Package {mvardlurt}

Type:	Package
Title:	Multivariate ARDL Unit Root Test
Version:	1.1.0
Date:	2026-09-30
Description:	Implements the multivariate autoregressive distributed lag (ARDL) unit root test of Sam, McNown, Goh and Goh (2025) <doi:10.1080/03796205.2024.2439101>. The test augments the ADF regression with the lagged level, the current difference and lagged differences of one or more covariates so that cointegration between the series under test and the covariates is taken into account. The t statistic on the lagged level of the series and the joint F statistic on the lagged levels of the covariates are bootstrapped with the respective null imposed (residual bootstrap), giving critical values and p-values. Provides automatic lag selection via AIC or BIC, diagnostic plots, and the four-case classification of the order of integration of the series.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/mvardlurt
BugReports:	https://github.com/muhammedalkhalaf/mvardlurt/issues
Encoding:	UTF-8
RoxygenNote:	7.3.2
Depends:	R (≥ 4.0.0)
Imports:	grDevices, graphics, stats, utils
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-09-30 23:13:49 UTC; root
Author:	Muhammad Alkhalaf [aut, cre, cph]
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-10-01 08:30:21 UTC

Multivariate ARDL Unit Root Test

Description

Implements the multivariate autoregressive distributed lag (ARDL) unit root test of Sam, McNown, Goh and Goh (2025). The test augments the ADF regression with the lagged level, the current difference and lagged differences of one or more covariates, so that a possible cointegrating relationship between the series under test and the covariates is taken into account. Critical values and p-values come from the residual bootstrap described in the paper.

Details

The main function is mvardlurt, which performs the multivariate ARDL unit root test. The package provides:

• Automatic lag selection via AIC or BIC on a common sample

• Residual bootstrap critical values and p-values with the null imposed (Section 4.2 of the paper)

• Three deterministic specifications (none, intercept, intercept + trend)

• One or several covariates (joint F test on their lagged levels)

• Four-case classification of the order of integration of y

• Comprehensive print and summary output

• Diagnostic plots

The test produces two statistics:

• t-statistic: Tests H_0: b_1 = 0 on the lagged level of y

• F-statistic: Tests H_0: \beta_2 = 0 jointly on the lagged levels of the covariates

The four cases of Section 3.2 of the paper are:

• Case I: neither rejected; y is I(1), no cointegration

• Case II: t rejected, F not rejected; y is I(0)

• Case III: t not rejected, F rejected; y is I(2)

• Case IV: both rejected; y is I(1) and cointegrated with x

Author(s)

Muhammad Alkhalaf

References

Sam, C. Y., McNown, R., Goh, S. K. and Goh, K. L. (2025). A multivariate autoregressive distributed lag unit root test. Studies in Economics and Econometrics, 49(1), 17-33. doi:10.1080/03796205.2024.2439101

See Also

mvardlurt

Examples

# Generate cointegrated data
set.seed(123)
n <- 200
x <- cumsum(rnorm(n))
y <- 0.5 * x + rnorm(n, sd = 0.5)

# Run the test
result <- mvardlurt(y, x, case = 3, fixlag = c(2, 2), nboot = 199)
print(result)

Create a Combined Diagnostic Plot for mvardlurt Objects

Description

Creates a 2x2 panel of diagnostic plots for the multivariate ARDL unit root test including residuals vs fitted, Q-Q plot, residuals over time, and ACF.

Usage

## S3 method for class 'mvardlurt'
autoplot(x, ...)

Arguments

Value

Invisibly returns x.

Author(s)

Muhammad Alkhalaf

See Also

mvardlurt, plot.mvardlurt

Examples

set.seed(123)
n <- 100
x <- cumsum(rnorm(n))
y <- 0.5 * x + rnorm(n, sd = 0.5)

result <- mvardlurt(y, x, fixlag = c(2, 2), nboot = 99)
autoplot.mvardlurt(result)

Multivariate ARDL Unit Root Test

Description

Implements the multivariate autoregressive distributed lag (ARDL) unit root test of Sam, McNown, Goh and Goh (2025). The test augments the ADF regression with the lagged level, the current difference and lagged differences of one or more covariates, so that a possible cointegrating relationship between the series under test and the covariates is taken into account. Critical values and p-values are obtained by the residual bootstrap of Section 4.2 of the paper.

Usage

mvardlurt(y, x, case = 3L, maxlag = 10L, ic = "aic",
          fixlag = NULL, nboot = 999L, level = 0.05,
          seed = NULL, boot = TRUE, reps = NULL)

Arguments

Details

The test regression is equation (1) of the paper,

\Delta y_t = c_1 + c_2 t + b_1 y_{t-1} + \beta_2' x_{t-1} + \sum_{i=1}^{p-1} \phi_i \Delta y_{t-i} + \sum_{j=1}^{q-1} \Phi_j' \Delta x_{t-j} + \omega' \Delta x_t + u_t,

estimated by OLS on t = \max(p, q) + 1, \ldots, n. The orders follow equation (1) of the paper: an ARDL(p, q) model contains p - 1 lagged differences of y and q - 1 lagged differences of x in addition to the contemporaneous difference of x, so p and q are at least 1 and ARDL(1, 1) contains no lagged differences at all. The paper is not consistent on this point: equations (1) and (24) sum to p - 1 and q - 1, whereas Section 4.1 describes the ARDL(1, 1) data generating process as having one lagged difference of each variable and Table 4 reports an ARDL(0, 2) model with no lagged difference of y. This package follows equation (1); the paper's ARDL(0, 2) corresponds to fixlag = c(1, 3) here, and fixlag = c(p, q) in versions before 1.1.0 corresponds to fixlag = c(p + 1, q + 1) in 1.1.0 (apart from the contemporaneous difference of x, which earlier versions omitted).

