Introduction to Transmission Channel Analysis

Overview

The SVARtca package implements Transmission Channel Analysis for structural vector autoregressive (SVAR) models, following the methodology of Wegner, Lieb, Smeekes and Wilms (2025).

TCA answers the question: How much of a shock’s total effect on a target variable passes through a particular intermediate variable (channel)?

The method works by:

Converting the VAR into a systems form representation.
Selectively blocking transmission paths in the systems form DAG.
Computing the difference between total and blocked effects.

Quick Start

library(SVARtca)

# Define a 4-variable VAR(1) 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(1.00, 0.30, 0.20, 0.10,
                  0.30, 1.50, 0.25, 0.15,
                  0.20, 0.25, 0.80, 0.10,
                  0.10, 0.15, 0.10, 0.60), K, K, byrow = TRUE)

Phi0 <- t(chol(Sigma))
var_names <- c("IntRate", "GDP", "Inflation", "Wages")

# Step 1: Build systems form
sf <- tca_systems_form(Phi0, list(A1), h = 20)

# Step 2: Run TCA
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     = var_names
)

print(result, target = 3)
#> 
#> TCA Results (mode: exhaustive_4way)
#> Shock from: IntRate | Horizon: 20
#> Response variable: Inflation
#> ---------------------------------------------------------------------- 
#>    h |        Total |     GDP only |   Wages only |  GDP & Wages |       Direct
#> ---------------------------------------------------------------------- 
#>    0 |     0.200000 |     0.040426 |     0.000000 |     0.000000 |     0.159574
#>    1 |    -0.025000 |     0.020023 |     0.004727 |     0.000273 |    -0.050023
#>    2 |    -0.161750 |    -0.019683 |     0.002645 |     0.002605 |    -0.147317
#>    4 |    -0.279529 |    -0.094721 |    -0.007391 |    -0.000064 |    -0.177354
#>    8 |    -0.253949 |    -0.138419 |    -0.016737 |    -0.015861 |    -0.082932
#>   12 |    -0.158588 |    -0.101321 |    -0.011863 |    -0.020979 |    -0.024425
#>   16 |    -0.086618 |    -0.057959 |    -0.005694 |    -0.017520 |    -0.005445
#>   20 |    -0.044372 |    -0.029408 |    -0.002182 |    -0.011907 |    -0.000876
#> ----------------------------------------------------------------------

Decomposition Modes

Overlapping

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

res_ov <- tca_analyze(
  from = 1, B = sf$B, Omega = sf$Omega,
  intermediates = c(2, 4), K = K, h = 20,
  order = 1:K, mode = "overlapping", var_names = var_names
)
print(res_ov, target = 3)
#> 
#> TCA Results (mode: overlapping)
#> Shock from: IntRate | Horizon: 20
#> Response variable: Inflation
#> ---------------------------------------------------------------------- 
#>    h |        Total |  Through GDP | Through Wage |       Direct
#> ---------------------------------------------------------------------- 
#>    0 |     0.200000 |     0.040426 |     0.000000 |     0.159574
#>    1 |    -0.025000 |     0.020296 |     0.005000 |    -0.050023
#>    2 |    -0.161750 |    -0.017078 |     0.005250 |    -0.147317
#>    4 |    -0.279529 |    -0.094784 |    -0.007454 |    -0.177354
#>    8 |    -0.253949 |    -0.154280 |    -0.032598 |    -0.082932
#>   12 |    -0.158588 |    -0.122300 |    -0.032842 |    -0.024425
#>   16 |    -0.086618 |    -0.075479 |    -0.023214 |    -0.005445
#>   20 |    -0.044372 |    -0.041314 |    -0.014088 |    -0.000876
#> ----------------------------------------------------------------------

Exhaustive 3-Way

Non-overlapping decomposition into: (1) through var1 inclusive, (2) through var2 only, (3) direct. The sum equals the total.

