Full text of Federal Open Market Committee Meeting Minutes, Transcripts, and Other Documents : Meeting, June 21-22, 2011 : Memo: Projections from EDO: Understanding the Current Outlook and the Great Recession in a DSGE Model

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Projections from EDO: Understanding the Current

Outlook and the Great Recession in a DSGE Model


Hess Chung, Michael T. Kiley, and Jean-Philippe Laforte

∗

June 1, 2011

1

The Outlook for 2011 to 2013

The EDO model projects economic growth a touch above trend and low inflation,
with monetary policy gradually lifting the federal funds rate over the next several
years. In particular, the current sizable shortfall of production relative to long-run
sustainable levels abates as the aversion to risk-taking apparent in the elevated level
of risk premia (and, implicitly, restrictions on credit availability) seen by EDO falls
back to historically typical levels. Inflation remains low as wage pressures are weak
relative to labor productivity, reflecting the declines in household wealth over the past
several years, low level of hours worked anticipated over the next few years, and the
rapid increases in productivity seen in 2009.
Conditional on the projected decline in risk premia, EDO projects that real GDP
will advance at a pace modestly above trend going forward– about 3 percent, on
average, over 2011-2013, as shown in figure 1. This improvement brings real activity
closer to EDO’s estimate of its long-run trend (as shown in figure 2). Moreover, the
above-trend pace of growth is accompanied by inflation just above 1 percent per year,
Hess Chung (hess.t.chung@frb.gov), Michael T. Kiley (michael.t.kiley@frb.gov), and JeanPhilippe Laforte (jean-philippe.laforte@frb.gov) are affiliated with the Division of Research and
Statistics of the Federal Reserve Board.
∗

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substantially below the target of 2 percent, reflecting the labor market slack apparent
in the output gap. Given these developments, the federal funds rate is projected to
remain near zero until late in 2011 and only rises gradually thereafter.1

Figure 1: Recent History and Forecasts
EDO Projection Summary

2010
Q4/Q4
Real GDP (a)
Credible set (c)

2011
Q4/Q4

2012
Q4/Q4

2013
Q4/Q4

2.8

3.2

3.0

2.8

2.8-2.8

1.8-4.0

1.5-4.4

1.3-4.2

1.1

1.1

1.1

.8-1.4

.6-1.6

.6-1.8

Core PCE Price index (a) 0.8
Credible set (c)
Federal Funds Rate (b)
Credible set (c)

.8-.8
0.2

0.5

1.7

2.4

.2-.2

.1-1.4

.6-3.0

1.0-3.9

(a) Q4/Q4 percent change, (b) Q4 level, (c) 68 percent

Black, solid line -- Data (through 2010Q4) and projections; Black, dashed line -- Steady-state or trend values
Contributions (bars): Red -- Financial; Blue -- Technology; Silver -- Monetary policy; Green -- Other

1
The EDO model has been shown to forecast as well as, or better than, alternatives in a number of
papers (e.g., Edge, Kiley, and Laforte (2010) and Wieland and Wolters (2010); however, forecasting
is very challenging, and models generally perform similar to, but not better than, simple time series
alternatives, or consensus forecasts.

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Figure 2: Deviation of Real GDP from its Long-run (Stochastic) Trend

The decomposition of the projections for these variables shown in figure 1 high­
lights the important role that the adverse shocks to financial conditions in 2008 and
early 2009 play in shaping the recession in that period and the projected recovery.
Specifically, the figures decompose the movements in real GDP, the federal funds rate,
and core inflation into the contributions from financial (risk premium) shocks, mone­
tary policy shocks, productivity movements, and other disturbances (largely markup,
or Phillips-curve, shocks); the first two are traditional “demand” shocks, and the
latter two are traditional “supply” shocks. As shown in the federal funds rate chart,
the need to accommodate the adverse impact of the tightening in financial conditions
(the red bars) is the most largest factor holding the federal funds rate at a low level
through the projection; indeed, monetary policy “shocks” were largely positive – that
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is, contractionary – in 2010, reflecting the binding zero-lower bound constraint. The
recovery in real GDP projected for 2011-13 is essentially entirely the result of the pro­
jected step-up in demand that should accompany lower risk premia, again illustrated
by the contribution of the red bars in the GDP chart.
Given that the outlook is largely driven by an unwinding of the factors that caused
the Great Recession, the next section walks through the EDO model’s accounting for
developments during that period.

2	

The Great Recession and the Prospects for Re­
covery

Real GDP in the second quarter of 2009 was more than 4 percent below its level a
year earlier, the sharpest four-quarter decline since the Great Depression. Over the
same four quarters, payroll employment fell more than 7 percent and hours worked
in the nonfarm business sector fell 8 percent.
What happened, and what is the outlook for 2011-2013?
The basic elements of the story are now well known (although the reasons why
these developments occurred are not fully understood). Specifically, a financial crisis
created sizable frictions all along the chain that connects savers and borrowers, and
these financial frictions led to a collapse in spending on intermediated purchases,
primarily investment-type expenditures; indeed, as emphasized elsewhere (e.g., Hall
(2010)), these financial frictions became apparent in a widening of the spread between
the costs of financial funds facing households and businesses and the rates on low-risk
assets (such as Treasury securities or federal funds).
For example, the left column of figure 3 reports three measures of financial condi­
tions facing households and firms: the spread between the average rate on personal
loans from banks and the federal funds rate since 1985; the spread between the earn­
ings yield on the S$P500 and the federal funds rate; and the spread between the
interest rate on a one-year adjustable-rate mortgage (ARM) and the yield on a oneyear U.S. Treasury note. Both the personal loan and earnings/price-based spreads
have cyclical patterns and jumped during 2008 as the financial crisis worsened. The
ARM spread increased sharply, as falling rates on low risk government obligations
were not accompanied by substantially lower mortgage rates for households, prob­

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ably reflecting both higher risk associated with loans backing home purchases and
imparied financial positions at mortgage lenders. Looking at earlier periods, other
notable developments are also apparent – for example, with the earnings/price spread
reaching lows in the late 1990s as equity prices soared, an easing in financial conditions
not apparent in a measure like the federal funds rate.

Figure 3: Financial Frictions
Data on Financial Conditions and EDO Risk Premia

Black, solid line -- Data (through 2010Q4); Black, dashed line -- EDO model estimate of financial friction

The right-hand column of figure 3 shows the movements in the indicators of fi­
nancial frictions just discussed along with several of the risk-premium measures from
EDO. these risk premia summarize EDO’s estimates of the role of financial frictions
and are derived from the model’s structure and the data on activity, inflation, and
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risk-free rates that inform EDO’s projections; as a result, the correspondence between
observable measures of frictions and EDO’s estimates provide a cross-check on the
model’s interpretation of the Great Recession. As shown in the top panel, the swings
in the spread between the rate on personal loans and the federal funds rate correlate
well with EDO’s aggregate risk premium. The middle panel reveals that the com­
bined aggregate risk premium and premium on business investment corresponds well
with the earnings-equity price/funds rate spread; and the model’s estimate of the risk
premium attached to residential investment moves with the ARM spread.
The role played by a widening in risk premiums during the Great Recession was
previously highlighted in figure 1. Specifically, the red bars show that the downturn
was driven by the rise in risk premiums; particularly significant was the aggregate risk
premium, shown in the upper right panel of figure 3. As shown in figure 4, an increase
in the aggregate risk premium (that is, a shift in demand away from real assets and
towards the nominal risk-free asset) causes the price level to fall, but, due to shortrun nominal rigidities, insufficiently to completely offset the shift.2 The resulting fall
in the demand for physical stocks lowers wages, hours and output throughout the
economy, with a particularly sharp fall in the capital-producing sector. With the
lower path for income, non-durable consumption growth falls below trend for several
quarters.
Turning to price developments, much of the high-frequency movements in infla­
tion since 2008 have reflected short-term noise, according to EDO, rather than the
effects of economic slack. In particular, the model would have expected only modest
downward pressure in the very short run on marginal costs following a transient risk
premium shock. As a result, other factors, most notably “markup” shocks, account
for a large portion of the weak readings on inflation seen in late 2008 (movements
which were subsequently reversed, consistent with the “noise” interpretation of such
shocks). Nonetheless, the low realizations of marginal cost associated with the strong
performance of productivity relative to wages, on net, since mid-2009, combined with
the sizable decline in wealth attributed to elevated risk premia for an extended period,
exert sizable influences on the outlook for prices, as can be seen in the blue and red
bars.

2

These responses are to a one standard deviation shock, and the responses are expressed at a
quarterly rate.

