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Federal Reserve Bank of Chicago
Subject:
Summary of Chicago Fed DSGE Model for Federal Reserve System Researchers
From:
Jeffrey R. Campbell
Date:
June 8, 2011
Jonas D.M. Fisher
Alejandro Justiniano
Overview
In this memo, we describe the Chicago Fed’s estimated dynamic stochastic
general equilibrium model. This framework yields a history of identified
structural shocks, which we apply to illuminate recent macroeconomic
developments. To aid the understanding of these results, we follow them with
summaries of the model’s structure, the data and methodology employed for
estimation, and the estimated model’s dynamic properties.
In several respects the Chicago Fed DSGE model resembles many other New
Keynesian frameworks. There is a single representative household that owns all
firms and provides the economy’s labor. Production uses capital, differentiated
labor inputs, and differentiated intermediate goods. The prices of all
differentiated inputs are “sticky”, so standard forward-looking Phillips curves
connect wage and price inflation with the marginal rate of substitution between
consumption and leisure and marginal cost, respectively. Other frictions include
investment adjustment costs and habit-based preferences. Monetary policy
follows a Taylor rule with shocks.
The model’s decomposition of the price of investment goods with respect to
consumption into two components distinguishes it from other similar
frameworks. We empirically identify the technological component with the
investment-consumption relative deflator from the NIPA, and we tie the financial
component to the High Yield-AAA corporate bond spread.
Another distinguishing feature of the Chicago model is the use of multiple price
indices. For this, alternative available indices of inflation are decomposed into a
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Table 1. Model Forecasts Q4 over Q4
Real GDP
Federal Funds Rate
PCE Core
Consumption
Investment
Marginal Cost
2010
2.71
0.19
0.80
1.63
10.40
0.93
2011
3.22
0.11
1.46
2.52
7.63
‐1.46
2012
4.40
1.24
0.68
2.68
11.25
‐0.81
2013
4.24
2.73
0.34
2.56
11.27
0.15
2014
4.17
3.44
0.54
2.68
10.34
0.56
single model-based measure of consumption goods and services inflation and
idiosyncratic (series specific) disturbances that allow for persistent deviations
from this common component. Estimation uses a factor model with the common
factor derived from the DSGE framework.
Forecasting Methodology
Constructing forecasts based on this model requires us to assign values to its
many parameters. We do so using Bayesian methods to update an
uninformative prior with data from 1987:Q1 through 2008:Q4. All of our
forecasts condition on the parameters equaling their values at the resulting
posterior’s mode. These parameter values together with the data yield a
posterior distribution of the economy’s state in the final sample quarter. For the
calculation of this initial state’s distribution, we add a sequence of forward
guidance shocks that signal the future path of the Federal Funds rate. These
shocks begin arriving in 2009:Q1 and continue to the present. We construct them
so that model-based expectations of the policy rate equal actual market-based
expectations for the first five quarters of each quarter’s forecast horizon. The
forecasts begin with 2011:Q3 and extend through 2014:Q4. Our plug for 2011:Q2
GDP growth was set at 2.4 percent.
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Figure 1. Quarterly
Model
Figure
1: Forecasts
Forecasts
starting 2011q3
Consumption
Real GDP
5
4.5
3
4
3.5
2.5
3
2
2.5
2
1.5
1.5
2011
2012
2013
2014
2011
Federal Funds Rate
5
2012
2013
2014
Investment
10
4
5
3
2
0
1
−5
0
2011
2012
2013
2014
2011
PCE Core
2012
2013
2014
Marginal Cost
3
0
2.5
2
−1
1.5
−2
1
−3
0.5
−4
0
2011
2012
2013
2014
2011
2012
2013
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Table 1 presents data from 2010 and forecasts for the following four years. The
first three rows correspond to three key macroeconomic observables, Real GDP
growth (Q4-over-Q4), the Federal Funds Rate (Q4 average), and growth of the
Core PCE deflator (Q4-over-Q4). The following rows report forecasts of
Q4-over-Q4 growth for three model-defined aggregates of importance:
Consumption of nondurable goods and non housing services, Investment in
durable goods, residential housing, and business equipment and structures, and
the marginal cost of production. Figure 1 complements this with
quarter-by-quarter data and forecasts of these series. The plots’ dashed grey
lines indicate the series’ long-run values.
