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HDGCvar

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HDGCvar provides tools for Granger-causality testing in high-dimensional vector autoregressive models. The package implements post-double-selection procedures for stationary VARs, as well as lag-augmented post-double-selection procedures for persistent, integrated, and cointegrated VARs. In the latter case, the lag augmentation is applied only to the Granger-causing variable or block being tested, rather than to the full high-dimensional system. This partial augmentation is the key feature that keeps the procedure feasible in high dimensions. The package also includes routines for testing multiple Granger-causality relations and for constructing Granger-causality networks.

Additional functionality is available for realized-volatility applications based on heterogeneous VAR specifications, including volatility spillover networks and networks of realized volatilities conditional on realized correlations.

Installation

You can install HDGCvar from GitHub with:

# install.packages("devtools")
devtools::install_github("Marga8/HDGCvar")

Related papers

The main functions in HDGCvar are based on two papers:

  1. Hecq, A., Margaritella, L. and Smeekes, S. (2023). “Granger Causality Testing in High-Dimensional VARs: A Post-Double-Selection Procedure.” Journal of Financial Econometrics, 21(3), 915–958.

  2. Hecq, A., Margaritella, L. and Smeekes, S. “Inference in Persistent High-Dimensional Vector Autoregressions.” Manuscript.

The stationary high-dimensional VAR and HVAR routines are based mainly on the first paper. The lag-augmented routines for persistent and potentially nonstationary systems are based on the second paper.

Background

Granger causality concerns conditional predictive content. A variable X is said to Granger-cause a variable Y, relative to an information set, if past values of X improve the prediction of Y after conditioning on the remaining available variables.

In high-dimensional systems, conditioning on a rich information set is attractive because it reduces the risk of detecting indirect or spurious predictive links. At the same time, standard VAR estimation becomes infeasible or unstable when the number of variables and lags is large relative to the sample size. HDGCvar addresses this problem by combining penalized regression, variable selection, and post-selection inference.

For persistent time series, additional complications arise because the system may contain unit roots, near unit roots, or cointegrating relations. The lag-augmented functions in HDGCvar allow users to perform inference in levels without first deciding, through pre-testing, which variables should be differenced. Importantly, the augmentation is restricted to the Granger-causing variable or block of variables entering the null hypothesis, not to all variables in the VAR.

Which function should I use?

Setting Main function(s)
Single Granger-causality test, stationary VAR HDGC_VAR_I0
Multiple Granger-causality tests, stationary VAR HDGC_VAR_multiple_I0, HDGC_VAR_multiple_pairs_I0
Full Granger-causality network, stationary VAR HDGC_VAR_all_I0
Single Granger-causality test, persistent VAR HDGC_VAR
Multiple Granger-causality tests, persistent VAR HDGC_VAR_multiple, HDGC_VAR_multiple_pairs
Full Granger-causality network, persistent VAR HDGC_VAR_all
Single Granger-causality test, realized-volatility HVAR HDGC_HVAR
Full realized-volatility spillover network HDGC_HVAR_all
Realized-volatility network conditional on realized correlations HDGC_HVAR_RV_RCoV_all

Stationary high-dimensional VARs

If the time series can be treated as stationary, use the stationary VAR functions:

  • HDGC_VAR_I0
  • HDGC_VAR_multiple_I0
  • HDGC_VAR_multiple_pairs_I0
  • HDGC_VAR_all_I0

The main inputs are:

  • GCpair or GCpairs: the Granger-causality relation(s) to test. Each relation specifies the Granger-caused variable, GCto, and the Granger-causing variable, GCfrom.
  • data: a data matrix or data frame with one time series per column.
  • p: the VAR lag length. If the lag length is not imposed directly, lags_upbound() can be used to estimate an empirical upper bound.
  • bound: a lower bound on the lasso penalty, expressed through the maximum number of variables that can be selected in each first-stage regression. The default is 0.5 * nrow(data).
  • parallel: whether to use parallel computing.
  • n_cores: the number of cores used when parallel = TRUE.

The function HDGC_VAR_all_I0 can be used to test all bivariate Granger-causality relations in a dataset and to construct a Granger-causality network.

Persistent and nonstationary high-dimensional VARs

If the system may contain persistent, integrated, or cointegrated series, use the lag-augmented functions:

  • HDGC_VAR
  • HDGC_VAR_multiple
  • HDGC_VAR_multiple_pairs
  • HDGC_VAR_all

These functions implement a lag-augmented post-double-selection procedure. The augmentation is restricted to the Granger-causing variable or block being tested. Hence, the procedure avoids the parameter proliferation that would arise from augmenting every variable in a high-dimensional VAR.

