large vector autoregressions with stochastic volatility ... · large vector autoregressions with...

Large Vector Autoregressions with Stochastic Volatilityand Flexible Priors

Andrea Carriero1, Todd Clark2, and Massimiliano Marcellino3

1Queen Mary, University of London

2Federal Reserve Bank of Cleveland

3Bocconi University and CEPR

June 2016

Todd Clark (FRBC) Large VARs June 2016 1 / 41

Introduction

Two VAR features helpful for forecasting and structural analysis:

Large variable set

Banbura, Giannone, and Reichlin (2010), Carriero, Clark, andMarcellino (2015), Giannone, Lenza, and Primiceri (2015) and Koop(2013)

Time variation in volatility

Clark (2011), Clark and Ravazzolo (2015), Cogley and Sargent (2005),D’Agostino, Gambetti and Giannone (2013), and Primiceri (2005)


Introduction

Few papers provide approaches for accommodating both features.Recent exceptions:

Koop and Korobilis (2013), Koop, et al. (2016): computationalshortcut using exponential smoothing of volatility

Carriero, Clark, and Marcellino (2016): single volatility factor andspecific prior that permits use of N-W steps


Introduction

Allowing large VARs with homoskedasticity requires symmetry oflikelihood and prior.

Homoskedastic VARs: SUR models w/ the same regressors in eachequation

Symmetry across equations Ñ likelihood has a Kronecker structure ÑOLS estimation equation by equation

With homoskedasticity, large BVARs require a specific prior structure,of conjugate N-W:

The coefficients of each equation feature the same prior variancematrix (up to a constant of proportionality).Priors are correlated across equations, with a correlation structureproportional to Σ.


Introduction

More general priors break symmetry and make large modelscomputationally difficult.

Priors more general than conjugate N-W break the Kroneckerstructure and symmetry.

Examples: prior with Litterman-style cross-variable shrinkage orNormal-diffuse prior

Model needs to be vectorized for estimation

Drawing the VAR coefficients from the conditional posterior involvesa variance matrix of dimension N2 ˆ lags.

SV also breaks symmetry and makes large models difficult

Each equation driven by a different volatility Ñ Model needs to bevectorized

Drawing the VAR coefficients involves a variance matrix of dimensionN2 ˆ lags.


Introduction

We develop a new estimation approach that makes tractable largemodels with asymmetric priors or SV

Algorithm exploits a simple triangularization of the VAR, whichpermits drawing VAR coefficients equation by equation

This reduces the computational complexity for estimating the VARmodel from N6 to N4, greatly speeding up estimation.

The triangularization can easily be inserted in any pre-existingalgorithm for estimation of BVARs.

Example code to be available on Carriero and Marcellino webpages

Estimation of large VARs with SV and flexible priors becomes feasible.


Introduction

Application 1: Structural analysis of BVAR-SV in 125 monthlyvariables

SV estimates: heterogeneity and yet much commonality

Impulse responses for a policy shock

Application 2: Out-of-sample forecasts from BVAR-SV in 20 monthlyvariables

Larger model forecasts better than smaller model

SV improves accuracy of both density and point forecasts


Outline

1 BVAR-SV specification and impediments to large models

2 Our estimation method for large BVARs

3 Application 1: Structural analysis with large BVAR-SV

4 Application 2: Out-of-sample forecasting

5 Conclusions


BVAR-SV Model

With N variables:

yt “ Π0 ` ΠpLqyt´1 ` vt

vt “ A´1Λ0.5t εt , εt „ iid Np0, INq; varpvtq ” Σt “ A´1ΛtA

´1

lnλj ,t “ lnλj ,t´1 ` ej ,t , j “ 1, . . . ,N

et „ iid Np0,Φq

Let Xt denote the pNp ` 1q-dimensional vector of regressors in eachequation

Collect parameter blocks and latent states:

Parameters: Θ “ tΠ,A,Φu

Latent states lnλj ,t , t “ 1, . . . ,T , j “ 1, . . . ,N


BVAR-SV Model: standard system estimation

Priors:

vecpΠq „ NpvecpµΠq,ΩΠq

A „ NpµA,ΩAq

Φ „ IW pdΦ ¨ Φ, dΦq

lnλj ,0 „ uninformative Gaussian


BVAR-SV Model: standard system estimation

Posteriors:

vecpΠq|A,ΛT , yT „ NpvecpµΠq,ΩΠq

A|Π,ΛT , yT „ NpµA,ΩAq

Φ|ΛT , yT „ IW ppdΦ ` T q ¨ Φ, dΦ ` T q,

Means and variances of conditional normal distributions takeGLS-based form, combining prior moments and likelihood moments