The economy’s long-run GDP growth rate – which we identify with potential
growth – equals 2.6 percent. Our forecast for 2011 only exceeds this by 62 basis
points, so an unfortunate series of negative shocks could easily send the
economy below potential. To place our forecasts from 2012 onward into context,
Figure 2 plots the level of GDP for the three recoveries in our sample. All have
been normalized so that their values equal zero at the NBER trough, and we
extend the current recovery with our forecast, given by the dashed line. It is
well-known that the ongoing recovery has been tepid when one considers the
depth of the most recent recession. The plot indicates that this comparison will
become less favorable over time.
Recall that we hard-wire the current values of monetary-policy news shocks to
match current market expectations. Currently, these date the tightening of
monetary policy at 2012:Q2. Thereafter, the forecast rate begins to rise as the
conventional Taylor rule dynamics take over. Although the forecasted path for
core PCE is nearly deflationary from 2012 through 2014, the Taylor rule sees
expected output growth as strong enough to merit the removal of the
extraordinary accommodation in place since 2008.
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5 of 22
-0.1
-10
-0.05
-5
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0
0.05
0.1
0.15
0.2
0.25
0.3
Log GDP (centered at trough)
Figure 2. Comparison of Recent Recoveries
0
5
10
Quarters After Trough of Recession
15
1991 Q1
2001 Q4
2009 Q2
Forecast
20
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Shock Decompositions
Our analysis identifies the structural shocks responsible for past fluctuations. To
summarize this information, we follow a suggestion of Charlie Evans: Fix an
object to be forecast, such as Q4-over-Q4 real GDP growth. Then, pick a date in
the past and forecast the object conditional on the information as of that date.
This is not a real-time forecast, because it uses revised data and model
parameters estimated with the complete sample. The model can be used to
decompose the forecast error into structural shocks. (A detailed explanation of
the forecast error decomposition procedure begins below on page 16) We
repeatedly advance the forecast date, decompose the forecast error, and finally
plot the results. In total, the model features nine structural shocks and four
idiosyncratic disturbances without structural interpretations. For parsimony’s
sake, we group the shocks according to a simple taxonomy.
Demand These are the structural non-policy shocks that move output and inflation
in the same direction. The model features two of them. One changes the
households rate of time discount, and the other alters the rate at which
consumption goods can be transformed into capital goods. We call these
the Time Discount and Marginal Efficiency of Investment shocks.
Supply Five shocks move real GDP and inflation in opposite directions on impact.
These supply shocks directly change
– Neutral Technology,
– Investment-Specific/Capital-Embodied Technology,
– Markups of Intermediate Goods Producers,
– Markups of Labor Unions, and
– Households’ Disutility from Labor
Policy The model’s monetary policy follows a Taylor rule with interest-rate
smoothing and an i.i.d. policy shock. Additionally, we have incorporated
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Table 2. The Model’s Decomposition of Business-Cycle Variance
Real GDP Growth
Federal Funds Rate
Core PCE Inflation
Demand
0.72
0.83
0.10
Supply
0.21
0.07
0.81
Monetary Policy
0.03
0.07
0.01
Residual
0.05
0.02
0.08
Note: For each variable, the table lists the fraction of variance at frequencies between 6
and 32 quarters attributable to shocks in the listed categories.
forward guidance shocks since 2008:Q4. These are revealed to the model’s
agents one to five quarters before they effect the federal funds rate, and
they allow our forecasting exercise to match model-based expectations
with information from futures markets.