Two statistics are computed (Section 3.3, equations (11) and (12) of the paper):

• the t statistic on y_{t-1} for H_0: b_1 = 0 against H_1: b_1 < 0 (left tailed);

• the F (Wald) statistic for the joint hypothesis H_0: \beta_2 = 0 on the lagged levels of all covariates (right tailed). With one covariate this equals the squared t statistic on x_{t-1}.

Their null distributions depend on nuisance parameters, so both are bootstrapped separately with the respective null imposed: the restricted regression (without y_{t-1} for the t test, without x_{t-1} for the F test) is estimated, its recentred residuals are resampled with replacement, a bootstrap series y^* is generated recursively from the restricted estimated equation with the observed x held fixed, and the unrestricted regression is re-estimated on (y^*, x). The contemporaneous difference of x, which equation (24) of the paper omits although equation (1) has it, is included in the restricted regressions and in the bootstrap, following equation (1). The initial values y^*_1, \ldots, y^*_{\max(p, q)} are set to the observed y, so the recursion starts at t = \max(p, q) + 1 with the observed history. Critical values are the percentiles of the nboot bootstrap statistics, and the bootstrap p-values are the proportions of bootstrap statistics at least as extreme as the observed ones.

The decision at significance level level classifies the series according to the four cases of Section 3.2 of the paper:

• Case I (t not rejected, F not rejected): nonstationary process, no cointegration; y is I(1).

• Case II (t rejected, F not rejected): stationary process; y is I(0).

• Case III (t not rejected, F rejected): degenerate lagged dependent variable; y is I(2).

• Case IV (both rejected): nonstationary process with cointegration between y and x; y is I(1).

The Monte Carlo experiments of the paper use a 5 percent level for both tests, which is the default here. In the empirical application of the paper the F statistic is reported as significant at the 10 percent level and cointegration is concluded (Section 6); set level = 0.10 to reproduce that convention. Before version 1.1.0 level was a confidence level (default 0.95); values of 0.5 or more are now rejected.

Lag orders are either fixed through fixlag or selected by AIC or BIC over p = 1, ..., maxlag and q = 1, ..., maxlag. All candidate models are estimated on the same observations (t = maxlag + 1, ..., n) so that the criteria are comparable; the selected model is then re-estimated on its own full sample.

Value

An object of class "mvardlurt" containing:

Author(s)

Muhammad Alkhalaf

References

Sam, C. Y., McNown, R., Goh, S. K. and Goh, K. L. (2025). A multivariate autoregressive distributed lag unit root test. Studies in Economics and Econometrics, 49(1), 17-33. doi:10.1080/03796205.2024.2439101

Examples

# Cointegrated data: y is I(1) through x (Case IV)
set.seed(123)
n <- 200
x <- cumsum(rnorm(n))
y <- 0.5 * x + rnorm(n, sd = 0.5)

result <- mvardlurt(y, x, case = 3, fixlag = c(2, 2), nboot = 199)
print(result)
summary(result)

# Automatic lag selection with two covariates
X <- cbind(x, z = cumsum(rnorm(n)))
result2 <- mvardlurt(y, X, maxlag = 3, nboot = 199)
result2$fstat

Plot Method for mvardlurt Objects

Description

Creates diagnostic plots for the multivariate ARDL unit root test.

Usage

## S3 method for class 'mvardlurt'
plot(x, which = c(1, 2, 3, 4),
     ask = (length(which) > 1 && dev.interactive()), ...)

Arguments

Value

Invisibly returns x.

Author(s)

Muhammad Alkhalaf

See Also

mvardlurt, autoplot.mvardlurt

Examples

set.seed(123)
n <- 100
x <- cumsum(rnorm(n))
y <- 0.5 * x + rnorm(n, sd = 0.5)

result <- mvardlurt(y, x, maxlag = 3, nboot = 99)

# Default diagnostic plots
plot(result, ask = FALSE)

# IC surface plot
plot(result, which = 6)

Methods for mvardlurt Objects

Description

Print, summary, and accessor methods for objects of class "mvardlurt".

Usage

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

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

## S3 method for class 'mvardlurt'
coef(object, ...)

## S3 method for class 'mvardlurt'
residuals(object, ...)

## S3 method for class 'mvardlurt'
fitted(object, ...)

Arguments

Value

print and summary invisibly return x/object.

coef returns a named numeric vector with b1, the beta2 coefficient of each covariate and the corresponding long-run multipliers lr_mult.

residuals returns the numeric vector of residuals from the fitted model.

fitted returns the numeric vector of fitted values.

Author(s)

Muhammad Alkhalaf

See Also

mvardlurt

Examples

set.seed(123)
n <- 100
x <- cumsum(rnorm(n))
y <- 0.5 * x + rnorm(n, sd = 0.5)

result <- mvardlurt(y, x, fixlag = c(2, 2), nboot = 99)

# Print method
print(result)

# Summary method
summary(result)

# Extract coefficients
coef(result)

# Extract residuals
head(residuals(result))

# Extract fitted values
head(fitted(result))