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
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?