Type:	Package
Title:	VAR Modeling for Heterogeneous Panels
Version:	1.1.0
Maintainer:	Lennart Empting <lennart.empting@vwl.uni-due.de>
Description:	Implements (1) panel cointegration rank tests, (2) estimators for panel vector autoregressive (VAR) models, and (3) identification methods for panel structural vector autoregressive (SVAR) models as described in the accompanying vignette. The implemented functions allow to account for cross-sectional dependence and for structural breaks in the deterministic terms of the VAR processes. Among the large set of functions, particularly noteworthy are those that implement (1) the correlation-augmented inverse normal test on the cointegration rank by Arsova and Oersal (2021, <doi:10.1016/j.ecosta.2020.05.002>), (2) the two-step estimator for pooled cointegrating vectors by Breitung (2005, <doi:10.1081/ETC-200067895>), and (3) the pooled identification based on independent component analysis by Herwartz and Wang (2024, <doi:10.1002/jae.3044>).
License:	MIT + file LICENSE
URL:	https://github.com/Lenni89/pvars
BugReports:	https://github.com/Lenni89/pvars/issues
Depends:	R (≥ 3.5.0), svars (≥ 1.3.4)
Imports:	clue, copula, DEoptim, expm, ggplot2, MASS, pbapply, reshape2, scales, stats, steadyICA, utils, vars
Suggests:	ggfortify, ggpubr, knitr, plm, RColorBrewer, testthat (≥ 2.1.0), tikzDevice, urca
Encoding:	UTF-8
LazyData:	TRUE
VignetteBuilder:	knitr
NeedsCompilation:	no
RoxygenNote:	7.3.2
Packaged:	2025-10-09 08:46:08 UTC; lennart
Author:	Lennart Empting [aut, cre, cph]
Repository:	CRAN
Date/Publication:	2025-10-15 19:30:02 UTC

pvars: VAR Modeling for Heterogeneous Panels

Description

This package implements (1) panel cointegration rank tests, (2) estimators for panel vector autoregressive (VAR) models, and (3) identification methods for panel structural vector autoregressive (SVAR) models as described in the accompanying vignette. The implemented functions allow to account for cross-sectional dependence and for structural breaks in the deterministic terms of the VAR processes.

Details

(1) The panel functions to determine the cointegration rank are:

• pcoint.JO panel Johansen procedures,

• pcoint.BR panel test with pooled two-step estimation,

• pcoint.SL panel Saikkonen-Luetkepohl procedures,

• pcoint.CAIN correlation-augmented inverse normal test.

(2) The panel functions to estimate the VAR models are:

• pvarx.VAR mean-group of a panel of VAR models,

• pvarx.VEC pooled cointegrating vectors in a panel VECM.

(3) The panel functions to retrieve structural impact matrices are:

• pid.chol identification of panel SVAR models using Cholesky decomposition to impose recursive causality,

• pid.grt identification of panel SVEC models by imposing long- and short-run restrictions,

• pid.iv identification of panel SVAR models by means of proxy variables,

• pid.dc independence-based identification of panel SVAR models using distance covariance (DC) statistic,

• pid.cvm independence-based identification of panel SVAR models using Cramer-von Mises (CVM) distance.

Supporting tools, such as the specification functions (speci.VAR, speci.factors) and the panel block bootstrap procedure (sboot.pmb), complement the panel VAR functions and complete this coherent approach to VAR modeling for heterogeneous panels within the vars ecosystem. The provided data sets further allow for the exact replication of the implemented literature.

Author(s)

Lennart Empting lennart.empting@vwl.uni-due.de (ORCID: 0009-0004-5068-4639)

See Also

Useful links:

• https://github.com/Lenni89/pvars

• Report bugs at https://github.com/Lenni89/pvars/issues

Data set on the Exchange Rate Pass-Through

Description

The data set ERPT consists of monthly observations for the logarithm of import prices lm^*, foreign prices lf^*, and the exchange rate against the US dollar llcusd. It covers the period January 1995 to March 2005 (T=123) for N=7 countries. The asterisk denotes the industry of the variables and can take values from 0 to 8:

• 0: Food and live animals chiefly for food

• 1: Beverages and tobacco

• 2: Crude materials (inedible, except fuels)

• 3: Mineral fuels, lubricants and related materials

• 4: Animal and vegetable oils, fats and waxes

• 5: Chemicals and related products

• 6: Manufactured goods classified chiefly by materials

• 7: Machines, transport equipment

• 8: Manufactured goods

Usage

data("ERPT")

Format

A long-format data panel of class 'data.frame', where the columns id_i and id_t indicate the country and month respectively.

Source

The prepared Eurostat data set is directly obtainable from the ZBW Journal Data Archive: doi:10.15456/jae.2022321.0717881037. This is open data under the CC BY 4.0 license.

References

Banerjee, A., and Carrion-i-Silvestre, J. L. (2015): "Cointegration in Panel Data with Structural Breaks and Cross-Section Dependence", Journal of Applied Econometrics, 30 (1), pp. 1-23.

See Also

Other data sets: EURO, EU_w, ICAP, MDEM, MERM, PCAP, PCIT

Data set on the Euro Monetary Policy Transmission

Description

The data set EURO is a list of 15 'data.frame' objects, each consisting of quarterly observations for

• the first-difference of log real GDP on national dl\_GDP or aggregated EA-level EA\_dl\_GDP,

• the annualized inflation of the (log) GDP deflator on national dl\_deflator or aggregated EA-level EA\_pi,

• the EA-wide short-term interest rate IR,

• the EA-wide option-adjusted bond spreads BBB,

• the first-difference of log real GDP in the remaining countries dl\_GDP\_EA,

• the weighted inflation in the remaining countries dl\_deflator\_EA,

• the inflation of a world commodity price index WCP,

• the US effective federal funds rate US\_FFR,

• the trade volume in percentage of GDP trade, and

• the government spending in percentage of GDP ge.

The data covers the period Q1 2001 to Q1 2020 (T=77) for the aggregate of the Euro area (EA, first element in list) and N=14 of its member countries (subsequent 14 elements in list).

Usage

data("EURO")

Format

A list-format data panel of class 'list' containing 15 'data.frame' objects with named time series.

Source

The prepared Eurostat data set is directly obtainable from the ZBW Journal Data Archive: doi:10.15456/jae.2024044.1425287131. This is open data under the CC BY 4.0 license in accordance with the deposit license of the ZBW Journal Data Archive.

References

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.

See Also

Other data sets: ERPT, EU_w, ICAP, MDEM, MERM, PCAP, PCIT

Weights for the Euro Monetary Policy Transmission

Description

The data set EU_w is a vector of 14 elements. These are weights for N=14 member countries of the Euro area, constructed as the average share of their respective real GDP over the sample period in Herwartz, Wang (2024).

Usage

data("EU_w")

Format

A numeric vector containing 14 named elements.

Source

The prepared Eurostat data set is directly obtainable from the ZBW Journal Data Archive: doi:10.15456/jae.2024044.1425287131. This is open data under the CC BY 4.0 license in accordance with the deposit license of the ZBW Journal Data Archive.

References

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.

See Also

Other data sets: ERPT, EURO, ICAP, MDEM, MERM, PCAP, PCIT

Data set on Infrastructure Capital Stocks

Description

The data set ICAP consists of annual observations for

• the real GDP Y in US dollars,

• the physical capital stocks K in US dollars,

• the physical capital stocks KBC30 via "backcasting" in US dollars,

• the number of workers LWDI providing the total labor force, and

• the average years of secondary education secondary of the population.

It further reports physical measures of infrastructure given by

• the electricity generation capacity EGC in megawatts,

• the number of main phone lines mlines,

• the kilometers of total roads troads,

• the number of cell phones lines cells,

• the kilometers of paved roads proads, and

• the kilometers of rails rails.

It covers the period 1960 to 2000 (T=41) for N=97 countries. The monetary values are given in US-Dollars at 2000 prices, i.e. constant PPP.

Usage

data("ICAP")

Format

A long-format data panel of class 'data.frame', where the columns id_i and id_t indicate the country and year respectively. Column COUNTRY contains the complete country names.

Source

The prepared data set is directly obtainable from the ZBW Journal Data Archive: doi:10.15456/jae.2022321.0717368489. This is open data under the CC BY 4.0 license.

References

Calderon, C., Moral-Benito, E., and Serven, L. (2015): "Is Infrastructure Capital Productive? A Dynamic Heterogeneous Approach", Journal of Applied Econometrics, 30 (2), pp. 177-198.

See Also

Other data sets: ERPT, EURO, EU_w, MDEM, MERM, PCAP, PCIT

Data set for the Monetary Demand Model

Description

The data set MDEM consists of annual observations for the nominal short-term interest rate R and the logarithm of the real money aggregate m1 and real GDP gdp. It covers the period 1957 to 1996 (T=40) for N=19 countries.

Usage

data("MDEM")

Format

A long-format data panel of class 'data.frame', where the columns id_i and id_t indicate the country and year respectively.

Source

The prepared data is sourced from OECD and IMF's International Financial Statistics of the year 1998, see the open terms of use. Employed by Carrion-i-Silvestre and Surdeanu (2011:24, Ch.6.1), it has been originally compiled and described in the unpublished appendix of Mark and Sul (2003). See the related working paper of Mark and Sul (1999, Appendix B).

References

Carrion-i-Silvestre, J. L., and Surdeanu L. (2011): "Panel Cointegration Rank Testing with Cross-Section Dependence", Studies in Nonlinear Dynamics & Econometrics, 15 (4), pp. 1-43.

Mark, N. C., and Sul, D. (1999): "A Computationally Simple Cointegration Vector Estimator for Panel Data", Working Paper, Department of Economics, Ohio State University.

Mark, N. C., and Sul, D. (2003): "Cointegration Vector Estimation by Panel DOLS and Long-Run Money Demand," Oxford Bulletin of Economics and Statistics, 65, pp. 655-680.

