A Guide to FRB/US

A Macroeconomic Model of the United States

Macroeconomic and Quantitative Studies 

Division of Research and Statistics

Federal Reserve Board

Washington, D.C. 20551

version 1.0, October 1996

Abstract: FRB/US is a large-scale quarterly econometric model of the U.S. economy, developed to

replace the MPS model. Most behavioral equations are based on specifications of optimizing behavior

containing explicit expectations of firms, households, and financial markets. Although expectations are

explicit, the empirical fits of the structural descriptions of macroeconomic behavior are comparable to

those of reduced-form time series models. In most instances, tests do not reject overidentifying restrictions

of rational expectations or the hypothesis of serially independent residuals. As modeled, private sector

expectations of policy constitute a major transmission channel of monetary policy.

Keywords: Macroeconomic models, private sector learning, rational expectations, vector autoregressions.



Edited by F. Brayton and P. Tinsley. Authors of sections are: F. Brayton (section 1); P. Tinsley (sections 2 and

4); A. Bomfim, D. Reifschneider, and P. von zur Muehlen (section 3); B. Tetlow and J. Williams (section 5). Other

members of the Macroeconomic and Quantitative Studies section who contributed to development of FRB/US and

provided comments on this summary are M. French, E. Mauskopf, and S. Schuh. In addition, D. Battenberg, T.

Grunwald, and T. Horvath provided significant technical assistance. Views presented are those of the editors and

authors and do not necessarily represent those of the Federal Reserve Board.
Contents

1 Introduction and Overview 1

1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Characteristics of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Simulation capabilities and properties . . . . .................... 5

1.5 Organization of remaining sections . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Types of Economic Behavior 7

2.1 Arbitrage equilibria. .................................. 8

2.2 Rational behavior under frictions. . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Heterogeneous households and firms. . . . . . . . . . . . . . . . . . . . . . . . . 14

3 A Bird's Eye View of FRB/US 15

3.1 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.3 Financial markets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

3.4 Foreign trade and government sectors . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Expectations 26

4.1 The scope of sectoral information. . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Long-run expectations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 The historical VAR. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

5 Full-System Properties 34

5.1 System responses to transitory shocks . . . . . . . . . . . . . . . . . . . . . . . . 34

5.2 System responses to permanent shocks . . . . . . . . . . . . . . . . . . . . . . . . 40

A Appendix: Testing the Theory 43

B References 45
1

1 Introduction and Overview

FRB/US is a new quarterly econometric model of the U.S. economy developed to replace

the MPS model. The new model has three distinctive features. First, the expectations of private

sectors are explicit, and these expectations, especially market perceptions of policy, constitute a

major transmission channel of monetary policy. Second, information assumed to be accessed by

private sectors can vary in scope and timing, and can include “perfect foresight” or learning from

past observations. Third, although expectations are explicit, the empirical fits of the structural

descriptions of macroeconomic behavior are comparable to those of reduced-form time series

models.

In contrast to the inability of MPS to disentangle forecast and response lags, the structure

of FRB/US parses dynamics into contributions of expectations and adjustment frictions. This

decomposition enables the new model to provide sharper interpretations of macroeconomic

developments and to examine the sensitivity of forecasts and policy scenarios to a range of

assumptions about how sectoral expectations are formed. However, the more explicit theoretical

structure has been achieved while also incorporating newer statistical techniques aimed at

improving the goodness of fit and reliability of empirical estimates.

Despite changes in structure and updated empirical estimates, FRB/US retains several notable

characteristics of MPS. One important similarity is the blend of long-run neoclassical conditions

for equilibrium with short-run sticky-price disequilibria where monetary policy actions have

significant short-run effects on the level of real activity.

FRB/US is a large-scale model, containing some 300 equations and identities. However, the

number of stochastic “core” equations or estimated descriptions of the economic behavior of

firms, households, and investors is much smaller, around 50 equations. In the current edition of

FRB/US (version 1.0), about half of these behavioral equations are based on formal specifications

of optimizing behavior containing explicit estimates of the forward expectations of firms and

households. Most of the discussion in this guide will focus on this subset of core equations

because the format of these structural equations is new. Although the explicit expectations format

may be extended to additional core equations in future model editions, most key macroeconomic

relationships in version 1.0 of FRB/US now include explicit expectations. These include the

estimated structural equations for aggregate consumption, two components of consumer durables,

residential construction, investment in producers' durable equipment, inventory investment, labor

hours, measures of aggregate price and labor cost, three long-term interest rates, and the value of

corporate equity.
2

1.1 Objectives

In the 25 years since the initial version of the MPS model was completed and brought into

operational use at the Board, the practice of macroeconomics has evolved substantially. Notable

developments include: greater emphasis on modeling expectations, especially the assumption that

expectations are rational or consistent with modeled outcomes; more extensive use of dynamic

optimization theory to characterize responses of households and firms to shocks; expansion of

the types of models in general use to include atheoretic vector autoregressions (VARs) and

theoretically-based, general equilibrium models of business cycles; and development of new

statistical techniques for estimation of long-run relationships among trending series and for testing

facets of equation performance.

Although the design of FRB/US draws on many of these developments, other adaptations and

new concepts were required in the construction of the model in order to meet a list of goals. The

main objectives that guided the development of FRB/US are:

1. Use: (a) The model should be suitable for both forecasting and policy simulations. (b)

It should also be able to run simulations of policy and other scenarios under a variety of

assumptions about how households, firms and financial markets form expectations, including

the extent of available information.

2. Conceptual design: (a) Expectations should be explicit. (b) Structural equations for

households, firms, and financial markets should be based on economic theories of optimizing

behavior.

3. Statistical implementation: (a) Estimation of equations in the model should be based on

modern time series techniques. (b) Equations should have satisfactory statistical properties,

including goodness of historical fit.

4. Simulation properties: For shocks that are not unusual in historical perspective, the model's

simulated responses should be close to those obtained from atheoretic models that do not

impose strong priors, such as VARs. The model should also be able to match established

rules of thumb regarding economic relationships under appropriate circumstances.

Some of these objectives are complementary: Explicit representation of expectations, 2a,

assists in identifying the contributions of alternative expectations, 1b. However, there are

well-known conflicts between the objective for more theoretical rigor, 2b, and the objectives

related to forecasting, 1a, and improved statistical properties, 3b. Indeed, major innovations in the

design of FRB/US have been motivated to improve the tradeoff between theoretical and empirical

properties.
3

1.2 Implementation

Key equations in FRB/US are based on four fundamental building blocks:

 arbitrage equilibria

 equilibrium planning

 dynamic adjustments

 forecasting (expectations formation)

The way these elements are combined in key equations varies. Equations for financial variables

are based on arbitrage equilibria. For example, arbitrage equates the rate of return on a

bond to a weighted average of expected future values of a short-term interest rate plus a term

premium. Because transactions costs are relatively small in financial markets, arbitrage is assumed

within each quarter. In contrast, nonfinancial variables are costly to adjust and, thus, move

only gradually to eliminate any disequilibria. Equations for nonfinancial variables are based on

equilibrium plans—values that would be desired in the absence of adjustment frictions—and

dynamic adjustments. The latter employ a general model of dynamic frictions to estimate the

optimal rate at which deviations from equilibria are eliminated. The approach to optimal dynamic

adjustments used in FRB/US is a generalization of a costly adjustment approach commonly used

in applied macroeconomics.

The main price equation in FRB/US is convenient for illustrating the structure of nonfinancial

equations. The equilibrium condition is derived from profit maximization and makes the planned

price level a markup over unit factor costs. A timeseries (cointegration) regression is used to

estimate the long-run weights on unit labor and energy costs. Once the equilibrium relationship

is specified, a generalized frictions model is estimated for price dynamics, including the degree

of cyclical variation of the markup. There are several ways to represent price adjustments, as

discussed in section 2, but for now it is sufficient to indicate that they depend on expectational

terms (future values of the equilibrium price level) and inertial terms (lags of the actual price

level).

Equations for financial and nonfinancial variables alike contain expectations of the private

sectors. As indicated above, expectations enter financial equations through the definition of

arbitrage equilibria. For nonfinancial equations, expectations may enter two ways. First,

components of the equilibrium plans of some equations are present values. For example, the

desired level of consumption is derived from the lifecycle model and, thus, depends on the current

value of tangible assets and current and expected future values of household income. Second, the

dynamic adjustment component of each nonfinancial equation depends on expectations because
4

households and firms aim, not at the current equilibrium value, but at the trajectory that the

equilibrium is expected to follow.

The last building block, forecasting, describes how sectoral expectations are formed. Here,

it is useful to distinguish between the assumption made about expectations when FRB/US was

estimated, and the various options for specifying expectations when the model is simulated.

For estimation purposes, sectoral expectations were derived from forecasts of small VARs.

Although the structure of the VARs varies across equations, the VARs have a similar organization.

Each contains a common set of variables—consumption price inflation, output gap, and the federal

funds rate—along with one or more sector-specific variables. Because each VAR contains an

equation for the federal funds rate, this form of expectations incorporates an average sample view

of how monetary policy was conducted historically.

For simulation purposes, several options for expectations are currently available. One is the

approach of VAR expectations just described. Under a second option, expectations are equal to

forecasts from the FRB/US model—an option labeled full-model expectations. Both types of

expectations can be viewed as reflecting rational behavior, but under different assumptions about

the scope of information available to individuals. Another dimension of expectations along which

FRB/US has flexible capabilities is the speed with which individuals learn about changes in the

economic environment. The specific application developed so far pertains to how quickly the

private sector catches on to changes in the long-run inflation objective of monetary policy—an

issue closely related to the topic of policy credibility.

1.3 Characteristics of equations

The economic behavior described by FRB/US can be summarized most easily by focusing on

three main sectors:

 Households choose equilibrium aggregate consumption based on the lifecycle model, an

approach motivated by utility maximization, but are assumed to be quite risk averse and,

thus, to discount the future heavily in computing expected income. The dynamic equation

for aggregate consumption contains sluggish adjustment of actual consumption toward its

equilibrium as well as modest effects of liquidity constraints. Investment in consumer

durables and residential construction varies with aggregate consumption as well as with real

interest rates and relative prices.

 Firms choose investment, inventories, labor hours, and prices based on profit maximization

under imperfect competition. Firms also are involved, along with households, in the

short-run determination of wages. Adjustment dynamics are estimated to be most rapid
5

for inventories and labor hours and slowest for wages and investment in producers' durable

equipment. The speed of adjustment of the aggregate price is intermediate. In addition to

its sensitivity to the cost of capital, investment in producers' durable equipment is modestly

sensitive to cash flow.

 Financial markets set bond rates, stock prices, and the exchange rate by standard arbitrage

conditions. Thus, bond yields depend on values of short-term interest rates expected to

prevail over the maturity of the bond, and the stock market valuation depends on expected

dividends. Term premiums in the bond equations vary countercyclically; the risk premium in

the equity market is modeled as a constant. All asset price equations have significant serial

correlation in their residuals, which is interpreted as an additional time-varying component

of term or risk premiums.

FRB/US also contains “traditional” equations—without explicit expectations—for imports,

exports, nonresidential construction, employment, labor force participation, and the relative price

of consumption.

