This paper introduces a new estimator of the fractional cointegrating vector between stationary long memory processes that is robust to low-frequency contamination such as level shifts, i.e., structural changes in the means of the series, and deterministic trends. In particular, the proposed medium band least squares (MBLS) estimator uses sample dependent trimming of frequencies in the vicinity of the origin to account for such contamination. Consistency and asymptotic normality of the MBLS estimator are established, a feasible inference procedure is proposed, and rigorous tools for assessing the cointegration strength and testing MBLS against the existing narrow band least squares estimator are developed. Finally, the asymptotic framework for the MBLS estimator is used to provide new perspectives on volatility factors in an empirical application to long-span realized variance series for S&P 500 equities.