i$ on $x_j$ can be calculated by combining the structural impulse responses of $\varepsilon_i$ together with non-structural impulse responses; thus, impulse responses are sufficient for the calculation of any transmission effect. All paths $p$ corresponding to the transmission channel $P_{\varepsilon_i, x_j}$ go through the intermediate variable $x_k$, through which the transmission channel $P_{\varepsilon_i, x_j}$ can be split. The first part consists of all paths connecting $\varepsilon_i$ to $x_k$. The effect through these paths -- the transmission effect -- is the total effect of $\varepsilon_i$ on $x_k$ obtained through the structural impulse response $\bm \varPhi_{k,i}$. The second part consists of all paths connecting $x_k$ to $x_j$ whose transmission effect corresponds to the total effect of a unit increase in variable $x_k$ on $x_j$; in other words, the reduced-form impulse response. Next, we prove these main findings using the potential outcomes framework.
In this section, we establish our main results on TCA. To this end, we use the potential outcomes framework as a convenient mathematical tool. In Section (ref), we introduce potential outcomes for TCA and show its equivalence with the graphical definitions and representation of TCA. In Section (ref), we then show that (i) the total effect of $\varepsilon_i$ on $x_j$ is the total path-specific effect of $\mathcal{P}_{\varepsilon_i, x_j}$, (ii) IRFs are sufficient statistics to compute transmission effects since structural IRFs can be combined with IRFs to Cholesky-orthogonalised shocks to obtain any transmission effect, and (iii) only $\varepsilon_i$, the shock of interest, needs to be structurally identified for transmission effects to be identified. Although transmission channels are generally determined by the chosen transmission matrix $\bm T$, we discuss in Section (ref) the conditions under which transmission effects remain invariant to the choice of some specific transmission matrices. All proofs are provided in Appendix (ref).
We introduce potential outcomes for TCA by adapting notation used in the mediation analysis literature danielCausalMediationAnalysis2015. We show that the potential outcomes framework and the more intuitive, graphical framework for TCA used in Section (ref) are equivalent. Throughout this section, we consider the effects from the structural shock $\varepsilon_i$ to the variable $x_j$.
Recall system (ref), $\bm x = \bm B\bm x + \bm \varOmega\bm \varepsilon$. Since $\bm B$ is lower-triangular with zeros on the diagonal, $x_j$ directly depends on the realisations of $x_i$ for $i<j$, which, in turn, directly depends on the realisations of $x_k$ for $k<i$. All variables may, additionally, depend on realisations of the shock of interest $\varepsilon_i$. However, shocks themselves are independent of all other variables and shocks. Thus, chains can be traced back from the variable of interest $x_j$ to the structural shock $\varepsilon_i$. Potential outcomes are statements about these chains, with the potential outcome for $x_j$ taking the form
where $\bm \epsilon^{(j)}$ is the potential outcomes assignment vector. The elements of this vector, $\epsilon_l$ for $l=1\dots \eta(j)$, are the realisations of the structural shock experienced by the $l$th nested chain, where $\eta(j)=2^{j-1}$ is the number of chains moving back from $x_j$ to $\varepsilon_i$.
As evident from the example, keeping track of the various dependencies and the corresponding shock realisations in the potential outcomes assignment vector $\bm \epsilon^{(j)}$ quickly leads to notational clutter. For notational convenience, we therefore introduce additional indexing notation to keep track of elements in the assignment vector that correspond to dependencies involving certain intermediate variables. Defining $H(j)=2^j-1$, the entries $H(k_1-1)+1$ to $H(k_1)$ in the assignment vector $\bm \epsilon^{(j)}$ correspond to the dependencies of the variable of interest $x_j$ on variable $x_{k_1}$, which we jointly collect in $\bm \epsilon^{(j\cdot k_1)}=(\epsilon^{(j)}_{H(k_1-1)+1}, ..., \epsilon^{(j)}_{H(k_1)})$. In the subset of dependencies of $x_j$ on $x_{k_1}$, we can further single out the dependencies on another variable $x_{k_2}$, $k_2 < k_1$, that is ranked higher up in the system. Then $\bm \epsilon^{(j\cdot k_1\cdot k_2)}$ collects the entries in the assignment vector corresponding to the dependencies of $x_j$ on $x_{k_1}$ and $x_{k_1}$ in turn on $x_{k_2}$. Continuing in this way, we thus define
Each potential outcomes assignment vector $\bm \epsilon^{(j)}$ defines a thought experiment in which some chains experience a shock while others may not. The most basic thought experiment is the one in which all $\epsilon_i=\xi$, implying that all chains experience the same shock realisation. A conceptually different thought experiment is one in which $\epsilon_k=0$, for some $1 \leq k \leq \eta(j)$, implying that some of the nested dependencies may experience a non-zero shock of size $\xi$, while others do not. Causality is then carried along a subset of all nested chains -- equivalently, along a subset of paths through the graph -- in other words, along a transmission channel. Throughout the paper we assume that a nested chain is either shut off ($\epsilon_k=0$) or is active and together with all other active chains experiences the same realised shock size ($\epsilon_l=\xi)$, as formalised in Assumption (ref).
