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There are some standard models for forecasting volatility (e.g., GARCH) and for forecasting returns (e.g., factor models). What kind of standard models exist for forecasting future correlation between securities?

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One common "model" is to assume the correlation to be constant, such as in a CCC-MVGARCH model. If you want a review of different multivariate GARCH models, you could look at:

Silvennoinen and Täräsvirta 2009, Multivariate Garch models, in Handbook of financial time series.

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