Type:	Package
Title:	Unit Root Tests for Bounded Time Series
Version:	1.0.1
Description:	Implements unit root tests for bounded time series following Cavaliere and Xu (2014) <doi:10.1016/j.jeconom.2013.08.012>. 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)
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-03-09 17:14:10 UTC; acad_
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:	2026-03-16 15:50:22 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:

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.2013.08.012

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

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

Augmented Dickey-Fuller normalized bias: T(\hat{\rho} - 1)

ADF-t

Augmented Dickey-Fuller t-statistic for \rho

MZ-alpha

Modified Phillips-Perron normalized bias

MZ-t

Modified Phillips-Perron t-statistic

MSB

Modified Sargan-Bhargava statistic

P-value Computation

P-values are computed by Monte Carlo simulation of bounded Brownian motion. 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.2013.08.012

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

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 optimal number of lags for the ADF regression using the Modified Akaike Information Criterion (MAIC) of Ng and Perron (2001).

Usage

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

Arguments

Details

The MAIC criterion is defined as:

MAIC(k) = \ln(\hat{\sigma}^2_k) + 2(k+1)/T

where \hat{\sigma}^2_k is the residual variance from the ADF regression with k lags.

This criterion provides better size properties than standard AIC for unit root testing.

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 a discretized reflected Brownian motion constrained between bounds, following the methodology of Cavaliere and Xu (2014).

Usage

simulate_bounded_bm(n, c_lower, c_upper = Inf)

Arguments

Details

The function simulates a standard Brownian motion and applies reflection at the boundaries. For two-sided bounds, both upper and lower reflections are applied. For one-sided bounds (c_upper = Inf), only lower reflection is used.

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.2013.08.012

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)