With regard to interactions among sectors, an important set of linkages describes the

transmission channels of monetary policy. As in MPS, key transmission channels operate through

medium- and long-term interest rates directly in equations for investment in producers' durable

equipment, residential construction, and consumer durables, and indirectly through effects of the

value of the stock market on aggregate consumption and effects of the exchange rate on exports and

imports. As indicated above, asset prices and bond rates are directly linked to expectations of future

federal funds rates which, in turn, are perceived to reflect policy responses to macroeconomic

indicators. In addition to forward funds rates, the transmission of monetary policy through sectoral

expectations is extended in FRB/US to private sector forecasts of future equilibrium values. For

example, under either VAR or full-model expectations, an increase in the current funds rate lowers

expected future output and income, thereby restraining current investment and consumption.1

1.4 Simulation capabilities and properties

FRB/US currently can be simulated with either VAR or full-model expectations, alternatives

that vary the scope of information available to the private sector. Although a common view is

that economic models have “stark” properties under the assumption of full-model expectations,

because of the extensive amount of information provided individuals under this assumption, this

1Note that the additional policy transmission channels identified in FRB/US do not necessarily imply larger policy

“multipliers” but may reflect only the explicit decomposition of estimated responses among lags due to frictions and

lags stemming from the formation of expectations, including learning.
6

characterization is not generally correct for FRB/US. The model contains significant adjustment

frictions that slow responses of nonfinancial variables, even to anticipated events. Also, properties

of FRB/US under full-model expectations can be similar to those under VAR expectations, if the

shock or change being simulated is not unusual in an historical context. One example is a transitory

change in the federal funds rate. Under either VAR or full-model expectations, output moves for a

period of time in the opposite direction of the interest rate change, as does inflation, and long-term

interest rates change by a fraction of the movement in the funds rate. In this instance, the VAR

contains the essential macroeconomic responses so VAR expectations are similar to full-model

expectations.

In contrast, unusual shifts can yield quite different outcomes under VAR and full-model

expectations—an example is a future change in fiscal policy that is perfectly anticipated under

full-model expectations but recognized only as it occurs under VAR expectations. In this instance,

macroeconomic variables move in advance of the fiscal change under full-model expectations, but

only after the policy change under VAR expectations.

A second dimension of FRB/US simulations that pertains to expectations is the speed at which

the private sector learns about changes in policy objectives, such as a shift in monetary policy that

seeks to reduce the rate of inflation. The private sector's perception of the inflation objective—as

distinct from the actual policy objective—is included explicitly in the structure of FRB/US.

Thus, consequences of a disinflationary policy can be compared under different assumptions

about the credibility of the shift. Under full credibility, perceptions of the inflation objective

respond immediately and the output cost of reducing inflation is relatively small. Alternatively,

if perceptions of the inflation target adjust more slowly—and only after policy actions to achieve a

lower rate of inflation are instigated—the output cost of reducing inflation is higher.

1.5 Organization of remaining sections

Four sections follow. Section 2 presents the specifications of economic behavior in FRB/US,

using a bond rate equation to illustrate the structure of the typical financial equation and the

aggregate price equation to portray the specification of adjustment dynamics in nonfinancial

equations. Section 3 reviews how these approaches have been applied to core behavioral equations.

Expectations are explicit in the structure of FRB/US and important to its properties. Section 4

discusses options for expectations that are currently used in simulations of FRB/US; develops

the concept of long-run expectations, which are termed expectation endpoints; and presents the

core equations of the vector autoregression used for VAR-based expectations. Finally, section 5

presents simulations of FRB/US to illustrate its responses to key types of shocks under different

assumptions about how expectations are formed.
7

2 Types of Economic Behavior

This section introduces the types of equations used to describe economic behavior in FRB/US.

Discussion of each equation type includes a brief summary of assumptions about economic

behavior that motivate the equation format; significant differences, if any, with specifications in

alternative macro models; and an example to illustrate how the equation is referenced in later

sections. There are only four standard types of equations, each describing a different activity by

individuals:

 Arbitrage equilibria. This category includes equations for bond yields and the price of equity,

based on standard formulations of efficient market pricing. Under the assumption that arbitrage

profits are zero, expected returns on assets traded in financial markets are related to an expected

baseline rate of return and term or risk premiums.

 Equilibrium planning. Variables not determined in financial markets are controlled by

individuals in a particular sector, such as consumption by households or fixed capital investment

by firms. Each sector selects equilibrium values for its own activities that would be undertaken

in the absence of frictions. These equilibrium settings are functions of variables not controlled

by individuals within the sector. Examples of predetermined explanatory variables are expected

income for household consumption and sales for business capital investment. Equilibrium

equations are essentially steady-state relationships and comprise the set of equations most likely

to be subject to functional and coefficient restrictions from economic theory.2

 Dynamic adjustments. Unlike financial market behavior, where asset prices are assumed to

reflect equilibrium valuations, activities in remaining sectors are subject to a variety of dynamic

frictions. A shift in an equilibrium setting for a variable controlled by a sector will initiate

a response to reach the new desired value. However, when responses are constrained, the

optimal approach to the revised equilibrium may be spread over many quarters. The format

of dynamic adjustment equations in FRB/US is dissimilar to that in other macroeconomic

models of rational behavior in that the extent of dynamic frictions is not imposed by priors

but determined by statistical testing; consequently, the empirical goodness of fit is comparable

to those of atheoretic time series models.

 Forecasting. Structural equations in FRB/US require forecasts of explanatory variables. In

the case of arbitrage equilibria, principal determinants of financial market yields, such as bond

rates, are multiperiod forecasts of the federal funds rate. For variables subject to dynamic

2The explicitness of equilibrium equations in FRB/US is useful in empirical checks of theoretical long-run

restrictions, using tests for trending variables developed in recent years by time series analysts, vid. Engle and Granger

(1987).
8

frictions, the formulation of optimal adjustment plans requires multiperiod forecasts of the

relevant equilibrium values. As indicated later, rational expectations are assumed in estimation

where sector expectations are generated by a VAR model of the economy. There are two

advantages in assuming rational expectations. First, the unobserved expectations that condition

actions of households and firms are replaced by forecasts from an explicit forecast model.

Second, the use of explicit expectations permits identification of frictions that impede sectoral

dynamic adjustments.3

The remainder of this section provides brief discussions of the arbitrage equilibrium equation

for the 10-year bond rate, the equilibrium and dynamic adjustment equations for the aggregate

price equation, and examples of additional constraints on behavior. Because sectoral expectations

require a forecast model of the economy, discussion of this topic is postponed until an overview of

the full model is presented.

2.1 Arbitrage equilibria.

The simplest form of economic behavior that is assumed to reflect rational expectations is

pricing of financial assets in auction markets. The assumption that auction market forecasts

are rational requires that the model's multiperiod forecasts of fundamental variables, such as a

representative short-term interest rate, should explain significant movements in the auction prices

of long-maturity assets.

Assuming bond valuations reflect rational forecasts of the future path of the federal funds rate,

the yield to maturity on a 10-year Treasury bond, r10, is a weighted moving average of the funds

rates expected over the next 40 quarters,

r10t = 10 + w1rt + w2r

e

t+1 + w3r

e

t+2 + ::: + w40r

e

t+39; (1) = 10 + 1:0 leads40(r

e

t+i):

The first term on the right hand side of the equal sign in equation 1 is the term premium for the

10-year bond, 10. The remaining explanatory variables are forecasts of the federal funds rate in

future periods. Current market forecasts based on information available in period t are denoted

by the superscript “e”. In this instance, no superscript is attached to the funds rate of the current

period because the quarterly bond rate is assumed to incorporate current-quarter information; thus,

3Of course, the assumption that expectations of sectors are rational may not be consistent with observed behavior.

Rational expectations (RE) impose very tight restrictions on the way that sectoral expectations influence sector

responses. In contrast to rejections of RE restrictions in many empirical macroeconomic studies, RE restrictions

are generally accepted in FRB/US.
9

rt is the realized funds rate in the current period, not a forecast.

The weights, wi, on expected funds rates over the 40-quarter maturity of the 10-year bond sum

to unity. This is indicated in the second line of equation 1 where the relevant weight sum (1.0 in

this equation) multiplies a condensed notation for the weighted average of 40-quarter funds rate

forecasts, leads40(.). For a discount bond, the relationship in equation 1 would be a simple moving

average with each funds rate forecast receiving the same weight of 1/40. However, in the current

example of a 10-year Treasury coupon bond, the quarterly weights decline geometrically at a 2%

rate consistent with applying an annual discount rate of 8% (approximately the sample mean of the

10-year bond rate) to future coupon payments.

Table 1: 10-Year Government Bond Rate Equation (r10)

equilibrium

relationship: r10 = .46 + 1.0 leads40(r

e

) - .79 leads40(x~e) + .85 lags1(~10). R2.98

SEE .32

properties: mean response lag to surprise = 0 quarters. span: 63q1-94q4

definitions: r10 - ten-year government bond rate.

r - federal funds rate.

x~ - aggregate output gap.

~10 - term premium residual.

The estimated equation for the 10-year bond rate in FRB/US is displayed in table 1.4 The joint

assumptions of rational expectations and the absence of frictions on portfolio adjustments leave

only the term premium to explain predictable movements in the bond equation residual. Some

of the predictable variation in the term premium is explained by a 40-quarter lead of expected

deviations of aggregate output from trend output, x~e

t+i

, where the negative coefficient sum, -.79,

indicates countercyclical movements in the term premium. Predictable variation in the term

premium residual, after accounting for the effect of expected output deviations, is approximated

by a one-quarter autoregressive lag, lags1(~10), where the notation for lags parallels that used for

leads.

As indicated in table 1, the mean lag response to unanticipated shocks is zero because the bond

rate is assumed to incorporate current-quarter news. Additional reported statistics include the

4The construction of expectations used in estimating model equations, such as the funds rate forecasts for the

10-year bond rate equation, is discussed in section 4.
10

proportion of sample variation in the 10-year bond rate that is explained by the estimated equation,

R2= .98, and the standard deviation of the quarterly equation residual, SEE = .32.

2.2 Rational behavior under frictions.

In contrast to financial markets, where quarterly outcomes are assumed to reflect equilibrium

valuations, behavior in most other sectors of FRB/US is constrained by costs of adjustment. This

section describes the basic features of behavior in FRB/US when actions planned by sectors are

subject to dynamic frictions. For concreteness, the aggregate price equation of FRB/US will

illustrate the basic features of rational behavior under frictions.

As indicated earlier, optimal behavior under frictions is described as a two-stage decision

process. In the first stage, a sector decides on the equilibrium setting of a variable under its

control. The equilibrium setting is the value that would be selected in the absence of frictions

and is derived from standard conditions for profit or utility maximization. In the case of the price

equation, profit maximization under imperfect competition requires the optimal (log) producer

price to be proportional to the (log) marginal cost of production. Because long-run production

is Cobb-Douglas in FRB/US, the marginal cost of production can be expressed as a weighted

average of unit labor and energy costs. As shown in the first equation of table 2, the long-run

elasticities of the aggregate price equilibrium with respect to unit costs of labor and energy are .98

and .02, respectively. To allow for possible cyclical variations in perceived demand elasticities and

marginal costs of production, the equilibrium condition also specifies that the optimal price margin

may vary with the aggregate unemployment rate. The negative coefficient of the unemployment

rate, u, indicates procyclical variation in the equilibrium price, with a decline in the unemployment

rate of one percentage point raising the level of the desired price markup by 0.3 percent.