Gibbs sampler for ppΘ,ΛT |yT q:

Draw from ppΘ|ΛT , yT q using conditional posteriors above

Draw from ppΛT |Θ, yT q using the mixture of normals approximationand multi-move algorithm of Kim, Shepard and Chib (1998)


BVAR-SV Model: impediments to standard systemestimation with a large model

Sampling the VAR coefficients involves drawinga NpNp ` 1q´dimensional vector rand, and computing

vecpΠmq “ ΩΠ

#

vec

˜

Tÿ

t“1

Xty1tΣ´1t

¸

` Ω´1Π vecpµ

Πq

+

`cholpΩΠqˆrand

(1)This calculation requires: i) computing ΩΠ by inverting

Ω´1Π “ Ω´1

Π `

Tÿ

t“1

pΣ´1t b XtX

1tq;

ii) computing its Cholesky factor cholpΩΠq; iii) multiplying thematrices obtained in i) and ii) by the vector in the curly brackets of(1) and the vector rand, respectively.Each operation requires OpN6q elementary operations, making thetotal computational complexity to draw Πm equal 4ˆ OpN6q.


Homoskedastic BVARs: similar impediments with flexiblepriors

Model:

yt “ Π0 ` ΠpLqyt´1 ` vt , vt „ iid Np0,Σq

Consider a general N-W prior:

vecpΠq „ NpvecpµΠq,ΩΠq; Σ „ IW pdΣ ¨ Σ, dΣq

Posterior:

vecpΠq|Σ, y „ NpvecpµΠq,ΩΠq; Σ|Π, y „ IW ppdΣ ` T q ¨ Σ, dΣ ` T q

Ω´1Π “ Ω´1

Π `

Tÿ

t“1

pΣ´1 b XtX1tq

Impediment to large models: Computational requirements withsystem variance ΩΠ that also exist with SV formulation


Homoskedastic BVARs: standard approach to making largemodels tractable

Following literature on large VARs, make the prior conjugate (andsymmetric) N-W.

vecpΠq|Σ „ NpvecpµΠq,Σb Ω0q

Prior for Π is conditional on Σ

Posterior variance simplifies and speeds up calculations:

Ω´1Π “ Σ´1 b

#

Ω´10 `

Tÿ

t“1

XtX1t

+

Kronecker structure permits manipulating the two terms in theKronecker product separately, reducing the computational complexityto N3


Homoskedastic BVARs: standard approach to making largemodels tractable

The conjugate (and symmetric) N-W form comes with someunappealing restrictions.

Issues discussed by Rothenberg (1963), Zellner (1973), Kadiyala andKarlsson (1993, 1997), and Sims and Zha (1998)

Rules out asymmetry in the prior across equations; coefficients ofeach equation feature the same prior variance matrix Ω0

Rules out one aspect of the Litterman (1986) prior: extra shrinkageon “other” lags vs. “own” lags

Σb Ω0 implies prior beliefs correlated across the equations of thereduced form VAR

Sims and Zha (1998) specify a prior featuring independence among thestructural equations, but does not achieve computational gains for anasymmetric prior on the reduced form.


Our estimation method for large VARs

Key to approach: In the Gibbs sampler, the posterior of the VARcoefficients Π is conditional on A and ΛT .

πpiq = the vector of coefficients for equation i contained in row i ofΠ, for the intercept and coefficients on lagged yt

Consider the decomposition vt “ A´1Λ0.5t εt :

»

—

—

–

v1,t

v2,t

...vN,t

fi

ffi

ffi

fl

“

»

—

—

–

1 0 ... 0a˚2,1 1 ...

... 1 0a˚N,1 ... a˚N,N´1 1

fi

ffi

ffi

fl

»

—

—

–

λ0.51,t 0 ... 0

0 λ0.52,t ...

