Bayesian multivariate

Beveridge–Nelson decomposition of

I(1) and I(2) series with (possible)

cointegration

Yasutomo Murasawa

JEA 2019 Spring Meeting

Konan University

1

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

2

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

3

(Multivariate) B–N decomposition

• Decompose I(1) series into

• random walk permanent component (=LR forecast)

• I(0) transitory component (=forecastable movement)

assuming a linear time series model (ARIMA, VAR,

VECM, etc.)

• For non-US data, often “unreasonably persistent”

transitory component (perhaps because of structural

breaks)

• Extension to I(2) series available

4

Multivariate B–N decomposition: Non-US data

Output gap in Japan

−0.10

−0.05

0.00

0.05

0.10

1990 2000 2010

I(1) log output

I(2) log output

Source: Murasawa (2015, Economics Letters)

5

Consumption Euler equation

Let

• yt : log output

• rt : real interest rate

Consumption Euler equation =⇒ dynamic IS curve:

Et(∆yt+1) =1

σ(rt − ρ)

Observations:

1. {rt} is I(1) =⇒ {∆yt} is I(1) =⇒ {yt} is I(2)

2. Cointegration between {∆yt} and {rt}3. Decompose {yt} (not {∆yt})

6

Contributions

1. Derive a formula for the multivariate B–N decomposition

of I(1) and I(2) series with cointegration

2. Develop a Bayesian method to obtain error bands for the

components

3. Apply the method to US data

=⇒ obtain a joint estimate of the natural rates (or their

permanent components) and gaps of

• output

• inflation

• interest

• unemployment

7

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

8

VAR model

Notations

• {xt,d} ∼ I(d)

• yt := (x ′t,1,∆x ′

t,2)′ ∼ CI(1, 1)

Steady state VAR model

Π(L)(yt −α− µt) = ut

{ut} ∼ WN

9

VECM representation

Let Π(1) = ΛΓ ′, where

• Λ: N × r loading matrix

• Γ : N × r cointegrating matrix

Steady state VECM

Φ(L)(∆yt − µ)︸ ︷︷ ︸VAR(p)

= −Λ [Γ ′yt−1 − β − δ(t − 1)]︸ ︷︷ ︸error correction term

+ut

where β := Γ ′α and δ := Γ ′µ.

10

State space representation

State vector

st :=

∆yt − µ

...

∆yt−p+1 − µΓ ′yt − β − δt

Note: Observable given the parameters

Linear state space model

st = Ast−1 + Bzt∆yt = µ+ Cst{zt} ∼ WN(IN)

11

State space representation

where

A :=

Φ1 . . . Φp −ΛI(p−1)N O(p−1)N×N O(p−1)N×r

Γ ′Φ1 . . . Γ ′Φp Ir − Γ ′Λ

B :=

Σ1/2

O(p−1)N×N

Γ ′Σ1/2

C :=

[IN ON×(p−1)N ON×r

]

12

Multivariate B–N decomposition

Theorem

B–N transitory component in xt,1

ct,1 = −C1(IpN+r − A)−1Ast

B–N transitory component in xt,2

ct,2 = C2(IpN+r − A)−2A2st

Note: Observable given the parameters =⇒

• Estimation of {st} (by state smoothing) is unnecessary.

• Easy to construct error bands.

13

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

14

Bayesian VECM

Recent developments:

1. Prior on the cointegrating space:

Strachan and Inder (2004), Koop, Leon-Gonzalez, and

Strachan (2010)

2. Prior on the steady state VECM:

Villani (2009)

3. Hierarchical prior for the VAR coefficients regarding the

tightness/shrinkage hyperparameter:

Giannone et al. (2015)

4. S–D density ratio to compute the Bayes factor for

choosing the cointegrating rank:

Koop, Leon-Gonzalez, and Strachan (2008)

15

Gaussian VECM

Gaussian VECM

∆yt − µ = Φ1(∆yt−1 − µ) + · · ·+Φp(∆yt−p − µ)︸ ︷︷ ︸VAR(p)

−Λ [Γ ′yt−1 − β − Γ ′µ(t − 1)]︸ ︷︷ ︸error correction term

+ut

{ut} ∼ INN

(0N ,P−1

)Notations

• ψ := (β′,µ′)′: steady state parameters

• Φ := [Φ1, . . . ,Φp]: VAR coefficients

• Y := [y0, . . . , yT ]: observations

16

Gibbs sampler

1. Steady state parameters: Draw ψ.

2. Repeat until (Φ,Λ,Γ ) satisfies the stationarity of {st}:2.1 VAR parameters: Draw (Φ,P).

2.2 Loading and cointegrating matrices: Draw Λ. Let

Λ∗ := (Λ−Λ0)[(Λ−Λ0)′(Λ−Λ0)]

−1/2

Draw Γ∗ given Λ∗. Discard Λ. Let

Γ := Γ∗(Γ′∗Γ∗)

−1/2

Λ := Λ0 +Λ∗(Γ′∗Γ∗)

1/2

3. Tightness hyperparameter: Draw ν.

4. B–N transitory components: Apply the formula given

(ψ,Φ,P,Λ,Γ ;Y ).

