Lenza.pdf

Prior selection for vector autoregressions

Domenico Giannone, Universitè Libre de Bruxelles Michele Lenza, European Central Bank Giorgio Primiceri, Northwestern University ECARES@20 Bruxelles, May 2012

Vector autoregression


VAR:

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)  




Flexible multivariate model Bridge between reduced-form and structural models

Problem: very densely parameterized   

High estimation uncertainty Overfitting Poor out-of-sample forecasting performance

Forecasting with VARs: an example




Quarterly macroeconomic data for the US (from 1960)       

GDP Consumption Investment Hours Wages GDP deflator Federal funds rate




p=5




Total #parameters=280  7x5x7 autoregressive coefficients + 7 constants + (7x8)/2 covariance of residuals

GDP growth

GDP growth and VAR forecasts (1-step ahead )

GDP growth and VAR forecast

Large information: curse of dimensionality


Also VAR models of moderate size can incur in serious issues with estimation error




However, recent developments in econometric theory and empirics have highlighted the relevance of looking at large information  Use of large cross-sections of data  Forni/Giannone/Hallin/Lippi/Reichlin and Stock/Watson

Bayesian VARs




Litterman (1980) and Doan, Litterman and Sims (1984)    

Informative priors Shrink towards naïve models Reduce estimation uncertainty Improve forecasting performance




Until very recently, BVARs remained a niche technique




Problems  - large information (too much shrinkage?) - not much guidance for the choice of priors - perceived as subjective

Bayesian VARs




De Mol, Giannone and Reichlin (2008)  Bayesian shrinkage and large information: turning the curse of dimensionality into a blessing  Typical “economic” comovement (think of macro and financial data) data conjure against priors and even large degree of shrinkage (needed to control estimation uncertainty) does not prevent extraction of sample information  Link to Principal Components/Factor models




Banbura, Giannone and Reichlin (2010) 

 

Application of the idea to VARs

Solution of the issues with large information! Still, lack of guidance for what concerns setting of the priors

Main points of this paper




Treat the informativeness of the prior as an unknown parameter  Conduct formal inference on it




Accurate out-of-sample forecasting performance  Point forecasts  Density forecasts




BVAR can be used for structural analysis  More accurate IRF than VAR

(Some) Related literature


Recent renewed interest in BVARs  Banbura, Giannone and Reichlin (2010)  Carriero, Kapetanios and Marcellino (2010a, b)  Clark (2010)  Christoffel, Coenen and Warne (2011)  Koop (2010)  Lenza, Pill and Reichlin (2010)  Stock and Watson (2009)  Wright (2010)  …




BVARs with DSGE priors  Del Negro and Schorfheide (2004)  Del Negro, Schorfheide, Smets and Wouters (2006)




Methodology  Very large literature in statistics on hierarchical models  Lopes, Moreira and Schmitt (1999)

Outline




BVARs




Forecasting with hierarchical models




Results  Macroeconomic forecasting  Structural VARs and impulse responses

BVAR

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)

BVAR

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ) Minnesota prior on

[C,B ,...,B ,Σ] 1

p

BVAR

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ) 1. Minnesota prior  Litterman (1980 and 1986)

2. Inverse-Wishart prior on

N-IW prior

Σ

3. Sum-of-coefficients prior  Doan, Litterman and Sims (1984)

4. Single-unit-root prior  Sims (1993)

1. A base prior: the Minnesota prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Shrink coefficients towards naïve model:




More precisely: 



y t = c + y t−1 + εt

[ ] 1 Σ V [(B ) ]= φ ⋅ s Ψ 1 Σ cov[(B ) , (B ) ]= φ ⋅ s Ψ

E (Bs )ij = 1 if s = 1 and i = j 2

s ij

ii

2

jj



2

s ij

r hm

ih

2

jj

if m = j and r = s

2. A simple prior on the covariance matrix

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Conjugate prior:

Σ ~ IW (Ψ, n + 2)

 E(Σ) = Ψ


Combined with MN: N-IW prior

Hyperparameters




Summary of hyperparameters to be chosen  ϕ: std of MN prior  µ: std of SoC prior  Ψ: scale of IW prior




How to chose them?

