<p>Many economic and financial time series exhibit concurrent changes in level and variability. Thus, when detecting mean shifts in time series, it is essential to account for the changing or smoothly evolving variability, as it confounds the detection ability of the methods. In this study, we introduce a scale-dependent, robust time-varying time-average variance constant (TAVC) estimator integrated with a multiscale moving sum (MOSUM) procedure for detecting mean shifts in time series data. We establish consistency of the time-varying TAVC estimator under mild regularity conditions, such as smoothly varying variance, heavy-tailed, and serially dependent errors. Furthermore, we derive the asymptotic distribution and detection accuracy of MOSUM procedure integrated with TAVC under both the null and alternative hypotheses. Extensive simulation studies show that the proposed method outperforms existing approaches in terms of location accuracy, false discovery control, and correct estimation of the number of change points across diverse noise structures. Empirical applications to U.S. gross national product growth and crude oil return series highlight its ability to uncover economically meaningful structural shifts, such as recession-expansion transitions, market shocks, and geopolitical crises. By combining statistical rigor with computational efficiency, the method offers a practical tool for structural change analysis in macroeconomic and financial data.</p>