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Learning and Time-Varying Macroeconomic
Volatility
Fabio Milani
University of California, Irvine
International Research Forum, ECB - June 26, 2008
Introduction
Strong evidence of changes in macro volatility over time
(The Great Moderation)
Kim and Nelson (1999), McConnell and Pérez-Quiròs (2000),
Stock and Watson (2002), Blanchard and Simon (2001)
Time-Varying Volatility
Conditional Standard Deviation (Inflation)
1.6
1.4
1.2
1
0.8
0.6
0.4
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
1995
2000
2005
Conditional Standard Deviation (Output Gap)
2
1.5
1
0.5
0
1960
1965
1970
1975
1980
1985
1990
Figure: Conditional Standard Deviation series for Inflation and Output
Gap
Introduction
Need to correctly model volatility
Sims and Zha (AER 2006): BVAR, Regime changes in
volatilities of shocks
Introduction
In DSGE Models?
Exogenous shocks with constant variance
(Smets and Wouters JEEA 2003, AER 2007, An and
Schorfheide ER 2007)
DSGE with Stochastic Volatility
Justiniano and Primiceri (AER forth.), Fernandez-Villaverde
and Rubio-Ramirez (RES 2007)
Time variation in the volatility of exogenous shocks
Introduction
But what explains the changing volatility?
Scope of the paper
Present a simple model with learning
The learning speed (gain coefficient) of the agents is
endogenous: it responds to previous forecast errors
Endogenous Time-Varying Volatility
Related: Branch and Evans (RED 2007), Lansing (2007),
Bullard and Singh (2007).
Results:
1
The changing gain induces endogenous time variation in the
volatilities of the macroeconomic variables the agents try to
learn
2
Evidence of time variation in endogenous gain from estimated
model
3
The econometrician can spuriously find evidence of stochastic
volatility if learning is not taken into account
The Model
Stylized New Keynesian Model
xt
bt πt+1 + κxt + ut
= βE
bt xt+1 − σ(it − E
bt πt+1 ) + gt
= E
it
= ρt it−1 + (1 − ρt )(χπ,t πt−1 + χx,t xt−1 ) + εt (3)
πt
Learning instead of RE
TV Monetary Policy
(1)
(2)
Expectations Formation
VAR to form inflation and output expectations
Perceived Law of Motion (VAR(1)):
Zt = at + bt Zt−1 + ηt
where Zt ≡ [πt , xt , it ]0
≈ Minimum State Variable solution
(4)
Learning
Coefficient Updating
φbt
Rt
= φbt−1 + gt,y Rt−1 Xt (Zt − Xt0 φbt−1 )
= Rt−1 +
0
gt,y (Xt−1 Xt−1
− Rt−1 )
where φbt = (at0 , vec(bt )0 )0 and Xt ≡ {1, Zt−1 }t−1
0 .
(5)
(6)
Endogenous Time-Varying Gain
Decreasing Gain if Forecast Errors are small
Switch to Constant Gain if Forecast Errors become large
PJ
−1
j=0 (|yt−j −Et−j−1 yt−j |)
t
if
< υty
J
PJ
gt,y =
g
j=0 (|yt−j −Et−j−1 yt−j |)
if
≥ υy ,
y
t
J
where y = π, x, i. (Decr. Gain reset to
1
)
g −1
y +t
Similar to Marcet-Nicolini (υt is m.a.d. of forecast errors)
Constant Gain is estimated
Which situations?
(7)
Questions:
1
Does the gain coefficient affect volatility? Can the model
generate time-varying volatility in inflation and in the output
gap?
2
Does the model fit U.S. data? Is there evidence of changes in
the gain over time?
3
Does the omission of learning imply that researchers
spuriously find stochastic volatility in the structural shocks?
4
Does the model-implied stochastic volatility resemble the SV
estimated from the data?
5
What are the effects of MP on the estimated Volatility?
1. Endogenous Gain and TV Volatility
4.5
Std. Infl
Std. Output Gap
4
3.5
3
2.5
2
1.5
1
0.5
0
0.05
0.1
0.15
Figure: Volatility of simulated Inflation and Output Gap as a function of
the constant gain coefficient.
1. Endogenous Gain and TV Volatility
Volatility typically increases in the gain
Simulation (10,000 periods)
Gain switches endogenously according to previous forecast
errors
1. Endogenous Gain and TV Volatility
Time−Varying Volatility (rolling standard deviation)
5
Std. Infl
Std. Gap
4
3
2
1
0
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
7000
8000
9000
10000
Endogenous Time−Varying Gain
0.2
TV gain (Infl)
TV gain (Gap)
0.15
0.1
0.05
0
0
1000
2000
3000
4000
5000
6000
Figure: Time-Varying Volatility with Time-Varying Endogenous Gain
Coefficient.
