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Growth Accounting with
Technological Revolutions
Andreas Hornstein

G

enerally, technological progress proceeds at a slow and measured
pace, with only incremental improvements seen in existing products
and technologies in the economy. At times, however, the pace accelerates, and the economy experiences a technological revolution during which
radically new products and technologies are introduced. Recent discussions suggest that the world economy is currently experiencing just such a revolution,
or paradigm shift, and that this revolution accounts for some of the observed
decline and rebound of productivity growth. For example, David (1991) argues
that the effect of information technologies on today’s economy is comparable
to the effects of the introduction of the dynamo and the subsequent availability of electric power in the late-nineteenth and early-twentieth centuries.
It is important to understand the effects of technological progress as reflected
in productivity growth because productivity growth determines the economy’s
long-run growth of output, consumption, and factor income such as wages.
In this article I consider one particular parable of a paradigm shift. This
story builds on three assumptions: first, that technological change is associated
with the introduction of new goods, in particular that new technologies are embodied in new machines; second, that production units learn about the newly
introduced technologies, that is, new technologies do not immediately attain
their full productivity potential, but instead productivity increases gradually for
some time; and third, that the experience which production units have with
existing technologies affects their ability to adopt new technologies.1
Andreas.Hornstein@rich.frb.org. I would like to thank Lawrence Santucci for research assistance and Mike Dotsey, Pierre Sarte, and Roy Webb for helpful comments. The views
expressed are the author’s and not necessarily those of the Federal Reserve Bank of Richmond
or the Federal Reserve System.
1 The ideas expressed in this article are based on work by Greenwood, Hercowitz, and
Krusell (1997), Hornstein and Krusell (1996), Greenwood and Jovanovic (1998), and Greenwood
and Yorukoglu (1997).

Federal Reserve Bank of Richmond Economic Quarterly Volume 85/3 Summer 1999

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Federal Reserve Bank of Richmond Economic Quarterly

In the following pages I summarize the available evidence in support of
these assumptions and then speculate on the possible implications of a paradigm
shift for future output and productivity growth based on a parametric version
of the standard neoclassical growth model. I find that all three assumptions
together can account for a substantial and long-lasting decline in measured
productivity and output growth during the initial stages of a technological revolution. This initial period is then followed by a long period of above-average
long-run growth. Unfortunately, the results depend crucially on how experience with existing technologies affects the ability to adopt new technologies,
a feature of the economy about which we know very little. An alternative
parameterization of this feature of the economy predicts that the effects of a
technological revolution on productivity and output growth might be negligible.
Finally, I reconsider the evidence on the slowdown of measured productivity
growth and find that it appears to be less dramatic if we calculate real output
numbers using a more reliable price index.

1.

SOME EVIDENCE ON TECHNOLOGICAL CHANGE

The Rate of Capital-Embodied Technological Change has
Accelerated in the Early ’70s
When people talk about a new technological revolution, they usually refer to
the more widespread use of computers: the application of computers makes
new products and services possible, it changes the way production processes
are organized, and it is no longer limited to a small fraction of the economy.
Unfortunately, many of these observations are anecdotal and provide only limited quantitative support for the impact of computers on the economy. There is
one observation, however, that we all make and that might well be quantified;
namely, that each new generation of PCs tends to do more things faster than
the previous generation, yet we do not have to pay more for these higherquality PCs. In short, for PCs the price-per-quality unit has been declining at
a dramatic rate. This observation applies not only to PCs but to many other
products, particularly producer-durable goods such as new capital goods.
While it is easy to say that new products are of better quality, it is difficult
to actually measure and compare quality across different goods. In an extensive
study, Gordon (1990) has constructed measures of the price of producer-durable
equipment that account for quality changes. The line labeled 1/q in Figure 1
graphs the price of new producer-durable equipment relative to the price of
nondurable consumption for the postwar U.S. economy.2 I identify the rate of
2 The series on the relative price of producer-durable equipment is from Greenwood, Hercowitz, and Krusell (1997). I have extrapolated the series from 1990 on using information on the
price of producer-durable equipment from the National Income Accounts. Consumption covers
nondurable goods and services, excluding housing services. Hornstein (1999) provides a complete
description of the data used.

A. Hornstein: Growth Accounting with Technological Revolutions

3

Figure 1 Measures of Embodied and Disembodied Technological Change

price decline with increased productivity in the capital goods producing sector
that is embodied in the new capital goods.3 In this figure it is apparent that
producer-durable equipment goods have become cheaper over time relative to
consumption goods and that the rate of price decline has accelerated in the
mid-’70s from 3 percent before 1973 to 4.3 percent after 1977. A substantial
part of the accelerated rate of price decline can be attributed to the fact that information technologies have gained more widespread application in the design
of producer-durable equipment.
Learning-by-Doing is an Important Feature of Production
New products or new plants do not attain their full potential at the time they
are introduced. Rather, we find that for some period of time productivity for
3 In general, relative prices may change because the technology changes (shift of the production possibility frontier, PPF) or because of simple substitution between goods (movements
along a PPF). Notice, however, that with an unchanged technology we would expect the relative
price of a good to fall only if relatively less of the good is produced. Yet we have not observed
a decline in the investment rate that should correspond to the decline in the relative price of
capital. Work that tries to account for substitution effects finds even more acceleration in the rate
of capital-embodied technological change (Hornstein and Krusell 1996).

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Federal Reserve Bank of Richmond Economic Quarterly

a new good or plant is increasing. This increase in productivity is attributed
to learning-by-doing (LBD); that is, firms acquire experience and improve
their efficiency in resource use in the process of producing a good. One can
think of this process as the accumulation of informational capital. This LBD
phenomenon is so widespread and uniform across industries that the management literature summarizes it with the “20 percent rule,” according to which
labor productivity increases by 20 percent for every doubling of cumulative
production (see, e.g., Hall and Howell [1985]).
One of the most frequently cited LBD examples is the case of the liberty
ships of World War II. The more ships a navy yard built, the smaller was the labor input required for the next vessel it built (Figure 2). A more recent example
of LBD is the production of dynamic random access memory (DRAM) chips
in the semiconductor industry. Figure 3 displays the time paths for the average
unit price and total shipments of successive generations of DRAM chips. This
figure displays two common features of LBD. First, productivity improvements
during the early stages of production are dramatic. Second, these improvements
are attained within a short period of time, occurring within the first three to five
years of production. Indeed, most of the productivity improvements have been
made once shipments of a chip generation reach their peak. Notice also that
during the first few years a new generation of chips is produced, the unit price is
higher than the one of the previous generation.4 The DRAM chip example also
points to an important feature for my discussion of accelerated capital-embodied
technological change: How much of the experience accumulated in the production of one generation of DRAM chips can be transferred to the production
of the next generation of chips? More generally, how much of the experience
accumulated for existing technologies can be applied to new technologies? The
answer to this question is still open. Evidence from the semiconductor industry
indicates that the transfer of experience is limited (Irwin and Klenow 1994).
New Technologies Diffuse Slowly Through the Economy
When a radically new technology becomes available, not everybody in the
economy will adopt this new technology simultaneously. For some time the
use of the old and new technology will coexist while firms continue to make
improvements in the old technology. This situation will occur since there are
costs to adopting new technologies such as learning costs. Potentially, a new
technology may be much more productive than the old technology, but initially
users of the new technology have to start with a low experience level relative
to that of old technologies.
4 My interpretation that a decline in average unit price reflects an increase in productivity
should be taken with a grain of salt since the market structure in the semiconductor industry is
only approximately competitive.

A. Hornstein: Growth Accounting with Technological Revolutions

5

Figure 2 Reductions in Man-Hours per Vessel with Increasing Production

Source: Lucas (1993).

The idea that new technologies diffuse slowly through the economy relates
to the observation that the use of new products diffuses slowly through the
economy. Experts have made this observation for a wide variety of products
from diesel locomotives to DRAM chips (Figures 3 and 4).5 David (1991)
makes a similar observation on the diffusion of the use of electrical power in
the late-nineteenth and early-twentieth centuries.

2.

TECHNOLOGY DIFFUSION IN A SIMPLE MODEL OF
CAPITAL-EMBODIED TECHNOLOGICAL CHANGE
AND LEARNING

The appearance of a new technology in the economy can significantly affect
output and productivity growth during the transitional period when the new
technology replaces the old technology. These effects come about because
learning introduces another kind of capital that is not measured, informational
5 Figure

3b plots shipments of DRAM chips from different generations, and Figure 4 plots
the numbers of diesels in use as a fraction of the total number of locomotives.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Dynamic Random Access Memory Semiconductors

Source: Irwin and Klenow (1994).

capital, and during the transitional period this capital stock can change significantly. This change in informational capital has real output growth effects, and
it creates a problem for measuring productivity growth.

A. Hornstein: Growth Accounting with Technological Revolutions

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Figure 4 Diesel Locomotion in the U.S. Railroad Industry,
1925–66: Diffusion

Source: Jovanovic and McDonald (1994).

Informational capital represents the economy’s experience with various
vintages of capital goods, and it is not part of our standard measure of capital.
Consequently, we do not measure changes of informational capital that occur
during transitional periods. In particular, after we correct for depreciation, we
assign the same value to capital from different vintages. So during transitional
periods when substantial investment in new technologies with lower experience
occurs, we tend to overestimate the contribution to output from investment in
these new technologies. Because we overestimate capital accumulation, we
underestimate total factor productivity growth. There is also a real effect of
learning, since output growth slows down in the transitional period. A feature
of this learning is that during the transitional period, production with new
technologies is relatively less efficient than production with old technologies.

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Federal Reserve Bank of Richmond Economic Quarterly

In the next section I will try to quantify the implications for output and
productivity growth measurement when a new technology is introduced in a
simple vintage capital model with learning. The structure of the model is very
mechanical and many of the elements discussed above are taken as exogenous.
The Solow Growth Model and Growth Accounting
I will start with the standard Solow growth model, which assumes a neoclassical
production structure and a constant savings and investment rate. Each period, a
homogeneous good yt is produced using a constant-returns-to-scale technology
with inputs capital kt and labor nt ,
yt = zt ktα n1−α ,
t

(1)

where the elasticity of output with respect to capital satisfies 0 < α < 1. For
simplicity I have assumed a Cobb-Douglas production function. Technological
change is represented through changes in total factor productivity zt and is
disembodied; that is, with the same inputs, output increases when total factor
productivity (TFP) increases. The economy’s endowment of labor is fixed,
nt = 1. The output good can be used for consumption ct or investment it :
ct + it = yt .

(2)

Investment is used to augment the capital stock and capital depreciates at a
constant rate δ:
kt+1 = (1 − δ)kt + it ,

(3)

and 0 < δ < 1. Expenditures on investment are assumed to be a constant
fraction σ of output,
it = σyt ,

(4)

and 0 < σ < 1.
Assume that TFP grows at a constant rate, zt+1 = γz zt and γz ≥ 1. It can
be easily verified that an equilibrium exists for this economy where output,
consumption, investment, and the capital stock all grow at constant rates. Such
an equilibrium is called a balanced growth path. For the following let gx denote
the gross growth rate of the variable x: that is, gx = xt /xt−1 . From the savings
equation (4), it follows that if both investment and output grow at a constant
rate, then they must grow at the same rate, gy = gi = g. In turn, the resource
constraint (2) shows that consumption must grow at that same rate gc = g.
Dividing the capital accumulation equation (3) by the capital stock kt subsequently shows that if the capital stock grows at a constant rate, it must grow at
the same rate as investment, gk = g. Finally, the production function (1) relates
the economy’s output growth rate to the growth of inputs and the exogenous

A. Hornstein: Growth Accounting with Technological Revolutions

9

productivity growth rate g = gy = γz gα = γz gα .6 From this expression one
k
can see that the economy’s growth rate on the balanced growth path increases
with the productivity growth rate and with the capital elasticity of output,
1/(1−α)
g = γz
.

(5)

We know that TFP in this economy is zt , but how can we measure TFP if
we do not observe zt ? In order to calculate the percentage change of TFP, take
the log of equation (1), take the first difference,7 and solve for the TFP growth
rate z,
ˆ
ˆ
zt = yt − αkt − (1 − α)ˆ t .
ˆ
ˆ
n
Here the measure of TFP growth requires observations on the growth rates of
output and inputs and knowledge of the parameter α. Solow’s (1957) important
insight was that in a competitive economy α can be measured through observations on factor income shares. Suppose that all markets in this economy are
competitive and that everybody has access to the technology represented by
(1). Then consider a firm that maximizes profits, sells the output good at a
price pt , and hires labor (capital) services at the wage rate wt (capital rental
rate ut ). In order to maximize profits, the firm will hire labor (capital) services
until the marginal revenue from the last unit of labor (capital) services hired
equals its price:
pt MPNt = pt (1 − α) zt ktα n−α = wt
t

(6a)

pt MPKt = pt αzt ktα−1 n1−α = ut .
t

(6b)

Multiplying each side of the equation with nt /pt yt (kt /pt yt ) shows that the labor
(capital) coefficient in the production function equals the share in total revenues
that goes to labor (capital):8
1 − α = wt nt /pt yt = snt
α = ut kt /pt yt = skt .
6 From equations (1), (3), and (4), it follows that a balanced growth path is associated with
a particular level of the capital stock in the initial period k0 . One can show that the economy
converges toward this balanced growth path if it starts with a different level of capital.
7 In the following, a hat denotes the net growth rate of a variable: for example, x = (x −
ˆt
t
xt−1 )/xt . For small changes in a variable, the first difference of the logs approximates the growth
rate; for example, xt = ln xt − ln xt−1 .
ˆ
8 Since the two coefficients sum to one, total payments to the two production factors capital
and labor exhaust revenues; that is, there are zero profits. This is not specific to the assumption
of a Cobb-Douglas production function. In general, profits are zero when production is constant
returns to scale and all markets are competitive.

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Federal Reserve Bank of Richmond Economic Quarterly

We can therefore measure productivity growth using observations on output
growth, input growth, and factor income shares. This measure of TFP growth
is the Solow residual:
ˆ
ˆ
ˆ
ˆ
zm = yt − skt kt − snt nt = zt .
ˆt

(7)

The Solow residual provides an accurate measure of disembodied technological
change not only for a Cobb-Douglas production structure but for any constantreturns-to-scale economy, as long as we are willing to assume that all markets
are competitive. Finally, note that the wage and capital rental rate equations (6a
and 6b) also imply that on a balanced growth path real wages wt /pt will grow
at the economywide growth rate g, which is determined by the productivity
growth rate, and that the real rental rate of capital is constant.
Capital-Embodied Technological Change
The secular decline of the relative price of producer-durable goods suggests that
a substantial part of technological progress is embodied in new capital goods. A
straightforward modification allows me to account for capital-embodied technological change in the Solow growth model. In the model described above the
homogeneous output good can be used for consumption or investment, and the
marginal rate of transformation between consumption and investment goods is
fixed. In particular I have assumed that one unit of the consumption good can
be transformed into qt units of the investment good and qt = 1. In order to
show that over time the economy becomes more efficient in the production
of capital goods, I simply assume that over time qt grows at a constant rate,
qt+1 = γq qt and γq ≥ 1. The resource constraint for the output good is now
ct + it /qt = yt .

(2a)

At the same time that the economy becomes more efficient in the production
of capital goods, the relative price of capital goods 1/qt declines. I continue to
measure output in terms of consumption goods and assume that expenditures on
investment goods in terms of consumption goods represent a constant fraction
of income:
it /qt = σyt .

(4a)

Analogous to the previous economy, there is a balanced growth path where
output, consumption, investment, and capital all grow at constant rates:
α
gy = gc = (γz γq )1/(1−α) and gi = gk = (γz γq )1/(1−α) .

(5a)

The measurement of TFP, that is, disembodied technological change, is
affected in two ways by the presence of capital-embodied technological change.
First, the capital stock measure is constructed as the cumulative sum of undepreciated past investment based on equation (3). Since changes in the quality of

A. Hornstein: Growth Accounting with Technological Revolutions

11

new capital goods are the hallmark of embodied technological change, we have
to use an appropriate price index that accounts for these quality changes when
we deflate nominal investment series to obtain real investment expenditures.
Second, because the relative price between consumption and investment goods
is changing over time, we have to decide whether we want to measure output
in terms of consumption or investment goods. Since ultimate well-being in the
economy depends on the availability of consumption goods, I decide to measure
output in terms of consumption goods. The line labeled z in Figure 1 displays
the measured TFP levels for the postwar U.S. economy. Here the measured
capital stock is adjusted for embodied technological change using data on the
relative price of durable goods.9 Notice that contrary to capital-embodied technological change, which was positive for all of the postwar period, measured
TFP does not represent a success story for the U.S. economy. Although TFP
was increasing rapidly in the late ’50s and ’60s, TFP stagnated in the early
’70s and has actually declined since the mid-’70s when the rate of embodied
technological change accelerated. Recently, starting in the ’90s, there has been
a slight recovery of TFP, but the apparent negative trend in the ’70s and ’80s
seems hard to rationalize.
Learning and Growth Accounting
The observed decline in measured TFP could simply be due to measurement
error; that is, there never was a decline in actual TFP. To make sense of this
explanation I provide a candidate for what has been mismeasured, and I argue
why the measurement problem got worse in the mid-’70s and why we now
observe a trend reversal. I suggest that the effective stock of capital has been
mismeasured. In particular, I consider the possibility that standard measures of
capital do not include informational capital in the economy. In the following I
introduce informational capital into the Solow growth model through a simple
model of learning. I show that even though measured capital does not include
informational capital, there is no measurement problem on the balanced growth
path; the measured capital stock may overestimate the effective capital stock
during transitional periods when there are significant changes in the economy’s
informational capital stock.
Assume that new capital goods do not immediately attain their full potential, but in the process of producing goods, more is learned about each capital
good and the efficiency with which it is used increases over time. We now have
to distinguish between different vintages of capital goods because a producer
has less experience with a capital good that is newly introduced than with a
capital good that has been around for some time. Let kt,a denote a capital good
9 The

measure of TFP is based on work by Greenwood, Hercowitz, and Krusell (1997) as
extended in Hornstein (1999). For a more detailed description see either of the two references.

