Bayesian multivariate Beveridge{Nelson decomposition of I(1)...
Transcript of Bayesian multivariate Beveridge{Nelson decomposition of I(1)...
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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
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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
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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
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(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
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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)
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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})
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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
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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
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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
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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.
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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)
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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
]
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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.
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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
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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)
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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
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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 ).
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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
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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
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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)
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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.
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Natural rate of inflation (trend inflation)
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Natural rate of interest
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Natural rate of unemployment
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Natural rate (level) of output
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Inflation rate gap
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Interest rate gap
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Unemployment rate gap
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Output gap
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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
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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
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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
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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
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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)
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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
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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
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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
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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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