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Figure 4: Impulse Responses: Aggregate risk-premium

−0.3

−0.4

−0.5

−0.6


−0.2

−0.3

−0.4


−0.2


Real Housing

Real GDP

−0.2


−0.1
Real Durables

Real Consumption

−0.1

0

0
−0.4

−0.6

−0.8

−1


−1

−1.5


−1.2

−0.7
5 10 15 20

5 10 15 20


5 10 15 20

5 10 15 20


0

−0.005

Hours

−1

−2

Fed Funds

−0.2
−0.4

−0.6

5 10 15 20


−0.04
−0.06

−0.08

−0.1

−0.12

−0.8

−3

Core PCE inflation

−0.02
Real Investment

−0.5


5 10 15 20


−0.01

−0.015

−0.02

−0.025

−0.03

−0.035

5 10 15 20


5 10 15 20


Our discussion of the exogenous disturbances that led to the recession may leave
some questioning the economic mechanisms through which risk premia affect demand
in EDO. To recap, EDO identifies the lead up to and the early stages of the recession
as associated with increasingly tight terms on financing residential investment, i.e.,
as driven by an increase in the risk premium on residential investment. However,
as the economic weakness broadened to include overall consumer spending and busi­
ness investment, the primary driver of the weakness centered on an increase in the
economywide risk premium.
Within the models interpretation of events, these shifts in fundamentals brought
about the weakness in economic activity. The increase in the risk premium asso­
ciated with residential investment directly depressed residential spending and real
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estate prices by raising the cost of capital for such spending, but the overall macroe­
conomic impact would have been fairly limited, according to EDO, if economywide
risk premiums had not risen as well. In this regard, future work may wish to inves­
tigate the mechanisms that could link the weakness in housing to the more general
macroeconomic fallout that followed the decline in house prices, perhaps through a
more sophisticated modeling of financial intermediation.
That said, the sharp increase in the economywide risk premium estimated for
the second half of 2008 through the middle of 2009 depressed consumer spending,
residential investment, and business investment through a range of channels. First,
this increase directly raised the cost of capital for residential and business investment
and consumer durable outlays. In addition, higher risk premiums lowered household
wealth (including equity claims on firms and the value of residential real estate), de­
pressing consumption of nondurables and services, and also led households topostpone
consumption. These declines in spending were further exacerbated by the weakening
in labor income.
All else equal, EDO would have expected a fairly strong recovery to have com­
menced after the first half of 2009, as risk premiums were projected to fall and mon­
etary policy would have been expected to provide continuous support to the recovery
in normal times. However, three conditions contributed to a more moderate recovery.
First, the zero lower bound limited the degree to which monetary policy could sup­
port the recovery. Second, risk premiums are estimated to have fallen more slowly
than expected, restraining the recovery in demand. Finally, the persistently slow re­
covery led to a modest downward adjustment in the models estimate of the economys
productive potential.

3

An Overview of Key Model Features

Figure 5 provides a graphical overview of the model. While similar to most related
models, EDO has a more detailed description of production and expenditure than
most other models.3
Specifically, the model possesses two final good sectors in order to capture key
long-run growth facts and to differentiate between the cyclical properties of different
3

Chung, Kiley, and Laforte (2011) provide much more detail regarding the model specification,
estimated parameters, and model propeties.

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Figure 5: Model Overview

categories of durable expenditure (e.g., housing, consumer durables, and nonresiden­
tial investment). For example, technological progress has been faster in the production
of business capital and consumer durables (such as computers and electronics).
The disaggregation of production (aggregate supply) leads naturally to some dis­
aggregation of expenditures (aggregate demand). We move beyond the typical model
with just two categories of (private domestic) demand (consumption and investment)
and distinguish between four categories of private demand: consumer non-durable
goods and non-housing services, consumer durable goods, residential investment, and
non-residential investment. The boxes surrounding the producers in the figure illus­
trate how we structure the sources of each demand category. Consumer non-durable
goods and services are sold directly to households; consumer durable goods, resi­
dential capital goods, and non-residential capital goods are intermediated through
capital-goods intermediaries (owned by the households), who then rent these capi­
tal stocks to households. Consumer non-durable goods and services and residential
capital goods are purchased (by households and residential capital goods owners,
respectively) from the first of economy’s two final goods producing sectors, while con­
sumer durable goods and non-residential capital goods are purchased (by consumer
durable and residential capital goods owners, respectively) from the second sector.
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In addition to consuming the non-durable goods and services that they purchase,
households supply labor to the intermediate goods-producing firms in both sectors of
the economy.
This remainder of this section provides an overview of the key properties of the
model. In particular, the model has five key features:
• Production of goods and services occurs in two sectors, with differential rates
of technological progress across sectors.
• A disaggregated specification of household preferences and firm production pro­
cesses that leads to separate modeling of nondurables and services consumption,
durables consumption, residential investment, and business investment.
• Risk premia associated with different investment decisions play a central role
in the model. These include A) an aggregate risk-premium, or natural rate of
interest, shock driving a wedge between the short-term policy rate and the in­
terest rate facing private decisionmakers (as in Smets and Wouters (2007)) and
B) fluctuations in the discount factor/risk premia facing the intermediaries fi­
nancing household (residential and consumer durable) and business investment.
• A new-Keynesian structure for price and wage dynamics.
• A monetary policy that reacts to inflation and a measure of resource utilization.

3.1

Two-sector production structure

It is well known (e.g., Edge, Kiley, and Laforte (2010)) that real outlays for business
investment and consumer durables have substantially outpaced those on other goods
and services, while the prices of these goods (relative to others) has fallen. For exam­
ple, real outlays on consumer durables have far outpaced those on other consumption,
while prices for consumer durables have been flat and those for other consumption
have risen substantially; as a result, the ratio of nominal outlays in the two categories
has been much more stable, although consumer durable outlays plummeted in the
Great Recession. Many models fail to account for this fact.
EDO accounts for this development by assuming that business investment and
consumer durables are produced in one sector and other goods and services in another
sector. Specifically, production by firm j in each sector s (where s equals kb for the
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sector producing business investment and consumer durables sector and cbi for the
sector producing other goods and services) is governed by a Cobb-Douglas production
function with sector-specific technologies:
s
Xts (j) = (Ztm Zts Lt (j))1−α (Ktu,nr,s (j))α , for s = cbi, kb.

(1)

In 1, Z m represents (labor-augmenting) aggregate technology, while Z s represents
(labor-augmenting) sector-specific technology; we assume that sector-specific techno­
logical change affects the business investment and consumer durables sector only; Ls is
labor input and K u,nr,s is capital input (that is, utilized non-residential business cap­
ital (and hence the nr and u terms in the superscript). Growth in this sector-specific
technology accounts for the long-run trends, while high-frequency fluctuations allow
the possibility that investment-specific technological change is an important source
of business cycle fluctuations.

3.2

The structure of demand

EDO differentiates between several categories of expenditure. Specifically, business in­
vestment spending determines non-residential capital used in production, and house­
holds value consumer nondurables goods and services, consumer durable goods, and
residential capital (e.g., housing). Differentiation across these categories is important,
as fluctuations in these categories of expenditure can differ notably, with the cycles
in housing and business investment, for example, occurring at different points over
the last three decades.
Valuations of these goods and services, in terms of household utility, is given by
the following utility function:
∞

E0

cnn
β t ς cnn ln(Etcnn (i)−hEt−1 (i))+ς cd ln(Ktcd (i))

t=0


+ς r ln(Ktr (i)) −ς l

(Lcbi (i)+Lkb (i))1+ν
t
t
,
1+ν

(2)

where E cnn represents expenditures on consumption of nondurable goods and ser­
vices, K cd and K r represent the stocks of consumer durables and residential capital
(housing), Lcbi + Lkb represents the sum of labor supplied to each productive sector
(with hours worked causing disutility), and the remaining terms represent parame­
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ters (such as the discount factor, relative value in utility of each service flow, and the
elasticity of labor supply).
By modeling preferences over these disaggregated categories of expenditure, EDO
attempts to account for the disparate forces driving consumption of nondurables and
durables, residential investment, and business investment – thereby speaking to issues
such as the surge in business investment in the second half of the 1990s or the housing
cycle the early 2000s recession and the most recent downturn. Many other models do
not distinguish between developments across these categories of spending.

3.3

Risk premia, financial shocks, and economic fluctuations

The structure of the EDO model implies that households value durable stocks accord­
ing to their expected returns, including any expected service flows, and according to
their risk characteristics, with a premium on assets which have high expected re­
turns in adverse states of the world. However, the behaviour of models such as EDO
is conventionally characterized under the assumption that this second component
is negligible. In the absence of risk adjustment, the model would then imply that
households adjust their portfolios until expected returns on all assets are equal.
Empirically, however, this risk adjustment may not be negligible and, moreover,
there may be a variety of factors, not explicitly modelled in EDO, which limit the
ability of households to arbitrage away expected return differentials across different
assets. To account for this possibility, EDO features several exogenous shocks to the
rates of return required by the household to hold the assets in question. Following
such a shock – an increase in the premium on a given asset, for example– households
will wish to alter their portfolio composition to favor the affected asset, leading to
changes in the prices of all assets and, ultimately, to changes in the expected path of
production underlying these claims.
The “sector-specific” risk shocks affect the composition of spending more than the
path of GDP itself. This occurs because a shock to these premia leads to sizable sub­
stitution across residential, consumer durable, and business investment; for example,
an increase in the risk premia on business investment leads households to shift away
from business investment and towards residential investment and consumer durables.
Consequently, it is intuitive that a large fraction of the non-cyclical, or idiosyncratic,
component of investment flows to physical stocks will be accounted for by movements

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in the associated premia.
Shocks to the required rate of return on the nominal risk-free asset play an es­
pecially large role in EDO. Following an increase in the premium, in the absence of
nominal rigidities, the households’ desire for higher real holdings of the risk-free asset
would be satisfied entirely by a fall in prices, i.e., the premium is a shock to the
natural rate of interest. Given nominal rigidities, however, the desire for higher riskfree savings must be off-set, in part, through a fall in real income, a decline which
is distributed across all spending components. Because this response is capable of
generating comovement across spending categories, the model naturally exploits such
shocks to explain the business cycle. Reflecting this role, we denote this shock as the
“aggregate risk-premium”.