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Federal Reserve Bank of Richmond Economic Quarterly

that is a years old at time t. If this capital good is employed with nt,a units of
labor, output yt,a is
α
yt,a = zt et,a kt,a n1−α ,
t,a

(1a)

where et,a is the experience index of a capital good that is a years old. For
simplicity I assume that maximal experience is one and convergence to it is
geometric at rate λ:
1 − et+1,a+1 = λ(1 − et,a ) for a = 1, 2, . . . ,

(8)

starting from some initial experience level 0 ≤ et,1 ≤ 1, and 0 < λ < 1.
I continue to assume that capital depreciates at rate δ:
kt+1,a+1 = (1 − δ)kt,a .

(3a)

Total output, employment, and investment are
∞

yt =

∞

yt,a , nt =
a=1

nt,a , and it = kt+1,1 ,

(9)

a=1

and I continue to assume that the markets for output, labor, and the different
capital vintages are all competitive. An attractive feature of this model is the
existence of an exact aggregate capital index. We can write aggregate output as
a Cobb-Douglas function of total employment and the aggregate capital index
¯
k :10
∞

¯ t
¯
yt = zt ktα n1−α , and kt =

e1/α kt,a .
t,a

(9a)

a=1

From this expression one can see how informational capital, et = {et,a : a =
1, 2, . . .}, affects aggregate output. Note that the usual measure of the
10 The aggregate capital index can be derived as follows. A profit-maximizing competitive
firm using vintage a capital goods hires labor until it equates the marginal revenue of labor with
α
its marginal cost, p(1 − α)zea ka n−α = w. Solving this expression for na defines the demand
a
for labor by firms using vintage a capital, na = [(1 − α)zea /(w/p)]1/α ka . The real wage w/p then
adjusts such that the total demand for labor is equal to the supply of labor:
∞

∞

na = [(1 − α)z/(w/p)]1/α

n=
a=1

¯
e1/α ka = [(1 − α)z/(w/p)]1/α k.
a
a=1

One can solve this expression for the equilibrium real wage, substitute it in the labor demand
¯
equation, and obtain the output of firms using vintage a capital as ya = ze1/α ka (n/k)1−α . Total
a
output is then
∞

¯
ya = zkα n1−α .

y=
a=0

A. Hornstein: Growth Accounting with Technological Revolutions

13

economy’s capital stock as the sum of undepreciated past investment does
not take into account the informational capital
∞
m
kt,a = (1 − δ)kt−1 + it−1 .

ktm =

(9b)

a=1

To close the model I identify what determines initial experience with a new
capital good. I assume that there is an externality, and experience with older
capital goods is partially transferrable to new capital goods according to the
following expression:
∞

et+1,1 =

θ
ρa−1 nt,a et,a ,
1 − ρ a=1

(8a)

with 0 < ρ < 1 and θ > 0. This formulation of the learning externality follows
Lucas (1993). The factor ρa measures the extent to which experience with vintage a contributes to initial experience with new capital goods. The larger that
ρ is, the more important is experience with existing capital goods. Since ρ < 1,
experience with older vintages is less important for the initial experience with
a new capital good. Notice also that I have assumed the contribution of vintage
a is weighted by how intensively this vintage is used, whereby I measure the
intensity of use by the share in employment.
The balanced growth path of this economy is very similar to the path of
the previous economy. Output, consumption, investment, and capital grow at
the same rates, and the initial experience e1 is constant. Because the initial
experience is constant, the informational capital does not change, eta = ea ,
and the exact aggregate capital index (9a) and the measured capital stock (9b)
grow at the same rate. Therefore, the Solow residual accurately reflects true
growth of TFP. If the economy is not on the balanced growth path, three things
happen. Initial experience and the informational capital changes over time,
changes in the measured capital stock do not accurately reflect changes in the
exact aggregate capital index, and the Solow residual mismeasures true TFP
growth.
The economy may not be on its balanced growth path for various reasons.
Here I consider the possibility that the acceleration of capital-embodied technological change in the mid-’70s was associated with a qualitative change in
the kind of technology used. Furthermore, the adoption of this new technology
proceeded gradually. To be more specific assume that at some time t0 this new
qualitatively different technology becomes available. From this point on I distinguish between vintages belonging to the old technology, i = 1, and vintages
belonging to the new technology, i = 2. This means that in any time period
t output, capital, employment, and experience are now indexed by the type of
i
technology i and its vintage a, {yit,a , kt,a , eit,a , nit,a }. I assume that the new technology is potentially better because capital-embodied technological progress

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Federal Reserve Bank of Richmond Economic Quarterly

2
1
proceeds at a higher rate for the new technology γq > γq and q20 = q10 . At
t
t
first, however, the new technology may be worse because the economy has
less experience with it. Since the new technology may be initially inferior, I
assume that the new technology diffuses slowly. In particular, only a fraction
ψt of total investment expenditures is used for the purchase of capital goods
with the new technology, and


= 0

ψt

for t < t0 ,
∈ (0, 1) for t = t0 + 1, . . . , t0 + T,

=1
for t > t0 + T,

(10)

and ψt increases monotonically. As before, initial experience ei for a new vintage of a technology i depends on the existing experience with older vintages
of that technology,
∞

eit+1,1 =

θ
ρa−1 nt,a eit,a .
1 − ρ a=1

(8b)

For completeness assume that the experience of a new technology vintage that
never existed is zero; that is, e2 = 0 for t − a < t0 .11
ta
In order to consider the quantitative implications of the diffusion of a new
technology, I select parameter values for the economy that are consistent with
observations on long-run growth, the evidence on the accelerated embodied
technological change, learning in the economy, and the diffusion of new technologies.
In the postwar U.S. economy, the average annual depreciation rate is about
10 percent, the average investment rate is about 20 percent, and the average
capital income share is about 30 percent. I assume that there is no disembodied
technological change such that we can interpret the output and measured TFP
growth rates as possible losses/gains due to the diffusion of a new technology. I
also assume that the new technology is implemented beginning in 1974 and that
it will take 40 years for all new investment to take the form of the new technology. This means that we have passed the midpoint of the diffusion process. The
parameterization of the diffusion process (T, ψt ) is consistent with observations
as discussed in Section 2. The rate of capital-embodied technological change
for the old and new technology corresponds to the average rate of decline for
the relative price of equipment before and after 1974. The parameterization of
the internal learning process (λ, e10 ,1 ) is based on Bahk and Gort (1993). We
t
know the least about learning externalities (ρ, θ). I simply assume that ρ = 0.8
and that in the years before 1974 the economy was on its balanced growth
path. With this observation I can recover the value of θ.
11 The

assumption that experience is not transferable across technologies is extreme, but
allowing for partial transferability changes the results insignificantly.

A. Hornstein: Growth Accounting with Technological Revolutions

15

Table 1 Parameter Values
Solow growth model

α = 0.3, δ = 0.1, σ = 0.20

Disembodied technological change

γz = 1.00

Capital-embodied technological
change

1
2
γq = 1.03 and γq = 1.04

Learning

λ = 0.7 and e1 ,1 = 0.8
t

Learning externality

ρ = 0.8 and θ = 12.11

Diffusion

T = 40 and ψt follows an S-shaped diffusion
(third-order polynomial)

0

The results are displayed in Figure 5. Panel a shows the gradual diffusion
of the new technology for investment and the capital stock. Since investment
adds to the existing capital stock, the diffusion of the new technology in the
total capital stock proceeds at a slower rate than it does relative to investment.
Panels c and d display measured TFP growth rates and output growth rates,
and we observe a long-lasting and substantial decline in measured TFP growth
and output growth (1 percentage point). This decline bottoms out in the mid’80s, and we are now in a recovery phase. According to this simulation we can
expect a considerable increase of the trend growth rates for measured TFP and
output for the next 20 years. Panel f shows that the effects of the lower output
growth are quite substantial in the sense that another 15 years have to pass
before the level of output catches up with the initial balanced growth path.12
Why do we get these big effects during the transitional period when the
new technology is adopted? The simple answer lies in the graph of initial
experience for the two technologies (panel b of Figure 5). Notice that initial
experience in the old technology is declining during the transitional phase. The
decline occurs because according to the specification of the learning externality
(8a), the contribution of a vintage is weighted by employment in that vintage.
During the transitional phase employment shifts from old to new technologies
and, with this learning specification, the economy tends to “forget” about the
old technology. Sizeable changes in output and measured TFP growth do not
occur, however, because investment in new technologies starts out with a low
experience; these changes make up only a small fraction of total investment
after all. Rather the big changes in output and measured TFP growth occur
because initial experience is falling for investment in old technologies, and this
investment contributes the most to total capital accumulation.
12 The results are sensitive with respect to technology spillovers ρ. If spillovers are unimportant (ρ = 0.5), then the decline in measured TFP growth is much more persistent, and output
growth does not overshoot very much.

16

Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 A Transition Path with Big Effects

A. Hornstein: Growth Accounting with Technological Revolutions

17

We can evaluate the importance of this effect by changing the specification of the learning externality (8a) such that we do not weight experience by
employment; that is, how much the experience of an old vintage contributes to
the initial experience of a new vintage is independent of how intensively the
old vintage is used:
∞

eit+1,1 =

θ
ρa−1 eit,a .
1 − ρ a=1

(8c)

The results of this change are displayed in Figure 6. Note that with this specification initial experience with the old technology remains constant at 0.8. As
we can see, the maximal reduction in measured TFP growth is now only 0.04
percentage points, as opposed to 1 percentage point previously, and there is
almost no decline in output growth; the maximal increase corresponds to the
balanced growth increase of about 0.5 percentage points.
I am not aware of any empirical work that has studied the quantitative
properties of the transfer of knowledge in the economy and that would allow
us to pick between the two learning specifications (8a) and (8c). Although I
find specification (8a) reasonable—in the sense that intensity of use should
matter for the transfer of knowledge—and although it is quite possible that an
economy “forgets” about old technologies if they are not used, I do not believe
that the process occurs as fast as implied by the specification above. If, as I
believe, the economy is not that forgetful, then specification (8c) may be a
good short- to medium-term approximation, and I would have to conclude that
the possible effects of a technological revolution are limited.

3.

RECONSIDERING THE MEASUREMENT OF
PRODUCTIVITY GROWTH

This article reviews the possible implications of a technological revolution
for the measurement of the U.S. economy’s productivity performance. I have
shown evidence for the acceleration of capital-embodied technological change
and at the same time a substantial decline of TFP, which represents disembodied technological change. I have argued that part of the decline in TFP can in
principle be attributed to a measurement problem associated with accumulating informational capital during a technological revolution. Unfortunately, the
process by which informational capital is accumulated in an economy is not
well understood, and any exercise that studies this aspect of the economy has
to be somewhat speculative in nature. I would like to conclude my discussion
of the U.S. economy’s productivity performance with one more observation.
Although this observation makes the description of productivity behavior even
more ambiguous, it seems to indicate that the performance of the U.S. economy
has not been as bad as Figure 1 suggests.

18

Federal Reserve Bank of Richmond Economic Quarterly

Figure 6 A Transition Path with Small Effects

A. Hornstein: Growth Accounting with Technological Revolutions

19

My discussion of the implications of a technological revolution has focused
on problems associated with the measurement of capital in a broad sense.
Part of the measurement problem is accounting for changes in the quality of
producer-durable goods, but for this part I have taken the view that Gordon’s
(1990) price index does account for most of the quality changes that occur for
producer-durable goods. I have also identified embodied technological progress
with the rate of decline of the price of producer-durable goods relative to consumption goods. At this point I should note that the quality of consumption
goods also changes over time, a process that in principle is no different from
that of producer-durable goods. But this means that for the construction of a
consumer price index one also has to be careful how one accounts for quality
change in new consumer goods. To the extent that our consumer price index
does not capture quality changes in goods, we will overestimate the rate of
price increase and underestimate the growth in real consumption.13
The diffusion of information technologies has certainly affected the quality
of consumer goods we are now able to purchase, an observation that is most
evident for consumer services. Take, for example, the services provided by the
financial sector: we are now able to obtain cash at conveniently located automatic teller machines, we can access our bank accounts and make transactions
from home, we can trade shares directly on the Internet without going through
a broker, etc. It has always been recognized that accounting for quality changes
is relatively more difficult for services than it is for commodities, a problem
that has probably been exacerbated through increasingly widespread use of the
new information technologies.14
A price index that overestimates the rate of price increase for consumer
goods has two implications for the productivity growth measures I have discussed in this article. First, since the rate of decline for the price of producerdurable goods relative to the price of consumer goods is overestimated, the rate
of embodied technological change is overestimated. Second, because output as
measured in terms of consumption goods is actually growing faster than the
consumption price index seems to indicate, the rate of disembodied technological change is underestimated. Can we say anything about the potential
magnitude of this measurement problem?
I have argued that the measurement problem is probably more relevant for
the consumption of services rather than the consumption of goods. If services
made up only a small fraction of consumption, the potential bias would probably
be small, but today expenditures on services excluding housing are about 50
percent higher than expenditures on nondurable goods. Since the price index
for nondurable consumption goods appears to be less subject to measurement
13 For

a discussion of the potential biases in the consumer price index, see Boskin et al.

(1996).
14 See

Griliches (1994) on the quality of output and price indexes for different industries.

20

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 Measures of Embodied and Disembodied Technological
Change Reconsidered

error than the price index for services, I recalculated the estimates for embodied and disembodied technological change using the price index for nondurable consumption goods only, rather than the price index for nondurable
goods and services (excluding housing) as shown in Figure 1. The revised
productivity series are graphed in Figure 7.
The alternative measure of real output mainly effects the measure of disembodied technological change as opposed to the measure of embodied technological change. Embodied technological change now proceeds at a slower rate, and
it does not accelerate as much in the mid-’70s.15 The effect on the measure of
disembodied technological change as reflected in TFP growth is more dramatic.
With the new measure of real output, TFP growth still stagnates starting in the
’70s, but there is no longer a secular decline. Notice also the strong recovery of
TFP since the early ’90s, although it remains to be seen whether this is a purely
cyclical upswing or whether it represents a change in the long-run growth path
for TFP. In conclusion, as is evident from Figures 1 and 7, the productivity
15 The

after 1977.

rate of price decline now accelerates from 2.7 percent before 1973 to 3.5 percent

A. Hornstein: Growth Accounting with Technological Revolutions

21

performance of the U.S. economy appears to be consistent with a wide range
of views, from pessimistic to guardedly optimistic. Clearly more work has to
be done.

REFERENCES
Bahk, Byong-Hyong, and Michael Gort. “Decomposing Learning by Doing in
New Plants,” Journal of Political Economy, vol. 101 (August 1993), pp.
561–83.
Boskin, Michael J., Ellen R. Dulberger, Robert J. Gordon, Zvi Griliches,
and Dale Jorgenson. “Toward a More Accurate Measure of the Cost of
Living.” Final Report to the Senate Finance Committee from the Advisory
Commission to Study the Consumer Price Index, 1996.
David, Paul. “Computer and Dynamo: The Modern Productivity Paradox in a
Not-Too-Distant Mirror,” in Technology and Productivity: The Challenge
for Economic Policy. Paris: Organization for Economic Cooperation and
Development, 1991, pp. 315– 47.
Gordon, Robert J. The Measurement of Durable Goods Prices. Chicago: The
University of Chicago Press, 1990.
Greenwood, Jeremy, and Boyan Jovanovic. “Accounting for Growth.”
Manuscript. 1998.
Greenwood, Jeremy, Zvi Hercowitz, and Per Krusell. “Long-Run Implications
of Investment-Specific Technological Change,” American Economic
Review, vol. 87 (June 1997), pp. 342–62.
Greenwood, Jeremy, and Mehmet Yorukoglu. “1974,” in Carnegie-Rochester
Conference Series on Public Policy, vol. 46 (June 1997), pp. 49–95.
Griliches, Zvi. “Productivity, R&D, and the Data Constraint,” American
Economic Review, vol. 84 (March 1994), pp. 1–23.
Hall, G., and S. Howell. “The Experience Curve from the Economist’s
Perspective,” Strategic Management Journal, vol. 6 (1985), pp. 197–212.
Hornstein, Andreas. “Reconsidering Investment-Specific Technological
Change.” Manuscript. 1999.
, and Per Krusell. “Can Technology Improvements Cause Productivity Slowdowns?” in Ben Bernanke and Julio Rotemberg, eds., NBER
Macroeconomics Annual 1996. Cambridge, Mass.: MIT Press, 1996, pp.
209–59.
Irwin, Douglas A., and Peter J. Klenow. “Learning-by-Doing Spillovers in
the Semiconductor Industry,” Journal of Political Economy, vol. 102
(December 1994), pp. 1200–27.

22

Federal Reserve Bank of Richmond Economic Quarterly

Jovanovic, Boyan, and Glenn M. MacDonald. “Competitive Diffusion,”
Journal of Political Economy, vol. 102 (February 1994), pp. 24–52.
Lucas, Robert E., Jr. “Making a Miracle,” Econometrica, vol. 61 (March
1993), pp. 251–72.
Solow, Robert. “Technical Change and the Aggregate Production Function,”
Review of Economics and Statistics, vol. 70 (1957), pp. 65–94.