Residual We group other shocks that are usually of small importance into a residual
category. These include a shock to the sum of government spending and
net exports.
Table 2 reports the fraction of business-cycle variance attributable to shocks in
each category for three key variables, the growth rate of Real GDP, the Federal
Funds Rate, and Core PCE Inflation. Three facts stand out here. First, demand
shocks dominate business cycles. Supply shocks account for only 21 percent of
the GDP growth rate’s total business-cycle variance, and the non-systematic part
of monetary policy shocks makes only a minor contribution. The accounting for
the Federal Funds Rate’s variance is similar. Perhaps this is unsurprising,
because we classify the shock that directly moves households’ rate of time
preference as “demand.” Nevertheless, supply shocks’ unimportance for the
Federal Funds Rate contrasts sharply with their dominance of inflation
fluctuations. The shock to intermediate goods’ firms optimal markup accounts
for about half of supply shocks 81 percent contribution, and the Hicks-neutral
technology shock accounts for another quarter of this.
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The Model’s Specification and Estimation
Our empirical work uses eleven variables, measured from 1987:Q1 through the
present:
• Growth of nominal per capita GDP,
• Growth of nominal per capita consumption, which sums Personal
Consumption Expenditures on Nondurable Goods and Services;
• Growth of nominal per capita investment, which sums Business Fixed
Investment, Residential Investment, Personal Consumption Expenditures
on Durable Goods, and Inventory Investment;
• Per capita hours worked in Nonfarm Business,
• Growth of nominal compensation per hour worked in Nonfarm Business,
• Growth of the implicit deflator for GDP,
• Growth of the implicit deflator for consumption, as defined above,
• Growth of the implicit deflator for investment, as defined above,
• Growth of the implicit deflator for core PCE,
• The interest rate on Federal Funds, and
• High Yield-AAA Corporate Bond Spread.
We do not directly use data on either government spending or net exports. Their
sum serves as a residual in the national income accounting identity. To construct
series measured per capita, we used the civilian non-institutional population 16
years and older. To eliminate level shifts associated with the decennial census,
we project that series onto a fourth-order polynomial in time.
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Our model confronts these data within the arena of a standard linear state-space
model. Given a vector of parameter values, θ, log-linearized equilibrium
conditions yield a first-order autoregression for the vector of model state
variables, ζt .
ζt = F (θ)ζt−1 + εt
εt ∼ N (0, Σ(θ))
Here, εt is a vector-valued innovation built from the model innovations
described above. Many of its elements identically equal zero. Table 3 lists the
elements of ζt . Habit puts lagged nondurable consumption into the list, and
investment adjustment costs place lagged investment there. Rules for indexing
prices and wages that cannot adjust freely require the state to include lags of
inflation and technology growth. The list includes the lagged policy rate because
it appears in the Taylor rule.
Gather the date t values of the eleven observable variables into the vector yt . The
model analogues to its elements can be calculated as linear functions of ζt and
ζt−1 . We suppose that the data equal these model series plus a vector of “errors”
vt .
yt = G(θ)ζt + H(θ)ζt−1 + vt
vt = Λ(ϕ)vt−1 + et
et ∼ N (0, D(ϕ))
Here, the vector ϕ parameterizes the stochastic process for vt . In our application,
the only non-zero elements of vt correspond to the observation equations for the
two consumption-based measures of inflation, the GDP deflator, and the High
Yield-AAA spread. The idiosyncratic disturbances in inflation fit the
high-frequency fluctuations in prices and thereby allow the price markup shocks
to fluctuate more persistently. These errors evolve independently of each other.