See Also

Other data sets: ERPT, EURO, EU_w, ICAP, MERM, PCAP, PCIT

Data set for the Monetary Exchange Rate Model

Description

The data set MERM consists of monthly observations for the log-ratios of prices p, money supply m, and industrial production y as well as the natural logarithm of nominal exchange rates against the dollar s. It covers the period January 1995 to December 2007 (T=156) for N=19 countries.

Usage

data("MERM")

Format

A long-format data panel of class 'data.frame', where the columns id_i and id_t indicate the country and month respectively.

Source

The prepared data set is directly obtainable from the journal website: doi:10.1016/j.ecosta.2016.10.001. Supplementary Raw Research Data. This is open data under the CC BY 4.0 license.

References

Oersal, D. D. K., and Arsova, A. (2017): "Meta-Analytic Panel Cointegrating Rank Tests for Dependent Panels", Econometrics and Statistics, 2, pp. 61-72.

See Also

Other data sets: ERPT, EURO, EU_w, ICAP, MDEM, PCAP, PCIT

Data set on Public Capital Stocks

Description

The data set PCAP consists of annual observations for

• the governmental capital stocks G and their logarithm g,

• the private capital stocks K and their logarithm k,

• the total hours worked L and their logarithm l, and

• the real GDP Y and its logarithm y.

It is constructed from the annual observations for

• the governmental investments IG,

• the private non-residential investments and capital stocks IB and B,

• the private housing investments and capital stocks IH and H, and

• the persons employed ET and hours worked per person HRS.

It covers the period 1960 to 2019 (T=60) for N=23 OECD countries. All monetary values are given in US-Dollars at 2005 prices, i.e. constant PPP.

Usage

data("PCAP")

Format

A long-format data panel of class 'data.frame', where the columns id_i and id_t indicate the country and year respectively.

Source

Own compilation based on data from PWT, Eurostat, and OECD's Economic Outlook. Capital stocks are derived by the Perpetual Inventory Method as described by Kamps (2006). This is open data under the CC BY 4.0 license.

References

Empting, L. F. T., and Herwartz, H. (2025): "Revisiting the 'Productivity of Public Capital': VAR Evidence on the Heterogeneous Dynamics in a New Panel of 23 OECD Countries".

Feenstra, R. C., Inklaar, R., and Timmer, M. P. (2015): "The Next Generation of the Penn World Table", American Economic Review, 105, pp. 3150-3182.

Kamps, C. (2006): "New Estimates of Government Net Capital Stocks for 22 OECD Countries, 1960-2001", IMF Staff Papers, 53, pp. 120-150.

See Also

Other data sets: ERPT, EURO, EU_w, ICAP, MDEM, MERM, PCIT

Examples

### Latex figure: public capital stocks ###
data(PCAP)
names_i = c("DEU", "FRA", "GRC", "ITA", "NLD")  # select 5 exemplary countries
idx_i  = PCAP$id_i %in% names_i
idx_pl = c(1, 3, 5, 9, 10)
pallet = c(RColorBrewer::brewer.pal(n=11, name="Spectral")[idx_pl], "#000000")
names(pallet) = c(names_i, "\\O23")
breaks = factor(c(names_i, "\\O23"), levels=c(names_i, "\\O23"))
events = data.frame(
  label   = paste0("\\quad \\textbf{ ",
            c("Oil Crisis", "Oil Crisis II", "Early 1990s", "Early 2000s", 
              "Great Recession", "European Debt Crisis", "COVID-19"), " }"), 
  t_start = c(1973, 1979, 1990, 2000, 2007, 2010, 2020),
  t_end   = c(1975, 1982, 1993, 2003, 2010, 2013, 2022))

# plot events #
library("ggplot2")
F.base <- ggplot() + 
  geom_hline(yintercept = 0, color="grey") +
  geom_rect(data=events, aes(xmin=t_start, xmax=t_end, ymin=-Inf, ymax=+Inf), 
            fill="black", alpha=0.2) +
  geom_text(data=events, aes(x=(t_start+t_end)/2, y=-Inf, label=label), 
            hjust=0, angle=90, colour='white') +
  scale_x_continuous(breaks = seq(1960, 2020, 10), limits = c(1960, 2022)) +
  theme_bw(base_size=10)

# add levels #
F.G <- F.base +
  geom_line(data=PCAP[idx_i, ], aes(x=id_t, y=G/1e+12, colour=id_i, group=id_i), 
            linewidth=2) +
  stat_summary(data=PCAP[idx_i, ], aes(x=id_t, y=G/1e+12, color="\\O23"), 
               fun=mean, geom="line", linewidth=0) +
  geom_text(data=events, aes(x=(t_start+t_end)/2, y=-Inf, label=label), 
            hjust=0, angle=90, colour='white', alpha=0.6) +
  scale_colour_manual(values=pallet, breaks=breaks) +
  labs(x=NULL, y="Trillion US-\\$ at 2005 prices", colour="Country", title=NULL) +
  guides(colour=guide_legend(override.aes = list(linewidth=2))) +
  theme(legend.position="inside", legend.position.inside=c(0.01, 0.99), 
        legend.justification = c(0, 1), 
        legend.title=element_text(size=8), legend.text=element_text(size=6), 
        legend.key.width = unit(0.35, "cm"), legend.key.height = unit(0.35, "cm"))

# add growth rates #
PCAP$gG = ave(PCAP$G, PCAP$id_i, FUN=function(x) 
  c(diff(x), NA)/x*100)  # beginning-of-the-year observations!
F.gG <- F.base +
  geom_line(data=PCAP[idx_i, ], aes(x=id_t, y=gG, colour=id_i, group=id_i), 
            linewidth=2) +
  stat_summary(data=PCAP, aes(x=id_t, y=gG, color="\\O23"), 
               fun=mean, geom="line", linewidth=2) +
  geom_text(data=events, aes(x=(t_start+t_end)/2, y=-Inf, label=label), 
            hjust=0, angle=90, colour='white', alpha=0.6) +
  scale_colour_manual(values=pallet, breaks=breaks, guide="none") +
  labs(x="Year", y="Growth in \\%", colour="Country", title=NULL)

# export to Latex #

library(tikzDevice)
textwidth = 15.5/2.54  # LaTeX textwidth from "cm" into "inch"
file_fig  = file.path(tempdir(), "Fig_G.tex")

tikz(file=file_fig, width=1.2*textwidth, height=0.8*textwidth)
 # gridExtra::grid.arrange(grobs=list(F.G, F.gG), layout_matrix=cbind(1:2))
 ggpubr::ggarrange(F.G, F.gG, ncol=1, nrow=2, align="v")
dev.off()

Data set on Personal and Corporate Income Tax

Description

The data set PCIT consists of quarterly observations for

• the average personal income tax rates APITR,

• the average corporate income tax rates ACITR,

• the logarithm of the personal income tax base PITB,

• the logarithm of the corporate income tax base CITB,

• the logarithm of government spending GOV,

• the logarithm of GDP divided by population RGDP, and

• the logarithm of government debt held by the public divided by the GDP deflator and population DEBT.

Moreover, the proxies for shocks to personal m\_PI and corporate m\_CI income taxes are prepended, where non-zero observations from the related narratively identified shock series T\_PI resp. T\_CI have been demeaned. The data set covers the period Q1 1950 to Q4 2006 (T=228) for the US.

Usage

data("PCIT")

Format

A time series data set of class 'data.frame', where the column id_t indicates the quarter of the year.

Source

The prepared data set is directly obtainable from openICPSR: doi:10.3886/E116190V1. Supplementary Research Data. This is open data under the CC BY 4.0 license.

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.

Mertens, K., and Ravn, M. O. (2019): "The Dynamic Effects of Personal and Corporate Income Tax Changes in the United States: Reply", American Economic Review, 109, pp. 2679-2691.

See Also

Other data sets: ERPT, EURO, EU_w, ICAP, MDEM, MERM, PCAP

Persistence Profiles

Description

Calculates persistence profiles for each of the r long-run relationships.

Usage

PP.system(x, n.ahead = 20)

PP.variable(x, n.ahead = 20, shock = NULL)

Arguments

Value

A list of class 'svarirf' holding the persistence profiles as a 'data.frame'.

Functions

• PP.system(): PP due to a system-wide shock

• PP.variable(): PP due to a structural or variable-specific shock

References

Lee, K., C., Pesaran, M. H. (1993): "Persistence Profiles and Business Cycle Fluctuations in a Disaggregated Model of UK Output Growth", Richerche Economiche, 47, pp. 293-322.

Pesaran, M. H., and Shin, Y. (1996): "Cointegration and Speed of Convergence to Equilibrium", Journal of Econometrics, 71, pp. 117-143.

Examples

data("PCAP")
names_k = c("g", "k", "l", "y") # variable names
names_i = levels(PCAP$id_i)     # country names
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
  
# estimate VAR for DNK under rank-restriction r=2 #
dim_r  = 2  # cointegrataion rank
R.t_D1 = list(t_break=c(23, 49))  # trend breaks
R.vecm = VECM(y=L.data$DNK, dim_p=2, dim_r=dim_r, type="Case4", t_D1=R.t_D1)

# define shocks #
shock1 = diag(4)   # 4 separate shocks
shock2 = cbind(c(1, 0,  0, 0),  # positive shock on "g"
               c(0, 0, -1, 0),  # negative shock on "l"
               c(0, 0,  1, 1))  # simultaneous shocks

# calculate persistence profiles #
R.ppv1 = PP.variable(R.vecm, n.ahead=50, shock=shock1)
R.ppv2 = PP.variable(R.vecm, n.ahead=50, shock=shock2)
R.ppsy = PP.system(R.vecm, n.ahead=50)

# edit plots #
library("ggplot2")
as.pplot(ppv1=plot(R.ppv1), n.rows=4)$F.plot + guides(color="none")
as.pplot(ppv2=plot(R.ppv2), n.rows=3, color_g="black") # reshape facet array
plot(R.ppsy, selection=list(1, c(1,4)))  # dismiss cross-term PP

Estimation of a Vector Error Correction Model

Description

Estimates a VECM under a given cointegration-rank restriction or cointegrating vectors.