Outline




BVARs




Forecasting with hierarchical models




Results  Macroeconomic forecasting  Structural VARs and impulse responses

Hierarchical model





Model  Likelihood:

p(Y | θ )

 Prior:

pλ (θ )

Our approach  Treat λ as an additional parameter:

p(θ | λ ) ≡ pλ (θ )

 Evaluate posterior of λ

p(λ |Y ) ∝ p(Y | λ ) p(λ ) Marginal likelihood

Hyperprior

Hyperparameters and hyperpriors




Summary of hyperparameters λ  ϕ: std of MN prior  µ: std of SoC prior  Ψ: scale of IW prior




Hyperpriors  ϕ ~ G ( mode = 0.2, std = 0.4)  µ ~ G ( mode = 1, std = 1)  Ψii ~ IG ( mode = .022, std = ∞ ), i = 1,…,n




Results very similar with flat hyperpriors

Three remarks

p(λ |Y ) ∝ p(Y | λ ) p(λ ) 1. If you just look at the posterior mode under flat hyperprior  Empirical Bayes  MLE of random coefficient model T

2.

p(Y | λ) = p(y1 | λ )

t−1 p y | y ∏ ( t , λ) t= 2

 Relation with forecasting

3.

p(Y | λ) =

∫ p(Y | θ ) p(θ | λ)

dθ

Θ



Available in closed form for VARs (with conjugate prior)

Outline




BVARs




Forecasting with hierarchical models




Results  Macroeconomic forecasting  Structural VARs and impulse responses

22-variable variable BVAR

Posterior of hyperparameter (λ)

RATS

7-variable BVAR 3-variable BVAR

22-variable variable BVAR

Posterior and prior of hyperparameter (λ)

Hyperprior 7-variable BVAR

3-variable BVAR

GDP growth

GDP growth and VAR forecast

GDP growth and BVAR forecast

Accuracy of point forecasts

Mean square forecast errors

Small OLS

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium OLS

13.07 10.74 23.03

BVAR

Large OLS

BVAR

8.97 77.32

9.76

GDP Deflator

2.33

1.54

3.65

1.52 15.14

1.31

Federal Funds Rates

1.67

1.11

2.25

1.08

6.62

1.08

Real GDP

5.15

4.21 17.35

3.65 152.5

4.99

GDP Deflator

2.28

1.60

4.94

1.56 54.48

1.14

Federal Funds Rates

0.63

0.37

0.94

0.32 64.34

0.40

Accuracy of point forecasts

Mean square forecast errors

Small OLS

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium OLS

13.07 10.74 23.03

BVAR

Large OLS

BVAR

8.97 77.32

9.76

GDP Deflator

2.33

1.54

3.65

1.52 15.14

1.31

Federal Funds Rates

1.67

1.11

2.25

1.08

6.62

1.08

Real GDP

5.15

4.21 17.35

3.65 152.5

4.99

GDP Deflator

2.28

1.60

4.94

1.56 54.48

1.14

Federal Funds Rates

0.63

0.37

0.94

0.32 64.34

0.40

Accuracy of point forecasts

Mean square forecast errors

Small OLS

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium OLS

13.07 10.74 23.03

BVAR

Large OLS

BVAR

8.97 77.32

9.76

GDP Deflator

2.33

1.54

3.65

1.52 15.14

1.31

Federal Funds Rates

1.67

1.11

2.25

1.08

6.62

1.08

Real GDP

5.15

4.21 17.35

3.65 152.5

4.99

GDP Deflator

2.28

1.60

4.94

1.56 54.48

1.14

Federal Funds Rates

0.63

0.37

0.94

0.32 64.34

0.40

Accuracy of point forecasts

Mean square forecast errors

Small OLS

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium OLS

13.07 10.74 23.03

BVAR

Large OLS

BVAR

8.97 77.32

9.76

GDP Deflator

2.33

1.54

3.65

1.52 15.14

1.31

Federal Funds Rates

1.67

1.11

2.25

1.08

6.62

1.08

Real GDP

5.15

4.21 17.35

3.65 152.5

4.99

GDP Deflator

2.28

1.60

4.94

1.56 54.48

1.14

Federal Funds Rates

0.63

0.37

0.94

0.32 64.34

0.40

BVAR and Dynamic Factor Model (DFM)




BVAR and DFM are intimately connected




Homogenous shrinkage on the data implies to shrink less the most important PC  Theory: De Mol, Giannone and Reichlin (2008),  Large BVAR: Banbura, Giannone and Reichlin (2010)




DFM great tool to forecast. BVAR and DFM comparable performance

Density forecasts




Simple MCMC algorithm for posterior evaluation  Draw λ from p ( λ | Y ) using the Metropolis algorithm  Draw ( β, Σ ) from p ( β, Σ | Y ), which is Normal-Inverse-Wishart

Density forecasts (1-step ahead)

Density forecasts (4-step ahead)

