Package 'mvardlurt' reference manual

Title:	Multivariate ARDL Unit Root Test
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.
Authors:	Muhammad Alkhalaf [aut, cre, cph] (ORCID: <https://orcid.org/0009-0002-2677-9246>)
Maintainer:	Muhammad Alkhalaf <[email protected]>
License:	GPL-3
Version:	1.1.1
Built:	2026-11-02 13:20:07 UTC
Source:	https://github.com/muhammedalkhalaf/mvardlurt

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

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

Arguments

x

An object of class "mvardlurt".

...

Additional arguments passed to plotting functions.

Value

Invisibly returns x.

Author(s)

Muhammad Alkhalaf

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

Arguments

y

A numeric vector or time series. The series under test.

x

A numeric vector, time series or numeric matrix with one column per covariate (the I(1) forcing variables of the paper).

case

Integer. Deterministic specification:

1: No deterministic terms
3: Intercept only (default)
5: Intercept and linear trend

maxlag

Integer. Maximum ARDL order for AIC/BIC selection. Default is 10. Must be between 1 and 12.

ic

Character. Information criterion for lag selection: "aic" (default) or "bic".

fixlag

Optional numeric vector of length 2, the fixed orders c(p, q) of the ARDL(p, q) model (see Details). Both must be at least 1. If provided, overrides automatic lag selection.

nboot

Integer. Number of bootstrap replications. Default is 999. Minimum is 99.

level

Numeric. Significance level of the two tests used for the case classification (0 to 1). Default is 0.05.

seed

Optional integer. If supplied, the random number generator is seeded with it for the bootstrap and the caller's RNG state is restored on exit. Default NULL leaves the RNG untouched.

boot

Logical. Whether to compute bootstrap critical values and p-values. Default is TRUE.

reps

Deprecated name for nboot, kept for backward compatibility.

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 lag orders p and q of the bootstrap are those of the test regression, either fixed through fixlag or selected by AIC or BIC on the observed data. They are held fixed in every bootstrap replication: the restricted regression is estimated with these orders and the unrestricted regression is re-estimated on $(y^*, x)$ with the same orders, and the lag selection is not repeated on the bootstrap samples. This follows Section 4.2 of the paper, where the restricted regression (24) is estimated with the lag length of the data generating process and equation (1) is re-estimated on $y^*_t$ and $x_t$ . When the orders are selected from the data, the bootstrap distribution therefore does not reflect the selection step. In a Monte Carlo experiment by the package author (n = 100, y and x independent random walks, 200 replications, 199 bootstrap replications, 5 percent level) the rejection frequencies of the bootstrap tests were 0.055 to 0.070 with fixed lag orders and 0.080 to 0.100 with AIC selection (Monte Carlo standard error about 0.02). Fixing the orders through fixlag avoids this source of size distortion.

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:

tstat

t statistic for $H_0: b_1 = 0$

fstat

F statistic for $H_0: \beta_2 = 0$

t_pval, f_pval

Bootstrap p-values (NA when boot = FALSE)

t_cv, f_cv

Bootstrap critical values at the 10, 5, 2.5 and 1 percent levels

t_boot, f_boot

The bootstrap distributions

b1, b1_se

Estimate and standard error of $b_1$

beta2, beta2_se

Estimates and standard errors of $\beta_2$ (one per covariate)

lr_mult

Long-run multipliers $-\beta_2 / b_1$

opt_p, opt_q

Orders of the ARDL(p, q) model used

case, casename

Deterministic case and its description

nboot

Number of bootstrap replications

nobs

Number of observations in the regression

aic, bic, r_squared, adj_r_squared

Fit statistics of the model

ic_table

Matrix of information criteria for all (p, q) candidates (NULL with fixlag)

decision

List with the decision at level: reject_t, reject_f, case_num, case_result, integration (the implied order of integration of y) and the significance codes

model

The fitted lm object, for inspection (its call refers to an internal data frame; to refit use lm(formula(object$model), data = object$data))

data

The data frame of the test regression

y, x

The data used (x as a matrix)

residuals

Residuals from the fitted model

Author(s)

Muhammad Alkhalaf

References

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

Arguments

x

An object of class "mvardlurt".

which

Integer vector indicating which plots to produce:

1: Residuals vs Fitted
2: Q-Q plot of residuals
3: Residuals over time
4: ACF of residuals
5: Time series of y and the covariates
6: Information criterion surface

Default is c(1, 2, 3, 4).

ask

Logical. If TRUE, prompt before each plot. Default is TRUE when multiple plots are requested in an interactive session.

...

Additional arguments passed to plotting functions.

Value

Invisibly returns x.

Author(s)

Muhammad Alkhalaf

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)
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, ...)
## 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

x

An object of class "mvardlurt".

object

An object of class "mvardlurt".

...

Additional arguments (ignored).

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

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))
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))

Package 'mvardlurt'

Help Index

Multivariate ARDL Unit Root Test

Description

Details

Author(s)

References

See Also

Examples

Create a Combined Diagnostic Plot for mvardlurt Objects

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Multivariate ARDL Unit Root Test

Description

Usage

Arguments

Details

Value

Author(s)

References

Examples

Plot Method for mvardlurt Objects

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Methods for mvardlurt Objects

Description

Usage

Arguments

Value

Author(s)

See Also

Examples