Usage

VECM(
  y,
  dim_p,
  x = NULL,
  dim_q = dim_p,
  dim_r = NULL,
  beta = NULL,
  type = c("Case1", "Case2", "Case3", "Case4", "Case5"),
  t_D1 = list(),
  t_D2 = list(),
  D1 = NULL,
  D2 = NULL
)

Arguments

Value

A list of class 'varx'.

References

Johansen, S. (1996): Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Advanced Texts in Econometrics, Oxford University Press, USA.

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Examples

### extend basic example in "vars" ###
library(vars)
data(Canada)
names_k = c("e", "U", "rw")  # names of endogenous variables
names_l = c("prod")  # names of exogenous variables
names_s = NULL  # optional shock names
x = Canada[ , names_l, drop=FALSE]
y = Canada[ , names_k, drop=FALSE]

# colnames of the restriction matrices are passed as shock names #
SR = matrix(NA, nrow=4, ncol=4, dimnames=list(NULL, names_s))
SR[4, 2] = 0
LR = matrix(NA, nrow=4, ncol=4, dimnames=list(NULL, names_s))
LR[1, 2:4] = 0
LR[2:4, 4] = 0

# estimate, identify, and plot the IRF #
R.vecm = VECM(y=y, dim_p=3, x=x, dim_q=3, dim_r=1, type="Case4")
R.grt  = id.grt(R.vecm, LR=LR, SR=SR)
R.irf  = irf(R.grt, n.ahead=50)
plot(R.irf)

Coerce into a 'pplot' object

Description

Coerce into a 'pplot' object. This function allows (1) to overlay and colorize multiple plots of IRF and confidence bands in a single 'ggplot' object and (2) to overwrite shock and variable names in verbatim LaTeX math mode suitable for the export via tikzDevice.

Usage

as.pplot(
  ...,
  names_k = NULL,
  names_s = NULL,
  names_g = NULL,
  color_g = NULL,
  shape_g = NULL,
  n.rows = NULL,
  scales = "free_y",
  Latex = FALSE
)

Arguments

Details

as.pplot is used as an intermediary in the 'pplot' functions to achieve compatibility with different classes. Equivalently to as.pvarx for panels of N VAR objects, as.pplot summarizes panels of G plot objects that have been returned from the 'plot' method for class 'svarirf' or 'sboot'. If the user wishes to extend the compatibility of the 'pplot' functions with further classes, she may simply specify accordant as.pplot-methods instead of altering the original 'pplot' functions.

Value

A list of class 'pplot'. Objects of this class contain the elements:

See Also

PP, irf.pvarx, pid.dc, and id.iv for further examples of edited plots, in particular for subset and reordered facet plots with reshaped facet dimensions.

Examples

### gallery of merged IRF plots ###

library("ggplot2")
data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names 
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
L.vars = lapply(L.data, FUN=function(x) vars::VAR(x, p=2, type="both"))
L.chol = lapply(L.vars, FUN=function(x) svars::id.chol(x))

# overlay all IRF to get an overview on the stability #
L.irf = lapply(L.chol, FUN=function(x) plot(irf(x, n.ahead=30)))
summary(as.pvarx(L.vars))
as.pplot(L.irf)

# overlay IRF of selected countries and quantiles of all countries #
F.mg  = plot(sboot.mg(L.chol, n.ahead=30), lowerq=0.05, upperq=0.95)
R.irf = as.pplot(MG=F.mg, L.irf[c("DEU", "FRA", "ITA", "JPN")])
plot(R.irf)  # emphasize MG-IRF in next step
R.irf = as.pplot(R.irf, color_g="black", shape_g=c(20, rep(NA, 4)))
R.irf$F.plot + guides(fill="none") + labs(color="Country", shape="Country")

# compare two mean-groups and their quantiles #
idx_nord = c(5, 6, 10, 17, 20)  # Nordic countries
R.irf = as.pplot(color_g=c("black", "blue"), 
  Other  = plot(sboot.mg(L.chol[-idx_nord])), 
  Nordic = plot(sboot.mg(L.chol[ idx_nord])))
plot(R.irf)
 
# compare different shock ordering for MG-IRF #
R.pid1 = pid.chol(L.vars)
R.pid2 = pid.chol(L.vars, order_k=4:1)
R.pid3 = pid.chol(L.vars, order_k=c(1,4,2,3))

R.pal = RColorBrewer::brewer.pal(n=8, name="Spectral")[c(8, 1, 4)]
R.irf = as.pplot(color_g=R.pal, shape_g=c(2, 3, 20), 
  GKLY = plot(irf(R.pid1, n.ahead=25)), 
  YLKG = plot(irf(R.pid2, n.ahead=25)), 
  GYKL = plot(irf(R.pid3, n.ahead=25)))
R.mg = as.pplot(color_g=R.pal, shape_g=c(2, 3, 20), 
  GKLY = plot(sboot.mg(R.pid1, n.ahead=25), lowerq=0.05, upperq=0.95), 
  YLKG = plot(sboot.mg(R.pid2, n.ahead=25), lowerq=0.05, upperq=0.95), 
  GYKL = plot(sboot.mg(R.pid3, n.ahead=25), lowerq=0.05, upperq=0.95))
         
# colorize and export a single sub-plot to Latex #
library("tikzDevice")
textwidth = 15.5/2.54  # LaTeX textwidth from "cm" into "inch"
file_fig  = file.path(tempdir(), "Fig_irf.tex")

R.irf = as.pplot(
  DEU = plot(irf(L.chol[["DEU"]], n.ahead=50), selection=list(4, 1)), 
  FRA = plot(irf(L.chol[["FRA"]], n.ahead=50), selection=list(4, 1)),
  color_g = c("black", "blue"),
  names_g = c("Germany", "France"),
  names_k = "y", 
  names_s = "\\epsilon_{ g }", 
  Latex   = TRUE)

tikz(file=file_fig, width=1.2*textwidth, height=0.8*textwidth)
  R.irf$F.plot + labs(color="Country") + theme_minimal()
dev.off()

Coerce into a 'pvarx' object

Description

Coerce into a 'pvarx' object. On top of the parent class 'pvarx', the child class 'pid' is imposed if the input object to be transformed contains a panel SVAR model.

Usage

as.pvarx(x, w = NULL, ...)

Arguments

Details

as.pvarx is used as an intermediary in the pvars functions to achieve compatibility with different classes of panel VAR objects. If the user wishes to extend this compatibility with further classes, she may simply specify accordant as.pvarx-methods instead of altering the original pvars function.

Value

A list of class 'pvarx'. Objects of this class contain the elements:

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names 
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
  
L.vars = lapply(L.data, FUN=function(x) vars::VAR(x, p=2, type="both"))
as.pvarx(L.vars)

Deterministic regressors in pvars

Description

pvars defines three kind of deterministic regressors, namely the conventional 'type', customized 'D', and period-specific 't_D'. While 'type' is a single character and 'D' a data matrix of dimension (n_{\bullet} \times (p+T)), the specifications for \tau in the list 't_D' are more complex and therefore preventively checked by as.t_D.

Usage

as.t_D(x, ...)

Arguments

Value

A list of class 't_D' specifying \tau. Objects of this class can exclusively contain the elements:

Reference Time Interval

The complete time series (i.e. including the presample) constitutes the reference time interval. Accordingly, 'D' contains p+T observations, and 't_D' contains the positions of activating periods \tau in 1,\ldots,(p+T). In a balanced panel p_i+T_i = T^*, the same date implies the same \tau in 1,\ldots,T^*, as shown in the example for pcoint.CAIN. However, in an unbalanced panel, the same date can imply different \tau across i in accordance with the individual time interval 1,\ldots,(p_i+T_i). Note that across the time series in 'L.data', it is the last observation in each data matrix that refers to the same date.

Conventional Type

An overview is given here and a detailed explanation in the package vignette.

• type (VAR) is specified in VAR models just as in vars' VAR, namely by a 'const', a linear 'trend', 'both', or 'none' of those.

• type_SL is used in the 'additive' SL procedure for testing the cointegration rank only, which removes the mean ('SL_mean') or mean and linear trend ('SL_trend') by GLS-detrending.

• type (VECM) is used in the 'innovative' Johansen procedure for testing the cointegration rank and estimating the VECM. In accordance with Juselius (2007, Ch.6.3), the available model specifications are: 'Case1' for none, 'Case2' for a constant in the cointegration relation, 'Case3' for an unrestricted constant, 'Case4' for a linear trend in the cointegration relation and an unrestricted constant, or 'Case5' for an unrestricted constant and linear trend.

References

Juselius, K. (2007): The Cointegrated VAR Model: Methodology and Applications, Advanced Texts in Econometrics, Oxford University Press, USA, 2nd ed.

Examples

t_D = list(t_impulse=c(10, 20, 35), t_shift=10)
as.t_D(t_D)

Coerce into a 'varx' object

Description

Coerce into a 'varx' object. On top of the parent class 'varx', the child class 'id' is imposed if the input object to be transformed contains an SVAR model.

Usage

as.varx(x, ...)

Arguments

Details

as.varx is used as an intermediary in the pvars functions to achieve compatibility with different classes of VAR objects. If the user wishes to extend this compatibility with further classes, she may simply specify accordant as.varx-methods instead of altering the original pvars function. Classes already covered by pvars are those of the vars ecosystem, in particular the classes

• 'varest' for reduced-form VAR estimates from VAR,

• 'vec2var' for reduced-form VECM estimates from vec2var,

• 'svarest' for structural VAR estimates from BQ,

• 'svecest' for structural VECM estimates from SVEC, and

• 'svars' for structural VAR estimates from svars' id.chol, id.cvm, or id.dc.

