Model Comparison in bvartools

bvartools comes with the functionality to set up and produce posterior draws for multiple models in an effort to reduce the time required for this potentially laborious process. This vignette illustrates how the package can be used to set up multiple models, produce prior specifications, obtain posterior draws and select the model with the best fit in a few steps.

Data

For this illustrations the data set E1 from Lütkepohl (2006) is used. It contains data on West German fixed investment, disposable income and consumption expenditures in billions of DM from 1960Q1 to 1982Q4. Like in the textbook only the log-differenced series up to 1978Q4 are used.

library(bvartools)

set.seed(123456) # Set seed for reproducibility

data("e1") # Load data
data <- diff(log(e1)) * 100 # Obtain log-differences

# Use date up to 1978Q4
data <- window(data, end = c(1978, 4))

# Plot
plot(data)

Set up models

Functions gen_var can be used to obtain a list of different model specifications. In the following example five models with an intercept and increasing lag orders are generated.

object <- gen_var(data, p = 0:4,
                  deterministic = "const",
                  iterations = 5000, burnin = 1000)

All objects use the same amounts of available observations to ensure consistency for the calculation of information criteria for model selection.

Priors

Function add_priors can be used to produce priors for each of the models in object models.

object <- add_priors(object,
                     coef = list(v_i = 0, v_i_det = 0),
                     sigma = list(df = "k", scale = 0.0001))

Estimation

Posterior draws can be obtained using function draw_posterior. The function allows to specify the number of CPUs, which are available for parallel computing.

object <- draw_posterior(object, mc.cores = 3)

If multiple models are estimated the function produces an object of class bvarlist, which is a list of objects of class bvar. Thus, each element of the list can be used for further analysis.

Model selection

If function summary is applied to an object of class bvarlist, it produces a table of information criteria for each specification. The information criteria are calculated based on the posterior draws of the respective model and calculated in the following way:

Log-likelihood: \(LL = \frac{1}{R} \sum_{i = 1}^{R} \left( \sum_{t = 1}^{T} -\frac{K}{2} \ln 2\pi - \frac{1}{2} \ln |\Sigma_t^{(i)}| -\frac{1}{2} (u_t^{{(i)}\prime} (\Sigma_t^{(i)})^{-1} u_t^{(i)} \right)\) for each draw \(i\) and \(u_t = y_t - \mu_t\);
Akaika information criterion: \(AIC = 2 (Kp + M (s + 1) + N) - 2 LL\);
Bayesian information criterion: \(BIC = ln(T) (Kp + M (s + 1) + N) - 2 LL\);
Hannan-Quinn information criterion: \(HQ = 2 ln(ln(T)) (Kp + M (s + 1) + N) - 2 LL\).

\(K\) is the number of endogenous variables and \(p\) the lag order of the model. If exogenous variables were used \(M\) is the number of stochastic exogenous regressors and \(s\) is the lag order for those variables. \(N\) is the number of deterministic terms.

summary(object)
#>   p s        LL      AIC      BIC       HQ
#> 1 0 0 -420.5985 843.1970 845.4597 844.0968
#> 2 1 0 -413.5109 835.0217 844.0724 838.6209
#> 3 2 0 -405.9225 825.8450 841.6837 832.1436
#> 4 3 0 -408.6497 837.2994 859.9262 846.2974
#> 5 4 0 -406.7910 839.5820 868.9968 851.2793

Since all information criteria have the lowest value for the model with \(p = 2\), the third element of object is used for further analyis.

plot(irf(object[[3]], impulse = "income", response = "cons", n.ahead = 10))

Model Comparison in bvartools

Franz X. Mohr

2026-09-11

Data

Set up models

Priors

Estimation

Model selection

Literature