Modern Bayesian StatisticsPart III: high-dimensional modeling

Example 4: Efficient Bayesian inference for FSVmodels

Hedibert Freitas Lopes1

hedibert.org

13a aMostra de EstatısticaIME-USP, October 2018

1Professor of Statistics and Econometrics at Insper, Sao Paulo. 1

Example 4: Efficient Bayesian inference for FSV modelsKastner, Fruhwirth-Schnatter & Lopes (2017)Journal of Computational and Graphical Statistics

For each point in time t = 1, . . . ,T ,

I m observed returns: yt = (y1t , . . . , ymt)′

I r unobserved factors: ft = (f1t , . . . , frt)′

I Volatilities: ht = (hUt , h

Vt ), hU

t = (h1t , . . . , hmt)′ and

hVt = (h1,m+1, . . . , hm+r,t)

′.

Our factor stochastic volatility model is

yt |ft ∼ N(Λft ,Ut)

ft ∼ N(0,Vt)

where

I Factor loadings: Λ is m × r

I Idiosyncratic variance: Ut = diag(exp(h1t , . . . , hmt))

I Factor variance: Vt = diag(exp(hm+1,t , . . . , hm+r,t))

I Log-volatilities:

hit = (1− φi )µi + φihi,t−1 + σiηit i = 1, . . . ,m

hit = φihi,t−1 + σiηit i = m + 1, . . . ,m + r . 2

Shallow/deep interweaving2

Trace plots (10,000) ACF(5,000,000)

Top: Standard samplerMiddle: Shallow interweavingBottom: Deep interweaving

2Yu and Meng (2011) To Center or not to Center: That is not the Question– An Ancillarity-Suffiency Interweaving Strategy (ASIS) for Boosting MCMCEfficiency, Journal of Computational and Graphical Statistics, 20, 531-570. 3

Inefficiency factor (based on Λ11)Estimated inefficiency factors for draws from p(Λ11|y [i ]), where y [i ],i ∈ {1, . . . , 100},denote artificially generated datasets whose underlyingparameters are identical.

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Application to exchange rate data

We analyze exchange rates with respect to EUR.

Data were obtained from the European Central Bank’s StatisticalData Warehouse and ranges from April 1, 2005 to August 6, 2015.

It contains m = 26 daily exchange rates on 2650 days.

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Posterior of factor loadings

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Posterior correlation matrices

R-package factorstochvol containing code to run the samplers described inthe article. Available athttps://cran.rproject.org/package=factorstochvol.

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Final remarks

I Model and prior are equally important

I Monte Carlo methods are here to stay

I Bayesian approach is the same across model complexity

I More flexibility to cycles between exploratory data analysis,modeling and inference

I Bayesian statistical learning in data-enriched environments

I It pays to be Bayes!

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Final remarks

I Model and prior are equally important

I Monte Carlo methods are here to stay

I Bayesian approach is the same across model complexity

I More flexibility to cycles between exploratory data analysis,modeling and inference

I Bayesian statistical learning in data-enriched environments

I It pays to be Bayes!

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