By transformation to 'varx', these VAR estimates can thus be subjected to pvars' bootstrap procedure sboot.mb and S3 methods such as summary and toLatex.

Value

A list of class 'varx'. Objects of this class contain the elements:

The following integers indicate the size of dimensions:

Some further elements required for the bootstrap functions are:

Examples

data("PCIT")
names_k = c("APITR", "ACITR", "PITB", "CITB", "GOV", "RGDP", "DEBT")

# estimate reduced-form VAR and coerce into 'varx' object #
R.vars = vars::VAR(PCIT[ , names_k], p=4, type="const")
as.varx(R.vars)

# identify structural VAR and coerce into 'id' object #
R.svar = svars::id.chol(R.vars, order_k=names_k)
as.varx(R.svar)

Test procedures for the cointegration rank

Description

Performs test procedures for the rank of cointegration in a single VAR model. The p-values are approximated by gamma distributions, whose moments are automatically adjusted to potential period-specific deterministic regressors and weakly exogenous regressors in the partial VECM.

Usage

coint.JO(
  y,
  dim_p,
  x = NULL,
  dim_q = dim_p,
  type = c("Case1", "Case2", "Case3", "Case4", "Case5"),
  t_D1 = NULL,
  t_D2 = NULL
)

coint.SL(y, dim_p, type_SL = c("SL_mean", "SL_trend"), t_D = NULL)

Arguments

Value

A list of class 'coint', which contains elements of length K for each r_{H0}=0,\ldots,K-1:

Functions

• coint.JO(): Johansen procedure.

• coint.SL(): (Trenkler)-Saikkonen-Luetkepohl procedure.

References

Johansen, S. (1988): "Statistical Analysis of Cointegration Vectors", Journal of Economic Dynamics and Control, 12, pp. 231-254.

Doornik, J. (1998): "Approximations to the Asymptotic Distributions of Cointegration Tests", Journal of Economic Surveys, 12, pp. 573-93.

Johansen, S., Mosconi, R., and Nielsen, B. (2000): "Cointegration Analysis in the Presence of Structural Breaks in the Deterministic Trend", Econometrics Journal, 3, pp. 216-249.

Kurita, T., Nielsen, B. (2019): "Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms", Econometrics, 7, pp. 1-35.

Saikkonen, P., and Luetkepohl, H. (2000): "Trend Adjustment Prior to Testing for the Cointegrating Rank of a Vector Autoregressive Process", Journal of Time Series Analysis, 21, pp. 435-456.

Trenkler, C. (2008): "Determining p-Values for Systems Cointegration Tests with a Prior Adjustment for Deterministic Terms", Computational Statistics, 23, pp. 19-39.

Trenkler, C., Saikkonen, P., and Luetkepohl, H. (2008): "Testing for the Cointegrating Rank of a VAR Process with Level Shift and Trend Break", Journal of Time Series Analysis, 29, pp. 331-358.

Examples

### reproduce basic example in "urca" ###
library("urca")
data(denmark)
sjd = denmark[ , c("LRM", "LRY", "IBO", "IDE")]

# rank test and estimation of the full VECM as in "urca" #
R.JOrank = coint.JO(y=sjd,      dim_p=2, type="Case2", t_D2=list(n.season=4))
R.JOvecm = VECM(y=sjd, dim_r=1, dim_p=2, type="Case2", t_D2=list(n.season=4))

# ... and of the partial VECM, i.e. after imposing weak exogeneity #
R.KNrank = coint.JO(y=sjd[ , c("LRM"), drop=FALSE],   dim_p=2,
                    x=sjd[ , c("LRY", "IBO", "IDE")], dim_q=2, 
                    type="Case2", t_D1=list(t_shift=36), t_D2=list(n.season=4))
R.KNvecm = VECM(y=sjd[ , c("LRM"), drop=FALSE],   dim_p=2,
                x=sjd[ , c("LRY", "IBO", "IDE")], dim_q=2, dim_r=1, 
                type="Case2", t_D1=list(t_shift=36), t_D2=list(n.season=4))

### reproduce Oersal,Arsova 2016:22, Tab.7.5 "France" ###
data("ERPT")
names_k = c("lpm5", "lfp5", "llcusd")  # variable names for "Chemicals and related products"
names_i = levels(ERPT$id_i)[c(1,6,2,5,4,3,7)]  # ordered country names
L.data  = sapply(names_i, FUN=function(i) 
   ts(ERPT[ERPT$id_i==i, names_k], start=c(1995, 1), frequency=12), 
   simplify=FALSE)
R.TSLrank = coint.SL(y=L.data$France, dim_p=3, type_SL="SL_trend", t_D=list(t_break=89))

Forecast Error Variance Decomposition

Description

Calculates the forecast error variance decomposition. Respects SVAR models of cases S \neq K, i.e. partially identified or excess shocks, too.

Usage

## S3 method for class 'id'
fevd(x, n.ahead = 10, ...)

Arguments

Value

A list of class 'svarfevd' holding the forecast error variance decomposition of each variables as a 'data.frame'.

References

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Jentsch, Lunsford (2022): "Asymptotically Valid Bootstrap Inference for Proxy SVARs", Journal of Business and Economic Statistics, 40, pp. 1876-1891.

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 proxy SVAR #
R.vars = vars::VAR(PCIT[ , names_k], p=dim_p, type="const")
R.idBL = id.iv(R.vars, iv=PCIT[-(1:dim_p), names_l], S2="MR", cov_u="OMEGA")
colnames(R.idBL$B) = names_s  # labeling

# calculate and plot FEVD under partial identification #
plot(fevd(R.idBL, n.ahead=20))

Identification of SVEC models by imposing long- and short-run restrictions

Description

Identifies an SVEC model by utilizing a scoring algorithm to impose long- and short-run restrictions. See the details of SVEC in vars.

Usage

id.grt(
  x,
  LR = NULL,
  SR = NULL,
  start = NULL,
  max.iter = 100,
  conv.crit = 1e-07,
  maxls = 1
)

Arguments

Value

List of class 'id'.

References

Amisano, G. and Giannini, C. (1997): Topics in Structural VAR Econometrics, Springer, 2nd ed.

Breitung, J., Brueggemann R., and Luetkepohl, H. (2004): "Structural Vector Autoregressive Modeling and Impulse Responses", in Applied Time Series Econometrics, ed. by H. Luetkepohl and M. Kraetzig, Cambridge University Press, Cambridge.

Johansen, S. (1996): Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Advanced Texts in Econometrics, Oxford University Press, USA.

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Pfaff, B. (2008): "VAR, SVAR and SVEC Models: Implementation within R Package vars", Journal of Statistical Software, 27, pp. 1-32.

See Also

... the original SVEC by Pfaff (2008) in vars. Note that id.grt is just a graftage, but allows for the additional model specifications in VECM and for the bootstrap procedures in sboot.mb, both provided by the pvars package.

Other identification functions: id.iv()

Examples

### reproduce basic example in "vars" ###
library(vars)
data("Canada")
names_k = c("prod", "e", "U", "rw")  # variable names
names_s = NULL  # optional shock names

# colnames of the restriction matrices are passed as shock names #
SR = matrix(NA, nrow=4, ncol=4, dimnames=list(names_k, names_s))
SR[4, 2] = 0
LR = matrix(NA, nrow=4, ncol=4, dimnames=list(names_k, names_s))
LR[1, 2:4] = 0
LR[2:4, 4] = 0

# estimate and identify SVECM #
R.vecm = VECM(y=Canada[ , names_k], dim_p=3, dim_r=1, type="Case4")
R.grt  = id.grt(R.vecm, LR=LR, SR=SR)

Identification of SVAR models by means of proxy variables

Description

Given an estimated VAR model, this function uses proxy variables to partially 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}.

In general, identification procedures determine B up to column ordering, scale, and sign. For a unique solution, id.iv follows the literature on proxy SVAR. The S columns in B = [B_1 : B_2] of the identified shocks \epsilon_{ts}, s=1,\ldots,S, are ordered first, and the variance \sigma^2_{\epsilon,s} = 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.varx and sboot.mb.

Usage

id.iv(x, iv, S2 = c("MR", "JL", "NQ"), cov_u = "OMEGA", R0 = NULL)

Arguments

Value

List of class 'id'.

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.

Lunsford, K. G. (2015): "Identifying Structural VARs with a Proxy Variable and a Test for a Weak Proxy", Working Paper, No 15-28, Federal Reserve Bank of Cleveland.

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

... the individual identification approaches by Lange et al. (2021) in svars.

Other identification functions: id.grt()

Examples

### reproduce Jentsch,Lunsford 2019:2668, Ch.III ###
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 under ordering "BLUE" of Fig.1 and 2 #
R.vars = vars::VAR(PCIT[ , names_k], p=dim_p, type="const")
R.idBL = id.iv(R.vars, iv=PCIT[-(1:dim_p), names_l], S2="MR", cov_u="OMEGA")
colnames(R.idBL$B) = names_s  # labeling

# estimate and identify under ordering "RED" of Fig.1 and 2 #
R.vars = vars::VAR(PCIT[ , names_k[c(2:1, 3:7)]], p=dim_p, type="const")
R.idRD = id.iv(R.vars, iv=PCIT[-(1:dim_p),names_l[2:1]], S2="MR", cov_u="OMEGA")
colnames(R.idRD$B) = names_s[2:1]  # labeling


# select minimal or full example #
is_min = TRUE
n.boot = ifelse(is_min, 5, 10000)

# bootstrap both under 1%-response normalization #
set.seed(2389)
R.norm = function(B) B / matrix(-diag(B), nrow(B), ncol(B), byrow=TRUE)
R.sbBL = sboot.mb(R.idBL, b.length=19, n.boot=n.boot, normf=R.norm)
R.sbRD = sboot.mb(R.idRD, b.length=19, n.boot=n.boot, normf=R.norm)

# plot IRF of Fig.1 and 2 with 68%-confidence levels #
library("ggplot2")
L.idx = list(BLUE1=c(1, 11, 5, 7, 3,  9)+0.1,
             RED1 =c(4, 12, 6, 8, 2, 10)+0.1, 
             RED2 =c(1, 11, 7, 5, 3,  9)+0.1, 
             BLUE2=c(4, 12, 8, 6, 2, 10)+0.1)
# Indexes to subset and reorder sub-plots in plot.sboot(), where 
# the 14 IRF-subplots in the 2D-panel are numbered as a 1D-sequence 
# vectorized by row. '+0.1' makes sub-setting robust against 
# truncation errors from as.integer(). In a given figure, the plots
# RED and BLUE display the same selection of IRF-subplots. 

