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Bayesian Unit Root Testing for Models with Maintained Trend

Shikhar Tyagi, Arvind Pandey, Bhupendra Singh, Vrijesh Tripathi

2026-07-28

Introduction

The BayesURTrend package implements the Bayesian unit root test for time series models with maintained polynomial trend components as proposed by Chaturvedi and Kumar (2005).

Unlike classical Augmented Dickey-Fuller (ADF) or Phillips-Perron (PP) tests which rely on asymptotic approximations, the Bayesian framework calculates exact finite-sample posterior probabilities and Bayes factors for unit root hypotheses against stationary alternatives.

Mathematical Model

Consider a time series \(\{y_t, t = 1, 2, \dots, T\}\) governed by: \[y_t = \mu_0 + \sum_{i=1}^p \mu_i t^i + u_t\] where the stochastic error term \(u_t\) follows an augmented \(\text{AR}(1)\) process: \[u_t = \rho u_{t-1} + \sum_{j=1}^k \psi_j \Delta y_{t-j} + \epsilon_t, \quad \epsilon_t \sim \text{iid } N(0, \sigma^2)\]

The unit root hypothesis is \(H_0: \rho = 1\) versus the stationary alternative \(H_1: \rho \in (a, 1)\) where \(-1 < a < 1\). Under \(H_0\), the polynomial trend degree reduces by 1, yielding a maintained trend in first differences.

Example Usage

Testing a Unit Root Process (Random Walk)

set.seed(123)
y_rw <- cumsum(rnorm(100))
res_rw <- bayes_ur_test(y_rw, p = 1, k = 1)
print(res_rw)
#> 
#>   Chaturvedi-Kumar Bayesian Unit Root Test with Maintained Trend 
#> 
#> data:   y_rw 
#> Posterior Odds Ratio (B01) = 6.873 
#> Bayes Factor (BF01)         = 6.873 
#> Posterior Prob P(H0|y)      = 0.873 
#> Posterior Prob P(H1|y)      = 0.127 
#> parameters: p = 1 , k = 1 , a = 0 , prior P(H0) = 0.5 
#> alternative hypothesis: stationary process with maintained trend (rho in (a, 1))
#> 
#> Conclusion: Strong evidence in favor of Unit Root (H0: rho = 1).
summary(res_rw)
#> =========================================================
#>       BAYESIAN UNIT ROOT TEST WITH MAINTAINED TREND      
#>               (Chaturvedi & Kumar, 2005)                
#> =========================================================
#> 
#> Data Series         :  y_rw 
#> Polynomial Trend (p):  1 
#> Augmentation Lags(k):  1 
#> Prior Interval (a,1): (0, 1)
#> Prior P(H0)         :  0.5 
#> 
#> --- HYPOTHESIS TESTING RESULTS ---
#> Posterior Odds Ratio (B01) : 6.87307
#> Bayes Factor (BF01)        : 6.87307
#> Posterior Probability P(H0): 0.87298
#> Posterior Probability P(H1): 0.12702
#> 
#> --- POSTERIOR SUMMARY FOR RHO (UNDER H1) ---
#>    Mean Std.Dev  Median    2.5%   97.5% 
#> 0.92868 0.04735 0.92956 0.82406 0.99488 
#> 
#> --- EVIDENCE INTERPRETATION (Kass & Raftery Scale) ---
#> Substantial evidence for Unit Root (H0).
plot(res_rw)

Testing a Stationary Process

set.seed(456)
y_stat <- numeric(100)
for (t in 2:100) {
  y_stat[t] <- 0.6 * y_stat[t - 1] + rnorm(1)
}

res_stat <- bayes_ur_test(y_stat, p = 1, k = 1)
print(res_stat)
#> 
#>   Chaturvedi-Kumar Bayesian Unit Root Test with Maintained Trend 
#> 
#> data:   y_stat 
#> Posterior Odds Ratio (B01) = 0.0149 
#> Bayes Factor (BF01)         = 0.0149 
#> Posterior Prob P(H0|y)      = 0.01468 
#> Posterior Prob P(H1|y)      = 0.9853 
#> parameters: p = 1 , k = 1 , a = 0 , prior P(H0) = 0.5 
#> alternative hypothesis: stationary process with maintained trend (rho in (a, 1))
#> 
#> Conclusion: Strong evidence in favor of Stationarity (H1: rho < 1).
summary(res_stat)
#> =========================================================
#>       BAYESIAN UNIT ROOT TEST WITH MAINTAINED TREND      
#>               (Chaturvedi & Kumar, 2005)                
#> =========================================================
#> 
#> Data Series         :  y_stat 
#> Polynomial Trend (p):  1 
#> Augmentation Lags(k):  1 
#> Prior Interval (a,1): (0, 1)
#> Prior P(H0)         :  0.5 
#> 
#> --- HYPOTHESIS TESTING RESULTS ---
#> Posterior Odds Ratio (B01) : 0.01490
#> Bayes Factor (BF01)        : 0.01490
#> Posterior Probability P(H0): 0.01468
#> Posterior Probability P(H1): 0.98532
#> 
#> --- POSTERIOR SUMMARY FOR RHO (UNDER H1) ---
#>    Mean Std.Dev  Median    2.5%   97.5% 
#> 0.60115 0.10115 0.59797 0.40203 0.80396 
#> 
#> --- EVIDENCE INTERPRETATION (Kass & Raftery Scale) ---
#> Strong evidence for Stationarity (H1).
plot(res_stat)

References

Chaturvedi, A., & Kumar, J. (2005). Bayesian unit root test for model with maintained trend. Statistics & Probability Letters, 74(1), 109–115. doi:10.1016/j.spl.2005.04.044
Schotman, P., & van Dijk, H. K. (1991). A Bayesian analysis of the unit root in real exchange rates. Journal of Econometrics, 49(1-2), 195–238. doi:10.1016/0304-4076(91)90038-F

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.