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Authorized for public release by the FOMC Secretariat on 1/13/2023
BOARD OF GOVERNORS OF THE FEDERAL RESERVE SYSTEM
DIVISION OF MONETARY AFFAIRS
FOMC SECRETARIAT
Date:
December 4, 2017
To:
Federal Open Market Committee
From:
James A. Clouse
Subject: Supporting Documents for DSGE Models Update
The attached documents support the update on the projections of the DSGE
models.
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The Current Outlook in EDO:
December 2017 FOMC Meeting
Class II FOMC – Restricted (FR)
Cristina Fuentes-Albero∗
November 30, 2017
1
The EDO Forecast from 2018 to 2020
The EDO models forecast is conditional on data through the third quarter of 2017 and on a preliminary Tealbook forecast for the fourth quarter of 2017.
Real GDP growth is 2.6 percent, on average, over the projection horizon, somewhat below its
long-run value of 3 percent. Inflation reaches the Committees 2 percent objective in the fourth
quarter of 2019 and then slightly overshoots the target thereafter. Below-trend real GDP growth is
driven by the slow fading of risk premium shocks and accommodative monetary policy. For inflation,
the EDO model interprets the weakness in inflation over the past few years as driven by negative
wage markup shocks and expects them to dissipate only gradually over the projection horizon.
The output gap is estimated to be currently negative 1 percent. The output gap closes very
slowly and remains at negative 0.3 percent by the end of 2020. The real natural rate of interest
is projected to increase from 1 percent in the fourth quarter of 2017 to 1.5 percent at the end of
2020, 0.6 percentage point below its steady-state value of 2.1 percent. According to the EDO model,
capital-specific risk premium shocksinferred from a combination of weaker-than-expected investment
and output data with stronger-than-expected consumption data over the past several yearshave been
holding down the output gap and the real natural rate. As these shocks slowly dissipate, the output
gap closes and the real natural rate rises. Consistent with the gradual return of inflation and the
output gap to their long-run values, the federal funds rate is projected to increase gradually over the
forecast horizon, reaching 3.5 percent by the end of 2020. At the end of the projection horizon, the
federal funds rate is still below its long-run value of 4.1 percent, reflecting the inertia in the policy
rule and the persistently negative output gap even at the end of the projection horizon.
∗ Cristina Fuentes-Albero is affiliated with the Division of Research and Statistics of the Federal Reserve Board.
Sections 2 and 3 contain background material on the EDO model, as in previous rounds. These sections were co-written
with Hess Chung and Jean-Philippe Laforte.
1
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Figure 1: Recent History and Forecasts
4
6
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.c
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~
·~ 1
e
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The EDO models projection of real GDP growth in this round is slower for the next three
years than it was in September 2017. The downward revision in the real GDP growth projection
is mostly driven by risk premium shocks. Core PCE inflation is, on average, 4 basis points lower
over the forecast horizon in this round than in September, also resulting from more negative wage
markup shocks. The output gap has revised down, on average, 13 basis points since September.
The projection of the real natural rate of interest has been revised down 18 basis points, on average,
since September. And, consistent with the lower inflation path, the path of the federal funds rate is
lower this round than in September.
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Figure 2: Recent History and Forecasts: Latent Variables
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An Overview of Key Model Features
Figure 3 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.1
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 categories of durable expenditure
(for example, housing, consumer durables, and nonresidential 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 disaggregation 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 nondurable goods and nonhousing services, consumer durable goods,
residential investment, and nonresidential investment. The boxes surrounding the producers in the
figure illustrate how we structure the sources of each demand category. Consumer nondurable goods
and services are sold directly to households; consumer durable goods, residential capital goods, and
nonresidential capital goods are intermediated through capital-goods intermediaries (owned by the
households), who then rent these capital stocks to households. Consumer nondurable 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 nonresidential capital goods are purchased (by consumer durable and residential
capital goods owners, respectively) from the second sector. In addition to consuming the nondurable
goods and services that they purchase, households supply labor to the intermediate goods-producing
firms in both sectors of the economy.
The remainder of this section provides an overview of the main properties of the model. In
1 Chung,
Kiley, and Laforte (2010) provide much more detail regarding the model specification, estimated parameters, and model properties.
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Figure 3: Model Overview
Residential Capital Rentals
Households
<
Sales of CBI Final Goods
Monop . Competitive Sales of CBI
Inter med . CBI
Intermed .
Goods Producers
Goods
Final CBI
Goods Producers
(Aggregators)
Sales of CBI
Final
Goods
)
)
Monop . Competitive. Sales of KB
Intermed. KB
Intermed .
Goods Producer s
Goods
Final KB
Goods Producers
(Aggregator s)
Sa les of KB
Fin a l
Goods
Consumer
Dur . Capital
Owners
)
Non-residential
Capital
Owners
particular, the model has five key features:
• A New-Keynesian structure for price and wage dynamics. Unemployment measures the difference between the amount workers are willing to be employed and firms’ employment demand.
As a result, unemployment is an indicator of wage and, hence, price pressures as in Gali (2011).
• Production of goods and services occurs in two sectors, with differential rates of technological
progress across sectors. In particular, productivity growth in the investment and consumer
durable goods sector exceeds that in the production of other goods and services, helping the
model match facts regarding long-run growth and relative price movements.
• A disaggregated specification of household preferences and firm production processes that
leads to separate modeling of nondurables and services consumption, durables consumption,
residential investment, and business investment.
• Risk premiums associated with different investment decisions play a central role in the model.
These include, first, an aggregate risk premium, or natural rate of interest, shock driving a
wedge between the short-term policy rate and the interest rate faced by private decisionmakers
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(as in Smets and Wouters (2007)) and, second, fluctuations in the discount factor/risk premiums faced by the intermediaries financing household (residential and consumer durable) and
business investment.
2.1
Two-sector production structure
It is well known (for example, Edge, Kiley, and Laforte (2008)) 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 example, 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 sector producing business investment and consumer
durables and cbi for the sector producing other goods and services) is governed by a Cobb-Douglas
production function with sector-specific technologies:
1−α
Xts (j) = (Ztm Zts Lts (j))
α
(Ktu,nr,s (j)) , 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 technological 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 nonresidential business capital (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 for the possibility that investment-specific technological change is a source of business cycle
fluctuations, as in Fisher (2006).
2.2
The structure of demand
EDO differentiates between several categories of expenditure. Specifically, business investment
spending determines nonresidential capital used in production, and households value consumer nondurables goods and services, consumer durable goods, and residential capital (for example, 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:
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E0
∞
X
cnn
β t ς cnn ln(Etcnn (i)−hEt−1
(i))+ς cd ln(Ktcd (i))
t=0
+ς r ln(Ktr (i)) −ΛtLpref ΘH
t
1
X Z
s=cbi,kb
ς l,s Lst (i)
1+σN
σN
1+
1+σh
0
⎫
⎬
di ,
⎭
(2)
where E cnn represents expenditures on consumption of nondurable goods and services, K cd and
K r represent the stocks of consumer durables and residential capital (housing), ΛLpref
represents a
t
labor supply shock, Θt is an endogenous preference shifter whose role is to reconcile the existence of
a long-run balance growth path with a small short-term wealth effect2 , Lcbi and Lkb represent the
labor supplied to each productive sector (with hours worked causing disutility), and the remaining
terms represent parameters (such as the discount factor, relative value in utility of each service flow,
and the elasticity of labor supply). Gali, Smets, and Wouters (2011) state that the introduction
of the endogenous preference shifter is key in order to match the joint behavior of the labor force,
consumption, and wages over the business cycle.
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 in the early 2000s recession and the most recent
downturn. Many other models do not distinguish between developments across these categories of
spending.
2.3
Risk premiums, financial shocks, and economic fluctuations
The structure of the EDO model implies that households value durable stocks according to their
expected returns, including any expected service flows, and according to their risk characteristics,
with a premium on assets that have high expected returns in adverse states of the world. However,
the behavior 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 modeled in EDO, that 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.
2 The
cnn , where Z =
endogenous preference shifter is defined as ΘH
t
t = Zt Λt
1−ν
Zt−1
Λcnn
t
and Λcnn
is the shadow price of
t
nondurable consumption. The importance of the short-term wealth effect is determined by the parameter ν ∈ (0, 1].
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The “sector specific” risk shocks affect the composition of spending more than the path of
GDP itself. This occurs because a shock to these premiums leads to sizable substitution across
residential, consumer durable, and business investment; for example, an increase in the risk premiums
on residential investment leads households to shift away from residential investment and toward
other types of productive investment. Consequently, it is intuitive that a large fraction of the noncyclical, or idiosyncratic, component of investment flows to physical stocks will be accounted for by
movements in the associated premiums.
Shocks to the required rate of return on the nominal risk-free asset play an especially 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,
that is, the premium is a shock to the natural rate of interest. Given nominal rigidities, however,
the desire for higher risk-free savings must be offset, in part, through a fall in real income, a decline
which is distributed across all spending components. Because this response is capable of generating
co-movement 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.”
Movements in financial markets and economic activity in recent years have made clear the role
that frictions in financial markets play in economic fluctuations. This role was apparent much earlier,
motivating a large body of research (for example, Bernanke, Gertler, and Gilchrist (1999)). While
the range of frameworks used to incorporate such frictions has varied across researchers studying
different questions, a common theme is that imperfections in financial markets —for example, related
to imperfect information on the outlook for investment projects or earnings of borrowers —drives a
wedge between the cost of riskless funds and the cost of funds facing households and firms. Much
of the literature on financial frictions has worked to develop frameworks in which risk premiums
fluctuate for endogenous reasons (for example, because of movements in the net worth of borrowers).
Because the risk-premium shocks induces a wedge between the short-term nominal risk-free rate and
the rate of return on the affected risky rates, these shocks may thus also be interpreted as a reflection
of financial frictions not explicitly modeled in EDO. The sector-specific risk premiums in EDO enter
the model in much the same way as does the exogenous component of risk premiums in models with
some endogenous mechanism (such as the financial accelerator framework used Boivin, Kiley, and
Mishkin (2010)), and the exogenous component is quantitatively the most significant one in that
research.3
2.4
Labor market dynamics in the EDO model
This version of the EDO model assumes that labor input consists of both employment and hours per
worker. Workers differ in the disutility they associate with employment. Moreover, the labor market
is characterized by monopolistic competition. As a result, unemployment arises in equilibrium – some
workers are willing to be employed at the prevailing wage rate, but cannot find employment because
firms are unwilling to hire additional workers at the prevailing wage.
3 Specifically, the risk premiums enter EDO to a first-order (log)linear approximation in the same way as in the
cited research if the parameter on net worth in the equation determining the borrowers cost of funds is set to zero; in
practice, this parameter is often fairly small in financial accelerator models.
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As emphasized by Gali (2011), this framework for unemployment is simple and implies that the
unemployment rate reflects wage pressures: When the unemployment rate is unusually high, the
prevailing wage rate exceeds the marginal rate of substitution between leisure and consumption,
implying that workers would prefer to work more.
The new preference specification and the incorporation of labor force participation in the information set impose discipline in the overall labor market dynamics of the EDO model. The estimated
short-run wealth effect on labor supply is relatively attenuated with respect to previous versions of
the EDO model. Therefore, the dynamics of both labor force participation and employment are
more aligned with the empirical evidence.
In addition, in our environment, nominal wage adjustment is sticky, and this slow adjustment
of wages implies that the economy can experience sizable swings in unemployment with only slow
wage adjustment. Our specific implementation of the wage adjustment process yields a relatively
standard New Keynesian wage Phillips curve. 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).
While the specific model on the labor market is suitable for discussion of the links between
employment and wage/price inflation, it leaves out many features of labor market dynamics. Most
notably, it does not consider separations, hires, and vacancies, and is hence not amenable to analysis
of issues related to the Beveridge curve.
The decline in employment during the Great Recession primarily reflected, according to the
EDO model, the weak demand that arose from elevated risk premiums that depressed spending,
as illustrated by the light blue and red bars in figure 1. The role played by these demand factors
in explaining the cyclical movements in employment is only determinant during the 1980s and
during the Great Recession. As apparent in figure 1, the most relevant drivers of employment in the
remaining of the sample are labor supply (preference) and markup shocks as shown by the blue bars.
Specifically, favorable supply developments in the labor market are estimated to have placed upward
pressure on employment until 2010; these developments have reversed, and some of the currently
low level for employment growth is, according to EDO, attributable to adverse labor market supply
developments. As discussed previously, these developments are simply exogenous within EDO and
are not informed by data on a range of labor market developments (such as gross worker flows and
vacancies).
2.5
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
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Phillips curve (governing price adjustment in both productive sectors) has the form
p,s
p,s
+ 0.76Et πt+1
+ .017mcst + θts
πtp,s = 0.22πt−1
(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
s
s
4wts = 0.014wt−1
+ 0.95Et 4wt+1
+ .012 mrstc,l − wts + θtw + adj. costs.
(4)
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.
The top right panel of figure 1 presents the decomposition of inflation fluctuations into the
exogenous disturbances that enter the EDO model. As can be seen, aggregate demand fluctuations,
including aggregate risk premiums and monetary policy surprises, contribute little to the fluctuations
in inflation according to the model. This is not surprising: In modern DSGE models, transitory
demand disturbances do not lead to an unmooring of inflation (so long as monetary policy responds
systematically to inflation and remains committed to price stability). In the short run, inflation
fluctuations primarily reflect transitory price and wage shocks, or markup shocks in the language of
EDO. Technological developments can also exert persistent pressure on costs, most notably during
and following the strong productivity performance of the second half of the 1990s, which is estimated
to have lowered marginal costs and inflation through the early 2000s. More recently, disappointing
labor productivity readings over the course of 2011 have led the model to infer sizable negative
technology shocks in both sectors, contributing noticeably to inflationary pressure over that period
(as illustrated by the blue bars in figure 1).
2.6
Monetary authority and a long-term interest rate
We now turn to the last 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
¯t
interest rate Rt to its target level R
Rt = (Rt−1 )
ρr
�
¯t
R
1−ρr
exp [rt ] ,
(5)
where the parameter ρr reflects the degree of interest rate smoothing, while rt represents a monetary
¯ t depends the deviation of output
policy shock. The central bank’s target nominal interest rate, R
from the level consistent with current technologies and “normal” (steady-state) utilization of capital
˜ pf , the “production function” output gap). Consumer price inflation also enters the
and labor (X
target. The target equation is
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pf ry Πc rπ
t
¯ t = X˜t
R∗ .
R
Πc∗
(6)
In equation (6), R∗ denotes the economy’s steady-state nominal interest rate, and ry and rπ denote
the weights in the feedback rule. Consumer price inflation, Πct , is the weighted average of inflation
in the nominal prices of the goods produced in each sector, Πp,cbi
and Πtp,kb :
t
Πct = (Πp,cbi
)1−wcd (Πp,kb
)wcd .
t
t
(7)
The parameter wcd is the share of the durable goods in nominal consumption expenditures.
The model also includes a long-term interest rate (RLt ), which is governed by the expectations
hypothesis subject to an exogenous term premiums shock:
RLt = Et ΠN
τ =0 Rτ ·
where
t.
(8)
is the exogenous term premium, governed by
Ln (
t)
�
= 1 − ρ Ln (
∗)
+ ρ Ln (
t−1 )
+ t .
(9)
In this version of EDO, the long-term interest rate plays no allocative role; nonetheless, the term
structure contains information on economic developments useful for forecasting (for example, Edge,
Kiley, and Laforte (2010)), and hence RL is included in the model and its estimation.
2.7
Summary of model specification
Our brief presentation of the model highlights several points. First, although our model considers
production and expenditure decisions in a bit more detail, it shares many similar features with other
DSGE models in the literature, such as imperfect competition, nominal price and wage rigidities, and
real frictions like adjustment costs and habit-persistence. The rich specification of structural shocks
(to aggregate and investment-specific productivity, aggregate and sector-specific risk premiums, and
markups) 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 13 structural shocks. It is most
convenient to summarize these shocks into five broad categories:
• Permanent technology shocks: This category consists of shocks to aggregate and investmentspecific (or fast-growing sector) technology.
• A labor supply shock: This shock affects the willingness to supply labor. As was apparent in our
earlier description of labor market dynamics and in the presentation of the structural drivers
below, this shock captures the dynamics of the labor force participation rate in the sample and
those of employment. While EDO labels such movements labor supply shocks, an alternative
interpretation would describe these as movements in the labor force and employment that
reflect structural features not otherwise captured by the model.
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• Financial, or intertemporal, shocks: This category consists of shocks to risk premiums. In
EDO, variation in risk premiums —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 (for
example, 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.
3
Estimation: Data and Properties
3.1
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, 13) 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.
The model is estimated using 13 data series over the sample period from 1984:Q4 to 2015:Q3.
The series are the following:
1. The growth rate of real gross domestic product (ΔGDP );
2. The growth rate of real consumption expenditure on nondurables 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 Consumption 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 );
9. Hours, which equals hours of all persons in the nonfarm business sector from the Bureau of
Labor Statistics (H);
10. Civilian employment-population ratio, defined as civilian employment from the Current Population Survey (household survey) divided by the noninstitutional population, age 16 and over
(N );
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11. Labor force participation rate;
12. The growth rate of real wages, as given by compensation per hour in the non-farm business
sector from the Bureau of Labor Statistics divided by the GDP price index (ΔRW ); and
13. The federal funds rate (R).
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.
3.2
Estimates of latent variable paths
Figures 4, 5, and 6 report estimates of the model’s persistent exogenous fundamentals (for example,
risk premiums and autonomous demand). These series have recognizable patterns for those familiar
with U.S. economic fluctuations. For example, the risk premiums jump at the end of 2008, reflecting
the financial crisis and the model’s identification of risk premiums, 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
premiums 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 mechanisms in our baseline approach as premature, pending ongoing
research on financial 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.