Outline




BVARs




Forecasting with hierarchical models




Results  Macroeconomic forecasting  Structural VARs and impulse responses

Structural BVARs

1. Estimate IRF using real data 

Not shown here

2. Simulation exercise to evaluate bias-variance trade-off

SVAR and accuracy of IRF




DSGE model as a data generating process  Justiniano, Primiceri and Tambalotti (2010) Slight variation of the Smets and Wouters model




GE “structural” model of the US economy based on    




HH maximizing utility Firms maximizing profits Policy setting the short-term nominal interest rate Many frictions

Model perturbed by many shocks, including a MP shock

SVAR and accuracy of IRF




Solution of log-linearized DSGE model

ξt = G(χ) ξt −1 + M (χ) ηt y t = Hξt + met


3000 data simulations with T = 200 quarters




For each simulation estimate VAR and BVAR  Identify MP shock  Identification consistent with the DSGE model the private sector is predetermined with respect to the monetary policy shock (as in Christiano, Eichenbaum, and Evans, 2005)

MSE(VAR) / MSE(BVAR)

Conclusions




Standard model  VAR




Standard priors  Naïve model / random walk / Minnesota prior  SoC prior




Set hyper-parameters by evaluating their posterior





Great tool for both forecasting and structural analysis No reason to use VARs as opposed to BVARs

Background slides

Why other priors (3 and 4)?

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ) 1. Minnesota prior  Litterman (1980 and 1986)

2. Inverse-Wishart prior on

N-IW prior

Σ

3. Sum-of-coefficients prior  Doan, Litterman and Sims (1984)

4. Single-unit-root prior  Sims (1993)

Why other priors (3 and 4)?

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ) 1. Minnesota prior  Litterman (1980 and 1986)

2. Inverse-Wishart prior on

N-IW prior

Σ

3. Sum-of-coefficients prior  Doan, Litterman and Sims (1984)

4. Single-unit-root prior  Sims (1993)

Why other priors?


Typical VAR estimation conditions on initial conditions  Treats them as carrying no info about model dynamics  No penalization for estimates of steady states or trends far away from initial conditions

Why other priors?


Typical VAR estimation conditions on initial conditions  Treats them as carrying no info about model dynamics  No penalization for estimates of steady states or trends far away from initial conditions

 Flat prior VARs imply large transient dynamics in the first part of the sample

Transient dynamics for the FFR - VAR(5) with 7 variables

Why sum-of-coefficients prior


Typical VAR estimation conditions on initial conditions  Treats them as carrying no info about model dynamics  No penalization for estimates of steady states or trends far away from initial conditions

 Flat prior VARs imply large transient dynamics in the first part of the sample  Deterministic component responsible for most low frequency variation in the data  Temporal heterogeneity: deterministic component behaves very differently in first and last part of the sample

Transient dynamics for the GDP - VAR(5) with 7 variables

Why other priors?


Typical VAR estimation conditions on initial conditions  Treats them as carrying no info about model dynamics  No penalization for estimates of steady states or trends far away from initial conditions

 Flat prior VARs imply large transient dynamics in the first part of the sample  Deterministic component responsible for most low frequency variation in the data  Temporal heterogeneity: deterministic component behaves very differently in first and last part of the sample

 Want a prior that favors temporal homogeneity

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components




Incorporate prior beliefs that a no-change forecast should be good at the beginning of the sample

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components




Incorporate prior beliefs that a no-change forecast should be good at the beginning of the sample




Down-weight importance of short-lived initial transients relative to long-lived smooth trends

3. Sum-of-coefficients prior

yt = C + B1 yt −1 + ... + B p yt − p + ε t

ε t ~ N (0, Σ )


Theil mixed estimation  Create observation for artificial time tj* such that

y j,t * = ... = y j,t * − p = j

j

1

µ

y j,0 ,

j = 1,...,n

Accuracy of point forecasts - The role of the sum-of-coefficients prior

Mean square forecast errors

Small BVAR (N-IW)

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium BVAR (N-IW)

11.66 10.74 10.41

BVAR

Large BVAR (N-IW)

BVAR

8.97 10.14

9.76

GDP Deflator

1.70

1.54

1.88

1.52

1.38

1.31

Federal Funds Rates

1.22

1.11

1.18

1.08

1.17

1.08

Real GDP

5.56

4.21

5.48

3.65

5.12

4.99

GDP Deflator

1.93

1.60

2.13

1.56

1.23

1.14

Federal Funds Rates

0.45

0.37

0.46

0.32

0.50

0.40

Accuracy of point forecasts - The role of the sum-of-coefficients prior

Mean square forecast errors

Small BVAR (N-IW)