We can now define transmission channels in potential outcome notation.
The causal effect of a potential outcomes assignment vector measures the effect of a thought experiment by comparing it to a baseline. Under Assumption (ref), a natural baseline is $\epsilon_i=0$ for all $i=1, \dots, \eta(j)$ such that no chain experiences a shock. If $\bm \epsilon^{(j)}$ is a transmission channel according to Definition (ref), then the transmission effect compares the reaction of $x_j$ under the thought experiment, $x_j^*(\bm \epsilon^{(j)})$, to the reaction of $x_j$ under the baseline, $x_j^*(\bm 0)$. This is the causal effect of $\bm \epsilon^{(j)}$ on $x_j$.
Finally, in Theorem (ref), we show that transmission channels defined in Definitions (ref) and (ref), and transmission effects defined in Definitions (ref) and (ref) are equivalent.
The previous section introduced potential outcomes and showed in Theorem (ref) that the potential outcomes and graphical framework for TCA are equivalent. In this section, we continue with the potential outcomes framework and use its equivalence with the graphical framework to derive three main results. We focus throughout on the (transmission) effect of the structural shock $\varepsilon_i$ on $x_j$.
First, since causal effects flow along paths in the associated graph $\mathcal{G}(\bm B, \bm \varOmega)$, intuitively, the total causal effect should be the effect along all paths connecting $\varepsilon_i$ and $x_j$, $\mathcal{P}_{\varepsilon_i, x_j}$. Theorem (ref) shows that this is the case.
Part (i) states that the total effect given by entry $\bm \varPhi_{j,i}$ in the impulse response matrix is equal to the total path-specific effect of the collection of all paths $\mathcal{Q}_1(\mathcal{P}_{\varepsilon_i, x_j})$, which is in turn equal to the causal effect of the potential outcomes assignment vector $\bm \epsilon^{(j)}=\bm 1_{\eta(j)}$ by Theorem (ref). Multiplication by $\xi$ simply accounts for the shock size. Part (ii) states that the total path-specific effect of each set in the partition $P^{(1)}_{\varepsilon_i, x_j}, \dots, P^{(k)}_{\varepsilon_i, x_j}$ of $\mathcal{P}_{\varepsilon_i, x_j}$ decomposes the total effect. Thus, since each set $P_{\varepsilon_i,x_j}^{(i)}$ in the partition is a transmission channel by Definition (ref), transmission effects of disjoint transmission channels decompose the total effect. Part (iii) implies the same statement for a set of potential outcomes assignment vectors $\bm e_1\dots \bm e_k$ such that $\sum_{m=1}^k \bm e_m = \xi\bm 1_{\eta(j)}$, which is the potential outcomes equivalent to the partition $P^{(1)}_{\varepsilon_i, x_j}, \dots, P^{(k)}_{\varepsilon_i, x_j}$.
While Theorem (ref) shows that total effects are decomposed by non-overlapping transmission channels, it says nothing about how the transmission effects can be obtained. This is the subject of Theorem (ref).
Theorem (ref) shows that the transmission effect of any transmission channel is a combination of the impulse response matrices $\bm \varPhi$ and $\tilde\bm \varPhi$; thus, IRFs are sufficient statistics for the calculation of transmission effects. This extends sufficiency results of IRFs for FEVDs kilianStructuralVectorAutoregressive2017, policy counterfactual mckayWhatCanTime2023 and policy optimality barnichonSufficientStatisticsApproach2023 analysis to TCA.