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Class II FOMC – Restricted (FR)
Figure 4: Model Estimates of Risk Premiums
Aggregate Risk
Capital Risk
2.5
2
4
1.5
2
1
0
0.5
-2
0
-4
-0.5
-6
-1
-8
1990
2000
2010
2020
1990
Housing Risk
2000
2010
2020
Durables Risk
6
40
4
20
2
0
0
-20
-2
-40
-4
-60
-6
1990
2000
2010
2020
1990
2000
2010
2020
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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Figure 5: Model Estimates of Key Supply-side Variables
Aggregate Tech
Capital Tech
3
2
2
1
1
0
0
-1
-1
-2
-2
1990
2000
2010
2020
2010
2020
1990
2000
2010
2020
Labor Pref
60
40
20
0
-20
1990
2000
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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Figure 6: Model Estimates of Selected Other Exogenous Drivers
Wage Markup
Exog Spending
50
10
5
0
0
-5
-10
-50
1990
2000
2010
2020
1990
2000
2010
2020
Black line: modal parameters. Red line: posterior median. Dark blue intervals: 68 percent credible
set. Light blue intervals: 95 percent credible set.
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References
[Bernanke, Gertler, and Gilchrist (1999)] Bernanke, B., M. Gertler, and S. Gilchrist. 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 13411393.
[Boivin, Kiley, and Mishkin (2010)] 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.
[Chung, Kiley, and Laforte (2010)] Chung, H., M. Kiley, and J.P. Laforte. 2010. Documentation of
the Estimated, Dynamic, Optimization-based (EDO) model of the U.S. economy: 2010 version.
Finance and Economics Discussion Series, 2010-29. Board of Governors of the Federal Reserve
System (U.S.).
[Edge, Kiley, and Laforte (2008)] Edge, R., M. Kiley, and J.P. Laforte. 2008. Natural rate measures
in an estimated DSGE model of the U.S. economy. Journal of Economic Dynamics and Control,
vol. 32(8), pages 2512-2535.
[Edge, Kiley, and Laforte (2010)] Edge, R., M. Kiley, and J.P. Laforte. 2010. A comparison of forecast performance between Federal Reserve staff forecasts, simple reduced-form models, and a
DSGE model. Journal of Applied Econometrics vol. 25(4), pages 720-754.
[Fisher (2006)] Fisher, Jonas D. M. 2006. The Dynamic Effects of Neutral and Investment-Specific
Technology Shocks. Journal of Political Economy, University of Chicago Press, vol. 114(3), pages
413-451.
[Gali (2011)] Gali, J. 2011. The Return Of The Wage Phillips Curve. Journal of the European
Economic Association, vol. 9(3), pages 436-461.
[Gali, Smets, and Wouters (2011)] Gali, J., F. Smets, and R. Wouters. 2011. Unemployment in an
Estimated New Keynesian Model. NBER Macroeconomics Annual vol. 26(1), pages 329-360.
[Smets and Wouters (2007)] Smets, F., and R. Wouters. 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.
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Research Directors Draft
December 1, 2017
Forecast Summary
The New York Fed model forecasts are obtained using data released through 2017Q3, augmented for 2017Q4 with the New York Fed staff forecasts (as of November 22) for real GDP
growth and core PCE inflation, and with values of the federal funds rate, the 10-year Treasury yield and the spread between Baa corporate bonds and 10-year Treasury yields based
on 2017Q4 averages up to November 22.
Based on this information, we project real GDP growth of 2.6 percent in 2017 on a Q4/Q4
basis, significantly stronger than the forecasts of 2.3 and 2 percent reported in September and
June respectively. This projection fully reflects the current New York Fed staff judgmental
forecast, which is somewhat more optimistic than the model’s unconditional assessment of
a 2.3 percent growth rate for this year. In 2018, GDP growth is anticipated to decline to
2 percent, the same as in September. Further into the future, however, the model forecasts
a very gradual strengthening of activity, with GDP growth expected to reach 2.2 percent
in 2019 and 2020, a slight improvement with respect to September. Consistent with this
somewhat more solid growth prospects, inflation is also forecast to be higher in the medium
term than expected in September, at 1.5 percent in both 2017 and 2018. However, its
progress towards the FOMCs longer-run goal of 2 percent will remain glacial according to
the model, with core PCE inflation only reaching 1.6 percent at the end of 2020.
Notwithstanding this modest improvement in the outlook, the output gap is currently
estimated to be somewhat larger in 2017Q4 than projected in September: -1.1 percent
compared to -0.9 percent. As in that round, the gap is expected to close very gradually
over the course of the next several years, shrinking to -0.6 percent at the end of 2020. The
natural rate of interest is also estimated to be somewhat lower at the end of 2017 than in
September, but it is expected to continue recovering gradually over the next three years, as
previously anticipated, reaching 1.3 percent at the end of 2020. The federal funds rate is
projected to increase alongside its natural counterpart, reaching 2.9 percent by the end of
2020. This path translates into approximately four rate hikes in 2018, two more in 2019 and
only one more in 2020.
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The projections for all the variables are surrounded by notable uncertainty. For instance,
the 68 percent posterior probability interval for GDP growth includes negative readings for
all three years between 2018 and 2020. In comparison, the posterior probability intervals
for inflation are tighter, with their upper bound never exceeding 3 percent throughout the
forecast horizon.
The model attributes the above average real GDP growth rate in 2017 to continued improvement in financial conditions, as captured by positive contributions of both the financial
and marginal efficiency of investment shocks. These positive forces were partly offset by
low TFP growth in the first half of the year, but this drag from productivity appears to
have abated in the last two quarters, contributing to the recent pickup in economic growth.
As for inflation, the model attributes its recent weakness to a confluence of several factors,
which continue to hold it below target over the forecast horizon. These factors include the
lingering effects of the financial headwinds that have hampered the recovery, whose impact
on inflation is estimated to be very persistent, as well as negative shocks to wage and price
markups, which in the model capture some of the more transitory influences on inflation
dynamics.
The Model and Its Transmission Mechanism
General Features of the Model
The New York Fed DSGE model is a medium-scale, one-sector dynamic stochastic general
equilibrium model which is based on the New Keynesian model with financial frictions used
in Del Negro et al. (2015). The core of the model is based on the work of Smets and
Wouters (2007) and Christiano et al. (2005): It builds on the neo-classical growth model
by adding nominal wage and price rigidities, variable capital utilization, costs of adjusting
investment, and habit formation in consumption. The model also includes credit frictions
as in the financial accelerator model developed by Bernanke et al. (1999), where the actual
implementation of the credit frictions follows closely Christiano et al. (2014); and it allows
for a time-varying inflation target following Del Negro and Schorfheide (2012). In contrast to
these papers, the model features both a deterministic and a stochastic trend in productivity.
Finally, it accounts for forward guidance in monetary policy by including anticipated policy
shocks as in Laseen and Svensson (2011). More details on the model are in the New York
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Fed DSGE Model Documentation, available upon request.
In this section, we briefly describe the microfoundations of the model, including the optimization problem of the economic agents and the nature of the exogenous processes. The
innovations to these processes, which we refer to as “shocks,” are the drivers of macroeconomic fluctuations. The model identifies these shocks by matching the model dynamics with
numerous quarterly data series: real GDP and GDI growth, real consumption growth, real
investment growth, real wage growth, hours worked, inflation as measured by the personal
consumption expenditures deflator and the GDP deflator, the federal funds rate (FFR),
the 10-year nominal Treasury bond yield, 10-year survey-based inflation expectations, the
Baa/10-year Treasury bond yield spread, and data on total factor productivity. In addition,
from 2008Q4 to 2015Q2, we use market expectations of future federal funds rates. Model
parameters are estimated from 1960Q1 to the present using Bayesian methods. structure of
the model, data sources, and results of the estimation procedure can be found in Del Negro
et al. (2015).
The economic units in the model are households, intermediate-goods producing firms,
banks, entrepreneurs, capital-goods producers and the government. (Figure 1 describes the
interactions among the various agents, the frictions and the shocks that affect the dynamics
of this economy.)
Households derive utility from leisure, supply labor services to firms, and set wages in
a monopolistically competitive fashion. random disturbance, which we call supply” shocks
(this shock is “leisure” capture exogenous movements in labor supply due and labor market
imperfections. The labor market is subject to frictions because of nominal wage rigidities.
In addition, we allow for exogenous disturbances to wage mark-ups, labeled “wage markup” shocks, which capture exogenous changes in the degree of competitiveness in the labor
market, or other exogenous movements in the labor supply.
Households, who discount future utility streams, also have to choose how much to consume and save. Their savings take the form of deposits to banks and purchases of government bills. Household preferences feature habit persistence, a characteristic that affects their
consumption smoothing decisions. In addition, “discount factor” shocks drive an exogenous
wedge between the change in the marginal utility of consumption and the riskless real return.
These shocks possibly capture phenomena like deleveraging, or increased risk aversion.
Monopolistically competitive firms produce intermediate goods, which a competitive firm
aggregates into the single final good that is used for both consumption and investment. The
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production function of intermediate producers is subject to “total factor productivity” (TFP)
shocks, which affect both the temporary and the permanent component of the level of total
factor productivity. Intermediate goods markets are subject to price rigidities. Together with
wage rigidities, this friction is quite important in allowing demand shocks to be a source of
business cycle fluctuations, as countercyclical mark-ups induce firms to produce less when
demand is low. Inflation evolves in the model according to a standard, forward-looking
New Keynesian Phillips curve with indexing, which determines inflation as a function of
marginal costs, expected future inflation, past inflation, and “price mark-up” shocks. The
latter capture exogenous changes in the degree of competitiveness in the intermediate goods
market. In practice, these shocks capture unmodeled inflation pressures, such as those arising
from fluctuations in commodity prices.
Financial intermediation involves two actors, banks and entrepreneurs, whose interaction
captures imperfections in financial markets. These actors should not be interpreted in a
literal sense, but rather as a device for modeling credit frictions. Banks take deposits from
households and lend to entrepreneurs. Entrepreneurs use their own wealth and the loans from
banks to acquire capital. They then choose the utilization level of capital and rent the capital
to intermediate good producers. Entrepreneurs are subject to idiosyncratic disturbances in
their ability to manage the capital. Consequently, entrepreneurs’ revenue may not be enough
to repay their loans, in which case they default. Banks protect against default risk by pooling
loans to all entrepreneurs and charging a spread over the deposit rate. Such spreads vary
endogenously as a function of the entrepreneurs’ leverage, but also exogenously depending
on the entrepreneurs’ riskiness. Specifically, mean-preserving changes in the volatility of
entrepreneurs’ idiosyncratic shocks lead to variations in the spread (to compensate banks for
changes in expected losses from individual defaults). We refer to these exogenous movements
as “spread” shocks. Spread shocks capture financial intermediation disturbances that affect
entrepreneurs’ borrowing costs. Faced with higher borrowing costs, entrepreneurs reduce
their demand for capital, and investment drops. With lower aggregate demand, there is
a contraction in hours worked and real wages. Wage rigidities imply that hours worked
fall even more (because nominal wages do not fall enough). Price rigidities mitigate price
contraction, further depressing aggregate demand.
Capital producers transform general output into capital goods, which they sell to the entrepreneurs. Their production function is subject to investment adjustment costs: producing
capital goods is more costly in periods of rapid investment growth. It is also subject to exogeNew York Fed DSGE Team, Research and Statistics
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nous changes in the “marginal efficiency of investment” (MEI). These MEI shocks capture
exogenous movements in the productivity of new investments in generating new capital. A
positive MEI shock implies that fewer resources are needed to build new capital, leading to
higher real activity and inflation, with an effect that persists over time. Such MEI shocks
reflect both changes in the relative price of investment versus that of consumption goods
(although the literature has shown the effect of these relative price changes to be small), and
most importantly financial market imperfections that are not reflected in movements of the
spread.
Finally, the government sector comprises a monetary authority that sets short-term interest rates according to a Taylor-type rule and a fiscal authority that sets public spending and
collects lump-sum taxes to balance the budget. Exogenous changes in government spending
are called “government” shocks; more generally, these shocks capture exogenous movements
in aggregate demand. All exogenous processes are assumed to follow independent AR(1)
processes with different degrees of persistence, except for mark-up shocks which have also a
moving-average component, disturbances to government spending which are allowed to be
correlated with total factor productivity disturbances, and exogenous disturbances to the
monetary policy rule, or “policy” shocks, which are assumed to be i.i.d.
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Figure 1: Model Structure
productivity shocks
Firms
wage
rigidities
utilization
capital
wage mark-up
shocks
intermediate goods
price
rigidities
mark-up
shocks
L
labor
MEI
shocks
Capital
Producers
investment
adjustment
costs
Final Goods
Producers
investment
Entrepreneurs
consumption
disc. factor
shocks
Banks
loans
credit
frictions
spread shocks
deposits
Households
bills
habit
persistence
Government
t
interest rate
policy
policy
shocks
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gov’t spending
shocks
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The Model’s Transmission Mechanism
In this section, we illustrate some of the key economic mechanisms at work in the model’s
equilibrium. We do so with the aid of the impulse response functions to the main shocks
hitting the economy, which we report in Figures 6 to 11.
We start with the shocks most closely associated with the Great Recession and the severe
financial crisis that characterized it: the discount factor shock and the spread shock. The
discount factor shock reflects a sudden desire by households to cut down on their consumption
and save more. This shift may capture the fact that households want to reduce their debt
level, or increased pessimism about future economic conditions. Figure 6 shows the impulse
responses of the variables used in the estimation to a one-standard-deviation innovation in
the discount factor shock. Such a shock results in a decline in consumption (fourth panel in
left column), and hence in aggregate demand, which leads to a fall in output growth (top
left panel), hours worked (top right panel), and real wage growth. The implied reduction in
marginal costs puts downward pressure on inflation (second and third rows). In addition, the
discount factor shock implies an increase in the credit spread (fifth panel in left row), which
weighs negatively on investment. Monetary policy typically attempts to mitigate the decline
in activity and inflation by lowering the FFR, but it cannot fully offset the macroeconomic
effects of the shock.
The other key shock, the spread shock, stems from an increase in the perceived riskiness
of borrowers, which induces banks to charge higher interest rates for loans, thereby widening
credit spreads. As a result of this increase in the expected cost of capital, entrepreneurs’
borrowing falls, hindering their ability to channel resources to the productive sector via
capital accumulation. Figure 7 shows the impulse responses to a one-standard-deviation
innovation in the spread shock. This leads to a reduction in investment and consequently
to a reduction in output growth (top left panel) and hours worked (top right panel). The
fall in the level of hours is fairly sharp in the first year and persists for many quarters
afterwards. Of course, the effects of this same shock on GDP growth, which roughly mirrors
the change in the level of hours, are more short-lived. Output growth returns to its steady
state level less than three years after the shock hits, but it barely moves above it after that,
implying no catch up of the level of GDP towards its previous trend (bottom left panel).
The persistent drop in the level of economic activity due to the spread shock also leads to
a prolonged decline in real marginal costs, and, via the New Keynesian Phillips curve, in
inflation. Finally, policymakers endogenously respond to the change in the inflation and real
7
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activity outlook by cutting the federal funds rate (right panel on the third row).
Similar considerations hold for the MEI shock, which represents a direct hit to the ‘technological’ ability of entrepreneurs to transform investment goods into productive capital,
rather than an increase in their funding cost. The impulse responses to MEI shocks, shown
in Figure 8, also feature a decrease in investment, output and hours worked, as well as in
real wages, although these are less persistent than in the case of spread shocks.
Another shock that plays an important role in the model is the stationary TFP shock
(the model features shocks to both the level and the growth rate of productivity – we discuss
here the former). As shown in Figure 9, a positive TFP shock has a large effect on output
growth, but it drives hours down on impact. This negative response of hours is due to the
presence of nominal rigidities, which prevent aggregate demand from expanding enough to
absorb the increased ability of the economy to supply output. With higher productivity,
marginal costs and thus the labor share fall, leading to lower inflation. These dynamics
make the TFP shock particularly suitable to account for the first phase of the recovery, in
which GDP growth was above trend, but hours and inflation remained weak.
The last shock that plays a relevant role in the current economic environment is the price
mark-up shock, whose impulse response is depicted in Figure 10. This shock is an exogenous
source of inflationary pressures, stemming from changes in the market power of intermediate
goods producers. As such, it leads to higher inflation and lower real activity, as producers
reduce supply to increase their desired markup. Compared to those of the other prominent
supply shock in the model, the TFP shock, the effects of markup-shocks are less persistent.
GDP growth falls on impact after mark-ups increase, but returns above average after about
one year, and the effect on the level of output is absorbed in a little over four years. Inflation
is sharply higher, but only for a few quarters, leading to a temporary spike in the nominal
interest rate, as monetary policy tries to limit the pass-through of the shock to inflation.
Unlike in the case of TFP shocks, however, hours fall immediately, mirroring the behavior
of output.
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Forecasts
2017
Real GDP
Growth (Q4/Q4)
Core PCE
Inflation (Q4/Q4)
Federal Funds
Rate (Q4)
Real Natural
Rate (Q4)
Output
Gap (Q4)
Dec.
2.3
(1.5,3.1)
1.3
(1.1,1.5)
1.2
(1.2,1.2)
0.3
(−0.9,1.6)
−1.2
(−2.5,0.0)
Sep.
2.3
(0.9,3.6)
1.2
(0.9,1.6)
1.4
(0.6,2.2)
0.5
(−1.0,2.0)
−0.9
(−2.3,0.6)
2017
Real GDP
Growth (Q4/Q4)
Core PCE
Inflation (Q4/Q4)
Federal Funds
Rate (Q4)
Real Natural
Rate (Q4)
Output
Gap (Q4)
Dec.
2.6
(2.6,2.6)
1.5
(1.5,1.5)
1.2
(1.2,1.2)
0.3
(−1.0,1.5)
−1.1
(−2.3,0.2)
Sep.
2.3
(1.3,3.2)
1.4
(1.2,1.6)
1.4
(0.6,2.2)
0.5
(−0.9,2.0)
−0.9
(−2.2,0.5)
Unconditional Forecast
2018
2019
Dec.
Sep.
Dec.
Sep.