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium BVAR (N-IW)

11.66 10.74 10.41

BVAR

Large BVAR (N-IW)

BVAR

8.97 10.14

9.76

GDP Deflator

1.70

1.54

1.88

1.52

1.38

1.31

Federal Funds Rates

1.22

1.11

1.18

1.08

1.17

1.08

Real GDP

5.56

4.21

5.48

3.65

5.12

4.99

GDP Deflator

1.93

1.60

2.13

1.56

1.23

1.14

Federal Funds Rates

0.45

0.37

0.46

0.32

0.50

0.40

(Some of) the literature




Litterman (1980)  maximizes out-of-sample fit on a pre-sample  λ = 0.2 (RATS default value)




De Mol, Giannone and Reichlin (2008)  Set λ to achieve a desired in-sample fit

Additional results




MSFE of BVAR with flat hyperpriors are very similar




Improve uniformly over ad hoc prior in RATS  Up to 50%




VAR in difference as inaccurate as VAR in levels

Additional results




MSFE of BVAR with flat hyperpriors are very similar




Improve uniformly over ad hoc prior in RATS  Up to 50%




VAR in difference as inaccurate as VAR in levels

Accuracy of point forecasts - Flat hyperpriors

Mean square forecast errors

Small BVAR (flat)

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

10.88 10.74

Medium BVAR (flat)

BVAR

Large BVAR (flat)

BVAR

8.92

8.97

9.71

9.76

GDP Deflator

1.45

1.54

1.43

1.52

1.31

1.31

Federal Funds Rates

1.12

1.11

1.08

1.08

1.08

1.08

Real GDP

4.70

4.21

3.61

3.65

5.11

4.99

GDP Deflator

1.37

1.60

1.33

1.56

1.13

1.14

Federal Funds Rates

0.35

0.37

0.31

0.32

0.40

0.40

Additional results




MSFE of BVAR with flat hyperpriors are very similar




Improve uniformly over ad hoc prior in RATS  Up to 50%




VAR in difference as inaccurate as VAR in levels

Accuracy of point forecasts

Mean square forecast errors

Small RATS

Real GDP 1 Quarter Ahead

1 Year Ahead

BVAR

Medium RATS

BVAR

Large RATS

BVAR

10.60 10.74

9.72

8.97

11.26

9.76

GDP Deflator

1.93

1.54

1.76

1.52

1.47

1.31

Federal Funds Rates

1.21

1.11

1.19

1.08

1.29

1.08

Real GDP

4.12

4.21

4.75

3.65

7.20

4.99

GDP Deflator

2.44

1.60

2.13

1.56

1.41

1.14

Federal Funds Rates

0.43

0.37

0.45

0.32

0.66

0.40

Additional results




MSFE of BVAR with flat hyperpriors are very similar




Improve uniformly over ad hoc prior in RATS  Up to 50%




VAR in difference as inaccurate as VAR in levels

Accuracy of point forecasts Remark: the OLS in differences is computed with a small shrinkage to avoid crazy patterns

Mean square forecast errors

Small OLS (diff)

Real GDP 1 Quarter Ahead

1 Year Ahead

GDP Deflator

BVAR

Medium OLS (diff)

BVAR

Large OLS (diff)

BVAR

12.68 10.74 1.90 1.54

16.37

8.97

60.70

9.76

2.49

1.52

5.63

1.31

Federal Funds Rates

1.49

1.11

1.70

1.08

5.01

1.08

Real GDP

5.84

4.21

8.27

3.65

28.44

4.99

GDP Deflator

1.59

1.60

2.63

1.56

4.19

1.14

Federal Funds Rates

0.41

0.37

0.60

0.32

3.65

0.40

Accuracy of point forecasts Remark: the OLS is computed with a small shrinkage to avoid crazy patterns

Mean square forecast errors

Small OLS Level

Real GDP 1 Quarter Ahead

1 Year Ahead

OLS Diff

13.62 12.68

Medium OLS Level

OLS Diff

Large OLS Level

OLS Diff

16.46 16.37 42.05 60.70

GDP Deflator

2.04

1.90

2.47

2.49

4.53

5.63

Federal Funds Rates

1.60

1.49

1.85

1.70

3.56

5.01

Real GDP

5.52

5.84

6.41

8.27 22.68 28.44

GDP Deflator

2.06

1.59

2.79

2.63

3.39

4.19

Federal Funds Rates

0.51

0.41

0.62

0.60

2.45

3.65

Q plots (1-step)

Q plots (4-step)