In this sense, we follow Boivin and Giannoni (2006) by making the model errors
“idiosyncratic”. The other notable feature of the observation equations concerns
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Table 3. Model State Variables
Symbol
Ct−1
It−1
p
πt−1
Description
Lagged Consumption
Lagged Investment
Lagged Price Inflation
Kt
At
at
at−1
Stock of Installed Capital
Hicks-Neutral Technology
Growth rate of At
Lagged Growth Rate of At
Zt
zt
zt−1
Investment-Specific Technology
Growth rate of Zt
Lagged Growth Rate of Zt
φt
bt
λw,t
Labor-Supply Shock
Discount Rate Shock
Employment Aggregator’s
Elasticity of Substitution
Intermediate Good Aggregator’s
Elasticity of Substitution
Marginal Efficiency of Investment Shock
Government Spending Share Shock
Lagged Nominal Interest Rate
Monetary Policy Shock
λp,t
µt
gt
Rt−1
εR,t
Disappears without
Habit-based Preferences
Investment Adjustment Costs
Indexing “stuck” prices
to lagged inflation
Autoregressive growth of At
Indexing “stuck” wages
to lagged labor productivity growth
Autoregressive growth of Zt
Indexing “stuck” wages
to lagged labor productivity growth
Time-varying Wage Markups
Time-varying Price Markups
Interest-rate Smoothing
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the GDP deflator. We model its growth as a share-weighted average of the
model’s consumption and investment deflators.
We denote the sample of all data observed with Y and the parameters governing
data generation with Θ = (θ, ϕ). The prior density for Θ is Π(Θ), which we
specify to be similar to that employed by Justiniano, Primiceri, and Tambalotti
(2011). Given Θ and a prior distribution for ζ0 , we can use the model solution
and the observation equations to calculate the conditional density of Y , F (Y |Θ).
To form the prior density of ζ0 , we apply the Kalman filter. The actual estimation
begins with 1987:Q1. Bayes rule then yields the posterior density up to a factor
of proportionality.
P (Θ|Y ) ∝ F (Y |Θ)Π(Θ)
We calculate our forecasts with the model’s parameter values set to this
posterior distribution’s mode.
Three Key Equations
This section summarizes the inferred parameters by reporting the estimates of
three key log-linearized equations: the Taylor Rule, the Price Phillips Curve, and
the Wage Phillips Curve.
Taylor Rule
3
5
X
1X
1.77
s
Rt = 0.84Rt−1 + (1 − 0.84) 1.63
πt−3 +
(yt − yt−4 ) + mp
+
ξt−s
,
t
4 j=0
4
s=1
Besides the lagged interest rate, the variables appearing on the right-hand side
are the four-quarter average of consumption inflation, the most recent
four-quarter output growth rate, the current monetary policy shock (mp
t ), and
s
the five previous quarters’ signals of the current monetary policy stance, ξt−s
for
s = 1, . . . , 4. (These signals play a prominent role in forecasting, but we do not
yet use them during estimation.) Note that
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• Holding the economy’s growth rate fixed, the long-run response of Rt to a
permanent one-percent increase in inflation is 1.63 percent. Thus, the
model satisfies the Taylor principle.
• Since the four-quarter growth rate of output replaces the usual output gap
in the rule, it is difficult to compare the estimated coefficient of 0.44 with
the typical calibrated output response of 0.5.
Price Phillips Curve
p
p
+ 0.019st + pt
+ 0.20πt−1
πtp = 0.80Et πt+1
Here, st represents intermediate goods producers’ common marginal cost.
• The associated Calvo probablity of an individual firm not updating its
price in a given quarter equals 0.86, which is well in line with other
calibrations.
• Producers unable to update their price with all current information are
allowed to index their prices to a convex combination of last quarter’s
p
in the
inflation rate with the steady-state inflation rate. This places πt−1
Phillips curve. The estimated weight on steady-state inflation is 0.76.