... ... 00 ... 0 λ0.5

N,t

fi

ffi

ffi

fl

»

—

—

–

ε1,t

ε2,t

...εN,t

fi

ffi

ffi

fl

,



Rewrite the VAR:

y1,t “ πp0q1 `

Nÿ

i“1

pÿ

l“1

πpiq1,l yi ,t´l ` λ

0.51,t ε1,t

y2,t “ πp0q2 `

Nÿ

i“1

pÿ

l“1

πpiq2,l yi ,t´l ` a˚2,1λ

0.51,t ε1,t ` λ

0.52,t ε2,t

...with the generic equation p˚q for variable j :

yj ,t´pa˚j ,1λ

0.51,t ε1,t`...`a˚j ,,j´1λ

0.5j´1,tεj´1,tq “ π

p0qj `

Nÿ

i“1

pÿ

l“1

πpiqj ,l yi ,t´l`λj ,tεj ,t

Consider estimating these equations in order from j “ 1 to j “ N

In the conditional posterior, the dependent variable of p˚q is known.

Dependent variablej = yj ´ aˆ the estimated residuals of all theprevious j ´ 1 equations.



Let y˚j ,t ” yj ,t ´ pa˚j ,1λ

0.51,t ε1,t ` ...` a˚j ,,j´1λ

0.5j´1,tεj´1,tq

The model is a system of standard generalized linear regression modelswith indep. Gaussian disturbances with mean 0 and variance λj ,t :

y˚j ,t “ πp0qj `

Nÿ

i“1

pÿ

l“1

πpiqj ,l yi ,t´l ` λj ,tεj ,t ,

Factorize the full conditional posterior distribution of Π:

ppΠ|A,ΛT , yq “ ppπpNq|πpN´1q, πpN´2q, . . . , πp1q,A,ΛT , yq

ˆppπpN´1q|πpN´2q, . . . , πp1q,A,ΛT , yq...

ˆppπp1q|A,ΛT , yq,



Our conditional posterior for the VAR coefficients:

ppΠtju|Πt1:j´1u,A,ΛT , yq9ppy |Πtju,Πt1:j´1u,A,ΛT qppΠ

tju|Πt1:j´1uq

ppy |Πtju,Πt1:j´1u,A,ΛT q “ the likelihood of equation j

ppΠtju|Πt1:j´1uq “ prior on the j-th equation, conditional on theprevious equations

With typical priors, the equation priors are independent:ppΠtju|Πt1:j´1uq “ ppΠtjuq

W/o independence, the moments of ppΠtju|Πt1:j´1uq can be obtainedfrom the joint prior.



Our conditional posterior for the VAR coefficients:

Draw the coefficient matrix Π in separate blocks Πtju obtained from:

Πtju|Πt1:j´1u,A,ΛT , y „ NpµΠtju ,ΩΠtjuq

µΠtju “ ΩΠtju

#

Ω´1ΠtjuµΠtju `

Tÿ

t“1

Xj ,tλ´1j ,t y

˚1j ,t

+

Ω´1Πtju “ Ω´1

Πtju `

Tÿ

t“1

Xj ,tλ´1j ,t X

1j ,t ,

where Ω´1Πtju and µ

Πtju “ the prior moments on the j-th equation,given by the j-th column of µ

Πand the j-th block on the diagonal of

Ω´1Π

Here Ω´1Π is block diagonal, as typical; this can be relaxed



Computational costs (not much):

Although we break the conditional posterior for Π into pieces, we arestill drawing from the conditional posterior for Π.

Our triangularization approach produces draws numerically identicalto those that would be obtained using system-wide estimation.

For the VAR coefficients, the ordering of variables does not matter.

Existing BVAR and BVAR-SV code can easily be modified to draw Πwith the triangularized system.



Computational benefits (significant):

Ω´1Πtju is of dimension pNp ` 1q square Ñ its manipulation only

involves operations of order OpN3q

With N equations, obtaining a draw for Π makes the totalcomputational complexity of order OpN4q

Compared to a standard algorithm, the complexity savings is N2

CPU savings rise quickly (more than quadratic rate) with the numberof variables.

With 20 variables and 13 lags of monthly data, the estimation of themodel using the traditional system-wide algorithm was about 261times slower.



Convergence and mixing

In a given unit of time, our triangular algorithm will always producemany more draws than the traditional system-wide algorithm.

This speed advantage will improve the precision of MCMC estimates:

Many more draws to use in averagesOr increased skip-sampling (preferable with large models) to reducecorrelation across retained draws


Application 1: large structural VAR with SV

Specification: BVAR-SV(13) in 125 monthly variables from thedataset of McCracken and Ng (2015)

Extending constant volatility analyses of (FAVAR) Bernanke, Boivinand Eliasz (2005) and (large BVAR) Banbura, Giannone, and Reichlin(2010)

VAR coefficient prior (asymmetric): independent Normal-Wishartprior, Minnesota form, with cross-variable shrinkage

Assessments:

Estimates of volatilities and comovement

Responses to monetary policy shock

For identification, the federal funds rate is ordered after slow-movingand before fast-moving variables.



