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Estimating DSGE models with observed
real–time expectation dataCentral Bank Macroeconomic Modeling Workshop,
Bangko Sentral ng Pilipinas
Michel Juillard1 Junior Maih2
October 19–20, 2010
The views expressed herein are ours and do not necessarily represent the
views of Banque de France or Norges Bank.
1Bank of France2Norges Bank
Introduction
In most DSGE models, agents have only past value of thevariables in their information set
When using DSGE models for forecasting, econometriciansoften use additional information, outside the model, aboutcurrent and next quarters
In this paper, we estimate a DSGE model in which agentsexploit existing real-time information on expectations of futureevents.
Use Smets and Wouters (2007) model of the US economy anddata from the Survey of Professional Forecasters.
Why estimate models with information about the future?
Using additional information potentially helps improveforecasts in short run
Assess the importance of publicly available information aboutthe future
real-time data potentially contains useful information aboutthe location of the structural parameters to estimate.
The estimated parameters affect both the economicimplications and the forecasting properties of models.
Proposed methodology
Exploit the forecasting framework of Maih(2010)
Agents allowed to react to anticipated events =⇒ informationset includes anticipated shocks
Information on anticipated events is used to back out the(implied) distribution of future shocks
Distribution of future shocks is fed to the filtering procedureto compute the likelihood
Results
Private agents do rely on future information.
Estimated parameter values and economic implicationssignificantly affected by existing future info.
Allowing private agents to exploit future info data improvesthe fit and forecasting properties of the model.
Ignoring info about the future, the estimated shocks appear tobe a mix of structural shocks and expected shocks.
Literature: Estimation using expectation data
Del Negro and Eusepi (2010): fit two DSGE models to USdata using observations on inflation expectations.
Expectations of SPF = the expectations of private agents
We treat SPF expectations as info available before privateagents form their own expectations and make decisions.
We condition the forecasts of private agents on theinformation set at their disposal.
Literature: News shocks
Beaudry and Portier (2006), Christiano, Illut, Motto andRostagno (2008), Fujiwara, Hirose, Shintani (2009)
News shocks are typically modeled as a hidden state
We solve the structural model under the assumption thatagents have some (observed) information about the future.
Literature: Conditional Forecast
VARs: Doan, Litterman and Sims (1984), Waggoner and Zha(1999), Andersson, Palmqvist and Waggoner (2008)
DSGE models: Christoffel, Coenen and Warne (2007), Benes,Binning and Lees (2008)
Relative entropy: Robertson, Tallman and Whiteman (2005)
Maih(2010): Conditional forecast techniques for DSGE:conditioning info is allowed to be central tendency (hardconditions), a truncated density (soft conditions), or a fulldistribution.
From future endogenous variables to future shocks (I)
DSGE model:EtFθ (yt , yt−1, yt+1, εt) = 0 (1)
withεt ∼ N (0, I )
Exogenous information about the future:
DtYt ∼ N (µt ,Ωt)
whereYt ≡
[y ′t , y
′
t+1, ..., y′
t+k
]
In practice, forecasters provide a mean forecast µt and maybe aconfidence interval.
From future endogenous variables to future shocks (II)
A solution takes the form
yt = Tyt−1 + Rηt
where
ηt ≡[(εtt)′
,(εtt+1
)′
, ...,(εtt+s−1
)′
]′
The dynamic model can in turn be written as
Yt =...Tyt−1 +Φηt
where
...T ≡
T
T 2
...T k
and ηt ≡[(ηt)
′, (ηt+1)
′, ..., (ηt+k)
′]′
From future endogenous variables to future shocks (III)
Information about future value of the variables implies
DtΦηt ∼ N (µt − Dt
...Tyt−1,Ωt)
We assume thatηt = M1tγ1t +M2tγ2t
where M1t is an orthonormal basis for the null space of matrix DtΦand M2t is an orthonormal basis for the column space of DtΦ.
From future endogenous variables to future shocks (IV)
We assume further that
γ1t ∼ N (0, I )
is a vector of disturbances that do not affect the restrictions.It follows that
γ2t ∼ N
[
(DtΦM2t)−1
(µt − Dt
...Tyt−1) , (DtΦM2t)
−1Ωt
(
(DtΦM2t)−1
)′
]
andηt = M2t (DtΦM2t)
−1 (µt − Dt
...Tyt−1) + ωtξt
with ξt ∼ N [0, I ] and
ωt =[
M2t (DtΦM2t)−1
Pt M1t
]
where Pt is such that Ωt = PtP′
t .
