Package {fbardl}

Type:	Package
Title:	Fourier Bootstrap ARDL Cointegration Test
Version:	1.1.0
Description:	Implements the Fourier Bootstrap Autoregressive Distributed Lag (FBARDL) bounds testing approach for cointegration analysis. Combines the Pesaran, Shin & Smith (2001) <doi:10.1002/jae.616> ARDL bounds testing framework with Fourier terms to capture structural breaks following Yilanci, Bozoklu & Gorus (2020) <doi:10.1016/j.scs.2020.102035>, and bootstrap critical values based on McNown, Sam & Goh (2018) <doi:10.1080/00036846.2017.1366643> and Bertelli, Vacca & Zoia (2022) <doi:10.1016/j.econmod.2022.105987>, with finite-sample bounds test critical values from Kripfganz and Schneider (2020) <doi:10.1111/obes.12377>. Features include automatic lag selection via AIC/BIC, optimal Fourier frequency selection by minimum SSR, long-run and short-run coefficient estimation, diagnostic tests, and dynamic multiplier analysis.
License:	GPL-3
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	stats
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown
LazyData:	true
Config/testthat/edition:	3
RoxygenNote:	7.3.3
URL:	https://github.com/muhammedalkhalaf/fbardl
BugReports:	https://github.com/muhammedalkhalaf/fbardl/issues
NeedsCompilation:	no
Packaged:	2026-09-28 17:31:59 UTC; root
Author:	Muhammad Alkhalaf [aut, cre, cph]
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-09-30 21:20:50 UTC

fbardl: Fourier Bootstrap ARDL Cointegration Test

Description

Implements the Fourier Bootstrap Autoregressive Distributed Lag (FBARDL) bounds testing approach for cointegration analysis. Combines the Pesaran, Shin & Smith (2001) doi:10.1002/jae.616 ARDL bounds testing framework with Fourier terms to capture structural breaks following Yilanci, Bozoklu & Gorus (2020) doi:10.1016/j.scs.2020.102035, and bootstrap critical values based on McNown, Sam & Goh (2018) doi:10.1080/00036846.2017.1366643 and Bertelli, Vacca & Zoia (2022) doi:10.1016/j.econmod.2022.105987. Features include automatic lag selection via AIC/BIC, optimal Fourier frequency selection by minimum SSR, long-run and short-run coefficient estimation, diagnostic tests, and dynamic multiplier analysis.

The fbardl package implements the Fourier Bootstrap ARDL bounds testing approach for cointegration analysis. It combines the Pesaran, Shin & Smith (2001) ARDL framework with Fourier terms to capture structural breaks, and provides bootstrap critical values for robust inference.

Main Function

• fbardl: Perform Fourier Bootstrap ARDL cointegration test

Test Types

• "fardl": Standard Fourier ARDL with PSS bounds test

• "fbardl_mcnown": Bootstrap ARDL (McNown, Sam & Goh, 2018)

• "fbardl_bvz": Bootstrap ARDL (Bertelli, Vacca & Zoia, 2022)

Data

• fbardl_data: Example dataset for demonstration

Author(s)

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

References

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326. doi:10.1002/jae.616

McNown, R., Sam, C. Y., & Goh, S. K. (2018). Bootstrapping the autoregressive distributed lag test for cointegration. Applied Economics, 50(13), 1509-1521. doi:10.1080/00036846.2017.1366643

See Also

Useful links:

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

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

Useful links:

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

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

Fourier Bootstrap ARDL Cointegration Test

Description

Performs the Fourier Bootstrap ARDL (FBARDL) bounds testing approach for cointegration analysis. This function combines the Pesaran, Shin & Smith (2001) ARDL bounds testing framework with Fourier terms to capture structural breaks, and provides bootstrap critical values for robust inference.

Usage

fbardl(
  formula,
  data,
  type = c("fardl", "fbardl_mcnown", "fbardl_bvz"),
  maxlag = 4,
  maxk = 5,
  ic = c("aic", "bic"),
  case = 3,
  reps = 999,
  fourier = TRUE,
  level = 0.95,
  horizon = 20,
  unconditional = FALSE
)

Arguments

Details

The FBARDL approach extends the standard ARDL bounds testing procedure by:

1. Incorporating Fourier terms to capture smooth structural breaks

2. Using bootstrap methods to generate finite-sample critical values

3. Implementing the McNown et al. (2018) procedure to detect degenerate cases

The PSS case sets the deterministic terms: a constant in all cases and a linear trend in cases 4 and 5. The overall F test restricts the lagged levels, together with the intercept in case 2 and the trend in case 4.