1.9
2.0
2.2
2.0
(−0.8,4.3) (−0.8,4.5) (−0.6,4.8) (−0.8,4.7)
1.3
1.3
1.4
1.5
(0.5,2.1)
(0.4,2.2)
(0.4,2.4)
(0.4,2.5)
2.1
2.1
2.6
2.6
(0.7,3.6)
(0.7,3.8)
(0.9,4.4)
(0.9,4.5)
0.8
0.9
1.1
1.1
(−0.9,2.5) (−0.8,2.6) (−0.8,2.9) (−0.7,3.0)
−1.2
−0.7
−0.9
−0.6
(−3.5,0.8) (−3.3,1.5) (−4.0,1.7) (−4.0,2.2)
2020
Dec.
Sep.
2.3
2.1
(−0.6,5.0) (−0.8,4.8)
1.5
1.6
(0.3,2.7)
(0.4,2.8)
2.9
2.9
(1.0,4.9)
(1.0,4.9)
1.3
1.2
(−0.7,3.2) (−0.7,3.2)
−0.7
−0.5
(−4.4,2.5) (−4.4,2.7)
Conditional Forecast
2018
2019
Dec.
Sep.
Dec.
Sep.
2.0
2.0
2.2
2.0
(−0.6,4.4) (−0.8,4.4) (−0.6,4.8) (−0.8,4.7)
1.5
1.3
1.5
1.5
(0.7,2.2)
(0.5,2.2)
(0.4,2.5)
(0.4,2.6)
2.2
2.1
2.6
2.6
(0.7,3.7)
(0.7,3.8)
(0.9,4.5)
(0.9,4.5)
0.8
0.9
1.1
1.1
(−0.9,2.5) (−0.9,2.6) (−0.8,2.9) (−0.7,3.0)
−1.0
−0.8
−0.8
−0.6
(−3.1,1.0) (−3.2,1.4) (−3.8,1.9) (−3.9,2.1)
2020
Dec.
Sep.
2.2
2.1
(−0.6,4.9) (−0.8,4.8)
1.6
1.7
(0.4,2.8)
(0.4,2.9)
2.9
2.9
(1.1,4.9)
(1.0,4.9)
1.3
1.2
(−0.6,3.2) (−0.7,3.2)
−0.6
−0.6
(−4.2,2.5) (−4.3,2.6)
*The unconditional forecasts use data up to 2017Q3, the quarter for which we have the most recent GDP release, as well as
the federal funds rate, 10-year Treasury yield, and spreads data for 2017Q4. In the conditional forecasts, we further include the
2017Q4 New York Fed projections for GDP growth and core PCE inflation as additional data points. Numbers in parentheses
indicate 68 percent probability intervals.
The table above presents annual forecasts for real GDP growth, core PCE inflation, the
real natural rate, and the output gap for 2017-2020, with 68 percent probability intervals.
We include two sets of forecasts. The unconditional forecasts use data up to 2017Q3, the
quarter for which we have the most recent GDP release. These forecasts also use federal funds
rate, 10-year Treasury yield, and spreads data for 2017Q4 by taking the average realizations
for the quarter up to the forecast date. In the conditional forecasts, we further include the
2017Q4 New York Fed staff projections as of November 22 for GDP growth (2.6 percent)
and core PCE inflation (1.5 percent) as additional data points. Treating the 2017Q4 staff
forecasts as data allows us to incorporate information about the current quarter into the
DSGE forecasts for the subsequent quarters. In addition to providing the current forecasts,
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the table reports the forecasts included in the DSGE memo forwarded to the FOMC in
advance of its December 2017 meeting.
Figure 2 presents quarterly forecasts, both unconditional (left panels) and conditional
(right panels). In the graphs, the black line represents data, the red line indicates the mean
forecast, and the shaded areas mark the 50, 60, 70, 80 and 90 percent probability intervals
for the forecasts, reflecting both parameter and shock uncertainty. Figure 3 compares the
current forecasts with the September forecasts.
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Figure 2: Forecasts
Unconditional
Conditional
Real GDP Growth
Real GDP Growth
+
7.5
Percent Q/Q Annualized
Percent Q/Q Annualized
7.5
5.0
2.5
0.0
-2.5
+
-5.0
+
-7.5
5.0
2.5
0.0
-2.5
-5.0
-7.5
2008 2010 2012 2014 2016 2018 2020 2022
2008 2010 2012 2014 2016 2018 2020 2022
Core PCE Inflation
Core PCE Inflation
Percent Q/Q Annualized
Percent Q/Q Annualized
4
3
2
1
0
4
+
+
+
+
3
+
+
+
+
-1-
-1-
-1-
-1-
2
1
0
2008 2010 2012 2014 2016 2018 2020 2022
2008 2010 2012 2014 2016 2018 2020 2022
Nominal FFR
Nominal FFR
Percent Annualized
6
Percent Annualized
+
5
4
3
2
1
6
+
+
+
+
5
+
+
+
+
4
+
+
+
+
3
+
+
+
2
+
+
+
1
2008 2010 2012 2014 2016 2018 2020 2022
-1-
2008 2010 2012 2014 2016 2018 2020 2022
Black lines indicate data, red lines indicate mean forecasts, and shaded areas mark the uncertainty associated with our forecast
as 50, 60, 70, 80, and 90 percent probability intervals.
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Figure 3: Change in Forecasts
Unconditional
Conditional
Real GDP Growth
Real GDP Growth
t
t
n
7.5
5.0
Percent Q/Q Annualized
Percent Q/Q Annualized
7.5
+-
2.5
0.0
-2.5
t
-5.0
t
t
-7.5
t
0.0
-2.5
-5.0
-7.5
2008 2010 2012 2014 2016 2018 2020 2022
Core PCE Inflation
4
4
Percent Q/Q Annualized
Percent Q/Q Annualized
t
2.5
Core PCE Inflation
3
2
1
0
3
2
1
0
2008 2010 2012 2014 2016 2018 2020 2022
2008 2010 2012 2014 2016 2018 2020 2022
Nominal FFR
6
t
t
5
t
t
4
t
t
3
t
t
2
t
t
Nominal FFR
6
Percent Annualized
Percent Annualized
t
5.0
2008 2010 2012 2014 2016 2018 2020 2022
1
t
/J
5
+
4
3
2
1
t
2008 2010 2012 2014 2016 2018 2020 2022
2008 2010 2012 2014 2016 2018 2020 2022
Solid (dashed) red and blue lines represent the mean and the 90 percent probability intervals of the current (previous) forecast.
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Interpreting the Forecasts
We use the shock decomposition shown in Figure 4 to interpret the forecasts. This figure
quantifies the relevance of the most important shocks for output growth, core PCE inflation,
and the federal funds rate (FFR) from 2007 onwards. In each of the three panels, the solid
line (black for realized data, red for mean forecast) shows the variable in deviation from
its steady state. The bars represent the contribution of each shock to the deviation of the
variable from steady state, computed as the counterfactual values (in deviations from the
mean) obtained when all other shocks are zero. Some of the shocks have been aggregated in
this decomposition. For example, the bars labeled “financial” (in purple) capture the effect
of shocks to the spread as well as to the discount factor.
Seen through the lens of this decomposition, the evolution of the economy over the past
few years and its forecast through 2020 can be described as follows. Between 2010 and
2014, persistent headwinds from the financial crisis, which are captured in the model by the
financial (purple) and MEI (azure) shocks, held back the pace of the recovery. These sources
of drag on the economy were also accompanied by a sequence of negative TFP shocks (orange
bars), as was also apparent from the extraordinarily weak readings on both TFP and labor
productivity over this period. During the course of 2014, the financial headwinds appeared
to be abating, providing positive contributions to GDP growth that helped to lift it over its
potential, hence also helping to close the output gap and increase the natural rate of interest
(Figure 5). However, this improvement in financial conditions suffered a setback since the
summer of 2015, pushing growth once again below steady state. More recently, monetary
policy shocks are estimated to be depressing growth and to continue to do so throughout the
forecast horizon, while financial and MEI shocks will provide a somewhat offsetting force.
The oscillations in the contribution of financial shocks to economic developments are
also evident in the historical decomposition of inflation, with the purple bars becoming
negative after the financial crisis and then contributing even more negatively beginning in
2011. Starting in 2016, these effects began to diminish very gradually, but are still projected
to keep inflation below steady state throughout the forecast horizon. In addition, the model
sees mark-up shocks (green bars), which capture the effect of exogenous changes in marginal
costs such as those connected with fluctuations in commodity prices, as a further negative
drag on inflation. This drag is especially pronounced in 2017 and it is projected to persist
throughout the forecast horizon. Beginning in 2011, inflation was also pulled down by
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negative government spending and TFP shocks.
In equilibrium, the negative impact of financial shocks on the economy is partly cushioned
by the endogenous response of monetary policy, in the form of a reduction in the policy
rate. In the case of financial shocks, for instance, this endogenous response is captured
by the purple bars in the interest rate panel, which indicate that the federal funds rate
was lowered throughout the recovery in response to the financial headwinds. In fact, this
endogenous adjustment of the policy instrument was decreasing during 2014, when the effects
of the headwinds were abating, but was dialed back up again in 2015 as financial conditions
deteriorated again. In addition, the negative impact of exogenous shocks can be offset
through expansionary monetary policy. In particular, forward guidance about the future path
of the federal funds rate (captured by anticipated policy shocks whose effects are included in
the yellow bars) played an important role in counteracting the financial headwinds between
2009 and 2013, lifting both output and inflation. However, the positive effect of this policy
accommodation on the level of output has been negligible over the most recent quarters, and
it is forecasted to be a drag on output growth over the forecast horizon.
Figure 5 shows the output gap—computed as the percent difference between output and
its “natural” level, namely the one that would prevail in the absence of nominal rigidities
and mark-up shocks—and the natural rate of interest through history. The natural rate
of interest is projected to increase slowly over time, reflecting the continuing restraining
effect of financial headwinds and lower productivity growth. This path for the real natural
rate is roughly in line with that for the real policy rate, implying that monetary policy is
not especially accommodative over the forecast horizon. However, policy is expected to be
slightly more accommodative in the near term, as indicated by the modest gap between the
ex-ante real interest rate and the real natural rate in 2017. The model’s estimate of the
output gap suggests that slack persists and will be absorbed only gradually over time. This
measure of underutilization of resources also reflects low marginal costs of production for
firms, a key driver of the inflation projections. The model’s estimate of firms marginal costs
suggests that these have not recovered much over the last few years, owing to the weakness
in real wage growth. The output gap thus closes only gradually, which explains the slow
return of inflation to target.
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Figure 4: Shock Decomposition
Percent Q/Q Annualized
(deviations from mean)
Real GDP Growth
3
0
-3
-6
-9
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Percent Q/Q Annualized
(deviations from mean)
Core PCE Inflation
0.5
0.0
-0.5
-1.0
-1.5
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
Nominal FFR
Percent Annualized
(deviations from mean)
2
1
0
-1
-2
-3
-4
--
t
t
t
t
t
t
Gov't
Financial
TFP
Mark-Up
Policy
MEI
+
+
+
2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022
The shock decomposition is presented for the conditional forecast. The solid lines (black for realized data, red for mean forecast)
show each variable in deviation from its steady state. The bars represent the shock contributions; specifically, the bars for each
shock represent the counterfactual values for the observables (in deviations from the mean) obtained by setting all other shocks
to zero.
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Figure 5: Output Gap and the Natural Interest Rate
Output Gap
10
5
0
-5
1960
1970
1980
1990
2000
2010
2020
Natural Rate & Ex-Ante Real Rate
9
Ex-Ante Real Rate
Real Natural Rate
Percent Annualized
6
i
3
0
-3
1960
1970
1980
1990
New York Fed DSGE Team, Research and Statistics
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2000
2010
2020
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Figure 6: Responses to a Discount Factor Shock
Hours Per Capita
Real GDP Growth
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
0.0
-0.1
-0.2
-0.3
10
20
30
40
10
20
10
20
30
40
10
Core PCE Inflation
~
30
40
10
Consumption growth per capita
t :
20
30
20
30
40
Real Investment per capita
0.0
-0.1
-0.2
-0.3
10
40
~V_ _
10
BAA - 10yr Treasury Spread
20
30
40
Long term inflation expectations
-0.005
-0.010
-0.015
-0.020
0.08
0.06
0.04
0.02
10
20
30
40
10
Long term interest rate expectations
kr
10
20
.
30
t10
20
30
20
30
40
Total Factor Productivity
0.00
-0.01
-0.02
-0.03
40
Real GDI Growth
0.0
-0.1
-0.2
-0.3
40
k-:::
-0.025
-0.050
-0.075
-0.100
20
20
Nominal FFR
;10
-0.01
-0.02
-0.03
-0.04
-0.05
30
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
0.00
-0.01
-0.02
-0.03
0.0
-0.1
-0.2
-0.3
40
GDP Deflator
Percent Change in Wages
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
30
40
~
t: ;_ _
10
1.0
0.8
0.6
0.4
0.2
0.0
0.0
New York Fed DSGE Team, Research and Statistics
Page 33 of 104
0.2
20
0.4
30
0.6
0.8
40
1.0
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Figure 7: Responses to a Spread Shock
Hours Per Capita
Real GDP Growth
0.01
0.00
-0.01
-0.02
-0.03
-0.04
0.00
-0.05
-0.10
-0.15
10
20
30
40
10
20
Percent Change in Wages
40
30
40
30
40
GDP Deflator
0.002
0.000
-0.002
-0.004
-0.006
0.001
0.000
-0.001
-0.002
10
20
30
40
10
Core PCE Inflation
20
Nominal FFR
0.005
0.000
-0.005
-0.010
-0.015
0.001
0.000
-0.001
-0.002
10
20
30
40
10
Consumption growth per capita
0.0
-0.1
-0.2
-0.3
10
20
30
20
Real Investment per capita
0.04
0.03
0.02
0.01
0.00
-0.01
40
V
BAA - 10yr Treasury Spread
0.05
0.04
0.03
0.02
0.01
0.00
30
+
10
20
30
40
Long term inflation expectations
0.0010
0.0005
0.0000
-0.0005
10
20
30
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.002
0.000
-0.002
-0.004
0.002
0.000
-0.002
-0.004
10
20
30
40
Real GDI Growth
0.01
0.00
-0.01
-0.02
-0.03
-0.04
10
20
30
40
10
1.0
0.8
0.6
0.4
0.2
0.0
0.0
New York Fed DSGE Team, Research and Statistics
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0.2
20
0.4
30
0.6
0.8
40
1.0
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Figure 8: Responses to an MEI Shock
Hours Per Capita
Real GDP Growth
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
L
0.2
0.0
-0.2
-0.4
10
20
30
~
40
10
20
kc<
10
b
0.007
0.006
0.005
0.004
0.003
0.002
0.001
20
30
. : ----
40
10
Core PCE Inflation
0.007
0.006
0.005
0.004
0.003
0.002
0.001
20
30
0.02
0.00
-0.02
-0.04
-0.06
40
V
~
10
30
40
V
h- "~
10
10
20
30
30
40
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
20
30
30
40
10
20
30
40
Total Factor Productivity
Real GDI Growth
10
40
t
40
0.012
0.009
0.006
0.003
0.000
20
30
20
0.004
0.003
0.002
0.001
Long term interest rate expectations
10
20
Long term inflation expectations
BAA - 10yr Treasury Spread
0.00
-0.01
-0.02
-0.03
-0.04
-0.05
40
Real Investment per capita
0.0
-0.5
-1.0
-1.5
20
30
· ·
10
Consumption growth per capita
0.025
0.000
-0.025
-0.050
20
Nominal FFR
.____b_ _---------10
40
GDP Deflator
Percent Change in Wages
0.005
0.000
-0.005
-0.010
-0.015
30
40
0.01
0.00
-0.01
-0.02
-0.03
r
1.0
0.8
0.6
0.4
0.2
0.0
0.0
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10
0.2
20
0.4
30
0.6
0.8
40
1.0
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Figure 9: Responses to a TFP Shock
Hours Per Capita
Real GDP Growth
0.4
0.3
0.2
0.1
0.0
L
~_____,;,...__f10
20
30
0.1
0.0
-0.1
-0.2
V:--r-: --:-+~
40
10
Percent Change in Wages
10
20
k::-:
10
30
40
10
+
30
40
10
10
20
~
10
30
40
10
30
0.000
-0.005
-0.010
-0.015
' - - - - -
10
20
30
40
L
10
+
20
30
20
0.6
0.4
0.2
0.0 1::.1........--- ~
40
10
20
+
+
+
10
40
30
40
Total Factor Productivity
Real GDI Growth
0.4
0.3
0.2
0.1
0.0
30
,:;..___k-=_ __
Long term interest rate expectations
-0.005
-0.010
-0.015
-0.020
20
Long term inflation expectations
20
40
0.20
0.15
0.10
0.05
0.00
BAA - 10yr Treasury Spread
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
30
20
Real Investment per capita
+
+
+
l
40
-0.01
-0.02
-0.03
-0.04
-0.05
-0.06
-0.07
Consumption growth per capita
0.3
0.2
0.1
0.0
30
20
Nominal FFR
+
20
40
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
Core PCE Inflation
-0.005
-0.010
-0.015
-0.020
-0.025
-0.030
-0.035
30
GDP Deflator
+
+
0.03
0.02
0.01
0.00
20
40
1.0
0.8
0.6
0.4
0.2
0.0
0.0
New York Fed DSGE Team, Research and Statistics
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0.2
0.4
-
30
0.6
40
0.8
1.0
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Figure 10: Responses to a Price Mark-up Shock
Real GDP Growth
Hours Per Capita
0.00
-0.03
-0.06
-0.09
-0.12
0.0
-0.1
-0.2
-0.3
10
20
30
40
10
20
Percent Change in Wages
20
30
40
10
20
Core PCE Inflation
~
40
30
40
Nominal FFR
0.06
0.05
0.04
0.03
0.02
0.01
0.00
:
10
20
30
40
~
10
v- i
10
0.05
0.00
-0.05
-0.10
-0.15
-0.20
-0.25
20
30
40
v::
10
20
+
+
20
30
40
Long term inflation expectations
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
30
+
10
BAA - 10yr Treasury Spread
t::-i- ~
20
Real Investment per capita
Consumption growth per capita
0.0000
-0.0025
-0.0050
-0.0075
-0.0100
30
0.15
0.10
0.05
0.00
10
0.00
-0.02
-0.04
-0.06
-0.08
-0.10
-0.12
40
GDP Deflator
0.00
-0.05
-0.10
-0.15
0.15
0.10
0.05
0.00
30
~L...::::::=====
f ~f =======
40
10
Long term interest rate expectations
20
30
40
Total Factor Productivity
0.00
-0.01
-0.02
-0.03
0.012
0.009
0.006
0.003
0.000
10
20
30
40
Real GDI Growth
0.00
-0.03
-0.06
-0.09
-0.12
10
20
30
40
10
1.0
0.8
0.6
0.4
0.2
0.0
0.0
New York Fed DSGE Team, Research and Statistics
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0.2
20
0.4
30
0.6
0.8
40
1.0
21
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Figure 11: Responses to a Monetary Policy Shock
Hours Per Capita
Real GDP Growth
0.4
0.3
0.2
0.1
0.0
l~~ = = t::-:h.