In the second decision stage, after selecting an equilibrium setting, the sector formulates an

optimal approach to the equilibrium price. A standard model of optimal price adjustment under

frictions is that by Rotemberg (1987), where costs associated with changing prices are proportional

to the squared deviation of the current price from last period's price, (pt  pt1)

2

. More generally,

this quadratic penalty on changing the level of sector activity is the standard approximation of

frictions used in all areas of applied macroeconomics, in part because it provides a rationale for

the familiar partial adjustment model in which a fixed fraction of the distance to the equilibrium is

eliminated in each period.5

The basic paradigm in FRB/US of adjustment dynamics employs a generalization of this

standard adjustment cost specification, which includes not only penalties for changing the level

5A recent historical review of dynamic friction specifications in empirical macroeconomics may be found in

Brayton and Tinsley (1995).
11

Table 2: Aggregate Price Equation (p)

equilibrium

relationship: p = :98(w  ) + :02pe  :003u:

remarks:  equilibrium condition includes also effects of farm and import prices.

dynamic

adjustment: pt = :10(pt1  p

t1) + .57lags2(pti ) + .43leads1(pet+i

). R2.88

SEE .0025

properties: mean response lag to surprise = 3.3 quarters. span: 63q1-94q4

remarks:  dynamic equation includes an accelerated

response to energy price inflation.

definitions: p - log price of final sales plus imports

less gov' t labor and indirect business taxes.

w - log compensation per hour (ECI).

 - log trend labor productivity.

pe - log crude energy price.

u - demographically-weighted unemployment rate.
12

of an action but also for changing its growth rate or for altering a moving average of recent

actions.6 There are three reasons to consider a more general description of dynamic frictions:

First, in contrast to equilibrium relationships, economic theory offers relatively little guidance on

the nature of dynamic frictions. Second, the standard prior that firms and households smooth only

the levels of macroeconomic aggregates is arbitrary and generally strongly rejected by postwar

data. Third, the extended model of frictions developed for FRB/US conveniently turns out to be a

restricted version of the familiar vector autoregression (VAR) developed by Sims (1980), enabling

FRB/US to take advantage of the data-oriented techniques of modern time series analysis while

estimating structural descriptions of rational behavior.

Under the generalized adjustment cost specification, the basic equation in FRB/US describing

rational adjustment under frictions contains three sets of regressors: a single regressor consisting

of the distance to the equilibrium that remains at the start of the current quarter, pt1  p

t1; a

second set of regressors consisting of lags of the dependent variable; and a third set of regressors

containing expected future changes in the desired equilibrium price. For convenient reference, the

dynamic adjustment equation reported in table 2 for the aggregate price level is reproduced here as

equation 2:

pt = :10(pt1  p



t1) + :57lags2(pti) + :43leads1(p

e

t+i): (2)

The first term after the equal sign indicates that 10% of the distance to the equilibrium price

level is eliminated in each quarter. Additional inertia is indicated in the next set of regressors

which includes two lags of the inflation rate, with a weight sum of .57. The coefficients of these

lagged terms would be zero if frictions were only associated with costs of changing the level of

the price. Thus, an implication of the joint presence of the initial level adjustment term and the

lagged inflation terms is that both the level and the inflation rate of the aggregate price index are

“sticky.” The third set of regressors are forecasts of changes in the equilibrium price over the

planning horizon of firms. As indicated by the subscript notation for forecasted leads, leads1(:),

the planning horizon is infinite because firms are assumed to be infinitely-lived entities.

In fact, the effective planning horizon of firms is considerably shortened by the rapid rate

of decay in weights assigned to distant forecasts of equilibrium price changes. Under rational

planning, there are two sources of influence on the relevance of future events. One is the standard

rate of return required by investors such as the discount factor applied to future earnings to derive

the current market value of equity. In the case of business firms, a fixed quarterly discount factor

of .98 is assumed, which is consistent with a postwar annual real rate of return to equity of about

6Issues in specifying and estimating more general descriptions of adjustment frictions are discussed in Tinsley

(1993) and Kozicki and Tinsley (1995).
13

Figure 1: Lead and Lag Response Weights of the Aggregate Price Equation

weights

0.0 0.02 0.04 0.06 0.08 0.10 0.12

Quarters

-11 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11

8%. The second and more important source of rapid decay in weights assigned to future periods

is the influence of frictions in constraining sectoral adjustments. Intuitively, if frictions are large

then the planning horizon must be lengthy because the equilibrium will not be reached quickly.

Conversely, if frictions are small, the required planning horizon will be short.

Lead and lag response weights of the estimated price equation are reproduced by the

downward-sloping solid lines in figure 1. These weights are obtained by a rearrangement of

the dynamic adjustment equation so the price in the current quarter, pt , is expressed as an

infinite distributed lead of future price equilibria, p

t+i

, and an infinite distributed lag of past price

equilibria, p

ti

.

7 Because the aggregate price eventually converges to the equilibrium price, the

combined sum of lead and lag weights is unity. Note that the lead and lag response weights

resemble a tent with the center pole anchored in the current quarter, designated by the origin. Both

the lead and lag response weights are approximately zero within twelve quarters, moving forwards

or backwards in time from the current quarter. The lead response weights are slightly more concave

downward due to the additional tilt introduced by the discount factor. As reported in table 2, the

mean response lag to an unanticipated shock (surprise) is 3.3 quarters; this estimate is the mean

lag defined by the left half of the weights in figure 1 that extend over past quarters. By contrast,

the response lag to a future event that has been perfectly anticipated (perfect foresight) is the mean

lag associated with the full distribution of lead and lag weights and is approximately zero.

7See derivations of response weights and additional discussion in von zur Muehlen (1996).
14

The alternative dotted line in figure 1 indicates the surprisingly modest effect on the response

weights of the aggregate price if the quarterly discount factor of the business sector were to be

reduced from .98 to .94, consistent with an increase of the annual discount rate from 8% to 24%.

8

Given the sizeable tripling of the discount rate, this exercise demonstrates that frictions on behavior

are generally the dominant influence in determining the effective planning horizons of rational

firms and households.

Another way to highlight differences in interpreting the dynamic responses of a rational pricing

model under frictions is to contrast the FRB/US equation in table 2 with the dynamic structure of

a standard reduced-form Phillips curve. If the distinction between actual and equilibrium prices

is suppressed except for the contribution of the unemployment rate to the equilibrium, then the

FRB/US aggregate price equation can be viewed as a two-sided Phillips curve with current inflation

depending on both past inflation and expected future inflation as well as on a cyclical indicator.

If it were further assumed that expected future inflation depended only on lagged inflation, then

the format of the price equation would resemble that of a simple Phillips curve in which inflation

is a function of inflation lags and a cyclical variable. Although the inflation lags appearing in

a typical Phillips curve are frequently identified only with inflation expectations, the discussion

of rational planning under frictions indicates that reduced-form inflation lags combine the effects

of both frictional inertia and expectations. An advantage of the rational adjustment format used

in FRB/US is that it provides an empirically tested separation of response lags attributable to

frictions on price adjustment from lags associated with forecasting. As implemented in FRB/US,

the forecasting lags are conditioned not only on lagged inflation but also on the business sector's

perceptions of response lags in other sectors, including those of the central bank.

2.3 Heterogeneous households and firms.

Although equations based on the assumption of identical individuals often provide reasonable

estimates of macroeconomic aggregates, in some cases there is no recourse but to acknowledge

measurable differences among individuals. This section discusses how the standard equation

formats are modified to accommodate heterogeneity among firms and households.

It is often possible to account for heterogeneity if there exists an observable proxy for a category

of constrained behavior. For example, in the case of imperfect capital markets, investment spending

of some firms may be limited to that which can be financed by retained earnings. Similarly,

consumption plans of some individuals may be constrained by current income, rather than wealth,

due to an inability to borrow on expected future labor income or an absence of tangible assets for

8As noted in section 3, an annual 25% discount rate is used in the household sector to discount expectations of

uncertain future income.
15

collateralized borrowing. In both examples, the net response can be modeled as a weighted average

of the proxy for liquidity-constrained behavior (either retained earnings or current income, in these

examples) and the behavior that would be predicted by the standard equation format for rational

adjustment under dynamic frictions.

Another way to accommodate heterogeneity is to disaggregate explanatory variables. For

example, unlike firms, households have finite lifetimes. Under the assumption of lifecycle

averaging of planned consumption, an individual's propensity to spend in each period out of the

sum of tangible and intangible wealth is approximately the inverse of the expected remaining

lifespan of the individual. Consequently, even for a stable age distribution of households, cyclical

or secular changes in the distributions of types of wealth over age cohorts will induce changes in the

aggregate propensity to spend from aggregate household wealth. For example, the expected present

value of labor income is more important for young households, who have lower propensities to

spend; by contrast, the expected present value of distributions from pension funds and social

security is more important for retirees who, in the absence of bequest motives, have higher

propensities to spend. Effects of heterogeneity in household spending responses are approximated

by including the composition of wealth, in addition to total wealth, in equilibrium equations for

household consumption.

3 A Bird's Eye View of FRB/US

This section presents the estimated structure of all equations that have been reformulated

using specifications required for either of the two primary forms of rational behavior in

FRB/US—arbitrage equilibria and dynamic adjustment subject to frictions. As described later

in section 4, these equations were estimated under the assumption of rational expectations,

implemented by representing sectoral expectations with VAR forecasts. For each equation, the

restrictions imposed by rational expectations were tested, as was the hypothesis that the equation

residuals were serially independent. Results of these tests, reported in Appendix A, generally

support the null hypotheses of rational expectations and serially independent residuals.

Before surveying individual estimated equations, a brief preview of three categories of dynamic

adjustment equations may be useful. First, as noted in section 2, the generic format for dynamic

adjustment under frictions expresses the first difference of a variable as a function of the gap

between the variable and its equilibrium value, lags of first differences of the variable, and leads

of expected first differences of its equilibrium. Three dynamic equations that use this standard

template are the quarterly adjustments in inventories, price, and wage.

A second category of dynamic adjustment formats is due to heterogeneous behavior, when

some households and firms follow the standard adjustment model and others behave according
16

to another criterion. Three equations exhibit heterogeneous behavior: aggregate consumption

includes effects of liquidity-constrained households; investment in producers' durable equipment

incorporates effects of liquidity-constrained firms; and labor hours include some workers whose

hours are costless to adjust. In each case, extra regressors are added to represent the alternative

behavior, and coefficient restrictions are imposed so that the proportions of each type of behavior

appearing in the aggregate adjustment equation sum to one.

A third dynamic adjustment category is also represented in investment equations. Although

investment decisions aim at achieving a desired capital stock, the estimated equations do not

include capital stocks among the explanatory variables. This omission is a consequence of a

current lack of data on stocks; nevertheless, even if stock data were available, doubts about its

quality might argue against its use. Investment equations have been modified to approximately

capture effects that arise when the underlying target is a stock. The modifications, which appear in

equations for investment in motor vehicles, other durables, residential construction, and producers'

durable equipment, involve the inclusion of extra growth rate variables which operate like the

output accelerator found in many traditional investment equations.