17

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

18

Data

Notations

• πt : CPI inflation rate

• rt : ex post real interest rate (3-month treasury bill)

• Ut : unemployment rate

• Yt : real GDP

Let

xt :=

πtrtUt

lnYt

, yt :=

πtrtUt

∆ lnYt

Estimation period: 1950Q1–2017Q4

19

Model specification

VECM

• N := 4

• p = 7

“Weakly informative” priors

• ψ ∼ Nr+N(0r+N , Ir+N)

• (Φ,P): Minnesota-type hierarchical truncated

normal–Wishart

• ν ∼ χ2(1)

• Λ ∼ NN×r (ON×r ; IN , 100IN)

• Γ ∼ MACGN×r (IN)

20

Bayesian computation

• Use ML estimate for initial values of (Φ,P,Λ,Γ ).

• Discard initial 1,000 draws. Use next 4,000 draws for

posterior inference.

• From the S–D density ratios (for r = 1, 2, 3 vs r = 0), we

find r = 2 with posterior probability (numerically) 1.

• Use posterior median for point estimate.

• Use posterior 2.5 and 97.5 percentiles for 95% error band.

21

Natural rate of inflation (trend inflation)

22

Natural rate of interest

23

Natural rate of unemployment

24

Natural rate (level) of output

25

Inflation rate gap

26

Interest rate gap

27

Unemployment rate gap

28

Output gap

29

Comparison: Inflation rate gap

−0.02

−0.01

0.00

0.01

0.02

0.03

1960 1980 2000 2020

I(1) log output

I(2) log output

I(2) log output with cointegration

30

Comparison: Interest rate gap

−0.02

0.00

0.02

1960 1980 2000 2020

I(1) log output

I(2) log output

I(2) log output with cointegration

31

Comparison: Unemployment rate gap

−0.02

0.00

0.02

1960 1980 2000 2020

I(1) log output

I(2) log output

I(2) log output with cointegration

32

Comparison: Output gap

−0.03

0.00

0.03

0.06

1960 1980 2000 2020

I(1) log output

I(2) log output

I(2) log output with cointegration

33

Interpretation

1. I(2) log output

=⇒ I(1) output growth rate

=⇒ Stochastic trend captures structural breaks

=⇒ “reasonable” transitory components

2. Cointegration

=⇒ VECM forecasts better than VAR

=⇒ “bigger” transitory components (=forecastable

movements) for all variables (not only for output)

34

Plan

Motivation and contributions

Multivariate B–N decomposition of I(1) and I(2) series with

cointegration

Estimation based on a Bayesian VECM

Application to US data

Conclusion

35

Summary and future plans

Contributions:

1. Derive a formula for the multivariate B–N decomposition

of I(1) and I(2) series with cointegration

2. Develop a Bayesian method to obtain error bands for the

components

3. Apply the method to US data

=⇒ obtain a joint estimate of the natural rates and gaps

To-do list:

1. Misspecified order of integration

2. Comparison with alternative methods

3. More interesting applications

36

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for vector autoregressions. Review of Economics and

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G. Koop, R. Leon-Gonzalez, and R. W. Strachan. Bayesian

inference in a cointegrating panel data model. In S. Chib,

W. Griffiths, G. Koop, and D. Terrell, editors, Bayesian

Econometrics, volume 23 of Advances in Econometrics,

pages 433–469. Emerald, 2008. doi:

10.1016/s0731-9053(08)23013-6.

G. Koop, R. Leon-Gonzalez, and R. W. Strachan. Efficient

posterior simulation for cointegrated models with priors on

the cointegration space. Econometric Reviews, 29:224–242,

2010. doi: 10.1080/07474930903382208.

Y. Murasawa. The multivariate Beveridge–Nelson

36

decomposition with I(1) and I(2) series. Economics Letters,

137:157–162, 2015. doi: 10.1016/j.econlet.2015.11.001.

R. W. Strachan and B. Inder. Bayesian analysis of the error

correction model. Journal of Econometrics, 123:307–325,

2004. doi: 10.1016/j.jeconom.2003.12.004.

M. Villani. Steady-state priors for vector autoregressions.

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10.1002/jae.1065.

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