Wage Phillips Curve
The Wage Phillips curve can be written as
p
p
w
+ jt−1 = βEt πt+1
+ πt+1
+ jt+1 − ιw (πtp + jt ) +κw xt +w
πtw +πtp +jt −ιw πt−1
t ,
where πtw and πtp correspond to inflation in real wages and consumption prices
respectively, jt = zt +
α
1−α µt
is the economy’s technologically determined
stochastic trend growth rate, with α equal to capital’s share in the production
function, zt the growth rate of neutral technology, and µt the growth rate of
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p
investment-specific technical change. The term πt−1
+ zt−1 + jt arises from
indexation of wages to a weighted average of last quarter’s
productivity-adjusted good’s price inflation and its steady state value. The
estimated weight on the steady state equals 0.64. The log-linearized expression
for the ratio of the marginal disutility of labor, expressed in consumption units,
to the real wage is
xt = bt + ψt + νlt − λt − wt ,
where bt and ψt are disturbances to the discount factor and the disutility of
working, respectively, lt hours,λt the marginal utility of consumption and wt the
real wage. Finally, w
t is a white noise wage markup shock.
Note that without indexation of wages to trend productivity, this equation says
that nominal wage inflation (adjusted by trend growth) depends positively on
future nominal wage inflation (also appropriately trend-adjusted), and increases
in the disutility of labor-real wage gap.
The estimated equation is given by
p
p
w
+jt+1 −0.32 (πtp + jt )]+0.0055xt +w
+ jt−1 = 0.9988×Et [πt+1
+πt+1
πtw +πtp +jt −0.32 πt−1
t ,
The estimated Calvo probability of a wage remaining unadjusted in a given
quarter underlying the estimate of κw = 0.0055 equals 0.73.
The Model’s Shocks
Our discussion of recent macroeconomic developments above featured the
following shocks prominently: The discount rate shock, the “financial” shock to
the marginal efficiency of investment, and both anticipated and unanticipated
monetary policy shocks. In this section, we provide greater detail on the model’s
responses to these four shocks.
Figure 3 plots responses to a discount rate shock that increases impatience and
tilts desired consumption profiles towards the present. The variables examined
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are real GDP, the Federal Funds Rate, inflation, consumption, investment, and
hours worked. The responses are scaled so that the change in GDP after 16
quarters equals one percent.
In a neoclassical economy, this shock would be contractionary on impact. Upon
becoming more impatient, the representative household would increase
consumption and decrease hours worked. To the extent that the production
technology is concave, interest rates and real wages would rise; and regardless
of the production technology both real GDP and investment would drop.
Increasing impatience instead expands activity in this New Keynesian economy.
As in the neoclassical case, consumption rises on impact. Habit causes the
consumption growth to persist for two more quarters. Adjustment costs
penalize the sharp contraction and recovery of investment from the neoclassical
model, so instead investment remains unchanged on impact. Market clearing
requires either a rise of the interest rate (to choke off the desired consumption
expansion) or an expansion of GDP. By construction, the Taylor rule prevents the
interest rate from rising unless the shock is inflationary or expansionary.
Therefore, GDP must rise. This in turn requires hours worked to increase. Two
model features overcome the neoclassical desire for more leisure. First, some of
the labor variants’ wages are sticky. For those, the household is obligated to
supply whatever hours firms demand. Second, the additional labor demand
raises the wages of labor variants with wage-setting opportunities. This rise in
wages pushes marginal cost up and lies behind the short-run increase in
inflation. After inflation has persisted for a few quarters, monetary policy
tightens and real rates rise.
Since the discount rate shock moves output and prices in the same direction, a
Keynesian analysis would label it a shift in “demand.” In the neoclassical sense,
it is also a demand shock, albeit a reduction in the demand for future goods. The
matching neoclassical supply shock in our model is to the marginal efficiency of
investment. A positive shock to it increases the supply of future goods. Figure 4
plots the responses to such a shock. Barro and King (1984) and Greenwood,
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Figure 3. Responses to a Discount Rate Shock
GDP level
8
6
6
4
4
2
2
0
0
5
10
15
Federal Funds Rate
8
20
0
0
Core PCE inflation
3
8
2
6
1.5
4
1
2
0.5
0
0
0
5
10
15
20
15
20
0
5
10
15
20
Hours level
Investment level
2
−2
10
Consumption level
10
2.5
5
8
0
6
−2
4
−4
2
−6
−8
0
5
10
15
20
0
0
5
10
15
20
Note: The impulse was scaled to yield a one percent increase in GDP after 16 quarters.
