Bayesian Inference of Structural Vector Autoregressions (SVAR) with the `bvartools` package

Posted in r var with tags r var svar vector autoregression bvartools -

The bvartools allows to perform Bayesian inference of Vector autoregressive (VAR) models, including structural VARs. This post guides through the Bayesian inference of SVAR models in R using the bvartools package.

Data

For this illustration we generate an artificial data set with three endogenous variables, which follows the data generating process

\[y_t = A_1 y_{t - 1} + B \epsilon_t,\]

where

\[ A_1 = \begin{bmatrix} 0.3 & 0.12 & 0.69 \\ 0 & 0.3 & 0.48 \\ 0.24 & 0.24 & 0.3 \end{bmatrix} \text{, } B = \begin{bmatrix} 1 & 0 & 0 \\ -0.14 & 1 & 0 \\ -0.06 & 0.39 & 1 \end{bmatrix} \text{ and } \epsilon_t \sim N(0, I_3). \]

# Reset random number generator for reproducibility
set.seed(24579)

tt <- 500 # Number of time series observations

# Coefficient matrix
A_1 <- matrix(c(0.3, 0, 0.24,
                0.12, 0.3, 0.24,
                0.69, 0.48, 0.3), 3)

# Structural coefficients
B <- diag(1, 3)
B[lower.tri(B)] <- c(-0.14, -0.06, 0.39)

# Generate series
sd_sigma <- 1
series <- matrix(rnorm(3, 0, sd_sigma), 3, tt + 1) # Raw series with zeros
for (i in 2:(tt + 1)){
  series[, i] <- A_1 %*% series[, i - 1] +  B %*% rnorm(3, 0, sd_sigma)
}

series <- ts(t(series)) # Convert to time series object
dimnames(series)[[2]] <- c("S1", "S2", "S3") # Rename variables

# Plot the series
plot.ts(series, main = "Artificial time series")

Model generation

library(bvartools)

# Generate basic model
mod <- gen_var(series,
               p = 1,
               deterministic = "none",
               structural = TRUE,
               iterations = 5000,
               burnin = 2000)

Prior specification

mod_w_prior <- add_priors(mod,
                          coef = list(v_i = 0),
                          sigma = list(shape = 3, rate = .0001))

Draw posteriors

post <- draw_posterior(mod_w_prior)

Evaluation

Summary statistics

summary(post)
## 
## Bayesian SVAR model with p = 1 
## 
## Model:
## 
## y ~ S1.01 + S2.01 + S3.01 + S1 + S2 + S3
## 
## Variable: S1 
## 
##         Mean      SD  Naive SD Time-series SD    2.5%    50%  97.5%
## S1.01 0.3293 0.02966 0.0004195      0.0004195 0.27280 0.3294 0.3884
## S2.01 0.1118 0.04009 0.0005670      0.0005670 0.03324 0.1115 0.1908
## S3.01 0.6810 0.04188 0.0005923      0.0005923 0.59815 0.6812 0.7624
## S1    1.0000 0.00000 0.0000000      0.0000000 1.00000 1.0000 1.0000
## S2    0.0000 0.00000 0.0000000      0.0000000 0.00000 0.0000 0.0000
## S3    0.0000 0.00000 0.0000000      0.0000000 0.00000 0.0000 0.0000
## 
## Variable: S2 
## 
##          Mean      SD  Naive SD Time-series SD      2.5%     50%  97.5%
## S1.01 0.05776 0.03355 0.0004745      0.0004745 -0.006956 0.05779 0.1239
## S2.01 0.30951 0.04081 0.0005771      0.0005771  0.230628 0.30994 0.3895
## S3.01 0.63676 0.05263 0.0007443      0.0007443  0.537193 0.63656 0.7400
## S1    0.18192 0.04636 0.0006556      0.0006556  0.092993 0.18148 0.2728
## S2    1.00000 0.00000 0.0000000      0.0000000  1.000000 1.00000 1.0000
## S3    0.00000 0.00000 0.0000000      0.0000000  0.000000 0.00000 0.0000
## 
## Variable: S3 
## 
##           Mean      SD  Naive SD Time-series SD     2.5%      50%   97.5%
## S1.01  0.25778 0.03293 0.0004657      0.0004657  0.19234  0.25788  0.3200
## S2.01  0.16723 0.04260 0.0006024      0.0006024  0.08265  0.16773  0.2490
## S3.01  0.19244 0.05762 0.0008148      0.0008148  0.07971  0.19282  0.3057
## S1     0.05055 0.04507 0.0006374      0.0005991 -0.03931  0.05031  0.1371
## S2    -0.31257 0.04345 0.0006145      0.0006145 -0.39792 -0.31234 -0.2277
## S3     1.00000 0.00000 0.0000000      0.0000000  1.00000  1.00000  1.0000
## 
## Variance-covariance matrix:
## 
##         Mean      SD  Naive SD Time-series SD   2.5%    50% 97.5%
## S1_S1 0.9880 0.06301 0.0008911      0.0009084 0.8745 0.9836 1.125
## S1_S2 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S1_S3 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S2_S1 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S2_S2 1.0375 0.06675 0.0009440      0.0009440 0.9135 1.0349 1.174
## S2_S3 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S3_S1 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S3_S2 0.0000 0.00000 0.0000000      0.0000000 0.0000 0.0000 0.000
## S3_S3 0.9862 0.06326 0.0008947      0.0009085 0.8692 0.9831 1.120

Compare with actual values

solve(B) %*% A_1
##         [,1]     [,2]     [,3]
## [1,] 0.30000 0.120000 0.690000
## [2,] 0.04200 0.316800 0.576600
## [3,] 0.24162 0.123648 0.116526
solve(B)
##        [,1]  [,2] [,3]
## [1,] 1.0000  0.00    0
## [2,] 0.1400  1.00    0
## [3,] 0.0054 -0.39    1

Plotting posterior distributions

t

plot(post, type = "boxplot")

Forecasts

pred_svar <- predict(post, "S1", n.ahead = 10)

plot(pred_svar)

Impulse response functions

irf_svar <- irf(post, n.ahead = 10, impulse = "S1", response = "S3", type = "sir")

plot(irf_svar)

Variance decomposition

fevd_svar <- fevd(post, n.ahead = 10, response = "S3", type = "sir")

plot(fevd_svar)

Compare with frequentist results

Estimate basic VAR

library(vars)

freq_var <- VAR(series, p = 1, type = "none")

Estimate structural coefficients

# Specify A0
a <- diag(1, 3)
a[lower.tri(a)] <- NA

freq_var_a <- SVAR(freq_var, Amat = a, max.iter = 1000)

freq_var_a
## 
## SVAR Estimation Results:
## ======================== 
## 
## 
## Estimated A matrix:
##         S1      S2 S3
## S1 1.00000  0.0000  0
## S2 0.18177  1.0000  0
## S3 0.05078 -0.3132  1

Impulse response function

freq_var_irf <- irf(freq_var_a, n.ahead = 10, impulse = "S1", response = "S3")

plot(freq_var_irf)

Literature

Chan, J., Koop, G., Poirier, D. J., & Tobias, J. L. (2019). Bayesian Econometric Methods (2nd ed.). Cambridge: University Press.

L├╝tkepohl, H. (2007). New Introduction to Multiple Time Series Analyis (2nd ed.). Berlin: Springer.

Pfaff, B. (2008). VAR, SVAR and SVEC models: Implementation within R package vars. Journal of Statistical Software, 27(4).