3.4

New-Keynesian Price and Wage Phillips Curves

As in most of the related literature, nominal prices and wages are both “sticky” in
EDO. This friction implies that nominal disturbances – that is, changes in monetary
policy – have effects on real economic activity. In addition, the presence of both
price and wage rigidities implies that stabilization of inflation is not, in general, the
best possible policy objective (although a primary role for price stability in policy
objectives remains).
Given the widespread use of the New-Keynesian Phillips curve, it is perhaps easiest
to consider the form of the price and wage Phillips curves in EDO at the estimated
parameters. The price Phillips curve (governing price adjustment in both productive
sectors) has the form:

p,s
p,s
p,s
s
πt = 0.22πt−1 + 0.76Et πt+1 + .017mcs + θt
t

(3)

where mc is marginal cost and θ is a markup shock. As the parameters indicate,
inflation is primarily forward-looking in EDO.
The wage (w) Phillips curve for each sector has the form:

)
(
c,l
s
s
s
s
w
△wt = 0.01△wt−1 + 0.95Et △wt+1 + .012 mrst − wt + θt + adj. costs.

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(4)

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where mrs represents the marginal rate of substitution between consumption and
leisure. Wages are primarily forward looking and relatively insensitive to the gap
between households’ valuation of time spent working and the wage.

3.5

The Monetary Policy Rule

The estimated monetary policy rule has standard features – the policy interest rate
responds inertially to inflation and a deviation of output from a trend level:

(
)
Rshock
.
rt = 0.76rt−1 + (1 − 0.76) 1.50△PtP CE + 1.20 (yt − trend) + δt
Rshock
Rshock
= ρRshock δt−1 + ǫR
δt
t

(5)
(6)

The long-run responses to the output gap and inflation are very similar to those
in the literature. The measure of trend output is based on a production-function
concept – that is, trend output is the level of output consistent with labor input
and the utilization of capital at long-run levels, given the current level of productive
capital; this output concept is a Divisia aggregate of production in the two sectors
discussed earlier.

3.6

Summary of Model Specification

To summarize, fluctuations in all economic variables are driven by eleven structural
shocks. It is most convenient to summarize these shocks into four broad categories:
• Permanent technology shocks: This category consists of shocks to aggregate
and investment-specific (or fast-growing sector) technology.
• Financial, or intertemporal, shocks: This category consists of shocks to risk
premia. In EDO, variation in risk premia – both the premium households’
receive relative to the federal funds rate on nominal bond holdings and the
additional variation in discount rates applied to the investment decisions of
capital intermediaries – are purely exogenous. Nonetheless, the specification
captures aspects of related models with more explicit financial sectors (e.g.,
Bernanke, Gertler, and Gilchrist (1999)).
• Monetary policy shocks.
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• Other shocks: This category is dominated by shocks to price and wage markups,
or Phillips curve shock; it als includes the shock to autonomous demand, which
is quantitatively not important in EDO.

4

Estimation: Data and Properties

4.1

Data

The empirical implementation of the model takes a log-linear approximation to the
first-order conditions and constraints that describe the economy’s equilibrium, casts
this resulting system in its state-space representation for the set of (in our case 12)
observable variables, uses the Kalman filter to evaluate the likelihood of the observed
variables, and forms the posterior distribution of the parameters of interest by com­
bining the likelihood function with a joint density characterizing some prior beliefs.
Since we do not have a closed-form solution of the posterior, we rely on Markov-Chain
Monte Carlo (MCMC) methods.
Because of the detailed modeling of demand, EDO can consider more data on
expenditure than other related models to inform its parameter estimates and pro­
jections. The model is estimated using 12 data series over the sample period from
1984:Q4 to 2011:Q1. The series are:
1. The growth rate of real gross domestic product (ΔGDP );
2. The growth rate of real consumption expenditure on non-durables and services
(ΔC);
3. The growth rate of real consumption expenditure on durables (ΔCD);
4. The growth rate of real residential investment expenditure (ΔRes);
5. The growth rate of real business investment expenditure (ΔI);
6. Consumer price inflation, as measured by the growth rate of the Personal Con­
sumption Expenditure (PCE) price index (ΔPC,total );
7. Consumer	 price inflation, as measured by the growth rate of the PCE price
index excluding food and energy prices (ΔPC,core );
8. Inflation for consumer durable goods, as measured by the growth rate of the
PCE price index for durable goods (ΔPcd );

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9. Hours, which equals hours of all persons in the non-farm business sector from
the Bureau of Labor Statistics (H);4
10.	 The growth rate of real wages, as given by compensation per hour in the nonfarm business sector from the Bureau of Labor Statistics divided by the GDP
price index (ΔRW );
11.	 The federal funds rate (R).
12.	 The yield on the 2-yr. U.S. Treasury security (RL).
Our implementation adds measurement error processes to the likelihood implied
by the model for all of the observed series used in estimation except the short-term
nominal interest rate series.
Figure 6 presents the observed data (in blue) and the observable data net of
the model’s estimated measurement error (in black), along 95 percent confidence
intervals. For series other than overall PCE price inflation, measurement error is a
moderate portion of movements in the series. The larger role for measurement error
in accounting for the path of PCE price inflation reflects the absence of separate
sectors for food and energy in the model.

4.2

Estimates of shocks and exogenous fundamentals

Figures 7 and 8 report modal estimates of the model’s structural shocks and the per­
sistent exogenous fundamentals (i.e., risk premia and autonomous demand). These
series have recognizable patterns for those familiar with U.S. economic fluctuations.
For example, the risk premia jump at the end of the sample, reflecting the finan­
cial crisis and the model’s identification of risk premia, both economy-wide and for
housing, as key drivers.
Of course, these stories from a glance at the exogenous drivers yield applications
for alternative versions of the EDO model and future model enhancements. For example, the exogenous risk premia can easily be made to have an endogenous component
following the approach of Bernanke, Gertler, and Gilchrist (1999) (and indeed we
have considered models of that type). At this point we view incorporation of such
4

We remove a low-frequency trend from hours via the Hodrick-Prescott filter with a smoothing
parameter of 128000; our model is not designed to capture low frequency trends in population growth
or labor force participation.

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Real Durables

1990

2000

2000

0
−0.5
−1

2010

−5
1990

0.5

2010

1990

2000

1990

0
−5

1990

2000

1990

1

0
−0.5
−1
1990

2000

2010

2000

2010

1990

2000

2010

1
0
−1

2010

Core Inflation

0
−1

2010

0.5

2010

−10

1990
Two−Year Treasury

2000

2000

2

5

PCE Inflation

Aggregate Hours
Investment Inflation

1990

−10

2010

1

−5

0

Real Wage

−2

Real Investment

Real Non−Durables

−1

Real Housing Expenditures

Real GDP

Figure 6: Smoothed Observables and Data

2000

2010

1990

2000

0.8
0.6
0.4
0.2
0
−0.2
−0.4

2010

1
0
−1

mechanisms in our baseline approach as premature, pending ongoing research on fi­
nancial frictions, banking, and intermediation in dynamic general equilibrium models.
Nonetheless, the EDO model captured the key financial disturbances during the last
several years in its current specification, and examining the endogenous factors that
explain these developments will be a topic of further study.

4.3

Variance Decompositions and impulse responses

We provide detailed variance decompositions and impulse response in Chung, Kiley,
and Laforte (2011), and only highlight the key results here.
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−2
1990

2000

−2
2000

2010

2000

0
−1
1990

2000

2010

2
1
0
−1
1990

2000

0
−0.2

2010

1

2010

−1

Funds Rate Shock
1990

Overall TFP

2010

−1

1990

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

1990

2000

2010

1
0
−1
−2

20
0
−20
−40

0.2
Risk−premium

Capital Risk−Premium

2000

−10

−1

1

Term Premium

Invest. Price Markup

Capital Goods Technology

1990

0

0.2

Non−Invest. Price Markup

−1

10

Durables Risk−Premium

Wage Markup

20

Housing Risk−Premium

Exog. Demand

Figure 7: Innovations to Exogenous Processes

0.5
0
−0.5

0.1
0
−0.1
−0.2

−1
1990

2000

2010

1990

2000

2010

Volatility in hours per capita is accounted for primarily by the economy-wide
risk premium and business investment risk premium shocks at horizons between one
and sixteen quarters. The large role for risk premia shocks in the forecast error
decomposition at business cycle horizons illustrates the importance of this type of
“demand” shock for volatility in the labor market. This result is notable, as hours
per capita is the series most like a “gap” variable in the model – that is, house per
capita shows persistent cyclical fluctuations about its trend value.
Volatility in aggregate GDP growth is accounted for primarily by the technol­
ogy shocks in each sector, although the economy-wide risk premium shock contributes
non-negligibly to the unconditional variance of GDP growth.
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Exog. Demand

1
0
−1
1990

1995

2000

2005

2010

−1
1995

2000

2005

2010

−2
−4
1985

1990

1995

2000

2005

2010

−2
1985

1990

1995

2000

2005

0
−2

2010

1990

1995

2000

2005

2010

1990

1995

2000

2005

2010

1990

1995

2000

2005

2010

1990

1995

2000

2005

2010

2
1
0
−1
−2
1985

Durables Risk−Premium

1990

2−y Term premium

Capital Risk−Premium

Housing Risk−Premium

1985

2

1985

TFP Tech.