The specific functional form $f(\cdot)$ of the transmission effects is application-dependent; Appendix (ref) gives computational algorithms to obtain the functional form and to compute the transmission effect. Note also that although the sufficiency result makes use of Cholesky-orthogonalised IRFs, the Cholesky orthogonalisation is not used to identify a structural shock; it is simply a computational tool. Only the impulse response matrix $\bm \varPhi$ is structural, while the matrix $\tilde\bm \varPhi$ contains impulse responses to Cholesky-orthogonalised shocks -- they are orthogonalised with respect to the transmission matrix $\bm T$ rather than a structural ordering. This means that $\tilde\bm \varPhi$ and thus, in turn, the transmission effects are sensitive to the ordering in $\bm T$. However, as Section (ref) shows, since $f(\cdot)$ often does not make use of all columns of $\tilde\bm \varPhi$, the transmission effect is often invariant to re-ordering of some variables; equivalently to re-ordering of some rows in the transmission matrix $\bm T$.
As the final step in our theoretical development, Theorem (ref) shows that TCA only requires the shock of interest to be identified.
TCA provides additional insights into what drives equilibrium dynamics due to a structural shock $\varepsilon_i$. This implies that TCA only requires the identification of $\varepsilon_i$, the structural intervention under investigation. This stands in contrast to counterfactual policy analysis (e.g. mckayWhatCanTime2023 or simsDOESMONETARYPOLICY2006), where in addition to $\varepsilon_i$ further identified shocks are needed to mimic the counterfactual policy rule in a different dynamic equilibrium.
Theorem (ref) and (ref) together imply that TCA is feasible whenever standard macroeconomic analysis of total dynamic causal effects is feasible. In addition, the theoretical properties established provide guidance for the practical computation of transmission effects either by aggregating path effects, or by calculating the relevant impulse responses. While both approaches are valid, their computational efficiency will depend on the specific application. Appendix (ref) goes into more details about the computational aspects and proposes an algorithm that exploits the established properties to yield a computationally efficient procedure.
TCA requires the researcher to choose a transmission matrix $\bm T$, see Definition (ref). With $K$ variables in the VARMA (ref), there are $K!$ choices. We, next, provide conditions under which transmission effects are invariant to the choice of $\bm T$, which offers guidance to practitioners on how an appropriate set of transmission matrices for the research question at hand can be specified.
Our first result identifies when edges in $\mathcal{G}(\bm B, \bm \varOmega)$ linking $\varepsilon_i$ to $x_j$ remain invariant under variable re-ordering in the transmission matrix.
Lemma (ref) states that edges connecting shock $\varepsilon_i$ to variable $x_j$ are invariant to re-ordering variables within the block before and after $x_j$ as long as variables from earlier time periods are not ordered after variables from later time periods. Intuitively, this is because the edge measures the causal effect of $\varepsilon_i$ on $x_j$ that does not go through variables ordered before $x_j$. This only depends on the variables that are ordered before and after, but not on the ordering within the respective groups.
Our second result explores when edges in the graph $\mathcal{G}(\bm B, \bm \varOmega)$ connecting variable $x_i$ to variable $x_j$ are invariant to re-ordering in the transmission matrix.
Lemma (ref) has a similar intuition as Lemma (ref). It states that an edge from a variable $x_i$ to another variable $x_j$ is invariant to any re-ordering of variables within the block of variables before $x_i$, between $x_i$ and $x_j$, and after $x_j$ as long as variables from earlier time periods are not ordered after variables from later time periods.
Note the analogy of the previous two lemmas to the invariance of the ordering in Cholesky identification schemes, where the ordering also only matters with respect to the shock and outcome variables. While the underlying conceptual framework here is different, as it does not relate to identification but to the flow of transmission effects, the analogy can be helpful in guiding the choice of a transmission matrix in practice.
Finally, we present invariance conditions on the IRFs needed to compute the transmission effects. This lemma is more general than Lemmas (ref) and (ref): if Lemmas (ref) and (ref) hold for each edge on the paths corresponding to the IRF, then Lemma (ref) holds too, but the reverse need not be true. Since structural IRFs are not subject to the choice of a transmission matrix $\bm T$, the invariance conditions in the IRF space only involve the Cholesky IRFs $\tilde\bm \varPhi_{i, j}$. Such conditions have been derived in Proposition 4.1 of christianoMonetaryPolicyShocks1999. Lemma (ref) presents these results in the context of TCA.
