Package 'SVARtca'

Transmission Channel Analysis in Structural VAR Models

Description

The SVARtca package implements Transmission Channel Analysis (TCA) for decomposing impulse response functions in structural vector autoregressive (SVAR) models. The package follows the methodology presented in Wegner, Lieb, Smeekes and Wilms (2025).

TCA enables researchers to decompose the total impulse response to a shock into contributions from different transmission channels. Supported channels include effects passing through specific intermediate variables, allowing analysts to distinguish direct effects from indirect effects that operate through identified transmission mechanisms.

Key features:

• Multiple decomposition modes: overlapping, exhaustive 3-way, and exhaustive 4-way.

• Support for VAR and SVAR models via integration with the vars package.

• Visualization of channel contributions using ggplot2.

• Diagnostic tools for validating decomposition accuracy.

Main Functions

tca_systems_form

Build the systems form representation (B and Omega matrices) from VAR coefficients and structural impact matrix.

tca_analyze

Run the main TCA analysis with specified decomposition mode.

tca_decompose_binary

Decompose IRF into "through" and "not_through" components for a single variable.

tca_validate_additivity

Validate that the binary decomposition satisfies exact additivity.

tca_from_var

Convenience wrapper to run TCA directly from a fitted VAR model object.

plot_tca

Create publication-quality visualizations of channel contributions.

Identification

The package supports two identification schemes:

• Cholesky: Lower-triangular Cholesky decomposition of the residual covariance matrix. Set Phi0 = t(chol(Sigma)).

• Manual: User-supplied structural impact matrix for alternative identifications (e.g., sign restrictions, zero restrictions).

Citation

If you use the SVARtca package in your research, please cite the original methodology paper:

Wegner, E., Lieb, L., Smeekes, S. and Wilms, I. (2025). Transmission Channel Analysis in Dynamic Models. arXiv:2405.18987, doi:10.48550/arXiv.2405.18987. https://github.com/enweg/tca-matlab-toolbox

Author(s)

Muhammad Alkhalaf

References

Wegner, E., Lieb, L., Smeekes, S. and Wilms, I. (2025). Transmission Channel Analysis in Dynamic Models. arXiv:2405.18987, doi:10.48550/arXiv.2405.18987.

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Plot TCA Channel Decomposition

Description

Creates a stacked bar chart (for exhaustive modes) or a line chart (for overlapping mode) showing channel contributions over horizons.

Usage

plot_tca(x, target = NULL, type = NULL, title = NULL,
  colors = NULL)

Arguments

x	A tca_result object from tca_analyze.
target	Integer index of the response variable to plot (default: first variable after shock).
type	Plot type: "bar" for stacked bar chart (default for exhaustive modes), "line" for overlaid lines (default for overlapping mode). If NULL, chosen automatically.
title	Custom plot title (optional).
colors	Named character vector of colours for channels (optional).

Value

A ggplot object.

Examples

K <- 4
A1 <- matrix(c(0.7,-0.1,0.05,-0.05, -0.3,0.6,0.10,-0.10,
                -0.2,0.1,0.70,0.05, -0.1,0.2,0.05,0.65), K, K, byrow=TRUE)
Sigma <- matrix(c(1,0.3,0.2,0.1, 0.3,1.5,0.25,0.15,
                   0.2,0.25,0.8,0.1, 0.1,0.15,0.1,0.6), K, K, byrow=TRUE)
Phi0 <- t(chol(Sigma))
sf <- tca_systems_form(Phi0, list(A1), h = 20)
res <- tca_analyze(from = 1, B = sf$B, Omega = sf$Omega,
                    intermediates = c(2, 4), K = K, h = 20,
                    order = 1:K, mode = "exhaustive_4way",
                    var_names = c("IntRate","GDP","Inflation","Wages"))
plot_tca(res, target = 3)

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Print TCA Result

Description

S3 print method for tca_result objects. Displays a formatted table of transmission channel contributions across horizons.

Usage

## S3 method for class 'tca_result'
print(x, target = NULL, ...)

Arguments

x	A tca_result object.
target	Integer index (original ordering) of the response variable to display. Default NULL, which displays variable 1 (the first variable), not the first intermediate.
...	Additional arguments (ignored).

Value

Invisibly returns x.

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Transmission Channel Analysis

Description

Decomposes impulse response functions into transmission channel contributions using the methodology of Wegner, Lieb, Smeekes and Wilms (2025), doi:10.48550/arXiv.2405.18987.

Three decomposition modes are supported:

"overlapping"

Each channel is through(j) = total - not_through(j). Channels may overlap, so their sum may differ from the total.

"exhaustive_3way"

(2 intermediates only) Non-overlapping: (1) through var1 inclusive, (2) through var2 only, (3) direct. Sum equals total.

"exhaustive_4way"

(2 intermediates only) Full inclusion-exclusion: (1) var1 only, (2) var2 only, (3) both, (4) direct. Sum equals total.

