Unit Root Tests for Bounded Time Series

Implements unit root tests for bounded time series following Cavaliere and Xu (2014) . Standard unit root tests (ADF, Phillips-Perron) have non-standard limiting distributions when the time series is bounded. This package provides modified ADF and M-type tests (MZ-alpha, MZ-t, MSB) with p-values computed via Monte Carlo simulation of bounded Brownian motion. Supports one-sided (lower bound only) and two-sided bounds, with automatic lag selection using the MAIC criterion of Ng and Perron (2001) .

boundedur

Unit root tests for bounded time series following Cavaliere and Xu (2014).

Overview

Standard unit root tests (ADF, Phillips-Perron) assume the time series is unbounded, but many economic variables are naturally bounded:

• Interest rates (non-negative, often with upper caps)
• Exchange rates (within currency bands)
• Capacity utilization (0-100%)
• Unemployment rates (0-100%)
• Price indices (non-negative)

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.

Installation

# From CRAN (when available)
install.packages("boundedur")

# Development version from GitHub
# install.packages("remotes")

Usage

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)

One-sided bounds

For series with only a lower bound (e.g., prices):

result <- boundedur(y, lbound = 0, ubound = Inf)

Select specific tests

# ADF tests only
result <- boundedur(y, lbound = 0, ubound = 10, test = "adf")

# Single test
result <- boundedur(y, lbound = 0, ubound = 10, test = "mz_t")

Control lag selection

# 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)

Available Tests

Methodology

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.

References

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.2013.08.026

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

License

GPL (>= 3)