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