Intuitively, since $\tilde\bm \varPhi_{\cdot, j}$ measures the effect of a unit increase in $x_j$ without increasing the variables ordered before $x_j$, all that matters is which variables are ordered before and after, but not the ordering within their respective groups. Thus, variables ordered before and after $x_j$ can be re-ordered in their respective groups as long as the time ordering is not broken.
Although Lemma (ref) is more general than Lemmas (ref) and (ref), some transmission channels are easier to analyse for invariance using the latter two. Take, for example, the transmission channel consisting of only the direct edge from a shock to a variable. Lemma (ref) directly provides conditions under which this transmission effect is invariant. Application of Lemma (ref), on the other hand, requires representing the single edge as a complicated function of IRFs, making the analysis of invariances much more difficult. Thus, both sets of Lemmas have their merit.
In this section, we empirically demonstrate the versatility of TCA in a variety of settings. Section (ref) demonstrates how TCA can be used to decompose monetary policy effects estimated using SVARs into contemporaneous and non-contemporaneous effects. Section (ref) applies TCA to government spending within a local projections (LP) framework decomposing the total effect of a government spending news shock into implementation and anticipation effects. Section (ref) shows how TCA can be used to decompose total impulse responses obtained from the linearised prototypical DSGE model by smetsShocksFrictionsUS2007. Appendix (ref) provides additional details, such as graphical representations of the specified transmission channels. All algorithms used for the computation of the transmission effects are implemented in the Julia package \href{https://github.com/enweg/TransmissionChannelAnalysis.jl}{TransmissionChannelAnalysis.jl} and the Matlab \href{https://github.com/enweg/tca-matlab-toolbox}{TCA Toolbox}. \footnote{The packages are available on GitHub: \href{https://github.com/enweg/TransmissionChannelAnalysis.jl}{https://github.com/enweg/TransmissionChannelAnalysis.jl} and \href{https://github.com/enweg/tca-matlab-toolbox}{https://github.com/enweg/tca-matlab-toolbox}. Replication code for the empirical applications considered in this section is also available on GitHub: \href{https://github.com/enweg/tca-replication-material}{https://github.com/enweg/tca-replication-material}.}
We use TCA to decompose the effects of monetary policy shocks into contemporaneous effects, related to a direct change in short-term interest rates, and effects that are not linked to direct changes of the main policy instrument, such as forward guidance. mckayWhatCanTime2023 argue that the romerNewMeasureMonetary2004 shock series predominantly drives short-term interest rates. In contrast, they argue that the gertlerMonetaryPolicySurprises2015 shock series moves long(er)-term interest rates and thus rather captures the non-contemporaneous components of monetary policy such as forward guidance. TCA provides a framework to quantify and assess this hypothesis by decomposing the total impulse responses to either monetary policy shock into direct implementation and non-contemporaneous effects. To conduct TCA, and to stay as close as possible to mckayWhatCanTime2023, we estimate an SVAR(4) in the federal funds rate $\mathrm{i}_t$, the output gap $\mathrm{x}_t$, inflation $\mathrm{\pi}_t$ and commodity prices $\mathrm{p}_t$ over the period 1969Q1 to 2007Q4. We identify the monetary policy shock using either RR or GK as an internal instrument plagborg-mollerLocalProjectionsVARs2021.\footnote{All data was obtained from the replication package of mckayWhatCanTime2023.}
We define the direct implementation channel of monetary policy as the effect of a monetary policy shock going through a contemporaneous adjustment in the federal funds rate. The non-contemporaneous effect is then defined as the complement; that is, the effect of the monetary policy shock not going through a contemporaneous adjustment in short-term interest rates. Since the federal funds rate is implicitly held constant on impact, the non-contemporaneous effect captures dimensions of monetary policy that relate to (expected) future changes of short-term interest rates, i.e.\ forward guidance.\footnote{Note that other effects might also be captured. We leave a thorough identification of forward guidance effects through a more precise definition of forward guidance channels to future research.}
Often, when a specific variable can be directly associated with the identified shock of interest and this specific variable can be straightforwardly related to the specification of the transmission channel the researcher wants to investigate, it helps guiding the specification of the transmission matrix. In this application the federal funds rate, as the policy instrument, is intuitively associated with monetary policy surprises. Ordering the federal funds rate first, allows for a straightforward distinction between the direct implementation channel, effects only going directly through the federal funds rate, and its complement, the non-contemporaneous channel, capturing transmission effects (on all other endogenous variables) not going directly through the federal funds rate. Results in Section (ref) indicate that all transmission matrices that order the federal funds rate first result in the same direct and non-contemporaneous transmission effects.