3.1 Households

Most households base consumption decisions on expected lifetime assets, which equal current

property wealth plus the present value of expected after-tax labor and transfer income. Risk

aversion causes future income flows to be discounted quite heavily with the consequence that

households do not act to fully offset government fiscal actions, as suggested by the Ricardian

equivalence hypothesis. Another characteristic of the household sector is that movements in labor

force participation are driven by changes in social norms in the long run, represented by time

trends, and by the availability of jobs in the short run. The real, after-tax wage has no effect on

labor supply. Thus, there are no effects of fiscal policy on the labor force, nor are consumption and

labor supply decisions intertwined.

Aggregate consumption (table 3). As in MPS, aggregate consumption is defined as the service

flow derived from the stock of consumer durables plus spending on nondurables and services. In

the steady state, aggregate consumption depends on total wealth—defined as the after-tax present

value of the sum of labor, property, and transfer income—and its composition. Expected future

income flows are discounted at a 25 percent annual rate in computing present values due to the

aversion of households to the uncertainty of future income. The composition of wealth matters

because the lifecycle hypothesis suggests that, absent strong bequest motives, the propensity to

spend out of expected lifetime resources at the individual level increases with age. Types of

income and assets are not evenly distributed over age groups; thus, consumption aggregated across
17

individuals varies with the composition of total wealth. Based on a linear approximation of the

logarithmic equilibrium condition, marginal propensities to consume out of categories of income

and tangible wealth are .51 for labor income, 1.05 for transfer income, .39 for property income,

.030 for corporate equities, and .075 for other net tangible assets.9 In addition to total wealth

and its composition, desired aggregate consumption also depends positively on the output gap,

approximating the effect of countercyclical variation in the perceived riskiness of future income

flows.

Table 3: Aggregate Consumption Equation (c)

equilibrium

relationship: c

 = 1:0v + :62strans  :15sprop + :52sstock + 1:28so + :013~x:

dynamic

adjustment: ct = :12(ct1  c



t1) + .17 lags1(cti) + .75 leads1(c

e

t+i) + .09yt:

span: 63q1-95q4 R2: .54 SEE: .0032 MRL a: 7.9 quarters

definitions: c - log consumption (including service flow of stock of durables).

Y - income (labor + transfer + property).

y - log Y . V - wealth = leads1(Y e). v - log V . strans - transfer wealth / V . sprop - property wealth / V . sstock - value of corp. equity / V . so - other net financial and tangible assets / V . x~ - aggregate output gap.

a

Mean response lag to a surprise.

The dynamic consumption equation is a weighted average of the behavior of lifecycle and

liquidity-constrained households. The share of income associated with the latter group is about 10

percent, based on the estimated coefficient on contemporaneous income growth. This direct effect

of income growth is in addition to the contribution of income growth in the VAR forecasting model

for expectations of target consumption that was used in estimation of the dynamic consumption

equation. Lifecycle consumers adjust spending sluggishly, with a mean response lag to shocks

(surprise) of about two years.

9The equilibrium equation contains both the present value of property income and a Flow of Funds estimate of

property wealth because the latter controls for differences between the market valuation of property assets and the

valuation of property income using a fixed discount rate. For a more detailed description of household consumption

and investment, see Reifschneider (1996).
18

Table 4: Household Investment Equations (cdv , cdo, and ih)

equilibrium

relationships: c

dv = 1:0c  :46(pdv  pc)  :41(pgas  pc)  :03rdv :

c

do = 1:0c  :56(pdo  pc)  :02rdo + :004t82:

ih = 1:0c

  :13rh  :003t47 + :003t88:

dynamic

adjustment: cdv;t = :30(cdv;t1  c

dv;t1) - .28 lags1(cdv;ti) + 3.22 leads1(cedv;t+i)

+ 7.46 lags4(c

dv;ti).

span: 63q1-94q4 R2: .43 SEE: .0054 MRL a: 3.3 quarters

cdo;t = :10(cdo;t1  c

do;t1) + .17 lags1(cdo;ti) + 2.15 leads1(cedo;t+i)

+ 1.12 lags4(c

do;ti).

span: 64q1-95q4 R2: .34 SEE: .0016 MRL a: 7.3 quarters

ih;t = :09(ih;t1  ih;t1) + .38 lags1(ih;ti) + 6.10leads1(ieh;t+i)

+ 4.15 lags4(ih;ti).

span: 63q1-95q4 R2: .60 SEE: .034 MRL a: 5.9 quarters

remark:  dynamic equation for ih also includes variables for deposit

disintermediation in 1966-7 and for credit controls in 1980.

definitions: cdv - log consumer expenditures on motor vehicles (constant dollars).

cdo - log consumer expenditures on other durables (constant dollars).

ih - log investment in residential construction (constant dollars).

c



- log aggregate consumption target (see table 3).

pdv and pdo - log prices of motor vehicles and other durables.

pc - log price of aggregate consumption.

pgas - log retail gasoline price, adjusted for vehicle fuel efficiency.

rdv , rdo and rh - costs of capital, motor vehicles, other durables, and housing.

t82, t47 and t88 - quarterly time trends starting 82q1, 47q1 and 88q1.

a

Mean response lag to a surprise.

Household investment (table 4). The equilibrium or long-run ratios of household investment

in motor vehicles, other durables, and housing to aggregate consumption are functions of relative

prices, user costs of capital, and time trends (for the equilibrium ratios of other durables and
19

housing). User costs incorporate effects of depreciation and the real rate of interest. The latter

is defined as a nominal interest rate—the auto loan rate in each of the durables equations and the

fixed-rate mortgage yield in the housing equation—less an expected consumer inflation measure

of corresponding maturity, adjusted for the marginal tax rate.

As noted earlier, the dynamic adjustment equation for each component of household investment

includes a modification of the standard rational adjustment format to account for the requirement

that investment decisions are aimed at achieving a desired stock of capital, in addition to optimal

rates of investment. This modification introduces accelerator effects to the investment equations

by adding lags and additional leads in the growth rates of desired investment as regressors.

As indicated by the large coefficient sums on the growth rates of equilibrium investment in

the dynamic adjustment equations of table 4, accelerator effects are most pronounced for two

categories of household investment, motor vehicles and housing.

3.2 Firms

Firms produce with a long-run Cobb-Douglas technology, setting equilibrium factor input

quantities (labor, capital, and energy) and the price of output to maximize profits. The production

function and first-order conditions for profit maximization define the equilibrium equations for

investment in producers' durable equipment, labor hours, and the price of output. Energy demand

is determined as an exogenous ratio to output rather than by a first-order condition. The firm sector

also includes equations for inventories and hourly labor compensation. The structure of the latter

is closely related to that of the price equation.

Investment in producers' durable equipment (table 5). The desired capital stock varies

proportionally with output and, based on the Cobb-Douglas production structure, inversely with

the user cost of capital. In the latter, financing costs are measured with weights of 0.8 on the cost

of debt and 0.2 on the cost of equity; the weights were chosen to maximize the fit of the equipment

equation. The ratio of target investment to target capital equals the sum of the depreciation rate and

the growth rate of output. In the equilibrium equation for investment, which is written in logs, the

log of the sum of depreciation and output growth has been linearized so that output growth—the

accelerator—appears separately.
20

Table 5: Business Investment Equations (ipd and ki)

equilibrium

relationships: ipd = 1:0xb  1:0rpd + 1:0zpd + 19:5xb:

k

i = 1:0xb:

dynamic

adjustment: ipd;t = :07(ipd;t2  ipd;t2) + .26 lags2(ipd;ti) + .47 leads1(iepd;t+i1)

+ .22 lags2(cfti).

span: 64q1-94q4 R2: .40 SEE: .0022 MRL a: 7.0 quarters

ki;t = :23(ki;t1  k

i;t1) + .47 lags3(ki;ti) + .53 leads1(ke

i;t+i).

span: 62q3-94q4 R2: .42 SEE: .0065 MRL a: 1.3 quarters

remarks:  dynamic equation for ipd is a weighted average of adjustment

model (.78) and cash flow model (.22).

 adjustment model component for ipd includes 1-quarter delivery lag.

definitions: ipd - log investment in producers' durable equipment (constant dollars).

ki - log stock of manufacturing and trade inventories (constant dollars).

xb - log output, business sector (constant dollars).

rpd - log user cost of capital, producer durables.

zpd - log(depreciation rate + mean of xb).

cf - log corporate cash flow (constant dollars).

a

Mean response lag to a surprise.

The dynamic investment equation augments the generalized adjustment model, which does not

fully capture cyclical investment fluctuations, with cash flow. The added variable is motivated

by recent empirical literature suggesting that some firms are constrained in their access to capital

markets. The resulting equation places 20 percent weight on the growth of cash flow and 80

percent weight on the standard adjustment specification. Considering only the adjustment part

of the equation, investment is found to be fairly sluggish, with a mean response lag of almost 2

years. The response includes a one-quarter delivery lag which was found to improve the fit of

the equation; the delivery lag causes the timing of the level correction term and the expectations

variables to be shifted back one quarter more than in the standard adjustment specification.

Inventory investment (table 5). The model's main inventory equation determines the stock

of manufacturing and trade inventories, where the inventory target stock, k, is proportional to the
21

output of the business sector, xb. Relative to the adjustments of most nonfinancial aggregates, firms

adjust inventory holdings to revised target levels rapidly with a mean lag response to unanticipated

shocks of 1.3 quarters.

Labor input (table 6). In MPS, the equilibrium level of aggregate labor hours was based on the

sum of labor requirements across the vintages of capital needed to produce a given level of output.

This condition, derived from an assumed putty-clay characteristic of capital, was approximated by

making target hours a function of output, total factor productivity (specified as a time trend), and

contributions of average capital and energy intensities of existing capacity to the labor-output ratio.

Coefficients based on capital and energy intensities were imposed, based on average factor shares,

without causing too much deterioration in the equation's goodness of fit. However, with revised

NIPA data, imposing the factor-intensity effects in FRB/US led to a substantial deterioration of

fit. Thus, the equilibrium condition has been simplified to make target hours a function of output

and a pair of time trends capturing the well-known slowdown of productivity growth in the early

1970s.10

Labor hours are modeled as heterogeneous with respect to adjustment costs. About one-third of

hours is estimated to respond immediately to changes in target hours, with the remainder following

the generalized adjustment cost model. Aggregated across both types of labor, the mean response

lag to a surprise is very short, less than a quarter.

Prices and Wages (table 6). The steady-state structures of the price and wage equations are

derived from the equilibrium Cobb-Douglas assumption that the share of income received by

capital is constant in the steady state. If the share of income received by energy producers is

stable, then constancy of the capital share implies constancy of the labor share. Price and wage

targets also vary procyclically. The two equilibrium conditions jointly determine the real wage and

the NAIRU. The latter is constant when defined by a demographically-weighted unemployment

rate and a bit less than 6 percent, currently, in terms of the civilian unemployment rate.

The generalized frictions model yields level error correction terms in the dynamic adjustment

equations for both price and wage. The level correction term in the price equation is a standard

feature of level price markup equations and captures variations in price margins. By contrast,

a level correction term in wage equations is less common, but consistent with models of wage

bargaining.