Two Approaches to
Macroeconomic Forecasting
Roy H. Webb

F

ollowing World War II, the quantity and quality of macroeconomic data
expanded dramatically. The most important factor was the regular publication of the National Income and Product Accounts, which contained
hundreds of consistently defined and measured statistics that summarized overall economic activity. As the data supply expanded, entrepreneurs realized that
a market existed for applying that increasingly inexpensive data to the needs
of individual firms and government agencies. And as the price of computing
power plummeted, it became feasible to use large statistical macroeconomic
models to process the data and produce valuable services. Businesses were
eager to have forecasts of aggregates like gross domestic product, and even
more eager for forecasts of narrowly defined components that were especially
relevant for their particular firms. Many government policymakers were also
enthusiastic at the prospect of obtaining forecasts that quantified the most likely
effects of policy actions.
In the 1960s large Keynesian macroeconomic models seemed to be natural
tools for meeting the demand for macroeconomic forecasts. Tinbergen (1939)
had laid much of the statistical groundwork, and Klein (1950) built an early
prototype Keynesian econometric model with 16 equations. By the end of the
1960s there were several competing models, each with hundreds of equations. A
few prominent economists questioned the logical foundations of these models,
however, and macroeconomic events of the 1970s intensified their concerns. At
the time, some economists tried to improve the existing large macroeconomic
models, but others argued for altogether different approaches. For example,
Sims (1980) first criticized several important aspects of the large models and
then suggested using vector autoregressive (VAR) models for macroeconomic
forecasting. While many economists today use VAR models, many others continue to forecast with traditional macroeconomic models.
Roy.Webb@rich.frb.org. The views expressed are the author’s and not necessarily those of
the Federal Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 85/3 Summer 1999

23

24

Federal Reserve Bank of Richmond Economic Quarterly

This article first describes in more detail the traditional and VAR approaches to forecasting. It then examines why both forecasting methods continue to be used. Briefly, each approach has its own strengths and weaknesses,
and even the best practice forecast is inevitably less precise than consumers
would like. This acknowledged imprecision of forecasts can be frustrating,
since forecasts are necessary for making decisions, and the alternative to a
formal forecast is an informal one that is subject to unexamined pitfalls and is
thus more likely to prove inaccurate.

1.

TRADITIONAL LARGE MACROECONOMIC MODELS

These models are often referred to as Keynesian since their basic design takes
as given the idea that prices fail to clear markets, at least in the short run. In
accord with that general principle, their exact specification can be thought of as
an elaboration of the textbook IS-LM model augmented with a Phillips curve.
A simple version of an empirical Keynesian model is given below:
Ct = α1 + β11 (Yt − Tt ) + ε1,t

(1)

e
It = α2 + β21 (Rt − πt+1 ) + ε2,t

(2)

Mt = α3 + β31 Yt + β32 Rt + ε3,t

(3)

Yt
+ ε4,t
Ytp

(4)

e
πt+1 = θ51 πt + θ52 πt−1

(5)

Y ≡ Ct + It + Gt .

(6)

πt = α4 + β41

Equation (1) is the consumption function, in which real consumer spending
C depends on real disposable income Y − T. In equation (2), business investment spending I is determined by the real interest rate R − π e . Equation (3)
represents real money demand M, which is determined by real GDP Y and
the nominal interest rate R.1 In equation (4), inflation is determined by GDP
relative to potential GDP Y p ; in this simple model, this equation plays the role
of the Phillips curve.2 And in equation (5), expected inflation π e during the
1 The same letter is used for GDP and personal income since in this simple model there are
no elements such as depreciation or indirect business taxes that prevent gross national product
from equaling national income or personal income.
2 In this article the role of the Phillips curve is to empirically relate the inflation rate and a
measure of slack in the economy. In a typical large Keynesian model, the Phillips curve would
be an equation that relates wage growth to the unemployment rate, with an additional equation
that relates wage growth to price changes and another relating the unemployment rate to GDP
relative to potential.

R. H. Webb: Two Approaches to Macroeconomic Forecasting

25

next period is assumed to be a simple weighted average of current inflation
and the previous period’s inflation. Equation (6) is the identity that defines real
GDP as the sum of consumer spending, investment spending, and government
spending G. In the stochastic equations, ε is an error term and α and β are
coefficients that can be estimated from macro data, usually by ordinary least
squares regressions. The Θ coefficients in equation (5) are assumed rather than
estimated.3
One can easily imagine more elaborate versions of this model. Each major aggregate can be divided several times. Thus consumption could be divided
into spending on durables, nondurables, and services, and spending on durables
could be further divided into purchases of autos, home appliances, and other
items. Also, in large models there would be equations that describe areas omitted from the simple model above, such as imports, exports, labor demand, and
wages. None of these additions changes the basic character of the Keynesian
model.
To use the model for forecasting, one must first estimate the model’s coefficients, usually by ordinary least squares. In practice, estimating the model
as written would not produce satisfactory results. This could be seen in several
ways, such as low R2 statistics for several equations, indicating that the model
fits the data poorly. There is an easy way to raise the statistics describing the
model’s fit, however. Most macroeconomic data series in the United States are
strongly serially correlated, so simply including one or more lags of the dependent variable in each equation will substantially boost the reported R2 values.
For example, estimating equation (2) above from 1983Q1 through 1998Q4
yields an R2 of 0.02, but adding the lagged dependent variable raised it to
0.97. What has happened is that investment has grown with the size of the
economy. The inclusion of any variable with an upward trend will raise the
reported R2 statistic. The lagged dependent variable is a convenient example of
a variable with an upward trend, but many other variables could serve equally
well. This example illustrates that simply looking at the statistical fit of an
equation may not be informative, and economists now understand that other
means are necessary to evaluate an empirical equation or model. At the time
the Keynesian models were being developed, however, this point was often not
appreciated.
Once the model’s coefficients have been estimated, a forecaster would
need future time paths for the model’s exogenous variables. In this case the
exogenous variables are those determined by government policy—G, T, and
M—and potential GDP, which is determined outside the model by technology.
And although the money supply is ultimately determined by monetary policy,

3 The

coefficients are assumed, rather than estimated, due to the problematic nature of existing data on actual expectations of inflation.

26

Federal Reserve Bank of Richmond Economic Quarterly

the Federal Reserve’s policy actions immediately affect the federal funds rate.
Thus rather than specifying a time path for the money supply, analysts would
estimate the money demand equation and then rearrange the terms in order to
put the interest rate on the left side. The future time path for short-term interest
rates then became a key input into the forecasting process, although its source
was rarely well documented.
Next, one could combine the estimated model with the recent data for
endogenous variables and future time paths for exogenous variables and produce a forecast. With most large Keynesian models that initial forecast would
require modification.4 The reason for modifying the forecast is to factor in
information that was not included in the model. For example, suppose that the
model predicted weak consumer spending for the current quarter, but an analyst
knew that retail sales grew rapidly in the first two months of the quarter. Or
suppose that the analyst observes that consumer spending had been more robust
than the model had predicted for the last several quarters. Also, the model’s
forecast might display some other property that the analyst did not believe,
such as a continuously falling ratio of consumer spending to GDP. These are
all examples of information that could lead an analyst to raise the forecast for
consumer spending above the model’s prediction. To change the forecast an
analyst would use “add factors,” which are additions to the constant terms in
the equations above. Thus if one wanted to boost predicted consumer spending
by $100 billion in a particular quarter, the analyst would add that amount to the
constant term for that quarter. In the model given above, there are four constant
terms represented by the α coefficients. To forecast ahead eight quarters, one
could consider 32 possible add factors that could modify the forecast. Add
factors have long been a key part of the process that uses Keynesian models to
produce forecasts and are still important. For example, an appendix to a recent
forecast by Data Resources, a leading econometric forecasting service that uses
a Keynesian model, lists over 10,000 potential add factors.

2.

CRITICISMS OF KEYNESIAN MODELS
FOR FORECASTING

One of the most critical components of an economywide model is the linkage
between nominal and real variables. The Phillips curve relation between wage
or price growth and unemployment rates provided that key linkage for Keynesian macroeconomic models. The Phillips curve was discovered, however, as
an empirical relationship. Thus when it was first incorporated in Keynesian
4 Not every Keynesian model required modification, however. Fair (1971), for example,
presented a model that has evolved over time but has not made use of the add factors defined
below.

R. H. Webb: Two Approaches to Macroeconomic Forecasting

27

models, it did not have a firm theoretical foundation in the sense that it was
not derived from a model of optimizing agents. Milton Friedman (1968) criticized the simple Phillips curve, similar to equation (5), at the time that it
appeared to be consistent with the unemployment and inflation rates that had
been observed in the 1950s and the 1960s. His concern was that the Phillips
curve may at times appear to give a stable relation between the amount of slack
in the economy and the inflation rate. But suppose that the Federal Reserve
were to ease monetary policy in an attempt to permanently raise output above
potential. The model above ignores the fact that people would eventually figure
out the new policy strategy, and thus, according to Friedman’s logic, an expectations formation equation such as (5) would no longer hold. In the long run,
he argued, an attempt to hold output above potential would fail; expectations
would fully adjust to the new policy and output would return to potential, but
inflation would be permanently higher.
Friedman’s verbal exposition was very influential, but it did not contain
a fully specified analytical model. Using a formal model that captured Friedman’s insight, Lucas (1972) introduced rational expectations to macroeconomic
analysis as a key element for constructing a dynamic macro model. Among the
important conclusions of that paper, he demonstrated that a Phillips curve could
fit previously observed data well but would not be valid if the monetary policy
process were to change. The book that contained the Lucas paper also contained
several papers that presented long-run Phillips curves from leading Keynesian
models; a representative result of those models was that a 4 percent rate of
unemployment corresponded to 3.5 percent inflation and that higher inflation
would give lower unemployment (Christ 1972).
Propositions in economics are rarely tested decisively. In this case, though,
it was soon clear that the simple Phillips curve was not a stable, dependable
relation. In the fourth quarter of 1972 the unemployment rate averaged 5.4 percent and consumer inflation over the previous four quarters was 3.3 percent. By
the third quarter of 1975, unemployment had risen to 8.9 percent; the inflation
rate, however, did not fall but instead rose to 11.0 percent.
In retrospect, one can identify many problems with the Keynesian models
of that period. Some could be resolved without making wholesale change to the
models. For example, most models were changed to incorporate a natural rate
of unemployment in the long run, thereby removing the permanent trade-off
between unemployment and inflation. Also, most large Keynesian models were
expanded to add an energy sector, so that exogenous oil price changes could
be factored in. But some of the criticisms called for a fundamental change in
the strategy of building and using macroeconomic models.
One of the most influential was the Lucas (1976) critique. Lucas focused
on the use of econometric models to predict the effects of government economic policy. Rather than thinking of individual policy actions in isolation,
he defined policy to mean a strategy in which specific actions are chosen in

28

Federal Reserve Bank of Richmond Economic Quarterly

order to achieve well-defined goals. As an example of this meaning of policy,
consider the possibility that the Federal Reserve changed interest rates during
the early 1960s in order to keep GDP close to potential and inflation low. That
behavior could be represented as a reaction function such as equation (7):
Rt = Rt−1 + β61

Yt
+ β62 πt + ε6,t .
Ytp

(7)

Now suppose that the reaction function changed in the late 1960s and that less
importance was placed on achieving a low rate of inflation. One can imagine
replacing equation (7) with the new reaction function; however, Lucas argued
that even with the new reaction function, a model would not give reliable policy advice. The reason is that the parameters of all the other equations reflect
choices that were made when the previous policy rule was in effect. Under
the new policy rule the parameters could well be significantly different in each
equation above. This result is easiest to see in equation (6), which describes
the formation of expectations of inflation in a manner that might be reasonable
for a period when the monetary authority was stabilizing inflation. Individuals
could do better, though, if the monetary policy strategy was in the process of
changing substantially. During that period an analyst who wanted to produce
reliable conditional forecasts would need to replace equation (6), even if the
model as a whole continued to provide useful short-term forecasts of overall
economic activity. As Lucas (1976, p. 20) put it, “the features which lead to
success in short-term forecasting are unrelated to quantitative policy evaluation,
. . . [T]he major econometric models are (well) designed to perform the former
task only, and . . . simulations using these models can, in principle, provide
no useful information as to the actual consequences of alternative economic
policies.”
This critique presented a difficult challenge for macroeconomic model
builders. Every macroeconomic model is a simplification of a very complex
economy, and the Keynesian models are no exception. One of the key elements
of Keynesian models is that prices do no adjust instantaneously to equate supply
and demand in every market. The reasons underlying sluggish price adjustment
are not usually modeled, however. Thus the models cannot answer the question
of to what extent, in response to a policy change, the sluggishness of price adjustment would change. The Lucas critique challenged the reliability of policy
advice from models that could not answer such a basic question.
Analysts continue to offer policy advice based on Keynesian models and
also other macroeconomic models that are subject to the Lucas critique. These
analysts are in effect discounting the relevance of the possibility that their
estimated coefficients could vary under the type of policy change analyzed by
Lucas. For a succinct example of the reasoning that would allow the use of
Keynesian models for policy analysis, consider the counterargument given by
Tobin (1981, p. 392), “Lucas’s famous ‘critique’ is a valid point . . . [but]

R. H. Webb: Two Approaches to Macroeconomic Forecasting

29

the critique is not so devastating that macroeconomic model-builders should
immediately close up shop. The public’s perception of policy regimes is not
so precise as to exclude considerable room for discretionary policy moves that
the public would see neither as surprises nor as signals of a systematic change
in regime. Moreover, behavioral ‘rules of thumb,’ though not permanent, may
persist long enough for the horizons of macroeconomic policy-makers.” Sims
(1982) gave a lengthier defense of traditional policy analysis.
Authors such as Lucas and Sargent (1979) and Sims (1980) also criticized
Keynesian models for not being based on intertemporal optimizing behavior
of individuals. At the time they recommended different strategies for model
building. Since that time, however, there have been notable improvements in
the economic theory embodied in Keynesian models. For example, in the Federal Reserve Board’s FRB/US model, it is possible to simulate the model under
the assumption that the expectations of individuals are the same as the entire
model’s forecasts (Brayton et al. 1997). And many modelers have successfully
derived individual equations from optimizing dynamic models. Still, Keynesian models continue to be based on unmodeled frictions such as sluggish price
adjustment. It is therefore not surprising that economists have explored alternative methods of forecasting and policy analysis. One important method was
proposed by Sims (1980) and is discussed in the next section.

3.

VAR MODELS

VAR models offer a very simple method of generating forecasts. Consider the
simplest reasonable forecast imaginable, extrapolating the recent past. In practice, a reasonably accurate forecast for many data series from the United States
over the past half century can be made by simply predicting that the growth
rate observed in the previous period will continue unchanged. One could do
better, though, by substituting a weighted average of recent growth rates for the
single period last observed. That weighted average would be an autoregressive
(AR) forecast, and these are often used by economists, at least as benchmarks.
Only slightly more complicated is the idea that, instead of thinking of an autoregressive forecast of a single variable, one could imagine an autoregressive
forecast of a vector of variables. The advantage of such a VAR relative to
simpler alternatives would be that it allowed for the possibility of multivariate
interaction. The simplest possible VAR is given below in equations (8) and (9),
with only two variables and only one lagged value used for each variable; one
can easily imagine using longer lag lengths and more variables:
Rt = a11 Rt−1 + a12 pt−1 + u1,t

(8)

pt = a21 Rt−1 + a22 pt−1 + u2,t .

(9)

Because of the extreme simplicity of the VAR model, it may seem unlikely to produce accurate forecasts. Robert Litterman (1986), however, issued a

30

Federal Reserve Bank of Richmond Economic Quarterly

series of forecasts from small VAR models that incorporated from six to eight
variables. The results, summarized in Table 1, are root mean squared errors
2
(RMSEs), that is, e =
t (At − Pt ) , where e is the RMSE, A is the actual
value of a macroeconomic variable, and P is the predicted value. One caveat
is that the data summarized in this table cover a relatively short time period,
and thus it is a statistically small sample. Over that period, in comparison with
forecasts from services using large Keynesian models, the VAR forecasts were
more accurate for real GNP more than one quarter ahead, less accurate for
inflation, and of comparable accuracy for nominal GNP and the interest rate.
In another study, Lupoletti and Webb (1986) also compared VAR forecasts
to those of commercial forecasting services over a longer time period than
in the previous comparison. A different caveat applies to their results, shown
in Table 2. They studied simulated forecasts versus actual forecasts from the
forecasting services. While the details5 of the simple model were not varied to
obtain more accurate forecasts, it is inevitable in such studies that if the VAR
forecasts had been significantly less accurate, then the results probably would
not have seemed novel enough to warrant publication. That said, their fivevariable VAR model produced forecasts that, for four and six quarters ahead,
were of comparable accuracy to those of the commercial forecasting services.
The commercial services predicted real and nominal GNP significantly more
accurately for one and two quarters ahead, which probably indicates the advantage of incorporating current data into a forecast by using add factors.6
The VAR model studied by Lupoletti and Webb has five variables, each
with six lags. With a constant term, each equation contains 31 coefficients to be
estimated—a large number relative to the length of postwar U.S. time series.
Although there are methods to reduce the effective number of coefficients
that need to be estimated, the number of coefficients still rises rapidly as the
number of variables is increased. Thus as a practical matter, any VAR model
will contain only a fairly small number of variables. As a result, a VAR model
will always ignore potentially valuable data. How, then, is it possible for them

5 In this case the authors could have changed the start date of the regressions used to
estimate the VAR model’s coefficients, the choice of variables (monetary base versus M1 or M2,
for example), or the number of lag lengths. In addition, this model was unrestricted, whereas
most VAR forecasters use restrictions to reduce the effective number of estimated coefficients;
experimenting with methods of restricting parameters would have lowered the average errors of
the VAR forecasts.
6 For example, an analyst might note that labor input, measured as employee hours, was
increasing rapidly in a quarter in which GDP was forecast to rise slowly. The unexpected increase in employee hours could indicate that labor demand had risen due to unexpectedly rapid
GDP growth. If other data were consistent with that line of reasoning, the analyst would then
increase the constant terms in the equations determining GDP for the current quarter and quite
possibly the next quarter as well. Since statistical agencies release important new data every week,
there are many such opportunities for skilled analysts to improve forecast accuracy by informally
incorporating the latest data.

