id.st: Identification of SVAR models by means of a smooth transition (ST) in covariance

Description

Given an estimated VAR model, this function uses a smooth transition in the covariance 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 pre-break covariance matrix $\Sigma_1=B B'$. The post-break covariance corresponds to $\Sigma_2=B\Lambda B'$ where $\Lambda$ is the estimated heteroskedasticity matrix.

Usage

id.st(
  x,
  c_lower = 0.3,
  c_upper = 0.7,
  c_step = 5,
  c_fix = NULL,
  transition_variable = NULL,
  gamma_lower = -3,
  gamma_upper = 2,
  gamma_step = 0.5,
  gamma_fix = NULL,
  nc = 1,
  max.iter = 5,
  crit = 0.001,
  restriction_matrix = NULL,
  lr_test = FALSE
)

Value

A list of class "svars" with elements

Lambda: Estimated heteroscedasticity matrix $\Lambda$
Lambda_SE: Matrix of standard errors of Lambda
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
n: Number of observations
Fish: Observed Fisher information matrix
Lik: Function value of likelihood
wald_statistic: Results of pairwise Wald tests
iteration: Number of GLS estimations
method: Method applied for identification
est_c: Structural break (number of observations)
est_g: Transition coefficient
transition_variable: Vector of transition variable
comb: Number of all grid combinations of gamma and c
transition_function: Vector of transition function
A_hat: Estimated VAR parameter via GLS
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
lr_test: Logical, whether a likelihood ratio test is performed
lRatioTest: Results of likelihood ratio test
VAR: Estimated input VAR object

Arguments

x: An object of class 'vars', 'vec2var', 'nlVar'. Estimated VAR object
c_lower: Numeric. Starting point for the algorithm to start searching for the volatility shift. Default is 0.3*(Total number of observations)
c_upper: Numeric. Ending point for the algorithm to stop searching for the volatility shift. Default is 0.7*(Total number of observations). Note that in case of a stochastic transition variable, the input requires an absolute value
c_step: Integer. Step width of c. Default is 5. Note that in case of a stochastic transition variable, the input requires an absolute value
c_fix: Numeric. If the transition point is known, it can be passed as an argument where transition point = Number of observations - c_fix
transition_variable: A numeric vector that represents the transition variable. By default (NULL), the time is used as transition variable. Note that c_lower,c_upper, c_step and/or c_fix have to be adjusted to the specified transition variable
gamma_lower: Numeric. Lower bound for gamma. Small values indicate a flat transition function. Default is -3
gamma_upper: Numeric. Upper bound for gamma. Large values indicate a steep transition function. Default is 2
gamma_step: Numeric. Step width of gamma. Default is 0.5
gamma_fix: Numeric. A fixed value for gamma, alternative to gamma found by the function
nc: Integer. Number of processor cores Note that the smooth transition model is computationally extremely demanding.
max.iter: Integer. Number of maximum GLS iterations
crit: Numeric. Critical value for the precision of the GLS estimation
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).
lr_test: Logical. Indicates whether the restricted model should be tested against the unrestricted model via a likelihood ratio test

References

Luetkepohl H., Netsunajev A., 2017. Structural vector autoregressions with smooth transition
in variances. Journal of Economic Dynamics and Control, 84, 43 - 57. ISSN 0165-1889.

Examples

Run this code

# \donttest{
# data contains quartlery observations from 1965Q1 to 2008Q2
# x = output gap
# pi = inflation
# i = interest rates
set.seed(23211)
v1 <- vars::VAR(USA, lag.max = 10, ic = "AIC" )
x1 <- id.st(v1, c_fix = 80, gamma_fix = 0)
summary(x1)
plot(x1)

# switching columns according to sign patter
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')

# Example with same data set as in Luetkepohl and Nestunajev 2017
v1 <- vars::VAR(LN, p = 3, type = 'const')
x1 <- id.st(v1, c_fix = 167, gamma_fix = -2.77)
summary(x1)
plot(x1)

# Using a lagged endogenous transition variable
# In this example inflation with two lags
inf <- LN[-c(1, 449, 450), 2]*(1/sd(LN[-c(1, 449, 450), 2]))
x1_inf <- id.st(v1, c_fix = 4.41, gamma_fix = 0.49, transition_variable = inf)
summary(x1_inf)
plot(x1_inf)

# }