Figure (ref) shows the total, direct implementation, and non-contemporaneous effects of the RR (top) and GK (bottom) shock, all normalised to a 25bp contemporaneous increase in the federal funds rate.\footnote{The scale of the impulse responses is determined by the chosen normalisation procedure, which makes a direct comparison in terms of absolute effects of the RR and GK shocks difficult. However, relative decompositions into transmission channels and qualitative conclusions about the shapes of the IRFs can be compared, since they are invariant to the chosen normalisation.} The total effect is depicted as a scatter-line, while the direct implementation and non-contemporaneous effects are depicted as stacked bars. The GK shock triggers dynamics in the federal funds rate and inflation that are clearly distinct from the dynamics induced by the RR shock. Additionally, the majority of the GK induced reactions are explained by non-contemporaneous effects, while only a small part of the reactions to an RR shock can be explained by non-contemporaneous effects. Thus, the results quantitatively confirm the qualitative discussion in mckayWhatCanTime2023. The RR shock series measures mostly contemporaneous effects of monetary policy, while the GK shock series identifies non-contemporaneous components of monetary policy that are likely linked to forward guidance.
In Section (ref) we defined the direct implementation channel of monetary policy as the effect of a policy that only goes through a contemporaneous change in short-term interest rates. In this section we build on this idea to specify a channel capturing the anticipation effects of government spending news shocks. The analysis is motivated by Ramey2011, rameyGovernmentSpendingMultipliers2018, rameyDefenseNewsShocks2016 who argue that many series of identified government spending shocks, such as war dates series rameyCostlyCapitalReallocation1998, do not correctly capture important anticipation effects. As their proposed series captures news about government military spending, we define the anticipation channel as the news shock effect not driven by the response of government military spending up to horizon $H$.
Here, Ramey's news series is directly associated with a news shock. Thus, similar to the federal funds rate in Section (ref), we order the news series first in the transmission matrix. Moreover, we order government military spending second, as we investigate news shock effects not going through actual government military spending, and, importantly, want to allow for the other variables to only partially adjust if either the anticipation or implementation channels are blocked. A similar ordering with the shock's associated variable first (potentially measuring news or expectations) and “implementing" variables afterwards can be adopted in many other potential applications of TCA, especially those investigating shocks to news or expectations. The results of Section (ref) then imply that all involved impulse response functions are invariant to re-ordering of the remaining variables and all transmission matrices ordering the news variable first and government military spending second result in identical transmission effects.
We estimate impulse responses using local projections based on quarterly data from 1890Q1 to 2015Q1. In line with the original papers, we identify the news shock recursively with the news measure ordered first by estimating the regression
where $\bm y_t=(\text{news}_t, \text{mil}_t, \text{gov}_t, \text{gdp}_t)$, $\text{news}_t$ is the news measure, $\text{mil}_t$ government military spending, $\text{gov}_t$ total government spending, and $\text{gdp}_t$ is GDP, all expressed in real terms and as percent of real potential GDP. Then, $\beta_i^h$ measures the impulse response of a news shock of one percent of real potential GDP on $y_{i, t+h}$. To estimate transmission channels we need additional, reduced-form impulse responses. These are estimated using local projections of the form
where $\tilde\gamma_i^h$ is the Cholesky impulse response of a government military spending shock on $y_{i, t+h}=x_{i+4h}$, following the ordering defined in the transmission matrix. Anticipation and implementation effects can now be computed as in Appendix (ref).