Usage

tca_analyze(from, B, Omega, intermediates, K, h, order,
  mode = "overlapping", var_names = NULL)

Arguments

from	Shock variable number (1-based).
B	Systems form B matrix (from tca_systems_form).
Omega	Systems form Omega matrix.
intermediates	Integer vector of intermediate variable numbers (1-based, original ordering).
K	Number of variables.
h	Maximum horizon.
order	Transmission ordering vector.
mode	Decomposition mode: "overlapping", "exhaustive_3way", or "exhaustive_4way".
var_names	Character vector of variable names (optional).

Value

A list of class "tca_result" with components:

irf_total

Matrix (h+1) x K of total IRFs.

irf_channels

Named list of channel IRF matrices, each (h+1) x K.

channel_names

Character vector of channel names.

mode

Decomposition mode used.

from

Shock variable number.

K

Number of variables.

h

Maximum horizon.

order

Transmission ordering.

var_names

Variable names.

References

Wegner, E., Lieb, L., Smeekes, S. and Wilms, I. (2025). Transmission Channel Analysis in Dynamic Models. arXiv:2405.18987, doi:10.48550/arXiv.2405.18987.

Examples

# Monetary policy model
K <- 4
A1 <- matrix(c(0.7,-0.1,0.05,-0.05, -0.3,0.6,0.10,-0.10,
                -0.2,0.1,0.70,0.05, -0.1,0.2,0.05,0.65), K, K, byrow=TRUE)
Sigma <- matrix(c(1,0.3,0.2,0.1, 0.3,1.5,0.25,0.15,
                   0.2,0.25,0.8,0.1, 0.1,0.15,0.1,0.6), K, K, byrow=TRUE)
Phi0 <- t(chol(Sigma))
sf <- tca_systems_form(Phi0, list(A1), h = 20)
result <- tca_analyze(from = 1, B = sf$B, Omega = sf$Omega,
                       intermediates = c(2, 4), K = K, h = 20,
                       order = 1:K, mode = "exhaustive_4way",
                       var_names = c("IntRate","GDP","Inflation","Wages"))
print(result)

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Binary Decomposition: Total = Through + Not-Through

Description

Decomposes the total IRF into the effect passing through a variable and the effect not passing through it. The not-through effect is obtained by blocking every node of the variable (NOT condition) and the through effect is defined as total - not_through, so the three matrices satisfy total = through + not_through by construction. Use tca_validate_additivity to check the decomposition against an independent computation of the through effect.

Usage

tca_decompose_binary(from, B, Omega, var_idx, K, h, order)

Arguments

from	Shock variable (1-based).
B	Systems form B matrix.
Omega	Systems form Omega matrix.
var_idx	Variable to decompose through (1-based).
K	Number of variables.
h	Maximum horizon.
order	Transmission ordering.

Value

A list with matrices total, through, not_through (each (h+1) x K).

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Run TCA from a VAR Estimation Object

Description

Extracts coefficient matrices and residual covariance from a fitted VAR model (from the vars package) and runs TCA.

Usage

tca_from_var(var_model, from, intermediates, h = 20,
  order = NULL, mode = "overlapping",
  identification = "cholesky", Phi0 = NULL)

Arguments

var_model	A fitted VAR model object from VAR.
from	Shock variable (integer or name).
intermediates	Integer vector or character vector of intermediate variable names.
h	Maximum horizon (default: 20).
order	Transmission ordering (default: variable ordering in the VAR).
mode	Decomposition mode: "overlapping", "exhaustive_3way", or "exhaustive_4way".
identification	Identification scheme: "cholesky" (default) or "manual".
Phi0	Manual impact matrix. Required if identification = "manual".

Value

A tca_result object (see tca_analyze).

Examples

library(vars)
data(Canada)
var_est <- VAR(Canada, p = 2, type = "const")
result <- tca_from_var(var_est, from = "e",
                        intermediates = c("prod", "rw"),
                        h = 20, mode = "exhaustive_4way")
plot_tca(result, target = "U")

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Build Complete Systems Form

Description

Constructs both B and Omega matrices for the systems form representation x = Bx + Omega*epsilon, following the methodology of Wegner, Lieb, Smeekes and Wilms (2025).

Usage

tca_systems_form(Phi0, As, h, order = NULL, Psis = NULL)

Arguments

Phi0	Structural impact matrix (K x K). For Cholesky identification, use t(chol(Sigma)).
As	List of VAR coefficient matrices (A_1, A_2, ..., A_p).
h	Maximum IRF horizon.
order	Transmission ordering vector (default: 1:K).
Psis	Optional list of reduced-form MA coefficient matrices (Psi_1, ..., Psi_q), each K x K, for VARMA or DSGE models. See Details. Default NULL (no MA component).

