id.garch: Identification of SVAR models through patterns of GARCH

Description

Given an estimated VAR model, this function uses GARCH-type variances to identify 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 decomposition of the least squares covariance matrix $\Sigma_u=B\Lambda_t B'$, where $\Lambda_t$ is the estimated conditional heteroskedasticity matrix.

Usage

id.garch(
  x,
  max.iter = 5,
  crit = 0.001,
  restriction_matrix = NULL,
  start_iter = 50
)

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 residuals
B_SE: Standard errors of matrix B
GARCH_parameter: Estimated GARCH parameters of univariate GARCH models
GARCH_SE: Standard errors of GARCH parameters
n: Number of observations
Fish: Observed Fisher information matrix
Lik: Function value of likelihood
iteration: Number of likelihood optimizations
method: Method applied for identification
A_hat: Estimated VAR parameter via GLS
type: Type of the VAR model, e.g. 'const'
restrictions: Number of specified restrictions
restriction_matrix: Specified restriction matrix
y: Data matrix
p: Number of lags
K: Dimension of the VAR
VAR: Estimated input VAR object
I_test: Results of a series of sequential tests on the number of heteroskedastic shocks present in the system as described in Luetkepohl and Milunovich (2016).

Arguments

x: An object of class 'vars', 'vec2var', 'nlVar'. Estimated VAR object
max.iter: Integer. Number of maximum likelihood optimizations
crit: Numeric. Critical value for the precision of the iterative procedure
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).
start_iter: Numeric. Number of random candidate initial values for univariate GRACH(1,1) optimization.

References

Normadin, M. & Phaneuf, L., 2004. Monetary Policy Shocks: Testing Identification Conditions under Time-Varying Conditional Volatility. Journal of Monetary Economics, 51(6), 1217-1243.

Lanne, M. & Saikkonen, P., 2007. A Multivariate Generalized Orthogonal Factor GARCH Model. Journal of Business & Economic Statistics, 25(1), 61-75.

Luetkepohl, H. & Milunovich, G. 2016. Testing for identification in SVAR-GARCH models. Journal of Economic Dynamics and Control, 73(C):241-258

Examples

Run this code

# \donttest{
# data contains quartlery observations from 1965Q1 to 2008Q2
# assumed structural break in 1979Q3
# x = output gap
# pi = inflation
# i = interest rates
set.seed(23211)
v1 <- vars::VAR(USA, lag.max = 10, ic = "AIC" )
x1 <- id.garch(v1)
summary(x1)

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

# Restrictions
# Assuming that the interest rate doesn't influence the output gap on impact
restMat <- matrix(rep(NA, 9), ncol = 3)
restMat[1,3] <- 0
x2 <- id.garch(v1, restriction_matrix = restMat)
summary(x2)


# }