R. H. Webb: Two Approaches to Macroeconomic Forecasting

31

Table 1 Average Forecast Errors from Forecasts Made
in the Early 1980s
Variable:
Forecast Horizon
(Quarters)

Chase

DRI

WEFA

BVAR

Real GNP:
1
2
4
8

2.4
2.6
2.7
2.0

2.0
2.3
2.5
2.0

3.1
2.6
2.4
1.7

2.8
2.1
1.9
1.3

GNP deflator:
1
2
4
8

1.4
1.0
1.4
2.2

1.4
1.1
1.4
2.2

1.9
1.5
1.7
2.4

2.5
2.5
3.3
4.1

Nominal GNP:
1
2
4
8

3.2
3.2
3.6
3.6

2.7
2.7
3.2
3.6

3.7
3.6
3.8
2.4

3.6
3.3
4.0
3.5

Treasury bill rate:
1
2
4
8

0.2
1.8
3.3
2.9

0.1
1.9
3.2
3.7

0.4
1.8
3.2
1.1

0.1
1.7
2.9
3.7

Notes: Data are root mean squared errors (RMSEs) from postsample forecasts. Forecasts are from
1980Q2 to 1985Q1. Forecasts of real GNP, the GNP deflator, and nominal GNP are percentage
changes from the previous quarter, and forecasts of the Treasury bill rate are cumulative changes
in the quarterly average level. Data are from McNees (1986). Forecasts from WEFA were made
in mid-quarter, and the others were made one month later.

to produce relatively accurate forecasts? One possibility is that there is only a
limited amount of information in all macroeconomic time series that is relevant
for forecasting broad aggregates like GDP or its price index and that a shrewdly
chosen VAR model can capture much of that information.
At best, then, a VAR model is a satisfactory approximation to an underlying structure that would be better approximated by a larger, more complex
model. That more complex model would include how government policymakers respond to economic events. The VAR approximation will be based on
the average response over a particular sample period. A forecast from a VAR
model will thus be an unconditional forecast in that it is not conditioned on
any particular sequence of policy actions but rather on the average behavior
of policymakers observed in the past. A forecast from a Keynesian model,

32

Federal Reserve Bank of Richmond Economic Quarterly

Table 2 Average Forecast Errors from Simulated Forecasts
Variable:
Forecast Horizon
(Quarters)

Chase

DRI

WEFA

VAR

Real GNP:
1
2
4
6

4.1
3.1
2.5
2.3

4.0
3.1
2.5
2.3

4.2
2.9
2.2
1.9

5.3
4.1
2.8
2.4

GNP deflator:
1
2
4
6

1.8
1.9
2.2
2.5

2.0
2.0
2.1
2.4

1.9
1.9
2.0
2.2

2.3
2.0
2.1
2.4

Nominal GNP:
1
2
4
6

5.1
4.1
3.5
3.3

4.6
3.6
3.0
2.7

4.9
3.8
3.0
2.6

6.0
4.7
3.3
3.1

Treasury bill rate:
1
2
4
6

1.5
2.2
2.9
3.5

1.4
2.1
2.6
3.2

.
.
.
.

1.3
2.1
2.8
3.5

.
.
.
.

.
.
.
.

Notes: Data are root mean squared errors (RMSEs) from postsample forecasts. Ranges for
RMSEs are: one-quarter forecasts, 1970:4–1983:4; two-quarter forecasts, 1971:1–1983:4; fourquarter forecasts, 1971:3–1983:4; and six-quarter forecasts, 1972:1–1983:4. The VAR forecasts
are simulated forecasts, as described in the text. Forecasts of real GNP, the GNP deflator, and
nominal GNP are cumulative percentage changes, and forecasts of the Treasury bill rate are for
its quarterly average level.

however, usually is based on a particular sequence of policy actions and is
referred to as a conditional forecast—that is, conditional on that particular
sequence. Despite the Lucas critique, many users of Keynesian models seek
to determine the consequences of possible policy actions by simulating their
model with different time paths of policy actions. But, although the Lucas
critique was discussed above in reference to Keynesian models, it is equally
valid for VAR models. To help emphasize this point, the next section reviews
some details of using a VAR model for forecasting.
Forecasting with VAR Models
To forecast with the VAR model summarized in equations (8) and (9), one
would estimate the aij coefficients, usually by ordinary least squares, and

R. H. Webb: Two Approaches to Macroeconomic Forecasting

33

calculate period t values based on data for period t − 1. One can then use the
period t forecasts to calculate forecasts for period t + 1; for example, inflation
forecasts in the above model would be
ˆ
ˆ
ˆ
pt+1 = a21 Rt + a22 pt + u2,t+1
ˆ
= (a21 a11 + a22 a21 )Rt−1 + (a21 a12 + a2 )pt−1 + a21 u1,t + a22 u2,t + u2,t+1 , (10)
ˆ
ˆ
ˆ
22
where the second line in (10) was obtained by taking the first line and substituting the right-hand sides of (8) and (9) for the estimated values of Rt and pt ,
respectively. The above procedure can be repeated as many times as needed to
produce as long a forecast as desired.
It is often assumed that the realizations of unknown error terms—u1,t , u2,t ,
and u2,t+1 —will all equal zero. One can discard that assumption to incorporate
information that was not used to estimate the model. Suppose the above model
uses monthly data, and at the beginning of a month one knows last month’s
average interest rate but not the inflation rate, which the Labor Department will
release two weeks later. One could simply substitute the realized interest rate
for the estimated rate in the calculations above; in equation (10) that would
mean plugging in the realized value of u1,t . Since the errors in a VAR are
usually contemporaneously correlated, a realization of u1,t will also provide
information about u2,t . Specifically, the variances and covariances of the error
terms are taken from the variance-covariance matrix that was estimated through
period t − 1 when the aij coefficients were estimated; the expected value of
u2,t is then the ratio of the estimated covariance of u1 and u2 to the estimated
variance of u1 times the realization of u1,t . This expected value of u2,t can
then also be included in equations (8) and (9) in order to forecast inflation in
periods t and t + 1. One can easily apply this basic method for forecasting with
a VAR, and the refinement for incorporating partial data for a period, to more
complicated models with longer lags, more variables, and deterministic terms
such as constants, time trends, and dummy variables.
With this background in mind, imagine that the true structure of the economy is given by the Keynesian model of equations (1) through (6) along with
the monetary reaction function (7). Now suppose that the VAR model represented by equations (8) and (9) is estimated. Algebraic manipulation7 yields
the estimated coefficients of the VAR model as functions of the underlying
structural coefficients and error terms in equations (8 ) and (9 ):
(8 )
πt = B1,t + A11 πt−1 + A12 Rt−1 + U1,t
7 In

brief, substitute equations (1) and (2) into (6), solve for Y, then substitute the resulting
expression for Y into equation (3), and rearrange terms so that πt is on the left. Next, solve
equations (4) and (7) for Y/Y p , equate the resulting expressions, and rearrange terms so that Rt
is on the left. The resulting two equations for πt and Rt can be solved for each variable as an
expression containing lagged values of π and R, exogenous variables, structural error terms, and
underlying structural coefficients.

34

Federal Reserve Bank of Richmond Economic Quarterly
Rt = B2,t + A21 πt−1 + A22 Rt−1 + U2,t ,

(9 )

where
B1,t = [(α1 + α2 − β11 Tt + Gt + (1 − β11 )(α3 − Mt ))/β21 θ51 + (β21 − α4
A11 = −
A12 =

β61
)]/δ
β41

θ52 πt−1
θ51 δ

β21 + (1 − β11 )β32
β21 δ

U1,t = [e1,t + e2,t + (1 − β11 )e3,t ]/β21 θ51 δ
B2,t = [α1 + α2 + α3 (1 − β11 ) +
A21 = −

β61
β41 β62 + β61
α4 + Gt − Tt − (1 − β11 )Mt ]
β41
β21 β41 θ51 δ
β21 θ51 (β41 β62 + β61 )
β21 β41 θ51 δ
A22 =

U2,t = [ε1,t + ε2,t + ε3,t (1 − β11 )]
δ = (1 −

1
δ

β41 β62 + β61
β61
+ ε4,t
+ ε5,t δ −1
β21 β41 θ51 δ
β41 δ

β21 + (1 − β11 )β32
β61
)(β62 +
).
β21
β41

Viewing the model as equations (8 ) and (9 ) reveals the problematic nature
of conditional forecasting with the model. Suppose an analyst wishes to study
the effect of a tighter monetary policy on the inflation rate by first obtaining
a baseline forecast from the VAR model and then raising the interest rate
prediction by a full percentage point for the next quarter. This step would be
accomplished by feeding in a particular nonzero value for u2,t+1 in equation
(10). However, note that in terms of the underlying structure, the error term
U2,t is a complicated composite of the five error terms from the equations of
the underlying model. Yet for policy analysis it would be necessary to identify
that composite error term as a monetary policy disturbance.8
An identification that ignores the distinction between VAR errors, the ui,t s,
and the underlying structural errors, such as the εj,t ’s in the example above,
can lead to absurd results. Suppose one simulates a tighter monetary policy
in the model presented above by forcing the VAR model to predict higher
interest rates; the outcome is a higher inflation prediction. The reason is that,
8 This

point is not new—see Cooley and LeRoy (1985).

R. H. Webb: Two Approaches to Macroeconomic Forecasting

35

in the quarterly macroeconomic time series of the last 50 years, the dominant
shocks to interest rates and inflation have been aggregate demand shocks, and
a positive aggregate demand shock raises interest rates, inflation, output, and
employment. The VAR model captures these correlations. Asking the model
to simulate a higher interest rate path will lead it to predict a higher inflation
path as well. Now a clever user can tinker with the model—adding variables,
changing the dates over which the model was estimated, and so forth—and
eventually develop a VAR model that yields a lower inflation path in response
to higher interest rates. At this point, though, the model would add little value
beyond reflecting the user’s prior beliefs.
To recap, VAR models are unsuited to conditional forecasting because a
VAR residual tends to be such a hodgepodge. In addition, the models are
vulnerable to the Lucas critique. Suppose that the monetary authority decided
to put a higher weight on its inflation target and a lower weight on its output
target and that its new reaction function could be represented by (7 ):
Rt = Rt−1 + (β61 − φ)

Yt
+ (β62 + φ)πt + ε6,t .
Ytp

(7 )

The interpretation of the VAR’s coefficients in terms of the underlying structural
coefficients would also change, with each instance of β61 changing to β61 − φ
and each instance of β62 changing to β62 + φ. Thus following a discrete change
in the monetary strategy, the VAR’s coefficients would be systematically biased
and even the accuracy of its unconditional forecasts would be compromised.
Some authors, including Sims (1982), have questioned whether large policy
changes in the United States have resulted in meaningful parameter instability
in reduced forms such as VARs. One of the most dramatic changes in estimated
coefficients in VAR equations for U.S. data occurred in an inflation equation.
Table 3 is reproduced from Webb (1995) and shows significant changes in an
inflation equation’s coefficients estimated in different subperiods.9 The subperiods, moreover, were determined by the author’s review of minutes of the
Federal Open Market Committee in order to find monetary policy actions that
could indicate a discrete change in the monetary strategy. The results are thus
consistent with the view that the monetary reaction function changed substantially in the mid-1960s and again in the early 1980s and that the changes in
the economic structure played havoc with a VAR price equation’s coefficients.
This section has thus presented two separate reasons for distrusting conditional forecasts from VAR models. First, their small size guarantees that
residuals will be complicated amalgamations, and no single residual can be
meaningfully interpreted as solely resulting from a policy action. Second,
9 Consider, for example, the sum of coefficients on the nominal variables—inflation, the
monetary base, and the nominal interest rate. In the early period the sum is 0.17, rising to 1.23
in the middle period, and then falling to 0.80 in the final period.

36

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Regression Results for Several Time Periods
2

1952Q2 to 1966Q4 R = −0.08
pt = 0.28 − 0.08pt−1 + 0.08pt−2 + 0.11pt−3 + 0.07rt−1 + 0.02ct−1 − 0.01mt−1 + 0.05yt−1
ˆ
(0.06) (−0.54)
(0.51)
(0.72)
(0.24)
(0.28)
(−0.01)
(0.76)
2

1967Q1 to 1981Q2 R = 0.57
pt = −2.78 + 0.30pt−1 − 0.04pt−2 + 0.04pt−3 + 0.33rt−1 + 0.02ct−1 + 0.60mt−1 − 0.08yt−1
ˆ
(−0.54) (2.30)
(−0.28)
(0.27)
(2.56)
(0.26)
(4.87)
(−1.52)
2

1981Q3 to 1990Q4 R = 0.51
pt = −8.87 + 0.21pt−1 + 0.09pt−2 + 0.20pt−3 + 0.20rt−1 + 0.10ct−1 + 0.10mt−1 − 0.15yt−1
ˆ
(−1.54) (1.16)
(0.53)
(1.15)
(1.07)
(1.68)
(1.02)
(−0.21)
2

1952Q2 to 1990Q4 R = 0.59
pt = −3.84 + 0.30pt−1 + 0.23pt−2 + 0.22pt−3 + 0.005rt−1 + 0.05ct−1 + 0.17mt−1 − 0.22yt−1
ˆ
(−1.42) (6.38)
(2.89)
(2.71)
(0.07)
(1.54)
(2.95)
(−0.59)
Note: Coefficients were estimated by ordinary least squares; t-statistics are in parentheses.

applying the Lucas critique to VAR models implies that a VAR model’s coefficients would be expected to change in response to a discrete policy change.
Several researchers who have recognized these deficiencies but were unwilling to give up the simplicity of the VAR approach have turned to structural
VARs, or SVARs.10 These models attempt to apply both economic theory that
is often loosely specified and statistical assumptions to a VAR in order to
interpret the residuals and conduct meaningful policy analysis. In many studies
key statistical assumptions are that the economy is accurately described by a
small number of equations containing stochastic error terms, and that these
structural errors are uncorrelated across equations. The economic restrictions
vary considerably from model to model; the common feature is that just enough
restrictions are introduced so that the reduced-form errors, such as in equations
(7 ) and (8 ) above, can be used to estimate the structural errors. For example,
two of the restrictions used in a widely cited paper by Blanchard (1989) were
(1) that reduced-form GDP errors were equal to structural aggregate demand
errors, and (2) that reduced-form unemployment errors, given output, were
equal to structural supply errors. After presenting those and other restrictions,
the author noted “There is an obvious arbitrariness to any set of identification
restrictions, and the discussion above is no exception” (p. 1150).
10 A

clear exposition of the SVAR approach is given by Sarte (1999).

R. H. Webb: Two Approaches to Macroeconomic Forecasting

37

It is often the case that a reader will find an identifying assumption of
an SVAR somewhat questionable. A major difficulty of the SVAR approach is
that there is no empirical method for testing a restriction. Moreover, if different
models give different results, there are no accepted performance measures that
can be used to identify superior performance. Since there are millions of possible SVARS that could be based on the last half century of U.S. macroeconomic
data, their results will not be persuasive to a wide audience until a method is
found to separate the best models from the rest.11

4.

FINAL THOUGHTS ON CONDITIONAL FORECASTING

This article has discussed two approaches to macroeconomic forecasting. Both
approaches have produced econometric models that fit observed data reasonably
well, and both have produced fairly accurate unconditional forecasts. The VAR
approach was found unsuitable for conditional forecasting and policy analysis.
There is a wide division within the economics profession on the usefulness
of large Keynesian models for policy analysis. At one extreme are those who
accept the Lucas critique as a fatal blow and accordingly see little value in using
Keynesian models for policy analysis. At the other extreme are analysts who
are comfortable with traditional Keynesian models. In the middle are many
economists with some degree of discomfort at using the existing Keynesian
models, in part due to the features that allow the models to fit the historical data
well but may not remain valid in the event of a significant policy change. But
policy analysis will continue, formally or informally, regardless of economists’
comfort with the models and with the strategies for using them. Decisions on
the setting of policy instruments will continue to be made and will be based
on some type of analysis.
One possibility is that policy analysis and economic forecasting will be seen
as two different problems requiring two different types of models. Economists
have constructed a large number of small structural models that can be quantified and used for policy analysis. A large number of statistical approaches to
forecasting are available as well. It is not necessary that the same model be
used for both.
Keynesian models, though, are still widely used for policy analysis, and
there are actions that model builders could take to enhance the persuasiveness
of their results. One would be to publish two forecasts on a routine basis—
the usual forecast with add factors incorporating the modelers’ judgment and
a mechanical forecast with no add factors. In that way a user could easily
distinguish the judgmental content from the pure model forecast. For example,
11 Other

authors have argued that SVAR results are not robust, including Cooley and Dwyer
(1998) and Cecchetti and Rich (1999).

38

Federal Reserve Bank of Richmond Economic Quarterly

if one wanted to determine the possible impact of a tax cut on consumption, one
would want to consider whether large add factors in a consumption equation
such as equation (1) above were needed to achieve satisfactory results.
It would also be helpful for forecast consumers to know how much a
model’s specification has changed over time. Of course one hopes that new
developments in economics are incorporated into models and that poorly performing specifications are discarded. As a result, some specification changes
are to be expected. But if one saw that the consumption equation of a large
model had been in a state of flux for several years, the numerous changes could
signify that the model’s analysis of a tax cut’s effect on consumption was based
on an unstable foundation.
In addition, it would be helpful to see more analysis of forecast errors.
At a minimum, each forecast should be accompanied by confidence intervals
for the most important variables stating the likely range of results. As the ex
post errors indicate in Tables 1 and 2, these confidence intervals could be quite
wide. For example, real GDP growth has averaged 2.8 percent over the last
30 years. In Table 2, the RMSE for four-quarter predictions of real growth
from the best commercial forecasting service was 2.2 percent. Thus if a model
predicted real growth to be the 2.8 percent average, and one used that RMSE
as an approximate standard deviation of future forecast errors, then one would
expect actual outcomes to be outside of a wide 0.6 to 5.0 percent range about
30 percent of the time. Now suppose that an exercise in policy analysis with
that model revealed a difference of 1.0 percent for real GDP growth over the
next year; a user might not consider that difference very meaningful, given the
relatively large imprecision of the model’s GDP forecast.
Finally, it would be especially helpful to have a detailed analysis of errors
in a manner relevant for policy analysis. For example, continuing with the predicted effect of a tax cut, the model’s predictions could be stated in the form
of a multiplier that related the tax cut to a predicted change in real growth.
That multiplier would be a random variable that could be statistically analyzed
in the context of the whole model, and the user could be told the sampling
distribution of that statistic. Also, one would want data on how well the model
predicted the effects of tax cuts that had actually occurred in the past.
The unifying theme of these recommendations is for model builders to
open the black box that generates forecasts. Until this supplementary information routinely accompanies the output of large forecasting models, many will
see an exercise in policy evaluation as having unknowable properties and value
it accordingly.