res_3w <- tca_analyze(
  from = 1, B = sf$B, Omega = sf$Omega,
  intermediates = c(2, 4), K = K, h = 20,
  order = 1:K, mode = "exhaustive_3way", var_names = var_names
)
print(res_3w, target = 3)
#> 
#> TCA Results (mode: exhaustive_3way)
#> Shock from: IntRate | Horizon: 20
#> Response variable: Inflation
#> ---------------------------------------------------------------------- 
#>    h |        Total | Through GDP  | Through Wage |       Direct
#> ---------------------------------------------------------------------- 
#>    0 |     0.200000 |     0.040426 |     0.000000 |     0.159574
#>    1 |    -0.025000 |     0.020296 |     0.004727 |    -0.050023
#>    2 |    -0.161750 |    -0.017078 |     0.002645 |    -0.147317
#>    4 |    -0.279529 |    -0.094784 |    -0.007391 |    -0.177354
#>    8 |    -0.253949 |    -0.154280 |    -0.016737 |    -0.082932
#>   12 |    -0.158588 |    -0.122300 |    -0.011863 |    -0.024425
#>   16 |    -0.086618 |    -0.075479 |    -0.005694 |    -0.005445
#>   20 |    -0.044372 |    -0.041314 |    -0.002182 |    -0.000876
#> ----------------------------------------------------------------------

Exhaustive 4-Way

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

res_4w <- 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 = var_names
)
print(res_4w, target = 3)
#> 
#> TCA Results (mode: exhaustive_4way)
#> Shock from: IntRate | Horizon: 20
#> Response variable: Inflation
#> ---------------------------------------------------------------------- 
#>    h |        Total |     GDP only |   Wages only |  GDP & Wages |       Direct
#> ---------------------------------------------------------------------- 
#>    0 |     0.200000 |     0.040426 |     0.000000 |     0.000000 |     0.159574
#>    1 |    -0.025000 |     0.020023 |     0.004727 |     0.000273 |    -0.050023
#>    2 |    -0.161750 |    -0.019683 |     0.002645 |     0.002605 |    -0.147317
#>    4 |    -0.279529 |    -0.094721 |    -0.007391 |    -0.000064 |    -0.177354
#>    8 |    -0.253949 |    -0.138419 |    -0.016737 |    -0.015861 |    -0.082932
#>   12 |    -0.158588 |    -0.101321 |    -0.011863 |    -0.020979 |    -0.024425
#>   16 |    -0.086618 |    -0.057959 |    -0.005694 |    -0.017520 |    -0.005445
#>   20 |    -0.044372 |    -0.029408 |    -0.002182 |    -0.011907 |    -0.000876
#> ----------------------------------------------------------------------

Plotting

plot_tca(res_4w, target = 3, type = "bar")

plot_tca(res_ov, target = 3, type = "line")

Validating Additivity

tca_validate_additivity() checks Theorem 2(ii) of the paper: for every variable j, the effect of the paths through j is recomputed with AND conditions (partitioning the paths by the first node of j they visit), added to the not-through effect and compared with the total effect. The residual is of the order of machine precision for a valid systems form. With pair, the inclusion-exclusion identity behind the 4-way mode is checked in the same way (this needs (h+1)^2 linear solves, so a shorter horizon is used here):

tca_validate_additivity(
  from = 1, B = sf$B, Omega = sf$Omega,
  K = K, h = 20, order = 1:K, var_names = var_names
)
#> ===== Binary Additivity Test =====
#>   IntRate: max |total - (through + not_through)| = 1.39e-17
#>   GDP: max |total - (through + not_through)| = 1.11e-16
#>   Inflation: max |total - (through + not_through)| = 1.11e-16
#>   Wages: max |total - (through + not_through)| = 1.11e-16
#> 
#> B strictly lower triangular: yes
#> Overall max |residual| = 1.11e-16 (tolerance 1.0e-10)
#> PASSED
sf8 <- tca_systems_form(Phi0, list(A1), h = 8, order = 1:K)
ok <- tca_validate_additivity(
  from = 1, B = sf8$B, Omega = sf8$Omega,
  K = K, h = 8, order = 1:K, var_names = var_names, pair = c(2, 4),
  verbose = FALSE
)
attr(ok, "pair_residual")
#> [1] 1.027824e-16

Using with the vars Package

If you have estimated a VAR using vars::VAR(), you can use the convenience wrapper:

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")

Reference

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