R.fig1 = as.pplot(
 BLUE=plot(R.sbBL, lowerq=0.16, upperq=0.84, selection=list(1, L.idx[[1]])),
 RED =plot(R.sbRD, lowerq=0.16, upperq=0.84, selection=list(1, L.idx[[2]])),
 names_g=c("APITR first", "ACITR first"), color_g=c("blue", "red"), n.rows=3)

R.fig2 = as.pplot(
 RED =plot(R.sbRD, lowerq=0.16, upperq=0.84, selection=list(1, L.idx[[3]])), 
 BLUE=plot(R.sbBL, lowerq=0.16, upperq=0.84, selection=list(1, L.idx[[4]])),
 names_g=c("ACITR first", "APITR first"), color_g=c("red", "blue"), n.rows=3)

R.fig1$F.plot + labs(x="Quarters", color="Ordering", fill="Ordering")
R.fig2$F.plot + labs(x="Quarters", color="Ordering", fill="Ordering")

Impulse Response Functions for panel SVAR models

Description

Calculates impulse response functions for panel VAR objects.

Usage

## S3 method for class 'pvarx'
irf(x, ..., n.ahead = 20, normf = NULL, w = NULL, MG_IRF = TRUE)

Arguments

Value

A list of class 'svarirf' holding the impulse response functions as a 'data.frame'.

References

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Gambacorta L., Hofmann B., and Peersman G. (2014): "The Effectiveness of Unconventional Monetary Policy at the Zero Lower Bound: A Cross-Country Analysis", Journal of Money, Credit and Banking, 46, pp. 615-642.

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

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)

# estimate and identify panel SVAR #
L.vars = lapply(L.data, FUN=function(x) vars::VAR(x, p=2, type="both"))
R.pid  = pid.chol(L.vars, order_k=names_k)

# calculate and plot MG-IRF #
library("ggplot2")
R.irf = irf(R.pid, n.ahead=60)
F.irf = plot(R.irf, selection=list(2:4, 1:2))
as.pplot(F.irf=F.irf, color_g="black", n.rows=3)$F.plot + guides(color="none")

Impulse Response Functions

Description

Calculates impulse response functions.

Usage

## S3 method for class 'varx'
irf(x, ..., n.ahead = 20, normf = NULL)

Arguments

Value

A list of class 'svarirf' holding the impulse response functions as a 'data.frame'.

References

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

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

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 proxy SVAR #
R.vars = vars::VAR(PCIT[ , names_k], p=dim_p, type="const")
R.idBL = id.iv(R.vars, iv=PCIT[-(1:dim_p), names_l], S2="MR", cov_u="OMEGA")
colnames(R.idBL$B) = names_s  # labeling

# calculate and plot normalized IRF #
R.norm = function(B) B / matrix(-diag(B), nrow(B), ncol(B), byrow=TRUE)
plot(irf(R.idBL, normf=R.norm))

Panel cointegration rank tests

Description

Performs test procedures for the rank of cointegration in a panel of VAR models. First, the chosen individual procedure is applied over all N individual entities for r_{H0}=0,\ldots,K-1. Then, the K \times N individual statistics and p-values are combined to panel test results on each r_{H0} using all combination approaches available for the chosen procedure.

Usage

pcoint.JO(
  L.data,
  lags,
  type = c("Case1", "Case2", "Case3", "Case4"),
  t_D1 = NULL,
  t_D2 = NULL,
  n.factors = FALSE
)

pcoint.BR(
  L.data,
  lags,
  type = c("Case1", "Case2", "Case3", "Case4"),
  t_D1 = NULL,
  t_D2 = NULL,
  n.iterations = FALSE
)

pcoint.SL(L.data, lags, type = "SL_trend", t_D = NULL, n.factors = FALSE)

pcoint.CAIN(L.data, lags, type = "SL_trend", t_D = NULL)

Arguments

Value

A list of class 'pcoint' with elements:

Functions

• pcoint.JO(): based on the Johansen procedure.

• pcoint.BR(): based on the pooled two-step estimation of the cointegrating vectors.

• pcoint.SL(): based on the Saikkonen-Luetkepohl procedure.

• pcoint.CAIN(): accounting for correlated probits between the individual SL-procedures.

References

Larsson, R., Lyhagen, J., and Lothgren, M. (2001): "Likelihood-based Cointegration Tests in Heterogeneous Panels", Econometrics Journal, 4, pp. 109-142.

Choi, I. (2001): "Unit Root Tests for Panel Data", Journal of International Money and Finance, 20, pp. 249-272.

Arsova, A., and Oersal, D. D. K. (2018): "Likelihood-based Panel Cointegration Test in the Presence of a Linear Time Trend and Cross-Sectional Dependence", Econometric Reviews, 37, pp. 1033-1050.

Breitung, J. (2005): "A Parametric Approach to the Estimation of Cointegration Vectors in Panel Data", Econometric Reviews, 24, pp. 151-173.

Oersal, D. D. K., and Droge, B. (2014): "Panel Cointegration Testing in the Presence of a Time Trend", Computational Statistics & Data Analysis, 76, pp. 377-390.

Oersal, D. D. K., and Arsova, A. (2017): "Meta-Analytic Cointegrating Rank Tests for Dependent Panels", Econometrics and Statistics, 2, pp. 61-72.

Arsova, A., and Oersal, D. D. K. (2018): "Likelihood-based Panel Cointegration Test in the Presence of a Linear Time Trend and Cross-Sectional Dependence", Econometric Reviews, 37, pp. 1033-1050.

Hartung, J. (1999): "A Note on Combining Dependent Tests of Significance", Biometrical Journal, 41, pp. 849-855.

Arsova, A., and Oersal, D. D. K. (2021): "A Panel Cointegrating Rank Test with Structural Breaks and Cross-Sectional Dependence", Econometrics and Statistics, 17, pp. 107-129.

Examples

### reproduce Oersal,Arsova 2017:67, Ch.5 ###
data("MERM")
names_k = colnames(MERM)[-(1:2)] # variable names
names_i = levels(MERM$id_i)      # country names
L.data  = sapply(names_i, FUN=function(i) 
   ts(MERM[MERM$id_i==i, names_k], start=c(1995, 1), frequency=12), 
   simplify=FALSE)

# Oersal,Arsova 2017:67, Tab.5 #
R.lags = c(2, 2, 2, 2, 1, 2, 2, 4, 2, 3, 2, 2, 2, 2, 2, 1, 1, 2, 2)
names(R.lags) = names_i  # individual lags by AIC (lag_max=4)
n.factors = 8  # number of common factors by Onatski's (2010) criterion
R.pcsl = pcoint.SL(L.data, lags=R.lags, n.factors=n.factors, type="SL_trend")
R.pcjo = pcoint.JO(L.data, lags=R.lags, n.factors=n.factors, type="Case4")

# Oersal,Arsova 2017:67, Tab.6 #
R.Ftsl = coint.SL(y=R.pcsl$CSD$Ft, dim_p=2, type_SL="SL_trend")  # lag-order by AIC
R.Ftjo = coint.JO(y=R.pcsl$CSD$Ft, dim_p=2, type="Case4")

### reproduce Oersal,Arsova 2016:13, Ch.6 ###
data("ERPT")
names_k = c("lpm5", "lfp5", "llcusd")  # variable names for "Chemicals and related products"
names_i = levels(ERPT$id_i)[c(1,6,2,5,4,3,7)]  # ordered country names
L.data  = sapply(names_i, FUN=function(i) 
   ts(ERPT[ERPT$id_i==i, names_k], start=c(1995, 1), frequency=12), 
   simplify=FALSE)

# Oersal,Arsova 2016:21, Tab.6 (only for individual results) #
R.lags = c(3, 3, 3, 4, 3, 3, 3); names(R.lags)=names_i  # lags of VAR model by MAIC
R.cain = pcoint.CAIN(L.data, lags=R.lags, type="SL_trend")
R.pcsl = pcoint.SL(L.data,   lags=R.lags, type="SL_trend")

# Oersal,Arsova 2016:22, Tab.7/8 #
R.lags = c(3, 3, 3, 4, 4, 3, 4); names(R.lags)=names_i  # lags of VAR model by MAIC
R.t_D  = list(t_break=89)  # a level shift and trend break in 2002_May for all countries
R.cain = pcoint.CAIN(L.data, lags=R.lags, t_D=R.t_D, type="SL_trend")

Recursive identification of panel SVAR models via Cholesky decomposition

Description

Given an estimated panel of VAR models, this function uses the Cholesky decomposition to 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}.

Matrix B_i corresponds to the decomposition of the least squares covariance matrix \Sigma_{u,i} = B_i B_i'.

Usage

pid.chol(x, order_k = NULL)

Arguments

Value

List of class 'pid' with elements:

References

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Sims, C. A. (2008): "Macroeconomics and Reality", Econometrica, 48, pp. 1-48.

See Also

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

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)

# estimate and identify panel SVAR #
L.vars = lapply(L.data, FUN=function(x) vars::VAR(x, p=2, type="both"))
R.pid  = pid.chol(L.vars, order_k=names_k)

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
)

Arguments

Value

List of class 'pid' with elements:

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.

See Also

... the individual id.cvm by Lange et al. (2021) in svars. Note that pid.cvm relies on a modification of their procedure and thus performs ICA on the pre-whitened shocks 'eps0' directly.