R. H. Webb: Two Approaches to Macroeconomic Forecasting

39

REFERENCES
Blanchard, Olivier Jean. “A Traditional Interpretation of Macroeconomic
Fluctuations,” American Economic Review, vol. 79 (December 1989), pp.
1146–64.
Brayton, Flint, Eileen Mauskopf, David Reifschneider, Peter Tinsley, and John
Williams. “The Role of Expectations in the FRB/US Macroeconomic
Model,” Federal Reserve Bulletin, vol. 83 (April 1997), pp. 227– 45.
Cecchetti, Stephen G., and Robert W. Rich. “Structural Estimates of the U.S.
Sacrifice Ratio,” Federal Reserve Bank of New York, 1999.
Christ, Carl. “Discussion,” in The Econometrics of Price Determination.
Washington: Board of Governors of the Federal Reserve System, 1972,
pp. 330–31.
Cooley, Thomas F., and Mark Dwyer. “Business Cycle Analysis without Much
Theory: A Look at Structural VARs,” Journal of Econometrics, vol. 83
(March–April 1998), pp. 57–88.
, and Stephan F. LeRoy. “Atheoretical Macroeconomics: A Critique,” Journal of Monetary Economics, vol. 16 (November 1985), pp.
283–308.
Fair, Ray. A Short-Run Forecasting Model of the United States Economy.
Lexington, Mass.: D.C. Heath, 1971.
Friedman, Milton. “The Role of Monetary Policy,” American Economic
Review, vol. 58 (March 1968), pp. 1–17.
Klein, Lawrence. Economic Fluctuations in the United States 1921–1941. New
York: Wiley, 1950.
Litterman, Robert. “Forecasting with Bayesian Vector Autoregressions—Five
Years of Experience,” Journal of Business and Economic Statistics, vol. 4
(January 1986), pp. 25–38.
Lucas, Robert. “Econometric Policy Evaluation: A Critique,” in Karl Brunner
and Allan H. Meltzer, eds., The Phillips Curve and Labor Markets,
Carnegie-Rochester Conference Series on Public Policy, vol. 1 (1976), pp.
19– 46.
. “Econometric Testing of the Natural Rate Hypothesis,” in The
Econometrics of Price Determination. Washington: Board of Governors
of the Federal Reserve System, 1972, pp. 50–59.
, and Thomas Sargent. “After Keynesian Macroeconomics,” Federal
Reserve Bank of Minneapolis Quarterly Review, vol. 3 (Spring 1979), pp.
1–16.

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Federal Reserve Bank of Richmond Economic Quarterly

Lupoletti, William, and Roy Webb. “Defining and Improving the Accuracy of
Macroeconomic Forecasts: Contributions from a VAR Model,” Journal of
Business, vol. 59 (April 1986), pp. 263–85.
McNees, Stephen. “Forecasting Accuracy of Alternative Techniques: A
Comparison of U.S. Macroeconomic Forecasts,” Journal of Business and
Economic Statistics, vol. 4 (January 1986), pp. 5–16.
Sarte, Pierre-Daniel G. “An Empirical Investigation of Fluctuations in Manufacturing Sales and Inventory within a Sticky-Price Framework,” Federal
Reserve Bank of Richmond Economic Quarterly, vol. 85 (Summer 1999),
pp. 61–84.
Sims, Christopher A. “Policy Analysis with Econometric Models,” Brookings
Papers on Economic Activity, 1:1982, pp. 107–165.
. “Macroeconomics and Reality,” Econometrica, vol. 48 (January
1980), pp. 1– 48.
Tinbergen, J. Business Cycles in the United States: 1912–1932. League of
Nations, 1939.
Tobin, James. “Comments,” in J. Kmenta and J. B. Ramsey, eds., Large-Scale
Macro-Econometric Models. New York: North-Holland, 1981, pp. 391–92.
Webb, Roy. “Forecasts of Inflation from VAR Models,” Journal of Forecasting,
vol. 14 (May 1995), pp. 267–85.

The Importance of
Systematic Monetary Policy
for Economic Activity
Michael Dotsey

H

ow the Federal Reserve reacts to economic activity has significant
implications for the way the economy responds to various shocks. Yet
the importance of these responses has received limited attention in the
economic literature. Much of the literature devoted to the economic effects of
monetary policy concentrates on the impact of random monetary policy shocks.
By contrast, this article analyzes the effects of the systematic, or predictable,
portion of policy. Specifically, I compare how different specifications of an
interest rate rule affect a model economy’s response to a technology shock and
a monetary policy shock. In the case of a technology shock, the central bank’s
adjustment of the interest rate is totally an endogenous response to economic
events. The experiments show that, when there are significant linkages between
real and nominal variables, the economy’s response to changes in technology
depends on the behavior of the monetary authority. With a monetary policy
shock—for example, an unexpected change in the interest rate—the effects of
that shock will depend on how the central bank subsequently reacts to changes
in inflation and output. In general, the way shocks propagate through the economic system is intimately linked to the systematic behavior of the monetary
authority.
The results of the experiments have a number of significant implications.
Most importantly, the specification of the interest rate rule, which dictates how
the monetary authority moves the interest rate in response to inflation and real
activity, fundamentally affects economic behavior. The economy’s behavior
Michael.Dotsey@rich.frb.org. I wish to thank Bob Hetzel, Andreas Hornstein, Pierre Sarte,
and Mark Watson for a number of useful suggestions. I have also greatly benefited from
many helpful discussions with Alex Wolman. The views expressed herein are the author’s
and do not necessarily represent the views of the Federal Reserve Bank of Richmond or the
Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 85/3 Summer 1999

41

42

Federal Reserve Bank of Richmond Economic Quarterly

may be very different depending on the parameters that govern how the central
bank reacts to inflation and the state of the economy, as well as the degree
of concern it has for interest rate smoothing. For example, the central bank’s
systematic behavior can alter the correlations between variables in the model.1
This type of policy effect calls into question whether changes in a policy instrument that are the result of a changing policy emphasis can be adequately
approximated as shocks to an unchanging policy rule.
The article’s emphasis on the effects of systematic monetary policy places it
in a long tradition dating back to Poole (1970), who discussed the implications
of different types of policy rules. In that paper, and in subsequent extensions to
a flexible-price rational expectations environment, the primary purpose was to
compare a policy that used a monetary instrument with one that employed an
interest rate instrument.2 An outcome of that literature was that the systematic
component of monetary policy was important. Of significance was the way that
informational frictions interacted with monetary policy, which allowed certain
types of feedback rules to improve the information content of the nominal
interest rate. This sharpening of information occurred only when the nominal
interest rate was determined endogenously, implying that the systematic portion mattered only when money was the instrument of policy. Futhermore, the
systematic portion mattered solely in the way that it affected expectations of
future policy and not because it affected the current money stock.
In other types of models, such as those of Fischer (1977) and Taylor (1980),
which included nominal frictions such as sticky prices and wages, an important
element was the effect that systematic monetary policy had on the economy.
In short, anticipated money mattered. But in these papers monetary policy was
largely depicted through changes in money, rather than interest rates.
Recently, there has been renewed interest in the effects of monetary policy
when the policy instrument is more accurately depicted as the interest rate.
These investigations share some of the same features of the earlier models of
Fischer and Taylor in that some nominal variables, usually prices, are assumed
to be sticky. That is, the price level only adjusts gradually to its long-run equilibrium value. Some notable examples of this research can be found in Batini
and Haldane (1997), McCallum and Nelson (1997), Rotemberg and Woodford
(1997), and Rudebusch and Svensson (1997). The concern of these papers is,
however, somewhat different than the one emphasized here. They concentrate
on both the welfare effects and the variability of output and inflation that
are induced by different forms of interest rate rules. In this article, I instead

1 Rotemberg and Woodford (1997) perform a detailed analysis of the effects that different
parameter values have on the second moments of various variables and on whether or not their
economy has a unique solution.
2 Prominent examples of this literature are Dotsey and King (1983, 1986) and Canzoneri,
Henderson, and Rogoff (1983).

M. Dotsey: Importance of Systematic Monetary Policy

43

emphasize the qualitatively different ways that a model economy behaves for
a variety of specifications of monetary policy.
Both types of investigations are important and complementary. Economic
welfare analysis is important because it is the primary concern of policy analysis. But welfare measures and variances cannot in themselves inform us whether
various rules yield similar forms of behavior that are just more or less volatile,
or if behavior is changed in more fundamental ways. On these key matters,
the article is more in the spirit of the work of Christiano and Gust (1999) and
McCallum (1999), who also investigate the differences in impulse response
functions when the feedback parameters of a given policy rule are varied.
Even so, they use models that are different from the one used here.
The article proceeds as follows. Section 1 sketches the underlying model
that is common to the analysis. The key feature of the model is the presence
of price rigidity. I also indicate how an economy with sticky or sluggishly
adjusting prices behaves when the money stock is held constant, and when
a policy rule that results in real business cycle behavior of real quantities is
implicitly followed. The latter policy rule essentially negates the real effects
of price stickiness by keeping the price level and the markup constant. This
exercise provides some intuition on how the model works. Section 2 describes
the form of the interest rate rules investigated. These rules derive from the
work of John Taylor (1993). Under the first rule, the monetary authority responds both to deviations in lagged inflation from target and to lagged output
from its steady-state value. The second rule adds a concern for smoothing the
behavior of the nominal interest rate. In Section 3, I analyze the response of
the model economy to a permanent increase in the level of technology. Section
4 investigates the effect of an unanticipated increase in the nominal interest
rate on the economy. Section 5 concludes.

1.

THE MODEL

For the purpose of this investigation, I use a framework that embeds sticky
prices into a dynamic stochastic model of the economy. Under flexible prices
and zero inflation, the underlying economy behaves as a classic real business
cycle model. The model is, therefore, of the new neoclassical synthesis variety
and displays features that are common to much of the current literature using
sticky-price models.3 Agents have preferences over consumption and leisure,
and rent productive factors to firms. For convenience, money is introduced via
a demand function rather than entering directly in utility or through a shopping time technology. Firms are monopolistically competitive and face a fixed
3 Examples

Grattan (1998).

of this literature are Goodfriend and King (1997), and Chari, Kehoe, and Mc-

44

Federal Reserve Bank of Richmond Economic Quarterly

schedule for changing prices. Specifically, one-quarter of the firms change their
price each period, and each firm can change its price only once a year. This type
of staggered time-dependent pricing behavior, referred to as a Taylor contract,
is a common methodology for introducing price stickiness into an otherwise
neoclassical model.
Consumers
Consumers maximize the following utility function:
∞

U = E0

β t [ln(Ct ) − χnζ ],
t

t=0
1

where C = [ 0 c(i)(ε−1)/ε di]ε/(ε−1) is an index of consumption and n is the
fraction of time spent in employment.
Consumers also face the following intertemporal budget constraint:
Pt Ct + Pt Kt+1 ≤ Wt nt + [rt + (1 − δ)]Pt Kt + Divt ,
1

where P = [ 0 p(i)1−ε di]1/(1−ε) is the price index associated with the aggregator C; W is the nominal wage rate; r is the rental rate on capital; δ is the rate
that capital depreciates; and Div are nominal dividend payments received from
firms.
The relevant first-order conditions for the representative consumer’s problem are given by
(1/Ct )(Wt /Pt ) = χςnς−1
t

(1a)

(1/Ct ) = βEt (1/Ct+1 )[rt+1 + (1 − δ)].

(1b)

and

Equation (1a) equates the marginal disutility of work with the value of additional earnings. An increase in wages implies that individuals will work harder.
Equation (1b) describes the optimal savings behavior of individuals. If the return to saving (r) rises, then households will consume less today, saving more
and consuming more in the future.
The demand for money, which is just assumed rather than derived from
optimizing behavior, is given by
ln(Mt /Pt ) = ln Yt − ηR Rt ,

(2)

where Y is the aggregator of goods produced in the economy and is the sum
of the consumption aggregator C and an analogous investment aggregator I.
The nominal interest rate is denoted R, and ηR is the interest semi-elasticity of
money demand. One could derive the money demand curve from a shopping
time technology without qualitatively affecting the results in the article.

M. Dotsey: Importance of Systematic Monetary Policy

45

Firms
There is a continuum of firms indexed by j that produce goods, y( j), using a
Cobb-Douglas technology that combines labor and capital according to
y( j) = at k( j)α n( j)1−α ,

(3)

where a is a technology shock that is the same for all firms. Each firm rents
capital and hires labor in economywide competitive factor markets. The costminimizing demands for each factor are given by
ψt at (1 − α)[kt ( j)/nt ( j)]α = Wt /Pt

(4a)

ψt at α[kt ( j)/nt ( j)]α−1 = rt ,

(4b)

and
where ψ is real marginal cost. The above conditions imply that capital-labor
ratios are equal across firms.
Although firms are competitors in factor markets, they have some monopoly power over their own product and face downward-sloping demand curves
of y( j) = ( p( j)/P)−ε Y, where p( j) is the price that firm j charges for its product.
This demand curve results from individuals minimizing the cost of purchasing the consumption index C and an analogous investment index. Firms are
allowed to adjust their price once every four periods and choose a price that
will maximize the expected value of the discounted stream of profits over that
period. Specifically, a firm sets its price in period t to
3

∆t+h φt+h ( j),

max Et
pt ( j)

h=0

where real profits at time t + h, φt+h ( j), are given by [ p∗ ( j)yt+h ( j) −
t
ψt+h Pt+h yt+h ( j)]/Pt+h , and ∆t+h is an appropriate discount factor that is related to the way in which individuals value future as opposed to current
consumption.4
The result of this maximization is that an adjusting firm’s relative price is
given by
ε
p∗ ( j)
t
=
Pt
ε−1

3
h
1+ε
Yt+h }
h=0 β Et {∆t+h ψt+h (Pt+h /Pt )
.
3
h E {∆
εY
t+h (Pt+h /Pt ) t+h }
h=0 β t

(5)

Furthermore, the symmetric nature of the economic environment implies that
all adjusting firms will choose the same price. One can see from equation (5)
that in a regime of zero inflation and constant marginal costs, firms would set
ε
their relative price p∗ ( j)/P as a constant markup over marginal cost of ε−1 . In
general, a firm’s pricing decision depends on future marginal costs, the future
4 Specifically,

the discount factor is the ratio of the marginal utility of consumption in period
t + h to the marginal utility of consumption in period t.

46

Federal Reserve Bank of Richmond Economic Quarterly

aggregate price level, future aggregate demand, and future discount rates. For
example, if a firm expects marginal costs to rise in the future, or if it expects
higher rates of inflation, it will choose a relatively higher current price for its
product.
The aggregate price level for the economy will depend on the prices the
various firms charge. Since all adjusting firms choose the same price, there will
be four different prices charged for the various individual goods. Each different
price is merely a function of when that price was last adjusted. The aggregate
price level is, therefore, given by
(1/(1−ε))

3

(1/4)( p∗ )1−ε
t−h

Pt =

.

(6)

h=0

Steady State and Calibration
An equilibrium in this economy is a vector of prices p∗ , wages, rental rates,
t−h
and quantities that solves the firm’s maximization problem, the consumers’
optimization problem, and one in which the goods, capital, and labor markets
clear. Furthermore, the pricing decisions of firms must be consistent with the
aggregate pricing relationship (6) and with the behavior of the monetary authority described in the next section. Although I will look at the economy’s
behavior when the Fed changes its policy rule, the above description of the
private sector will remain invariant across policy experiments.
The baseline steady state is solved for the following parameterization. Labor’s share, 1 − α, is set at 2/3, ζ = 9/5, β = 0.984, ε = 10, δ = 0.025, ηR = 0,
and agents spend 20 percent of their time working. These parameter values
imply a steady-state ratio of I/Y of 18 percent, and a value of χ = 18.47. The
choice of ζ = 9/5 implies a labor supply elasticity of 1.25, which agrees with
recent work by Mulligan (1999). A value of ε = 10 implies a steady-state
markup of 11 percent, which is consistent with the empirical work in Basu and
Fernald (1997) and Basu and Kimball (1997). The interest sensitivity of money
demand is set at zero. The demand for money is generally acknowledged to
be fairly interest insensitive and zero is simply the extreme case. Since the
ensuing analysis concentrates on interest rate rules, the value of this parameter
is unimportant.
The economy is buffeted by a technology shock modeled as a random
walk and assumed to have a standard deviation of 1 percent. Thus, increases
in technology have a permanent effect on the economy. This specification is
consistent with the assumptions of much of the empirical work in this area.
The Model under Constant Money Growth
In this section, I analyze the response of the model economy to a technology shock under a constant money growth rule. As a preliminary matter, it is

M. Dotsey: Importance of Systematic Monetary Policy

47

worthwhile to recall how a standard real business cycle (RBC) model would
behave when subjected to such shocks. The behavior of real variables in the
baseline RBC model is closely mimicked in this model with a rule that keeps
the price markup and the inflation rate close to their steady-state values. The
behavior of the economy under such a rule is of independent interest as well,
because some recent work indicates that a constant markup would be a feature
of optimal monetary policy (e.g., King and Wolman [1999]).
The reason this policy produces a response in real variables very much
like that obtained in a model with flexible prices can be seen by examining
equation (5). If prices were flexible, then each firm would choose the same
price, and relative prices would equal unity. Real marginal cost would then be
−1
, which is exactly the steady-state value of marginal cost under staggered
price setting and zero inflation. If the steady-state inflation rate were zero, then
stabilizing real marginal cost at its steady-state value would imply that firms
would have no desire to deviate from steady-state behavior and would keep
their relative price constant at one. Thus, in an environment of zero average
inflation, stabilizing marginal cost or the markup leads to firm behavior that
replicates what firms would do in a world of flexible prices. In short, when
inflation rates are close to zero, one can find a policy that virtually replicates
the behavior found in a flexible price model.
Figures 1a and 1b show the deviations of output, money stock, price level,
nominal interest rate, and inflation from their steady-state values in response
to a permanent increase in the level of technology under a rule that keeps
the markup approximately constant (it varies by less than 0.0002 percent from
its steady-state value of 0.11). Output initially jumps by 1.2 percent and then
gradually increases to its new steady-state value. The money supply grows
one-for-one with output, and given an income elasticity of one and an interest
elasticity of zero, its behavior is consistent with prices growing at their steadystate rate of 2 percent. Consequently, inflation remains at its steady-state rate.
The slight uptick in the nominal interest rate is, therefore, entirely due to a
small increase in the real rate of interest.
In contrast, Figures 2a and 2b depict the behavior of the economy in
response to the same shock but with money supply growth kept at its steadystate rate of 2 percent. From the money demand curve, equation (2), it is clear
that nominal income growth cannot deviate from steady state. If prices were
flexible, they would fall by enough so that output would behave as shown
in Figure 1a. But, because 75 percent of the firms are unable to adjust their
prices, the price level declines by much less, and as a result the response of real
output is damped. As additional firms adjust their price over time, the price
level falls and output eventually reaches its new steady state. Falling prices
imply disinflation and a decline in the nominal interest rate. This behavior is
shown in Figure 2b.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 1 Constant Markup and Technology Shock

By analyzing the various interest rate rules, it will become evident that
they differ in their ability to produce the type of output behavior associated
with flexible prices. The above discussion should help in clarifying why that
is the case.

