Type:	Package
Title:	Multiple Granger Causality Tests for Time Series and Panel Data
Version:	1.1.1
Description:	Comprehensive suite of Granger causality tests for time series and panel data. For time series: Toda-Yamamoto (1995) <doi:10.1016/0304-4076(94)01616-8>, Fourier-based tests with single frequency (Enders and Jones, 2016) <doi:10.1515/snde-2014-0101> and cumulative frequencies (Nazlioglu et al., 2019) <doi:10.1080/1540496X.2018.1434072>, quantile causality tests (Cai et al., 2023) <doi:10.1016/j.frl.2023.104327>, and Bootstrap Fourier Granger Causality in Quantiles (Cheng et al., 2021) <doi:10.1007/s12076-020-00263-0>. For panel data: Panel Fourier Toda-Yamamoto (Yilanci and Gorus, 2020) <doi:10.1007/s11356-020-10092-9> and Panel Quantile Causality tests (Wang and Nguyen, 2022) <doi:10.1080/1331677X.2021.1952089>, as well as Group-Mean and Pooled Fully Modified OLS estimators for panel cointegrating polynomial regressions (Wagner and Reichold, 2023) <doi:10.1080/07474938.2023.2178141>. All tests include bootstrap inference for robust p-values.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/caustests
BugReports:	https://github.com/muhammedalkhalaf/caustests/issues
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 3.5.0)
Imports:	stats, quantreg
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-04-04 14:51:37 UTC; SYSTEM
Author:	Muhammad Alkhalaf [aut, cre, cph]
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-04-05 06:50:06 UTC

caustests: Multiple Granger Causality Tests for Time Series and Panel Data

Description

Comprehensive suite of Granger causality tests for time series and panel data including standard Toda-Yamamoto, Fourier-based tests, quantile causality tests, panel Fourier Toda-Yamamoto, panel quantile causality, and Group-Mean/Pooled FM-OLS for panel cointegrating polynomial regressions.

Time Series Functions

• caustests: Perform Granger causality tests (time series)

Panel Data Functions

• xtpcaus: Panel Granger causality tests (PFTY and PQC)

• xtpcmg: Panel cointegrating polynomial regressions (FM-OLS)

Available Tests (Time Series)

1. Toda-Yamamoto (1995) - Robust to integration order

2. Single Fourier Granger (Enders & Jones, 2016)

3. Single Fourier Toda-Yamamoto (Nazlioglu et al., 2016)

4. Cumulative Fourier Granger (Enders & Jones, 2019)

5. Cumulative Fourier Toda-Yamamoto (Nazlioglu et al., 2019)

6. Quantile Toda-Yamamoto (Cai et al., 2023)

7. Bootstrap Fourier Granger in Quantiles (Cheng et al., 2021)

Available Tests (Panel Data)

1. Panel Fourier Toda-Yamamoto (Yilanci and Gorus, 2020)

2. Panel Quantile Causality (Wang and Nguyen, 2022)

3. Group-Mean FM-OLS (Wagner and Reichold, 2023)

4. Pooled FM-OLS (de Jong and Wagner, 2022)

Data

• caustests_data: Example time series dataset

• grunfeld_panel: Example panel dataset for xtpcaus

• grunfeld_cmg: Example panel dataset for xtpcmg

Author(s)

Maintainer: Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

References

Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1-2), 225-250. doi:10.1016/0304-4076(94)01616-8

Enders, W., & Jones, P. (2016). Grain prices, oil prices, and multiple smooth breaks in a VAR. Studies in Nonlinear Dynamics & Econometrics, 20(4), 399-419. doi:10.1515/snde-2014-0101

Nazlioglu, S., Gormus, N. A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168-175. doi:10.1016/j.eneco.2016.09.009

Nazlioglu, S., Soytas, U., & Gormus, N. A. (2019). Oil prices and monetary policy in emerging markets: Structural shifts in causal linkages. Emerging Markets Finance and Trade, 55(1), 105-117. doi:10.1080/1540496X.2018.1434072

Cai, Y., Chang, T., Xiang, Y., & Chang, H. L. (2023). Testing Granger causality in quantiles between the stock and the foreign exchange markets of Japan. Finance Research Letters, 58, 104327. doi:10.1016/j.frl.2023.104327

Cheng, S. C., Hsueh, H. P., Ranjbar, O., Wang, M. C., & Chang, T. (2021). Bootstrap Fourier Granger causality test in quantiles and the asymmetric causal relationship between CO2 emissions and economic growth. Letters in Spatial and Resource Sciences, 14, 31-49. doi:10.1007/s12076-020-00263-0