Conditional state–space representation
[ytηt
]
=
[RStM2t (DtΦM2t)
−1
M2t (DtΦM2t)−1
]
µt
+
[T − RStM2t (DtΦM2t)
−1Dt
...T 0
−M2t (DtΦM2t)−1
Dt
...T 0
] [yt−1
ηt−1
]
+
[RStωt
ωt
]
ξt
When µt = Dt
...Tyt−1, the mean of the information provided
by the SPF matches the mean of the unconditional forecasts
When Ωt = DtΦΦ′D ′
t , the uncertainty of the restrictions ofthe SPF matches the uncertainty of the agents.
Conditional state–space representationApply Maih(2010) and from the unconditional state equation
yt = T (θ) yt−1 + R (θ) ηt
we get the conditional state equation[
ytηt
]
︸ ︷︷ ︸
≡αt
= bt (µt) + Tt
[yt−1
ηt−1
]
+ Rt (Ωt)ξt , ξt ∼ N [0, I ]
to which we add a measurement equation
y∗t = Ztαt + ǫt , ǫt ∼ N (0,Ht)
Only current variables (y∗t ) are observable in the measurementequation.
Conditioning information does not enter the model throughthe measurement equation.
Benefit: Model comparison properties preserved when varyingthe information set.
Smets and Wouters (2007) model
Medium size model estimated on US data,1966:1 to 2004:4
Nominal rigidities on prices and wages (Calvo)
Real rigidities (consumption habits and investment adjustmentcost)
Shocks on TFP, risk premium, investment-specific technology,wage mark-up, price mark-up, exogenous spending, monetarypolicy
Observed variables: growth rates for real GDP, realconsumption, real investment, real wage, and GDP deflator;log of hours worked in level
Data in the Survey of Professional Forecasters
SPF expectation data on GDP growth (1968Q4:2004Q4),Inflation(GDP deflator, 1968Q4:2004Q4) and consumptiongrowth(1981Q3:2004Q4)
anticipation horizon: uniform prior over [0,6]
1 2 3 4 5 6
GDP growth NA 0.2731 0.2570 0.2375 0.2321 0.0760
Consumption growth 0.2063 0.2571 0.1453 0.2297 0.2136 -0.0635
Inflation NA 0.8221 0.7557 0.6741 0.5976 0.5340
Table: Correlations between actual data and predictions of the Survey ofProfessional Forecasters for quarters 1 to 6
Estimation results I
Prior distr. SW Infl Infl+GDP Infl+Cons Infl+GDP+Cons
MDD(Laplace) -922.40 -912.57 -910.89 -910.19 -911.27
100(
β−1− 1
)
gamm 0.1444 0.1267 0.1361 0.1976 0.1876
l norm 0.7259 0.3268 -0.3762 -0.4926 -1.1931ϕ norm 5.4880 5.5785 5.2430 4.8944 4.5337σc norm 1.4219 1.4553 1.4467 1.2232 1.2971λ beta 0.7063 0.6677 0.6616 0.6944 0.6713ξw beta 0.7343 0.7009 0.6947 0.7891 0.7865σl norm 1.8749 1.9793 1.9215 2.2561 2.1381ιw beta 0.5983 0.7617 0.7621 0.5163 0.5310ξp beta 0.6542 0.5593 0.5518 0.7087 0.6958ιp beta 0.2187 0.5375 0.5113 0.6429 0.6492ψ beta 0.5453 0.5979 0.6282 0.5492 0.5733Φ norm 1.6097 1.6065 1.6136 1.5468 1.5577α norm 0.1910 0.1930 0.1953 0.1899 0.1966100 (γ − 1) norm 0.4344 0.4379 0.4230 0.3047 0.3018rπ norm 2.0217 1.9379 1.8513 1.9978 2.0139ρ beta 0.8145 0.8036 0.7862 0.8338 0.8338ry norm 0.0881 0.0486 0.0227 0.1286 0.1226r∆y norm 0.2223 0.2171 0.2221 0.2178 0.2211100 (π − 1) gamm 0.7652 0.7548 0.7081 0.6688 0.6373Anticipation unif 0.0000 2.0000 2.3093 1.9271 2.0000
Estimation results II
Prior distr. SW Infl Infl+GDP Infl+Cons Infl+GDP+Cons
ρa beta 0.9607 0.9634 0.9580 0.9881 0.9826ρb beta 0.1833 0.5865 0.5371 0.7552 0.7531ρg beta 0.9761 0.9766 0.9783 0.9636 0.9672ρI beta 0.7032 0.8647 0.8604 0.8939 0.8844ρr beta 0.1227 0.1307 0.1615 0.0969 0.0959ρp beta 0.9078 0.9782 0.9683 0.2631 0.2665ρw beta 0.9743 0.9194 0.9169 0.8385 0.8255µp beta 0.7438 0.7767 0.7564 0.4687 0.4620µw beta 0.8929 0.5738 0.5518 0.4617 0.4465ρga norm 0.5232 0.5194 0.5135 0.5362 0.5455std(ηa) invg 0.4529 0.4543 0.4503 0.4544 0.4515