M. Dotsey: Importance of Systematic Monetary Policy

49

Figure 2 Constant Money Growth and Technology Shock

2.

MONETARY POLICY

For studying the effects that the systematic part of monetary policy has on the
transmission of the various shocks to the economy, I shall be fundamentally

50

Federal Reserve Bank of Richmond Economic Quarterly

concerned with two basic types of policy rules. These rules employ an interest
rate instrument and fall into the category broadly labeled as Taylor (1993) type
rules. Both rules are backward-looking and allow the central bank to respond
to deviations of past inflation from its target and past output levels from the
steady-state level of output. However, one rule implies interest smoothing on
the part of the monetary authority. Specifically, the first rule is given by
Rt = r + π ∗ + 0.75(πt−1 − π ∗ ) + 0.6(Yt−1 − Yt−1 ),

(7)

where π ∗ is the inflation target of 2 percent, and Y is the steady-state level
of output. Under this rule, the central bank responds only to readily available information when adjusting the nominal rate of interest. When inflation
is running above target or output is above trend, the central bank tightens
monetary policy by raising the nominal interest rate. This type of rule restores
the inflation rate to 2 percent after the shock’s effects dissipate. The rule differs
from the original one proposed by Taylor in that it includes a response to last
quarter’s lagged inflation rather than current yearly inflation, and the coefficient on inflation is somewhat smaller than that initially specified by Taylor
(1993). This specification is adopted for two reasons. First, as emphasized by
McCallum (1997), the elements of a feedback rule should involve only variables
that are readily observable. Although contemporaneous output and inflation are
observed in the stylistic setting of the model, in practice these variables may
be observed only with a lag.5 Second, the rule specified in (7) is explosive for
the parameters chosen by Taylor. In the above lagged specification, explosive
behavior results if the monetary authority responds too aggressively to both
inflation and output.6
The second rule is similar to the first but adds a degree of interest rate
smoothing. The actual interest rate can be thought of as a weighted average of
some target that depends on the state of the economy and last period’s nominal
interest rate. The greater the weight on the nominal interest rate, the more
concerned the monetary authority is for smoothing the interest rate. This rule
is given by7
Rt = r + π ∗ + 0.75Rt−1 + 0.75(πt−1 − π ∗ ) + 0.15(Yt−1 − Yt−1 ).

(8)

5 In actuality the Fed does not observe potential or steady-state output either. It must respond
to estimates.
6 If the interest rate rule was specified in terms of the deviation of a four-quarter average of
inflation from target, then a coefficient on inflation’s deviation from target of 1.5 and a coefficient
on output’s deviation from a potential of 0.5 would produce well-behaved economic responses to
shocks. For a detailed discussion of issues regarding determinacy and instability, see Rotemberg
and Woodford (1997) and Christiano and Gust (1999).
7 I initially tried to use the same coefficient on output as in the first rule, but the behavior
of the economy was erratic. Scaling down the coefficient on lagged real activity produced more
reasonable behavior.

M. Dotsey: Importance of Systematic Monetary Policy

51

Contrary to many theoretical and empirical studies, the model experiments
I run in the ensuing section take a far-from-typical perspective concerning the
effects of monetary policy than is usually taken in theoretical and empirical
studies. Standard investigations attempt to determine how the economy reacts to policy shocks represented as unexpected disturbances to either money
growth rates or the interest rate set by the Fed. While those analyses tackle
an interesting problem, only recently have economists begun to analyze the
economic effects of the systematic component of policy. By concentrating on
the sensitivity of the economy’s responses to various shocks under different
policies, the article has a different emphasis from much of the recent work
on systematic policy. The analysis is, therefore, similar in emphasis to recent
papers by McCallum (1999) and Christiano and Gust (1999).

3.

A COMPARISON OF THE POLICY RULES

This section analyzes the way the model economy reacts to a technology shock
under the two different interest rate rules. These responses are depicted in
Figures 3 and 4, where as before all changes represent deviations from steadystate values. Figures 3a and 3b and Figures 4a and 4b refer to rules 1 and 2,
respectively. The differences across the policy rules are striking, especially
when one also considers the behavior depicted in Figures 1 and 2. Although
output increases on impact under each rule, the magnitude of the increase varies
greatly across policy regimes, ranging from less than 0.4 percent for a money
growth rule to approximately 2.5 percent under the Taylor rule that uses interest
rate smoothing. The impulse response for output under the Taylor rule that does
not employ interest rate smoothing is closest to the response shown in Figures
1 and 2 for a standard RBC model.
The nominal behavior of the economy is also very different under the various rules. Under the first rule the price level barely moves on impact but then
falls as the effect of the technology shock works its way through the economy.
This behavior is in sharp contrast to that associated with the constant money
growth rule in which the price level declines on impact. It is, therefore, not
staggered price setting that is responsible for the initial stickiness of the price
level but the specification of the interest rate rule. Because the nominal interest
rate responds only to lagged variables, it doesn’t react initially. Consequently,
all the money demanded at the initial interest rate is supplied, and there is no
need for price adjustment to equilibrate the money market. With output slightly
below its new steady-state value, the nominal interest begins to decline, and it
continues to decline in response to falling prices. It is important to emphasize
that the decline in the interest rate does not represent an easing of policy but
rather an endogenous response to an economic shock. That is, the central bank
is not attempting to independently stimulate the economy.

52

Federal Reserve Bank of Richmond Economic Quarterly

Figure 3 Taylor Rule and Technology Shock

Under the second rule the economy booms. Output rises by an extraordinary 2.5 percent, and with it is an accompanying increase in marginal cost
as firms must bid up the wage to induce additional labor supply. The increase
in marginal cost implies that adjusting firms will raise their prices. In contrast

M. Dotsey: Importance of Systematic Monetary Policy

53

Figure 4 Smooth Taylor Rule and Technology Shock

to the previous example, the economy experiences inflation in response to the
increase in technology. Because prices will be rising over time, the current
period is a relatively good time to consume, and output demand is high as
well. The increase in inflation as well as the increase in output above its new

54

Federal Reserve Bank of Richmond Economic Quarterly

steady-state level causes the central bank to raise interest rates. As in the previous case, the subsequent rise in the interest rate should not be interpreted as
an attempt to shock an overheated economy but simply as the central bank’s
usual response to strong economic growth. The endogenous rise in the interest
rates, as we shall see in the next section, is responsible for the dramatic fall
in economic activity. The initial overshooting is subsequently corrected, and
output then gradually approaches its new steady-state level. It is important to
note that under the two rules the marked difference in the impulse response
functions is not due to the somewhat smaller coefficient on lagged output in
the second rule. If that coefficient were increased to 0.6, then the response of
the economy would be similar but the volatility, or saw-toothed behavior, of
the variables would be more pronounced.
The difference in the functions is due to the interest rate smoothing present
in the second policy rule. Under the first policy rule, any increase in inflation
is aggressively reacted to because the monetary authority does not have to
take into account the past level of the interest rate. Knowing the relatively
aggressive nature of policy, individuals and firms expect less inflation, creating
less pressure to raise prices. The subsequent downward path of prices makes
postponing purchases optimal. As a result, there is less demand pressure in
response to the shock. Output does not rise to its new steady-state level on impact, and there is no upward pressure on marginal cost. Under the second policy
rule, the monetary authority will be less aggressive, so prices are expected to
rise. Such expectations spur consumers to purchase goods today, resulting in
relatively strong aggregate demand. The economy booms and the combined
effect of expected inflation and upward pressure on marginal cost causes firms
to raise prices.
There are a number of points to take away from the analysis presented in
this section. First and foremost is that the systematic component of monetary
policy is key in determining the economy’s reaction to shocks. For example,
with no interest rate smoothing, inflation and prices are negatively correlated
with output, while they are positively correlated when the monetary authority
smooths the interest rate. As a consequence of sticky prices, both positive
and negative correlations between real and nominal variables are possible. The
type of correlation observed may be entirely due to the systematic behavior of
monetary policy and have nothing to do with the structure of the economy.

4.

A FURTHER INVESTIGATION OF TAYLOR RULES

In this section I illustrate the sensitivity of the model economy’s responses
under the two policy rules to a transitory tightening of monetary policy as
reflected in a 100 basis point increase in the nominal interest rate. As with the
case of the technology shock, the responses are very different. These responses

M. Dotsey: Importance of Systematic Monetary Policy

55

are displayed in Figures 5 and 6, with 5a and 5b depicting the response under
the first rule and 6a and 6b reflecting behavior under the second rule.
Under rule 1 output actually rises on impact, while under the interest rate
smoothing rule output falls. This difference in behavior occurs because the unexpected rise in the nominal rate under the first rule will accommodate modest
inflation. As long as inflation doesn’t accelerate—behavior the rule is designed
to prevent—the nominal rate will gradually return to steady state, and there
will be upward movement in prices as well as strong economic growth. The
economy only suffers a mild recession four quarters into the future.
The presence of interest rate smoothing in this case means that any upward
movement in real economic activity or inflation will drive the interest rate even
higher. Rather than acting as an anchor as in the previous section, the interest
smoothing term implies a much more aggressive response to nominal growth.
Because today’s interest rate is high, all things being equal, the next period’s
interest rate will be high as well. Individuals and firms understand the nature
of the rule, and, therefore, an increase in nominal activity is inconsistent with
interest rate behavior under this rule. Output, prices, and inflation decline immediately in response to the rise in the nominal interest rate. It is noteworthy
that the nominal interest is more volatile under the policy rule that reflects a
concern for interest rate smoothing.
Thus, if the Fed were to significantly and periodically alter its reaction
to the past behavior of interest rates, policy would appear to operate with
variable impact effects and variable lagged effects. It would do so not because
the changes in the policy rule are reflected in small quantitative differences in
the economy’s response to policy shocks but because these changes in policy
may actually lead to an economy that qualitatively responds in a different way
altogether.
Admittedly, the behavioral changes analyzed may be severe and the model
economy may not reflect important elements of actual behavior, but the experiments in this and the preceding section send a strong message that the form
of the policy rule is far from innocuous.

5.

CONCLUSION

The basic conclusion of this article is that money matters. More to the point,
monetary policy matters, and specifically the systematic part of monetary policy
matters. While most studies have devoted a great deal of effort to understanding
and quantifying the economic effects of monetary policy shocks, my results
indicate that it may be equally if not more important to determine the appropriate design of a policy rule. From my own perspective, which is influenced
by numerous (or perhaps endless) policy debates, what is typically discussed
is not what monetary disturbance should impact the economy but what response

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 5 Taylor Rule and Policy Shock

should policy have to the economy. Significant tightenings of policy are generally not an attempt to shock the economy but the Fed’s realization that inflation
and expected inflation have risen and that tightening is appropriate. The degree of the response may, and probably does, vary in different periods. And it

M. Dotsey: Importance of Systematic Monetary Policy

57

Figure 6 Smooth Taylor Rule and Policy Shock

may be inappropriate to model these changes as shocks to an unvarying rule.
As I have shown, the sign of correlations among economic variables can differ
across rules. That type of behavior would not be captured by appending a shock
to a given policy rule. The message from the above exercises is that it may be

58

Federal Reserve Bank of Richmond Economic Quarterly

more appropriate to model the coefficients in the response function as random,
rather than attaching some randomness to an invariant rule.

REFERENCES
Basu, Susanto, and John Fernald. “Returns to Scale in U.S. Production:
Estimates and Implications,” Journal of Political Economy, vol. 105 (April
1997), pp. 249–83.
Basu, Susanto, and Miles S. Kimball. “Cyclical Productivity with Unobserved
Input Variation,” NBER Working Paper 5915. February 1997.
Batini, Nicoletta, and Andrew G. Haldane. “Forward-Looking Rules for
Monetary Policy,” in John B. Taylor, ed., Monetary Policy Rules. Chicago:
University of Chicago Press, 1999.
Canzoneri, Matthew, Dale Henderson, and Kenneth Rogoff. “The Information
Content of Interest Rates and the Effectiveness of Monetary Policy Rules,”
Quarterly Journal of Economics, vol. 98 (November 1983), pp. 545–66.
Chari, V. V., Patrick J. Kehoe, and Ellen R. McGrattan. “Sticky Price Models
of the Business Cycle: Can the Contract Multiplier Solve the Persistence
Problem?” Staff Report 217. Minneapolis: Federal Reserve Bank of
Minneapolis, May 1998.
Christiano, Lawrence J., and Christopher J. Gust. “Taylor Rules in a Limited
Participation Model,” NBER Working Paper 7017. March 1999.
Dotsey, Michael, and Robert G. King. “Informational Implications of Interest
Rate Rules,” American Economic Review, vol. 76 (March 1986), pp.
33–42.
. “Monetary Instruments and Policy Rules in a Rational Expectations Environment,” Journal of Monetary Economics, vol. 12 (September
1983), pp. 357–82.
Fischer, Stanley. “Long-Term Contracts, Rational Expectations, and the
Optimal Money Supply Rule,” Journal of Political Economy, vol. 85
(February 1977), pp. 191–205.
Goodfriend, Marvin S., and Robert G. King. “The New Neoclassical Synthesis
and the Role of Monetary Policy,” NBER Macroeconomics Annual 1997.
Cambridge, Mass.: MIT Press, 1997.
King, Robert G., and Alexander L. Wolman. “What Should the Monetary
Authority Do When Prices are Sticky?” in John B. Taylor, ed., Monetary
Policy Rules. Chicago: University of Chicago Press, 1999.
McCallum, Bennett T. “Analysis of the Monetary Transmission Mechanism:
Methodological Issues.” Manuscript. February 1999.

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. “Comment” [on “An Optimization-Based Econometric Framework
for the Evaluation of Monetary Policy”]. NBER Macroeconomics Annual
1997. Cambridge, Mass.: MIT Press, 1997, pp. 355–59.
, and Edward Nelson. “Performance of Operational Policy Rules
in an Estimated Semiclassical Structural Model,” in John B. Taylor, ed.,
Monetary Policy Rules. Chicago: University of Chicago Press, 1999.
Mulligan, Casey B. “Substitution over Time: Another Look at Life Cycle
Labor Supply,” NBER Macroeconomics Annual 1998. Cambridge, Mass.:
MIT Press, 1999.
Poole, William W. “Optimal Choice of Monetary Policy Instruments in a
Simple Stochastic Model,” Quarterly Journal of Economics, vol. 84 (May
1970), pp. 197–216.
Rotemberg, Julio J., and Michael Woodford. “Interest Rate Rules in an
Estimated Sticky Price Model,” in John B. Taylor, ed., Monetary Policy
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Rudebusch, Glenn D., and Lars E. O. Svensson. “Policy Rules for Inflation
Targeting,” in John B. Taylor, ed., Monetary Policy Rules. Chicago:
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Taylor, John B. “Discretion versus Policy Rules in Practice,” CarnegieRochester Conference Series on Public Policy, vol. 39 (December 1993),
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An Empirical Investigation
of Fluctuations in
Manufacturing Sales
and Inventory within a
Sticky-Price Framework
Pierre-Daniel G. Sarte

T

he macroeconomics literature has recently witnessed a resurgence of
interest in issues related to nominal price rigidities. In particular, advances in computational methods have allowed for the analysis of fully
articulated quantitative general equilibrium models with inflexible prices.1 Because nominal price rigidities create predictable variations in sales, these models
provide a natural setting for the study of inventory behavior. Specifically, firms
that face increasing marginal costs wish to smooth production and, given predictable variations in sales, can naturally use inventories to accommodate any
difference between a smooth production volume and sales.
Hornstein and Sarte (1998) study the implications of sticky prices for inventory behavior under different assumptions about the nature of the driving
process. Regardless of whether the economy is driven by nominal demand or
real supply shocks, the authors find that an equilibrium model with inflexible
prices can replicate the main stylized facts of inventory behavior. Namely, production is more volatile than sales while inventory investment is positively correlated with sales at business cycle frequencies. More importantly, their study
also makes specific predictions about the dynamic adjustment of inventories
and sales to these shocks. In response to a permanent positive money growth

Pierre.Sarte@rich.frb.org. I wish to thank Andreas Hornstein and Mark Watson for helpful
discussions. I also wish to thank Yash Mehra, Wenli Li, and Roy Webb for thoughtful comments. The opinions expressed herein are the author’s and do not necessarily represent those
of the Federal Reserve Bank of Richmond or the Federal Reserve System. Any errors are,
of course, my own.
1 See

Goodfriend and King (1997) for a survey of recent work.