The additional input is:

  • d: the number of augmenting lags added to the Granger-causing variable or block. It should be chosen to reflect the maximum suspected order of integration. Using a value of d larger than strictly needed is conservative, although it may reduce power.

In many macroeconomic and financial applications, d = 2 is a conservative default. When p <= d, the package returns a warning because, in some designs, the first-stage post-double-selection regressions may not contain enough own-lag information to rule out spurious-regression issues. This warning should be interpreted as a diagnostic rather than as an automatic rejection of the specification. In particular, p = 2 and d = 2 is a feasible lag-augmented specification when the series are at most I(1).

The lag length p can be imposed by the user or selected using:

lags_upbound(data, p_max = 10)

This function estimates an empirical upper bound for the lag length using information criteria applied to univariate autoregressions.

Realized-volatility HVARs

HDGCvar also includes functions for Granger-causality testing among realized volatilities using heterogeneous VAR specifications. These models use daily, weekly, and monthly volatility components, following the heterogeneous autoregressive structure of Corsi (2009).

For realized-volatility applications, use:

  • HDGC_HVAR
  • HDGC_HVAR_multiple
  • HDGC_HVAR_all

The main inputs are the same as for the stationary VAR functions, except that the HVAR lag structure is fixed by construction. The option log = TRUE can be used to log-transform realized-volatility series, which is typically recommended.

The HVAR functions return different versions of the LM test, including the asymptotic chi-square version, the finite-sample corrected F-test, and a heteroskedasticity-robust asymptotic chi-square version.

Realized volatilities conditional on realized correlations

To condition realized-volatility spillover networks on realized correlations, use:

  • HDGC_HVAR_RV_RCoV_all
  • HDGC_HVAR_RVCOV
  • HDGC_HVAR_multiple_RVCOV

The main inputs are:

  • realized_variances: a matrix of realized variances.
  • realized_covariances: a matrix of realized covariances.
  • fisher_transf: whether to Fisher-transform realized correlations. The default is TRUE.
  • log: whether to log-transform the realized-volatility series. The default is TRUE.
  • bound: the selection bound used in the first-stage regressions.
  • parallel: whether to use parallel computing.
  • n_cores: the number of cores used when parallel = TRUE.

Plotting Granger-causality networks

Networks created by HDGC_VAR_all_I0, HDGC_VAR_all, HDGC_HVAR_all, or HDGC_HVAR_RV_RCoV_all can be plotted with:

  • Plot_GC_all

The main inputs are:

  • Comb: the output of one of the network-estimation functions.
  • Stat_type: the statistic used to determine edges. The default is "FS_cor", the finite-sample corrected F-test. Alternatives include "Asymp" and "Asymp_Robust".
  • alpha: the significance level. The default is 0.01.
  • multip_corr: optional multiple-testing correction using stats::p.adjust() or the APF/FDR procedure.
  • cluster: optional network clustering using igraph::cluster_edge_betweenness().
  • ...: additional plotting arguments passed to igraph.

Data

The package contains three example datasets:

  • sample_dataset_I0: a simulated stationary high-dimensional VAR dataset.
  • sample_dataset_I1: a simulated integrated dataset obtained from the stationary system by inverse differencing.
  • sample_RV: simulated realized-volatility series generated from heterogeneous autoregressive dynamics.

These datasets are intended for illustrating the main functions and plotting routines.

Example

The following example simulates a persistent high-dimensional VAR system, selects an empirical lag upper bound, tests one Granger-causality relation, estimates the full network, and plots it.

library(HDGCvar)
library(igraph)
#> 
#> Attaching package: 'igraph'
#> The following objects are masked from 'package:stats':
#> 
#>     decompose, spectrum
#> The following object is masked from 'package:base':
#> 
#>     union

# Simulate a stationary VAR(1)
SimulVAR <- function(T_, g) {
  coef1 <- matrix(NA, nrow = g, ncol = g)

  for (i in seq_len(g)) {
    for (j in seq_len(g)) {
      coef1[i, j] <- ((-1)^(abs(i - j))) * (0.4^(abs(i - j) + 1))
    }
  }

  presample <- 1
  T_new <- T_ + presample

  eps <- matrix(rnorm(ncol(coef1) * T_new, 0, 1), nrow = ncol(coef1))
  X <- matrix(nrow = ncol(coef1), ncol = T_new)
  X[, 1] <- eps[, 1]

  for (t in 2:T_new) {
    X[, t] <- coef1 %*% X[, t - 1] + eps[, t]
  }

  t(X[, (1 + presample):T_new])
}

# Simulate persistent data by inverse differencing
set.seed(123)

g <- 20
dataset_I0 <- as.matrix(SimulVAR(100, g))
dataset <- as.matrix(diffinv(dataset_I0))
colnames(dataset) <- paste("Var", seq_len(g))

