pid.iv: Identification of panel SVAR models by means of proxy...

In pvars: VAR Modeling for Heterogeneous Panels

Identification of panel SVAR models by means of proxy variables

Description

Given an estimated panel of VAR models, this function uses proxy variables to partially identify 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}.

In general, identification procedures determine B_i up to column ordering, scale, and sign. For a unique solution, pid.iv follows the literature on proxy SVAR. The S columns in B_i = [B_{i,1} : B_{i,2}] of the identified shocks \epsilon_{its}, s=1,\ldots,S, are ordered first, and the variance \sigma^2_{\epsilon,is} = 1 is normalized to unity (see e.g. Lunsford 2015:6, Eq. 9). Further, the sign is fixed to a positive correlation between proxy and shock series. A normalization of the impulsed shock that may fix the size of the impact response in the IRF can be imposed subsequently via 'normf' in irf.pvarx and sboot.pmb.

Usage

pid.iv(
  x,
  iv,
  S2 = c("MR", "JL", "NQ"),
  cov_u = "OMEGA",
  R0 = NULL,
  combine = c("pool", "indiv")
)

Arguments

x	An object of class 'pvarx' or a list of VAR objects that will be coerced to 'varx'. Estimated panel of VAR objects.
iv	List. A single 'data.frame' of the L common proxy time series m_t or a list of N 'data.frame' objects of the L individual proxy time series m_{it}. The proxies must have the same succession l = 1,\ldots,L in each individual 'data.frame'.
S2	Character. Identification within multiple proxies m_{it} via 'MR' for lower-triangular [I_S : -B_{i,11} B_{i,12}^{-1} ] B_{i,1} by Mertens, Ravn (2013), via 'JL' for chol(\Sigma_{mu,i} \Sigma_{u,i}^{-1} \Sigma_{um,i}) by Jentsch, Lunsford (2021), or via 'NQ' for the nearest orthogonal matrix from svd decomposition by Empting et al. (2025). In case of S=L=1 proxy, all three choices provide identical results on B_{i,1}. In case of combine='pool', the argument is automatically fixed to 'NQ'.
cov_u	Character. Selection of the estimated residual covariance matrices \hat{\Sigma}_{u,i} to be used in the identification procedure. Either 'OMEGA' (the default) for \hat{U}_i \hat{U}_i'/T_i as used in Mertens, Ravn (2013) and Jentsch, Lunsford (2021) or 'SIGMA' for \hat{U}_i \hat{U}_i'/(T_i - n_{zi}), which corrects for the number of regressors n_{zi}. Both character options refer to the name of the respective estimate in the 'varx' objects.
R0	Matrix. A (L \times S) selection matrix for 'NQ' that governs the attribution of the L proxies to their specific S structural shock series. If NULL (the default), R0 = I_S will be used such that the S=L columns of B_{i,1} are one-by-one estimated from the S=L proxy series 'iv' available.
combine	Character. The combination of the individual reduced-form residuals via 'pool' for the pooled shocks by Empting et al. (2025) using a common orthogonal matrix or via 'indiv' for individual-specific B_i \forall i using strictly separated identification runs.

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 pooled identification procedure has been applied subsequently. These are still the unrotated baseline shocks! NULL if 'indiv' identifications have been used.
Q	Matrix. The orthogonal matrix suggested by the pooled 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.

References

Mertens, K., and Ravn, M. O. (2013): "The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States", American Economic Review, 103, pp. 1212-1247.

Jentsch, C., and Lunsford, K. G. (2019): "The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States: Comment", American Economic Review, 109, pp. 2655-2678.

Jentsch, C., and Lunsford, K. G. (2021): "Asymptotically Valid Bootstrap Inference for Proxy SVARs", Journal of Business and Economic Statistics, 40, pp. 1876-1891.

Empting, L. F. T., Maxand, S., Oeztuerk, S., and Wagner, K. (2025): "Inference in Panel SVARs with Cross-Sectional Dependence of Unknown Form".

See Also

Other panel identification functions: pid.chol(), pid.cvm(), pid.dc(), pid.grt()

Examples

data("PCIT")
names_k = c("APITR", "ACITR", "PITB", "CITB", "GOV", "RGDP", "DEBT")
names_l = c("m_PI", "m_CI")  # proxy names
names_s = paste0("epsilon[ ", c("PI", "CI"), " ]")  # shock names
dim_p   = 4  # lag-order

# estimate and identify panel SVAR #
L.vars = list(USA = vars::VAR(PCIT[ , names_k], p=dim_p, type="const"))
L.iv   = list(USA = PCIT[-(1:dim_p), names_l])
R.pid  = pid.iv(L.vars, iv=L.iv, S2="NQ", cov_u="SIGMA", combine="pool")
colnames(R.pid$B)[1:2] = names_s  # labeling

pvars documentation built on Nov. 5, 2025, 6:57 p.m.