Computation:

Model includes 203,250 VAR coefficients

On a 3.5 GHz Intel Core i7 processor, our algorithm produces 5000draws (after discarding 500 burning in) in just above 7 hours

The traditional system-based algorithm would be extremely difficult,just for memory requirements: the covariance matrix of the 203,250coefficients would require about 330 GB of RAM



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Figure 8: Posterior distribution of volatilities (diagonal elements of Σt ), fast variables.



Results on volatilities:

Substantial homogeneity in the volatility patterns of variablesbelonging to the same group, such as IP components

Heterogeneity across groups of variables

Principal component analysis on the posterior mean of Φ indicatesmacroeconomic volatility is primarily driven by two shocks

The Great Moderation is evident in most series; the effects of therecent crisis are more heterogeneous.

Volatilities of real variables and financial variables go back to lowerlevels after the peak associated with the crisis.

Volatilities of inflation measures have tended to remain elevatedfollowing the crisis.



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HWI

12 24 36 48 60

10 -3

-2

-1

0HWIURATIO

12 24 36 48 60

10 -3

-3-2-10

CLF16OV

12 24 36 48 60

10 -3

-8-6-4-20

CE16OV

12 24 36 48 600

0.2

0.4

0.6UNRATE

12 24 36 48 600

0.5

1

UEMPMEAN

12 24 36 48 600

0.02

0.04UEMPLT5

12 24 36 48 600

0.020.040.060.08

UEMP5TO14

12 24 36 48 600

0.050.10.15

UEMP15OV

12 24 36 48 600

0.05

0.1

UEMP15T26

12 24 36 48 600

0.1

0.2UEMP27OV

12 24 36 48 60

00.050.10.15

CLAIMSx

12 24 36 48 60

10 -3

-10

-5

0PAYEMS

12 24 36 48 60

-0.02

-0.01

0USGOOD

12 24 36 48 60

-0.02-0.01

00.01

CES1021000001

12 24 36 48 60-0.04

-0.02

0USCONS

12 24 36 48 60

10 -3

-15-10-50

MANEMP

12 24 36 48 60

-0.02

-0.01

0DMANEMP

12 24 36 48 60

10 -3

-8-6-4-20

NDMANEMP

12 24 36 48 60

10 -3

-8-6-4-20

SRVPRD

12 24 36 48 60

10 -3

-10

-5

0USTPU

12 24 36 48 60

10 -3

-10

-5

0USWTRADE

12 24 36 48 60

10 -3

-8-6-4-20

USTRADE

12 24 36 48 60

10 -3

-6-4-20

USFIRE

12 24 36 48 60

10 -3

-10

-5

0USGOVT

12 24 36 48 60

-0.1

-0.05

0CES0600000007

12 24 36 48 60

-0.1-0.05

0

AWOTMAN

12 24 36 48 60-0.15-0.1

-0.050

AWHMAN

12 24 36 48 60

10 -3

-10-50

BUSINVx

12 24 36 48 60

10 -3

-5051015

ISRATIOx

12 24 36 48 60

10 -3

-1

0

1CES0600000008

12 24 36 48 60

10 -3

-2

0

2CES2000000008

12 24 36 48 60

10 -3

-1

0

1CES3000000008

12 24 36 48 60

10 -3

-3-2-101

PPIFGS

12 24 36 48 60

10 -3

-2

0

2PPIFCG

12 24 36 48 60

10 -3

-2

0

2

PPIITM

12 24 36 48 60-0.01

0

0.01PPICRM

12 24 36 48 60

-0.02

0

0.02OILPRICEx

12 24 36 48 60

10 -3

-5051015

PPICMM

12 24 36 48 60

10 -3

-1

0

1CPIAUCSL

12 24 36 48 60

10 -3

-2-101

