Introduction to bvartools

knitr::opts_chunk$set( collapse = TRUE, comment = "#>" ) Introduction The package bvartools implements some common functions used for Bayesian inference for mulitvariate time series models. It should give researchers maximum freedom in setting up an MCMC algorithm in R and keep calculation time limited at the same time. This is achieved by implementing posterior simulation functions in C++. Its main features are • The bvar and bvec function collects the output of a Gibbs sampler in standardised objects, which can be used for further analyses • Further functions such as predict, irf, fevd for forecasting, impulse response analysis and forecast error variance decomposition, respectively. • Computationally intensive functions - such as for posterior simulation - are written in C++ using the RcppArmadillo package of Eddelbuettel and Sanderson (2014).[^cpp] This vignette provides the code to set up and estimate a basic Bayesian VAR (BVAR) model with the bvartools package. For this illustration the dataset E1 from Lütkepohl (2007) is used. It contains data on West German fixed investment, disposable income and consumption expenditures in billions of DM from 1960Q1 to 1982Q4. library(bvartools) data("e1") e1 <- diff(log(e1)) plot(e1) # Plot the series The gen_var function produces the inputs y and x for the BVAR estimator, where y is a matrix of dependent variables and x is the matrix of regressors for the model $$y_t = A x_t + u_t,$$ with$u_t \sim N(0, \Sigma)$. data <- gen_var(e1, p = 2, deterministic = "const") y <- data$Y[, 1:73] x <- data$Z[, 1:73] As in Lütkepohl (2007) only the first 73 observations are used. Estimation Frequentist estimator We calculate frequentist VAR estimates using the standard formula$y x' (x x')^{-1}$to obtain a benchmark for the Bayesian estimator. The parameters are obtained by OLS: A_freq <- tcrossprod(y, x) %*% solve(tcrossprod(x)) # Calculate estimates round(A_freq, 3) # Round estimates and print And$\Sigma\$ is calculated by

u_freq <- y - A_freq %*% x u_sigma_freq <- tcrossprod(u_freq) / (ncol(y) - nrow(x)) round(u_sigma_freq * 10^4, 2)

These are the same values as in Lütkepohl (2007).

Bayesian estimator

The following code is a Gibbs sampler for a simple VAR model with non-informative priors.

# Reset random number generator for reproducibility set.seed(1234567) iter <- 10000 # Number of iterations of the Gibbs sampler burnin <- 5000 # Number of burn-in draws store <- iter - burnin t <- ncol(y) # Number of observations k <- nrow(y) # Number of endogenous variables m <- k * nrow(x) # Number of estimated coefficients # Set (uninformative) priors a_mu_prior <- matrix(0, m) # Vector of prior parameter means a_v_i_prior <- diag(0, m) # Inverse of the prior covariance matrix u_sigma_df_prior <- 0 # Prior degrees of freedom u_sigma_scale_prior <- diag(0, k) # Prior covariance matrix u_sigma_df_post <- t + u_sigma_df_prior # Posterior degrees of freedom # Initial values u_sigma_i <- diag(.00001, k) u_sigma <- solve(u_sigma_i) # Data containers for posterior draws draws_a <- matrix(NA, m, store) draws_sigma <- matrix(NA, k^2, store) # Start Gibbs sampler for (draw in 1:iter) { # Draw conditional mean parameters a <- post_normal(y, x, u_sigma_i, a_mu_prior, a_v_i_prior) # Draw variance-covariance matrix u <- y - matrix(a, k) %*% x # Obtain residuals u_sigma_scale_post <- solve(u_sigma_scale_prior + tcrossprod(u)) u_sigma_i <- matrix(rWishart(1, u_sigma_df_post, u_sigma_scale_post)[,, 1], k) u_sigma <- solve(u_sigma_i) # Invert Sigma_i to obtain Sigma # Store draws if (draw > burnin) { draws_a[, draw - burnin] <- a draws_sigma[, draw - burnin] <- u_sigma } }

After the Gibbs sampler has finished, point estimates can be obtained as the mean of the posterior draws:

A <- rowMeans(draws_a) # Obtain means for every row A <- matrix(A, k) # Transform mean vector into a matrix A <- round(A, 3) # Round values dimnames(A) <- list(dimnames(y)[], dimnames(x)[]) # Rename matrix dimensions A # Print
Sigma <- rowMeans(draws_sigma) # Obtain means for every row Sigma <- matrix(Sigma, k) # Transform mean vector into a matrix Sigma <- round(Sigma * 10^4, 2) # Round values dimnames(Sigma) <- list(dimnames(y)[], dimnames(y)[]) # Rename matrix dimensions Sigma # Print

The means of the coefficient draws are very close to the results of the frequentist estimatior, which would be expected with non-informative priors.

bvar objects

The bvar function can be used to collect relevant output of the Gibbs sampler into a standardised object, which can be used by further functions such as predict to obtain forecasts or irf for impulse respons analysis.

bvar_est <- bvar(y = y, x = x, A = draws_a[1:18,], C = draws_a[19:21, ], Sigma = draws_sigma)

Posterior draws can be thinned with function thin:

bvar_est <- thin(bvar_est, thin = 5)

Forecasts

Forecasts with credible bands can be obtained with the function predict. If the model contains deterministic terms, new values can be provided in the argument new_D. If no values are provided, the function sets them to zero. The number of rows of new_D must be the same as the argument n.ahead.

bvar_pred <- predict(bvar_est, n.ahead = 10, new_D = rep(1, 10)) plot(bvar_pred)

Impulse response analysis

Currently, bvartools supports forecast error, orthogonalised, and generalised impulse response functions.

Forecast error impulse response

FEIR <- irf(bvar_est, impulse = "income", response = "cons", n.ahead = 8) plot(FEIR, main = "Forecast Error Impulse Response", xlab = "Period", ylab = "Response")

Orthogonalised impulse response

OIR <- irf(bvar_est, impulse = "income", response = "cons", n.ahead = 8, type = "oir") plot(OIR, main = "Orthogonalised Impulse Response", xlab = "Period", ylab = "Response")

Generalised impulse response

GIR <- irf(bvar_est, impulse = "income", response = "cons", n.ahead = 8, type = "gir") plot(GIR, main = "Generalised Impulse Response", xlab = "Period", ylab = "Response")

Forecast error variance decomposition

Default forecast error variance decomposition (FEVD) is based on orthogonalised impulse responses (OIR).

bvar_fevd_oir <- fevd(bvar_est, response = "cons") plot(bvar_fevd_oir, main = "OIR-based FEVD of consumption")

It is also possible to calculate FEVDs, which are based on generalised impulse responses (GIR). Note that these do not automatically add up to unity.

bvar_fevd_gir <- fevd(bvar_est, response = "cons", type = "gir") plot(bvar_fevd_gir, main = "GIR-based FEVD of consumption")

Eddelbuettel, D., & Sanderson C. (2014). RcppArmadillo: Accelerating R with high-performance C++ linear algebra. Computational Statistics and Data Analysis, 71, 1054-1063. https://doi.org/10.1016/j.csda.2013.02.005

Koop, G., Pesaran, M. H., & Potter, S.M. (1996). Impulse response analysis in nonlinear multivariate models. Journal of Econometrics 74(1), 119-147. https://doi.org/10.1016/0304-4076(95)01753-4

Lütkepohl, H. (2007). New introduction to multiple time series analysis (2nd ed.). Berlin: Springer.

Pesaran, H. H., & Shin, Y. (1998). Generalized impulse response analysis in linear multivariate models. Economics Letters, 58(1), 17-29. https://doi.org/10.1016/S0165-1765(97)00214-0

Sanderson, C., & Curtin, R. (2016). Armadillo: a template-based C++ library for linear algebra. Journal of Open Source Software, 1(2), 26. https://doi.org/10.21105/joss.00026

[^cpp]: RcppArmadillo is the Rcpp bridge to the open source 'Armadillo' library of Sanderson and Curtin (2016).