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R implementation of the multivariate autoregressive distributed lag (ARDL) unit root test of Sam, McNown, Goh and Goh (2025).
The test regression is equation (1) of the paper,
dy[t] = c1 + c2 t + b1 y[t-1] + beta2' x[t-1] + sum_{i=1}^{p-1} phi_i dy[t-i]
+ sum_{j=1}^{q-1} Phi_j' dx[t-j] + omega' dx[t] + u[t]Two statistics are computed: the t statistic on y[t-1]
(H0: b1 = 0) and the F statistic on the lagged levels of the covariates
(H0: beta2 = 0). Their null distributions depend on nuisance parameters,
so both are bootstrapped separately with the null imposed (residual
bootstrap of Section 4.2 of the paper): the restricted regression is
estimated, its recentred residuals are resampled, a bootstrap series y*
is generated recursively from the restricted equation with the observed
x held fixed, and the unrestricted regression is re-estimated on (y*,
x). Bootstrap critical values at 10, 5, 2.5 and 1 percent and bootstrap
p-values are reported.
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 (p, q >= 1), following equation (1) of the paper. The paper is not
consistent on this point (Section 4.1 and Table 4 count lagged
differences directly); the paper’s ARDL(0, 2) corresponds to
fixlag = c(1, 3) here. The contemporaneous difference of x,
omitted in equation (24) of the paper, is kept in the restricted
regressions and in the bootstrap.
Install from CRAN:
install.packages("mvardlurt")library(mvardlurt)
# Generate example data with cointegration
set.seed(123)
n <- 200
x <- cumsum(rnorm(n)) # I(1) process
y <- 0.5 * x + rnorm(n, sd = 0.5) # Cointegrated with x
# Run the test (lags selected by AIC, 999 bootstrap replications)
result <- mvardlurt(y, x, case = 3, nboot = 999, seed = 1)
print(result)
summary(result)
# Several covariates: the F test is joint on all lagged levels
X <- cbind(x, z = cumsum(rnorm(n)))
result2 <- mvardlurt(y, X, fixlag = c(2, 2), nboot = 999)
# Diagnostic plots
plot(result)The two tests are judged at the significance level level
(default 0.05, as in the paper’s simulations; in the paper’s empirical
application the F statistic is reported as significant at the 10 percent
level and cointegration is concluded, Section 6). Before 1.1.0
level was a confidence level (default 0.95).
| Case | t-test (b1 = 0) | F-test (beta2 = 0) | y is | Interpretation |
|---|---|---|---|---|
| I | Not rejected | Not rejected | I(1) | Nonstationary, no cointegration |
| II | Rejected | Not rejected | I(0) | Stationary process |
| III | Not rejected | Rejected | I(2) | Degenerate lagged dependent variable |
| IV | Rejected | Rejected | I(1) | Nonstationary, cointegrated with x |
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
GPL-3
Muhammad Alkhalaf
These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.