Federal Reserve Bank of Richmond Economic Quarterly Volume 85/3 Summer 1999

61

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Federal Reserve Bank of Richmond Economic Quarterly

innovation, both sales and inventories contemporaneously rise before gradually
returning to the steady state. In contrast, a permanent positive technology shock
leads to a rise in sales and a fall in inventories on impact. As time passes by,
sales increase monotonically and eventually reach a new higher steady-state
level.
In this article, we estimate a structural vector autoregression (SVAR), where
money is constrained to be neutral in the long run, in order to gauge the degree
to which these theoretical dynamic adjustment paths hold in the data. Using
manufacturing data, we find that the impulse response of sales and inventories
to nominal shocks is generally consistent with the predictions of a sticky-price
model. Furthermore, both sales and inventories also behave as predicted in the
long run in response to a technology shock. Contrary to theory, however, we find
that inventories contemporaneously rise in response to a positive innovation in
technology. In all cases, the data indicate significantly more sluggishness in the
dynamic adjustment of sales and inventories to shocks than implied by current
models with sticky prices. The latter finding is consistent with earlier work by
Feldstein and Auerbach (1976), as well as Blinder and Maccini (1991), using
stock-adjustment equations. More recently, Ramey and West (1997) also find
that the inventory:sales relationship is unusually sluggish. They are able to explain this result by appealing either to persistent shocks to the cost of production
or to a strong accelerator motive within a linear quadratic framework.
Although the earlier analysis in Hornstein and Sarte (1998) makes specific
predictions regarding the dynamic response of sales and inventories to various
shocks, it does not assess the relative importance of these shocks as sources
of fluctuations. Here we use our estimated VAR to acquire some insight into
the significance of both real and nominal shocks in generating fluctuations
in sales and the inventory:sales ratio. We find that nominal shocks generally
contribute little to the forecast error variance in the latter variables at both short
and long horizons. Instead, consistent with earlier work such as King, Plosser,
Stock, and Watson (1991), fluctuations in real variables tend to be dominated by
real disturbances. Moreover, these empirical findings tend to hold consistently
throughout different historical episodes at the business cycle frequency. One
exception concerns monetary disturbances that play a noticeably more important
role in generating inventory:sales ratio fluctuations in the early 1990s.
This article is organized as follows. We first set up and motivate an empirical model that is consistent with generic restrictions implied by an equilibrium
model of inventory behavior. In particular, we assume that money is neutral in
the long run and that the inventory:sales ratio is a stationary process without
trend. Note that we do not impose any a priori restrictions that are directly
tied to the assumption of sticky prices. The next section examines various
integration and cointegration properties of the data under consideration. We
then analyze the impulse responses of sales and the inventory:sales ratio to
various shocks. We also try to gauge the relative importance of these shocks

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

63

as sources of fluctuations in the latter variables. After that, we offer some cautionary remarks regarding the specific empirical implementation in this article.
The final section concludes the analysis.

1.

INVENTORY FLUCTUATIONS:
THEORETICAL MOTIVATION

To set the stage and notation for the econometric specification, we will provide some theoretical background on the behavior of inventories. The basic
framework we have in mind is one in which firms use inventories to smooth
production in a setting with staggered nominal prices.2 The assumption of
inflexible price adjustment provides a natural role for production smoothing as
the underlying factor driving inventory behavior. In particular, nominal price
rigidity creates predictable variations in sales. Suppose, for instance, that the
nominal price set by a given firm is fixed over some time interval. If the
general price level increases over that time interval, then the firm’s relative
price correspondingly falls and its sales rise, all else being equal. Given this
rising sales path, the firm also attempts to minimize total production costs by
keeping production relatively smooth. Inventories can then be used to make up
for the differences between production and sales. In addition to identifying this
sticky-price motive, we, like Khan (1987), assume that firms may also hold
inventories to avoid costly stock-outs.
Within the context of this framework, the dynamic adjustment of inventories and sales to various shocks will generally depend on how preferences and
technology are specified. In the long run, however, the model exhibits basic
neoclassical properties that can be used for the purposes of identification. One
of these properties suggests that money is neutral and, moreover, that changes
in the steady-state level of sales ultimately arise from innovations in technology.
With this in mind, we let the long-run component of the sales process evolve
according to
s∗ = δs + s∗ + Φs (L)at ,
t
t−1

(1)

where s∗ denotes the log level of sales and at captures shocks to technology.
t
The lag polynomial Φs (L), as well as all other polynomials described below, is
assumed to have absolutely summable coefficients with roots lying outside the
unit circle. Observe that equation (1) implicitly assumes that the sales process
possesses a unit root. We formally test this assumption later in this article.
In principle, the steady-state level of inventories can be thought of as
being determined by the two forces we described previously. Note that in a
2 See

Hornstein and Sarte (1998) for details of the model.

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Federal Reserve Bank of Richmond Economic Quarterly

model with rigid prices, firms naturally wish to hold inventories to accommodate any difference between predictable variations in sales and a smooth
production volume. Moreover, by using inventories to avoid costly stock-outs,
firms generally target some appropriate inventory:sales ratio in the long run.
Although the short- and medium-run dynamics of inventories typically depend
on both these forces, Hornstein and Sarte (1998) note that in the steady state, the
level of inventories reflects almost exclusively the stock-out avoidance motive.
Accordingly, we may express long-run inventories as
n∗ = s∗ + ξ,
t
t

(2)

where n∗ denotes the log level of inventories and ξ is some target invent
tory:sales ratio. It immediately follows from (1) and (2) that inventories and
sales share a common stochastic trend whose growth rate is δs + Φs (L)at .
Furthermore, as we make clear below, the inventory:sales ratio becomes a
stationary stochastic process.
Since in this article we are partly interested in how monetary shocks affect
the dynamics of inventories and sales, we must specify our beliefs about the
behavior of money. To this end, we let the long-run component of money evolve
according to
m∗ = δm + Φm (L)[at , ηt ] ,
t

(3)

where m∗ is the log level of money and ηt denotes money innovations. Note
t
that, as in Gali (1999), we allow monetary policy to respond permanently to
long-run changes in technology, at . This assumption captures the idea that the
Federal Reserve reacts to permanent real changes in the economic environment
in its effort to keep prices stable. We further assume that at and ηt are serially
and mutually uncorrelated shocks.
While we have assumed that long-run changes in sales are ultimately determined by technological considerations, sales may actually respond to a variety
of economic shocks in the short run. More specifically, the level of sales, st ,
may deviate temporarily from its long-run value because of money shocks or
transitory real demand shocks. Such real shocks may include temporary changes
in tastes, for instance. Therefore, a complete process for sales can be described
as
st = s∗ + ψs (L)[at , ηt , et ] ,
t

(4)

where et captures a mixture of temporary real demand shocks. These are assumed to be serially uncorrelated as well as uncorrelated with at and ηt . In
principle, the fact that st depends on all shocks in the model allows for flexible short-run dynamics. The aim of our empirical exercise is, in part, to gauge
whether these short-run dynamics are consistent with the predictions of a model
with nominal price rigidities. Taking the first difference in equation (4) and

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

65

substituting equation (1) into it yields
∆st = δs + Φs (L)at + (1 − L)ψs (L)[at , ηt , et ] ,

(5)

which represents one of the structural equations to be estimated.
As in (4), one generally expects the level of inventories to be sensitive to
all shocks in the short run. Consequently, we may write the following stochastic
process for inventories:
nt = n∗ + ψn (L)[at , ηt , et ] .
t

(6)

Note that the theoretical framework we have been using predicts a testable
cointegrating restriction. In particular, while (1) and (2) suggest that both inventories and sales are integrated of order one (often denoted I(1)), these equations
combined with (4) also suggest that the difference between inventories and sales
is stationary (or I(0)). Formally, we can use (6) along with equations (1), (2),
and (4) to show that
nt − st = ξ + {ψn (L) − ψs (L)}[at , ηt , et ] .

(7)

The above equation clearly indicates that the inventory:sales ratio will deviate
from its long-run value at high and medium frequency. By construction, these
deviations are never permanent.
To complete the econometric specification, we allow monetary policy to
respond to various shocks not only in the long run but also in the short run.
The latter assumption along with equation (3) yields
∆mt = δm + Φm (L)[at , ηt ] + (1 − L)ψm (L)[at , ηt , et ] .

(8)

At this point, we wish to stress that the identifying restrictions made in this
section are, in fact, quite generic and unrelated to the notion of sticky prices
per se. Therefore, if the results below turn out to be consistent with the notion
of nominal rigidities, this outcome will not be as a direct consequence of the
identifying strategy used. It remains that different identification strategies may
yield different results. Because our restrictions are relatively general, however,
they encompass a broad range of models.3

2.

ECONOMETRIC METHOD AND DATA ANALYSIS

Using Long-Run Restrictions for the Purpose of Identification
We can summarize our model thus far in the form of a vector moving average,
Y t = T(L)εt ,
3 See

(9)

Cooley and Dwyer (1998) for a thorough discussion of the pitfalls associated with the
identification of SVARs.

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Federal Reserve Bank of Richmond Economic Quarterly

where Yt = (∆st , ∆mt , nt −st ) and εt = (at , ηt , et ). The matrix polynomial T(L)
consists of the polynomials Φa (L), Φm (L), ψs (L), ψn (L), and ψm (L) in equations
(1) through (8). In addition, embedded in T(L) are long-run restrictions implied
by our model that can be used to identify each of the three structural shocks.
Specifically, the matrix of long-run multipliers, T(1), may be written as


a11 0
0
(10)
T(1) =  a12 a21 0  .
a31 a32 a33
Thus, the first row of T(1) reflects our restriction that only technology shocks
alter the level of sales in the long run (in the steady state, sales should equal
production). Money, therefore, is neutral, and we can appropriately constrain
the estimation of the sales growth equation to identify technology shocks. To
see how to impose the restrictions contained in T(1), note first that under the
assumption that T(L) is invertible, T(1)−1 is also lower block triangular. In
estimating the sales growth equation, therefore, it suffices to set the long-run
elasticity of ∆st with respect to both ∆mt and nt − st to zero.
Real transitory demand shocks cannot, by definition, have long-run effects
on any of the variables in the model. As the second row of T(1) suggests, this
restriction, already imposed in estimating the sales growth regression, can be
used to identify money shocks. In other words, except for permanent changes
in technology, long-run changes in money are associated only with their own
innovations, as equation (3) illustrates. To uncover money innovations, therefore, we estimate the money growth equation subject to the restriction that the
long-run elasticity of ∆mt with respect to nt − st be set to zero. It remains that
the long-run elasticity of ∆mt with respect to ∆st will generally not be zero.
To account for the presence of this contemporaneous endogenous variable in
the money growth regression, we use the fact that the structural disturbances
are assumed to be mutually uncorrelated and use the residual from the sales
growth regression as an instrument. The econometric methodology used here,
therefore, follows that of Shapiro and Watson (1988), Blanchard and Quah
(1989), King, Plosser, Stock, and Watson (1991), as well as many others.
It follows that the last remaining innovation captures real temporary demand shocks. In particular, the third row of T(1) suggests that the latter shocks
can simply be uncovered by estimating the inventory:sales ratio equation without any restrictions. We use the residuals from both the sales growth and money
growth regressions to instrument for ∆st and ∆mt in this last regression.
Cointegration Properties of the Data
Before proceeding with the estimation, we first investigate the cointegrating
restriction implied by (7). As with the majority of the empirical literature on
inventory behavior, this article focuses mainly on manufacturing inventories.

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

67

Figure 1

More specifically, the notion of production smoothing applies best to manufactured goods, as pointed out in Hornstein (1998). In Section 4, we shall take
the research one step further by showing that the econometric specification
above may be ill-suited to both the retail and service sectors. We add one cautionary note, however, regarding our assumption that money may respond to
long-run innovations in technology (recall equation [3]). In all likelihood, this
assumption is most relevant for aggregate shocks rather than sectoral shocks.
Our model does not allow us to disentangle these shocks. Consequently, shocks
captured by at should be interpreted as a linear combination of both aggregate
and sectoral innovations.

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Table 1 Cointegration Statistics—1947:1–1998:3
a. Results from Unrestricted Levels Vector Autoregression:
Largest Eigenvalues of Estimated Companion Matrix
VAR(6) with constant and trend
Real

Imaginary

Modulus

0.98
0.98
0.85
0.85
0.63
0.63

0.04
−0.04
0.15
−0.15
0.47
−0.47

0.98
0.98
0.86
0.86
0.78
0.78

b. Multivariate Unit-Root Statistics: Stock and Watson q f Statistic
H0: 3 unit roots vs. H1: at most 2 unit roots
Number of Lags
1
2
3
4
5

f
q τ (3, 2) statistic
−42.51
−45.10
−42.27
−34.68
−31.89

P value
2.75
1.75
2.75
9.75
14.75

H0: 2 unit roots vs. H1: at most 1 unit root
Number of Lags
1
2
3
4
5

f
q τ (2, 1) statistic
−8.99
−11.94
−9.39
−8.69
−7.87

P value
83.25
67.50
81.25
85.00
88.75

Figure 1 shows the logarithms of money, as defined by M1 (i.e., currency
and demand deposits), manufacturing inventories, and sales of finished goods.
The data are quarterly U.S. observations spanning the period 1947:1 to 1998:3.
Early figures for M1 were obtained from the Monetary Statistics of the United
States since they were unavailable from the Board of Governors dataset. The inventory and sales data were downloaded from the National Income and Products
Accounts on February 19, 1999. Regressions were run over the period 1948:3
to 1998:3 to allow for six lags. The plots of the variables display familiar, clear
upward trends with inventories being the most volatile component. Note that
inventories and sales indeed seem to share the same trend over the period considered. Figure 1 also plots the logarithm of the inventory:sales ratio, (n − s),
which appears relatively stable. One possible exception concerns the period

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

69

Table 1 Cointegration Statistics—1947:1–1998:3 (cont.)
Johansen’s Likelihood Ratio Statistics
2(L1 − L0 )
L
H0: h = 0 vs. H1: No restrictions: −T
H0: h = 0 vs. H1: h = 1 :

= 41.97

29.51

−T log(1 − λ1 ) = 25.99

H0: h = 1 vs. H1: No restrictions: −T
H0: h = 1 vs. H1: h = 2 :

3
i=1 log(1 − λi )

5% critical value

3
i=2 log(1 − λi )

20.77

= 15.98

15.20

−T log(1 − λ2 ) = 12.95

14.03

Notes: T = 201, where T is the sample size, λ1 = 0.1213, λ2 = 0.0624, and λ3 = 0.0148, where
the λi ’s refer to the square of the canonical correlations.

Saikkonen’s Estimator for Cointegrated Regressions
Variable

Null Hypothesis

Estimates

s

−1

−1

m

0

0

n

1

0.95 (0.02)

Wald Test for the Cointegrating Vector (−1, 0, 1): χ2 = 12.08.
[1]

beginning in the early 1990s in which this ratio seems to have started to fall.4
On the whole, however, it would be difficult to argue that the inventory:sales
ratio does not fluctuate around a constant mean. Alternatively, Figure 1 loosely
suggests that inventories and sales are cointegrated.
A univariate analysis of the three variables plotted in Figure 1 suggests
that they can each be characterized as an I(1) process with positive drift. Our
concern, however, is mostly with a multivariate analysis of the relationship
described by (9). Accordingly, Tables 1a and 1b present a number of statistics
that relate to the three-variable system, Y t = (st , mt , nt ) .
Panel a of Table 1 shows the largest eigenvalues of the companion matrix
associated with a VAR(6) estimated with a constant and a linear trend. Under
the assumption that only one cointegrating restriction links the variables in Y t ,
the companion matrix should have two unit eigenvalues corresponding to two
common stochastic trends. All other eigenvalues should be less than one in
modulus. These results follow directly from Stock and Watson’s (1988) common trends representation. The point estimates displayed in Table 1a indeed
4 Regressions

were also run over the period 1947:1 to 1990:1 to check for robustness with
respect to this feature of the data. Our empirical results, however, were largely unaffected.

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Federal Reserve Bank of Richmond Economic Quarterly

support the hypothesis of two common trends or, alternatively, that there exists
a single cointegrating restriction in our three-variable system.
Panel b presents more formal tests of cointegration developed by both
f
Stock and Watson (1988) and Johansen (1988). Stock and Watson’s q τ (k, m)
statistic tests the null of k unit roots against the alternative of m, (m < k), unit
roots using Stock and Watson’s (1989) dynamic Ordinary Least Squares (OLS)
procedure. Specifically, if there are n variables and h cointegrating vectors,
the procedure estimates h regression equations containing a constant, n − h
regressors in levels, as well as leads and lags of the first differences in these
regressors as right-hand-side variables. The τ subscript indicates that a linear trend is included in the regressions. In panel b of Table 1, we note that
f
the q τ (3, 2) statistic is consistent with rejecting the null of no cointegrating
restrictions against the alternative of at least one cointegrating restriction. In
particular, the P values are generally small regardless of the number of lags
f
used in the dynamic OLS equations. In addition, the q τ (2, 1) statistic suggests
rejecting the alternative of two cointegrating restrictions against the null of one
cointegrating vector. Put together, these results provide evidence of only one
cointegrating vector in our three-variable system.
Panel b also presents results obtained from Johansen’s Likelihood Ratio
Trace and Maximum Eigenvalue statistics. For these statistics, we can think of
the number of unit roots as the number of variables less the number of cointegrating relations. Consider first the likelihood ratio test for the null of zero
cointegrating relation against the alternative of three cointegrating relations. For
this test, the likelihood ratio statistic, 2(L1 −L0 ), is 41.97, which is greater than
29.51. Therefore, the null hypothesis is rejected at the 5 percent significance
level. Similarly, the test statistic for the null of zero cointegrating restriction
against the alternative of one restriction is 25.99 > 20.77. It follows that the
null hypothesis of no cointegration is rejected by this second test as well.
To see whether a second cointegrating relation potentially exists, consider
the likelihood ratio test for the null of h = 1 against the alternative of h = 3.
In this case, the test statistic is 15.98 > 15.20 so that the null hypothesis is, in
fact, rejected at the 5 percent significance level. However, the likelihood ratio
test statistic for the null of one cointegrating relation against the alternative of
two relations is 12.95 < 14.03. Therefore, although the Johansen tests generally
suggest one cointegrating relation, they also offer conflicting evidence as to the
presence of a second cointegrating restriction.
Finally, panel b gives an estimate of the cointegrating relation associated
with the vector of variables (st , mt , nt ) using Saikkonen’s (1991) procedure.5
5 This

procedure is essentially that of dynamic OLS. In this case, the regression involves
the level of sales as the dependent variable; as right-hand-side variables it involves the level of
inventories, a constant but no deterministic trend, as well as leads and lags of the differences in
inventories.