0.5
0.4
0.3
0.2
0.1
0.0
10
20
30
40
10
Percent Change in Wages
0.015
0.010
0.005
0.000
10
20
30
40
C'½
l .
10
-0.025
-0.050
-0.075
-0.100
-0.125
20
30
c:::::
V
10
Core PCE Inflation
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
40
10
30
~
10
30
v-
40
10
20
30
40
40
l . .
10
20
30
40
20
30
40
6'
+
+
10
20
30
40
Total Factor Productivity
Real GDI Growth
0.4
0.3
0.2
0.1
0.0
30
Long term inflation expectations
0.04
0.03
0.02
0.01
0.00
10
20
+
Long term interest rate expectations
0.000
-0.005
-0.010
-0.015
40
+
0.0010
0.0005
0.0000
-0.0005
-0.0010
20
30
Nominal FFR
0.6
0.4
0.2
0.0
20
40
Real Investment per capita
BAA - 10yr Treasury Spread
0.005
0.000
-0.005
-0.010
20
10
Consumption growth per capita
0.4
0.3
0.2
0.1
0.0
30
GDP Deflator
0.005
0.004
0.003
0.002
0.001
0.000
-0.001
~
l
20
40
l
1.0
0.8
0.6
0.4
0.2
0.0
0.0
New York Fed DSGE Team, Research and Statistics
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+
+
+
+
+
10
0.2
20
0.4
30
0.6
0.8
40
1.0
22
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New York Fed DSGE Model: Research Directors Draft
December 1, 2017
References
Bernanke, B. S., M. Gertler, and S. Gilchrist (1999): “The Financial Accelerator
in a Quantitative Business Cycle Framework,” in Handbook of Macroeconomics, ed. by
J. B. Taylor and M. Woodford, Amsterdam: North-Holland, vol. 1C, chap. 21, 1341–93.
Christiano, L. J., M. Eichenbaum, and C. L. Evans (2005): “Nominal Rigidities and
the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113,
1–45.
Christiano, L. J., R. Motto, and M. Rostagno (2014): “Risk Shocks,” American
Economic Review, 104, 27–65.
Del Negro, M., M. P. Giannoni, and F. Schorfheide (2015): “Inflation in the Great
Recession and New Keynesian Models,” American Economic Journal: Macroeconomics,
7, 168–196.
Del Negro, M. and F. Schorfheide (2012): “DSGE Model-Based Forecasting,” Federal
Reserve Bank of New York Working Paper.
Laseen, S. and L. E. Svensson (2011): “Anticipated Alternative Policy-Rate Paths in
Policy Simulations,” International Journal of Central Banking, 7, 1–35.
Smets, F. and R. Wouters (2007): “Shocks and Frictions in US Business Cycles: A
Bayesian DSGE Approach,” American Economic Review, 97, 586 – 606.
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Detailed Philadelphia (PRISM) Forecast Overview
December 2017
Keith Sill
Forecast Summary
The FRB Philadelphia DSGE model denoted PRISM, projects that real GDP growth will
run at a fairly strong pace over the forecast horizon with real output growth peaking at a bit
under 3.4 percent in mid-2019. Core PCE inflation edges up to reach 2 percent at the end of
2020. The funds rate rises to 2.2 percent in 2018Q4 and rises steadily to reach 3.6 percent at the
end of 2020. The current gap between the level of output and its trend level remains significant
in the estimated model and, absent any shocks, the model continues to predict a fairly rapid
recovery to the trend level. The relatively slow pace of growth and low inflation that have
characterized U.S. economic performance over the past few years require the presence of shocks
to offset the strength of the model’s internal propagation channels.
The Current Forecast and Shock Identification
The PRISM model is an estimated New Keynesian DSGE model with sticky wages,
sticky prices, investment adjustment costs, and habit persistence. The model is similar to the
Smets & Wouters 2007 model and is described more fully in Schorfheide, Sill, and Kryshko
2010. Unlike in that paper though, we estimate PRISM directly on core PCE inflation rather
than projecting core inflation as a non-modeled variable. Details on the model and its estimation
are available in a Technical Appendix that was distributed for the June 2011 FOMC meeting or
is available on request.
The current forecasts for real GDP growth, core PCE inflation, and the federal funds rate
are shown in Figures 1a-1c along with 68 percent probability coverage intervals. The forecast
uses data through 2017Q3 supplemented by a 2017Q4 nowcast. The model takes the 2017Q4
nowcast for output growth of 2.6 percent as given and the projection begins with 2018Q1.
PRISM anticipates that output growth rises to a 3 percent pace in 2018Q1, with growth then
edging up to 3.3 percent in mid-2019. Overall, the growth forecast for this round is similar to the
September projection. While output growth is fairly robust going forward, core PCE inflation
stays contained and runs at a pace slightly below the 2 percent target until mid-2020 when it
reaches 2 percent. Based on the 68 percent coverage interval, the model sees a minimal chance of
deflation or recession (measured as negative quarters of real GDP growth) over the next 3 years.
The federal funds rate is determined by an estimated policy rule and the funds rate rises from 1.2
percent in 2017Q4, 2.2 percent in 2018Q4, 3.1 percent in 2019Q4, and 3.6 percent in 2020Q4.
This path for the funds rate is similar to that in the September projection.
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The key factors driving the projection are shown in the forecast shock decompositions
(Figures 2a-2e) and the smoothed estimates of the model’s primary shocks (shown in Figure 3,
where they are normalized by standard deviation). Over the last few quarters real GDP growth
has been running close to the model’s trend rate. Positive shocks to government spending,
investment, and labor supply have offset negative contributions from the model’s other shocks
(TFP, markups, financial, and monetary policy). As these shocks unwind, output growth
maintains edges up to a slightly above steady state pace over the next few years. Over the course
of the recession and recovery PRISM estimated a series of large positive shocks to leisure
(negative shocks to labor supply) that have a persistent effect on hours worked and so pushed
hours well below steady state. As these shocks unwind hours worked continue to rebound over
the forecast horizon and so support higher output growth. Similarly, the unwinding of investment
shocks contribute to output growth over the forecast horizon.
After strong performance in early 2016, consumption growth (Figure 2d) pulled back to a
below trend pace through mid 2017. This was largely driven by negative contributions from TFP
shocks and investment shocks. Consumption is projected to gradually rise toward trend over the
next three years. A gradual unwinding of investment shocks and higher interest rates keeps
consumption growth below steady state until 2020. Financial shocks that boost consumption in
turn weaken investment growth (Figure 2d-e). However, strong investment shocks pushed
investment growth to an above-trend pace in the second half of 2016 and into 2017. The model
now forecasts above-trend growth in investment (gross private domestic + durable goods
consumption) in 2018 as the gradual unwinding of MEI shocks (see Figures 2e and 3) are
partially offset by the effects of financial shocks: the unwinding of the discount factor shocks
leads to a downward pull on investment growth over the next three years.
The forecast for core PCE inflation continues to be a story of upward pressure from the
unwinding of negative labor supply shocks and MEI shocks being offset by downward pressure
from the waning of discount factor shocks. Negative discount factor shocks have a strong and
persistent negative effect on marginal cost and inflation in the estimated model. But labor supply
shocks that push down aggregate hours also serve to put upward pressure on the real wage and
hence marginal cost. The effect is persistent -- as the labor supply shocks unwind over the
forecast horizon they exert a waning upward push to inflation. On balance the effect of these
opposing forces keep inflation close to, or slightly below, target over the next 3 years.
The federal funds rate is projected to rise fairly quickly over the forecast horizon. The
model attributes the current level of the funds rate primarily to a combination of monetary
policy, discount factor and MEI shock dynamics. Looking ahead, the positive contribution from
labor supply shocks is more than offset by discount factor shock dynamics over the medium
term, but as these shocks wane the funds rate gradually rises to 3.6 percent by the end of 2020.
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References
Schorfheide, Frank, Keith Sill, and Maxym Kryshko. 2010. “DSGE model-based forecasting of
non-modelled variables.” International Journal of Forecasting, 26(2): 348-373.
Smets, Frank, and Rafael Wouters. 2007. “Shocks and Frictions in U.S. Business Cycles: A
Bayesian DSGE Approach.” American Economic Review, 97(3): 586-606.
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Figure 1a
Real GDP Growth
10
8
......
./
6
_
.......
_,,,,,,
4
2
--
0
........
_
-2
-4
-6
-8
-10
2010
2012
2014
2016
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2018
2020
2022
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Figure 1b
Core PCE Inflation
6
5
4
3
2
1
0
-1
2010
2012
2014
2016
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Figure 1c
Fed Funds Rate
8
/
6
/
/
...,
/
I
I
4
/
I
I
I
I
2
I
0
-2
-4
2010
2012
2014
2016
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Figure 2a
Shock Decompositions
Real GDP Growth
percent
8
8
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
-6
Date
1-
tech
-
gov
-
mei
-
mrkp
-
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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labor
-
fin
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Figure 2b
Shock Decompositions
Core PCE Inflation
4
4
3
3
.... - ...................................
percent
2
1
1
0
0
-1
-1
-2
-2
-3
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
-3
,_ - - - - - Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
2
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 2c
Shock Decompositions
Fed Funds Rate
6
6
4
4
percent
2
0
I
I
I
I
I
I
I
I
~~v-
I I I I I
I I l l f ll\ 11111 11 11 '11, ,
, I I I I I I
I I I I I I
1 11
__ ,
1
0
-2
-2
-4
-4
I
I
I
I
I
I
I
:
I
I
I
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
--- - -Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
I
2
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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labor
fin
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Figure 2d
Shock Decompositions
Real Consumption Growth
percent
6
6
4
4
2
2
0
0
-2
-2
-4
-4
-6
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
-6
Date
1-
tech
-
gov
-
mei
-
mrkp
-
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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-
fin
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Figure 2e
Shock Decompositions
Real Investment Growth
percent
30
30
20
20
10
10
0
0
-10
-10
-20
2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020
-20
,_ - - - - - Date
tech
gov
mei
mrkp
shocks:
TFP:
Gov:
MEI:
MrkUp:
Labor:
Fin:
Mpol:
Total factor productivity growth shock
Government spending shock
Marginal efficiency of investment shock
Price markup shock
Labor supply shock
Discount factor shock
Monetary policy shock
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Figure 3
Smoothed Shock Estimates for Conditional Forecast Model
(normalized by standard deviation)
4
labor shock
discount factor shock
5
2
0
0
-2
-4
-5
2010 2012 2014 2016 2018
2
2010 2012 2014 2016 2018
TFP shock
mei shock
2
0
0
-2
-2
-4
2010 2012 2014 2016 2018
2010 2012 2014 2016 2018
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Impulse Responses to TFP shock
output growth
consumption growth
1
0.5
0
1
IS:: I IS: I
I~
1----== I
2-3 =-:2____
0.5
0
5
10
15
0
0
investment growth
2
0
-2
0
-0.05
10
15
aggregate hours
0.5
0
0
5
10
15
-0.5
0
inflation
0.05
5
5
10
15
nominal rate
0.05
0
I
0
5
10
15
-0.05
I
0
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Impulse Response to Leisure Shock
output growth
consumption growth
2
0
-2
2
1-== I IC I
I==:3 Is ~-=_
LJ
0
0
5
10
15
-2
0
investment growth
5
0
-5
0.2
0
10
15
aggregate hours
0
-1
0
5
10
15
-2
0
inflation
0.4
5
5
10
15
nominal rate
0.4
0.2
I ______:__]
0
5
10
15
0
I
0
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Impulse Responses to MEI Shock
output growth
consumption growth
2
0.2
0
0
-2
-~• I
0
5
10
15
-0.2
investment growth
1
0
0.5
1----== • I -~
0
5
10
15
0
0
inflation
-0.1
10
15
5
10
15
nominal rate
0.1
0
5
aggregate hours
10
-10
IZJ
0
0.4
I=----=---:J I~
0.2
0
5
10
15
0
0
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Impulse Responses to Financial Shock
output growth
consumption growth
1
0
-1
2
-~
0
0
5
10
15
-2
1:::__
0
investment growth
-5
0.2
0
I
15
0.5
-~
-~
1----=--=J 1----=-3
0
0
5
10
15
-0.5
0
inflation
0.4
10
aggregate hours
5
0
5
5
10
15
nominal rate
1
0.5
0
5
10
15
0
0
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Impulse Responses to Price Markup Shock
output growth
consumption growth
0.5
0
-0.5
0.5
1==- 1c I
IL J 1=-z:1
0
0
5
10
15
-0.5
0
investment growth
1
0
-1
15
aggregate hours
-0.1
0
5
10
15
-0.2
0
inflation
-1
10
0
5
10
15
nominal rate
1
0
5
0.5
I\ • I-~
0
0
5
10
15
-0.5
0
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Impulse Responses to Unanticipated Monetary Policy Shock
output growth
consumption growth
0.5
0
-0.5
0.5
123 I/ : I
123 1::3
I~ I 1==--- I
0
0
5
10
15
-0.5
0
investment growth
1
0
-1
0
-0.1
10
15
aggregate hours
0.2
0
0
5
10
15
-0.2
0
inflation
0.1
5
5
10
15
nominal rate
1
0
0
5
10
15
-1
0
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Impulse Responses to Govt Spending Shock
output growth
2
0
-2
consumption growth
0.5
I\
11~ I
1: 3 1~
S: I --=:__J
0
0
5
10
15
-0.5
0
investment growth
0.2
0
-0.2
0.01
0
10
15
aggregate hours
0.4
0.2
0
5
10
15
0
0
inflation
0.02
5
5
10
15
nominal rate
0.04
0.02
I
0
5
10
15
0
I
0
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Research Directors’ Guide to
the Chicago Fed DSGE Model*
Jeffrey R. Campbell
Filippo Ferroni
Jonas D. M. Fisher
Leonardo Melosi
November 30, 2017
This guide describes the construction and estimation of the Chicago Fed’s DSGE
model, which we use both for internal forecasting and for creating our contributions
to the System DSGE memo distributed quarterly to the FOMC. The model has been
in use and under ongoing development since 2010. Originally, it was largely based
on Justiniano, Primiceri, and Tambalotti (2010). We published results based on
simulations from the estimated model in Campbell, Evans, Fisher, and Justiniano
(2012) and in Campbell, Fisher, Justiniano, and Melosi (2016).
The model contains many features familiar from other DSGE analyses of
monetary policy and bussiness cycles. External habit in preferences, i-dot costs of
adjusting investment, price and wage stickiness based on Calvo’s (1983) adjustment
probabilities, and partial indexation of unadjusted prices and wages using recently
observed price and wage inflation. The features which distinguish our analysis from
many otherwise similar undertakings are
• Forward Guidance Shocks: An interest-rate rule which depends on recent
(and expected future) inflation and output and is subject to stochastic
disturbances governs our model economy’s monetary policy rate. Standard
analysis prior to the great recession restricted the stochastic disturbances to be
unforecastable. Our model deviates from this historical standard by including
forward guidance shocks, as in Laséen and Svensson (2011). A j-quarter ahead
forward guidance shock revealed to the public at time t influences the interestrate rule’s stochastic intercept only at time t + j. Each period, the model’s
monetary authority reveals a vector of these shocks with one element for each
quarter from the present until the end of the forward guidance horizon. The
*
The views expressed herein are the authors’. They do not necessarily represent those of the
Federal Reserve Bank of Chicago, the Federal Reserve System, or its Board of Governors.
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vector’s elements may be correlated with each other, so the monetary authority
could routinely reveal persistent shifts in the interest-rate rule’s stochastic
intercept. However, the forward guidance shocks are serially uncorrelated
over time, as is required for them to match the definition of “news.”
• Investment-Specific Technological Change: As in the Real Business
Cycle models from which modern DSGE models decend (King, Plosser, and
Rebelo, 1988a), stochastic trend productivity growth both short-run and longrun fluctuations. Our model features two such stochastic trends, one to
Hicks-neutral productivity (King, Plosser, and Rebelo, 1988b) and one to
the technology for converting consumption goods into investment goods (as
in Fisher (2006)). This investment-specific technological change allows our
model to reproduce the dynamics of the relative price of investment goods
to consumption goods, which is a necessary input into the formula we use to
create Fisher-ideal chain-weighted index of real GDP.
• A Mixed Calibration-Bayesian Estimation Empirical Strategy:
Bayesian estimation of structural business cycle models attempts to match all
features of the data’s probability distribution using the model’s parameters.
Since no structural model embodies Platonic “truth,” this exercise inevitably
requires trading off between the model’s ability to replicate first moments
with its fidelty to the business cycles in second moments. Since the criteria for
this tradeoff are not always clear, we adopt an alternative “first-moments-first”
strategy. This selects the values of model parameters which govern the model’s
steady-state growth path, such as the growth rates of Hicks-neutral and
investment-specific technology, to match estimates of selected first moments.
These parameter choices are then fixed for Bayesian estimation, which chooses
values for model parameters which only influence second moments, such as
technology innovation variances. (Since we employ a log linear solution of our
model and all shocks to its primitives have Gaussian distributions, our analysis
has no non-trivial implications for third and higher moments of the data.)
