Restrict VAR coefficients to zero

Applies zero restrictions to the VAR coefficient matrix \(\beta\) by setting the corresponding prior variances in Omega_beta to a value near zero (prior means are zero by default in the Minnesota prior).

Usage

restrict_beta(x, restriction_matrix)

Arguments

x

A steady-state bvar object that has been passed through priors.

restriction_matrix

A numeric matrix of dimension k*p x k where entries of 0 indicate coefficients to be restricted to zero and entries of 1 indicate unrestricted coefficients.

Value

The steady-state bvar object with the restriction matrix stored in setup and the prior covariance matrix Omega_beta updated accordingly.

Details

The steady-state BVAR model takes the form

$$y_t = \Psi d_t + \Pi_1(y_{t-1}-\Psi d_{t-1})+\dots+\Pi_p(y_{t-p}-\Psi d_{t-p})+u_t$$

where \(y_t\) is an \(k\)-dimensional vector of endogenous variables at time \(t\), and \(d_t\) is a \(q\)-dimensional vector of deterministic (exogenous) variables at time \(t\). Here \(\Pi_\ell\) for \(\ell=1,\dots,p\) is a \((k \times k)\) matrix of autoregressive parameters, and \(\Psi\) is a \((k \times q)\) matrix of steady-state parameters. One can stack the (transposed) \(\Pi_i\) matrices in the \((kp \times k)\) matrix \(\beta\) $$\beta=\begin{bmatrix}\Pi'_1 \\ \vdots \\\Pi'_p\end{bmatrix}$$

This function puts zero restrictions on the elements of \(\beta\).

Examples

yt <- matrix(rnorm(50), 25, 2)

bvar_obj <- bvar(data = yt)

bvar_obj <- setup(bvar_obj, p=2)

bvar_obj <- priors(bvar_obj,
                   theta_Psi = rep(0, 2),
                   Omega_Psi = diag(0.1, 2, 2))
                   
p <- bvar_obj$setup$p
k <- bvar_obj$setup$k

restriction_matrix <- matrix(1,k*p,k, dimnames=list(NULL,c("y1","y2")))

#restrict beta so y2 does not granger-cause y1

restriction_matrix[2, 1] <- 0
restriction_matrix[4, 1] <- 0

print(restriction_matrix)
#>      y1 y2
#> [1,]  1  1
#> [2,]  0  1
#> [3,]  1  1
#> [4,]  0  1

bvar_obj <- restrict_beta(bvar_obj, restriction_matrix)