# Visual inspection
ts.plot(dataset)

# Select an empirical upper bound for the lag length
selected_lag <- lags_upbound(dataset, p_max = 10)
selected_lag
#> [1] 2

# Test whether Var 5 Granger-causes Var 1
interest_variables <- list("GCto" = "Var 1", "GCfrom" = "Var 5")

HDGC_VAR(
  GCpair = interest_variables,
  data = dataset,
  p = selected_lag,
  d = 2,
  bound = 0.5 * nrow(dataset),
  parallel = TRUE,
  n_cores = NULL
)
#> Warning in HDGC_VAR(GCpair = interest_variables, data = dataset, p = selected_lag, : To avoid spurious regression problems in the post-double-selection steps,
#>             unless you are certain that your series are maximum I(1), you might want to consider increasing the lag length p to be larger than d
#> $tests
#>             Asymp    FS_cor
#> LM_stat 2.2245113 0.8919124
#> p_value 0.3288164 0.4141173
#> 
#> $selections
#>  Var 1 l1  Var 2 l1  Var 3 l1  Var 4 l1  Var 6 l1  Var 7 l1  Var 8 l1  Var 9 l1 
#>      TRUE      TRUE      TRUE      TRUE      TRUE     FALSE      TRUE      TRUE 
#> Var 10 l1 Var 11 l1 Var 12 l1 Var 13 l1 Var 14 l1 Var 15 l1 Var 16 l1 Var 17 l1 
#>     FALSE     FALSE     FALSE      TRUE     FALSE     FALSE      TRUE     FALSE 
#> Var 18 l1 Var 19 l1 Var 20 l1  Var 1 l2  Var 2 l2  Var 3 l2  Var 4 l2  Var 6 l2 
#>     FALSE     FALSE      TRUE      TRUE      TRUE     FALSE     FALSE      TRUE 
#>  Var 7 l2  Var 8 l2  Var 9 l2 Var 10 l2 Var 11 l2 Var 12 l2 Var 13 l2 Var 14 l2 
#>     FALSE      TRUE     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE 
#> Var 15 l2 Var 16 l2 Var 17 l2 Var 18 l2 Var 19 l2 Var 20 l2 
#>      TRUE      TRUE      TRUE     FALSE     FALSE     FALSE

# Test multiple Granger-causality relations
mult_interest_variables <- list(
  list("GCto" = "Var 7", "GCfrom" = "Var 19"),
  list("GCto" = "Var 4", "GCfrom" = "Var 16")
)

HDGC_VAR_multiple(
  data = dataset,
  GCpairs = mult_interest_variables,
  p = selected_lag,
  d = 2,
  bound = 0.5 * nrow(dataset),
  parallel = TRUE,
  n_cores = NULL
)
#> $tests
#> , , GCtests = Var 19 -> Var 7
#> 
#>          type
#> stat          Asymp    FS_cor
#>   LM_stat 0.8564206 0.3340397
#>   p_value 0.6516743 0.7170851
#> 
#> , , GCtests = Var 16 -> Var 4
#> 
#>          type
#> stat          Asymp    FS_cor
#>   LM_stat 0.2839762 0.1174474
#>   p_value 0.8676316 0.8893403
#> 
#> 
#> $selections
#> $selections$`Var 19 -> Var 7`
#>  Var 1 l1  Var 2 l1  Var 3 l1  Var 4 l1  Var 5 l1  Var 6 l1  Var 7 l1  Var 8 l1 
#>     FALSE      TRUE     FALSE      TRUE     FALSE     FALSE      TRUE      TRUE 
#>  Var 9 l1 Var 10 l1 Var 11 l1 Var 12 l1 Var 13 l1 Var 14 l1 Var 15 l1 Var 16 l1 
#>     FALSE     FALSE      TRUE      TRUE     FALSE     FALSE      TRUE      TRUE 
#> Var 17 l1 Var 18 l1 Var 20 l1  Var 1 l2  Var 2 l2  Var 3 l2  Var 4 l2  Var 5 l2 
#>     FALSE      TRUE      TRUE     FALSE     FALSE     FALSE      TRUE      TRUE 
#>  Var 6 l2  Var 7 l2  Var 8 l2  Var 9 l2 Var 10 l2 Var 11 l2 Var 12 l2 Var 13 l2 
#>      TRUE      TRUE     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE 
#> Var 14 l2 Var 15 l2 Var 16 l2 Var 17 l2 Var 18 l2 Var 20 l2 
#>     FALSE      TRUE      TRUE     FALSE      TRUE      TRUE 
#> 
#> $selections$`Var 16 -> Var 4`
#>  Var 1 l1  Var 2 l1  Var 3 l1  Var 4 l1  Var 5 l1  Var 6 l1  Var 7 l1  Var 8 l1 
#>     FALSE      TRUE     FALSE      TRUE      TRUE      TRUE     FALSE      TRUE 
#>  Var 9 l1 Var 10 l1 Var 11 l1 Var 12 l1 Var 13 l1 Var 14 l1 Var 15 l1 Var 17 l1 
#>     FALSE      TRUE     FALSE     FALSE     FALSE      TRUE      TRUE     FALSE 
#> Var 18 l1 Var 19 l1 Var 20 l1  Var 1 l2  Var 2 l2  Var 3 l2  Var 4 l2  Var 5 l2 
#>     FALSE     FALSE     FALSE     FALSE     FALSE     FALSE      TRUE     FALSE 
#>  Var 6 l2  Var 7 l2  Var 8 l2  Var 9 l2 Var 10 l2 Var 11 l2 Var 12 l2 Var 13 l2 
#>     FALSE     FALSE     FALSE     FALSE     FALSE      TRUE      TRUE      TRUE 
#> Var 14 l2 Var 15 l2 Var 17 l2 Var 18 l2 Var 19 l2 Var 20 l2 
#>     FALSE      TRUE     FALSE     FALSE     FALSE     FALSE