std(ηb) invg 0.2416 0.0707 0.0721 0.0714 0.0697std(ηg ) invg 0.5213 0.5210 0.5284 0.5101 0.5213
std(ηI ) invg 0.4552 0.2436 0.2523 0.2173 0.2365std(ηr ) invg 0.2389 0.2403 0.2461 0.2319 0.2322std(ηp) invg 0.1398 0.0918 0.0935 0.1732 0.1719std(ηw ) invg 0.2465 0.1340 0.1360 0.0990 0.1024
Forecasting performance
0 2 4 6 80
0.5
1
InflInfl+ConsInfl+GDPInfl+GDP+Cons
2 4 6 8
0.9
1
1.1
GDP growth
2 4 6 8
0.8
1
Consumption growth
2 4 6 8
1
1.05
1.1
Investment growth
2 4 6 81
2
3
Hours worked
2 4 6 81
1.5
2
Inflation
2 4 6 8
0.95
1
Wage growth
2 4 6 81
1.5
2
Fed funds rate
Variance decomposition
TFP Risk Gov Invest Mon. pol price mkp wage mkp1-step ahead, in percent
GDP 24.86 14.31 38.35 5.71 11.16 2.93 2.67Inflation 6.63 2.05 0.39 3.61 4.95 48.82 33.54Fed Funds rate 9.00 13.71 2.79 1.20 66.21 4.57 2.53
8-step ahead, in percentGDP 32.76 3.69 8.42 10.63 5.37 10.24 28.89Inflation 7.13 2.58 0.91 8.68 8.68 21.83 50.19Fed Funds rate 13.04 12.17 3.20 23.61 21.22 6.62 20.14
∞-step ahead, in percentGDP 35.47 1.15 3.76 6.48 1.72 13.89 37.52Inflation 7.21 2.46 1.42 10.09 8.60 21.33 48.89Fed Funds rate 12.15 9.41 4.86 31.54 16.39 6.33 19.32
Table: Variance decompositions for the model with conditioninginformation on Inflation
Impulse Responses: technology shock
0 5 10 15 200
0.5
1output
UnanticipatedAnticipated
0 5 10 15 200
0.5Consumption
0 5 10 15 200
1
2investment
0 5 10 15 20−0.5
0
0.5hours
0 5 10 15 200
0.2
0.4real wages
0 5 10 15 20−0.1
0
0.1inflation
0 5 10 15 20−0.1
0
0.1interest rate
0 5 10 15 20−0.2
0
0.2capacity utilization
Impulse Responses: government spending shock
0 5 10 15 20−0.5
0
0.5output
UnanticipatedAnticipated
0 5 10 15 20−0.4
−0.2
0Consumption
0 5 10 15 20−1
−0.5
0investment
0 5 10 15 20−0.5
0
0.5hours
0 5 10 15 20−0.05
0
0.05real wages
0 5 10 15 200
0.02
0.04inflation
0 5 10 15 20−0.05
0
0.05interest rate
0 5 10 15 20−0.2
0
0.2capacity utilization
Impulse Responses: monetary policy shock
0 5 10 15 20−0.5
0
0.5output
UnanticipatedAnticipated
0 5 10 15 20−0.5
0
0.5Consumption
0 5 10 15 20−1
0
1investment
0 5 10 15 20−0.5
0
0.5hours
0 5 10 15 20−0.2
−0.1
0real wages
0 5 10 15 20−0.1
0
0.1inflation
0 5 10 15 20−0.2
0
0.2interest rate
0 5 10 15 20−0.2
0
0.2capacity utilization
Scaled revisions of expected shocks at horizon 2
1970 1980 1990 2000−4
−2
0
2
TFP
1970 1980 1990 2000
−2
0
2
Risk Premium
1970 1980 1990 2000
−2
0
2
Gov spend
1970 1980 1990 2000−4
−2
0
2
Invest specific tech
1970 1980 1990 2000−4−2
024
Monetary policy
1970 1980 1990 2000
−2
0
2
4Price makup
1970 1980 1990 2000
−2
0
2
4
Wage markup
Conclusion
Framework for estimating DSGE models in presence ofreal–time information on expectations
Apply the technique to the Smets and Wouters (2007) model
Private agents do rely on future information.
Estimated parameter values and economic implicationssignificantly affected by existing future info.
Allowing private agents to exploit future info data improvesthe fit of the model
Future work: compare forecast performance when only theeconometrician knows about the future