The dynamics of wages and prices are strongly interrelated because the target wage is largely

determined by the price level and, reciprocally, the target price is a function primarily of the wage

level. With regard to adjustment speeds, wages are estimated to be more sluggish than prices. The

mean response lag in the wage equation (8.7 quarters) is more than twice the mean response lag

10A simulation option is available to make labor productivity respond gradually to movements in relative factor

prices, similar to the corresponding MPS equation.
22

Table 6: Aggregate Labor Hours, Wages, and Prices (h, w, and p)

equilibrium

relationship: h = 1:0xg  :0069t47 + :0042t73: w = 1:0 + 1:02pg  :02pe  :01u:

p = :98(w  ) + :02pe  :003u:

remark:  equilibrium condition for p also includes effects of farm and import prices.

dynamic

adjustment: ht = :15(ht1  h

t1) + .38 lags1(hti) + .41 leads1(het+i).

+ .31 h

t

- .12 lags1(h

ti).

span: 63q1-94q4 R2: .76 SEE: .0046 MRL a: 0.7 quarters

wt = :03(wt1  w

t1) + .71 lags3(wti) + .29 leads1(wet+i).

span: 63q1-94q4 R2: .82 SEE: .0028 MRL a: 8.7 quarters

pt = :10(pt1  p



t1) + .57 lags2(pti) + .43 leads1(p

e

t+i).

span: 63q1-94q4 R2: .88 SEE: .0025 MRL a: 3.3 quarters

remarks:  dynamic equation for h is a weighted average of standard adjustment

model (.69) and immediate response model (.31) .

 dynamic equation for w also includes variables for wage and price controls,

employer social insurance contributions, and the minimum wage.

 dynamic equation for p also includes an accelerated

response to energy price inflation.

definitions: h - log hours, nonfarm business sector (employees and self-employed).

w - log compensation per hour (ECI).

p - log price of final sales plus imports

less gov' t labor and indirect business taxes.

xg - log output, nonfarm business sector plus oil imports

less housing product (constant dollars).

t47 and t73- quarterly time trends starting 47q1 and 73q1.

 - log trend labor productivity.

pg - log price of xg less indirect business taxes.

pe - log crude energy price.

u - demographically-weighted unemployment rate. a

Mean response lag to a surprise.
23

(3.3 quarters) in the price equation. Wages are also less responsive to forward expectations, with

a coefficient sum of .29 on the expected rate of growth of the equilibrium wage compared with an

expectations coefficient sum of .43 in the price equation. The dynamic wage and price equations

contain inflation-neutrality restrictions to insure that the equilibrium real wage and NAIRU are

independent of the rate of inflation.11

Other equations. Nonresidential construction is assumed to move with business output and

a time trend. In the current version, no effects of interest rates or tax policy are included.

Employment is determined by a standard error-correction equation in which the employment target

depends on aggregate hours and trends in the workweek.

3.3 Financial markets

Equations for three long-term interest rates and the stock market comprise the core of the financial

market sector of FRB/US. Unlike nonfinancial behavior, where frictions make it too costly to move

immediately to equilibrium values, asset prices are assumed to be in equilibrium continuously.

Long-term bond rates (table 7) are determined according to the expectations theory of the

term structure. The theory posits that, up to a term premium, the yield on a long-term bond is

given by the expected future path of short-term interests rates. Abstracting from term premia, the

yield on a 5-year government bond is a weighted average of expected federal funds rates over the

next 20 quarters.12 The yields on the other two long-term bonds—the 10-year government bond

and the Moody AAA corporate bond—are modeled in an analogous way.

Term premia for all three bond equations are estimated to vary negatively with a weighted

average of the output gap expected over the maturity of the bond. One can interpret this negative

relationship to mean that investors require larger risk premiums when they expect a deterioration

in the average performance of the economy over their investment horizon. Consistent with this

notion, the sensitivity of the term premium to expected economic conditions increases with the

maturity of the bond.

11These restrictions set the sum of the coefficients on the lead and lag inflation terms to one. Analogous restrictions

were not imposed on other adjustment equations, because they explain the behavior of real variables whose steady-state

growth rates are likely to have relatively small variations.

12The weights attached to each future value decline about 2 percent per quarter. The implied average duration of

the 5-year government bond rate is approximately 4 years. The 10-year government bond and the corporate bond have

durations of 7 and 12 years, respectively.
24

Table 7: Financial Sector Equations (r5, r10, rcb, and vs)

5-year gov' t

bond rate a: r5;t = :34 + 1:0 leads20(r

e

t+i)  :62 leads20(~xet+i) + :83 lag1(~5;ti)

span: 63q1-94q4 R2: .97 SEE: .47 MRL b: 0 quarters

10-year gov' t

bond rate a: r10;t = :46 + 1:0 leads40(r

e

t+i)  :79 leads40(~xet+i) + :85 lag1(~10;ti)

span: 63q1-94q4 R2: .99 SEE: .32 MRL b: 0 quarters

corporate

bond rate a: rcb;t = 1:21 + 1:0 leads120(r

e

t+i)  1:21 leads120(~xet+i) + :87 lag1(~30;t)

span: 63q1-94q4 R2: .99 SEE: .27 MRL b: 0 quarters

stock market

wealth: vs;t  pg;t = 4:7 + dt + 50 leads1(de

t+i)  50( (rcb;t=400)leads120(pec;t+i) )

span: 65q1-95q4 R2: .97 SEE: .20 MRL b: 0 quarters

definitions: r - federal funds rate.

x~ - output gap.

~5, ~10, and ~30 - term premium residuals for r5, r10, and rcb. vs - log stock market wealth (current dollars, flow of funds accounts).

d - log national income dividends (constant dollars, deflated by pg ).

pg - log price, business sector output.c pc - inflation rate, household consumption price.c

a

For the three bond equations, the reported SEE and R2are computed after

adjustment for first-order serial correlation of the term-premium residuals.

b

Mean response lag to a surprise. c

Price indexes divided by 100 before taking logarithms.

Stock market wealth (table 7). Similar to the log linearized model in Campbell and Shiller

(1989), the real value of the stock market is determined by expectations of the future flow of real

dividend payments. Future expected dividends are discounted by the expected opportunity yield

on corporate bonds, measured by the current corporate bond rate less expected consumer inflation

rates over a 30-year horizon. Both the stream of expected real dividends and the real corporate

bond rate are multiplied by a normalization factor due to linearization about the sample average

of the real return on equity. As noted in the fourth equation of table 7, on a quarterly basis, the
25

normalization factor is approximately 1/.02 = 50. Additional compensation for equity ownership,

apart from the cyclical risk premium already embedded in the corporate bond rate, is captured

by freely estimating the intercept of the equity equation. However, the substantial residual serial

correlation of this equation suggests the additional risk premium required by households for equity

holding is not constant over time.

Mortgage and car loan rates. The equilibrium relationships for the mortgage rate and the

loan rate on new cars are based on the 10-year and 5-year government bond rates, respectively.

In the short run, movements in the mortgage and new car loan rates reflect partial adjustments to

long-run equilibria; the mortgage rate also varies countercyclically.

3.4 Foreign trade and government sectors

In the current model version, these sectors are constructed using unrestricted error correction

regressions and not the restricted rational adjustment specifications described in section 2. A brief

description follows of the theoretical motivations of the main equations in these sectors.

The equations describing the long-run determinants of real exports and imports are standard.

Real exports depend on foreign GDP and the real exchange rate. The latter is defined by

open-interest parity arbitrage with an expected long-term real rate of interest and a country risk

premium, which is a function of U.S. net foreign indebtedness. Real nonoil imports are a function

of domestic GDP and the relative price of imports. Each trade equation contains a time trend

and is formulated as an error correction; long-run income elasticities are constrained to unity and

long-run price elasticities to minus unity.

The government sector of FRB/US is disaggregated into two tiers: federal and state & local.

Most variables in this sector are either exogenous or defined through identities. However, tax

payments are endogenous, as are transfers to persons, which are estimated to have components that

vary countercyclically, and net interest payments, which are functions of stocks of debt outstanding

and interest rates.
26

4 Expectations

Expectations of future events are important determinants of private sector behavior in FRB/US,

especially over multiperiod horizons as the initial restraining influences of frictions dissipate. This

section discusses the formulation of explicit expectations in FRB/US and options that may be used

in hypothetical scenarios as well as those used in estimation.

In estimating the structural descriptions of behavior described in section 3, explicit expectations

of the desired equilibria of firms and households were required to identify the frictions that inhibit

dynamic adjustments. Although explicit expectations need not be rational, the assumption that

sectors formulate rational expectations using a condensed model of the economy was imposed

in estimation and not empirically rejected in most instances. Rational expectations (RE) denote

forecasts of variables that are consistent with expected outcomes of the modeled mechanisms that

generate these variables.13

Although the assumption of rational expectations has proved to be a useful organizing principle

in estimating the model—as was the assumption that financial auction markets are efficient—there

are obviously instances when such an assumption is unlikely to provide a good prediction of

short-run behavior, as in the aftermath of an unusual event which is not well understood by

individuals in any sector, such as the prospect of war or the collapse of a market. Also, prior

reasoning cannot indicate the level of detailed information about the general economy that is

needed to make rational decisions in a particular sector.

Therefore, several options exist (or are being developed) for FRB/US that vary two dimensions

in testing or imposing rational expectations in estimation and simulation. One dimension is the

scope of rational expectations, where sector forecasts may be consistent for summary aggregates

but not necessarily for all forecast components. The other is the speed of rational expectations

formation where, in the most typical case, private sector perceptions may be consistent with policy

goals in the long run but not necessarily over short forecast horizons.

Discussion in this section is organized as follows: The first subsection outlines the main

options that specify the scope of information shared by sectors. The second subsection discusses

specifications of long-run expectations in FRB/US, including private sector perceptions of long-run

policy goals. Finally, a third subsection describes the VAR model used in the estimation of FRB/US

to generate sector expectations.

13Note that rational behavior by individuals, in the sense of optimal dynamic planning, need not imply rational

expectations. As discussed by Sargent (1993), both rational planning and consistent forecasts are required for rational

expectations. The first condition of rational behavior is always imposed in FRB/US structural equations, whereas the

second condition of consistent perceptions by firms and households is a simulation option. Other options that limit

the information or computing ability of individuals are examples of bounded or limited rationality. Not all bounds on

rationality admit rational expectations, even in the long run.
27

4.1 The scope of sectoral information.

The two main options for the scope of information shared in common by firms, households,

and investors are:

Full-model expectations. Under full-model expectations, all sectors use the same forecast model

of the economy. The common forecast model is a closed version of FRB/US, meaning that that

the behavioral responses of all sectors—including policy responses—are specified.14 The forecast

of a variable appearing anywhere in the model will be equal to the forecast generated by the full

FRB/US model. The full-model expectations option is the conventional assumption regarding

shared information in RE models and provides a useful benchmark of policy effects under complete

information and perfect foresight of selected future events.

