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Unit root tests for bounded time series following Cavaliere and Xu (2014).
Standard unit root tests (ADF, Phillips-Perron) assume the time series is unbounded, but many economic variables are naturally bounded:
When bounds are binding or nearly so, standard tests have non-standard limiting distributions, leading to incorrect inference. The boundedur package implements the modified tests of Cavaliere and Xu (2014), which properly account for bounds.
# From CRAN (when available)
install.packages("boundedur")
# Development version from GitHub
# install.packages("remotes")library(boundedur)
# Generate bounded random walk (e.g., interest rate between 0 and 10%)
set.seed(123)
n <- 200
y <- numeric(n)
y[1] <- 5
for (i in 2:n) {
y[i] <- y[i-1] + rnorm(1, 0, 0.3)
y[i] <- max(0, min(10, y[i])) # Reflect at bounds
}
# Test for unit root with known bounds
result <- boundedur(y, lbound = 0, ubound = 10)
print(result)
summary(result)For series with only a lower bound (e.g., prices):
result <- boundedur(y, lbound = 0, ubound = Inf)# ADF tests only
result <- boundedur(y, lbound = 0, ubound = 10, test = "adf")
# Single test
result <- boundedur(y, lbound = 0, ubound = 10, test = "mz_t")# Automatic (MAIC criterion, default)
result <- boundedur(y, lbound = 0, ubound = 10)
# Manual lag specification
result <- boundedur(y, lbound = 0, ubound = 10, lags = 4)
# Custom maximum lag
result <- boundedur(y, lbound = 0, ubound = 10, maxlag = 12)| Test | Description |
|---|---|
adf_alpha |
ADF normalized bias: T(ρ̂ - 1) |
adf_t |
ADF t-statistic |
mz_alpha |
Modified Phillips-Perron normalized bias |
mz_t |
Modified Phillips-Perron t-statistic |
msb |
Modified Sargan-Bhargava |
The tests modify standard unit root statistics to account for the effect of bounds on the limiting distribution. Under the null hypothesis of a unit root, the process is constrained by bounds and follows a reflected Brownian motion rather than standard Brownian motion.
P-values are computed via Monte Carlo simulation of the bounded Brownian motion null distribution, with bound parameters estimated from the data.
Cavaliere, G., & Xu, F. (2014). Testing for unit roots in bounded time series. Journal of Econometrics, 178(2), 259-272. doi:10.1016/j.jeconom.2013.08.012
Ng, S., & Perron, P. (2001). Lag length selection and the construction of unit root tests with good size and power. Econometrica, 69(6), 1519-1554. doi:10.1111/1468-0262.00256
GPL (>= 3)
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