Title: Multiscale Hidden Markov Models For Covariance Prediction. Abstract: This paper presents a novel variant of hierarchical hidden Markov models (HMMs), the multiscale hidden Markov model (MSHMM), and an associated spectral estimation and prediction scheme that is consistent, finds global optima, and is computationally efficient. Our MSHMM is a generative model of multiple HMMs evolving at different rates where the observation is a result of the additive emissions of the HMMs. While estimation is relatively straightforward, prediction for the MSHMM poses a unique challenge, which we address in this paper.  Further, we show that spectral estimation of the MSHMM outperforms standard  methods of predicting the asset covariance of stock prices, a widely addressed problem that is multiscale, non-stationary, and requires processing huge amounts of data.