Yilanci, V. and Gorus, M.S. (2020). Does economic globalization have predictive power for ecological footprint in MENA counties? A panel causality test with a Fourier function. Environmental Science and Pollution Research, 27, 40552-40562. doi:10.1007/s11356-020-10092-9

Wang, K.M. and Nguyen, T.B. (2022). A quantile panel-type analysis of income inequality and healthcare expenditure. Economic Research, 35(1), 873-893. doi:10.1080/1331677X.2021.1952089

Wagner, M. and Reichold, K. (2023). Panel cointegrating polynomial regressions: group-mean fully modified OLS estimation and inference. Econometric Reviews, 42(4), 358-392. doi:10.1080/07474938.2023.2178141

de Jong, R.M. and Wagner, M. (2022). Panel cointegrating polynomial regressions. Annals of Applied Statistics, 16(1), 416-442. doi:10.1214/21-AOAS1536

See Also

Useful links:

• https://github.com/muhammedalkhalaf/caustests

• Report bugs at https://github.com/muhammedalkhalaf/caustests/issues

Multiple Granger Causality Tests

Description

Performs various Granger causality tests including Toda-Yamamoto, Fourier-based tests (single and cumulative frequency), and quantile causality tests with bootstrap inference.

Usage

caustests(
  data,
  test,
  pmax = 8,
  ic = 1,
  nboot = 1000,
  kmax = 3,
  dmax = NULL,
  quantiles = seq(0.1, 0.9, 0.1),
  verbose = TRUE
)

Arguments

Details

The package implements seven Granger causality tests:

Test 1: Toda-Yamamoto (1995) Standard Granger causality in levels using VAR with extra lags equal to the maximum integration order (dmax). This approach is robust to unknown integration and cointegration properties.

Tests 2-3: Single Fourier Frequency Incorporate a single Fourier frequency to capture smooth structural breaks. Test 2 uses first differences, Test 3 uses levels (Toda-Yamamoto style).

Tests 4-5: Cumulative Fourier Frequency Use cumulative Fourier frequencies (1 to k) for more flexible break patterns. Test 4 uses first differences, Test 5 uses levels.

Test 6: Quantile Toda-Yamamoto Extends Toda-Yamamoto to quantile regression, allowing causality analysis across different quantiles of the conditional distribution.

Test 7: Bootstrap Fourier Granger Causality in Quantiles (BFGC-Q) Combines Fourier flexibility with quantile regression for robust inference under structural breaks and across quantiles.

Value

An object of class "caustests" containing:

References

Toda, H. Y., & Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1-2), 225-250. doi:10.1016/0304-4076(94)01616-8

Enders, W., & Jones, P. (2016). Grain prices, oil prices, and multiple smooth breaks in a VAR. Studies in Nonlinear Dynamics & Econometrics, 20(4), 399-419. doi:10.1515/snde-2014-0101

Nazlioglu, S., Gormus, N. A., & Soytas, U. (2016). Oil prices and real estate investment trusts (REITs): Gradual-shift causality and volatility transmission analysis. Energy Economics, 60, 168-175. doi:10.1016/j.eneco.2016.09.009

Nazlioglu, S., Soytas, U., & Gormus, N. A. (2019). Oil prices and monetary policy in emerging markets: Structural shifts in causal linkages. Emerging Markets Finance and Trade, 55(1), 105-117. doi:10.1080/1540496X.2018.1434072

Cai, Y., Chang, T., Xiang, Y., & Chang, H. L. (2023). Testing Granger causality in quantiles between the stock and the foreign exchange markets of Japan. Finance Research Letters, 58, 104327. doi:10.1016/j.frl.2023.104327

Cheng, S. C., Hsueh, H. P., Ranjbar, O., Wang, M. C., & Chang, T. (2021). Bootstrap Fourier Granger causality test in quantiles and the asymmetric causal relationship between CO2 emissions and economic growth. Letters in Spatial and Resource Sciences, 14, 31-49. doi:10.1007/s12076-020-00263-0

Examples

# Load example data
data(caustests_data)


# Test 1: Toda-Yamamoto test
result1 <- caustests(caustests_data, test = 1, nboot = 199)
print(result1)
summary(result1)

# Test 3: Single Fourier Toda-Yamamoto
result3 <- caustests(caustests_data, test = 3, kmax = 2, nboot = 199)
print(result3)

# Test 6: Quantile causality (fewer quantiles for speed)
result6 <- caustests(caustests_data, test = 6, 
                     quantiles = c(0.25, 0.50, 0.75), nboot = 199)
print(result6)

Example Time Series Dataset for Causality Tests

Description

A simulated dataset containing three time series variables for demonstrating Granger causality tests. The data includes one dependent variable (Y) and two potential causal variables (X1, X2) with known causal relationships.