Figure (ref) shows the total, implementation, and anticipation effects of the government (military) spending news shock on GDP, total government spending, and government military spending. The total effect is shown as a black scatter-line with the anticipation and implementation effect shown as stacked bars in blue and yellow, respectively. The response of both, GDP and government spending, is predominantly driven by anticipation effects. Implementation effects are small (but positive) as long as defence spending is increasing. Interestingly, anticipation and implementation effects start to offset each other at longer horizons; a pattern that is hidden in total impulse responses but that becomes apparent when using TCA. Reconciling these findings with theory is outside the scope of this paper. A crucial difference to the related literature, which uses military spending news as exogenous measure to identify overall government spending news shocks, is that we can attribute observed (implementation) effects solely to changes in military spending. This could possibly suggest that implementation effects of military spending and civilian government spending are different (c.f. Perotti2014). In contrast, anticipation effects are much less tangibly defined and, by construction, could relate to broader economic expectations not only related to future military spending. Further investigation goes beyond the scope of this paper and is left for future research. A sub-sample analysis could be a start to shed some more light on the drivers of this finding (e.g. AcaraiEtAl2023).
In the previous sections we studied transmission channels using time series models typically used in empirical macroeconomics. However, TCA can also be applied in other dynamic models often used in macroeconomic analysis, such as DSGEs. Since DSGE models consist of many interrelated equilibrium equations, TCA may provide important insights into the exact mechanisms behind total dynamic effects.
We analyse the role of wages in the transmission of monetary policy shocks using the linearised smetsShocksFrictionsUS2007 model,\footnote{Replication files including the standard parameterisation of the smetsShocksFrictionsUS2007 model were obtained from pfeiferDSGEcollection.} including the following endogenous variables: policy rate ($r_t$), labour hours ($l_t$), wage growth ($w_t$), consumption growth ($c_t$), investment growth ($i_t$), output growth ($y_t$), and realised inflation ($\pi_t$). As required for TCA, we re-formulate the linearised DSGE in SVARMA form using the method of morrisVARMARepresentationDSGE2016.
We differentiate two channels through which a contractionary monetary policy shock affects inflation. First, higher interest rates discourage consumption and investment and hence reduce aggregate demand which puts prices gradually under downward pressure. We refer to this as the demand channel. Second, in response to weaker demand, firms reduce labour hours, which gradually puts wages under downward pressure. Lower wages reduce marginal costs, which eventually influences firms' price-setting decisions. We refer to this as the wage channel.
To investigate the quantitative importance of the demand and wage channel, we use TCA and decompose the total dynamic effect of a monetary policy shock on inflation into two mutually exclusive effects: a broad wage channel capturing all effects that go through wages $w_t$ in at least one period, and a demand channel capturing all effects that do not go through wages in any period.
Since we decompose the effect on inflation, we order inflation last. Additionally, and in line with the previous applications, we order the shock's associated variable -- the interest rate -- first. This leaves us with the decision of the relative ordering of wages $w_t$ and the remaining endogenous variables $(l_t, c_t, i_t, y_t)$, as our results in Section (ref) imply that the relative ordering inside the group $(l_t, c_t, i_t, y_t)$ does not matter. We explore two choices. First, labelled as `channel 1', we use a transmission matrix that orders wages second to last, allowing the demand channel to contemporaneously feed into the wage channel, in line with the discussion above. Second, in `channel 2' we consider a transmission matrix that orders wages second, allowing the wage channel to contemporaneously feed into the demand channel.
Figure (ref) shows the resulting transmission effects. Each horizon shows two sets of stacked bars, where the bars on the left show the decomposition obtained by ordering wages second to last in the transmission matrix, and the bars to the right show the decomposition obtained by ordering wages second in the transmission matrix. The blue bars correspond to the effect through the demand channel and the red to the effect through the wage channel, and add up to the total effect shown as a black scatter-line.
The two considered transmission matrices yield nearly identical effects. This suggests that the contemporaneous interaction between the wage and demand channel is likely small. In both cases the initial reduction of inflation is completely driven by the demand channel and thus by demand-suppressing factors, with the wage channel adding slight inflationary pressures. Moreover, wages become more important over time, and the negative total response of inflation between periods two and ten is driven by effects going through the wage channel -- the humped shaped response is entirely explained by the wage channel. This confirms and assigns a quantitative value to the model's qualitative causal chain discussed above, in which wages play a delayed role, with suppression in demand, due to the monetary policy shock, feeding into wages, and eventually into prices.