The guide proceeds as follows. The next section presents the model economy’s
primitives, while Section 2 presents the agents’ first-order conditions. Section 3
gives the formulas used to remove nominal and technological trends from model
variables and thereby induce model stationarity, and Sections 4 and 5 discuss
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the stationary economy’s steady state and the log linearization of its equilibrium
necessary conditions around it. Section 6 discusses measurement issues which arise
when comparing model-generated data with data measured by the BEA and BLS.
Section 7 describes our mixed Calibration-Bayesian Estimation empirical strategy
and presents the resulting parameter values we use for model simulations and
forecasting.
1
The Model’s Primitives
Eight kinds of agents populate the model economy:
• Households,
• Investment producers,
• Competitive final goods producers,
• Monopolistically-competitive differentiated goods producers,
• Labor Packers,
• Monopolistically-competitive guilds,
• a Fiscal Authority and
• a Monetary Authority.
These agents interact with each other in markets for
• final goods used for consumption
• investment goods used to augment the stock of productive capital
• differentiated intermediate goods
• capital services
• raw labor
• differentiated labor
• composite labor
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• government bonds
• privately-issued bonds, and
• state-contingent claims.
The households have preferences over streams of an aggregate consumption good,
leisure, and the real value of the fiscal authority’s bonds in their portfolios. Our
specification for preferences displays balanced growth. They also feature external
habit in consumption; which creates a channel for the endogenous propagation of
shocks. Our bonds-in-the-utility-function preferences follow those of Fisher (2015),
and they allow us to incorporate a persistent spread between the monetary policy
rate and the return on productive capital. The aggregate consumption good has a
single alternative use, as the only input into the linear production function operated
by investment producers. These firms sell their output to the households. In turn,
households produce capital services from their capital stocks, which they then sell
to differentiated goods producers. Producers of final goods operate a constantreturns-to-scale technology with a constant elasticity of substitution between its
inputs, which are differentiated goods produced by the monopolistically-competitive
firms. These firms operate technologies with affine cost curves (a constant fixed cost
and linear marginal cost), which employs capital services and composite labor as
inputs. The labor packers produce composite labor using a constant-returns-toscale technology with a constant elasticity of substitution between its inputs, the
differentiated labor sold by guilds. Each of these produces differentiated labor from
the raw labor provided by the households with a linear technology, and they sell their
outputs to the labor packers. There is a nominal unit of account, called the “dollar.”
The fiscal authority issues one-period nominally risk-free bonds, provides public
goods through government spending, and assesses lump-sum taxes on households.
The monetary authority sets the interest rate on the fiscal authority’s one-period
bond according to an interest-rate rule.
All non-financial trade is denominated in dollars, and all private agents take
prices as given with two exceptions: the monopolistically-competitive differentiatedgoods producers and guilds. These choose output prices to maximize the current
value of expected future profits taking as given their demand curves and all relevant
input prices. Financial markets are complete, but all securities excepting equities
in differentiated-goods producers are in zero net supply. These producers’ profits
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and losses are rebated to the households (who own the firms’ equities) lump-sum
period-by-period, as are the profits and losses of the guilds. Given both a process
for government spending and taxes and a rule for the monetary authority’s interest
rate choice, a competitive equilibrium consists of allocations and prices that are
consistent with households’ utility maximization, firms’ profit maximization, guilds’
profit maximization, and market clearing.
The economy is subject to stochastic disturbances in technology, preferences,
and government policy. Without nominal rigidities, the economy’s real allocations
in competitive equilibrium can be separated from inflation and other dollardenominated variables. Specifically, monetary policy only influences inflation. To
connect real and nominal variables in the model and thereby consider the impact
of monetary policy on the business cycle, we introduce Calvo-style wage and price
setting. That is, nature endows both differentiated goods producers and guilds with
stochastic opportunities to incorporate all available information into their nominal
price choices. Those producers and guilds without such a opportunity must set their
prices according to simple indexing formulas. These two pricing frictions create two
forward-looking Phillips curves, one for prices and another for wages, which form
the core of the new Keynesian approach to monetary policy analysis.
The model economy is stochastic and features complete markets in statecontingent claims. To place these features on a sound footing, we base all shocks on
a general Markovian stochastic process st . Denote the history of this vector from
an initial period 0 through τ with sτ (s0 , s1 , . . . , sτ ). The support of sτ is Στ , and
the probability density of sτ given st for some t < τ is s (sτ st ). (The Hebrew letter
s, pronounced “samekh,” corresponds to the Greek letter σ.) All model shocks are
implicit functions of st , and all endogenous variables are implicit functions of st . We
refer to all such implicit functions as “state-contingent sequences.” We use braces to
denote such a sequence. For example, {Xt } represents the state-contingent sequence
for a generic variable Xt .
1.1
Households
Our model’s households are the ultimate owners of all assets in positive net supply
(the capital stock, differentiated goods producers, and guilds). They provide labor
and divide their current after-tax income (from wages and assets) between current
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consumption, investment in productive capital, and purchases of financial assets,
both those issued by the government and those issued by other households. The
individual household divides its current resources between consumption and the
available vehicles for intertemporal substitution (capital and financial assets) to
maximize a discounted sum of current and expected future felicity.
ª
Et Q β τ εbt+τ ‰Ut+τ + εst+τ L ‰
τ =0
Bt+τ
’’
Pt+τ Rt+τ
with
Ut =
1
(1−γc )
›(Ct − %C̄t−1 )(1 − Ht1+γh )”
1 − γc
(1)
The function L(�) is strictly increasing, concave, and differentiable everywhere
on [0, ª). In particular, Lœ (0) exists and is finite. Without loss of generality, we
set Lœ (0) to one. The argument of L(�) equals the real value of government bonds
in the household’s portfolio: their period t + 1 redemption value Bt divided by their
nominal yield Rt expressed in units of the consumption good with the nominal price
index Pt . The time-varying coefficient multiplying this felicity from bond holdings,
εst , is the liquidity preference shock introduced by Fisher (2015). A separate shock
influences the household’s discounting of future utility to the present, εbt . Specifically,
the household discounts a certain utility in t + τ back to t with β τ Et [εbt+τ ~εbt ]. In
logarithms, these two preference shocks follow independent autoregressive processes.
ln εbt = (1 − ρb ) ln εb⁄ + ρb ln εbt−1 + ηtb , ηtb N(0, σb )
(2)
ln εst = (1 − ρs ) ln εs⁄ + ρs ln εst−1 + ηts , ηts N(0, σs ).
(3)
A household’s wealth at the beginning of period t consists of its nominal
government bond holdings, Bt , its net holdings of privately-issued financial assets,
and its capital stock Kt−1 . The household chooses a rate of capital utilization ut , and
the capital services resulting from this choice equal ut Kt−1 . The cost of increasing
utilization is higher depreciation. An increasing, convex and differentiable function
δ(U ) gives the capital depreciation rate. We specify this as
δ(u) = δ0 + δ1 (u − uƒ ) +
δ2
2
(u − uƒ ) .
2
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A household can augment its capital stock with investment, It . Investment
requires paying adjustment costs of the “i-dot” form introduced by Christiano,
Eichenbaum, and Evans (2005). Also, an investment demand shock alters the
efficiency of investment in augmenting the capital stock. Altogether, if the
household’s investment in the previous period was It−1 , and it purchases It units
of the investment good today, then the stock of capital available in the next period
is
Kt = (1 − δ(ut )) Kt−1 + εit „1 − S „
AK
t−1 It
‚‚ It .
K
At It−1
(4)
In (4), AK
t equals the productivity level of capital goods production, described in
more detail below, and εit is the investment demand shock. In logarithms, this
follows a first-order autoregression with a normally-distributed innovation.
ln εit = (1 − ρi ) ln εi⁄ + ρi ln εit−1 + ηti , ηti N(0, σi )
1.2
(5)
Production
The producers of investment goods use a linear technology to transform the final
good into investment goods. The technological rate of exchange from the final good
to the investment good in period t is AIt . We denote Δ ln AIt with ωt , which we call the
investment-specific technology shock and which follows first-order autogregression
with normally distributed innovations.
ωt = (1 − ρω )ωƒ + ρω ωt−1 + ηtω , ηtω N(0, σω2 )
(6)
Investment goods producers are perfectly competitive.
Final good producers also operate a constant-returns-to-scale technology; which
takes as inputs the products of the differentiated goods producers. To specify this,
let Yit denote the quantity of good i purchased by the representative final good
producer in period t, for i > [0, 1]. The representative final good producer’s output
then equals
Yt „S
1
0
1
p
1+λ
t
Yit
1+λpt
di‚
.
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With this technology, the elasticity of substitution between any two differentiated
products equals 1 + 1~λpt in period t. Although this is constant across products
within a time period, it varies stochastically over time according to an ARMA(1, 1)
in logarithms.
p
p
− θp ηt−1
+ ηtp , ηtp N(0, σp )
ln λpt = (1 − ρp ) ln λpƒ + ρp ln λt−1
(7)
Given nominal prices for the intermediate goods Pit , it is a standard exercise to
show that the final goods producers’ marginal cost equals
Pt = „S
1
0
−
Pit
1
p
λ
t
−λpt
di‚
(8)
Just like investment goods firms, the final goods’ producers are perfectly
competitive. Therefore, profit maximization and positive final goods output together
require the competitive output price to equal Pt . Therefore, we can define inflation
of the nominal final good price as πt ln(Pt ~Pt−1 ).
The intermediate goods producers each use the technology
1−α
α
Yit = (Kite ) ›AtY Hitd ”
− At Φ
(9)
Here, Kite and Hitd are the capital services and labor services used by firm i, and
AYt is the level of neutral technology. Its growth rate, νt ln(AYt ~AYt−1 ), follows a
first-order autogregression.
νt = (1 − ρν ) ν⁄ + ρv νt−1 + ηtν , ηtν N(0, σν ),
(10)
The final term in (9) represents the fixed costs of production. These grow with
α
At AtY ›AtI ” 1−α .
(11)
We demonstrate below that At is the stochastic trend in equilibrium output and
consumption, measured in units of the final good. We denote its growth rate with
zt = ν t +
α
ωt
1−α
(12)
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Similarly, define
I
AK
t A t At
(13)
In the specification of the capital accumulation technology, we labelled AK
t the
“productivity level of capital goods production.” We demonstrate below that this
is indeed the case with the definition in (13).
Each intermediate goods producer chooses prices subject to a Calvo (1983)
pricing scheme. With probability ζp > [0, 1], producer i has the opportunity to
set Pit without constraints. With the complementary probability, Pit is set with the
indexing rule
p
Pit = Pit−1 πt−1
πƒ
ι
1−ιp
.
(14)
In (14), πƒ is the gross rate of price growth along the steady-state growth path, and
ιp > [0, 1].1
1.3
Labor Markets
Households’ hours worked pass through two intermediaries, guilds and labor packers,
in their transformation into labor services used by the intermediate goods producers.
The guilds take the households’ homogeneous hours as their only input and produce
differentiated labor services. These are then sold to the labor packers, who assemble
the guilds’ services into composite labor services.
The labor packers operate a constant-returns-to-scale technology with a constant
elasticity of substitution between the guilds’ differentiated labor services. For its
specification, let Hit denote the hours of differenziated labor purchased from guild
i at time t by the representative labor packer. Then that packer’s production of
composite labor services, Hts are given by
Hts
1
= ‰S (Hit )
1
1+λw
t
1+λw
t
di’
.
0
As with the final good producer’s technology, an ARMA(1, 1) in logarithms governs
To model firms’ price-setting opportunities as functions of st , define a random variable upt
which is independent over time and uniformly distributed on [0, 1]. Then, firm i gets a pricesetting opportunity if either upt C ζp and i > [upt − ζp , upt ] or if upt < ζp and i > [0, upt ] 8 [1 + upt − ζp , 1].
1
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the constant elasticity of substitution between any two guilds’ labor services.
w
w
w
w w
2
ln λw
t = (1 − ρw ) ln λƒ + ρw ln λt−1 − θw ηt−1 + ηt , ηt N(0, σw )
(15)
Just as with the final goods producers, we can easily show that the labor packers’
marginal cost equals
1
− λ1w
Wt = ‰S (Wit )
t
−λw
t
di’
.
(16)
0
Here, Wit is the nominal price charged by guild i per hour of differentiated labor.
Since labor packers are perfectly competitive, their profit maximization and positive
output together require that the price of composite labor services equals their
marginal cost.
Each guild produces it’s differentiated labor service using a linear technology
with the household’s hours worked as its only input. A Calvo (1983) pricing
scheme similar to that of the differentiated goods producers constrains their nominal
prices. Guild i has an unconstrained opportunity to choose its nominal price with
probability ζw > [0, 1]. With the complementary probability, Wit is set with an
indexing rule based on πt−1 and last period’s trend growth rate, zt−1 .
Wit = Wit−1 (πt−1 ezt−1 ) w (πƒ ezƒ )
ι
1−ιw
.
(17)
α
In (17), zƒ νƒ + 1−α
ωƒ is the unconditional mean of zt and ιw > [0, 1].
1.4
Fiscal and Monetary Policy
The model economy hosts two policy authorities, each of which follows exogenouslyspecified rules that receive stochastic disturbances. The fiscal authority issues bonds,
Bt , collects lump-sum taxes Tt , and buys “wasteful” public goods Gt . Its periodby-period budget constraint is
Gt + Bt−1 = Tt +
Bt
.
Rt
(18)
The left-hand side gives the government’s uses of funds, public goods spending and
the retirement of existing debt. The left-hand side gives the sources of funds, taxes
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and the proceeds of new debt issuance at the interest rate Rt . We assume that the
fiscal authority keeps its budget balanced period-by-period, so Bt = 0. Furthermore,
the fiscal authority sets public goods expenditure equal to a stochastic share of
output, expressed in consumption units.
Gt = (1 − 1~gt )Yt ,
(19)
ln gt = (1 − ρg ) ln sgƒ + ρg ln gt−1 + ηtg , ηtg N(0, σg2 ).
(20)
with
The monetary authority sets the nominal interest rate on government bonds, Rt .
For this, it employs a Taylor rule with interest-rate smoothing and forward guidance
shocks.
M
j
ln Rt = ρR ln Rt−1 + (1 − ρR ) ln Rtn + Q ξt−j
.
(21)
j =0
The monetary policy disturbances in (21) are ξt0 , ξt1−1 , . . . , ξtM−M . The public learns
the value of ξtj−j in period t − j. The conventional unforecastable shock to current
monetary policy is ξt0 , while for j C 1, these disturbances are forward guidance
shocks. We gather all monetary shocks revealed at time t into the vector ε1t . This is
normally distributed and i.i.d. across time. However, its elements may be correlated
with each other. That is,
ε1t ›ξt0 , ξt1 , . . . , ξtM ” N(0, Σ1 ).
(22)
The off-diagonal elements of Σ1 are not necessarily zero, so forward-guidance shocks
need not randomly impact expected future monetary policy at two adjacent dates
independently. Current economic circumstances influence Rt through the notional
interest rate, Rtn .
ln Rtn = ln rƒ + ln πtƒ +
1
1
φ1
φ2
Et Q (ln πt+j − ln πtƒ ) + Et Q (ln Yt+j − ln y ƒ − ln At+j ) .
4 j=−2
4 j=−2
(23)
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The constant rƒ equals the real interest rate along a steady-state growth path, and πtƒ
is the central bank’s intermediate target for inflation. We call this the inflation-drift
shock. it follows a first-order autoregression with a normally-distributed innovation.
Its unconditional mean equals πƒ , the inflation rate on a steady-state growth path.
ƒ
ln πtƒ = (1 − ρπ )πƒ + ρπ ln πt−1
+ ηtπ , ηtπ N(0, σπ2 )
(24)
Allowing πtƒ to change over time enables our model to capture the persistent
decline in inflation from the early 1990s through the early 2000s engineered by
the Greenspan FOMC.
1.5
Other Financial Markets and Equilibrium Definition
All households participate in the market for nominal risk-free government debt.
Additionally, they can buy and sell two classes of privately issued assets without
restriction. The first is one-period nominal risk-free private debt. We denote the
P
value of household’s net holdings of such debt at the beginning of period t with Bt−1
and the interest rate on such debt issued in period t maturing in t + 1 with RtP+1 .
The second asset class consists of a complete set of real state-contingent claims. As
of the end of period t, the household’s ownership of securities that pay off one unit
of the aggregate consumption good in period τ if history sτ occurs is Qt (sτ ), and
the nominal price of such a security in the same period is Jt (sτ ).
We define an equilibrium for our economy in the usual way: Households maximize
their utility given all prices, taxes, and dividends from both producers and guilds;
final goods producers and labor packers maximize profits taking their input and
output prices as given; differentiated goods producers and guilds maximize the
market value of their dividend streams taking as given all input and financial-market
prices; differentiated goods producers and guilds produce to satisfy demand at their
posted prices; and otherwise all product, labor, and financial markets clear.
2
First Order Conditions
In this section we present the first-order conditions associated with the optimization
problems that the agents in our model solve.