Usage

caustests_data

Format

A data frame with 200 observations and 3 variables:

Y

Dependent variable, generated as AR(2) plus causal effects from X1

X1

First explanatory variable, AR(1) process

X2

Second explanatory variable, independent AR(1) process

Details

The data generating process is:

• X1 and X2 are independent AR(1) processes

• Y depends on its own lags plus lagged values of X1 (but not X2)

• This creates a true causal relationship from X1 to Y

• There is no true causality from X2 to Y or from Y to X1/X2

This allows users to verify that the causality tests correctly identify the causal direction X1 => Y while finding no significant causality in other directions (with appropriate sample sizes and test settings).

Source

Simulated data for package demonstration

Examples

data(caustests_data)
head(caustests_data)
summary(caustests_data)

# Check correlations
cor(caustests_data)

Example Panel Data for xtpcmg

Description

Returns the Grunfeld (1958) balanced panel dataset for examples.

Usage

grunfeld_cmg()

Value

A data frame with columns firm, year, invest, and mvalue.

Examples

dat <- grunfeld_cmg()
head(dat)

Example Panel Data for xtpcaus

Description

Returns a small balanced panel dataset (subset of Grunfeld 1958) for use in examples and testing.

Usage

grunfeld_panel()

Value

A data frame with columns firm, year, invest, and mvalue.

Examples

dat <- grunfeld_panel()
head(dat)

Plot Quantile Causality Results

Description

Creates diagnostic plots for quantile causality tests (tests 6-7).

Usage

## S3 method for class 'caustests'
plot(x, which = 1, type = "both", ...)

Arguments

Value

Invisibly returns the plotted data.

Examples

data(caustests_data)
result <- caustests(caustests_data, test = 6,
                    quantiles = c(0.25, 0.50, 0.75), nboot = 199)
plot(result)

Print Method for xtpcaus Objects

Description

Print Method for xtpcaus Objects

Usage

## S3 method for class 'xtpcaus'
print(x, ...)

Arguments

Value

Invisibly returns x.

Print Method for xtpcmg Objects

Description

Print Method for xtpcmg Objects

Usage

## S3 method for class 'xtpcmg'
print(x, ...)

Arguments

Value

Invisibly returns x.

Summary Method for xtpcaus Objects

Description

Summary Method for xtpcaus Objects

Usage

## S3 method for class 'xtpcaus'
summary(object, ...)

Arguments

Value

Invisibly returns object.

Summary Method for xtpcmg Objects

Description

Summary Method for xtpcmg Objects

Usage

## S3 method for class 'xtpcmg'
summary(object, ...)

Arguments

Value

Invisibly returns object.

Panel Granger Causality Tests

Description

Tests whether x Granger-causes y in a balanced panel using either the Panel Fourier Toda-Yamamoto (PFTY) test or the Panel Quantile Causality (PQC) test.

Usage

xtpcaus(
  data,
  y,
  x,
  panel_id,
  time_id,
  test = c("pfty", "pqc"),
  pmax = 4L,
  dmax = 1L,
  nboot = 499L,
  kmax = 3L,
  ic = c("aic", "bic"),
  quantiles = c(0.1, 0.25, 0.5, 0.75, 0.9),
  seed = -1L
)

Arguments

Value

An object of class "xtpcaus" containing:

test

Character. "pfty" or "pqc".

N

Integer. Number of panel units.

TT

Integer. Number of time periods.

nboot

Integer. Number of bootstrap replications.

y

Character. Name of the y variable.

x

Character. Name of the x variable.

For PFTY:

fisher

Numeric. Fisher panel statistic.

fisher_df

Integer. Degrees of freedom (2*N).

fisher_pv

Numeric. Fisher p-value.

wbar

Numeric. Average individual Wald statistic.

zbar

Numeric. Dumitrescu-Hurlin Z-bar statistic.

zbar_pv

Numeric. Z-bar p-value.

ind_wald

Numeric vector. Individual Wald statistics (length N).

ind_freq

Integer vector. Optimal Fourier frequencies (length N).

ind_pval_b

Numeric vector. Bootstrap p-values (length N).

ind_lags

Integer vector. Selected lag orders (length N).