We develop a new coherent framework for analysing transmission channels in a large class of dynamic models by connecting them to a graphical representation. Within the graphical representation, transmission channels are defined as paths from the shock of interest to the outcome variable; transmission effects are the effects flowing along these paths. We show that this graphical representation is equivalent to a potential outcomes representation which we use to prove that impulse responses are sufficient for computing all transmission effects.
Current methods to analyse transmission channels are purely based on qualitative inspections of various impulse responses, which cannot be used to precisely define and to quantitatively compare the importance of transmission channels. Our framework paves the way for a formal quantitative analysis of transmission channels, thereby extending the empirical macroeconomic toolbox. Of particular relevance for applied work is that TCA does not require any additional identification requirements beyond the identification of the structural shock of interest.
A crucial component of TCA is a precise definition of the transmission channel of interest. We show how this can be formalised through the transmission matrix which defines the ceteris paribus ordering of the variables in equilibrium; this ordering is entirely defined by the research question. We also show that in some cases, a partial ordering is sufficient. This is also the case in our empirical applications, in which we show how TCA can shed a quantitative light on the channels through which policy operates in a variety of macroeconom(etr)ic settings.
In some applications, the desired transmission channel can be implemented through multiple transmission matrices, resulting in different but qualitatively similar versions of the transmission channel -- as in Section (ref). If the set of possible transmission matrices is small, the analysis can be applied to each matrix and the results can be compared. If, on the other hand, the set of possible transmission matrices is large, and results in Section (ref) cannot be used to further reduce this set, manually comparing the results obtained using each transmission matrix may be infeasible. In these circumstances it can be of benefit to plot a range, such as the minimum and maximum at each horizon across all definitions of the channel. Although such an approach loses a lot of detailed information, it gives a quick overview of how much the various transmission channels differ, and is more informative than horizon-wise means or medians, which are difficult to interpret.
While our proposed framework is applicable to a large set of commonly used dynamic macroeconomic models, including SVARs, DSGEs, and local projections, it is currently not applicable to general state space models, which are used to model dynamics via a system of latent variables. Although some state space representations, such as those used for DSGE models, can be rewritten into SVARMA form, this rewriting cannot accommodate all such models; in particular, one cannot apply our framework directly to dynamic factor models. In state space models, transmission channels and effects can still be defined in an equivalent way using either the graphical or the potential outcomes representation. However, this definition comes with statistical identification problems that are beyond the scope of this paper. We therefore leave this extension for future research.
Our proposed framework relies on linearity of the dynamic model and is thus inherently subject to potential shortcomings of linear models. This might be considered a limitation as some evidence points to nonlinear effects of policies. For example, the effects of monetary expansions and contractions appear to be distinct coverAsymmetricEffectsPositive1992, angristSemiparametricEstimatesMonetary2018, tenreyroPushingStringUS2016. This may, for instance, be driven by the difference in the transmission of an expansionary and a contractionary shock. Under the current setup, TCA cannot differentiate between expansionary and contractionary shocks -- it simply analyses the average transmission effects unconditional of the sign. However, most of macroeconomic analysis still relies on linear dynamic models, and TCA is applicable in a very wide class of these. TCA in its current form can therefore already complement the vast majority of empirical macroeconomic analyses. In addition, linear estimands are often robust to non-linearities, and moving towards non-linearity can often do more harm than good kolesarDynamicCausalEffects2024. Furthermore, extending TCA to nonlinear models, particularly when allowing for interaction effects, requires a complete rethink of how to define transmission channels. While this is another exciting avenue for future research, it is clearly far outside the scope of the paper.
{\bf Acknowledgements.} We thank Sumanta Basu, Martin Ellison, Tobias Hartl, James McNeil, Geert Mesters, Jose Luis Montiel Olea, and seminar participants at Cornell University, the Deutsche Bundesbank Vo-Seminar, EC$^2$ 2024 conference, European Seminar on Bayesian Econometrics 2023, Maastricht MILE seminar, the NESG conference, and VTSS Workshop for Junior Researchers (2025) for their helpful comments and discussions. The last author was financially supported by the Dutch Research Council (NWO) under grant number VI.Vidi.211.032. Remaining errors are our own.
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