−2
1985

Capital−specific Tech.

Risk−premium

Figure 8: Exogenous Drivers

20
0
−20
−40
1985
1

0.5

0
1985

Volatility in core inflation is accounted for primarily by the markup shocks in
the short run and technology shocks in the long run.
Volatility in the federal funds rate is accounted for primarily by the econo­
mywide risk premium.
Volatility in expenditures on consumer non-durables and non-housing
services is, in the near horizon, accounted for predominantly by economy-wide and
non-residential investment specific risk-premia shocks.
Volatilities in expenditures on consumer durables, residential invest­
ment, and non-residential investment are, in the near horizon, accounted for
predominantly by their own sector specific risk-premium shocks.
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With regard to impulse responses, we previously highlight the responses to the
most important shock, the aggregate risk premium, in figure 4. As we noted, this
shock looks like a traditional demand shock, with an increase in the risk premium
lowering real GDP, hours worked, and inflation; monetary policy offsets these nega­
tive effects somewhat by becoming more accommodative. As for responses to other
disturbances, the impulse responses to a monetary policy innovation captures the con­
ventional wisdom regarding the effects of such shocks. In particular, both household
and business expenditures on durables (consumer durables, residential investment,
and nonresidential investment) respond strongly (and with a hump-shape) to a con­
tractionary policy shock, with more muted responses by nondurables and services
consumption; each measure of inflation responds gradually, albeit more quickly than
in some analyses based on vector autoregressions (VARs).5
Shocks to sectoral risk premia principally depress spending in the associated cate­
gory of expenditure (e.g., an increase in the residential risk premium lowers residential
investment), with offsetting positive effects on other spending (which is “crowded in”).
Following an economy-wide technology shock, output rises gradually to its long­
run level; hours respond relatively little to the shock (in comparison to, for example,
output), reflecting both the influence of stick prices and wages and the offsetting
income and substitution effects of such a shock on households willingness to supply
labor.

References
Bernanke, B., M. Gertler, and S. Gilchrist. 1999. The financial accelerator in a quan­
titative business cycle framework, In: John B. Taylor and Michael Woodford, Editor(s), Handbook of Macroeconomics, Elsevier, 1999, Volume 1, Part 3, Pages
1341-1393.
Boivin, J., M. Kiley, and F.S. Mishkin. 2010. How Has the Monetary Transmission
Mechanism Evolved Over Time? In B. Friedman and M. Woodford, eds., The
Handbook of Monetary Economics, Elsevier.
Edge, R., Kiley, M., Laforte, J.P., 2010. A comparison of forecast performance be­
5

This difference between VAR-based and DSGE-model based impulse responses has been high­
lighted elsewhere – for example, in the survey of Boivin, Kiley, and Mishkin (2010).

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tween Federal Reserve staff forecasts, simple reduced-form models, and a DSGE
model. Journal of Applied Econometrics.
Chung H., Kiley, M., Laforte, J.P., 2011. Using the Great Recession to Understand
the EDO Model of the U.S. Economy: Documentation of the 2011 Model Version.
Forthcoming in Federal Reserve Board Finance and Economics Discussion Paper
Series.
Hall, Robert E., 2010. Why Does the Economy Fall to Pieces af­
ter a Financial Crisis?
Journal of Economic Perspectives 24 4 3-20
http://www.aeaweb.org/articles.php?doi=10.1257/jep.24.4.3
Smets, F., Wouters, R., 2007. Shocks and Frictions in the US Busines Cycles: A
Bayesian DSGE Approach. American Economic Review, American Economic As­
sociation, vol. 97(3), pages 586-606, June.
Wieland, Volker and Wolters, Maik H, 2010. ”The Diversity of Forecasts from Macroe­
conomic Models of the U.S. Economy,” CEPR Discussion Papers 7870, C.E.P.R.
Discussion Papers.

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Estimated Dynamic Optimization-based (EDO) Model
This appendix provides documentation for the EDO model, that is, the Estimated
Dynamic Optimization-based model developed at the Federal Reserve Board; for
further details, see Chung, Kiley, and Laforte (2010).1

1

Overview

The EDO model builds on the Smets and Wouters (2007) model. Households have
preferences over nondurable consumption services, durable consumption services,
housing services, and leisure and feature internal habit in each service flow. Pro­
duction occurs in two sectors that experience different (stochastic) rates of techno­
logical progress, thereby allowing the model to match the faster rate of growth (in
real, or constant dollar, terms) for some expenditure components (like nonresidential
investment); growth is balanced in nominal, rather than real, terms. Expenditures
on nondurable consumption, durable consumption, residential investment, nonresi­
dential investment, and the remainder of demand are each modeled, with the last
category exogenous.
Individuals’ wages and firms’ prices are sticky in the sense of Rotemberg, with
indexation to a weighted average of long-run inflation and lagged inflation. This
structure, for the two productive sectors of the economy, yield four Phillips curves:
two for wage inflation and two for price inflation. The deviation of marginal cost
from its steady-state value plays its usual role in the price Phillips curve. In the
wage Phillips curve, the deviation of the wage from the marginal rate of substitution
between consumption and leisure is the driving fundamental.
A simple monetary policy reaction function governs monetary policy choices. The
Federal Funds rate responds to it’s value in the previous quarter, the current and
lagged value of the output gap (defined as the deviation of output from its long-run
1

Available at http://www.federalreserve.gov/pubs/feds/2010/201029/201029abs.html.

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Beveridge-Nelson (stochastic) trend), and the deviation of inflation from the assumed

objective of 2 percent (at an annual rate).
The exogenous shocks/processes in the model include the monetary policy shock,
the growth rates of economywide and investment-specific technologies, financial shocks
which include an economywide risk premium and risk premia that affect the interme­
diaries for consumer durable, residential investment, and nonresidential investment,
and other shocks including autonomous aggregate demand and price and wage markup
shocks.
The model is estimated (using Bayesian methods) over the sample period 1984Q4
to 2008Q4. The data used in estimation include the following: Real GDP; Real
consumption of nondurables and services excluding housing; Real consumption of
durables; Real residential investment; Real business investment; Aggregate hours
worked in the nonfarm business sector (per capita); PCE price inflation; core PCE
price inflation; Percent change in PCE durables price index; Compensation per hour
divided by GDP price index; and federal funds rate. Each expenditure series is mea­
sured in per capita terms, using the (smoothed) civilian noninstitutional population
over the age of 16. We remove a very smooth trend from hours per capita prior to
estimation.

2

The Structure of the Model

The model possesses two final good sectors in order to capture key long-run growth
facts and to differentiate between the cyclical properties of different categories of
durable expenditure (e.g., housing, consumer durables, and nonresidential invest­
ment). For example, technological progress has been faster in the production of
business capital and consumer durables (such as computers and electronics). Edge,
Kiley, and Laforte (2008 and 2010) discuss this motivation in greater detail. The first
sector is the slow-growing sector—called “CBI” because most of these goods are used

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for consumption (C) and because they are produced by the business and institutions
(BI) sector—and the second is the fast-growing sector—called “KB” because these
goods are used for capital (K) accumulation and are produced by the business (B)
sector. The goods are produced in two stages by intermediate- and then final-goods
producing firms. As in most new-Keynesian models, the introduction of intermediate
and final goods producers facilitates the specification of nominal rigidities.
The disaggregation of production (aggregate supply) leads naturally to some dis­
aggregation of expenditures (aggregate demand). We move beyond the typical model
with just two categories of (private domestic) demand (consumption and invest­
ment) and distinguish between four categories of private demand: consumer non­
durable goods and non-housing services, consumer durable goods, residential invest­
ment, and non-residential investment. Consumer non-durable goods and services are
sold directly to households; consumer durable goods, residential capital goods, and
non-residential capital goods are intermediated through capital-goods intermediaries
(owned by the households), who then rent these capital stocks to households. Con­
sumer non-durable goods and services and residential capital goods are purchased
(by households and residential capital goods owners, respectively) from the first of
economy’s two final goods producing sectors, while consumer durable goods and non­
residential capital goods are purchased (by consumer durable and residential capital
goods owners, respectively) from the second sector. In addition to consuming the
non-durable goods and services that they purchase, households supply labor to the
intermediate goods-producing firms in both sectors of the economy.
This remainder of this section provides an overview of the decisions made by each
of the agents in our economy. Given some of the broad similarities between our model
and others, our presentation is selective.
The Final Goods Producers’ Problem. The economy produces two final
goods and services: slow-growing “consumption” goods and services, Xtcbi , and fastgrowing “capital” goods, Xtkb . These final goods are produced by aggregating (accord­

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ing to a Dixit-Stiglitz technology) an infinite number of sector-specific differentiated
intermediate inputs, Xts (j) for s = cbi, kb, distributed over the unit interval. The
representative firm in each of the consumption and capital goods producing sectors
chooses the optimal level of each intermediate input, taking as given the prices for
each of the differentiated intermediate inputs, Pts (j), to solve the cost-minimization
problem:
1

min


1
s
{Xt (j)}j=0

0

(
Pts (j)Xts (j)dj subject to


1
0

Θs −1
t
Θs
t

(Xts (j))


s
t
)
ΘΘ−1
s
t
dj

≥ Xts , for s = cbi, kb.

