id.cv: Identification of SVAR models based on Changes in volatility (CV)

Description

Given an estimated VAR model, this function applies changes in volatility 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 unconditional heteroskedasticity matrix.

Usage

id.cv(
  x,
  SB,
  SB2 = NULL,
  start = NULL,
  end = NULL,
  frequency = NULL,
  format = NULL,
  dateVector = NULL,
  max.iter = 50,
  crit = 0.001,
  restriction_matrix = NULL
)

Value

A list of class "svars" with elements

Lambda: Estimated unconditional 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 sequential Wald-type identification test on equal eigenvalues as described in Luetkepohl et. al. (2021). In case of more than two regimes, pairwise Wald-type tests of equal diagonal elements in the Lambda matrices are performed.
iteration: Number of GLS estimations
method: Method applied for identification
SB: Structural break (number of observations)
A_hat: Estimated VAR parameter via GLS
type: Type of the VAR model, e.g. 'const'
SBcharacter: Structural break (date; if provided in function arguments)
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

Arguments

x: An object of class 'vars', 'vec2var', 'nlVar'. Estimated VAR object
SB: Integer, vector or date character. The structural break is specified either by an integer (number of observations in the pre-break period), a vector of ts() frequencies if a ts object is used in the VAR or a date character. If a date character is provided, either a date vector containing the whole time line in the corresponding format (see examples) or common time parameters need to be provided
SB2: Integer, vector or date character. Optional if the model should be estimated with two volatility regimes. The structural break is specified either by an integer (number of observations in the pre-break period), a vector of ts() frequencies if a ts object is used in the VAR or a date character. If a date character is provided, either a date vector containing the whole time line in the corresponding format (see examples) or common time parameters need to be provided
start: Character. Start of the time series (only if dateVector is empty)
end: Character. End of the time series (only if dateVector is empty)
frequency: Character. Frequency of the time series (only if dateVector is empty)
format: Character. Date format (only if dateVector is empty)
dateVector: Vector. Vector of time periods containing SB in corresponding format
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).

References

Rigobon, R., 2003. Identification through Heteroskedasticity. The Review of Economics and Statistics, 85, 777-792.

Herwartz, H. & Ploedt, M., 2016. Simulation Evidence on Theory-based and Statistical Identification under Volatility Breaks. Oxford Bulletin of Economics and Statistics, 78, 94-112.

Luetkepohl, H. & Meitz, M. & Netsunajev, A. & and Saikkonen, P., 2021. Testing identification via heteroskedasticity in structural vector autoregressive models. Econometrics Journal, 24, 1-22.

Examples

Run this code

#' # 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.cv(v1, SB = 59)
summary(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')

# 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.cv(v1, SB = 59, restriction_matrix = restMat)
summary(x2)

# In alternative Form
restMat <- diag(rep(1,9))
restMat[7,7]= 0
x2 <- id.cv(v1, SB = 59, restriction_matrix = restMat)
summary(x2)

#Structural brake via Dates
# given that time series vector with dates is available
dateVector = seq(as.Date("1965/1/1"), as.Date("2008/7/1"), "quarter")
x3 <- id.cv(v1, SB = "1979-07-01", format = "%Y-%m-%d", dateVector = dateVector)
summary(x3)

# or pass sequence arguments directly
x4 <- id.cv(v1, SB = "1979-07-01", format = "%Y-%m-%d", start = "1965-01-01", end = "2008-07-01",
frequency = "quarter")
summary(x4)

# or provide ts date format (For quarterly, monthly, weekly and daily frequencies only)
x5 <- id.cv(v1, SB = c(1979, 3))
summary(x5)

#-----# Example with three covariance regimes

x6 <- id.cv(v1, SB = 59, SB2 = 110)
summary(x6)