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2.1
Households
Given initial financial asset holdings holdings, a stock of productive capital,
investment in the previous period (which influences investment adjustment costs),
and the external habit stock; households’ choices of consumption, capital investment,
capital utilization, hours worked, and financial investments maximize utility subject
to the constraints of the capital accumulation and utilization technology and a
sequence of one-period budget constraints. To specify these budget constraints,
denote the nominal wage-per-hour paid by labor guilds to households with Wth , the
nominal rental rate for capital services with Rtk , the nominal price of investment
goods with PtI , and the dividends paid by labor guilds added to those paid by
differentiated good producers with Dt . With this notation, writing the period t
budget constraint with uses of funds on the left and sources of funds on the right
yields
Ct +
PtI It
Bt
BP
Tt Bt−1 BtP−1 Wth Ht Rtk ut Kt−1 Dt
+
+ Pt +
B
+
+
+
+
Pt
Rt Pt Rt Pt Pt
Pt
Pt
Pt
Pt
Pt
(25)
Denote the Lagrange multiplier on (25) with β t Λ1t , and that on the capital
accumulation constraint in (4) with β t Λ2t . With these definitions, the first-order
conditions for a household’s utility maximization problem are
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Λ1t = εbt ›(Ct − %C¯t−1 )(1 − εht Ht1+γh )”
−γc
(1 − εht Ht1+γh )
Wth
−γc
= (1 + γh )εtb ›(Ct − %C̄t−1 )(1 − εht Ht1+γh )” (Ct − %C̄t−1 )εht Htγh
Pt
1
Λ1
Bt
Λt
εs
− εbt+q Lœ (
) t
= βEt t+1
Rt Pt
Rt Pt Rt Pt
Pt+1
Λ1t
Λ1t+1
Λ1t
=
βE
t
Pt+1
RtP Pt
Λ2t = βE Λ1t+1
k
Rt+1
ut+1
+ Λ2t+1 (1 − δ(ut+1 ))
Pt+1
Λt1 Rtk
= Λ2t δ œ (ut )
Pt
Λ1t = εit Λ2t ‰(1 − St (�)) − Stœ (�)
it
it−1
’
+βEt εit+1 e(1−γC )zt+1 λ2t+1 Stœ+1 (�)
i2t+1
i2t
In equilibrium, C¯t = Ct always.
2.2
2.2.1
Goods Sector
Final Goods Producers
The nominal marginal cost of final goods producers equals the right-hand side of
(8). A producer of final goods maximizes profit by shutting down if Pt is less than
this marginal cost and can make an arbitrarily large profit if Pt exceeds it. When
(8) holds, an individual final goods producer’s output is indeterminate.
Final goods producers’ demand for intermediate goods takes the familiar
constant-elasticity form. If we use Yt to denote total final goods output, then the
amount of differentiated good i demanded by final goods producers is
Pit −
Yit = Yt ‰ ’
Pt
p
1+λ
t
p
λ
t
.
Given the choice of a reset price, we wish to calculate the overall price level.
All intermediate goods producers with a price-setting opportunity choose P˜t . The
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remaining producers use the price-indexing rule in (14). The aggregate price level
is given by
1
λp,t −1
Pt = (1 − ζp )P̃t
+ ζp ›(πt−1 ) (π⁄ )
ιp
1−ιp
Pt−1 ”
1
λp,t −1
λp,t −1
where P˜t is the optimal reset price
2.2.2
Intermediate Goods Producers
Intermediate goods producers’ cost minimization reads as follows:
d
e
+ Rtk Ki,t
max Wt Ht,i
e
Ht,i ,Ki,t
1−α
e α
d
”
s.t. Yt,i = εat (Kt,i
) ›Ayt Ht,i
− At Φ
We get the following optimal capital-labor ratio.
α Wt (Kite )s
=
d
1 − α Rtk
Ht,i
Notice how for each firm, the idiosyncratic capital to labor ratio is not a function of
any firm-specific component. Hence, each firm has the same capital to labor ratio.
In equilibrium,
Kte = ut Kt−1
To find the marginal cost, we differentiate the variable part of production with
respect to output, and substitute in the capital-labor ratio.
−(1−α)
M Ct,i = (εat )−1 (Ayt )
Wt1−α Rtkα α−α (1 − α)−(1−α)
Again, notice that each firm as the same marginal cost.
Given cost minimization, a differentiated goods producer with an opportunity to
adjust its nominal price does so to maximize the present-discounted value of profits
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earned until the next opportunity to adjust prices arrives. Formally,
ª
max Et Q ζps
P̃t,i
s=0
β s Λ1t+s Pt
y
P̃t,i Xt,s
− M Ct+s Yt+s,i
1
Λt Pt+s
λp,t
˜ 1−λp,t
y Pt,i
s.t. Yt (i) = „Xt,s
‚
Yt
Pt
¢
¨
s=0
¨ 1
y
where Xt,s = ¦ s ιp
1−ιp
¨
s = 1, . . . , ª
¨
¤ Ll=1 πt+l−1 π⁄
£
¨
¨
§
¨
¨
¥
This problem leads to the following price-setting equation for firms that are allowed
to reoptimize their price:
0=
<
s 1
@
s β Λt+s Pt
@
E t Q ζp 1
Yit+s @λp,t+s M Ct+s
Λt Pt+s
@
s=0
ª
>
=
A
A
− Xt,s P̃it A
A
?
It can be shown that the producers that are allowed to reoptimize choose the
same price. So henceforth, P̃it = P̃t .
2.2.3
Investment Goods Producers
Characterizing the profit-maximizing choices of investment goods and final goods
producers is straightforward. If PtI > Pt ~AIt , then each investment goods producer
can make infinite profit by choosing an arbitrarily large output. On the other
hand, if PtI < Pt ~AIt , then investment goods producers maximize profits with zero
production. Finally, their profit-maximizing production is indeterminate when
PtI = Pt ~AIt .
(26)
−1
The relative price of investment to consumption is equal to (AIt ) . Making
this substitution into the household f.o.c and noting that Pt YtI is an intermediate
input that will not be reflected in the aggregate resource constraint, it suffices to
−1
substitute the relative price (AIt ) in the constraint for the household.
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2.3
Labor Sector
2.3.1
Labor Packers
The labor packers choose the the labor inputs supplied by guilds, pack them into a
composite labor service to be sold to the intermediate goods producers. Formally,
labor packers’ problem reads as follows:
max
Wt Hts − S
s
Ht ,Hit
1
Wit Hit di
0
s.t. S
1
1
1+λw,t
Hit
0
1+λw,t
= Hts
di
We obtain the following labor demand equation for guild i:
Wit −
Hit = ‰
’
Wt
1+λw,t
λw,t
Ht
(27)
As for the goods sector, we can show that aggregate wage is given by the following
equation:
−λ1
Wt = (1 − ζw )W̃t
w,t
+ ζw ›(ezt−1 πt−1 ) w (π⁄ ez⁄ )
ι
1−ιw
−λw,t
−λ1
Wt−1 ”
w,t
˜ is the optimal reset wage for guilds.
where W
2.3.2
Guilds
Each guild with an opportunity to set its nominal price does so to maximize the
current value of the stream of dividends returned to the household. Formally, their
problem reads
ª
max Et Q ζws „
W̃it
s=0
s.t. Hit+s =
β s Λ1t+s Pt
˜ it − Wth+s Hit+s
‚ Xtl+s W
Λt1 Pt+s
l ˜
Wit f
™ Xt,s
−
1+λw,t+s
λw,t+s
Ht+s
f Wt+s Ł
¢
¨
s=0
¨ 1
l
where Xt,s = ¦ s
1−ι
w
A
¨
(πeγ )ιw s = 1, . . . , ª
‘
¨ Lj=1 −πt+j−1 At+j−1
¤
t+j−2
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£
¨
¨
§
¨
¨
¥
Authorized for public release by the FOMC Secretariat on 1/13/2023
W̃t is the optimal reset wage. This optimal wage is chosen by the guilds who are
allowed, with probability ζw , to change their prices in a given period. Also, we index
the nominal wage inflation rate with ιw .
This maximization problem gives a wage-setting equation that reads as follows:
ª
0 = Et Q ζws
s=0
1
β s Λt+s
Pt
1
h
l ˜
›(1 + λw,t+s )Wt+s
Hit+s
− Xt,s
Wit ”
1
Λt Pt+s
λw,t+s
It can be shown that the guilds that are allowed to reoptimize choose the same wage.
So henceforth, W̃it = W̃t .
3
Detrending
To remove nominal and real trends, we deflate nominal variables by their matching
price deflators, and we detrend any resulting real variables influenced permanently
by technological change. All scaled versions of variables are the lower-case
counterparts.
ct =
kt =
wt =
p̃t =
yt =
rtk =
Ct
At
Kt
At AIt
Wt
A t Pt
P̃t
Pt
Yt
At
Rtk AtI
Pt
it =
kte =
w̃t =
πt =
mct =
wth =
λ1t = Λ1t Aγt C
It
At AIt
Kte
At AIt
˜t
W
At P t
Pt
Pt−1
M Ct
Pt
Wth
A t Pt
λt2 = Λt2 AtγC AIt
εts = Aγt C εst
3.1
Detrended Equations
The detrended equations describing our model are listed in the following sections.
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Households’ FOC
−γc
ct−1
h 1+γh
›1
”
›1 − εth ht1+γh ”
’
−
ε
h
t t
ezt
−γc
ct−1
ct−1
(1+σ )
λ1t wth = (1 + γh )εbt ‰ct − % zt ’ −1 − εht ht h ‘ ‰ct − % zt ’ εht hγt h
e
e
1
1
−γ
z
t+1
C
λ e
λt
= βEt t+1
P
πt+1
Rt
1
λ1
εb εs
λt
− Lœ (0) t t = βEt t+1 e−zt+1 γC
Rt
Rt
πt+1
i2
it
œ
’ + βEt εit+1 e(1−γC )zt+1 λ2t+1 St+1
(�) t+1
λ1t = εit λt2 ‰(1 − St (�)) − Stœ (�)
it−1
i2t
λ1t = εtb ‰ct − %
k
λ2t = βEt e−γC zt+1 −ωt+1 ›λ1t+1 rt+1
ut+1 + λ2t+1 (1 − δ(ut+1 ))”
λ1t rtk = λt2 δ œ (ut )
kt = (1 − δ(ut )) kt−1 e−zt −ωt + εit (1 − S(�)) it
kte = ut kt−1 e−zt −ωt
Final Goods Price Index
1
1−λp,t
1 = (1 − ζp )p̃t
1
1−λp,t
ιp
+ ζp (πt−1
π ⁄(1−ιp ) πt−1 ) 1−λp,t
Intermediate Goods Firms: Capital-Labor Ratio
kte
α wt
=
d
ht 1 − α rtk
Intermediate Goods Firms: Real Marginal Costs
α
mct =
wt1−α (rtk )
εat αα (1 − α)1−α
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Intermediate Goods Firms: Price-Setting Equation
ª
0 =Et Q ζps β s λ1t+s
s=0
ỹt,t+s
At+s 1−γC
p
λp,t+s mct+s − X̃t,s
‰
p̃t
’
λp,t+s − 1 At
where
p
X̃t,s
¢̈
¨
¨ 1
1−ιp
ι
= ¦ Lsj=1 πt+j−1
πp
¨
¨ Ls πt+j ⁄
j=1
¤̈
£
¨
¨
¨
§
¨
s = 1, . . . , ª ¨
¥̈
s=0
ỹt,t+s denotes the time t + j output sold by the producers that have optimized at
time t the last time they have reoptimized. Since it can be shown that optimizing
producers all choose the same price, then we do not have to carry the i-subscript.
Labor Packers: Aggregate Wage Index
−λ1
wt = (1 − ζw )w̃t
w,t
+ ζw ›e
ιw zt−1 −zt (1−ιw )z⁄
e
− 1
πtι−1 πt−1 π⁄1−ιw wt−1 ” λw,t
−λw,t
Guilds: Wage-Setting Equation
ª
0 =Et Q ζws βλ1t+s ‰
s=0
At+s 1−γC h̃t,t+s
h
l
›(1 + λw,t+s )wt+s
’
− X̃t,s
w̃t ”
At
λw,t+s
where
l
X̃t,s
¢̈
¨
¨ 1
= ¦ Lsj=1 ›πt+j−1 ezt+j−1 ”1−ιw (πγ)ιw
¨
z
¨
Lsj=1 πt+j e t+j
¤̈
£
¨
¨
¨
§
s = 1, . . . , ª ¨
¨
¥̈
s=0
h̃t,t+s denotes the time t + j labor supplied by the guild that have optimized at time
t the last time they have reoptimized. Since it can be shown that optimizing guilds
all choose the same wage, then we do not have to carry the i-subscript.
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Monetary Authority
ψ1
ψ2 =1−ρR
<
@
1
1
M
4
4 A
π
y
t+j
t+j
A
ρR @
@r⁄ πt⁄ „ M ⁄ ‚ „ M ⁄ ‚ A
ξt−j,j
Rt = Rt−1
M
@
A
j=−2 y
j=−2 πt
j=0
@
A
>
?
The Aggregate Resource Constraint
yt
=ct + it
gt
Production Function
yt =εat (kte ) (hdt )1−α − Φ
α
Labor Market Clearing Condition
ht = htd
4
Steady State
We normalize most shocks and the utilization rate:
uƒ =1
εi =1
εa =1
εb =1
Next, we set the following restriction on adjustment costs:
S(�⁄ ) 0
S œ (�⁄ ) 0
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4.1
Prices and Interest Rates
Given β, z⁄ , γC , and π⁄ , we can solve for the steady-state nominal interest rate on
private bonds R⁄P by using the FOC on private bonds:
R⁄P =
π⁄
(βe−γC z⁄ )
(28)
From the definition of δ(u), we have
δ(1) =δ0
δ œ (1) =δ1 .
Next, given ω⁄ , δ0 , and the above, we can solve for the real return on capital r⁄k
using the FOC on capital:
r⁄k =
4.2
eγC z⁄ +ω⁄
− (1 − δ0 )
β
(29)
Ratios
Moving to the production side, we can use the aggregate price equation to solve for
p̃⁄ :
p̃⁄ = 1
Using this result and given λp,⁄ , we can use the price Phillips curve to solve for mc⁄ :
mc⁄ =
1
1 + λp,⁄
(30)
Given values for α and εa⁄ , we can use the marginal cost equation to solve for
w⁄ :
1
w⁄ = ›mc⁄ αα (1 − α)1−α (r⁄k )−α ” 1−α
(31)
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The definition of effective capital gives us a value for k⁄e in terms of k⁄ :
k⁄e = k⁄ e−z⁄ −ω⁄
Calculating y⁄ using the labor share of output 1 − α:
y⁄ =
w⁄ h⁄
1−α
Using capital shares based off our value of α, we can calculate the output to
capital ratio as follows:
y⁄ r⁄k
=
k⁄e α
y⁄ −z⁄ −ω⁄ r⁄k
=e
α
k⁄
Using the capital accumulation equation, we can get a value for
i⁄
k⁄
i⁄
= 1 − (1 − δ0 )e−z⁄ −ω⁄
k⁄
Using the resource constraint, we can get
c⁄
k⁄ :
c⁄
i⁄
y⁄
=
g −
k⁄ k⁄ sƒ k⁄
These ratios will give us the remaining steady-state levels and ratios:
k⁄ =y⁄ ‰
c⁄ =
4.3
y⁄ −1
’
k⁄
i⁄ =
c⁄
k⁄
k⁄
i⁄
k⁄
k⁄
g⁄ =gy y⁄
Liquidity Premium
Using the aggregate wage equation, we can get the following for w̃⁄ :
w̃⁄ = w⁄
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Combining this result with the wage Phillips curve, we get the following:
w⁄h =
w⁄
1 + λw,⁄
We can use the FOC for consumption and the labor supply to pin down εh and
λ1⁄
% −γc
(1+γ )
ε c⁄ ‰1 − z ’ −1 − εh h⁄ h ‘ − λ1⁄ = 0
e
b
(1−γ )
−(1 + γh )εb c⁄ c ‰1 −
−γc
% (1−γc )
h (1+γh )
’
−1
−
ε
h
‘
εh hγ⁄h + λ1⁄ w⁄h = 0
⁄
z
ε
Finally, the government bond rate is calculated from
λ1⁄ − εb⁄ εs⁄ = βR⁄
π⁄
π⁄
− εb⁄ εs⁄ −γ z 1 = R⁄
−γ
z
C
βe
βe C λ⁄
´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶
λ1⁄ −γC z
e
π⁄
R⁄P
Noting that R⁄P =
π⁄
βe−γC z
we can write
R⁄P − R⁄ εb⁄ εs⁄
= 1 .
R⁄P
λ⁄
This is the liquidity premium in steady state.
5
Log Linearization
Hatted variables refer to log deviations from steady-state (x̂ = ln › xx⁄t ”):
ln εjt = ρj ln εjt−1 + ηtj
In the cases of zt , ωt , and νt , we have that x̂ = xt − x⁄ as these variables are already
in logs.