CPIAPPSL

12 24 36 48 60

10 -3

-20246

CPITRNSL

12 24 36 48 60

10 -4

-10-505

CPIMEDSL

12 24 36 48 60

10 -3

-1012

CUSR0000SAC

12 24 36 48 60

10 -3

-1012

CUUR0000SAD

12 24 36 48 60

10 -4

-15-10-505

CUSR0000SAS

12 24 36 48 60

10 -3

-1

0

1CPIULFSL

12 24 36 48 60

10 -3

-101

CUUR0000SA0L2

12 24 36 48 60

10 -3

-1

0

1CUSR0000SA0L5

12 24 36 48 60

10 -4

-10-505

PCEPI

12 24 36 48 60

10 -3

-1

0

1DDURRG3M086SBEA

12 24 36 48 60

10 -3

-2

0

2DNDGRG3M086SBEA

12 24 36 48 60

10 -4

-10-505

DSERRG3M086SBEA

Figure 9: Impulse responses to a monetary policy shock: slow variables.

12 24 36 48 600

0.5

1FEDFUNDS

12 24 36 48 60

-0.06-0.04-0.02

0HOUST

12 24 36 48 60

-0.06-0.04-0.02

0HOUSTNE

12 24 36 48 60-0.08-0.06-0.04-0.02

00.02

HOUSTMW

12 24 36 48 60-0.06-0.04-0.02

0HOUSTS

12 24 36 48 60-0.08-0.06-0.04-0.02

00.02

HOUSTW

12 24 36 48 60-0.03-0.02-0.01

0

AMDMNOx

12 24 36 48 60-0.06

-0.04

-0.02

0AMDMUOx

12 24 36 48 60-0.03-0.02-0.01

00.01

S&P 500

12 24 36 48 60

-0.02

0

0.02S&P: indust

12 24 36 48 60-0.1

0

0.1S&P div yield

12 24 36 48 60

-0.020

0.020.040.06

S&P PE ratio

12 24 36 48 60

00.010.020.03

EXSZUSx

12 24 36 48 60-0.01

00.010.020.03

EXJPUSx

12 24 36 48 60

-0.04-0.03-0.02-0.01

EXUSUKx

12 24 36 48 60

10 -3

-50510

EXCAUSx

12 24 36 48 60

00.20.40.60.8

CP3Mx

12 24 36 48 60

00.20.40.6

TB3MS

12 24 36 48 60

00.20.40.6

TB6MS

12 24 36 48 60

00.20.40.6

GS1

12 24 36 48 600

0.2

0.4

GS5

12 24 36 48 60

0.10.20.30.4

GS10

12 24 36 48 600.10.20.30.4

AAA

12 24 36 48 600.10.20.30.40.5

BAA

12 24 36 48 60-0.4

-0.2

0COMPAPFFx

12 24 36 48 60

-0.4

-0.2

0TB3SMFFM

12 24 36 48 60

-0.4

-0.2

0TB6SMFFM

12 24 36 48 60

-0.4

-0.2

0T1YFFM

12 24 36 48 60-0.8-0.6-0.4-0.20

T5YFFM

12 24 36 48 60-0.8-0.6-0.4-0.20

0.2T10YFFM

12 24 36 48 60

-0.8-0.6-0.4-0.20

0.2AAAFFM

12 24 36 48 60-0.8-0.6-0.4-0.20

0.2BAAFFM

12 24 36 48 60

10 -3

-4

-2

0

2M1SL

12 24 36 48 60

10 -3

-1

0

1M2SL

12 24 36 48 60-0.02

-0.01

0M2REAL

12 24 36 48 60

10 -3

-4-2024

AMBSL

12 24 36 48 60-0.01

00.010.02

TOTRESNS

12 24 36 48 60

-0.1

0

0.1

NONBORRES

12 24 36 48 60

10 -3

-4

-2

0

BUSLOANS

12 24 36 48 60

10 -3

-3-2-101

REALLN

12 24 36 48 60

10 -3

-2

0

2NONREVSL

12 24 36 48 60

10 -3

-4

-2

0CONSPI

12 24 36 48 60

10 -3

-4

-2

0

MZMSL

12 24 36 48 60

10 -3

-10-505

DTCOLNVHFNM

12 24 36 48 60

10 -3

-6-4-2024

DTCTHFNM

12 24 36 48 60

10 -3

-20246

INVEST

12 24 36 48 60-2

-1

0NAPMPI

12 24 36 48 60-1.5-1

-0.50

NAPMEI

12 24 36 48 60-1.5-1

-0.50

NAPM

12 24 36 48 60-2

-1

0NAPMNOI

12 24 36 48 60-1.5-1

-0.50

0.5NAPMSDI

12 24 36 48 60

-1-0.50

0.5NAPMII

12 24 36 48 60-1.5-1

-0.50

0.5NAPMPRI

Figure 10: Impulse responses to a monetary policy shock: fast variables.