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

Examples

# 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))

Independence-based identification of panel SVAR models using distance covariance (DC) statistic

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 distance covariance (Szekely et al., 2007), determines least dependent structural shocks. The algorithm described in Matteson and Tsay (2013) is applied to calculate the matrix B_i.

Usage

pid.dc(
  x,
  combine = c("group", "pool", "indiv"),
  n.factors = NULL,
  n.iterations = 100,
  PIT = FALSE
)

Arguments

Value

List of class 'pid' with elements:

Notes on the Reproduction in "Examples"

The reproduction of Herwartz and Wang (HW, 2024:630) serves as an exemplary application and unit-test of the implementation by pvars. While vars' VAR employs equation-wise lm with the QR-decomposition of the regressor matrix X, HW2024 and accordingly the reproduction by pvarx.VAR both calculate X'(XX')^{-1} for the multivariate least-squares estimation of A_i. Moreover, both use steadyICA for identification such that the reproduction result for the pooled rotation matrix Q is close to HW2024, the mean absolute difference between both 4x4 matrices is less than 0.0032. Note that the single EA-Model is estimated and identified the same way, which can be extracted as a separate 'varx' object from the trivial panel object by $L.varx[[1]] and even bootstrapped by sboot.mb.

Some differences remain such that the example does not exactly reproduce the results in HW2024. To account for the n exogenous and deterministic regressors in slot $D, pvarx.VAR calculates \Sigma_{u,i} with the degrees of freedom T-Kp_i-n instead of HW2024's T-Kp_i-1. Moreover, the confidence bands for the IRF are based on pvars' panel moving-block- instead of HW2024's wild bootstrap. The responses of real GDP and of inflation are not scaled by 0.01, unlike in HW2024. Note that both bootstrap procedures keep D fixed over their iterations.

References

Calhoun, V. D., Adali, T., Pearlson, G. D., and Pekar, J. J. (2001): "A Method for Making Group Inferences from Functional MRI Data using Independent Component Analysis", Human Brain Mapping, 16, pp. 673-690.

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

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.

Matteson, D. S., and Tsay, R. S. (2017): "Independent Component Analysis via Distance Covariance", Journal of the American Statistical Association, 112, pp. 623-637.

Risk, B., Matteson, D. S., Ruppert, D., Eloyan, A., and Caffo, B. S. (2014): "An Evaluation of Independent Component Analyses with an Application to Resting-State fMRI", Biometrics, 70, pp. 224-236.

Szekely, G. J., Rizzo, M. L., and Bakirov, N. K. (2007): "Measuring and Testing Dependence by Correlation of Distances", Annals of Statistics, 35, pp. 2769-2794.

See Also

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

Examples

### replicate Herwartz,Wang 2024:630, Ch.4 ###

# select minimal or full example #
is_min = TRUE
n.boot = ifelse(is_min, 5, 1000)
idx_i  = ifelse(is_min, 1, 1:14)

# load and prepare data #
data("EURO")
names_i = names(EURO[idx_i+1])  # country names (#1 is EA-wide aggregated data)
names_s = paste0("epsilon[ ", c(1:2, "m", "f"), " ]")  # shock names
idx_k   = 1:4   # endogenous variables in individual data matrices
idx_t   = 1:(nrow(EURO[[1]])-1)  # periods from 2001Q1 to 2019Q4 
trend2  = idx_t^2

# panel SVARX model, Ch.4.1 #
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]))
R.lags = c(1,2,1,2,2,2,2,2,1,2,2,2,2,1)[idx_i]; names(R.lags) = names_i
R.pvar = pvarx.VAR(L.data, lags=R.lags, type="both", D=L.exog)
R.pid  = pid.dc(R.pvar, combine="pool")
print(R.pid)  # suggests e3 and e4 to be MP and financial shocks, respectively.
colnames(R.pid$B) = names_s  # accordant labeling

# EA-wide SVARX model, Ch.4.2 #
R.data = EURO[[1]][idx_t, idx_k]
R.exog = cbind(trend2, EURO[[1]][idx_t, 5:6])
R.varx = pvarx.VAR(list(EA=R.data), lags=2, type="both", D=list(EA=R.exog))
R.id   = pid.dc(R.varx, combine="indiv")$L.varx$EA
colnames(R.id$B) = names_s  # labeling

# comparison of IRF without confidence bands, Ch.4.3.1 #
data("EU_w")  # GDP weights with the same ordering names_i as L.varx in R.pid
R.norm = function(B) B / matrix(diag(B), nrow(B), ncol(B), byrow=TRUE) * 25
R.irf  = as.pplot(
  EA=plot(irf(R.id,  normf=R.norm), selection=list(idx_k, 3:4)),
  MG=plot(irf(R.pid, normf=R.norm, w=EU_w), selection=list(idx_k, 3:4)),
  color_g=c("#3B4992FF", "#008B45FF"), shape_g=16:17, n.rows=length(idx_k))
plot(R.irf)

# comparison of IRF with confidence bands, Ch.4.3.1 #
R.boot_EA = sboot.mb(R.id, b.length=8, n.boot=n.boot, n.cores=2, normf=R.norm)
R.boot_MG = sboot.pmb(R.pid, b.dim=c(8, FALSE), n.boot=n.boot, n.cores=2, 
                      normf=R.norm, w=EU_w)
R.irf = as.pplot(
  EA=plot(R.boot_EA, lowerq=0.16, upperq=0.84, selection=list(idx_k, 3:4)),
  MG=plot(R.boot_MG, lowerq=0.16, upperq=0.84, selection=list(idx_k, 3:4)),
  color_g=c("#3B4992FF", "#008B45FF"), shape_g=16:17, n.rows=length(idx_k))
plot(R.irf)

Identification of panel SVEC models by imposing long- and short-run restrictions

Description

Identifies a panel of SVEC models by utilizing a scoring algorithm to impose long- and short-run restrictions. See the details of SVEC in vars.

Usage

pid.grt(
  x,
  LR = NULL,
  SR = NULL,
  start = NULL,
  max.iter = 100,
  conv.crit = 1e-07,
  maxls = 1
)

Arguments

Value

List of class 'pid' with elements:

References

Amisano, G. and Giannini, C. (1997): Topics in Structural VAR Econometrics, Springer, 2nd ed.

Breitung, J., Brueggemann R., and Luetkepohl, H. (2004): "Structural Vector Autoregressive Modeling and Impulse Responses", in Applied Time Series Econometrics, ed. by H. Luetkepohl and M. Kraetzig, Cambridge University Press, Cambridge.

Johansen, S. (1996): Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Advanced Texts in Econometrics, Oxford University Press, USA.

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Pfaff, B. (2008): "VAR, SVAR and SVEC Models: Implementation within R Package vars", Journal of Statistical Software, 27, pp. 1-32.

See Also

... the original SVEC by Pfaff (2008) in vars. Note that pid.grt relies on this underlying procedure, but allows for the additional model specifications in pvarx.VEC and for the bootstrap procedures in sboot.pmb, both provided by the pvars package.

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

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names
names_s = NULL                   # optional shock names
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)

# colnames of the restriction matrices are passed as shock names #
SR = matrix(NA, nrow=4, ncol=4, dimnames=list(names_k, names_s))
SR[1, 2] = 0
SR[3, 4] = 0
LR = matrix(NA, nrow=4, ncol=4, dimnames=list(names_k, names_s))
LR[ , 3:4] = 0

# estimate and identify panel SVECM #
R.pvec = pvarx.VEC(L.data, lags=2, dim_r=2, type="Case4")
R.pid  = pid.grt(R.pvec, LR=LR, SR=SR)

Identification of panel SVAR models by means of proxy variables

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}.

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

Value

List of class 'pid' with elements:

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

Estimation of VAR models for heterogeneous panels

Description

Performs the (pooled) mean-group estimation of a panel VAR model. First, VAR models are estimated for all N individual entities. Then, their (pooled) mean-group estimate is calculated for each coefficient.

Usage

pvarx.VAR(
  L.data,
  lags,
  type = c("const", "trend", "both", "none"),
  t_D = NULL,
  D = NULL,
  n.factors = FALSE,
  n.iterations = FALSE
)

pvarx.VEC(
  L.data,
  lags,
  dim_r,
  type = c("Case1", "Case2", "Case3", "Case4", "Case5"),
  t_D1 = NULL,
  t_D2 = NULL,
  D1 = NULL,
  D2 = NULL,
  idx_pool = NULL,
  n.factors = FALSE,
  n.iterations = FALSE
)

Arguments

Value

A list of class 'pvarx' with the elements:

Functions

• pvarx.VAR(): Mean Group (MG) of VAR models in levels.

• pvarx.VEC(): (Pooled) Mean Group (PMG) of VECM.

References

Canova, F., and Ciccarelli, M. (2013): "Panel Vector Autoregressive Models: A Survey", Advances in Econometrics, 32, pp. 205-246.

Hsiao, C. (2014): Analysis of Panel Data, Econometric Society Monographs, Cambridge University Press, 3rd ed.

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

Pesaran, M. H., and Smith R. J. (1995): "Estimating Long-Run Relationships from Dynamic Heterogeneous Panels", Journal of Econometrics, 68, pp. 79-113.

Rebucci, A. (2010): "Estimating VARs with Long Stationary Heterogeneous Panels: A Comparison of the Fixed Effect and the Mean Group Estimators", Economic Modelling, 27, pp. 1183-1198.

Ahn, S. K., and Reinsel (1990): "Estimation for Partially Nonstationary Multivariate Autoregressive Models", Journal of the American Statistical Association, 85, pp. 813-823.

Breitung, J. (2005): "A Parametric Approach to the Estimation of Cointegration Vectors in Panel Data", Econometric Reviews, 24, pp. 151-173.

Johansen, S. (1996): Likelihood-based Inference in Cointegrated Vector Autoregressive Models, Advanced Texts in Econometrics, Oxford University Press, USA.

