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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
D. Giannone, M. Lenza, and G. E. Primiceri. Prior selection
for vector autoregressions. Review of Economics and
Statistics, 97:436–451, 2015. doi: 10.1162/rest a 00483.
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.
Journal of Applied Econometrics, 24:630–650, 2009. doi:
10.1002/jae.1065.
36