Details

Note on Psis. For a pure VAR or SVAR model leave Psis at its default (NULL, treated as an empty list): the moving-average dynamics are generated by (I - B)^{-1}. Only VARMA or DSGE models with a moving-average component pass Psis. The matrices in Psis must be the reduced-form MA coefficient matrices, that is, the coefficients on the lagged reduced-form errors u_t = Phi0 %*% epsilon_t in y_t = A_1 y_{t-1} + ... + u_t + Psi_1 u_{t-1} + .... If the model is written with structural MA matrices on the structural shocks, y_t = ... + Phi0 %*% (epsilon_t + Psi^s_1 epsilon_{t-1} + ...), convert them first: Psi_j = Phi0 %*% Psi^s_j %*% solve(Phi0). Passing the structural matrices directly gives wrong impulse responses at horizons 1 and above, because makeOmega_systems multiplies each Psi_j by Phi0 on the right, so that Psi_j %*% Phi0 is the horizon-j structural MA coefficient.

Value

A list with components:

B

Systems form B matrix (K*(h+1) x K*(h+1)).

Omega

Systems form Omega matrix (K*(h+1) x K*(h+1)).

References

Wegner, E., Lieb, L., Smeekes, S. and Wilms, I. (2025). Transmission Channel Analysis in Dynamic Models. arXiv:2405.18987, doi:10.48550/arXiv.2405.18987. https://github.com/enweg/tca-matlab-toolbox

Examples

# 2-variable VAR(1)
Phi0 <- matrix(c(1, 0.3, 0, 0.95), 2, 2)
As <- list(matrix(c(0.5, -0.1, 0.2, 0.4), 2, 2))
sf <- tca_systems_form(Phi0, As, h = 10)
dim(sf$B)  # 22 x 22

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Validate Binary Additivity

Description

Checks Theorem 2(ii) of Wegner, Lieb, Smeekes and Wilms (2025): the effects of disjoint transmission channels sum to the total effect. For every variable j, the effect of the paths passing through j (at any horizon) is computed independently of the identity through = total - not_through, by partitioning those paths by the first node of j they visit and summing the AND-conditioned effects of the parts (see tca_analyze for the AND and NOT conditions). This independent through effect plus the NOT-conditioned not-through effect is then compared with the total effect. The residual is evaluated at all response variables other than j itself, because the AND condition sets the effect on the conditioning node to zero while the effect on j counts entirely as "through j" in the total - not_through convention.

If pair is given, the inclusion-exclusion identity used by the "exhaustive_4way" mode of tca_analyze, through(v1 and v2) = through(v1) + through(v2) - through(v1 or v2), is also checked: the left-hand side is computed directly with AND conditions on both variables (a double first-passage sum of (h+1)^2 linear solves, so keep h moderate), the right-hand side from the independent single-variable through effects and total - not_through(v1, v2). The residual is evaluated at all response variables other than v1 and v2.

The function also reports whether B is strictly lower triangular (entries on and above the diagonal at most 1e-12 times the largest absolute entry), which the systems form requires for the graph to be acyclic. Before version 1.0.3 the residual was computed as total - ((total - not_through) + not_through), which is zero by construction and could not detect any error.

Usage

tca_validate_additivity(from, B, Omega, K, h, order,
  var_names = NULL, verbose = TRUE, pair = NULL, tol = 1e-10)

Arguments

from	Shock variable (1-based).
B	Systems form B matrix.
Omega	Systems form Omega matrix.
K	Number of variables.
h	Maximum horizon.
order	Transmission ordering.
var_names	Character vector of variable names (optional).
verbose	Logical; print results? Default TRUE.
pair	Optional integer vector of two distinct variable numbers (1-based) for which the inclusion-exclusion identity is checked.
tol	Tolerance for the maximum absolute residual. Default 1e-10.

Value

Invisibly returns a logical, TRUE if B is strictly lower triangular and every residual is below tol, with attributes residuals (named numeric vector, the maximum absolute additivity residual for each variable), pair_residual (the maximum absolute inclusion-exclusion residual, or NA if pair is NULL), max_residual (the largest of these) and lower_triangular (logical).

References

Wegner, E., Lieb, L., Smeekes, S. and Wilms, I. (2025). Transmission Channel Analysis in Dynamic Models. arXiv:2405.18987, doi:10.48550/arXiv.2405.18987.

Examples

Phi0 <- matrix(c(1, 0.3, 0, 0.95), 2, 2)
As <- list(matrix(c(0.5, -0.1, 0.2, 0.4), 2, 2))
sf <- tca_systems_form(Phi0, As, h = 5)
ok <- tca_validate_additivity(1, sf$B, sf$Omega, K = 2, h = 5,
                              order = 1:2, pair = c(1, 2))
attr(ok, "max_residual")
# A backward edge breaks the acyclic structure and is detected
Bc <- sf$B; Bc[1, 4] <- 0.2
tca_validate_additivity(1, Bc, sf$Omega, K = 2, h = 5, order = 1:2)

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