sequential learning in dynamic graphical model hao wang, craig reeson department of statistical...
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Sequential learning in dynamic graphical model
Hao Wang, Craig ReesonDepartment of Statistical Science, Duke University
Carlos CarvalhoBooth School of Business, The University of Chicago
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
Daily return of a portfolio (S&P500)
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)
Uncorrelated residuals? An exploratory analysis on 100 stocks
Possible signals Index explains a lots
Seeking structure to relax uncorrelated assumption
Perhaps too simple
Perhaps too complex
Sparse signals
Structures: Gaussian graphical model
Graph exhibits conditional independencies ~ missing edges
International exchange rates example, p=11Carvalho, Massam, West, Biometrika, 2007
No edge:No edge:
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)
-- 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
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
Graphical model adaptation
• AIM: historical data gradually lose relevance to inference of current graphs
• Residual sample covariance matrices
Graphical model uncertainty
Challenges: Interesting graphs?
graphsGraphical model search
Jones et al (2005) Stat Sci: static modelsMCMC Metropolis Hasting Shotgun stochastic search
Scott & Carvalho (2008): Feature inclusion
Challenges: Interesting graphs?
graphs
Keys:
>> Analytic evaluation of posterior probability of any graph …
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
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?
Time-varying sparsity
Performance of correlation matrix prediction
Performance on portfolio optimization
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