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Households’ First Order Conditions
%
1
ez
(ĉt−1 − ẑt )
% ĉt + γc
1 − ez
1 − e%z
%
1 − γc
ez
1
b
h
h
λ̂t + ŵt − ε̂t − ε̂t −
ĉt + (1 − γc )
(ĉt−1 − ẑt )
1 − e%z
1 − e%z
εˆbt − λ̂t1 − γc
h
εh h1+γ
⁄
‚ ĥt = 0
h 2
)
(1 − εh h1+γ
⁄
R⁄
RP − R⁄
λ̂1t = ⁄ P (ε̂st + ε̂bt ) + P (R̂t + Et [(λ̂1t+1 − π̂t+1 − γC ẑt+1 ])
R⁄
R⁄
(32)
(33)
− „γh + γc (1 + γh )
λ̂1t = Et λ̂1t+1 − γC ẑt+1 + R̂t − π̂t+1
ˆ 1t = ›ln εti + λ̂t2 ” − S œœ (ˆıt − ı̂t−1 ) + βe(1−γC )γ S œœ Et (ı̂t+1 − ı̂t )
λ
1
k
ˆ t+1
ˆ 2t = βe−γC z⁄ −ω⁄ λ1⁄ u⁄ r⁄k Et ›−γC ẑt+1 − ω̂t+1 + λ
λ⁄2 λ
+ r̂t+1
+ ût+1 ” +
+ βe−γC z⁄ −ω⁄ (1 − δ0 )λ2⁄ Et ›−γC ẑt+1 − ω̂t+1 + λ̂2t+1 ” − λ2⁄ δ1 u⁄ Et ût+1
δ2
λ̂1t = λ̂2t + u⁄ ût − r̂tk
δ1
εi⁄ i⁄
εi i⁄
k̂t = ‰1 −
’ ›k̂t−1 − ẑt − ω̂t ” + ⁄ ›ε̂it + ı̂t ” − δ1 u⁄ e−z⁄ −ω⁄ ût
k⁄
k⁄
k̂te = ût + k̂t−1 − ẑt − ω̂t
(34)
(35)
(36)
(37)
(38)
(39)
(40)
Capital-Labor Ratio
k̂te = ŵt − r̂tk + ĥtd
(41)
Real Marginal Costs
à t = (1 − α) ŵt + αr̂tk − εˆta
mc
(42)
The New Keynesian Phillips Curve for Inflation
(1 − βζp e(1−γC )z⁄ )(1 − ζp )
λp,⁄ ˆ
Ãt +
λp,t + mc
(1−γ
)z
⁄
C
1 + λp,⁄
(1 + βιp e
)ζp
ιp
βe(1−γC )z⁄
+
π̂
+
Et π̂t+1
t−1
1 + βιp e(1−γC )z⁄
1 + βιp e(1−γC )z⁄
π̂t =
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Wage Mark-Up
h
µ̂w
t = ŵt − ŵt
(44)
The New Keynesian Phillips Curve for Wages
ŵt =
1
1 + βe(1−γC
ŵ +
)z⁄ t−1
βe(1−γC )z⁄
βe(1−γC )z⁄
ŵ
+
(Et π̂t+1 + Et ẑt+1 )+
t+1
1 + βe(1−γC )z⁄
1 + βe(1−γC )z⁄
(45)
1 + ιw βe(1−γC )z⁄
ιw
(π̂
+
ẑ
)
−
(π̂t + ẑt )+
t−1
t−1
1 + βe(1−γC )z⁄
1 + βe(1−γC )z⁄
λw,⁄ ˆ
1 − βζw e(1−γC )z⁄ 1 − ζw
λw,t − µ̂w
t
(1−γ
)z
ζw
1 + λw,⁄
1 + βe C ⁄
The Aggregate Resource Constraint
y⁄
c⁄
i⁄
(ŷt − ĝt ) =
ĉt +
ı̂t
c⁄ + i⁄
c⁄ + i⁄
g⁄
(46)
The Production Function
yˆt =
1
›ln εat + αk̂te + (1 − α) ĥtd ”
mc⁄
(47)
Labor Market Clearing Condition
ĥt = ĥdt
(48)
Monetary Authority’s Reaction Function
1
1
M
ˆ t−1 + (1 − ρR ) (1 − ψ1 ) π̂t⁄ + ψ1 „ Q π̂t+j ‚ + ψ2 „ Q ŷt+j ‚ + Q ξˆt−j,j
R̂t = ρR R
4 j=−2
4 j=−2
j=0
(49)
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6
Measurement
6.1
National Income Accounts
The model economy’s basic structure, with the representative household consuming
a single good and accumulating capital using a different good, differs in some
important ways from the accounting conventions of the Bureau of Economic Analysis
(BEA) underlying the National Income and Product Accounts (NIPA). In particular
1. The BEA treats household purchases of long-lived goods inconsistently. If
classifies purchases of residential structures as investment and treats the service
flow from their stock as part of Personal Consumption Expenditures (PCE) on
services. The BEA classifies households purchases of all other durable goods
as consumption expenditures. No service flow from the stock of household
durables enters measures of current consumption. In the model, all long-lived
investments add to the productive capital stock.
2. The BEA treats all government purchases as government consumption.
However, government at all levels makes purchases of investment goods on
behalf of the populace. In the model, these should be treated as additions to
the single stock of productive capital.
3. The BEA sums PCE and private expenditures on productive capital (Business
Fixed Investment and Residential Investment), with government spending,
inventory investment, and net exports to create Gross Domestic Product. The
model features only the first three of these.
To bridge these differences, we create four model consistent NIPA measures from
the BEA NIPA data.
1. Model-consistent GDP. Since the model’s capital stock includes both the stock
of household durable goods and the stock of government-purchased capital, a
model-consistent GDP series should include the value of both stocks’ service
flows. To construct these, we followed a five-step procedure.
(a) We begin by estimating a constant (by assumption) service-flow rate by
dividing the nominal value of housing services from NIPA Table 2.4.5
by the beginning-of-year value of the residential housing stock from the
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BEA’s Fixed Asset Table 1.1. We use annual data and average from
1947 through 2014. The resulting estimate is 0.096. That is, the annual
value of housing services equals approximately 10 percent of the housing
stock’s value each year.
(b) In the second step, we estimate estimate constant (by assumption) depreciation rates for residential structures, durable goods, and government
capital. We constructed these by first dividing observations of value lost
to depreciation over a calendar year by the end-of-year stocks. Both
variables were taken from the BEA’s Fixed Asset Tables. (Table 1.1 for
the stocks and Table 1.3 for the deprecation values.) We then averaged
these ratios from 1947 through 2014. The resulting estimates are 0.021,
0.194, and 0.044 for the three durable stocks.
(c) In the third step, we calculated the average rates of real price depreciation
for the three stocks. For this, we began with the nominal values and
implicit deflators for PCE Nondurable Goods and PCE Services from
NIPA Table 1.2. We used these series and the Fisher-ideal formula to
produce a chain-weighted implicit deflator for PCE Nondurable Goods
and Services. Then, we calculated the price for each of the three
durable good’s stocks in consumption units as the ratio of the implicit
deflator taken from Fixed Asset Table 1.2 to this deflator. Finally, we
calculated average growth rates for these series from 1947 through 2014.
The resulting estimates equal 0.0029, −0.0223, and 0.0146 for residential
housing, household durable goods, and government-purchased capital.
(d) The fourth combines the previous steps’ calculations to estimate constant
(by assumption) service-flow rates for household durable goods and
government-purchased capital. To implement this, we assumed that all
three stocks yield the same financial return along a steady-state growth
path. These returns sum the per-unit service flow with the appropriately
depreciated value of the initial investment. This delivers two equations
in two unknowns, the two unknown service-flow rates. The resulting
estimates are 0.29 and 0.12 for household durable goods and governmentpurchased capital.
(e) The fifth and final step uses the annual service-flow rates to calculate real
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and nominal service flows from the real and nominal stocks of durable
goods and government-purchased capital reported in Fixed Asset Table
1.1. This delivers an annual series. Since the stocks are measured as of
the end of the calendar year, we interpret these as the service flow values
in the next year’s first quarter. We create quarterly data by linearly
interpollating between these values.
With these real and nominal service flow series in hand, we create nominal
model-consistent GDP by summing the BEA’s definition of nominal GDP
with the nominal values of the two service flows. We create the analogous
series for model-consistent real GDP by applying the Fisher ideal formula to
the nominal values and price indices for these three components.
2. Model-consistent Investment. The nominal version of this series sums nominal
Business Fixed Investment, Residential Investment, PCE Durable Goods, and
government investment expenditures. The first three of these come from NIPA
Table 1.1.5, while government investment expenditures sums Federal Defense,
Federal Nondefense, and State and Local expenditures from NIPA Table 1.5.5.
We construct the analogous series for real Model-consistent Investment by
combining these series with their real chain-weighted counterparts found in
NIPA Tables 1.1.3 and 1.5.3 using the Fisher ideal formula. By construction,
this produces an implicit deflator for Model-consistent investment as well.
3. Model-consistent Consumption. The nominal version of this series sums
nominal PCE Nondurable Goods, PCE Services, and the series for nominal
services from the durable goods stock. The first two of thse come from
NIPA Table 1.1.5. We construct the analogous series for real Modelconsistent consumption by combining these series with their real chainweighted counterparts using the Fisher ideal formula. The two real PCE series
come from NIPA Table 1.1.3. Again, this produces an implicit deflator for
Model-consistent consumption as a by-product.
4. Model-consistent Government Purchases. Conceptually, the model’s measure
of Government Purchases includes all expenditures not otherwise classified as
Investment or Consumption: Inventory Investment, Net Exports, and actual
Government Purchases. We construct the nominal version of this series simply
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by subtracting nominal Model-consistent Investment and Consumption from
nominal Model-consistent GDP. We calculate the analogous real series using
“chain subtraction.” This applies the Fisher ideal formula to Model-consistent
GDP and the negatives of Model-consistent Consumption and Investment.
Our empirical analysis requires us to compare model-consistent series measured
from the NIPA data with their counterparts from the model’s solution. To do this,
we begin by solving the log-linearized system above, and then we feed the model
specific paths for all exogenous shocks starting from a particular initial condition.
for a given such simulation, the growth rates of Model-consistent Consumption and
Investment equal
Δ ln Ct = z⁄ + Δĉt + zt and
Δ ln It = z⁄ + ω⁄ + Δît + zt + ωt
The measurement of GDP growth in the model is substantially more complicated,
because the variables Yt and yt denote model output in consumption units. In
contrast, we mimic the BEA by using a chain-weighted Fisher ideal index to measure
model-consistent GDP. Therefore, we construct an analogus chain-weighted GDP
index from model data. Since such an ideal index is invariant to the units with
which nominal prices are measured, we can normalize the price of consumption to
equal one and employ the prices of investment goods and government purchases
relative to current consumption. Our model identifies the first of these relative
prices as with investment-specific technology. However, the model characterizes
only government purchases in consumption units, because private agents do not
care about their division into “real” purchases and their relative price. For this
reason, we use a simple autoregression to characterize the evolution of the price of
government services in consumption units. Denote this price in quarter t with Ptg .
We construct this for the US economy by dividing the Fisher-ideal price index for
model-consistent government purchases by that for model-consistent consumption.
Then, our model for its evolution is
ln(Ptg ~Ptt−1 ) = µg + θgg1 ln(Ptg−1 ~Ptg−2 ) + θgg2 ln(Ptg−2 ~Ptg−3 ) + εgg
t .
(50)
g
2
Here, εgg
t N(0, σgg ). Given an arbitrary normalization of Pt to one for some time
period, simulations from (??) can be used to construct simulated values of Ptg for
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all other time periods. With these and a simulation from the model of all other
variables in hand, we can calculate the simulation’s values for Fisher ideal GDP
growth using
Qt
Qt−1
¼
Q̇Pt Q̇Lt ,
(51)
where the Paasche and Laspeyres indices of quantity growth are
Ct + PtI It + PtG (Gt ~PtG )
and
Ct−1 + PtI It−1 + PtG (Gt−1 ~Ptg−1 )
Ct + PtI−1 It + PtG−1 (Gt ~PtG )
.
Ct−1 + PtI−1 It−1 + PtG−1 (Gt−1 ~PtG−1 )
Q̇Pt
(52)
Q̇Lt
(53)
In both (52) and (53), PtI is the relative price of investment to consumption. In
equilibrium, this always equals AIt .
The above gives a complete recipe for simulating the growth of model-consistent
real GDP growth. However, we also embody its insights into our estimation with a
log-linear approximation. For this, we start by removing stochastic trends from all
variables in (52) and (53), and we proceed by taking a log-linear approximation of
the resulting expression. Details are available from the authors upon request.
6.2
Hours Worked Measurement
Empirical work using DSGE models like our own typically measure labor input with
hours worked per capita, constructed directly from BLS measures of hours worked
and the civilian non-institutional population over age 16. However, this measure
corresponds poorly with business cycle models because it contains underlying low
frequency variation. This fact led us to construct a new measure of hours for the
model using labor market trends produced for the FRB/US model and for the
Chicago Fed’s in-house labor market analysis.
We begin with a multiplicative decomposition of hours worked per capita into
hours per worker, the employment rate of those in the labor force, and the laborforce participation rate. The BLS provides CPS-based measures of the last two rates
for the US as a whole. However, its measure of hours per worker comes from the
Establishment Survey and covers only the private business sector. If we use hours
per worker in the business sector to approximate hours per worker in the economy
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as a whole, then we can measure hours per capita as
Ht HtE EtC LC
t
= -----.
C
Pt EtE LC
P
t
t
Here, Ht and Pt equal total hours worked and the total population, HtE ~EtE equals
hours per worker measured with the Establishment survey, EtC ~LC
t equals one
C
C
minus the CPS based unemployment rate, and Lt ~Pt equals the CPS based laborforce participation rate. Our measure of model-relevant hours worked deflates each
component on the right-hand side by an exogenously measured trend. The trend for
the unemployment rate comes from the Chicago Fed’s Microeconomics team, while
those for hours per worker and labor-force participation come from the FRB/US
model files.
6.3
Inflation
Our empirical analysis compares model predictions of price inflation, wage inflation,
inflation in the price of investment goods relative to consumption goods, and
inflation expectations with their observed values from the U.S. economy. We
describe our implementations of these comparisons sequentially below.
6.3.1
Price Inflation
Our model directly characterizes the inflation rate for Model-consistent Consumption. In principle, this is close to the FOMC’s preferred inflation rate, that for
the implicit deflator of PCE. However, in practice the match between the two
inflation rates is poor. In the data, short-run movements in food and energy prices
substantially influences the short-run evolution of PCE inflation. Our model lacks
such a volatile sector, so if we ask it to match observed short-run inflation dynamics,
it will attribute those to transitory shocks to intermediate goods’ producers’ desired
markups driven by λpt .
To avoid this outcome, we adopt a different strategy for matching model and data
inflation rates, which follows that of Justiniano, Primiceri, and Tambalotti (2013).
This relates three observable inflation rates – core CPI inflation, core PCE inflation,
and market-based PCE inflation – to Model-consistent consumption inflation using
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auxiliary observation equations. For core PCE inflation, this equation is
πtp1 = π⁄ + π⁄p1 + β1p1 π̂t + β2p1 πtD + εtp1
(54)
In (54) as elsewhere, π⁄ equals the long-run inflation rate. The constant π⁄1 is
an adjustment to this long-run inflation rate which accounts for possible long-run
differences between realized inflation and the FOMC’s goal of πƒ . The right-hand
side’s inflation rates, π̂t and πtD equal Model-consistent consumption inflation and
PCE Durables inflation. We refer to the coefficients multiplying them, β1p1 and
β2p1 , as the inflation loadings. We include PCE Durables inflation on the righthand side of (54) because the principle adjustment required to transform Modelconsistent inflation into core PCE inflation is the replacement of the price index
for durable goods services with that for durable goods purchases. The disturbance
term εpt1 follows a first-order autoregression with autocorrelation ϕp1 and normally
distributed innovations with mean zero and standard deviation σp1 .
The other two observed inflation measures, market-based PCE inflation and core
CPI inflation, have identically specified observation equations. We use p2 and p3 in
superscripts to denote these equations parameters and error terms, and we use the
same expressions as subscripts to denote the parameters governing the evolution of
p2
p3
their error terms. We assume that the error terms εp1
t , εt , and εt are independent
of each other at all leads and lags.
To produce forecasts of inflation with these these three observation equations, we
must forecast their right-hand side variables. The model itself gives forecasts of π̂t .
The forecasts of durable goods inflation come from a second-order autoregression.
πtD = θ0D + θ1D πtD−1 + εD
t
(55)
Its innovation is normally distributed and serially uncorrelated with standard
deviation σD .
6.3.2
Wage Inflation
Although observed wage inflation does not feature the same short-run variability
as does price inflation, it does include the influences of persistent demographic
labor-market trends which we removed ex ante from our measure of hours worked.
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Therefore, we follow the same general strategy of relating observed measures of wage
inflation to the model’s predicted wage inflation with a error-augmented observation
equation. For this, we employ two measures of compensation per hour, Earnings per
Hour and Total Compensation per Hour. In parallel with our notation for inflation
measures, we use w1 and w2 to denote these two wage measures of wage inflation.
The observation equation for Earnings per Hour is
1
πtw1 = z⁄ + π⁄w1 + β w1 π̂tw + εw
t
(56)
Just as with the price inflation measurement errors, εw1
follows a first-order
t
autoregression with autocorrelation ϕw1 and innovation standard deviation σw1 . The
observation equation for Total Compensation per Hour is analogous to (56).
6.3.3
Relative Price Inflation
To empirically ground investment-specific technological change in the model, we use
an error-augmented observation equation to relate the relative price of investment
to consumption, both model-consistent measures constructed from NIPA and Fixed
Asset tables as described above, with the model’s growth rate of the rate of
technological transformation between these two goods, ωt .
C~I
πt
= ωt + εt
C~I
Here, we use the superscript C~I to indicate that the variables characterize the price
C~I
of Consumption relative to Investment. The measurement error εt follows a firstorder autoregression with autocorrelation ϕC~I and normally-distributed innovations
with standard deviation σC~I .
6.3.4
Inflation Expectations
We also discipline our model’s inferences about the state of the economy by
comparing expectations of one-yea and 10-year inflation from the Survey of
Professional Forecasters with the analogous expectations from our model. Just
as with all of the other inflation measures, we allow these two sets of expectations
to differ from each other by including serially correlated measurement errors. The
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observation equations are
πte4
=
π⁄ + π⁄e4
1 4
+ Q Et [π̂t+i ] + εe4
t
4 i=1
πte40 = π⁄ + π⁄e40 +
1 40
Q Et [π̂t+i ] + εte40
40 i=1
The two measurement errors follow mutually-independent first-order autoregressions
with autocorrelations ϕe4 and ϕe40 and innovation standard deviations σe4 and σe40 .
6.4
Interest Rates and Monetary Policy Shocks
Since our model features forward guidance shocks, it has non-trivial implications
for the current policy rate as well as for expected future policy rates. We use two
distinct but complementary approaches to disciplining the parameters governing
their realizations, the elements of Σ1 , using data. The first method compares the
model’s monetary policy shocks to high-frequency interest-rate innovations informed
by event studies, such as that of Gürkaynak, Sack, and Swanson (2005). Those
authors applied a factor structure to innovations in implied expected interest rates
from futures prices around FOMC policy announcement dates. Specifically, they
show that the vector of M implied interest rate changes following an FOMC policy
announcement, Δr, can be written as
Δr = Λf + η
Where f is a 2 × 1 vector of factors, Λ is a M × 2 matrix of factor loadings, and
η is an M × 1 vector of mutually independent shocks. Denoting the 2 × 2 diagonal
variance covariance matrix of f with Σf and the M ×M diagonal variance-covariance
matrix of η with Ψ, we can express the observed variance-covariance matrix of Δr
as ΛΣf Λœ + Ψ.