Because FRB/US is nonlinear, full-model expectations are obtained by iterative, numerical

simulations under the assumption that future shocks are either known (perfect foresight) or equal

to zero.15 In discussing the method of solving for full-model expectations, the 10-year bond rate

equation is again used as an example. The bond rate requires 40-quarter forecasts of the funds rate

and the output gap (trend deviation).

r10 = :46 + 1:0leads40(r

e)  :79leads40(~xe) + :85lag1(~10): (3)

The full-model expectations solution proceeds by iterative revisions of expectations.16 Assume

that initial “guesses” by investors of 40-quarter forecasts of the funds rates and the output gap are

available for each quarter of the forecast horizon. These will define investor forecasts of the bond

rate in each forecast period. However, after also solving the full model, where each sector uses

its own “guesses” to initialize forecasts of explanatory variables, the forecasts of the funds rate

and output generated by the full model will not generally match the initial investor guesses used

to construct the bond rate forecasts. Similarly, the initial bond rate guesses used in other equations

of the model, such as the guesses by the business sector to forecast the cost of capital for fixed

investment, will generally not match the bond rate forecasts produced by equation 3. So, in the

14Obviously, individuals in private sectors cannot formulate rational forward plans without a description of expected

policies. For purposes of discussion in this section, monetary policy is summarized by an equation describing the

responses of the federal funds rate to recent movements in inflation and the trend deviation in output. Other ways to

characterize monetary policy are discussed in section 5.

15The practice of setting future residuals to zero (thus ignoring nonzero expectations of nonlinear functions of

residuals) has become a convention in macroeconomic constructions of full-model expectations, which are often

termed model-consistent expectations. The bias of expectations generally is small because FRB/US is approximately

linear for small shocks and, indeed, for many purposes may be viewed as a large, restricted VAR.

16The following is a simplified description of the Fair-Taylor method of solving RE models, which is an iterative

Jacobi method of solving systems of dynamic equations.
28

next iteration, each sector uses the multiperiod forecast of the model to replace its initial guesses.

This iterative cycle of updating sectoral expectations by model forecast solutions from the previous

iteration and then solving the full model for a new multiperiod forecast sequence continues until

sectoral expectations exactly match the full-model forecast. Since FRB/US is approximately linear

for small shocks, this iterative process will converge to a unique forecast, if initial guesses are not

too distant from the solution.

This cursory description ignores an important issue involving the forecast horizon. Note that

the bond rate in the last period of the forecast horizon requires 40 more quarters of funds rates

and output gap forecasts. But these variables, in turn, require accompanying bond rate forecasts.

This second sequence of more distant bond rates requires 40 additional quarters of funds rates and

output gap forecasts, and so on. This infinite recursive sequence is approximated by solving the

model for a large number of periods, say 25 years (100 quarters). In practical terms, the impact

of distant variables on present actions dissipates rather quickly for most variables. For example,

as noted in section 2, the lead weights in the aggregate price equation are approximately zero at

the end of a three-year forecast horizon even though the theoretical planning horizon of firms is

infinite. Although the effective planning horizon for most variables does not exceed three or four

years, notable exceptions are long-maturity instruments such as bonds and equity.

VAR expectations. Under this option, a small vector autoregression (VAR) model of summary

macroeconomic aggregates replaces the full FRB/US model as the description of the economy that

conditions sectoral forecasts. The same VAR model is used by all sectors, although disaggregated

information may be added to provide additional local information within a sector. The VAR

model most commonly used for expectations in FRB/US is the historical VAR. The historical

VAR provides an average-history summary of the dynamic behavior of the economy and was

used to generate sectoral expectations in estimation of FRB/US arbitrage and dynamic adjustment

equations.17 Using the same description of policy, general multiplier properties are similar for

the historical VAR and the full FRB/US model. This correspondence suggests that the many

restrictions in FRB/US required for structural interpretations are not noticeably inconsistent with

historical data, reinforcing the results of direct statistical tests of RE restrictions shown in Appendix

A. Given the well-behaved and generally white noise residuals of equations in FRB/US under

historical VAR expectations, the historical VAR is the standard expectations option for short-term

forecast analysis.

If a hypothetical monetary policy is simulated that deviates markedly from the policy responses

captured by the historical VAR, it is unlikely that forecasts of summary aggregates from the

17Conditions for maximum likelihood properties of estimated friction parameters based on VAR expectations are

discussed in Brayton and Tinsley (1995) and Kozicki and Tinsley (1995).
29

historical VAR model and from FRB/US using historical VAR expectations would agree. In this

instance, tests of simulated behavior from the full model would be likely to reject a hypothesis that

the historical VAR expectations were rational.18

Given the linear structure of VAR models, VAR expectations can be directly obtained by

analytical solutions rather than numerical iterations. Thus, the calculation of sectoral VAR

expectations, even for infinite forecast horizons, is significantly less computationally demanding

than full-model expectations. The matrix manipulations19 required for VAR expectations typically

require only a few seconds on a Sparc10 computer, whereas a standard forecast simulation using

full-model expectations can take 25-30 minutes. In the example of the bond rate, equation 3, the

funds rate and output gap are explicit variables in the historical VAR. The resultant funds rate and

output gap forecasts will be the same for all sectors since all sectors use the same VAR model to

generate expectations.

4.2 Long-run expectations.

As noted in previous sections, long-horizon forecasts of the federal funds rate and inflation

rate are important determinants of the equity price and bond rates in the current period.

Thus, long-horizon expectations constitute a significant policy transmission channel in FRB/US.

Short-run policy actions, such as fund rate alterations, can have very different effects depending on

how the funds rate movements influence private sector perceptions of long-run policy objectives.

Long-horizon expectations, succinctly called expectation endpoints in FRB/US, are generally

defined by the steady-state values of equilibrium planning equations. However, the endpoints of the

nominal funds rate and consumer inflation rate are explicitly defined in FRB/US, in part to indicate

the role of policy in determining these variables but also to better capture historical shifts in private

sector perceptions of the endpoints of these variables. These shifting endpoint expectations apply

to both full-model expectations and VAR expectations.

A default assumption in many RE models is symmetric policy information, where the plans of

policymakers are known by all sectors of the economy. By contrast, asymmetric policy information

is also an option in FRB/US, where private sectors either do not know policy objectives or are

sceptical of policy announcements. Explicit endpoint expectations allow the often fuzzy topic

18Another option under development for VAR expectations is a virtual VAR, which is a miniaturization of any closed

version of FRB/US. If a hypothetical alteration in the full model changes dynamic relationships among summary

aggregates included in the VAR, the reduced-form implications of these changes are captured by the virtual VAR.

Virtual or “mapped” VARs are iteratively estimated from the outcomes of full model simulations using virtual VAR

expectations. This expectations option is very computer-intensive and the subject of ongoing work.

19These are simple manipulations of a matrix containing the VAR coefficients. The VAR model is transformed into

a first-order autoregressive format. A two-period forecast requires the square of the matrix; a three-period forecast

requires the cube of the matrix, and so on.
30

of policy “credibility” to be precisely framed in FRB/US. If endpoint expectations are based on

symmetric policy information, then the inflation endpoint of private sector forecasts will be equal

to the long-run inflation goal of a fully credible policy. If policy information is asymmetric then

the long-run inflation goal of policy is not known or believed and must be inferred by the private

sector from observable indicators, such as past rates of inflation.

Endpoints are defined, usually implicitly, in conventional macroeconomic models as either

constants (if variables are detrended) or moving averages (if variables contain a random walk

component). The differences in summary statistics and one-period prediction errors between

equations estimated with these two endpoint alternatives are often small or insignificant, but

profound differences appear in long-horizon forecasts, as shown below. A third category of shifting

endpoints, developed for FRB/US, draws on private sector perceptions of long-run expectations.

The remainder of this subsection indicates typical effects of alternative endpoints on long-horizon

forecasts and reviews the historical measurements used in FRB/US to represent shifting endpoint

perceptions of firms, households, and investors.

Constant and moving-average endpoints. To illustrate long-horizon forecast effects of the two

endpoints used in conventional forecast models, alternative forecasts of the funds rate are displayed

in the two panels of figure 2. Forecasts in the top panel are generated by a four-lag autoregression

in the level of the funds rate. This model is appropriate if the variable is without a trend (stationary).

Note that both forecasts shown in the top panel, one starting from the high level of interest rates in

1980:Q1 and the other starting from the relatively low level in 1986:Q4, converge to and remain

at a common forecast. This constant endpoint is approximately the mean of the funds rate in the

sample used to estimate the autoregression in the level of the funds rate.

Forecasts in the second panel of figure 2 are generated by a four-lag autoregression in the

first-difference of the funds rate. This model is often selected if the variable contains a random

walk component (making it nonstationary).20 In this panel, each forecast rapidly converges to

and remains at a constant that is near the level of the funds rate at the start of the forecast. This

is because, in a first-difference autoregression, the endpoint moves over time and is defined by a

weighted moving average of funds rates in the periods immediately prior to the start of the forecast.

A characteristic of the forecasts in both panels of figure 2 is that endpoints are reached rather

quickly. Typically, in linear forecasting models of the funds rate, the funds rate endpoint is reached

by the fifth or sixth year of the forecast horizon.21 Consequently, in the case of long-maturity

20Many empirical studies in macroeconomics and macrofinance assume postwar nominal interest rates and inflation

rates contain random walk components.

21Obviously, closure is faster for forecasts that start near the endpoint and slower for initial forecasts that are farther

away, such as the forecast in the top panel of figure 2 that begins in 1980:Q1.
31

Figure 2: Autoregressive funds rate forecasts with alternative endpoints

1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 2

4

6

8

10

12

14

16

18

20

22

CONSTANT ENDPOINTS

RFF

1976 1978 1980 1982 1984 1986 1988 1990 1992 1994 2

4

6

8

10

12

14

16

18

20

22

MOVING AVERAGE ENDPOINTS

RFF

bonds such as the 10-year bond equation discussed in section 2, the selection of the expected

funds rate endpoint will determine much of the variation in the predicted bond rate. As can be

inferred from the behavior of the funds rate predictions in figure 2, bond rate predictions from

the autoregression in the level of the funds rate understate historical bond movements because the

long-horizon forecasts of the funds rate converge to the fixed endpoint. On the other hand, bond

rate predictions from the autoregression in the first-difference of the funds rate overstate historical

bond rate variation because the moving-average endpoint is too sensitive to recent levels of the

funds rate.

A shifting endpoint for the funds rate. The combination of the rational expectations assumption

that bond rates are an average of expected funds rates over the appropriate maturity horizon, as

discussed in section 2, and the fact that forward rates in the second five years of a 10-year bond

appear to be dominated by the funds rate endpoint suggests that an average of distant forward rates

in the observed term structure will provide a direct estimate of investors' time-varying perceptions

of the funds rate endpoint. The historical time series in FRB/US of the perceived funds rate

endpoint is based on forward rates in the 10-30 year segment of the term structure. For future
32

reference, this expectations endpoint for the funds rate is denoted by the subscript convention, r1.

The empirical fit of bond rate equations using this endpoint, r1, is substantially better

than those using fixed or moving-average endpoints.22 However, use of this shifting endpoint

measurement merely postpones the task of selecting a model that can predict investor perceptions

of the nominal interest rate endpoint.