(1)
The term Θs is the stochastic elasticity of substitution between the differentiated
t
intermediate goods inputs used in the production of the consumption or capital goods
s
s
sectors. Letting θt ≡ ln Θt −ln Θs denote the log-deviation of Θs from its steady-state
∗
t

value of Θs , we assume that
∗
s
θt = cθ,s , for s = cbi, kb,
t

(2)

where cθ,s is a shock process. A stochastic elasticity of substitution introduces tran­
t
sitory markup shocks into the pricing decisions of intermediate-goods producers.
The Intermediate Goods Producers’ Problem. The intermediate goods
entering each final goods technology are produced by aggregating (according to a
Dixit-Stiglitz technology) an infinite number of differentiated labor inputs, Ls (j) for
t
s = cbi, kb, distributed over the unit interval and combining this aggregate labor
input (via a Cobb-Douglas production function) with utilized non-residential capital,
Ktu,nr,s . Each intermediate-good producing firm effectively solves three problems:
two factor-input cost-minimization problems (over differentiated labor inputs and
the aggregate labor and capital) and one price-setting profit-maximization problem.
In its first cost-minimization problem, an intermediate goods producing firm
chooses the optimal level of each type of differential labor input, taking as given

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the wages for each of the differentiated types of labor, Wts (i), to solve:
1

min

s
{Lt (i,j)}1
i=0

0

1

Wts (i)Ls (i, j)di subject to
t

0

(Ls (i, j))
t

Θl −1
t
Θl
t

di

l
Θt
Θl −1
t

s
≥ Lt (j), for s = cbi, kb.

(3)
The term Θl is the stochastic elasticity of substitution between the differentiated labor
t
l
inputs. Letting θt ≡ ln Θl − ln Θl denote the log-deviation of Θl from its steady-state
∗
t
t

value of Θl , we assume that
∗
θ,l
l
θt = c t .

(4)

where cθ,l is a shock process. A stochastic elasticity of substitution introduces tran­
t
sitory wage markup shocks into the wage decisions of households.
In its second cost-minimization problem, an intermediate-goods producing firm
chooses the optimal levels of aggregated labor input and utilized capital, taking as
given the wage, Wts , for aggregated labor, Ls (which is generated by the cost function
t
nr,s
derived the previous problem), and the rental rate, Rt , on utilized capital, Ktu,nr,s ,

to solve:

{

min

u,nr,s
Ls (j),Kt
(j)
t

}

nr,s
s
Wts Lt (j) + Rt Ktu,nr,s (j)

subject to (Ztm Zts Ls (j))1−α (Ktu,nr,s (j))α ≥ Xts (j), for s = cbi, kb, with Ztcbi ≡ 1. (5)
t
The parameter α is the elasticity of output with respect to capital, while the Zt vari­
ables denote the level of productivity. The level of productivity has two components.
The first, Ztm , is common to both sectors and thus represents the level of economywide technology. The second, Zts , is sector specific; we normalize Ztcbi to one, while
Ztkb is not restricted.
The exogenous productivity terms contain a unit root, that is, they exhibit per­
manent movements in their levels. We assume that the stochastic processes Ztm and
Ztkb evolve according to
z,n
n
z,n
z,n
ln Ztn − ln Zt−1 = ln Γz,n = ln (Γ∗ · exp[ct ]) = ln Γ∗ + cz,n , n = kb, m
t
t

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where Γz,n and cz,n are the steady-state and stochastic components of Γz,n . The
t
t
∗
stochastic component cz,n is an i.i.d shock process.
t
The unit-root in technology in both sectors yields a non-trivial Beveridge-Nelson
permanent/transitory decomposition. The presence of capital-specific technological
progress allows the model to generate differential trend growth rates in the economy’s
two production sectors. In line with historical experience, we assume a more rapid
rate of technological progress in capital goods production by calibrating Γz,kb > 1,
∗
where (as is the case for all model variables) an asterisk on a variable denotes its
steady-state value.
In its price-setting (or profit-maximization) problem, an intermediate goods pro­
ducing firm chooses its optimal nominal price and the quantity it will supply con­
sistent with that price. In doing so it takes as given the marginal cost, M Cts (j),
of producing a unit of output, Xts (j), the aggregate price level for its sector, Pts ,
and households’ valuation of a unit of nominal profits income in each period, which
is given by Λcnn /Ptcbi where Λcnn denotes the marginal utility of non-durables and
t
t
non-housing services consumption. Specifically, firms solve:
max
s

s
{Pt (j ),Xt (j)}∞
t=0

E0

∞
0
t=0

βt

Λcnn
t
{Pts (j)Xts (j)−M Cts (j)Xts (j)

cbi
Pt

(
)2
100 · χp Pts (j)
p p,s
p
p,s
− η Πt−1 −(1−η )Π∗
Pts Xts
−
s
2
Pt−1 (j)
s

s
s
subject to Xτ (j) = (Pτs (j)/Pτs)−Θτ Xτ for τ = 0, 1, . . . , ∞ and s = cbi, kb.

(7)


The profit function reflects price-setting adjustment costs (the size which depend on
the parameter χp and the lagged and steady-state inflation rate). The constraint
against which the firm maximizes its profits is the demand curve it faces for its differ­
entiated good, which derives from the final goods producing firm’s cost-minimization
problem. This type of price-setting decision delivers a new-Keynesian Phillips curve.
Because adjustment costs potentially depend upon lagged inflation, the Phillips curve
can take the “hybrid” form in which inflation is linked to its own lead and lag as well
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as marginal cost.

The Capital Owners’ Problem. We now shift from producers’ decisions to
spending decisions. There exists a unit mass of non-residential capital owners (indi­
vidually denoted by k, with k distributed over the unit interval) who choose invest­
ment in non-residential capital, Etnr , the stock of non-residential capital, Ktnr (which
is linked to the investment decision via the capital accumulation identity), and the
amount and utilization of non-residential capital in each production sector, Ktnr,cbi ,
Utcbi , Ktnr,kb , and Utkb . (Recall, that the firm’s choice variables in equation 5 is uti­
lized capital Ktu,nr,s = Uts Ktnr,s .) The mathematical representation of this decision is
described by the following maximization problem (in which capital owners take as
nr
given the rental rate on non-residential capital, Rt , the price of non-residential cap­

ital goods, Ptkb , and households’ valuation of nominal capital income in each period,
Λcnn /Ptcbi , and the exogenous risk premium specific to non-residential investment,
t
Anr ):
τ
max

nr,kb
nr,cbi
nr
nr
cbi
kb
{Et (k),Kt+1 (k),Kt
(k),Kt
(k)Ut (k),Ut (k)}∞
t=0

cnn
 {
Λt
nr
nr
Rt Utcbi (k)Ktnr,cbi (k)+Rt Utkb (k)Ktnr,kb (k)−Ptkb Etnr (k )

nr P cbi
Aτ t
t=0
( cbi 1+ψ
)
( kb 1+ψ
)
}
Ut (k)
Ut (k)
−1
−1
nr nr,cbi
nr nr,kb
−κ
−κ
Qt Kt
Qt Kt
1+ψ
1 + ψ

subject to


E0

∞
0

βt

nr
nr
nr
Kτ +1 (k) = (1− δ nr )Kτ (k)+Eτ (k)

100· χnr
−
2

nr
nr
Eτ (k)− Eτ −1 (k)Γx,kb
t
nr
Kτ


nr,cbi
nr,kb
nr

Kτ
(k)+Kτ (k) = Kτ (k) for τ = 0, 1, . . . , ∞.

2
nr
Kτ and

(8)

The parameter δ nr in the capital-accumulation constraint denotes the depreciation
rate for non-residential capital, while the parameter χnr governs how quickly invest­
x,kb
nr
nr
ment adjustment costs increase when (Eτ (k) − Eτ −1 (k)Γt ) rises above zero; note

that these adjustment costs include a term for the stochastic growth rate of the trend
in the level of the output in sector KB, Γx,kb equal to Γz,m Γz,kb . The variable Anr is
t
t
t
t
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a stochastic element reflecting a risk premium on non-residential investment. Letting

anr ≡ ln Anr denote the log-deviation of Anr from its steady-state value of unity, we
t
t
t
assume that:
a,nr
nr
at = ρnr anr + ct .
t−1

(9)

Higher rates of utilization incur a cost (reflected in the last two terms in the capital
owner’s profit function). We assume that utilization is unity in the steady-state,
nr
implying κ = R∗ /Qnr .
∗

The time-variation in utilization, along with the imperfect competition in product
and labor markets, implies that direct measurement of total factor productivity may
not provide an accurate estimate of technology; as a result, the EDO model can
deliver smoother estimates of technology that might be implied by a real-business­
cycle model.
The problems solved by the consumer durables and residential capital owners are
slightly simpler than the non-residential capital owner’s problems. Since utilization
rates are not variable for these types of capital, their owners make only investment and
capital accumulation decisions. Taking as given the rental rate on consumer durables
cd
capital, Rt , the price of consumer-durable goods, Ptkb , and households’ valuation of

nominal capital income, Λcnn /Ptcbi , and the exogenous risk premia specific to consumer
t
durables investment, Acd , the capital owner chooses investment in consumer durables,
τ
Itcd , and its implied capital stock, Ktcd , to solve:
max

E0

cd
cd
{Et (k),Kt+1 (k)}∞ }
t=0

∞
0
t=0

}
Λcnn { cd cd
β cdt cbi Rt Kt (k) − Ptkb Etcd (k)
Aτ Pt
t

subject to
100
cd
cd
cd
Kτ +1 (k) = (1−δ cd )Kτ (k)+Eτ (k)−

· χcd
2

for τ = 0, 1, . . . , ∞.