2. Bayesian Estimation
Gain switches from decreasing to constant
Constant Gain jointly estimated in the system
Metropolis-Hastings
Quarterly U.S. data, 1960:I-2006:I, data from 1954 to 1959 to
initialize learning algorithm
Uniform priors for gains
2. Bayesian Estimation: Priors
Description
Inverse IES
Slope PC
Discount Rate
Interest-Rate Smooth
Feedback to Infl.
Feedback to Output
Interest-Rate Smooth
Feedback to Infl.
Feedback to Output
Std. MP shock
Std. gt
Std. ut
Constant Gain infl.
Constant Gain gap
Constant Gain FFR
Param.
σ −1
κ
β
ρpre79
χπ,pre79
χx,pre79
ρpost79
χπ,post79
χx,post79
σε
σg
σu
gπ
gx
gi
Range
R+
R+
.99
[0, 1]
R
R
[0, 1]
R
R
R+
R+
R+
[0, 0.3]
[0, 0.3]
[0, 0.3]
Table 1 - Prior Distributions.
Prior Distribution
Distr. Mean
95% Int.
G
1
[.12, 2.78]
G
.25
[.03, .7]
−
.99
−
B
.8
[.46, .99]
N
1.5
[.51, 2.48]
N
.5
[.01, .99]
B
.8
[.46, .99]
N
1.5
[.51, 2.48]
N
.5
[.01, .99]
IG
1
[.34, 2.81]
IG
1
[.34, 2.81]
IG
1
[.34, 2.81]
U
.15
[.007, .294]
U
.15
[.007, .294]
U
.15
[.007, .294]
2. Bayesian Estimation: Results
Description
Inverse IES
Slope PC
Discount Factor
IRS pre-79
Feedback Infl. pre79
Feedback Gap pre79
IRS post-79
Feedback Infl. post79
Feedback Gap post79
Autoregr. Cost-push shock
Autoregr. Demand shock
Std. Cost-push shock
Std. Demand shock
Std. MP shock
Constant gain (Infl.)
Decreasing gain (Infl.)
Constant gain (Gap)
Decreasing gain (Gap)
Constant gain (FFR)
Decreasing gain (FFR)
Parameter
σ −1
κ
β
ρpre79
χπ,pre−79
χx,pre−79
ρpost79
χπ,post−79
χx,post−79
ρu
ρg
σu
σg
σε
gπ
t −1
gx
t −1
gi
t −1
Posterior Distribution
Mean 95% Post. Prob. Int.
6.04
[4.17-9.14]
0.021
[0.0026-0.054]
0.99
0.937
[0.85-0.99]
1.30
[0.83-1.81]
0.66
[0.29-1.13]
0.93
[0.88-0.97]
1.66
[1.19-2.11]
0.48
[0.07-0.85]
0.39
[0.27-0.49]
0.85
[0.78-0.92]
0.89
[0.81-0.98]
0.65
[0.59-0.72]
0.97
[0.88-1.07]
0.082
[0.078-0.09]
0.073
[0.06-0.082]
0.003
[0,0.023]
-
Table 2 - Posterior Distributions: baseline case with J = 4.
2. Bayesian Estimation: Time-Varying Gain
Endogenous Time−Varying Gain − Inflation
0.08
0.06
0.04
0.02
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2000
2005
Endogenous Time−Varying Gain − Output Gap
0.08
0.06
0.04
0.02
0
1960
1965
1970
1975
1980
1985
1990
1995
Figure: Endogenous Time-Varying Gain Coefficients (estimated constant
gain). Baseline Case
Is it a good idea to use this learning rule?
Is it dominated by alternatives?
Endogenous TV Gain
Decreasing Gain
Constant Gain
Inflation
0.94
0.97
0.98
Output Gap
0.88
1.00
0.91
Table 6 - RMSEs.
Optimality Tests.
bt+1,t ) = α + β Y
bt+1,t + ut+1
It+1,t ≡ 1(Yt+1,t < Y
Back out Loss Function
(8)
2. Bayesian Estimation: Time-Varying Gain
Endogenous Time−Varying Gain − Inflation
0.08
0.06
0.04
0.02
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2000
2005
Endogenous Time−Varying Gain − Output Gap
0.08
0.06
0.04
0.02
0
1960
1965
1970
1975
1980
1985
1990
1995
Figure: Endogenous Time-Varying Gain Coefficients (estimated constant
gain). Case with J = 20
2. Bayesian Estimation: Time-Varying Gain
0.035
Posterior Distribution gπ
0.03
Prior Distribution
0.025
0.02
0.015
0.01
0.005
0
0.05
0.06
0.07
0.08
0.09
0.1
0.02
Posterior Distribution gx
Prior Distribution
0.015
0.01
0.005
0
0.05
0.06
0.07
0.08
0.09
0.1
Figure: Constant Gain Coefficients: Prior and Posterior Distributions.
2. Bayesian Estimation: Time-Varying Gain
Endogenous Time−Varying Gain − Inflation
0.1
0.08
0.06
0.04
0.02
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
2000
2005
Endogenous Time−Varying Gain − Output Gap
0.06
0.05
0.04
0.03
0.02
0.01
0
1960
1965
1970
1975
1980
1985
1990
1995
Figure: Endogenous Time-Varying Gain Coefficients (Case with low and
high constant gain coefficients only).