Pesaran, M. H., Shin, Y, and Smith R. J. (1999): "Pooled Mean Group Estimation of Dynamic Heterogeneous Panels", Journal of the American Statistical Association, 94, pp. 621-634.

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names 
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
R.lags = c(2, 4, 2, 3, 2, 4, 4, 2, 2, 3, 3, 3, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 4)
names(R.lags) = names_i

### MG of VAR by OLS ###
R.t_D  = list(t_shift=10)  # common level shift for all countries 
R.pvar = pvarx.VAR(L.data, lags=R.lags, type="both", t_D=R.t_D)
R.pirf = irf(R.pvar, n.ahead=50)  # MG of individual forecast-error IRF
plot(R.pirf)

### Pooled MG of rank-restricted VAR ###
R.pvec = pvarx.VEC(L.data, lags=R.lags, dim_r=2, idx_pool=1:4, type="Case4")
R.pirf = irf(R.pvec, n.ahead=50)  # MG of individual forecast-error IRF
plot(R.pirf)

Bootstrap for JB normality test

Description

Bootstraps the distribution of the Jarque-Bera test for individual VAR and VECM as described by Kilian, Demiroglu (2000).

Usage

rboot.normality(x, n.boot = 1000, n.cores = 1, fix_beta = FALSE)

Arguments

Value

A list of class 'rboot' with elements:

References

Jarque, C. M. and Bera, A. K. (1987): "A Test for Normality of Observations and Regression Residuals", International Statistical Review, 55, pp. 163-172.

Kilian, L. and Demiroglu, U. (2000): "Residual-Based Tests for Normality in Autoregressions: Asymptotic Theory and Simulation Evidence", Journal of Business and Economic Statistics, 32, pp. 40-50.

See Also

... the normality.test by Pfaff (2008) in vars, which relies on theoretical distributions. Just as this asymptotic version, the bootstrapped version is computed by using the residuals that are standardized by a Cholesky decomposition of the residual covariance matrix. Therefore, the results of the multivariate test depend on the ordering of the variables in the VAR model.

Examples

# select minimal or full example #
is_min = TRUE
n.boot = ifelse(is_min, 50, 5000)

# prepare the data, estimate and test the VAR model #
set.seed(23211)
library("vars")
data("Canada")
exogen = cbind(qtrend=(1:nrow(Canada))^2)  # quadratic trend
R.vars = VAR(Canada, p=2, type="both", exogen=exogen)
R.norm = rboot.normality(x=R.vars, n.boot=n.boot, n.cores=1)

# density plot #
library("ggplot2")
R.data = data.frame(t(R.norm$sim[ , "MULTI", ]))
R.args = list(df=2*R.vars$K)
F.density = ggplot() +
  stat_density(data=R.data, aes(x=JB, color="bootstrap"), geom="line") +
  stat_function(fun=dchisq, args=R.args, n=500, aes(color="theoretical")) +
  labs(x="JB statistic", y="Density", color="Distribution", title=NULL) +
  theme_bw()
plot(F.density)

Bootstrap with residual moving blocks for individual SVAR models

Description

Calculates confidence bands for impulse response functions via recursive-design bootstrap.

Usage

sboot.mb(
  x,
  b.length = 1,
  n.ahead = 20,
  n.boot = 500,
  n.cores = 1,
  fix_beta = TRUE,
  deltas = cumprod((100:0)/100),
  normf = NULL
)

Arguments

Value

A list of class 'sboot2' with elements:

References

Breitung, J., Brueggemann R., and Luetkepohl, H. (2004): "Structural Vector Autoregressive Modeling and Impulse Responses", in Applied Time Series Econometrics, ed. by H. Luetkepohl and M. Kraetzig, Cambridge University Press, Cambridge.

Brueggemann R., Jentsch, C., and Trenkler, C. (2016): "Inference in VARs with Conditional Heteroskedasticity of Unknown Form", Journal of Econometrics, 191, pp. 69-85.

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

Kilian, L. (1998): "Small-Sample Confidence Intervals for Impulse Response Functions", Review of Economics and Statistics, 80, pp. 218-230.

Luetkepohl, H. (2005): New Introduction to Multiple Time Series Analysis, Springer, 2nd ed.

See Also

mb.boot, irf, and the panel counterpart sboot.pmb.

Examples

# select minimal or full example #
is_min = TRUE
n.boot = ifelse(is_min, 5, 500)

# use 'b.length=1' to conduct basic "vars" bootstraps #
set.seed(23211)
data("Canada")
R.vars = vars::VAR(Canada, p=2, type="const")
R.svar = svars::id.chol(R.vars)
R.boot = sboot.mb(R.svar, b.length=1, n.boot=n.boot, n.ahead=30, n.cores=1)
summary(R.boot, idx_par="A", level=0.9)  # VAR coefficients with 90%-confidence intervals 
plot(R.boot, lowerq = c(0.05, 0.1, 0.16), upperq = c(0.95, 0.9, 0.84))

# second step of bootstrap-after-bootstrap #
R.bab = sboot.mb(R.boot, b.length=1, n.boot=n.boot, n.ahead=30, n.cores=1)
summary(R.bab, idx_par="A", level=0.9)  # VAR coefficients with 90%-confidence intervals 
plot(R.bab, lowerq = c(0.05, 0.1, 0.16), upperq = c(0.95, 0.9, 0.84))

# conduct bootstraps for Blanchard-Quah type SVAR from "vars" #
set.seed(23211)
data("Canada")
R.vars = vars::VAR(Canada, p=2, type="const")
R.svar = vars::BQ(R.vars)
R.boot = sboot.mb(R.svar, b.length=1, n.boot=n.boot, n.ahead=30, n.cores=1)
summary(R.boot, idx_par="B", level=0.9)  # impact matrix with 90%-confidence intervals 
plot(R.boot, lowerq = c(0.05, 0.1), upperq = c(0.95, 0.9), cumulative=2:3) 
# impulse responses of the second and third variable are accumulated

# set 'args_id' to CvM defaults of "svars" bootstraps #
set.seed(23211)
data("USA")
R.vars = vars::VAR(USA, lag.max=10, ic="AIC")
R.cob  = copula::indepTestSim(R.vars$obs, R.vars$K, verbose=FALSE)
R.svar = svars::id.cvm(R.vars, dd=R.cob)

R.varx = as.varx(R.svar, dd=R.cob, itermax=300, steptol=200, iter2=50)
R.boot = sboot.mb(R.varx, b.length=15, n.boot=n.boot, n.ahead=30, n.cores=1)
plot(R.boot, lowerq = c(0.05, 0.1, 0.16), upperq = c(0.95, 0.9, 0.84))

Mean group inference for panel SVAR models

Description

Calculates confidence bands for impulse response functions via mean group inference. The function does not perform bootstraps, but coerces the panel VAR object to class 'sboot' and, therewith, gives a distributional overview on the parameter heterogeneity.

Usage

sboot.mg(x, n.ahead = 20, normf = NULL, idx_i = NULL)

Arguments

Details

MG inference presumes the individual estimates to be the empirical variation around a common parameter. In case of heterogeneous lag-orders p_i, specifically the 'summary' of VAR coefficient matrices fills \hat{A}_{ij} = 0_{K \times K} for p_i < j \le max(p_1,\ldots,p_N) in accordance with the finite order VAR(p_i).

Value

A list of class 'sboot2' with elements:

References

Pesaran, M. H., and Smith R. J. (1995): "Estimating Long-Run Relationships from Dynamic Heterogeneous Panels", Journal of Econometrics, 68, pp. 79-113.

See Also

For an actual panel bootstrap procedure see sboot.pmb.

Examples

data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names 
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
R.lags = c(2, 4, 2, 3, 2, 4, 4, 2, 2, 3, 3, 3, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 4)
names(R.lags) = names_i
idx_nord = c("DNK", "FIN", "ISL", "SWE")

R.pvec = pvarx.VEC(L.data, lags=R.lags, dim_r=2, type="Case4")
R.pid  = pid.chol(R.pvec)
R.boot = sboot.mg(R.pid, idx_i=idx_nord)
plot(R.boot, lowerq=c(0, 0.25), upperq=c(1, 0.75))
summary(as.pvarx(R.pid$L.varx[idx_nord]))

# suppress imprecise results of restricted cointegrating coefficients #
dim_r = R.pvec$args_pvarx$dim_r
R.boot$beta$sim[ , 1:dim_r, ] = diag(dim_r)  # for normalized beta
summary(R.boot, idx_par="beta", level=0.95)

Bootstrap with residual panel blocks for panel SVAR models

Description

Calculates confidence bands for impulse response functions via recursive-design bootstrap.

Usage

sboot.pmb(
  x,
  b.dim = c(1, 1),
  n.ahead = 20,
  n.boot = 500,
  n.cores = 1,
  fix_beta = TRUE,
  deltas = cumprod((100:0)/100),
  normf = NULL,
  w = NULL,
  MG_IRF = TRUE
)

Arguments

Details

In case of heterogeneous lag-orders p_i or sample sizes T_i, the initial periods are fixed in accordance with the usage of presamples. Only the (K \times T_{min} \times N) array of the T_{min} = min(T_1,\ldots,T_N) last residuals is resampled.

Value

A list of class 'sboot2' with elements:

References

Brueggemann R., Jentsch, C., and Trenkler, C. (2016): "Inference in VARs with Conditional Heteroskedasticity of Unknown Form", Journal of Econometrics, 191, pp. 69-85.

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

Kapetanios, G. (2008): "A Bootstrap Procedure for Panel Data Sets with many Cross-sectional Units", The Econometrics Journal, 11, pp.377-395.

Kilian, L. (1998): "Small-Sample Confidence Intervals for Impulse Response Functions", Review of Economics and Statistics, 80, pp. 218-230.

Gambacorta L., Hofmann B., and Peersman G. (2014): "The Effectiveness of Unconventional Monetary Policy at the Zero Lower Bound: A Cross-Country Analysis", Journal of Money, Credit and Banking, 46, pp. 615-642.

