id.ngml: Non-Gaussian maximum likelihood (NGML) identification of SVAR models

Description

Given an estimated VAR model, this function applies identification by means of a non-Gaussian likelihood for the structural impact matrix B of the corresponding SVAR model $$y_t=c_t+A_1 y_{t-1}+...+A_p y_{t-p}+u_t =c_t+A_1 y_{t-1}+...+A_p y_{t-p}+B \epsilon_t.$$ Matrix B corresponds to the unique decomposition of the least squares covariance matrix $\Sigma_u=B B'$ if the vector of structural shocks $\epsilon_t$ contains at most one Gaussian shock (Comon, 94). A likelihood function of independent t-distributed structural shocks $\epsilon_t=B^{-1}u_t$ is maximized with respect to the entries of B and the degrees of freedom of the t-distribution (Lanne et al., 2017).

Usage

id.ngml(x, stage3 = FALSE, restriction_matrix = NULL)

Value

A list of class "svars" with elements

B: Estimated structural impact matrix B, i.e. unique decomposition of the covariance matrix of reduced form errors
sigma: Estimated scale of the standardized matrix B_stand, i.e. $B=B_stand*diag(\sigma_1,...,\sigma_K)$
sigma_SE: Standard errors of the scale
df: Estimated degrees of freedom
df_SE: Standard errors of the degrees of freedom
Fish: Observed Fisher information matrix
A_hat: Estimated VAR parameter via ML
B_stand: Estimated standardized structural impact matrix
B_stand_SE: Standard errors of standardized matrix B_stand
Lik: Function value of likelihood
method: Method applied for identification
n: Number of observations
type: Type of the VAR model, e.g. 'const'
y: Data matrix
p: Number of lags
K: Dimension of the VAR
restrictions: Number of specified restrictions
restriction_matrix: Specified restriction matrix
stage3: Logical, whether Stage 3 is performed
VAR: Estimated input VAR object

Arguments

x: An object of class 'vars', 'vec2var', 'nlVar'. Estimated VAR object
stage3: Logical. If stage3="TRUE", the VAR parameters are estimated via non-gaussian maximum likelihood (computationally demanding)
restriction_matrix: Matrix. A matrix containing presupposed entries for matrix B, NA if no restriction is imposed (entries to be estimated). Alternatively, a K^2*K^2 matrix can be passed, where ones on the diagonal designate unrestricted and zeros restricted coefficients. (as suggested in Luetkepohl, 2017, section 5.2.1).

References

Lanne, M., Meitz, M., Saikkonen, P., 2017. Identification and estimation of non-Gaussian structural vector autoregressions. J. Econometrics 196 (2), 288-304.
Comon, P., 1994. Independent component analysis, A new concept?, Signal Processing, 36, 287-314

Examples

Run this code

# data contains quarterly observations from 1965Q1 to 2008Q3
# x = output gap
# pi = inflation
# i = interest rates
set.seed(23211)
v1 <- vars::VAR(USA, lag.max = 10, ic = "AIC" )
x1 <- id.ngml(v1)
summary(x1)

# switching columns according to sign pattern
x1$B <- x1$B[,c(3,2,1)]
x1$B[,3] <- x1$B[,3]*(-1)

# impulse response analysis
i1 <- irf(x1, n.ahead = 30)
plot(i1, scales = 'free_y')