# Estimate the full Granger-causality network
network <- HDGC_VAR_all(
  data = dataset,
  p = selected_lag,
  d = 2,
  bound = 0.5 * nrow(dataset),
  parallel = TRUE,
  n_cores = NULL
)

# Plot the estimated network
Plot_GC_all(
  network,
  Stat_type = "FS_cor",
  alpha = 0.01,
  multip_corr = list(FALSE),
  directed = TRUE,
  layout = layout.circle,
  main = "Granger-causality network",
  edge.arrow.size = 0.2,
  vertex.size = 5,
  vertex.color = "lightblue",
  vertex.frame.color = "blue",
  vertex.label.color = "black",
  vertex.label.cex = 0.6,
  vertex.label.dist = 1,
  edge.curved = 0,
  cluster = list(TRUE, 5, "black", 0.8, 1, 0)
)
#> Warning: Same attribute for columns and rows, row names are ignored

Notes on choosing p and d

The lag length p controls the dynamic order of the VAR, while d controls the number of augmenting lags used to guard against unknown persistence. These augmenting lags are added only to the Granger-causing variable or block being tested. The value of d should reflect the maximum suspected order of integration. For example, if the variables are at most I(1), then d = 2 is a conservative over-augmentation that can also provide a safeguard against near-I(2)-type persistence, although possibly at the cost of some power.

In the post-double-selection step, the first-stage regressions should not be spurious. If p <= d, the package returns a warning because this can be problematic in some nonstationary designs. The warning does not automatically imply that the result is invalid; it signals that the user should check whether the chosen combination of p and d is appropriate for the application. The specification p = 2, d = 2, for example, is appropriate when the simulated or empirical series are at most I(1).

References

  • Belloni, A., Chernozhukov, V. and Hansen, C. (2014). “Inference on treatment effects after selection among high-dimensional controls.” The Review of Economic Studies, 81(2), 608–650.

  • Corsi, F. (2009). “A simple approximate long-memory model of realized volatility.” Journal of Financial Econometrics, 7(2), 174–196.

  • Granger, C. W. J. (1969). “Investigating causal relations by econometric models and cross-spectral methods.” Econometrica, 37(3), 424–438.

  • Hecq, A., Margaritella, L. and Smeekes, S. (2023). “Granger Causality Testing in High-Dimensional VARs: A Post-Double-Selection Procedure.” Journal of Financial Econometrics, 21(3), 915–958.

  • Hecq, A., Margaritella, L. and Smeekes, S. “Inference in Persistent High-Dimensional Vector Autoregressions.” Manuscript.

  • Newman, M. E. J. and Girvan, M. (2004). “Finding and evaluating community structure in networks.” Physical Review E, 69(2), 026113.

  • Quatto, P., Margaritella, L., et al. (2020). “Brain networks construction using Bayes FDR and average power function.” Statistical Methods in Medical Research, 29(3), 866–878.

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Granger causality testing in High Dimensional Vector Autoregressive Models

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