TransmissionChannelAnalysis.jl

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TransmissionChannelAnalysis.jl provides a suite of functions for conducting transmission channel analysis (TCA) in Julia. It includes both standard methods—which are sufficient for most users—and more advanced, customisable techniques.

This overview focuses on the standard methods. Advanced options are described in the Advanced Section, and implementation details can be found in the Internals Section.

Before performing any transmission analysis, add TransmissionChannelAnalysis.jl to Julia and load it.

using Pkg; 
Pkg.add("TransmissionChannelAnalysis")
using TransmissionChannelAnalysis

A typical transmission channel analysis workflow consists of the following steps:

  1. Define and estimate a model
  2. Compute total effects
  3. Construct a transmission matrix
  4. Define transmission channels
  5. Compute transmission effects
  6. Visualise results

The sections that follow document the functions available for each of these steps.

1. Defining and estimating a model.

The first step in transmission channel analysis is to define a model. This model will be used to derive total dynamic causal effects (impulse response functions) and to decompose these effects into effects along transmission channels. Currently supported are the following model types:

  • Vector Autoregressions (VARs) for baseline dynamic modelling, estimated via OLS without structural identification.
  • Structural VARs (SVARs) with recursive and internal identification schemes.
  • Local projections with recursive and external instrument identification schemes.

Each model class extends the abstract type Model. Detailed descriptions of each model follow.

Vector Autoregressions (VARs)

A VAR model can be defined using the VAR type and its constructor as shown below:

VAR(data::DataFrame, p::Int;
    trend_exponents::Vector{<:Number} = [0])

Constructs a VAR model object with data, lag length p, and specified time trend exponents. Coefficients and residuals are uninitialised but can be estimed using fit!.

Arguments

  • data::DataFrame: Dataset to construct lag matrix and outcomes
  • p::Int: Lag length

Keyword Arguments

  • trend_exponents::Vector{<:Number}: Exponents of time trends (default: [0])

The constructor requires a dataset of type DataFrame, with each column representing a variable and each row an observation, ordered from earliest to latest. For example:

# data is some dataset of type DataFrame
p = 2  # lag length
model = VAR(data, p)

You can include deterministic trends by specifying the trend_exponents keyword-argument:

p = 2
model = VAR(data, p; trend_exponents = 0:1)

Here, 0:1 indicates a constant (\(t^0\)) and a linear trend (\(t^1\)). For constant, linear, and quadratic trends use 0:2, or [0,2] to skip the linear term.

To estimate the VAR via OLS, call fit!:

fit!(model)

Retrieve estimates with coeffs, fitted, and residuals:

# All coefficients
coeffs(model)
# Excluding constants/trends
coeffs(model, true)
# Fitted values
fitted(mdoel)
# Residuals
residuals(model)

To select the optimal lag length automatically, use fit_and_select!. This method fits VAR models for every lag length from 0 up to the specified maximum, evaluates each using the chosen information criterion (AIC by default), and returns the model with the lowest criterion value along with a table of lag lengths and their corresponding scores:

model = VAR(data, 20)
model_best, ic_table = fit_and_select!(model)

fit_and_select! supports AIC (aic), BIC/SIC (bic/sic), and HQC (hqc). For example, to use the Bayesian Information Criteria (BIC), call:

model = VAR(data, 20)
model_best, ic_table = fit_and_select!(model, bic)

To compute information criteria for an already estimated model over its own sample, call aic, bic, sic, or hqc on the model. Note these values may differ from those returned by fit_and_select!, which evaluates ICs over a common sample across all models:

aic(model)  # computes the AIC of the model
bic(model)  # computes the BIC of the model
sic(model)  # computes the SIC of the model
hqc(model)  # computes the HQC of the model

Structural Vector Autoregressions (SVARs)

An SVAR can be defined using the SVAR class and its constructor as documented below:

SVAR(data::DataFrame, 
     p::Int;
     trend_exponents::Vector{<:Number} = [0])

Constructs a SVAR model object with data, lag length p, and specified time trend exponents. Coefficients and residuals are uninitialised but can be estimed using fit! and an appropriate AbstractIdentificationMethod.

Arguments

  • data::DataFrame: Dataset used for the estimation
  • p::Int: Lag length

Keyword Arguments

  • trend_exponents::Vector{<:Number}: Exponents of time trends (default: [0])

The constructor requires a dataset of type DataFrame, with each column representing a variable and each row an observation, ordered from earliest to latest. For example:

p = 4  # lag length
model = SVAR(data, p)

You may include deterministic trends as for the VAR case, using the trend_exponents keyword-argument:

model = SVAR(data, p; trend_exponents=0:1)

To estimate the structural model, an identification method must be provided. Only recursive identification (Recursive) is supported for full structural identification. Note that InternalInstrument methods can be used to identify impulse responses to a single shock; these are documented in a later section. Fit the model as follows:

fit!(model, Recursive())

After estimation, coefficients, fitted values, and residuals can be obtained analogenously to the VAR using coeffs, fitted, and residuals, respectively.