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Hercowitz, and Huffman (1988) consider the analogous responses from a
neoclassical model. Increasing the marginal efficiency of investment raises
investment, hours worked, GDP, and the real interest rate but decreases
consumption. Two aspects of our model stop consumption from falling on the
same shock’s impact. First, habit-based preferences penalize an immediate
decrease in consumption. Second, monetary policy responds to the shock only
slowly, so real interest rates actually fall slightly on impact. The investment
adjustment costs give the responses of GDP, hours, and investment their hump
shape. Although this shock changes the economy’s technology for intertemporal
substitution – and therefore deserves the neoclassical label “supply” – it makes
prices and output move in the same direction. For this reason, it falls into our
Keynesian taxonomy’s “demand” category.
As noted above, monetary policy shocks partially offset negative demand shocks
in 2010. Figures 6 and 5 present the impulse response functions for two of these,
the unanticipated “contemporaneous” shock and the shock revealed to all
agents five quarters in advance. The responses to the unanticipated shock are
standard, but those following an anticipated shock require more explanation. At
the announcement date, the expected value of the policy rate five quarters hence
increases by 300 basis points. Because both Phillips curves are forward looking,
this expected contraction causes both prices and quantities to fall. This
anticipated weakness then feeds through the Taylor rule to create a gradual
easing of policy. When the anticipated tightening arrives, it mostly offsets the
prior endogenous easing. The policy rate rises only 85 basis points above its pre-shock
value. These responses are clearly worthy of further study.
Shock Decomposition Methodology
We credit Charles Evans with the original ideas behind this decomposition. For
the shock decomposition, we set the model’s parameters to their values at the
ˆ Using all available data we use the Kalman
posterior distribution’s mode, θ.
smoother to extract sequences of estimated states {ζˆt }Tt=1 and a innovations
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Figure 4. Responses to a Marginal Efficiency of Investment Shock
GDP level
2
Federal Funds Rate
2
1.5
1.5
1
1
0.5
0.5
0
0
0
5
10
15
20
−0.5
0
5
Core PCE inflation
0.5
10
15
20
Consumption level
1
0.4
0.5
0.3
0
0.2
0.1
0
5
10
15
20
0
5
10
15
20
Hours level
Investment level
15
−0.5
2
1.5
10
1
5
0.5
0
−5
0
0
5
10
15
20
−0.5
0
5
10
15
20
Note: The impulse was scaled to yield a one percent increase in GDP after 16 quarters.
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Figure 5. Responses to an Unanticipated Monetary Policy Shock
GDP level
2.5
Federal Funds Rate
2
2
0
1.5
−2
1
−4
0.5
0
0
5
10
15
20
Core PCE inflation
1.5
−6
0
5
10
15
20
Consumption level
1.5
1
1
0.5
0.5
0
0
0
5
10
15
20
−0.5
5
Investment level
8
10
15
20
Hours level
2.5
6
2
4
1.5
2
1
0
−2
0
0.5
0
5
10
15
0
20
0
5
10
15
20
Note: The impulse was scaled to yield a one percent decrease in GDP 16 quarters after that date.
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Figure 6. Responses to an Anticipated Monetary Policy Shock
GDP level
0
−0.5
0
−1
−1
−1.5
−2
−2
0
5
10
15
20
−3
0
10
15
20
Consumption level
0.5
0.5
0
−1
−0.5
0
5
10
15
20
−1
−0.5
−2
−1
−4
−1.5
5
10
15
5
10
15
20
0
0
0
0
Hours level
Investment level
2
−6
5
Core PCE inflation
0
1.5
Federal Funds Rate
1
20
−2
0
5
10
15
20
Note: The monetary policy shock is revealed to all agents five quarters before its realization, and
the impulse was scaled to yield a one percent decrease in GDP 16 quarters after that date.