(

)2
cd
cd
Eτ (k)− Eτ −1 (k)Γx,kb
τ
cd
Kτ
cd
Kτ
(10)

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The residential capital owner’s decision is analogous:
max

E0

r
r
∞
{Et (k),Kt+1 (k)}t=0 }

∞
0
t=0

}
Λcnn { r r
β r t cbi Rt Kt (k) − Ptcbi Etr (k)
Aτ Pt
t

subject to
· χr
2

100
r
r
r
Kτ +1 (k) = (1−δ r )Kτ (k)+Eτ (k)−

(

)2
r
r
Eτ (k)−Eτ −1 (k)Γx,cbi
τ
cd
Kτ
cd
Kτ

for τ = 0, 1, . . . , ∞.

(11)

The notation for the consumer durables and residential capital stock problems paral­
lels that of non-residential capital. In particular, the asset-specific risk premia shocks,
Acd and Ar , follow an autoregressive process similar to that given in equation (9).
t
t
The Households’ Problem. The final group of private agents in the model are
households who make both expenditure and labor-supply decisions. Households derive
utility from four sources: their purchases of the consumer non-durable goods and nonhousing services, the flow of services from their rental of consumer-durable capital, the
flow of services from their rental of residential capital, and their leisure time, which is
equal to what remains of their time endowment after labor is supplied to the market.
Preferences are separable over all arguments of the utility function. The utility that
households derive from the three components of goods and services consumption is
influenced by the habit stock for each of these consumption components, a feature
that has been shown to be important for consumption dynamics in similar models.
A household’s habit stock for its consumption of non-durable goods and non-housing
cnn
services is equal to a factor h multiplied by its consumption last period Et−1 . Its

habit stock for the other components of consumption is defined similarly.
Each household chooses its purchases of consumer non-durable goods and services,
Etcnn , the quantities of residential and consumer durable capital it wishes to rent, Ktr
and Ktcd , its holdings of bonds, Bt , its wage for each sector, Wtcbi and Wtkb , and the
supply of labor consistent with each wage, Lcbi and Lkb . This decision is made subject
t
t
to the household’s budget constraint, which reflects the costs of adjusting wages and
the mix of labor supplied to each sector, as well as the demand curve the household
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faces for its differentiated labor. Specifically, the ith household solves:

max
∞
cnn
s
{Et (i),Ktcd (i),Ktr (i),{Wts (i),Lt (i)}s=cbi,kb ,Bt+1 (i)}t=0
∞
0 {
cnn
cd
E0 β t ς cnn ln(Etcnn (i)− hEt−1 (i))+ς cd ln(Ktcd (i)−hKt−1 (i))
t=0

+ς

r

r
ln(Ktr (i)−hKt−1 (i))

−ς

cbi
kb
1+ν
l (Lt (i)+Lt (i))

}

1+ν

subject to:
0
Bτ +1 (i)
s
cnn
= Bτ (i) +
Wτs (i)Lτ (i)+CapitalandProfitsIncome τ (i)−Pτcbi Eτ (i)
Rτ Ωτ
s=cbi,kb
)2
0 100 · χw ( W s (j)
τ
w w,s
cd cd
r
r
w
w
−Rτ Kτ (i) − Rτ Kτ (i) −
−η Πτ −1 −(1− η )Π∗
Wτs Ls
τ
2
Wτs−1 (j)
s=cbi,kb

(
) ( cbi
)2 kb

cbi
kb
Lτ (i) Lτ −1 Lτ
100 · χl Lcbi · Wτcbi
L∗ · Wτkb
∗
−
+
−
.
cbi
Lkb (i) Lkb
Lτ

2
Lcbi + Lkb Lcbi + Lkb
∗
∗
∗
τ
∗
τ −1
(
r−Θl
(
r−Θl
cbi
Lτ (i) = Wτcbi (i)/Wτcbi t Lcbi , and Lkb (i) = Wτkb (i)/Wτkb t Lkb
,
τ
τ
τ
for τ = 0, 1, . . . , ∞.

(12)

In the utility function the parameter β is the household’s discount factor, ν denotes
its inverse labor supply elasticity, while ς cnn , ς cd , ς r , and ς l are scale parameter that
tie down the ratios between the household’s consumption components.
The stationary, unit-mean, stochastic variable Ωt represents an aggregate riskpremium shock that drives a wedge between the policy short-term interest rate and
the return to bonds received by a household. Letting ωt ≡ ln Ωt − ln Ω∗ denote the
log-deviation of Ωt from its steady-state value of Ω∗ , we assume that
ωt = ρω ωt−1 + cω .
t

(13)

The variable cω is a shock process, and ρω represents the persistence of Ωt .
t
The household’s budget constraint reflects wage setting adjustment costs, which
depend on the parameter χw and the lagged and steady-state wage inflation rate, and
the costs in changing the mix of labor supplied to each sector, which depend on the
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parameter χl . The costs incurred by households when the mix of labor input across
sectors changes may be important for sectoral comovements.
Gross Domestic Product. The demand and production aspects of the model
are closed through the exogenous process for demand other than private domestic
X
demand and the GDP identity. XtHG represents exogenous demand (i.e., GDP other
than private domestic demand, the aggregate of Etcnn , Etcd , Etr , and Etnr ). Exogenous
demand is assumed to follow the process:
)
(
X
X HG + cHG .
X HG − ln X HG = ρHG ln XtHG − ln X∗
X
ln Xt
t
∗
We assume that the exogenous demand impinges on each sector symmetrically, and
specifically that the percent deviation of exogenous demand proportionally affects
demand for each sector’s (s = cbi, kb) output via the share of exogenous demand in
X
total demand, ωHG . (In this formulation, XtH G represents the level of expenditure
relative to the stochastic long-run trend, i.e., the model assumes balanced growth, so
exogenous demand for each sector fluctuates around its long-run trend; for example,
the long-run trend for sector KB is given by Ztm Ztkb ).
The rate of change of Gross Domestic Product (real GDP) equals the Divisia
(share-weighted) aggregate of production in the two sectors (and of final spending
across each expenditures category), as given by the identity:
(
Htgdp =

Xtcbi
cbi
Xt−1

cbi cbi
)P∗ X∗ (

kb kb
)P∗ X∗
Xtkb
kb

Xt−1


1
cbi cbi
k
kb
P∗ X∗ +P∗ b X∗

.

(14)

Monetary Authority. We now turn to the last important agent in our model,
the monetary authority. It sets monetary policy in accordance with an Taylor-type
interest-rate feedback rule. Policymakers smoothly adjust the actual interest rate Rt
¯
to its target level Rt
Rt = (Rt−1 )φ

r

(

¯
Rt

r1−φr

exp [cr ] ,
t

(15)

where the parameter φr reflects the degree of interest rate smoothing, while cr rep­
t
¯
resents a monetary policy shock. The central bank’s target nominal interest rate, Rt
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˜
depends the deviation of output from its stochastic trend (X bn , the output gap as
defined by Beveridge and Nelson (1981))
	 t
∞
0
0
gdp
gdp
˜ bn = Et
Xt
Hτ −
Hτ
.	
τ =−∞

(16)

τ =−∞

In equation 16, the deterministic, or steady-state, levels of growth are suppressed.
Consumer price inflation and the change in the output gap also enter the target. The
target equation is:
(

¯
˜
R t = Xt

) y(
bn φ

˜
Xt

bn

˜ bn
/Xt−1

)φΔy ( Πc )φπ
t

Πc
∗

R∗ .	