Our model has implications for this same variance covariance matrix. For this,
use the model’s solution to express the changes in current and future expected
interest rates following monetary policy shocks as Δr = Γ1 ε1 . Here, ε1 is the vector
which collects the current monetary policy shock with M − 1 forward guidance
shocks, and Γ1 is an M × M matrix. In general, Γ1 does not simply equal the
identity matrix, because current and future inflation and output gaps respond to the
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monetary policy shocks and thereby influence future monetary policy “indirectly”
through the interest rate rule. Given this solution for Δr, we can calculate its
variance-covariance matrix as Γ1 Σ1 Γœ1 . Equating these two expressions and solving
for Σ1 yields
œ
œ−1
Σ1 = Γ−1
1 (ΛΣf Λ + Ψ) Γ1 .
The second approach to disciplining Σ1 is more traditional: directly compare
quarterly observations of the current policy rate and expected future interest rates
– from market prices, surveys of market participants, or both – with their implied
values from the model given a particular realization of the vector of monetary policy
shocks. We use both methods in the estimation procedure described below.
7
Calibration and Bayesian Estimation
As we noted in the introduction, we follow a two-stage approach to the estimation
of our model’s parameters. In a calibration stage, we set the values of selected
parameters so that the model has empirically-sensible implications for long-run
averages from the U.S. economy. In this stage, we also enforce several normalizations
and a judgemental restriction on one of the measurement error variances. In
the second stage, we estimate the model’s remaining parameters using standard
Bayesian methods.
We employ standard prior distributions, but those governing monetary policy
shocks deserve further elaboration. Our estimation requires the variance-covariance
matrix of monetary policy shocks to be consistent with the factor-structure of
interest rate innovations used by Gürkaynak, Sack, and Swanson (2005), as described
above. Therefore, we parameterize Σ1 in terms of Λ, Σf , Ψ, and the model
parameters which influence Γ1 . We then center our priors for Λ, Σf , and Ψ at their
estimates from event-studies. However, we do not require our estimates to equal
their prior values. Our Bayesian estimation procedure employs quarterly data on
expected future interest rates, the posterior likelihood function includes Λ, Σf , and
Ψ as free parameters. It is well known that Λ and Σf are not separately identified,
so we impose two scale normalizations and one rotation normalization on Λ. The
rotation normalization requires that the first factor, which we label “Factor A”, is
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the only factor influence the current policy rate. That is, the second factor, “Factor
B” influences only future policy rates. Gürkaynak, Sack, and Swanson (2005) call
Factors A and B the “target” and “path” factors.
Our estimation’s sample period begins in the first quarter of 1993 and ends in the
fourth quarter of 2016. Of course, the FOMC substantially changed its operating
procedures in the aftermath of its persistent stay at the Zero Lower Bound, so it
would be unwise to imagine the data from this entire period being generated from
our model with time-invariant parameters. For this reason, we estimate the model
twice. For the first sample, which runs from 1993Q1 through 2008Q3, we estimate
all model parameters while allowing for four quarters of forward guidance. For
the second sample, we estimate the parameters governing monetary policy shocks
allowing for ten quarters of forward guidance, adjust the average rate of Hicksneutral productivity growth to bring potential GDP growth rate from its first-sample
value of 3 percent down to 2 percent, and hold all other model parameters fixed at
their first-sample posterior-mode values.
We report the results of our two-stage two-sample estimation in a series of tables.
Table 1 reports our most notable calibration targets. The long-run policy rate equals
1.1 percent on a quarterly basis. We target a two percent growth rate of per capita
GDP. Given an average population growth rate of one percent per year, this implies
that our potential GDP growth rate equals three percent. The other empirical
moments we target are a nominal investment to output ratio of 26 percent and
nominal government purchases to output ratio of 15 percent. Finally, we target a
capital to output ratio of approximately 10 on a quarterly basis.
Table 2 lists the parameters which we calibrate along with their given values.
The table includes many more parameters than there are targets in Table 1. This is
because Table 1 omitted calibration targets which map one-to-one with particular
parameter values. For example, we calibrate the steady-state capital depreciation
rate (δ0 ) using standard methods applied to data from the Fixed Asset tables.
It is also because Table 2 lists several parameters which are normalized prior to
estimation. Most notable among these are the three factor loadings listed at the
table’s bottom.
Tables 3 and 4 report prior distributions and posterior modes for the model’s
remaining paramters, for the first and second samples respectively.
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Table 1: Calibration Targets
Description
Fixed Interest Rate (quarterly, gross)
Per-Capita Steady-State Output Growth Rate (quarterly)
Investment to Output Ratio
Capital to Output Ratio
Fraction of final good output spent on public goods
Expression Value
R
1.011
Yt+1 ~Yt
1.005
It ~Yt
0.260
Kt ~Yt
10.763
Gt ~Yt
0.153
Table 2: First Sample Calibrated Parameters
Parameter
Discount Factor
Steady-State Measured TFP Growth (quarterly)
Investment-Specific Technology Growth Rate
Elasticity of Output w.r.t Capital Services
Steady-State Wage Markup
Steady-State Price Markup
Steady-State Scale of the Economy
Steady-State Inflation Rate (quarterly)
Steady-State Depreciation Rate
Steady-State Marginal Depreciation Cost
Nominal Output over Nominal Private Purchases
Std. Dev Long-Run Inflation Expectations Measurement Error
Long-Run Inflation Expectations (Constant CPI Adjustment)
Average Earnings Constant
Average Total Compensation Constant
Loading Compensation
Loading Core PCE
Constant for Relative Price Inflation
Loading 0 Factor A
Loading 0 Factor B
Loading 4 Factor B
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Symbol Value
β
0.986
z⁄
0.489
ω⁄
0.371
α
0.401
λw
1.500
⁄
p
λ⁄
1.500
H⁄
1.000
π⁄
0.500
δ0
0.016
δ1
0.039
g⁄
0.847
σe40
0.010
e40
π⁄
0.122
w1
π⁄
-0.237
π⁄w2
-0.202
w2
β1
1.000
p1
β1
1.000
π⁄G
0.252
λ0,1
0.981
λ0,2
0.000
λ4,2
0.951
Authorized for public release by the FOMC Secretariat on 1/13/2023
Table 3: First Sample Estimated Parameters
Parameter
Prior
Mean
Std.Dev
Posterior
Mode
ιp
ιw
λ
γH
ζp
ζw
ϕ
γc
ψ1
ψ2
ρR
G
B
B
B
N
B
B
G
N
G
G
B
1.0000
0.5000
0.5000
0.7500
0.6000
0.8000
0.7500
3.0000
1.5000
1.7000
0.2500
0.8000
0.150
0.150
0.150
0.025
0.050
0.050
0.050
0.750
0.375
0.150
0.100
0.100
0.499
0.280
0.082
0.790
0.591
0.833
0.904
4.326
1.915
1.833
0.488
0.791
ρb
ρπ
ρg
ρi
ρs
ρλp
ρλw
ρν
ρω
B
B
B
B
B
B
B
B
B
0.5000
0.9900
0.6000
0.5000
0.6000
0.6000
0.5000
0.3000
0.3500
0.250
0.010
0.100
0.100
0.200
0.200
0.150
0.150
0.100
0.850
0.998
0.920
0.759
0.841
0.687
0.668
0.496
0.407
θλp
θλw
B
B
0.4000
0.4000
0.200
0.200
0.608
0.306
σb
σπ
σg
U
I
U
0.5000
0.0150
1.0000
2.000
0.0075
2.000
1.187
0.094
2.500
Symbol Density
Depreciation Curve
Active Price Indexation Rate
Active Wage Indexation Rate
External Habit Weight
Labor Supply Elasticity
Price Stickiness Probability
Wage Stickiness Probability
Adjustment Cost of Investment
Elasticity of Intertemporal Substitution
Interest Rate Response to Inflation
Interest Rate Response to Output
Interest Rate Smoothing Coefficient
Autoregressive Coefficients of Shocks
Discount Factor
Inflation Drift
Exogenous Spending
Investment
Liquidity Preference
Price Markup
Wage Markup
Neutral Technology
Investment Specific Technology
Moving Average Coefficients of Shocks
Price Markup
Wage Markup
Standard Deviations of Innovations
Discount Factor
Inflation Drift
Exogenous Spending
δ2
δ1
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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First Sample Estimated Parameters (Continued)
Parameter
Symbol Density
Investment
Liquidity Preference
Price Markup
Wage Markup
Neutral Technology
Investment Specific Technology
Relative Price of Cons to Inv
Monetary Policy
Unanticipated
1Q Ahead
2Q Ahead
3Q Ahead
4Q Ahead
Total Earnings
Loading 1
Standard Deviation
AR(1) Coefficient
Total Compensation
Standard Deviation
AR(1) Coefficient
Core PCE
Constant
Loading 2
Standard Deviation
AR(1) Coefficient
Market-Based Core PCE
Constant
Loading 1
Loading 2
Standard Deviation
Prior
Mean
Std.Dev
Posterior
Mode
σi
σs
σλp
σ λw
σν
σω
σ ci
I
U
I
I
U
I
I
0.2000
0.5000
0.1000
0.1000
0.5000
0.2000
0.0500
0.200
2.000
1.000
1.000
0.250
0.100
2.000
0.618
0.390
0.069
0.031
0.504
0.183
0.215
σµ0
σµ1
σµ2
σµ3
σµ4
N
N
N
N
N
0.0050
0.0050
0.0050
0.0050
0.0050
0.0025
0.0025
0.0025
0.0025
0.0025
0.012
0.012
0.007
0.009
0.010
β1w1
σw1
ϕw1
N
I
B
0.8000
0.0500
0.4000
0.100
0.100
0.100
0.824
0.147
0.624
σw2
ϕw2
I
B
0.0500
0.4000
0.100
0.100
0.169
0.343
π⁄p1
β2p1
σp1
ϕp1
N
N
I
B
-0.1000
0.0000
0.0500
0.2000
0.100
1.000
0.100
0.100
-0.087
0.014
0.048
0.091
π⁄p2
β1p2
β2p2
σp2
N
N
N
I
-0.1000
1.0000
0.0000
0.0500
0.100
0.100
1.000
0.100
-0.123
1.102
0.028
0.039
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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First Sample Estimated Parameters (Continued)
Parameter
Symbol Density
AR(1) Coefficient
Core CPI
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
PCE Durable Goods Inflation
Constant
1st Lag Coefficient
2nd Lag Coefficient
Standard Deviation
Relative Price Inflation
1st Lag Coefficient
2nd Lag Coefficient
Standard Deviation
Factor A
Loading 1
Loading 2
Loading 3
Loading 4
Standard Deviation
Factor B
Loading 1
Loading 2
Loading 3
Standard Deviation
Prior
Mean
Std.Dev
Posterior
Mode
ϕp2
B
0.2000
0.100
0.128
π⁄p3
β1p3
β2p3
σp3
ϕp3
N
N
N
I
B
0.0500
1.0000
0.0000
0.1000
0.4000
0.100
0.100
1.000
0.100
0.200
0.047
0.804
0.119
0.076
0.597
θ⁄D
θD1
θD2
σD
N
N
N
I
-0.3500
0.4500
0.4000
0.2000
0.100
0.200
0.200
2.000
-0.356
0.430
0.362
0.287
θG1
θG2
σG
N
N
I
0.0000
-0.1000
0.5000
0.500
0.500
2.000
0.279
0.006
0.811
λ1,1
λ2,1
λ3,1
λ4,1
σF1
N
N
N
N
N
0.6839
0.5224
0.4314
0.3243
0.1000
0.200
0.200
0.200
0.200
0.0750
1.256
0.857
0.361
0.032
0.041
λ1,2
λ2,2
λ3,2
σF2
N
N
N
N
0.3310
0.6525
0.8059
0.1000
0.200
0.200
0.200
0.0750
0.698
1.162
1.199
0.072
Notes: Distributions (N) Normal, (G) Gamma, (B) Beta, (I) Inverse-gamma-1, (U) Uniform
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Table 4: Second Sample Estimated Parameters
Parameter
Total Earnings
Constant
Loading 1
Standard Deviation
AR(1) Coefficient
Total Compensation
Constant
Standard Deviation
AR(1) Coefficient
Core PCE
Loading 2
Standard Deviation
AR(1) Coefficient
Market PCE
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
CPI
Constant
Loading 1
Loading 2
Standard Deviation
AR(1) Coefficient
Durable Goods Inflation
Constant
Standard Deviation
Relative Price Inflation
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Symbol
Prior
Mean
Std.Dev
Posterior
Mode
β⁄w1
β1w1
σw1
ϕw1
-0.2370
0.8242
0.1468
0.6239
0.200
0.200
0.100
0.200
-0.096
0.252
0.183
0.529
β⁄w2
σw2
ϕw2
-0.2023
0.1687
0.3430
0.200
0.100
0.200
-0.142
0.233
0.351
β2p1
σp1
ϕp1
0.0281
0.0481
0.0913
0.100
0.100
0.150
0.232
0.143
0.256
π⁄p2
β1p2
β2p2
σp2
ϕp2
-0.1230
1.1022
0.0139
0.0755
0.5972
0.100
0.150
0.100
0.100
0.150
-0.114
0.358
0.219
0.121
0.527
π⁄p3
β1p3
β2p3
σp3
ϕp3
0.0475
0.8039
0.1192
0.0388
0.1278
0.100
0.150
0.100
0.100
0.150
-0.022
0.305
0.206
0.085
0.220
π⁄D
σD
-0.4500
0.5000
0.200
0.150
-0.463
0.291
Authorized for public release by the FOMC Secretariat on 1/13/2023
Second Sample Estimated Parameters (Continued)
Parameter
Constant
Standard Deviation
Factor A
Loading 0
Loading 1
Loading 2
Loading 3
Loading 4
Loading 6
Loading 7
Loading 8
Loading 9
Loading 10
Standard Deviation
Factor B
Loading 0
Loading 1
Loading 2
Loading 3
Loading 4
Loading 5
Loading 6
Loading 7
Loading 9
Standard Deviation
Standard Deviations of Monetary Policy Innovations
Unanticipated
1Q Ahead
2Q Ahead
3Q Ahead
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Symbol
Prior
Mean
Std.Dev
Posterior
Mode
π⁄G
σG
0.8900
0.8143
0.400
0.080
-0.146
1.023
λ0,1
λ1,1
λ2,1
λ3,1
λ4,1
λ6,1
λ7,1
λ8,1
λ9,1
λ10,1
σF1
0.0180
0.0574
0.1941
0.3996
0.6520
1.2266
1.5237
1.8139
2.0914
2.3523
0.0442
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.250
0.100
0.158
0.356
0.496
0.623
0.820
0.990
1.122
1.102
1.087
2.515
0.072
λ0,2
λ1,2
λ2,2
λ3,2
λ4,2
λ5,2
λ6,2
λ7,2
λ9,2
σF2
-0.0181
0.2211
0.3679
0.4424
0.4612
0.4370
0.3817
0.3032
0.1074
0.0334
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.300
0.100
0.029
0.039
0.070
0.095
0.123
0.138
0.167
0.184
0.229
0.429
σµ0
σµ1
σµ2
σµ3
0.0061
0.0021
0.0004
0.0019
0.005
0.005
0.005
0.005
0.011
0.010
0.010
0.009
Authorized for public release by the FOMC Secretariat on 1/13/2023
Second Sample Estimated Parameters (Continued)
Parameter
4Q Ahead
5Q Ahead
6Q Ahead
7Q Ahead
8Q Ahead
9Q Ahead
10Q Ahead
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Symbol
Prior
Mean
Std.Dev
Posterior
Mode
σµ4
σµ5
σµ6
σµ7
σµ8
σµ9
σµ10
0.0001
0.0025
0.0019
0.0011
0.0001
0.0014
0.0028
0.005
0.005
0.005
0.005
0.005
0.005
0.005
0.010
0.010
0.010
0.010
0.009
0.010
0.010
Authorized for public release by the FOMC Secretariat on 1/13/2023
References
Calvo, G. A. (1983). Staggered Prices in a Utility-Maximizing Framework. Journal
of Monetary Economics 12, 383–398. 1, 9, 10
Campbell, J. R., C. L. Evans, J. D. Fisher, and A. Justiniano (2012). Macroeconomic
effects of Federal Reserve forward guidance. Brookings Papers on Economic
Activity Spring, 1–54. 1
Campbell, J. R., J. D. M. Fisher, A. Justiniano, and L. Melosi (2016).
Forward guidance and macroeconomic outcomes since the financial crisis. In
M. Eichenbaum and J. Parker (Eds.), NBER Macroeconomics Annual 2016,
Volume 31. University of Chicago Press. 1
Christiano, L., M. Eichenbaum, and C. Evans (2005). Nominal rigidities and
the dynamic effects of a shock to monetary policy. Journal of Political
Economy 113 (1), 1–45. 7
Fisher, J. (2015, March-April). On the structural interpretation of the smets-wouters
“risk premium” shock. Journal of Money, Credit and Banking 47 (2-3), 511–516.
4, 6
Fisher, J. D. M. (2006). The dynamic effects of neutral and investment-specific
technology shocks. Journal of Political Economy 114 (3), 413–451. 2
Gürkaynak, R., B. Sack, and E. Swanson (2005). Do actions speak louder than
words? International Journal of Central Banking 1 (1), 55–93. 35, 36, 37
Justiniano, A., G. E. Primiceri, and A. Tambalotti (2010). Investment shocks and
business cycles. Journal of Monetary Economics 57 (2), 132 – 145. 1
Justiniano, A., G. E. Primiceri, and A. Tambalotti (2013, April). Is there a tradeoff between inflation and output stabilization? American Economic Journal:
Macroeconomics 5 (2), 1–31. 32
King, R. G., C. I. Plosser, and S. T. Rebelo (1988a). Production, growth and
business cycles I. the basic neoclassical model. Journal of Monetary Economics 21,
195–232. 2
45
Page 103 of 104
Authorized for public release by the FOMC Secretariat on 1/13/2023
King, R. G., C. I. Plosser, and S. T. Rebelo (1988b). Production, growth and
business cycles II. new directions. Journal of Monetary Economics 21, 309–341.
2
Laséen, S. and L. E. Svensson (2011). Anticipated alternative policy-rate paths in
policy simulations. International Journal of Central Banking. 1
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@FedFRASER