2. Bayesian Estimation: Forecast Errors
Forecast Errors Inflation
4
3
2
1
0
1960
1965
1970
1975
1980
1985
1990
Forecast Errors Output Gap
1995
2000
2005
1965
1970
1975
1980
1985
1990
Forecast Errors FFR
1995
2000
2005
1965
1970
1975
1980
1995
2000
2005
4
3
2
1
0
1960
10
5
0
1960
1985
1990
Figure: Forecast errors for inflation, output gap, and federal funds rate
(absolute values).
2. Bayesian Estimation: Forecast Errors
Inflation
3
Mean Absolute Forecast Error
νπ
t
2
1
0
1960
1965
1970
1975
1980
1985
Output Gap
1990
3
1995
2000
2005
Mean
Absolute Forecast Error
νxt
2
1
0
1960
1965
1970
1975
1980
1985
FFR
1990
6
1995
2000
2005
Mean
Absolute Forecast Error
νit
4
2
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure: Rolling Mean Absolute Forecast errors vs. Updated νt for
inflation, output gap, and federal funds rate series.
3. If learning is neglected:
The volatility of shocks may be overestimated
Possible to spuriously find Stochastic Volatility
3. Test for ARCH/GARCH Effects
Inflation
Output Gap
Endogenous TV Gain
J=4
J = 20
ARCH(1) GARCH(1,1) ARCH(1) GARCH(1,1)
0.517
0.61
0.48
0.56
0.785
0.89
0.85
0.90
No Learning
ARCH(1)
0.05
GARCH(1,1)
0.06
0.045
0.05
Table 7 - Test for the existence of ARCH/GARCH effects (5%
significance): proportion of rejections of the null hypothesis of no
ARCH/GARCH effects.
4. Volatility
0.012
Max. Std. Inflation eq. Residuals
0.01
0.008
0.006
0.004
0.002
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
0.012
Max. Std. Output Gap eq. Residuals
0.01
0.008
0.006
0.004
0.002
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure: Maximum rolling Standard Deviation of residuals across
simulations: Kernel Density Estimation.
4. The Great Moderation
Endogenous TV Gain
Baseline J = 20 CG
Data
Std. Infl. 1985−2006
Std. Infl. 1960−1984
0.39
0.42
0.43
1.00
0.35
(Std. OutputGap 1985−2006)
(Std. Output Gap 1960−1984)
0.42
0.52
0.54
1.00
0.50
Ratio
Ratio
No Learning
Table 8 - The Great Moderation: ratio of standard deviations for inflation
and output gap in the second versus the first part of the simulated
samples (median across simulations).
5. Monetary Policy, Learning, and Volatility
Simulation for χπ = [0, ..., 5]:
Related: Benati-Surico (2007)
1
Fraction of Switches to a Constant Gain
0.8
0.6
0.4
0.2
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
χ
5 π
0.08
Average Gain in Sample
0.07
0.06
0.05
0.04
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
χ
5 π
0.9
% Rejections no ARCH Effects
0.8
0.7
0.6
0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
χπ
Figure: Effects of Monetary Policy on Volatility.
5. Bernanke - Great Moderation Speech
I am not convinced that the decline in macroeconomic volatility of
the past two decades was primarily the result of good luck.
changes in monetary policy could conceivably affect the size and
frequency of shocks hitting the economy, at least as an
econometrician would measure those shocks
changes in inflation expectations, which are ultimately the
product of the monetary policy regime, can also be confused
with truly exogenous shocks in conventional econometric
analyses.
some of the effects of improved monetary policies may have been
misidentified as exogenous changes in economic structure or in the
distribution of economic shocks.
6. TV Volatility: Learning or Exogenous Shocks?
Test ARCH/GARCH in DSGE Model Innovations now
Output Gap
Inflation
DSGE-RE
ARCH
ARCH
DSGE-TV Gain
ARCH
No ARCH
6. TV Volatility: Learning or Exogenous Shocks?
Innovation in Inflation Equation: Rolling Std.
2
Under Learning/TV Gain
Under RE
1.5
1
0.5
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Innovation in Output Gap Equation: Rolling Std.
1.4
Under Learning/TV Gain
Under RE
1.2
1
0.8
0.6
0.4
0.2
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Figure: Rolling Std. estimated innovations under RE and Learning
Conclusions
Strong Evidence of Stochastic Volatility in the economy
Usually Exogenous
Learning with endogenous TV gain (depends on previous
forecast errors) ⇒ Endogenous Stochastic Volatility
Gain often larger in pre-1984 sample
Overestimation of TV in volatility of exogenous shocks.
Future Directions
How much volatility can learning explain? (estimate DSGE
model with learning and TV volatility).
More serious attempt to match volatility series in the data.
Different ways to model endogenous gain/ Optimality
Interactions Policy/Learning/Volatility
@FedFRASER