See Also

For the the individual counterpart see sboot.mb.

Examples

# select minimal or full example #
is_min = TRUE
n.boot = ifelse(is_min, 5, 500)

# prepare data panel #
data("PCAP")
names_k = c("g", "k", "l", "y")  # variable names
names_i = levels(PCAP$id_i)      # country names 
L.data  = sapply(names_i, FUN=function(i) 
  ts(PCAP[PCAP$id_i==i, names_k], start=1960, end=2019, frequency=1), 
  simplify=FALSE)
R.lags = c(2, 4, 2, 3, 2, 4, 4, 2, 2, 3, 3, 3, 2, 4, 4, 2, 2, 2, 4, 2, 2, 2, 4)
names(R.lags) = names_i

# estimate, identify, and bootstrap #
R.pvar = pvarx.VAR(L.data, lags=R.lags, type="both")
R.pid  = pid.chol(R.pvar)
R.boot = sboot.pmb(R.pid, n.boot=n.boot)
summary(R.boot, idx_par="A", level=0.95)  # VAR coefficients with 95%-confidence intervals
plot(R.boot, lowerq = c(0.05, 0.1, 0.16), upperq = c(0.95, 0.9, 0.84))

# second step of bootstrap-after-bootstrap #
R.bab = sboot.pmb(R.boot, n.boot=n.boot)
summary(R.bab, idx_par="A", level=0.95)  # VAR coefficients with 95%-confidence intervals
plot(R.bab, lowerq = c(0.05, 0.1, 0.16), upperq = c(0.95, 0.9, 0.84))

Criteria on the lag-order and break period(s)

Description

Determines the lag-order p and break period(s) \tau jointly via information criteria on the OLS-estimated VAR model for a given number of breaks. These m breaks are common to all K equations of the system and partial, as pertaining the deterministic term only.

Usage

speci.VAR(
  x,
  lag_set = 1:10,
  dim_m = FALSE,
  trim = 0.15,
  type_break = "const",
  add_dummy = FALSE,
  n.cores = 1
)

Arguments

Details

The literature on structural breaks in time series deals mostly with the determination of the number m and position \tau of breaks (e.g. Bai, Perron 1998 and 2003), but leaves the lag-order p aside. For example, under a given p, Luetkepohl et al. (2004) use a full-rank VAR in levels to determine m=1 common break period \tau_1 and subsequently perform cointegration analysis with coint.SL (which actually provides p-values for up to m=2). Note yet that the lag-order of a VECM is usually determined via information criteria of a full-rank VAR in levels alike.

speci.VAR combines Bai, Perron (2003) and Approach 3 of Yang (2002) into a global minimization of information criteria on the pair (p,\tau). Specifically, Yang (2002:378, Ch.2.2) estimates all candidate VAR models by OLS and then determines their optimal lag-order p^* and m=1 break period \tau^* jointly via the global minimum of the information criteria. Bai and Perron (2003, Ch.3) determine \tau^* = (\tau_1^*, \ldots, \tau_m^*) of multiple breaks via the minimum sum of squared residuals from a single-equation model (K=1). They use dynamic programming to reduce the number of least-squares operations. Although adapting their streamlined set of admissible combinations for \tau, speci.VAR yet resorts to (parallelized brute-force) OLS estimation of all candidate VAR models and therewith circumvents issues of correct initialization and iterative updating for the models with partial breaks.

Value

A list of class 'speci', which contains the elements:

References

Bai, J., and Perron, P. (1998): "Estimating and Testing Linear Models with Multiple Structural Changes", Econometrica, 66, pp. 47-78.

Bai, J., and Perron, P. (2003): "Computation and Analysis of Multiple Structural Change Models", Journal of Applied Econometrics, 18, pp. 1-22.

Luetkepohl, H., Saikkonen, P., and Trenkler, C. (2004): "Testing for the Cointegrating Rank of a VAR Process with Level Shift at Unknown Time", Econometrica, 72, pp. 647-662.

Yang, M. (2002): "Lag Length and Mean Break in Stationary VAR Models", Econometrics Journal, 5, pp. 374-386.

See Also

Other specification functions: speci.factors()

Examples

### extend basic example in "urca" ###
library("urca")
library("vars")
data("denmark")
sjd = denmark[, c("LRM", "LRY", "IBO", "IDE")]

# use the single lag-order p=2 to determine only the break period #
R.vars  = VAR(sjd, type="both", p=1, season=4)
R.speci = speci.VAR(R.vars, lag_set=2, dim_m=1, trim=3, add_dummy=FALSE)

library("ggfortify")
autoplot(ts(R.speci$df[3:5], start=1+R.speci$args_speci$trim), 
 main="For a single 'p', all IC just reflect the variation of det(UU').")
print(R.speci)

# perform cointegration test procedure with detrending #
R.t_D   = list(t_shift=8, n.season=4)
R.coint = coint.SL(sjd, dim_p=2, type_SL="SL_trend", t_D=R.t_D)
summary(R.coint)

# m=1: line plot #
library("ggplot2")
R.speci1 = speci.VAR(R.vars, lag_set=1:5, dim_m=1, trim=6)
R.values = c("#BDD7E7", "#6BAED6", "#3182BD", "#08519C", "#08306B")
F.line   = ggplot(R.speci1$df) +
  geom_line( aes(x=tau_1, y=HQC, color=as.factor(p), group=as.factor(p))) +
  geom_point(aes(x=tau_1, y=HQC, color=as.factor(p), group=as.factor(p))) +
  geom_point(x=R.speci1$selection["tau_1", "HQC"], 
             y=min(R.speci1$df$HQC), color="red") +
  scale_x_continuous(limits=c(1, nrow(sjd))) +
  scale_color_manual(values=R.values) +
  labs(x=expression(tau), y="HQ Criterion", color="Lag order", title=NULL) +
  theme_bw()
plot(F.line)

# m=2: discrete heat map #
R.speci2 = speci.VAR(R.vars, lag_set=2, dim_m=2, trim=3)
dim_T    = nrow(sjd)  # total sample size
F.heat   = ggplot(R.speci2$df) +
  geom_point(aes(x=tau_1, y=tau_2, color=AIC), size=3) +
  geom_abline(intercept=0, slope=-1, color="grey") +
  scale_x_continuous(limits=c(1, dim_T), expand=c(0, 0)) +
  scale_y_reverse(limits=c(dim_T, 1), expand=c(0, 0)) +
  scale_color_continuous(type="viridis") +
  labs(x=expression(tau[1]), y=expression(tau[2]), color="AIC", title=NULL) +
  theme_bw()
plot(F.heat)

Criteria on the number of common factors

Description

Determines the number of factors in an approximate factor model for a data panel, where both dimensions (T \times KN) are large, and calculates the factor time series and corresponding list of N idiosyncratic components. See Corona et al. (2017) for an overview and further details.

Usage

speci.factors(
  L.data,
  k_max = 20,
  n.iterations = 4,
  differenced = FALSE,
  centered = FALSE,
  scaled = FALSE,
  n.factors = NULL
)

Arguments

Details

If differenced is TRUE, the approximate factor model is estimated as proposed by Bai, Ng (2004). If all data transformations are selected, the estimation results are identical to the objects in $CSD for PANIC analyses in 'pcoint' objects.

Value

A list of class 'speci', which contains the elements:

References

Ahn, S., and Horenstein, A. (2013): "Eigenvalue Ratio Test for the Number of Factors", Econometrica, 81, pp. 1203-1227.

Bai, J. (2004): "Estimating Cross-Section Common Stochastic Trends in Nonstationary Panel Data", Journal of Econometrics, 122, pp. 137-183.

Bai, J., and Ng, S. (2002): "Determining the Number of Factors in Approximate Factor Models", Econometrica, 70, pp. 191-221.

Bai, J., and Ng, S. (2004): "A PANIC Attack on Unit Roots and Cointegration", Econometrica, 72, pp. 1127-117.

Corona, F., Poncela, P., and Ruiz, E. (2017): "Determining the Number of Factors after Stationary Univariate Transformations", Empirical Economics, 53, pp. 351-372.

Onatski, A. (2010): "Determining the Number of Factors from Empirical Distribution of Eigenvalues", Review of Econometrics and Statistics, 92, pp. 1004-1016.

See Also

Other specification functions: speci.VAR()

Examples

### reproduce Oersal,Arsova 2017:67, Ch.5 ###
data("MERM")
names_k = colnames(MERM)[-(1:2)] # variable names
names_i = levels(MERM$id_i)      # country names
L.data  = sapply(names_i, FUN=function(i) 
   ts(MERM[MERM$id_i==i, names_k], start=c(1995, 1), frequency=12), 
   simplify=FALSE)

R.fac1 = speci.factors(L.data, k_max=20, n.iterations=4)
R.fac0 = speci.factors(L.data, k_max=20, n.iterations=4, 
   differenced=TRUE, centered=TRUE, scaled=TRUE, n.factors=8)
   
# scree plot #
library("ggplot2")
pal = c("#999999", RColorBrewer::brewer.pal(n=8, name="Spectral"))
lvl = levels(R.fac0$eigenvals$scree)
F.scree = ggplot(R.fac0$eigenvals[1:20, ]) +
  geom_col(aes(x=n, y=share, fill=scree), color="black", width=0.75) +
  scale_fill_manual(values=pal, breaks=lvl, guide="none") +
  labs(x="Component number", y="Share on total variance", title=NULL) +
  theme_bw()
plot(F.scree)

# factor plot (comp. Oersal,Arsova 2017:71, Fig.4) #
library("ggfortify")
Ft = ts(R.fac0$Ft, start=c(1995, 1), frequency=12)
F.factors = autoplot(Ft, facets=FALSE, size=1.5) + 
  scale_color_brewer(palette="Spectral") +
  labs(x="Year", y=NULL, color="Factor", title=NULL) +
  theme_bw()
plot(F.factors)