Package {boundedur}

Type:	Package
Title:	Unit Root Tests for Bounded Time Series
Version:	1.0.3
Description:	Implements unit root tests for bounded time series following Cavaliere and Xu (2014) <doi:10.1016/j.jeconom.2013.08.026>. 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) <doi:10.1111/1468-0262.00256>.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/boundedur
BugReports:	https://github.com/muhammedalkhalaf/boundedur/issues
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	stats
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
Config/roxygen2/version:	8.1.0
NeedsCompilation:	no
Packaged:	2026-09-28 15:26:52 UTC; root
Author:	Muhammad Alkhalaf [aut, cre, cph], Giuseppe Cavaliere [ctb] (Original methodology), Fang Xu [ctb] (Original methodology)
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-09-28 20:00:26 UTC

boundedur: Unit Root Tests for Bounded Time Series

Description

The boundedur package implements unit root tests for bounded time series following the methodology of Cavaliere and Xu (2014). Standard unit root tests (ADF, Phillips-Perron) have non-standard limiting distributions when the time series is constrained to lie within bounds. This package provides modified tests with p-values computed via Monte Carlo simulation of bounded Brownian motion.

Main Functions

boundedur

Main function to perform bounded unit root tests

simulate_bounded_bm

Simulate bounded Brownian motion

select_lag_maic

Select optimal lag using MAIC criterion

Available Tests

• ADF-alpha: Augmented Dickey-Fuller normalized bias test

• ADF-t: Augmented Dickey-Fuller t-statistic test

• MZ-alpha: Modified Phillips-Perron normalized bias test

• MZ-t: Modified Phillips-Perron t-statistic test

• MSB: Modified Sargan-Bhargava test

Author(s)

Maintainer: Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

Authors:

• Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID) [copyright holder]

Other contributors:

• Giuseppe Cavaliere (Original methodology) [contributor]

• Fang Xu (Original methodology) [contributor]

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

See Also

Useful links:

• https://github.com/muhammedalkhalaf/boundedur

• Report bugs at https://github.com/muhammedalkhalaf/boundedur/issues

Unit Root Tests for Bounded Time Series

Description

Performs unit root tests for time series constrained within known bounds, following the methodology of Cavaliere and Xu (2014). Provides modified ADF and M-type test statistics with p-values computed via Monte Carlo simulation of bounded Brownian motion.

Usage

boundedur(
  y,
  lbound,
  ubound = Inf,
  test = c("all", "adf", "adf_alpha", "adf_t", "mz_alpha", "mz_t", "msb"),
  lags = NULL,
  maxlag = NULL,
  detrend = c("constant", "none"),
  nsim = 499,
  nstep = NULL,
  seed = NULL
)

Arguments

Details

Standard unit root tests assume the series is unbounded, leading to non-standard limiting distributions when bounds are present. This function implements the bounded unit root tests of Cavaliere and Xu (2014), which account for the effect of bounds on the limiting distribution.

The null hypothesis is that the series has a unit root while respecting the bounds. The alternative is stationarity.

Value

An object of class "boundedur" containing:

Test Statistics

ADF-alpha

T(\hat{\rho} - 1)/\hat\alpha(1) from the ADF regression of the de-meaned series (no deterministic terms)

ADF-t

The t-statistic for \rho - 1 = 0 in that regression

MZ-alpha

(T^{-1}\hat X_T^2 - T^{-1}\hat X_0^2 - s^2_{AR}) / (2T^{-2}\sum \hat X_{t-1}^2)

MZ-t

MZ_\alpha \times MSB

MSB

(T^{-2}\sum \hat X_{t-1}^2 / s^2_{AR})^{1/2}

All five reject for small values; ADF-alpha and MZ-alpha, and ADF-t and MZ-t, share the same limiting distribution (Theorem 1).

P-value Computation

P-values follow Algorithm 1 of Cavaliere and Xu (2014): the bound parameters are estimated by \hat c = (b - X_0)/(s_{AR} T^{1/2}), and the null distribution is simulated from a random walk regulated at \hat c and \bar c. The number of replications (nsim) controls accuracy; larger values give more precise p-values but increase computation time.

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

Examples

# Generate bounded random walk (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, 0.5)
  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, nsim = 199)
print(result)
summary(result)

# One-sided bound (e.g., price level, lower bound = 0)
result_lower <- boundedur(y, lbound = 0, ubound = Inf, nsim = 199)

Select Optimal Lag using MAIC Criterion

Description

Selects the number of lagged differences in the ADF regression with the Modified Akaike Information Criterion (MAIC) of Ng and Perron (2001).

Usage

select_lag_maic(y, maxlag = NULL, detrend = "constant")

Arguments

Details

For k = 0, \ldots, k_{max}, the ADF regression of \Delta \hat X_t on \hat X_{t-1} and k lagged differences is estimated on the common sample t = k_{max} + 2, \ldots, T, and

MAIC(k) = \ln(\hat{\sigma}^2_k) + 2(\tau_T(k) + k)/(T - k_{max} - 1),

with \tau_T(k) = \hat\sigma_k^{-2} \hat\beta_0^2 \sum \hat X_{t-1}^2 and \hat\beta_0 the coefficient on \hat X_{t-1}.

Value

A list with class "lag_selection" containing:

References

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

Examples

# Generate random walk
set.seed(123)
y <- cumsum(rnorm(200))

# Select lag
lag_sel <- select_lag_maic(y)
print(lag_sel)

Simulate Bounded Brownian Motion

Description

Simulates the discretized regulated Brownian motion of Algorithm 1 in Cavaliere and Xu (2014).

Usage

simulate_bounded_bm(n, c_lower, c_upper = Inf)

Arguments

Details

The path follows the recursion (4.11) of Cavaliere and Xu (2014): X_t = X_{t-1} + n^{-1/2}\varepsilon_t, set to the bound whenever it would cross it, with X_0 = 0 and i.i.d. standard normal \varepsilon_t.

The standardized bound parameters c_lower and c_upper are computed from the original bounds as:

c = (b - X_0) / (\sigma \sqrt{T})

where b is the bound, X_0 is the initial value, \sigma is the long-run standard deviation, and T is the sample size.

Value

A numeric vector of length n + 1 containing the simulated bounded Brownian motion path, starting at 0.

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

Examples

# Simulate bounded Brownian motion with two-sided bounds
set.seed(123)
bm <- simulate_bounded_bm(n = 1000, c_lower = -2, c_upper = 2)
plot(bm, type = "l", main = "Bounded Brownian Motion")
abline(h = c(-2, 2), col = "red", lty = 2)

# One-sided bound (lower only)
bm_lower <- simulate_bounded_bm(n = 1000, c_lower = -1, c_upper = Inf)