In this paper, we study the coverage rate of the confidence interval proposed by Bai (2010) focusing on breaks with small magnitude and explore the potential cause of undercoverage. Following Elliott and Muller (2007), we propose the construction of confidence intervals with correct coverage rate by inverting a test statistic that is invariant to the magnitude of break. We show that our confdence intervals have coverage rates close to the nominal level via Monte Carlo Simulations. Recently Kim (2010) suggested a asymptotically valid confidence intervals based on the limiting distributions he derived in the paper. We compare the relative performances our confindence intervals to that of Kim's (2010) in terms of length of the interval and the coverage rate.