For PQC:

quantiles

Numeric vector. Quantiles tested.

wald_xy

Numeric vector. Wald statistics per quantile (x => y).

pval_xy

Numeric vector. Bootstrap p-values per quantile (x => y).

wald_yx

Numeric vector. Wald statistics per quantile (y => x).

pval_yx

Numeric vector. Bootstrap p-values per quantile (y => x).

supwald_xy

Numeric. Sup-Wald statistic for x => y.

supwald_yx

Numeric. Sup-Wald statistic for y => x.

p_opt

Integer. Selected optimal lag.

References

Chuang, C.C., Kuan, C.M. and Lin, H.Y. (2009). Causality in quantiles and dynamic stock return-volume relations. Journal of Banking and Finance, 33(7), 1351–1360. doi:10.1016/j.jbankfin.2009.02.013

Emirmahmutoglu, F. and Kose, N. (2011). Testing for Granger causality in heterogeneous mixed panels. Economic Modelling, 28(3), 870–876. doi:10.1016/j.econmod.2010.10.018

Toda, H.Y. and Yamamoto, T. (1995). Statistical inference in vector autoregressions with possibly integrated processes. Journal of Econometrics, 66(1–2), 225–250. doi:10.1016/0304-4076(94)01616-8

Wang, K.M. and Nguyen, T.B. (2022). A quantile panel-type analysis of income inequality and healthcare expenditure. Economic Research, 35(1), 873–893. doi:10.1080/1331677X.2021.1952089

Yilanci, V. and Gorus, M.S. (2020). Does economic globalization have predictive power for ecological footprint. Environmental Science and Pollution Research, 27, 40552–40562. doi:10.1007/s11356-020-10092-9

Examples

dat <- grunfeld_panel()
# PFTY test (quick with few bootstrap reps)

res <- xtpcaus(dat, y = "invest", x = "mvalue",
               panel_id = "firm", time_id = "year",
               test = "pfty", pmax = 2L, dmax = 1L,
               nboot = 99L, kmax = 2L, seed = 42L)
print(res)


# PQC test

res2 <- xtpcaus(dat, y = "invest", x = "mvalue",
                panel_id = "firm", time_id = "year",
                test = "pqc", pmax = 2L, nboot = 99L,
                quantiles = c(0.25, 0.50, 0.75), seed = 42L)
print(res2)

Panel Cointegrating Polynomial Regressions via FM-OLS

Description

Estimates a polynomial cointegrating regression in a panel using either Group-Mean FM-OLS (Wagner & Reichold 2023) or Pooled FM-OLS (de Jong & Wagner 2022). Models the long-run relationship:

Usage

xtpcmg(
  data,
  y,
  x,
  panel_id,
  time_id,
  model = c("mg", "pmg"),
  q = 2L,
  controls = NULL,
  trend = 1L,
  kernel = "ba",
  bw = "And91",
  effects = "oneway",
  corr_rob = FALSE
)

Arguments

Details

y_{it} = \alpha_i + \beta_1 x_{it} + \beta_2 x_{it}^2 [+ \beta_3 x_{it}^3] [+ \gamma z_{it}] + u_{it}

where x_{it} and z_{it} are I(1) processes.

Value

An object of class "xtpcmg" with elements: coefficients (named numeric vector), vcov (variance-covariance matrix), se (standard errors), tstat (t-statistics), pvalue (two-sided p-values), model (estimator type: mg or pmg), q (polynomial degree), N (number of panel units), TT (number of time periods), y (dependent variable name), x (polynomial variable name), tp (turning point estimate, quadratic models only), tp_se (delta-method SE for turning point), tp_lo and tp_hi (95% CI bounds), ind_coef (individual FM-OLS estimates, MG model), swamy_s and swamy_p (Swamy slope homogeneity test).

References

Andrews, D.W.K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3), 817–858. doi:10.2307/2938229

de Jong, R.M. and Wagner, M. (2022). Panel cointegrating polynomial regressions. Annals of Applied Statistics, 16(1), 416–442. doi:10.1214/21-AOAS1536

Wagner, M. and Reichold, K. (2023). Panel cointegrating polynomial regressions. Econometric Reviews, 42(9–10), 782–827. doi:10.1080/07474938.2023.2178141

Examples

dat <- grunfeld_cmg()

# Group-Mean FM-OLS (quadratic EKC-type model)
res <- xtpcmg(dat, y = "invest", x = "mvalue",
              panel_id = "firm", time_id = "year",
              model = "mg", q = 2L)
print(res)
summary(res)

# Pooled FM-OLS
res2 <- xtpcmg(dat, y = "invest", x = "mvalue",
               panel_id = "firm", time_id = "year",
               model = "pmg", q = 2L)
print(res2)