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

71

Although the Wald Statistic suggests rejecting the null hypothesis that the
cointegrating vector is proportional to (−1, 0, 1), the point estimates are broadly
consistent with the notion that the inventory:sales ratio is stationary.

3.

ESTIMATION OF A THREE-VARIABLE SYSTEM

The results presented in this section are based on the estimation of the Vector
Error Correction Model (VECM) implied by equation (9). Each regression
equation is estimated using six lags of ∆st , ∆mt , the error-correction term
nt − st , as well as a constant. As we indicated earlier, the triangular nature of
the long-run multiplier matrix and the assumption that the structural error terms
are mutually uncorrelated allows us to recursively estimate each equation in the
system. In estimating the money growth equation, the residual from the sales
growth regression was used to instrument for contemporaneous endogenous
variables. Similarly, in estimating the inventory:sales ratio equation, the residual from the money growth regression was added to the list of instruments.6
Estimated Structural Impulse Responses
Figure 2 displays the estimated impulse response function obtained from the
system summarized by (9). The 95 percent confidence bands also displayed
in Figure 2 were computed using Monte Carlo simulations. These simulations
were carried out by using draws from the normal distribution for the technology,
money growth, and temporary real demand innovations. One thousand Monte
Carlo draws were completed.
We now interpret these impulse response functions in terms of a productionsmoothing model with nominal rigidities. Let us first focus our attention on the
effect of a money growth innovation. In a framework with staggered prices,
Hornstein and Sarte (1998) suggest that in response to a money growth shock,
sales should contemporaneously rise before gradually reverting back to the
steady state. To see why this is true, note that a firm that does not adjust its
price following an increase in nominal demand naturally sees its sales rise on
impact. Moreover, its relative price continues to decline as long as its nominal
price remains fixed. These results occur because other firms eventually increase
their price so that the price level rises. Firms that do adjust their price immediately following the money growth innovation set their price high enough so
that their sales initially fall. In the aggregate, however, the latter firms typically
represent a small fraction of the total number of firms and aggregate sales
initially rise. Looking at the point estimates of the sales response to a money
6 See

Shapiro and Watson (1988) for details of how to estimate just-identified SVARs using
an instrumental variables approach.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2

innovation in Figure 2a, we see that sales actually fall when the shock occurs.
However, immediately following this initial response, sales increase before reverting back to the steady state. This dynamic adjustment in sales, therefore, is
almost compatible with the predicted response in Hornstein and Sarte (1998).
The main difference lies in the contemporaneous response that appears negative
in the data. On the one hand, this difference may be evidence that a relatively
nontrivial fraction of firms actually do adjust their price at the time of the
shock. On the other hand, the upper bound of the confidence interval suggests

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

73

a positive initial response of sales as expected. Furthermore, the subsequent
dynamic adjustment in sales is consistent with that of a sticky-price model.
We now turn to the dynamic response of the inventory:sales ratio to a
money growth innovation. In theory, the combination of production smoothing and sticky-price forces predicts that inventories should rise on impact in
response to a positive nominal shock. Because the inventory:sales ratio is constant in the steady state, and nominal shocks have no long-run effect on sales,
inventories then gradually fall back so as to meet some target inventory:sales
ratio. Alternatively, changes in the inventory:sales ratio fall back to zero. To
understand the nature of this dynamic adjustment, recall that a firm that does
not adjust its price in response to a nominal shock initially experiences a rise
in sales. Afterwards, sales continue to rise as long as its price remains unchanged. Given that this firm also smooths production over its pricing cycle,
it must initially increase production by more than sales. This large initial increase in production effectively allows output to grow relatively slowly over
the remainder of the firm’s pricing cycle. Therefore, firms that keep their price
fixed following a money growth shock increase their inventory holdings at
the outset. Now, what about firms that do change their price at the time the
shock occurs? Since these firms also smooth production over their pricing cycle, they initially reduce output by less than the fall in sales they experience.
Consequently, inventory holdings increase for the latter firms as well. In the
aggregate, therefore, inventory holdings should unambiguously rise on impact
in response to a positive nominal demand shock. Looking at the response of
the inventory:sales ratio to a money shock in Figure 2b, we see that it rises on
impact by approximately 1 percent. Since sales contemporaneously fall by 0.4
percent in response to the same shock, the level of inventories does indeed rise
at the outset by about 0.6 percent as suggested by our sticky-price framework.7
When examining the dynamic adjustment of sales and inventories to a
technology shock, Hornstein and Sarte (1998) suggest that total sales should
contemporaneously rise in response to a technology shock. This result is mainly
driven by the firms that respond to the innovation. In particular, a productivity
increase implies a fall in the marginal cost of production. Firms that immediately respond to the shock, therefore, lower their price and see their sales
increase. Furthermore, during the transition, aggregate sales continue to rise
monotonically to a higher steady state as more firms also reduce their price.
The sales response in Figure 2c indeed broadly suggests that sales first increase
in response to a technology shock and eventually reach a new higher steadystate level. The dynamic adjustment, however, is not monotonic. Specifically,
sales appear to overshoot the new steady state twice during the early portion

7 Letting

n:s denote the inventory:sales ratio, observe that the change in inventories is then
given by ∆n = ∆n:s + ∆s = 0.01 − 0.004 = 0.06.

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 2

of the transition phase. This oscillatory impulse response in sales is somewhat
difficult to reconcile with a standard sticky-price model. It may suggest that
some firms find it difficult to know exactly where to set a new price following
the shock. In particular, the overshooting suggests that these firms may initially
set their price too low. The subsequent corrective rise in price that would then
occur causes a temporary decline in sales. It remains that, as expected, sales
ultimately rise in the long run relative to their initial level.

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

75

As with the dynamic adjustment to a monetary innovation, the response of
inventories to a technology shock hinges on the production-smoothing behavior
of firms. Consider first the behavior of firms that adjust their price immediately
following the shock. As we have just seen, these firms initially lower their price
so that their sales at first increase. However, these firms then face declining
sales over the remainder of their pricing cycle. This result stems from the fact
that, following the initial adjustment, their price remains fixed while the price
level continues to fall. Therefore, firms that adjust their price on impact also
raise production but by a lesser amount than the initial sales increase. These
firms consequently experience a fall in inventory holdings.
For the firms that do not adjust their price at the time the shock occurs,
sales initially decrease as adjusting firms cause the aggregate price level to fall.
Since these firms anticipate further declines in sales while their price remains
fixed, they reduce production on impact by more than the initial decline in
sales. Therefore, inventory holdings contemporaneously fall for the latter firms
as well. It follows that aggregate inventory holdings should unambiguously
decline immediately following the technology shock. As sales eventually rise
to a higher steady state, inventories should then rise by the same amount in the
long run to keep the inventory:sales ratio constant.
When we examine the inventory:sales ratio response to a technology shock
in Figure 2d, we see that it falls by approximately 0.8 percent on impact in
response to the innovation. Given the 1.2 percent rise in sales that contemporaneously follows the same technology shock, inventories then rise by about
0.4 percent at the time the shock hits. Since, on the contrary, a framework with
sticky prices predicts an unambiguous initial decline in inventory holdings, the
implied initial reaction of inventories in the data represents evidence against
such a framework. However, we note that the lower bound of the 95 percent
confidence interval for the impact response of sales is relatively small at about
0.35 percent. Because of the contemporaneous fall in the inventory:sales ratio
by 0.8 percent, the level of inventories would also fall if we were to use the
lower bound on the contemporaneous sales response. The latter observation
mitigates the evidence against a sticky-price framework implied by the point
estimates.
Thus far, the dynamic adjustment of sales and the inventory:sales ratio
to both nominal demand and technology shocks are roughly consistent with
what might have been predicted from a rigid price framework. The two main
exceptions are (1) the implied initial response of inventories to technology
shocks, and (2) the extremely sluggish dynamic adjustment of both sales and
the inventory:sales ratio to shocks, as shown in Figure 2. In the case of the
inventory:sales ratio’s response to a money innovation, for instance, the halflife of the impulse is approximately 25 quarters or more than six years. While
typical sticky-price models deliver nowhere near this kind of sluggishness
in real variables, Ramey and West (1997) also note that the inventory:sales

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relationship exhibits a very high degree of persistence. In fact, these findings
turn out to be a reflection of a well-known problem in the empirical literature on
inventory behavior. Specifically, Feldstein and Auerbach (1976) point out early
on the incongruity inherent to the notion that firms may take years to adjust to
a sales shock, while the widest swings in inventory levels seldom amount to
more than a few days’ production. More recently, Blinder and Maccini (1991,
p. 81) write that “one major difficulty with stock-adjustment models is that
adjustment speeds turn out to be extremely low,” a comment referring to the
estimation of stock-adjustment equations generally.8 They further note that “a
natural reaction is that the slow estimated adjustment speeds must be an artifact
of econometric biases. One potential source of such bias is omitted variables.”
As with the estimation of stock-adjustment equations, we should be conscious
that the structural equations we estimate may also be subject to the latter source
of bias.
Forecast Error Variance Decompositions
Having investigated the way in which the variables in (9) empirically respond
to various structural shocks, we now wish to gauge the importance of each
of these shocks in determining short-run variations in the data. We have seen
that the dynamic adjustment of sales and the inventory:sales ratio, and hence
inventories, to a money shock is generally consistent with the predictions of a
sticky-price framework in which firms also smooth production. In some sense,
however, this concept may be of secondary importance to a monetary policymaker if money shocks only play a small role in determining real variables.
King, Plosser, Stock, and Watson (1991), for example, present compelling
evidence to that effect in the case of aggregate variables. Of primary importance
is the role played by each structural shock in determining short- and mediumrun fluctuations in the data, as reflected when decomposing the variance of the
k-step-ahead forecast errors.
Consider the moving-average process given by (9) and let T(L) = T0 +
T1 L + T2 L2 + . . . + Tk Lk + . . . , Lj xt = xt−j , while E(εt εt ) = Σε . Then, we
may write the k-step-ahead forecast error in Y as
k

Y t+k − Et−1 Y t+k =

Tk εt+k−j .

(11)

j=0

For our purposes, what we wish to assess is the fraction of variance in
the left-hand side of equation (11) that is attributable to each of the structural
shocks. In other words, we ask the question: To the degree that the actual
data differ from the optimal forecast, which of the structural shocks is most

8 See

Lovell (1961) for instance.

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

77

Table 2 Decompositions of Forecast Error Variance
a. Fraction of Sales Forecast Error Variance Attributed to Shocks
Horizon

Technology Shock

1
4
8
12
16
20
∞

0.70
0.92
0.97
0.99
0.99
0.99
1.00

(0.21)
(0.17)
(0.13)
(0.11)
(0.09)
(0.07)
(0.00)

Money Shock
0.22
0.06
0.02
0.01
0.01
0.01
0.00

(0.16)
(0.12)
(0.09)
(0.08)
(0.07)
(0.05)
(0.00)

Real Demand
Shock
0.08
0.02
0.01
0.00
0.00
0.00
0.00

(0.13)
(0.12)
(0.07)
(0.05)
(0.05)
(0.02)
(0.00)

b. Fraction of Inventory:Sales Ratio Forecast Error Variance
Attributed to Shocks
Horizon
1
4
8
12
16
20
∞

Technology Shock
0.90
0.85
0.79
0.75
0.73
0.72
0.69

(0.27)
(0.26)
(0.22)
(0.22)
(0.21)
(0.21)
(0.20)

Money Shock
0.08
0.01
0.01
0.05
0.06
0.09
0.14

(0.26)
(0.25)
(0.25)
(0.24)
(0.25)
(0.24)
(0.24)

Real Demand
Shock
0.02
0.14
0.20
0.20
0.21
0.18
0.17

(0.21)
(0.24)
(0.24)
(0.24)
(0.24)
(0.24)
(0.25)

responsible for this difference? Note that our identifying restrictions imply that
100 percent of the sales forecast error variance is explained by the technology
shock at the infinite horizon. At shorter horizons, however, both nominal and
real demand disturbances are allowed to contribute to fluctuations in sales. Table
2, panel a, shows that in fact, this contribution is relatively minor.9 At the onequarter horizon, technology shocks already explain 70 percent of the forecast
error variance in sales. The bulk of the remaining variance is attributable to
money growth shocks while real demand disturbances play a very small role.
At the four-quarter horizon, technology shocks account for 92 percent of the
variation in sales. At the three-year horizon, virtually all of the forecast error
fluctuations in sales can be explained by technology shocks.
Focusing on fluctuations in the inventory:sales ratio, we again find that they
tend to be dominated by real disturbances. In this case, however, it is interesting
that as the forecast horizon lengthens, the important role played by technology
9 Standard

errors are in parentheses.

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Federal Reserve Bank of Richmond Economic Quarterly

innovations diminishes somewhat at the expense of real demand disturbances.
Also, by contrast to sales above, forecast errors in the inventory:sales ratio
are not restricted to be uniquely driven by real disturbances in the long run.
As a result, we find that at the infinite horizon, monetary disturbances explain
approximately 14 percent of the forecast error in the inventory:sales ratio. While
this number may not be too consequential, it is slightly larger than most other
findings concerning the role of nominal shocks in determining the behavior of
real variables. Gali (1992), for example, finds that after 20 quarters, money
supply shocks only explain 9 percent of the variation in aggregate output.
Historical Decompositions
The variance decompositions in Table 2 show the relative importance of each
structural shock in explaining variations in both sales and the inventory:sales
ratio on average. It is also interesting to note that these shocks may matter
more or less during various historical episodes. Figures 3 and 4 plot the historical forecast error decompositions in sales and the inventory:sales ratio at
the 12-quarter horizon. This 12-quarter horizon concept of the business cycle
is adopted from King, Plosser, Stock, and Watson (1991).
Figure 3 confirms that while money shocks have historically played a small
role in explaining fluctuations in sales, technology shocks have played a more
substantial role. Interestingly, this finding appears to remain consistent throughout the entire sample period considered. Temporary real demand disturbances
take on relatively more importance in explaining sales fluctuations in the 1990s.
On the whole, the largest forecast errors occur in the mid-1970s and, as might
have been expected, coincide with the oil price shock of 1973.
The latter observation also applies to the forecast errors in the inventory:sales ratio as suggested by Figure 4. Again we note that money generally
plays a small role in driving inventory:sales ratio fluctuations, as implied by the
variance decompositions in Table 2. In contrast, we also find that the importance
of the monetary component, even if small on average, noticeably increases in
the early 1990s. Put another way, Figure 4 suggests that even if monetary
fluctuations have traditionally represented a small portion of fluctuations in the
inventory:sales ratio, this does not imply that monetary disturbances are always
unimportant.

4.

CAUTIONARY REMARKS

An important part of the empirical analysis above has been the assumption
that the inventory:sales ratio is stationary around a constant mean. As we have
seen, various cointegration tests have generally confirmed this hypothesis for
manufacturing inventories. Moreover, the notion of a stationary ratio is typically explained on the grounds that stock-outs are costly and, therefore, that

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory
Figure 3 Historical Forecast-Error Decomposition: Sales

79

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Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Historical Forecast-Error Decomposition: N:S Ratio

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

81

Figure 5 Logarithm of the Inventory:Sales Ratio (n−s)

firms generally try to meet some target inventory:sales ratio in the long run.
The fall in the inventory:sales ratio that begins in the early 1990s (Figure 1)
is sometimes taken as evidence of the just-in-time inventory method taking
hold in the United States. The inventory:sales ratio in both the wholesale and
retail sectors, however, reveals a much different story. Figure 5 suggests that
for much of the period under consideration, the inventory:sales ratio in both
these sectors has actually trended upwards. Both the ratios seem to stabilize
in the early 1990s, perhaps again because of the widespread emergence of the
just-in-time method. Nevertheless, it remains that much of the increase in the
inventory:sales ratio up to the early 1990s, in both the wholesale and retail
sectors, represents somewhat of a puzzle.

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To explain this puzzle, one can speculate that, over time, consumers have
gained easier access to a wide variety of goods through improved means of
communication and transportation. As a result, back orders for any one business are less likely to arise since consumers can simply acquire the same
goods elsewhere. So not having goods on hand more readily results in lost
sales, which effectively drives up the cost of stock-outs and, consequently, inventory:sales ratios. Alternatively, consistent improvements in technology may
simply have reduced storing costs over time. This would have made it easier for wholesalers and retailers to avoid stock-outs and is directly consistent
with increasing inventory:sales ratios. Food products, for instance, have become increasingly storable because of consistent innovations in preservatives
technology. Whatever the case may be, Figure 5 makes it clear that traditional
theories of inventory behavior need to be amended to account for the data in the
wholesale and retail sectors. Perhaps a focus away from production smoothing
is even necessary.

5.

CONCLUSIONS

We have used an SVAR to acquire some insight into the dynamic responses of
manufacturing sales and inventories to both nominal demand and real supply
shocks. We assumed that money is neutral in the long run and, moreover, that
the inventory:sales ratio can be properly characterized as a stationary process
without trend. We then found that the estimated dynamic adjustments to nominal demand and real supply shocks are generally consistent with those of an
equilibrium model of inventory behavior with inflexible prices. However, the
degree of sluggishness exhibited by both sales and the inventory:sales ratio,
and hence inventories, in response to these shocks is much greater than that
suggested by current sticky-price models. The latter findings confirm earlier
observations by Blinder and Maccini (1991) and, more recently, Ramey and
West (1997).
We also used our empirical framework to gauge the relative importance
of both nominal and real disturbances as sources of fluctuations in the manufacturing sector. The results indicate that nominal shocks generally contribute
little to the forecast error variance in both sales and the inventory:sales ratio
at all horizons. Instead, fluctuations in real variables are mainly driven by real
disturbances. In addition, the latter results appeared to hold consistently at the
business cycle frequency throughout the sample period under consideration.

P.-D. G. Sarte: Fluctuations in Manufacturing Sales and Inventory

83

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