Package 'caustests'

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

data	A data frame or matrix with time series variables (columns).
test	Integer 1-7 specifying the test type: 1: Toda-Yamamoto (1995) 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 Causality in Quantiles (Cheng et al., 2021)
pmax	Maximum lag order for model selection (default: 8).
ic	Information criterion: 1 for AIC, 2 for SBC/BIC (default: 1).
nboot	Number of bootstrap replications (default: 1000).
kmax	Maximum Fourier frequency (default: 3, used for tests 2-5, 7).
dmax	Extra lags for Toda-Yamamoto augmentation. If NULL, automatically set to 0 for tests 2, 4 (differences) and 1 for tests 1, 3, 5, 6, 7 (levels).
quantiles	Numeric vector of quantiles for tests 6-7 (default: seq(0.1, 0.9, 0.1)).
verbose	Logical; print progress messages (default: TRUE).

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:

results	Data frame with test results for each direction
test	Test number used
test_name	Name of the test
pmax	Maximum lag considered
ic	Information criterion used
nboot	Number of bootstrap replications
kmax	Maximum Fourier frequency
dmax	Augmentation lags
quantiles	Quantiles used (for tests 6-7)
quantile_results	Detailed quantile results (for tests 6-7)

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

x	An object of class "caustests" from test 6 or 7.
which	Which direction to plot (default: 1, first direction).
type	Plot type: "wald" for Wald statistics, "pval" for p-values, or "both" (default).
...	Additional arguments passed to plot.

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

x	An object of class "xtpcaus".
...	Additional arguments (ignored).

Value

Invisibly returns x.

---

Print Method for xtpcmg Objects

Description

Print Method for xtpcmg Objects

Usage

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

Arguments

x	An object of class "xtpcmg".
...	Additional arguments (ignored).

Value

Invisibly returns x.

---

Summary Method for xtpcaus Objects

Description

Summary Method for xtpcaus Objects

Usage

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

Arguments

object	An object of class "xtpcaus".
...	Additional arguments (ignored).

Value

Invisibly returns object.

---

Summary Method for xtpcmg Objects

Description

Summary Method for xtpcmg Objects

Usage

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

Arguments

object	An object of class "xtpcmg".
...	Additional arguments (ignored).

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

data	A data frame in long format.
y	Character. Name of the dependent (caused) variable.
x	Character. Name of the independent (causing) variable.
panel_id	Character. Name of the panel identifier variable.
time_id	Character. Name of the time variable.
test	Character. Test type: "pfty" for Panel Fourier Toda-Yamamoto or "pqc" for Panel Quantile Causality.
pmax	Integer. Maximum lag order for selection. Default is 4.
dmax	Integer. Maximum integration order for Toda-Yamamoto augmentation. Default is 1.
nboot	Integer. Number of bootstrap replications. Minimum 99. Default is 499.
kmax	Integer. Maximum Fourier frequency (PFTY only). Default is 3.
ic	Character. Information criterion: "aic" or "bic". Default is "aic".
quantiles	Numeric vector. Quantile grid for PQC test (values strictly between 0 and 1). Default is c(0.1, 0.25, 0.50, 0.75, 0.90).
seed	Integer. Random seed for bootstrap. -1 means no seed. Default is -1.

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

data	A data frame in long format.
y	Character. Name of the dependent variable.
x	Character. Name of the polynomial (I(1)) regressor.
panel_id	Character. Name of the panel identifier variable.
time_id	Character. Name of the time variable.
model	Character. Estimator: "mg" for Group-Mean FM-OLS (default) or "pmg" for Pooled FM-OLS.
q	Integer. Polynomial degree: 2 (quadratic, default) or 3 (cubic).
controls	Character vector. Names of additional I(1) control variables. Default is NULL (no controls).
trend	Integer. Deterministic trend type: 1 for demeaning only (default), 2 for demeaning and detrending.
kernel	Character. HAC kernel: "ba" (Bartlett, default), "pa" (Parzen), "qs" (Quadratic Spectral), "tr" (truncated), "bo" (Bohman).
bw	Character or numeric. Bandwidth for HAC estimation. "And91" (default) uses Andrews (1991) automatic selection. A numeric value sets the bandwidth directly.
effects	Character. For Pooled FM-OLS: "oneway" (default) for one-way demeaning or "twoways" for two-way demeaning.
corr_rob	Logical. For Group-Mean FM-OLS: if TRUE, use cross-sectionally robust VCV. Default is FALSE.

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)

---