A shifting endpoint for inflation. By the Fisher identity, the expected endpoint of the nominal funds

rate is a weighted average of the expected real rate endpoint (determined by the marginal product

of capital) and the expected inflation endpoint.23 Thus, a source of sizeable movements in the

nominal interest rate endpoint is the shifting of investor perceptions of the endpoint of expected

inflation. Although survey estimates of individual perceptions of the inflation endpoint such as

the Philadelphia survey of 10-year inflation expectations are available in recent years, none are

available prior to the major shift in policy in late 1979. In order to estimate a longer historical

series and to provide a behavioral description of investors' evolving perceptions of endpoints,

Kozicki and Tinsley (1996) develop an investor learning model where individuals sequentially test

for statistically significant shifts in the endpoint of expected inflation, given a null hypothesis

of an unchanged endpoint. Learning is nonlinear with faster responses to signals of a large

change in the inflation endpoint than to signals of small changes. However, movements in the

aggregate perception of the inflation endpoint are smoothed because the rate of learning varies

among individuals in the economy. The inflation endpoint constructed by this learning model is

quite similar to the (discontinued) Hoey survey estimates of inflation expected in the second five

years of a 10-year horizon and accounts for most of the sample variability of the shifting nominal

interest rate endpoint. The Kozicki-Tinsley series of the inflation rate endpoint is spliced with the

Philadelphia estimate of expected 10-year inflation to provide the FRB/US historical estimate of

the inflation rate endpoint, 1, perceived by the private sector; note that this need not be equal to

the long-run policy target for inflation.

4.3 The historical VAR.

Under VAR expectations, all sectors share a condensed description of the aggregate economy

represented by a three-variable vector autoregression in aggregate output, inflation, and the federal

funds rate (where the latter is selected as a summary indicator of monetary policy). The historical

22Explicit contrasts of bond rate predictions from funds rate forecast models with alternative endpoints are presented

in Kozicki and Tinsley (1996).

23Under the assumption that investors arbitrage after-tax real rates, adjusted for differential risk premia, the weights

are functions of the tax rate on investor earnings.
33

VAR is an average-history estimate of reduced form relationships among these three variables over

the 32 year sample beginning in 1963.

Estimated equations of the historical VAR are listed in table 8 for the three summary

aggregates: the federal funds rate, r; consumption inflation, ; and the trend deviation in aggregate

output (the “output gap”), x~.

Table 8: The Historical VAR

r = .03 lag1(  1) + .12 lag1(x~  0) - .05 lag1(r  r1) SEE 1.14

+ .33 lag3() + .22 lag3(~x) - .27 lag3 (r). R2.30

 = - .17 lag1(  1) + .13 lag1(x~  0) - .01 lag1(r  r1) SEE 1.13

- .27 lag3() - .17 lag3(~x) + .02 lag3 (r). R2.26

~x = -.02 lag1(  1) - .04 lag1(x~  0) - .21 lag1(r  r1) SEE 1.12

+ .09 lag3() + .19 lag3(~x) + .08 lag3 (r). R2.33

remarks:  span 1963q1 - 1994q4.

definitions: r - federal funds rate.

 - inflation rate of personal consumption deflator (chain weights).

x~ - trend deviation of output.

The equations in table 8 differ from those in conventional VARs due to the presence of explicit

endpoints for each of the variables in the historical VAR. As discussed above, each endpoint

represents private sector perceptions of the long run outcome for that variable. The format of

the equations in the historical VAR enforces this view where, intuitively, all terms on the left-hand

and right-hand sides of the VAR equations are zero in the long run. Thus, in the long run, the funds

rate will reach the funds rate endpoint, r1; the inflation rate will attain the inflation rate endpoint,

1; and the output gap will converge to its endpoint, which is zero.24

In contrast to analysis of individual equations, the average speed of adjustment of variables

toward long run values in a fully interdependent system, such as the historical VAR, is the same

for all variables in the system. This is evident in table 8, since each endpoint deviation appears in

24Further discussion of endpoint-deviation formulations of VARs may be found in Brayton and Tinsley (1995) and

Kozicki and Tinsley (1996).
34

all equations; thus, if one endpoint deviation is nonzero, then all endpoint deviations generally will

be nonzero.

Unlike the structural equations in FRB/US, the equations in VAR models are reduced forms so

direct behavioral interpretations are ordinarily not possible. However, under the assumption that

the funds rate is the variable most responsive to “news”, the funds rate equation may be interpreted

as an average-history representation of policy responses to shocks and observed movements of

inflation and output. Under this interpretation, the first equation in table 8 indicates that historical

policy generally increased the funds rate if either inflation or output were above their endpoints or

the funds rate was below its endpoint.25

5 Full-System Properties

This section provides an overview of the system properties of FRB/US and demonstrates

through a few examples how the model can be used to analyze a rich set of forecast and policy

questions, including cases where the public has imperfect knowledge of policy objectives or where

monetary or fiscal policy lack credibility. As discussed in previous sections, the dynamic behavior

of sectors depends significantly on expectations of households, firms, and financial markets,

including anticipations of policy. Assumptions about how expectations are formed, such as the

scope of information or the speed of learning, can be tailored in FRB/US to address a wide range

of issues.

Unless otherwise indicated, simulations in this section assume that monetary policy responds

to economic conditions according to the VAR equation for the federal funds rate in table 8 that

reflects the average-sample behavior of historical policy. In figures below, this simulated monetary

policy is referenced as the average historical policy. To highlight the role of private sector

expectations in the transmission of policy effects, simulation results are shown for two cases of

expectations formation. Under VAR expectations, firms and households use the same estimated

VAR to generate forecasts of the future that was used in estimating the FRB/US equations. Under

full-model expectations, expected values of future variables equal the values forecast by the full

FRB/US model.

5.1 System responses to transitory shocks

To provide a brief introduction of the system properties under different assumptions regarding

expectation formation, the initial simulations indicate responses of the economy to transitory

25Note this interpretation is based on symmetric policy information under the implicit assumption that the endpoints

of operational policy in the funds rate equation are the same as those perceived by private sectors in the remainder of

the model.
35

shocks in the federal funds rate and in aggregate demand (government spending). Beyond showing

representative dynamic responses of key macroeconomic variables in FRB/US, these exercises

demonstrate under what conditions VAR expectations differ from full-model expectations and how

such differences affect macroeconomic outcomes. The general rule that can be extracted from

these (and similar) simulations of transitory shocks is:

Rule of Thumb 1 For transitory shocks, the dynamic response of most main macroeconomic

aggregates differs between VAR expectations and full-model expectations to the extent the shocks

applied to the system deviate from average historical experience.

VAR expectations are based on the historical behavior of macroeconomic aggregates. For

shocks that are not unusual in an historical perspective, the summary macroeconomic responses

provided by the VAR contain most of the information needed to predict the responses of the full

FRB/US model. In contrast, when the simulation experiment strays from the typical pattern of

shocks in the economy, differences between VAR and full-model expectations emerge.

Figure 3 shows the responses of inflation, output, the federal funds rate, and the 10-year

government bond rate to a one-quarter, 100-basis point positive shock to the VAR equation for

the federal funds rate.26 In this and most of the following figures, the economy's response under

full-model expectations is shown as the solid line, while the dashed line represents the results under

VAR expectations. Results in all instances are displayed as deviations from a baseline forecast. As

the figure shows, the presence of lagged endogenous variables in the funds rate equation amplifies

the initial impulse, and the funds rate stays above baseline for about two years. The higher level

of the funds rate brings about a decline in aggregate demand and downward pressure on prices.

Although not shown in the figure, eventually all variables return to their baseline values as the

effects of the shock wear off.

26In order to simplify the design of the transitory shocks, each simulation of this type is based on the assumption

that the long-run inflation objective of monetary policy is unchanged, as are private perceptions of the policy objective.
36

Figure 3

One-quarter, 100-basis-point shock to the federal funds rate

Average Historical Policy

(deviations from baseline, per cent)

full-model expectations (solid); VAR expectations (dashed) 

-0.40

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

02 4 6 8 years

consumption price inflation

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

02 4 6 8 years

output gap

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

02 4 6 8 years

nominal federal funds rate

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

02 4 6 8 years

10-year government bond rate

The responses to the interest rate shock under full-model and VAR expectations are quite

similar, except for the response of inflation which is somewhat damped under full-model

expectations. The relatively minor differences between the dynamic responses under the two types

of expectations can be traced to two sources. First, the dynamics of the VAR model, while close,

are not identical to the dynamics of the full FRB/US system in this instance, introducing some bias

in VAR expectations. To some degree, agents are not using the right model to form expectations of

future events. Second, a one-period difference in timing of some responses is due to the assumption

under VAR expectations that anticipations generally are formed at the beginning of the quarter and

do not depend on contemporaneous information, while full-model expectations implicitly take into

consideration observations in the current period.
37

Figure 4

Four-quarter shock to government spending equal to 1% of GDP

Average Historical Policy

(deviations from baseline, per cent)

Full-model expectations (solid); VAR expectations (dashed)

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

02 4 6 8 years

consumption price inflation

-1.0

-0.5

0.0

0.5

1.0

1.5

02 4 6 8 years

output gap

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

02 4 6 8 years

nominal federal funds rate

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

02 4 6 8 years

10-year government bond rate

Figure 4 shows the response to an increase in government purchases, equal to one percent of

GDP, lasting one year. In this experiment, an additional source of expectations bias is introduced

into VAR expectations: The hypothesized duration of the demand shock does not correspond to

the average historical serial correlation of output. Historically, deviations of output from potential

tend to amplify themselves initially and then to die out gradually. Under VAR expectations, firms

and households do not “see through” the four-quarter design of the shock and expect output (and

inflation) to follow typical historical patterns. The perception of a high level of future activity and

inflation drives up prices today. Once the shock ends, expectations of future activity and inflation

are revised downward, dampening wage and price inflation. By contrast, under figure 4's example

of full-model expectations, firms and households have perfect foresight about this shock and know

the spending impulse will only last one year. This leads to a smaller rise in the bond rate and

no discernable rise in inflation. In fact, for nearly all of the first eight years, inflation is below

baseline. The fall in inflation is due to effects of future low activity and inflation and also to effects

of the appreciation of the exchange rate caused by the rise in bond yields. Agents make ex post

expectational errors using VAR expectations; but, if the true duration of the shock is not known
38

beforehand, the responses under VAR expectations may be considered reasonable.

The government purchases scenario shown in figure 4 demonstrates how deviations of

hypothetical shocks from historical behavior introduce a wedge between VAR expectations and

full-model expectations. Figure 5 illustrates another way differences can arise between the two

types of expectations—a shift in the actual responsiveness of monetary policy to output and

inflation deviations that is not reflected in VAR expectations, at least over the simulation interval.

The policy used in this simulation is one estimated over the shorter sample period from 1979 to

1995 (termed the post-1970's policy) and is more aggressive in combatting output and inflation

deviations than is the average historical policy.

Figure 5

One-quarter, 100-basis-point shock to the federal funds rate

Post-1970s Policy

(deviations from baseline, per cent)

full-model expectations (solid); VAR expectations (dashed) 

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

02 4 6 8 years

consumption price inflation

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

02 4 6 8 years

output gap

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

02 4 6 8 years

nominal federal funds rate

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

02 4 6 8 years

10-year government bond rate

The simulations reported in figure 5 repeat the interest rate increase shown in figure 3, but

with actual monetary policy determined by the average post-1970s policy. In this example, the

revised policy is captured by full-model expectations but not by VAR expectations. Differences

in responses under VAR expectations from those under full-model expectations are larger than

in figure 3, because an additional source of expectations bias has been introduced due to the
39

misperception of monetary policy under VAR expectations, which continue to be based on the

average historical characterization of monetary policy. In particular, under VAR expectations the

federal funds rate is anticipated to persist at an elevated level far longer than the actual policy

entails. This leads to an overly pessimistic view of future output and prices, and an exaggerated

response of output and inflation to the shock.

