Sequential learning in dynamic graphical model

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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 (SP500)
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How to forecast index model
Common index
Uncorrelated error terms
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
Index explains a lots
Possible signals
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Seeking structure to relax uncorrelated
assumption
Perhaps too simple
Perhaps too complex
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Structures Gaussian graphical model

• Graph exhibits conditional independencies
• missing edges

International exchange rates example,
p11 Carvalho, Massam, West, Biometrika, 2007
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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 QW 1987 Q et al 1990s)
Multivariate stochastic volatility
Variance matrix discounting model for
Conjugate, closed-form sequential
learning/updating and forecasting
(Quintana 1987 QW 1987 Q et al 1990s)
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Random regression vector and sequential
forecasting
1-step covariance forecasts
Mild assumption
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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
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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 t1

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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