# All coefficients (A0, A_plus)
coeffs(model)
# Excluding constants/trends
coeffs(model, true)
# Fitted values
fitted(mdoel)
# Residuals
residuals(model)

You can select the lag length automatically via fit_and_select!, which first determines the best reduced‑form VAR up to the maximum lag and then applies the chosen identification method:

Using AIC (default) with a maximum of twenty lags:

model = SVAR(data, 20)
model_best, ic_table = fit_and_select!(model, Recursive())

Or using HQC for model selection:

model = SVAR(data, 20)
model_best, ic_table = fit_and_select!(model, Recursive(), hqc)

All VAR information‑criterion functions may be supplied to choose the optimal SVAR.

Local Projections

A local projection model can be specified using the LP class and its constructor:

LP(data::DataFrame,
   treatment::Union{Symbol, Int},
   p::Int,
   horizons::Union{Int, AbstractVector{<:Int}};
   include_constant::Bool=true)

Constructs a LP model object for estimating local projections with specified horizons, treatment variable, and lag length p.

All variables before the treatment variable in the dataset are used as contemporaneous controls.

Arguments

  • data: the time series dataset
  • treatment: the treatment variable (column index or name)
  • p: number of lags to include
  • horizons: forecast horizons to compute IRFs for

Keyword Arguments

  • include_constant: whether to include a constant in the regressors

The constructor requires:

  • A dataset of type DataFrame
  • A treatment variable (either the shock itself or the instrumented variable)
  • The number of lags (p)
  • A vector of horizons (horizons)

For example, to estimate horizons \(0\)–\(20\) with 4 lags for treatment variable Y1:

model = LP(data, :Y1, 4, 0:20)

By default, each regression includes a constant term. To omit the constant, set the include_constant keyword-argument to false:

model = LP(data, :Y1, 4, 0:20; include_constant=false)

Use the fit! method to estimate the projections:

fit!(model)

By default, fit! applies a recursive (conditional ignorability) assumption. These estimates are structural only if the true shock satisfies this assumption; otherwise, they should be treated as reduced‐form.

Available identification schemes:

To use an external instrument, first instantiate ExternalInstrument with the treatment name and an instrument dataset, then supply it to fit!:

method = ExternalInstrument(:Y1, data_instruments)
fit!(model, method)

After estimation, extract the results (note the differing dimensions) via:

  • coeffs: (response variable × regressors × horizons)
  • fitted, residuals: (observations × variables × horizons)
# All coefficients
coeffs(model)
# Excluding the constant
coeffs(model, true)
# Fitted values 
fitted(model)
# Residuals
residuals(model)

To select the lag order automatically, use fit_and_select!, which fits VARs internally to choose the optimal p:

All VAR information‑criterion functions (e.g., aic, bic, sic, hqc) may be used for LP lag selection. For example, using BIC:

model = LP(data, :Y1, 20, 0:20)
model_best, ic_table = fit_and_select!(model, Recursive(), bic)

2. Obtaining the effects – structural IRFs

Once a model is defined, you can compute total dynamic causal effects—impulse response functions (IRFs)—using the IRF method. The IRF function always requires a maximum horizon (max_horizon) for computation. For reduced‐form models, you may also supply an IdentificationMethod to derive structural IRFs.

Details on IRF computation for all models are provided below.

VARs

Obtain reduced‐form IRFs from a VAR model by specifying the maximal horizon:

max_horizon = 20
irf_obj = IRF(model, max_horizon)

Here, irf_obj is of type IRF holding both the IRF array and related metadata:

Printing irf_obj displays a 3D array with dimensions:

  1. Response variable
  2. Shock
  3. Horizon (starting at 0 for contemporaneous effect)

To retrieve the raw IRF array, call:

irf_obj.irfs

To compute structural IRFs, supply an IdentificationMethod. Supported methods include recursive (Recursive) and internal‐instrument (InternalInstrument) schemes:

For a recursive identification:

irf_obj = IRF(model, Recursive(), max_horizon)

To use an internal instrument, specify a normalising_variable, then optionally override the instrument (default: first variable) or the normalising_horizon (default: 0 for contemporaneous effect):

normalising_variable = :Y1
method = InternalInstrument(normalising_variable)
irf_obj = IRF(model, method, max_horizon)
# Changing the instrument to be Y2 and the normalising variable to be Y3
method = InternalInstrument(:Y3; instrument=:Y2)
# Changing the normalising horizon to period 1
method = InternalInstrument(:Y3; instrument=:Y2, normalising_horizon=1)

SVARs

Impulse response functions can be obtained from an SVAR using the IRF function, analogous to the VAR case. Since an SVAR is already structurally identified, IRF returns structural IRFs directly:

# model is a SVAR
irf_obj = IRF(model, max_horizon)  # will be structural IRFs
Using SVARs and Internal Instruments

The IRF method for SVARs requires a fully identified SVAR. Internal instruments (via InternalInstrument) cannot be applied to an SVAR directly. To use internal instruments for IRF identification, first define a reduced‐form VAR, apply the instrument there, and then compute IRFs:

model = VAR(data, p);
normalising_variable = :Y2;
method = InternalInstrument(normalising_variable);
irf_obj = IRF(model, method, max_horizon)

Local Projections

Impulse response functions can be obtained from an estimated LP model using IRF and providing an identification method. Available methods are:

For example, using Recursive:

# model is an LP model
irf_obj = IRF(model, Recursive(), max_horizon)

Or using external instruments

method = ExternalInstrument(:Y1, data_instruments)
irf_obj = IRF(model, method, max_horizon)

3. Defining a transmission matrix.

The next step in a transmission channel analysis is to define the transmission matrix, which specifies the ceteris paribus ordering of variables in the chosen equilibrium representation. The simplest way is to list the variables in the desired order as a vector of Symbol. For example, if your data contains Y1, Y2, Y3, and Y4, and you wish to order them as Y4, Y2, Y1, Y3, define:

transmission_order = [:Y3, :Y2, :Y1, :Y3]

Note: The ordering must be a vector of Symbol not String.

Advanced methods for constructing transmission matrices are covered in the Advanced Methods Section.

4. Definining transmission channels.

With the transmission matrix in place, you can specify transmission channels of interest via the helper functions through and not_through. These functions return a transmission condition Q that you can combine to define complex transmission channels. More advanced methods for defining transmission channels are documented in the Advanced Methods Section.

through

Use through to define channels that pass through specified variables at given horizons.

Examples (for any model — VAR, SVAR, or LP — with transmission_order or a transmission matrix as described above):

  1. Channel through Y1 at horizons 0-3:
q = through(model, :Y1, 0:0, transmission_order)
  1. Channel through Y1 and Y2 contemporaneously:
q = through(model, [:Y1, :Y2], 0:0, transmission_order)
  1. Channel through Y1 at horizon 0 and Y2 at horizons 0–1:
q = through(model, [:Y1, :Y2], [0:0, 0:1])

not_through

Use not_through to define channels that do not go through specified variables at given horizons.

Examples (for any model — VAR, SVAR, or LP — with transmission_order a transmission matrix as described above):

  1. Channel not through Y1 at horizons 0-20:
q = not_through(model, :Y1, 0:20, transmission_order)
  1. Channel not through Y1 and not through Y2 at horizons 0-20:
q = not_through(model, [:Y1, :Y2], 0:20, transmission_order)
  1. Channel not through Y1 contemporaneously and not through Y2 at horizon 1:
q = not_through(model, [:Y1, :Y2], [0:0, 1:1], transmission_order)

Combining through and not_through

Both through and not_through return a transmission condition Q, which can be combined using logical operators:

  • AND (&): both conditions must be satisfied.
  • OR (|): at least one condition must be satisfied.
  • NOT (!): negates a condition.

Examples:

  1. Channel through Y1 contemporaneously but not through Y2 contemporaneously:
q1 = through(model, :Y1, 0:0, transmission_order)
q2 = not_through(model, :Y2, 0:0, transmission_order)
q = q1 & q2
  1. Channel through Y1 contemporaneously or not through Y2 contemporaneously:
q1 = through(model, :Y1, 0:0, transmission_order)
q2 = not_through(model, :Y2, 0:0, transmission_order)
q = q1 | q2
  1. Channel not through Y1 contemporaneously (both ways are equivalent):
q1 = through(model, :Y1, 0:0, transmission_order)
q = !q1
q = not_through(model, :Y1, 0:0, transmission_order)
  1. Channel through Y1 in at least one period between horizons 0-20:
q = not_through(model, :Y1, 0:20, transmission_order)
q = !q

By combining through and not_through with logical operators, you can flexibly construct any complex transmission channel you require. 

5. Obtaining transimssion effects.

Once you have defined a model (VAR, SVAR, or LP), the transmission method computes transmission effects along your specified channel. Below are the details for each model type.

For VAR and LP, or for SVAR models that have not yet been structurally estimated, you must provide an identification method (since at least one structural shock is required). As outlined above, this can be Recursive for SVAR and LP, InternalInstrument for SVAR, or ExternalInstrument for LP.

For example, to compute the effects of the first structural shock up to max_horizon, using a transmission matrix transmission_order and a transmission condition q:

effects = transmission(
    model,               # VAR, SVAR, or LP
    method,              # identification method
    1,                   # shock index
    q,                   # transmission condition
    transmission_order,  # transmission matrix
    max_horizon          # maximum horizon
)

The returned effects array has dimensions:

  1. Response variables (in original data order)
  2. Transmission effect of the chosen shock
  3. Horizons (starting at 0)

If your SVAR model is already structurally identified (fitted with a structural identification scheme), you may omit the identification argument:

effects = transmission(model, 1, q, transmission_order, max_horizon)