pid.cvm: Independence-based identification of panel SVAR models via Cramer-von Mises (CVM) distance

Description

Given an estimated panel of VAR models, this function applies independence-based identification for the structural impact matrix $B_i$ of the corresponding SVAR model $$y_{it} = c_{it} + A_{i1} y_{i,t-1} + ... + A_{i,p_i} y_{i,t-p_i} + u_{it}$$ $$ = c_{it} + A_{i1} y_{i,t-1} + ... + A_{i,p_i} y_{i,t-p_i} + B_i \epsilon_{it}.$$ Matrix $B_i$ corresponds to the unique decomposition of the least squares covariance matrix $\Sigma_{u,i} = B_i B_i'$ if the vector of structural shocks $\epsilon_{it}$ contains at most one Gaussian shock (Comon, 1994). A nonparametric dependence measure, the Cramer-von Mises distance (Genest and Remillard, 2004), determines least dependent structural shocks. The minimum is obtained by a two step optimization algorithm similar to the technique described in Herwartz and Ploedt (2016).

Usage

pid.cvm(
  x,
  combine = c("group", "pool", "indiv"),
  n.factors = NULL,
  dd = NULL,
  itermax = 500,
  steptol = 100,
  iter2 = 75
)

Value

List of class 'pid' with elements:

A: Matrix. The lined-up coefficient matrices $A_j, j=1,\ldots,p$ for the lagged variables in the panel VAR.
B: Matrix. Mean group of the estimated structural impact matrices $B_i$, i.e. the unique decomposition of the covariance matrices of reduced-form errors.
L.varx: List of 'varx' objects for the individual estimation results to which the structural impact matrices $B_i$ have been added.
eps0: Matrix. The combined whitened residuals $\epsilon_{0}$ to which the ICA procedure has been applied subsequently. These are still the unrotated baseline shocks! NULL if 'indiv' identifications have been used.
ICA: List of objects resulting from the underlying ICA procedure. NULL if 'indiv' identifications have been used.
rotation_angles: Numeric vector. The rotation angles suggested by the combined identification procedure. NULL if 'indiv' identifications have been used.
args_pid: List of characters and integers indicating the identification methods and specifications that have been used.
args_pvarx: List of characters and integers indicating the estimator and specifications that have been used.

Arguments

x: An object of class 'pvarx' or a list of VAR objects that will be coerced to 'varx'. Estimated panel of VAR objects.
combine: Character. The combination of the individual reduced-form residuals via 'group' for the group ICA by Calhoun et al. (2001) using common structural shocks, via 'pool' for the pooled shocks by Herwartz and Wang (2024) using a common rotation matrix, or via 'indiv' for individual-specific $B_i \forall i$ using strictly separated identification runs.
n.factors: Integer. Number of common structural shocks across all individuals if the group ICA is selected.
dd: Object of class 'indepTestDist' generated by 'indepTest' from package 'copula'. Simulated independent sample(s) of the same size as the data. If the sample sizes $T_i - p_i$ differ across '$i$', the strictly separated identification requires a list of $N$ individual 'indepTestDist'-objects with respective sample sizes. If NULL (the default), a suitable object will be calculated during the call of pid.cvm.
itermax: Integer. Maximum number of iterations for DEoptim.
steptol: Numeric. Tolerance for steps without improvement for DEoptim.
iter2: Integer. Number of iterations for the second optimization.

References

Comon, P. (1994): "Independent Component Analysis: A new Concept?", Signal Processing, 36, pp. 287-314.

Genest, C., and Remillard, B. (2004): "Tests of Independence and Randomness based on the Empirical Copula Process", Test, 13, pp. 335-370.

Herwartz, H., and Wang, S. (2024): "Statistical Identification in Panel Structural Vector Autoregressive Models based on Independence Criteria", Journal of Applied Econometrics, 39 (4), pp. 620-639.

Herwartz, H. (2018): "Hodges Lehmann detection of structural shocks - An Analysis of macroeconomic Dynamics in the Euro Area", Oxford Bulletin of Economics and Statistics, 80, pp. 736-754.

Herwartz, H., and Ploedt, M. (2016): "The Macroeconomic Effects of Oil Price Shocks: Evidence from a Statistical Identification Approach", Journal of International Money and Finance, 61, pp. 30-44.

Examples

Run this code

# \donttest{
# select minimal or full example #
is_min = TRUE
idx_i  = ifelse(is_min, 1, 1:14)

# load and prepare data #
data("EURO")
data("EU_w")
names_i = names(EURO[idx_i+1])  # country names (#1 is EA-wide aggregated data)
idx_k   = 1:4   # endogenous variables in individual data matrices
idx_t   = 1:76  # periods from 2001Q1 to 2019Q4
trend2  = idx_t^2

# individual VARX models with common lag-order p=2 #
L.data = lapply(EURO[idx_i+1], FUN=function(x) x[idx_t, idx_k])
L.exog = lapply(EURO[idx_i+1], FUN=function(x) cbind(trend2, x[idx_t, 5:10]))
L.vars = sapply(names_i, FUN=function(i) 
  vars::VAR(L.data[[i]], p=2, type="both", exogen=L.exog[[i]]), 
  simplify=FALSE)

# identify under common orthogonal matrix (with pooled sample size (T-p)*N) #
S.pind = copula::indepTestSim(n=(76-2)*length(names_i), p=length(idx_k), N=100)
R.pcvm = pid.cvm(L.vars, dd=S.pind, combine="pool")
R.irf  = irf(R.pcvm, n.ahead=50, w=EU_w)
plot(R.irf, selection=list(1:2, 3:4))

# identify individually (with same sample size T-p for all 'i') #
S.pind = copula::indepTestSim(n=(76-2), p=length(idx_k), N=100)
R.pcvm = pid.cvm(L.vars, dd=S.pind, combine="indiv")
R.irf  = irf(R.pcvm, n.ahead=50, w=EU_w)
plot(R.irf, selection=list(1:2, 3:4))
# }