3. Sum-of-coefficients prior

yt = C + B1 yt −1 + ... + B p yt − p + ε t

ε t ~ N (0, Σ )


Theil mixed estimation  Create observation for artificial time tj* such that

y j,t * = ... = y j,t * − p = j

j

1

µ

y j,0 ,

j = 1,...,n

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Theil mixed estimation  Create observation for artificial time tj* such that

y j,t * = ... = y j,t * − p = j

j

1

µ

y j,0 ,

j = 1,...,n

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Theil mixed estimation  Create observation for artificial time tj* such that

y j,t * = ... = y j,t * − p = j

j

1

µ

y j,0 ,

j = 1,...,n

 It is essentially a prior on the sum of coefficients

Π ≡ B1 + ...+ B p − I  Introduces correlation among coefficients on a given variable in a given equation

Robustness with respect to the prior




The hierarchical prior structure implies that the unconditional prior for the parameter has a mixed distribution

p (θ ) =



∫ p (θ | λ ) p ( λ ) d λ

Mixed distributions have generally fatter tails than each of the component distributions p (θ|λ). Fat tailed distributions allow for robust inference.  When the prior has tails flatter than the tails of the likelihood the posterior is less sensitive to extreme discrepancies between prior and likelihood (Berger, 1985; Berger and Berliner, 1986)

1. A base prior: the Minnesota prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Shrink coefficients towards naïve model:




More precisely: 



[ ] 1 V [(B ) ]= φ ⋅ s

E (Bs )ij = 1 if s = 1 and i = j 2

s ij

2

Σii Ψjj

y t = c + y t−1 + εt

BVAR: Additional priors

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ) 1. Minnesota prior  Litterman (1980 and 1986)

2. Inverse-Wishart prior on

N-IW prior

Σ

3. Sum-of-coefficients prior  Doan, Litterman and Sims (1984)

4. Single-unit-root prior  Sims (1993)

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components




Incorporate prior beliefs that a no-change forecast should be good at the beginning of the sample

3. Sum-of-coefficients prior

y t = C + B1 y t−1 + ...+ B p y t− p + εt

εt ~ N (0,Σ)


Express disbelief in models with too much explanatory power for complex deterministic components




Incorporate prior beliefs that a no-change forecast should be good at the beginning of the sample




Down-weight importance of short-lived initial transients relative to long-lived smooth trends

Databases Variable

Transformation

Real GDP

Log-levels

GDP Deflator

Log-levels

Consumers Prices (CPI) - All items

Log-levels

Real spot market price index, BLS & CRB, all commodities

Log-levels

Industrial Production

Log-levels

Total non-farm employment

Log-levels

Unemployment

Levels

Real private consumption

Log-levels

Real residential investment

Log-levels

Real non-residential investment

Log-levels

Real private investment

Log-levels

Personal consumption expenditures price Index

Log-levels

Gross private domestic investment price Index

Log-levels

Capacity utilization – manufacturing

Levels

University of Michigan index of consumer expectations

Levels

Total hours worked - business sector

Log-levels

Real Compensation per hour

Log-levels

Federal Funds Rate

Levels

Bond Rate - 1 year maturity

Levels

Bond Rate - 5 year maturity

Levels

Standard and Poor 500 index

Log-levels

Nominal Effective Exchange rate

Log-levels

M2

Log-levels

Small

Medium

Large

BVAR and Dynamic Factor Model (DFM) One quarter ahead MSE relative to AR in differences

Variable Real GDP GDP Deflator Consumers Prices (CPI) - All items Real spot market price index, BLS & CRB, all commodities Industrial Production Total non-farm employment Unemployment Real private consumption Real residential investment Real non-residential investment

DFM

0.81 1.04 0.90 0.94 0.98 0.92 0.87 0.94 0.69 1.05

BVAR Small

BVAR Medium

BVAR Large

0.91 1.03

0.75 1.05

0.83 0.87 0.89 0.92 0.87 0.78 0.82 0.85 0.71 0.80

0.90

0.60

Real private investment Personal consumption expenditures price Index Gross private domestic investment price Index Capacity utilization – manufacturing University of Michigan index of consumer expectations Total hours worked - business sector Real Compensation per hour Federal Funds Rate Bond Rate - 1 year maturity Bond Rate - 5 year maturity Standard and Poor 500 index Nominal Effective Exchange rate M2

0.96 0.95 0.94 0.89 1.00 0.99 0.92 0.94 1.02 0.97 1.02 0.86

0.79

0.89 0.83 0.77

0.89 0.75 0.74 0.77 0.87 0.81 0.76 0.98 0.97 0.98 0.96 0.92

Bias