Results on impulse responses to FFR shock:

The patterns of impulse responses align with typical structuralmodels: significant deterioration in real activity, very limited pricepuzzle, a significant deterioration in stock prices, and a less thanproportional increase in the entire term structure

Inclusion of SV does not affect substantially the VAR coefficientestimates with respect to Banbura, Giannone and Reichlin (2010)

But it matters for inference and time variation in variancecontributions and shares


Application 2: forecasts of 20 monthly variables

Variables in baseline specification

Real Personal Income PPI: CommoditiesReal PCE PCE Price IndexReal M&T Sales Federal Funds RateIP Index Housing StartsCapacity Utilization: Manufacturing S&P 500Unemployment Rate U.S.-U.K. exchange rateAll Employees: Total nonfarm Spread, 1y Treasury-Fed fundsHours: Manufacturing Spread, 10y Treasury-Fed fundsAvg. Hourly Earnings: Goods Spread, Baa-Fed fundsPPI: Finished Goods ISM: New Orders Index

Samples:

Estimation sample begins with 1960:3

Forecast evaluation sample is 1970:3 to 2014:5.



Four models:

3-variable BVAR, homoskedastic: growth rate of IP (∆ ln IP), PCEinflation (∆ lnPECEPI ), fed funds rate (FFR)

3-variable BVAR-SV

20-variable BVAR, homoskedastic

20-variable BVAR-SV



Drivers of forecast gains:

Direct effects:

SV improves density forecasts by capturing time variation in errorvariances.Use of a larger dataset should improve point forecasts by improving theconditional means.

Interactions:

A better point forecast improves the density forecast by bettercentering the predictive density.SV improves the point forecasts by making parameter estimates moreefficient (GLS).This efficiency also helps the predictive densities.



step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.98

1

1.02

1.04

1.06

1.08

1.1

RMSE (ratio to benchmark)INDPRO

step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

RMSE (ratio to benchmark)PCEPI

step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

RMSE (ratio to benchmark)FEDFUNDS

homo20homo3SV3SV20

Figure 14: Point forecasts: relative RMSE of di§erent models. Black line (benchmark, marker:

crosses) is a homoschedastic VAR with 20 variables, red line (marker: squares) is a homoschedastic

VAR with 3 variables, blue line (marker: circles) is heteroschedastic VAR with 3 variables, purple

line (marker: diamonds) is heteroschedastic VAR with 20 variables.


Application 2: forecasts of 20 monthly variablesstep-ahead

1 2 3 4 5 6 7 8 9 10 11 120.98

1

1.02

1.04

1.06

1.08

1.1


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08


homo20homo3SV3SV20







step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.98

1

1.02

1.04

1.06

1.08

1.1


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.96

0.97

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08


homo20homo3SV3SV20







step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

mean log-score (deviation from benchmark)INDPRO

step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

mean log-score (deviation from benchmark)PCEPI

step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

mean log-score (deviation from benchmark)FEDFUNDS

homo20homo3SV3SV20

Figure 15: Point forecasts: Log-score gains of di§erent models vs benchmark. Black line (bench-

mark, marker: crosses) is a homoschedastic VAR with 20 variables, red line (marker: squares) is a

homoschedastic VAR with 3 variables, blue line (marker: circles) is heteroschedastic VAR with 3

variables, purple line (marker: diamonds) is heteroschedastic VAR with 20 variables.


Application 2: forecasts of 20 monthly variablesstep-ahead

1 2 3 4 5 6 7 8 9 10 11 12-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


homo20homo3SV3SV20







step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2


step-ahead1 2 3 4 5 6 7 8 9 10 11 12

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


homo20homo3SV3SV20






Conclusions

We develop a new approach that makes feasible fully Bayesianinference of large BVARs with SV.

Also makes feasible the use of asymmetric priors (independent N-Wpriors) with SV or constant volatility, in large models

The method is based on a straightforward triangularization of thesystem, and it is very simple to implement by modifying existing codefor drawing VAR coefficients.

The algorithm ensures computational gains of order N2 and yieldsbetter mixing and convergence properties.


large vector autoregressions with stochastic volatility ... · large vector autoregressions with...

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