The estimation and forecasting of dynamically varying covariance matrices plays a crucial role in a host of different economic and financial decision making processes. This paper proposes multivariate volatility models that use the asymptotic theory of high-frequency covariance estimates to endogenously deal with time-variation in the precision of the estimates, by allowing the degree of measurement error attenuation to vary over time. The models allow for increased responsiveness when recent estimates are precise, and limit the impact of noisy high-frequency estimates on forecasts. I illustrate the working of the models in two financial applications. First, I show in a portfolio allocation setting that the forecasts from the new models result in more efficient portfolios and significantly reduced turnover for which a risk-averse investor would be willing to pay up to 180 basis points per year. Second, realized beta forecasts from the dynamic attenuation models lead to improved hedging decisions.