Figure 6

Anticipated future four-quarter shock to government spending

Average Historical Policy

(deviations from baseline, per cent)

Full-model expectations (solid); VAR expectations (dashed)

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

02 4 6 8 years

consumption price inflation

-1.0

-0.5

0.0

0.5

1.0

1.5

02 4 6 8 years

output gap

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

02 4 6 8 years

nominal federal funds rate

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

02 4 6 8 years

10-year government bond rate

Finally, implications of perfect foresight are indicated by contrasting the effects of a future

shock that is foreseen in one case and unexpected in the other. Figure 6 shows the same four-quarter

government purchases shock as in figure 4, with the exception that it occurs one year in the future

(at the date designated by the vertical line). For full-model expectations, the shock is assumed to be

foreseen in advance whereas under VAR expectations it is unexpected and there is no reaction until

the shock occurs. In both cases, monetary policy follows the average historical policy. For VAR

expectations, the responses are identical to those in figure 4 aside from a one-year delay. Under

full-model expectations, inflation and output rise with the announcement of the future spending

increase. The initial rise in inflation is due to labor and product markets that are foreseen to
40

be tighter in the future. Once the fiscal expansion is in full swing, however, inflation begins to

fall because of the anticipated weakening of output after the end of the temporary increase in

government spending. Also, long bond rates rise upon the announcement of the policy change

under full-model expectations, due to the anticipated rise in the federal funds rate.

5.2 System responses to permanent shocks

Thus far, simulations have illustrated the effects of expectation formation on system dynamics

for temporary disturbances. Discussion now turns to analysis of a permanent change in monetary

policy, where the experiment is a policy that aims to reduce the inflation rate permanently by one

percentage point within ten years. Any number of funds rate paths can achieve this objective; in

the simulations that follow, the funds rates set by policy are consistent with the planned reduction

in inflation but otherwise respond to movements in observed output and inflation using the average

post-1970's responses discussed above. We consider two cases of policy credibility. In the first

case of “perfect credibility,” the private sector fully believes that the announced disinflationary

policy will occur as planned. In the second case of “learning,” the private sector only slowly

adjusts its views about the probability that the full disinflationary program will be carried out. In

the latter case, the rate of adjustment in the inflation endpoint is 5% per quarter, so that long-run

inflation expectations will have fallen by one-half of one percentage point after 3 1/2 years.27 These

simulations provide the basis for a second rule of thumb:

Rule of Thumb 2 The cost of disinflation, in terms of lost output and employment, is decreasing

in the degree of credibility of the policy.

Figure 7 shows the consequences of a credible policy of disinflation. As with the transitory

monetary policy shift of figure 3, there is little difference between the outcomes under VAR and

full-model expectations. With the change in the policy taken as known, the information contained

in the VAR is sufficient to understand the responses of the full FRB/US model. Sacrifice ratios

(cumulative annual increase in the unemployment rate divided by the percentage point decrease in

the inflation rate) are also similar. For VAR expectations, the sacrifice ratio is 1.3; for full-model

expectations it is 1.7.28

27This rate of inflation endpoint learning is consistent with the fall in long-run expectations measured by surveys

during the disinflation of the 1980's.

28Sacrifice ratios are computed at the end of the tenth year of the simulations.
41

Figure 7

Permanent disinflation of one percentage point

Post-1970s Policy

(deviations from baseline, per cent)

Full-model expectations (solid); VAR expectations (dashed) 

Instantaneous recognition of change in target inflation

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

02 4 6 8 years

consumption price inflation

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

02 4 6 8 years

output gap

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

02 4 6 8 years

nominal federal funds rate

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

02 4 6 8 years

10-year government bond rate

The assumption of perfect credibility of the disinflationary policy is removed in figure 8.29

Inflation declines more gradually in this case, as the dampening effects of the credible policy on

expected inflation are attenuated in the case of gradual learning. Also, the rapid decline in bond

rates in the case of perfect credibility is absent under imperfect credibility. Bond traders, like all

agents in the economy, only gradually adjust their views about the long-run objectives of policy.

The higher real interest rates generated by this disinflationary policy lead to losses of output that are

significantly greater than those under perfect credibility. In terms of the sacrifice ratio, the effect

of imperfect credibility is to increase the cost of disinflation from 1.3 to 2.6 for VAR expectations

and from 1.7 to 2.3 for full-model expectations.

The credibility of monetary policy is not the only aspect of policy that affects the output and

employment cost of disinflation in FRB/US. Also important is the speed at which policy attempts

to reduce inflation, with the cost being higher the faster is the desired reduction

29For ease of comparison, the figure also repeats the simulated responses found under full credibility. Unlike the

other figures, the federal funds rate is plotted in real terms in figure 8.
42

Figure 8

Disinflation, with and without learning

Post-1970s Policy

(deviations from baseline, per cent)

Full-model expectations, instantaneous recognition (thick solid)

Full-model expectations, learning, 5% rate (dashed) 

VAR expectations, instantaneous recognition (dotted)

 VAR expectations, learning, 5% rate (thin solid)

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

02 4 6 8 years

consumption price inflation

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

02 4 6 8 years

output gap

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

02 4 6 8 years

real federal funds rate

-1.4

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

02 4 6 8 years

10-year government bond rate

in inflation. In a model where inflation depends only on past observations, the cost of a permanent

disinflation is invariant to the speed at which the disinflation occurs. In a model where inflation

depends on the expected future values, this invariance disappears. A credible policy to reduce

inflation that affects variables in the future will also affect the present. Thus, an effective way of

reducing current inflation at little cost in terms of lost output is to “announce” a restrictive future

policy that reduces expectations of inflation. The private sector is not being “fooled” because

policy must generate a reduction in output below potential at some point to be consistent with

the reduction in inflationary expectations. The key is that much of the reduction in inflation is

accomplished through lower inflation expectations, as opposed to operating only through reduced

aggregate demand. The effect of the speed of disinflation on the sacrifice ratio is illustrated

by comparing outcomes for the two “average” descriptions of historical policy. Under VAR

expectations with full endpoint credibility, the sacrifice ratio falls from 1.3 to 1.1 if the more

gradual average historical policy is substituted for the average post-1970's policy used in figures 7

and 8.
43

Even though FRB/US is an empirically estimated model with well-behaved statistical

properties, the simulations reported in this section demonstrate that the structural design of

FRB/US is suitable for analyses aimed at a broad range of macroeconomic policy questions.

The model has the flexibility to examine policy issues under different assumptions about policy

credibility and the extent of economic information upon which expectations are based.

A Appendix: Testing the Theory

The goodness of fit of the main structural equations in the model is summarized by the

proportions of explained variation, R2, and the standard deviations of equation residuals, SEE,

reported in tables 1 through 7 in the main text. This appendix presents two additional empirical

tests directed at assessing the adequacy of the theoretical specifications of dynamic adjustments

and the assumption of rational expectations.

The first is a test for serial independence of the residuals to determine if the generalized

adjustment cost specifications are able to describe dynamic behavior adequately or if significant

correlations in the data remain unexplained. The test in the first column of numbers in table A1

indicates the significance of autocorrelations of an equation residual with any of its first twelve lags.

The entries in this column are rejection probabilities (p-values) of the null hypothesis of serially

independent residuals. A p-value of .05 (.01) or less indicates rejection of the serial independence

hypothesis with at least a 95% (99%) level of confidence. The entries in this column suggest that

eight (ten) of the twelve equations examined have white noise residuals.

The second test examines coefficient restrictions imposed by the VAR-based implementation

of rational expectations (RE). Using the example of the aggregate price equation, the FRB/US

price equation presumes that firms use the VAR to generate predictions of the equilibrium price.

These predictions are then weighted by the lead response weights of the price equation (shown

earlier in figure 1) to determine the estimated price change in the current period. The potential

information in the VAR consists of lagged values of all variables included in the VAR model.

Although this information is organized by the VAR to produce minimum mean square errors in

predicting the equilibrium price, it may be that firms prefer to organize this information in some

other way, such as rule-of-thumb extrapolations. The test in the second column of numbers in table

A1 augments the FRB/US dynamic adjustment equation with lagged values of the variables in the

sectoral VAR as additional regressors. If the additional VAR regressors are statistically significant

then the p-values in the second column will be low, indicating that firms are not using rational

expectations (at least as defined by VAR forecasts) in their dynamic adjustment equations. Again,

a p-value of .01 or less indicates rejection of RE restrictions with at least a 99% level of confidence.

The entries in the second column suggest that rational expectations restrictions are not rejected for
44

ten of the twelve equations examined.

Table A1: Tests for Serially Independent Residuals and RE Restrictions a

serially independent RE

equation residuals restrictions

aggregate consumption .24 .30

consumer durables, motor vehicles .28 .12

other consumer durables .28 .19

residential investment .01 .01

producers' durable equipment .67 .72

inventory investment .24 < .01

labor hours .02 .09

aggregate price .20 .71

wage .04 .27

5-year Treasury bond rate .01 .78

10-year Treasury bond rate .07 .68

corporate bond rate .37 .91

atable entries are rejection probabilities (p-values); a low p-value indicates rejection of a null

hypothesis (white noise residuals or RE restrictions).
45

B REFERENCES

Brayton, F. and P. Tinsley, “Polynomial Generalization of Dynamic Frictions in Structural Macro

Models,” FRB staff working paper, June 1995.

Campbell, J. and R. Shiller, “The Dividend-Price Ratio and Expectations of Future Dividends and

Discount Factors,” Review of Financial Studies, 1(3), Fall 1989, 195-228.

Engel, R. and C. Granger, “Cointegration and Error Correction: Representation, Estimation, and

Testing,” Econometrica, 55(2), March 1987, 251-76.

Kozicki, S. and P. Tinsley, “Specification and Estimation of Rational Error Correction Models,”

FRB staff working paper, December 1995.

Kozicki, S. and P. Tinsley, “Moving Endpoints in the Term Structure of Interest Rates,” FRB staff

working paper, May 1996.

Reifschneider, D., “Household Consumption and Investment in the FRB/US Model”, August 1996.

Rotemberg, J., “The New Keynesian Microfoundations,” S. Fischer (ed.), NBER Macroeconomics

Annual 1987, Cambridge: MIT Press, 1987, 69-104.

Sargent, T., Bounded Rationality in Macroeconomics, Oxford, Clarendon Press, 1993.

Sims, C., “Macroeconomics and Reality,” Econometrica, 48(1), January 1980, 1-48.

Tinsley, P. “Fitting Both Data and Theories: Polynomial Adjustment Costs and Error-Correction

Decision Rules,” FRB FEDS Working Paper 93-21, 1993.

von zur Muehlen, P., “Lead and Lag Weight Distributions in the FRB/US Model,” FRB staff

working paper, May 1996.