(17)

In equation (17), R∗ denotes the economy’s steady-state nominal interest rate and
φy , φΔy , and φπ denote the weights in the feedback rule. Consumer price inflation,
Πc , is the weighted average of inflation in the nominal prices of the goods produced
t
in each sector, Πp,cbi and Πp,kb :
t
t
p,cbi
p,kb
Πc = (Πt )1−wcd (Πt )wcd .	
t

(18)

The parameter wcd is the share of the durable goods in nominal consumption
expenditures.
Structural Shocks. The rich specification of structural shocks (to aggregate
and investment-specific productivity, aggregate and sector-specific risk premiums,
and mark-ups) and adjustment costs allows our model to be brought to the data with
some chance of finding empirical validation.
Within EDO, fluctuations in all economic variables are driven by eleven structural
shocks. It is most convenient to summarize these shocks into four broad categories:
•	 Permanent technology shocks: This category consists of shocks to aggregate
and investment-specific (or fast-growing sector) technology.
•	 Financial, or intertemporal, shocks: This category consists of shocks to risk
premia. In EDO, variation in risk premia – both the premium households’
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receive relative to the federal funds rate on nominal bond holdings and the
additional variation in discount rates applied to the investment decisions of
capital intermediaries – are purely exogenous. Nonetheless, the specification
captures important aspects of related models with more explicit financial sectors
(e.g., Bernanke, Gertler, and Gilchrist (1999)), as we discuss in our presentation
of the model’s properties below.
•	 Markup shocks: This category includes the price and wage markup shocks.
•	 Other demand shocks: This category includes the shock to autonomous demand
and a monetary policy shock.
Market Clearing. There are a number of market clearing conditions that must
be satisfied in our model. Market clearing in the slow-growing “consumption” goods
and fast-growing “capital” goods sectors, given price- and wage-adjustment costs and
variable utilization costs, implies that
Xtcbi

1

1

( rα
X
+ Etr (k)dk + XtHG ∗ Ztm ∗ Ztkb
0
0
)2
100 · χp ( p,kb p p,cbi
p,cbi
Πt −η Πt−1 −(1− η p )Π∗
+
Ptcbi Xtcbi
2

( cbi 1+ψ
)
)2
100·χw
 ( w,cbi w w,cbi
Ut (k) − 1
w
w,cbi
cbi cbi
Πt − η Πt−1 − (1−η )Π∗
+	
Wt Lt	 −κ
Ptcbi Ktnr,cbi
2	
1+ψ
(19)
Etcnn	(i)di

=

and
1

1

X
Etcd (k)dk + Etnr (k)dk + XtHG ∗ Ztm ∗ Ztkb
0
0
)2
100 · χp ( p,kb p p,kb
p
p,kb
Πt − η Πt−1 −(1−η )Π∗
++
Ptkb Xtkb
2
( kb 1+ψ
)
)2
Ut (k) −1
100·χw ( w,kb w w,kb
w,kb
kb kb
w
Πt −η Πt−1 −(1−η )Π∗
+	
Wt Lt	 −κ
Ptkb Ktnr,kb(20)
.
2
1+ψ

Xtkb =

The market clearing conditions for the labor and non-residential capital supplied and

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demanded in sector s are given by

Ls (i) =
t

1
0

Ls (i, j)dj and
t

1
0

s
U (k)t Ktnr,s (k)dk =

1
0

Ktu,nr,s (j)dj ∀ i ∈ [0, 1] and for s = cbi, kb.
(21)

The market clearing conditions for consumer durables and residential capital are
1
0

Ktcd (k)dk =

1
0

Ktcd (i)di and

1
0

Ktr (k)dk =

1
0

Ktr (i)di.

(22)

The identities for inflation include:
w,s
s
s
Wts (i) = Πw,s (i)Wt−1 (i) and Wts = Πt Wt−1 ∀ i ∈ [0, 1] and for s = cbi, kb, and (23)
t
s
Pts (j) = Πp,s (j)Pt−1 (j)
t

3

s
and Pts = Πp,s Pt−1
t

∀ j ∈ [0, 1] and for s = cbi, kb.

(24)

Solution and Estimation

We estimate the log-linearized, symmetric and stationary version of the model de­
scribed above. The log-linearization of our model equations is performed symbolically
by the software that we use to parse the model into its estimable form. The steadystate solution to the symmetric and stationary version of the model is an input into the
model’s estimation. The empirical implementation of the model takes a log-linear ap­
proximation to the first-order conditions and constraints that describe the economy’s
equilibrium, casts this resulting system in its state-space representation for the set of
(in our case 11) observable variables, uses the Kalman filter to evaluate the likelihood
of the observed variables, and forms the posterior distribution of the parameters of
interest by combining the likelihood function with a joint density characterizing some
prior beliefs. Since we do not have a closed-form solution of the posterior, we rely on
Markov-Chain Monte Carlo (MCMC) methods.
Data. The model is estimated using data over the sample period from 1984:Q4
to 2008:Q4. There are 11 data series: real gross domestic product; real consumption
expenditure on non-durables and services excluding housing services; real consump­
tion expenditure on durables; real residential investment expenditure; real business
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investment expenditure; the Personal Consumption Expenditure (PCE) price index;
the PCE price index excluding food and energy prices; the PCE price index for
durable goods; real compensation per hour in the non-farm business sector (that is,
nominal compensation deflated by the GDP price index); detrended hours of work
in the non-farm business sector; and the federal funds rate.2 Our implementation
adds measurement error processes to the likelihood implied by the model for all of
the observed series used in estimation except the nominal interest rate series.
Calibrated Parameters. The calibrated structural parameters of the model are
presented in Table 1. Some important determinants of steady-state behavior were cal­
ibrated to yields growth rates of GDP and associated price indexes that corresponded
to “conventional” wisdom in policy circles, even though slight deviations from such
values would have been preferred (in a “statistically significant” way) to our calibrated
values. In other cases, parameters were calibrated based on how informative the data
were likely to be on the parameter and/or identification and overparameterization
issues.
The standard deviations of the measurement errors for observable variables are
reported in Table 2. These standard deviations were calibrated to ensure a moderate
contribution of such errors to the overall variability of the data (according to our
model) while also preserving desirable forecast properties.
Estimated Parameters. The first three columns of Table 3 and 4 outline our
assumptions about the prior distributions of the estimated parameters, the remaining
columns describe the parameters’ posterior distributions.
First, consider the estimated parameters related to household and business spend­
ing decisions. The habit-persistence parameter is moderate, near 0.6.3 Investment
2

We remove a low-frequency trend from hours via the Hodrick-Prescott filter with a smoothing

parameter of 64000, because our model is not designed to capture low frequency trends in population
growth or labor force participation.
3
See Kiley (2010a) for a discussion of issues related to identification of the habit parameter using
frequentist techniques.

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adjustment costs are large for residential investment but small for business invest­
ment. This finding highlights once advantage of our disaggregated approach. In
addition, this result is importantly driven by the inclusion of inventory investment in
business investment; this is a very cyclically important component of GDP and was
an important element in early investigations of dynamic general equilbrium models
(e.g., Kydland and Prescott (1982)), but is typically ignored in similar DSGE models.
The estimated value of the inverse of the labor supply elasticity implies quite
elastic labor supply. We also find a role for the sectoral adjustment costs to labor:
In our multisector setup, shocks to productivity or preferences in one sector of the
economy result in strong shifts of labor towards that sector, which conflicts with the
high degree of sectoral co-movement in the data.
Finally, adjustment costs to prices and wages are both estimated to be important.
Our estimate of the price adjustment cost is equivalent to a Calvo pricing setting
where a bit more than half of the firms cannot update their prices each period. The
estimated quadratic costs in wages imply a slightly larger frequency of adjustments
for the suppliers of labor. We also find only a modest role for lagged inflation in our
adjustment cost specification (around 1/4), equivalent to modest indexation to lagged
inflation in other sticky-price specifications. This differs from some other estimates,
perhaps because of the focus on a more recent post-1983 sample (similar to results in
Kiley (2007) and Laforte (2007)).

4

Variance Decompositions

We have computed forecast error variance decompositions at various (quarterly) hori­
zons at the posterior mode of the parameter estimates for key variables and shocks.
Volatility in aggregate GDP growth is accounted for primarily by the technol­
ogy shocks in each sector, although the economy-wide risk premium shock contributes
non-negligibly to the unconditional variance of GDP growth.

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Volatility in hours per capita is accounted for primarily by the economy-wide

risk premium and business investment risk premium shocks at horizons between one
and sixteen quarters. Technology shocks in each sector contribute appreciably to the
unconditional variance. The large role for risk premia shocks in the forecast error
decomposition at business cycle horizons illustrates the importance of this type of
“demand” shock for volatility in the labor market. This result is notable, as hours
per capita is the series most like a “gap” variable in the model – that is, house per
capita shows persistent cyclical fluctuations about its trend value.
Volatility in core inflation is accounted for primarily by the markup shocks in
the short run and technology shocks in the long run.
Volatility in the federal funds rate is accounted for primarily by the econo­
mywide risk premium.
Volatility in expenditures on consumer non-durables and non-housing
services is, in the near horizon, accounted for predominantly by economy-wide and
non-residential investment specific risk-premia shocks. In the far horizon, volatility
is accounted for primarily by capital-specific and economy-wide technology shocks.
Volatilities in expenditures on consumer durables, residential invest­
ment, and non-residential investment are, in the near horizon, accounted for
predominantly by their own sector specific risk-premium shocks. At farther horizons,
their volatilities are accounted for by capital-specific technology shocks.