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{ˆ
εt }Tt=1 . By construction, these satisfy the estimated transition equation for the
state.
ˆ ζˆt−1 + εˆt ,
ζˆt = F (θ)
To keep this discussion simple, we henceforth suppose that the “error” shocks in
vt equal zero. Incorporating them into the analysis changes the actual
calculations only little.
For concreteness, suppose that the forecasted object of interest is Q4-over-Q4
GDP growth for 2010. We position ourselves in 2009:Q4 and calculate
2009:Q4
ˆ ζˆ2009:Q4
ζˆ2010:Q1
≡ F (θ)
2009:Q4
ˆ ζˆ2009:Q4
ζˆ2010:Q2
≡ F (θ)
2010:Q1
ˆ ζˆ2009:Q4
= F 2 (θ)
..
.
2009:Q4
ˆ ζˆ2009:Q4
ζˆ2010:Q4
≡ F (θ)
2010:Q3
These are the “expectations” of the model’s states in each quarter of 2010
conditional on the state at the end of 2009 equalling its estimated value.
With these “state forecasts” in hand, we can construct corresponding forecast
errors by comparing them with their “realized values” from the Kalman
smoother. For the period t state forecasted in 2009:Q4, we denote these with
ηˆt2009:Q4 = ζˆt − ζˆt2009:Q4 .
These forecast errors are related to the structural shocks by
ηˆt2009:Q4 =
t−2009:Q4
X
ˆ ε2009:Q4+j .
F j−1 (θ)ˆ
j=1
ˆ 2009:Q4 for
The shock decomposition is based on four alternative forecasts, ζ(ι)
t
t = 2010:Q1, . . . , 2010:Q4 and ι ∈ {D, S, M, R}. Here, ι indexes one of the four
groups of structural shocks. For these, let εˆ(ι)t denote a version of εˆt with all
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shocks except those in group ι set to zero. With these, we construct
ˆ 2009:Q4 ≡ F (θ)
ˆ ζˆ2009:Q4 + εˆ(ι)2010:Q1 ,
ζ(ι)
2010:Q1
..
.
2009:Q4
ˆ ζˆ2009:Q4 + εˆ(ι)2010:Q4 ,
ζˆ2010:Q4
≡ F (θ)
2010:Q3
and
ˆ 2009:Q4 .
ηˆ(ι)2009:Q4
≡ ζˆt − ζ(ι)
t
t
By construction,
ηˆt2009:Q4 =
X
ηˆ(ι)2009:Q4
.
t
ι∈{D,S,M,R}
That is, each forecast error can be written as the sum of contributions from each
of the shock groups. Using the observation equations, we transform these into
components of the forecast error for observable variables.
With this completed, we can then move the forecast date forward to 2010:Q1.
The decomposition for that date proceeds similarly, except that we treat growth
in 2010:Q1 as data.
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Bibliography
Barro, R. J. and R. G. King (1984). Time-separable preferences and
intertemporal-substitution models of business cycles. The Quarterly Journal of
Economics 99(4), pp. 817–839.
Boivin, J. and M. Giannoni (2006). DSGE models in a data-rich environment.
Working Paper 12772, National Bureau of Economic Research.
Greenwood, J., Z. Hercowitz, and G. W. Huffman (1988). Investment, capacity
utilization, and the real business cycle. The American Economic Review 78(3),
pp. 402–417.
Justiniano, A., G. E. Primiceri, and A. Tambalotti (2011). Investment shocks and
the relative price of investment. Review of Economic Dynamics 14(1), 101–121.
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