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Sequential learning in dynamic graphical model Hao Wang, Craig Reeson Department of Statistical Science, Duke University Carlos Carvalho Booth School of Business, The University of Chicago

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Motivating example: forecasting stock return covariance matrix Observe p- vector stock return time series Interested in forecast conditional covariance matrix WHY? Buy dollar stock i Expected return Risks

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Daily return of a portfolio (S&P500)

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How to forecast: index model Common index Uncorrelated error terms Covariance structure Assumption: stocks move together only because of common movement with indexes (e.g. market)

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Uncorrelated residuals? An exploratory analysis on 100 stocks Possible signals Index explains a lots

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Seeking structure to relax uncorrelated assumption Perhaps too simple Perhaps too complex Sparse signals

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Structures: Gaussian graphical model Graph exhibits conditional independencies ~ missing edges International exchange rates example, p=11 Carvalho, Massam, West, Biometrika, 2007 No edge:

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Dynamic matrix-variate models Example: Core class of matrix-variate DLMs Multivariate stochastic volatility: Variance matrix discounting model for Conjugate, closed-form sequential learning/updating and forecasting (Quintana 1987; Q&W 1987; Q et al 1990s) Multivariate stochastic volatility: Variance matrix discounting model for Conjugate, closed-form sequential learning/updating and forecasting (Quintana 1987; Q&W 1987; Q et al 1990s)

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-- Global structure: stochastic change of indexes affecting return of all assets, e.g. SV model -- Local structure: local dependences not captured by index, e.g. graphical model -- Dynamic structure: adaptively relating low dimension index to high dimension returns e.g. DLM

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Random regression vector and sequential forecasting 1-step covariance forecasts : Mild assumption: 1-step covariance forecasts : Variance from graphical structured error terms Variance from regression vector Analytic updates

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Graphical model adaptation AIM: historical data gradually lose relevance to inference of current graphs Residual sample covariance matrices

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Graphical model uncertainty Challenges: Interesting graphs? graphs Graphical model search Jones et al (2005) Stat Sci: static models MCMC Metropolis Hasting Shotgun stochastic search Scott & Carvalho (2008): Feature inclusion Challenges: Interesting graphs? graphs Keys: >> Analytic evaluation of posterior probability of any graph …

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Sequential model search Time t-1, N top graphs At time t, evaluate posterior of top N graphs from time t-1 Random choose one graph from N graphs according to their new posteriors Shotgun stochastic search Stop searching when model averaged covariance matrix estimates does not differ much between the last two steps, and proceed to time t+1

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100 stock example Monthly returns of randomly selected 100 stocks, 01/1989 – 12/2008 Two index model Capital asset pricing model: market Fama-French model: market, size effect, book-to-price effect, about 60 monthly moving window How sparse signals help?

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Time-varying sparsity

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Performance of correlation matrix prediction

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Performance on portfolio optimization

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Bottom line For either set of regression variables we chose, we will perhaps be better off by identifying sparse signals than assuming uncorrelated/fully correlated residuals

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