With type = "fardl" the F and t statistics are compared with the finite-sample critical values and approximate p-values of Kripfganz and Schneider (2020), computed from their response surface coefficients (Stata package ardl) for the sample size, the number of regressors and the number of short-run coefficients (the Fourier terms included). These bounds do not account for the Fourier terms; the F test on the lagged regressors has no tabulated distribution and is reported with the bootstrap types only.

The bootstrap types follow the Stata module fbardl 1.3.0. Data are generated recursively under the null from the restricted equilibrium-correction equation for y and the equations for \Delta x, resampling the residual pairs. "fbardl_mcnown" uses one null (all lagged levels, plus the restricted deterministic term) for the three statistics and an unrestricted \Delta x equation; "fbardl_bvz" uses a separate null for each statistic, a marginal \Delta x equation without the lagged level of y, and recentred residuals.

The procedure involves three main steps:

1. Selection of optimal Fourier frequency k* by minimum SSR

2. Selection of lag orders (p, q) by AIC or BIC

3. Cointegration testing with bootstrap or PSS critical values

Three test statistics are computed:

• F.overall: Joint test on all lagged level variables

• t.dependent: t-test on lagged dependent variable

• F.independent: Joint test on lagged independent variables

Value

An object of class "fbardl" containing:

coefficients

Named vector of estimated coefficients

std.errors

Standard errors of coefficients

t.values

t-statistics

p.values

p-values

long.run

Long-run coefficient estimates with standard errors

short.run

Short-run coefficient estimates

ecm.coef

Error correction coefficient (speed of adjustment)

best.p

Selected lag order for dependent variable

best.q

Selected lag orders for independent variables

best.kstar

Selected Fourier frequency

F.overall

F-statistic for overall cointegration test

t.dependent

t-statistic on lagged dependent variable

F.independent

F-statistic on lagged independent variables

cointegration

Cointegration test results with critical values

diagnostics

Diagnostic test results

model.fit

Model fit statistics (R2, AIC, BIC, etc.)

residuals

Model residuals

fitted.values

Fitted values

nobs

Number of observations

call

The matched call

References

Kripfganz, S. and Schneider, D. C. (2020). Response surface regressions for critical value bounds and approximate p-values in equilibrium correction models. Oxford Bulletin of Economics and Statistics, 82(6), 1456-1481. doi:10.1111/obes.12377

Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326. doi:10.1002/jae.616

McNown, R., Sam, C. Y., & Goh, S. K. (2018). Bootstrapping the autoregressive distributed lag test for cointegration. Applied Economics, 50(13), 1509-1521. doi:10.1080/00036846.2017.1366643

Yilanci, V., Bozoklu, S., & Gorus, M. S. (2020). Are BRICS countries pollution havens? Evidence from a bootstrap ARDL bounds testing approach with a Fourier function. Sustainable Cities and Society, 55, 102035. doi:10.1016/j.scs.2020.102035

Kripfganz, S., & Schneider, D. C. ( 2020). Response surface regressions for critical value bounds and approximate p-values in equilibrium correction models. Oxford Bulletin of Economics and Statistics, 82(6), 1456-1481. doi:10.1111/obes.12377

Examples

# Load example data
data(fbardl_data)

# Basic Fourier ARDL test
result <- fbardl(y ~ x1 + x2, data = fbardl_data, type = "fardl")
summary(result)

# Bootstrap ARDL (McNown approach)
result_boot <- fbardl(y ~ x1 + x2, data = fbardl_data,
                      type = "fbardl_mcnown", reps = 499)
summary(result_boot)

# Without Fourier terms
result_nofourier <- fbardl(y ~ x1 + x2, data = fbardl_data,
                           fourier = FALSE)

Example Data for Fourier Bootstrap ARDL Analysis

Description

A simulated time series dataset suitable for demonstrating the Fourier Bootstrap ARDL cointegration testing procedure. The data contains a dependent variable y and two independent variables x1 and x2 with a cointegrating relationship and structural breaks.

Usage

fbardl_data

Format

A data frame with 150 observations and 3 variables:

y

Dependent variable (simulated I(1) series)

x1

First independent variable (simulated I(1) series)

x2

Second independent variable (simulated I(1) series)

Details

The data is generated from a data-generating process (DGP) that includes:

• A long-run cointegrating relationship: y_t = 2 + 0.8 x_{1t} - 0.5 x_{2t} + u_t

• Short-run dynamics with AR(1) errors

• A structural break modeled by Fourier terms

• Error correction mechanism with adjustment speed of -0.3

This dataset is designed to produce clear cointegration test results when analyzed with the fbardl function.

Source

Simulated data for package demonstration.

See Also

fbardl

Examples

data(fbardl_data)
head(fbardl_data)
summary(fbardl_data)

# Plot the series
ts.plot(ts(fbardl_data), col = 1:3, lty = 1:3)
legend("topleft", colnames(fbardl_data), col = 1:3, lty = 1:3)