5

Impulse Responses

We now turn to the impulse responses of some of the key observable variables to the
exogenous shocks that drive fluctuations in the model. In each case we consider unit
shocks; the reader is referred to the reported estimates of the standard deviation of
the shocks for information that will scale these responses to units consistent with a
standard deviation shock. Expenditure variables are reported as percent deviations

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from initial values (in natural log points); inflation variables and the federal funds

rate are reported at quarterly (not annual) rates.
The impulse responses to a monetary policy innovation (shown in figure 4) cap­
tures the conventional wisdom regarding the effects of such shocks. In particular, both
household and business expenditures on durables (consumer durables, residential in­
vestment, and nonresidential investment) respond strongly (and with a hump-shape)
to a contractionary policy shock, with more muted responses by nondurables and ser­
vices consumption; each measure of inflation responds gradually, albeit more quickly
than in some analyses based on vector autoregressions (VARs). (This difference be­
tween VAR-based and DSGE-model based impulse responses has been highlighted
elsewhere – for example, in the survey of Boivin, Kiley, and Mishkin (2010)).
Figures 1 to 6 present the impulse responses of key variables to the model’s four
risk premia shocks (Ωt , Anr , Acd , and Ar ), the autonomous spending shock (X HG ),
t
t
t
price and wage mark-up shocks (Θcbi , Θcbi , and Θl ), and technology shocks (Γz,m and
t
t
t
t
Γz,kb ).
t
The aggregate risk premium shock (figure 1) depresses spending across the board,
lowering hours appreciably; inflation and the federal funds rate fall in response. (As
in the model of Smets and Wouters (2007), the aggregate risk premium drives down
the flexible-price nominal interest rate one-for-one, and hence the downward move
in the nominal funds rate facilitates moving the economy toward its flexible price
outcome).
Shocks to sectoral risk premia (figures 9, 10 and 11) principally depress spending
in the associated category of expenditure, with offsetting positive effects on other
spending (which is “crowded in”).
The impulse responses to a capital-specific technology shock (shown in figure 5)
are a touch more gradual, as the embodied component of this type of technological
progress implies a need for nonresidential capital accumulation. (In addition, the longrun responses of nonresidential investment and consumer durables are much larger

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than those of other spending, reflecting the biased nature of this technology shock).

Following an economy-wide technology shock (figure 6), output rises gradually
to its long-run level; hours respond relatively little to the shock (in comparison to,
for example, output, reflecting both the influence of stick prices and wages and the
offsetting income and substitution effects of such a shock on households willingness
to supply labor.

References
Bernanke, B., Gertler, M., and Gilchrist, S. 1999. The financial accelerator in a
quantitative business cycle framework, In: John B. Taylor and Michael Woodford,
Editor(s), Handbook of Macroeconomics, Elsevier, 1999, Volume 1, Part 3, Pages
1341-1393.
Beveridge, S. and Nelson, C.R. 1981. A new approach to the decomposition of eco­
nomic time series into permanent and transitory components with particular at­
tention to measurement of the business cycle, Journal of Monetary Economics 7,
151-174.
Boivin, J., Kiley, M., and Mishkin, F.S. 2010. How Has the Monetary Transmission
Mechanism Evolved Over Time? Forthcoming in B. Friedman and M. Woodford,
eds., The Handbook of Monetary Economics, Elsevier.
Chung, H., Kiley, M., and Laforte, J.P. 2010. Documentation of the Estimated, Dy­
namic, Optimization-based (EDO) Model of the U.S. Economy: 2010 Version. Fi­
nance and Economics Discussion Series 2010-29, Board of Governors of the Federal
Reserve System.
Edge, R., Kiley, M.,and Laforte, J.P. 2008. Natural rate measures in an estimated
DSGE model of the U.S. economy. Journal of Economic Dynamics and Control
32:25122535.
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Edge, R., Kiley, M., and Laforte, J.P. 2010. A comparison of forecast performance
between Federal Reserve staff forecasts, simple reduced-form models, and a DSGE
model. Forthcoming in Journal of Applied Econometrics.
Kiley, M. 2007. A Quantitative Comparison of Sticky-Price and Sticky-Information
Models of Price Setting. Journal of Money, Credit, and Banking 39, 101-25.
Kiley, M. 2010a. Habit Persistence, Non-separability between Consumption and
Leisure, or Rule-of-Thumb Consumers: Which Accounts for the Predictability of
Consumption Growth? Forthcoming in The Review of Economics and Statistics.
Kiley, M. 2010b. Output Gaps. Forthcoming in Federal Reserve Board Finance and
Economics Discussion Series (FEDS).
Kydland, F.E. and Prescott, E.C. 1982. Time-to-build and Aggregate Fluctuations.
Econometrica, 50, 6,1345 - 1370.
Laforte, J. 2007. Pricing Models: A Bayesian DSGE Approach to the U.S. Economy.
Journal of Money, Credit, and Banking 39, 127-54.
Smets, F. and Wouters, R. 2007. Shocks and Frictions in the US Busines Cycles:
A Bayesian DSGE Approach. American Economic Review, American Economic
Association, vol. 97(3), pages 586-606, June.

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Table 1: Calibrated Parameters

β

α

ψ

0.990 0.260

1

δ nr

δ cd

δr

cbi
kb
Θ∗ , Θ∗ , Θl
∗

0.030 0.055 0.004

Γz,m
∗

ωHG

Πc
∗

1.000 1.011

7.000

Γz,kb
∗

0.20

1.005

Table 2: Measurement Errors on Observable Variables

M EΔgdp

M EΔcns

M EΔcd

M EΔres

M EΔbi

0.3

0.1

1.5

1.5

1.5

M EΔppce

M EΔpcorepce

M EΔpcd

M Eh

M Erw

0.5

0.05

0.2

0.3

0.3

Table 3: Prior and Posterior Distributions of the Behavioral and Policy Parameters

Prior Distribution
Parameter Type Mean

Posterior Distribution

S.D.

Mode

S.D.

10th perc. 50th perc. 90th perc.

h

N

0.000

0.3300

0.6024

0.0350

0.5917

0.6392

0.6807

ν

G

2.000

1.0000

0.1918

0.2514

0.1409

0.3860

0.7701

χp

G

4.000

1.0000

2.5028

1.0797

2.2321

3.2782

4.8710

χl

G

4.000

1.0000

3.8424

1.9715

1.9764

3.9778

6.8915

χw

G

4.000

1.0000

2.1868

1.0576

2.1997

3.3348

4.8769

χnr

G

4.000

1.0000

0.2411

0.0911

0.2239

0.3180

0.4504

χcd

G

4.000

1.0000

0.3702

0.5521

0.4485

0.9534

1.8840

χr

G

4.000

1.0000

8.6694

2.3585

7.4588

9.9908

13.3231

ηp

N

0.000

0.5000

0.3006

0.1343

0.2325

0.4056

0.5779

ηw

N

0.000

0.5000

0.2542

0.1318

0.0823

0.2505

0.4207

φπ

N

1.500

0.0625

1.4562

0.0606

1.3776

1.4548

1.5331

φy

N

0.250

0.1250

0.2096

0.0283

0.1769

0.2101

0.2486

φ�y

N

0.000

0.1250

0.3310

0.0936

0.2104

0.3273

0.4488

φr

N

0.500

0.2500

0.6593

0.0453

0.5949

0.6559

0.7116

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Table 4: Prior and Posterior Distributions of the Parameters corresponding to the
Exogenous Processes
Posterior Distribution

Prior Distribution
Parameter Type

Mean

S.D.

Mode

S.D.

ρω

N

0.000

0.3300

0.7930

0.0364

0.7579

0.8070

0.8502

ρnr

N

0.000

0.3300

0.8297

0.0302

0.8076

0.8496

0.8836

ρcd

N

0.000

0.3300

-0.2110

0.1422

-0.4099

-0.2412

-0.0469

ρHG

B

0.500

0.0150

0.9173

0.1637

0.4577

0.6821

0.8969

ρr

N

0.000

0.3300

0.8328

0.0285

0.7914

0.8324

0.8637

σω

I

1.000

2.0000

0.3742

0.0597

0.3234

0.3881

0.4737

σHG

I

1.000

2.0000

1.4573

0.3374

0.5267

0.7994

1.3940

σθ,l

I

1.000

2.0000

1.5877

0.7145

1.6168

2.4055

3.4337

σr

I

0.200

2.0000

0.1572

0.0134

0.1437

0.1595

0.1778

σz,k

I

0.250

2.0000

0.8771

0.1321

0.7181

0.8748

1.0533

σz,m

I

0.250

2.0000

0.4036

0.0663

0.3751

0.4551

0.5437

σθ,cbi

I

0.200

2.0000

0.3125

0.1576

0.2845

0.4296

0.6678

σθ,kb

I

0.200

2.0000

0.4621

0.2747

0.3926

0.6584

1.0556

σa,r

I

1.000

2.0000

0.4921

0.1562

0.4102

0.5433

0.7742

σa,cd

I

1.000

2.0000

7.2703

11.9676

8.8443

18.8741

38.5473

σa,nr

I

1.000

2.0000

0.4788

0.0866

0.3984

0.4922

0.6190

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10th perc. 50th perc. 90th perc.

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Figure 1: Impulse Responses: Risk-premium


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Figure 2: Impulse Responses: Exog. Demand


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Figure 3: Impulse Responses: Wage Markup


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 4: Impulse Responses: Funds Rate


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 5: Impulse Responses: Capital Goods Technology


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 6: Impulse Responses: Overall TFP


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 7: Impulse Responses: Non-Invest. Price Markup


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 8: Impulse Responses: Invest. Price Markup


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 9: Impulse Responses: Housing Risk-Premium


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 10: Impulse Responses: Durables Risk-Premium


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Authorized for public release by the FOMC Secretariat on 08/04/2017

Figure 11: Impulse Responses: Capital Risk-Premium


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