Full text of Economic Quarterly (Federal Reserve Bank of Richmond) : Winter 1995, Vol. 81, No. 1

View original document
The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Is “High” Capacity
Utilization Inﬂationary?
Mary G. Finn

C

apacity utilization in U.S. industry features prominently in discussions
of inﬂation. This prominence derives from the widely held viewpoint
that “high” rates of capacity utilization are tantamount to resourceshortage conditions or “bottlenecks” that inevitably erupt into price inﬂation.
For instance, an article in Citicorp’s Economic Week (January 18, 1994) argues:
“In the past, a utilization rate swinging up toward the 84%–85% range was a
source of much anxiety. Usually when the rate got that high, production bottlenecks started to appear. . . . Shortages developed. And soon, key price indexes
were shooting up.” In the American Banker (January 12, 1994), Stephen Davies
points out: “Economists say that, historically, there has been a connection between a healthier industrial sector and rising prices. That’s because factories
start to run into bottlenecks in which supplies and labor are short” (p. 1). And in
an article in Barron’s (June 20, 1994), Gene Epstein states, “Capacity utilization
should remain below 85%, the assumed inﬂationary danger zone” (p. 48). A
concomitant viewpoint is that when capacity utilization is “low,” the economy
is in the inﬂationary safe zone. The threshold deﬁning “high” and “low” rates
of capacity utilization is often 85 percent, as the above quotations exemplify.
The purpose of the present study is to outline the theory underlying these
popular views and to evaluate it in terms of its ability to explain the facts about
the U.S. economy. To accomplish this task, the article proceeds as follows. Section 1 isolates and discusses the core features of the theory. Section 2 presents
the evidence on the relationship between capacity utilization and inﬂation and
related evidence on the linkage between cyclical GDP and inﬂation. Section
3 assesses the theory in terms of its ability to explain the evidence. Finally,
Section 4 concludes with a summary and suggestions for future research.

The author thanks Michael Dotsey, Thomas Humphrey, Peter Ireland, Anatoli Kuprianov,
Jeffrey Lacker and Alan Stockman for many helpful suggestions and comments. The analysis
and conclusions are those of the author and do not necessarily reﬂect the views of the Federal
Reserve Bank of Richmond or the Federal Reserve System.

Federal Reserve Bank of Richmond Economic Quarterly Volume 81 Winter 1995

1

2

Federal Reserve Bank of Richmond Economic Quarterly

1.

THE THEORY

The theory supporting the view that high rates of capacity utilization are inﬂationary has not been fully articulated. Yet, it seems to be a variant of traditional
Keynesian theory. Figure 1 illustrates the key elements of this theory. In this
ﬁgure, P is the general price level, Y is aggregate real output, Y ∗ is the fullemployment level of output, Y c is the capacity level of output, D1 and D2 denote
alternative aggregate demand curves, and S is the aggregate supply curve.
Figure 1 shows that the intersection between aggregate demand and supply
determines the price level and output. It is a snapshot of the economy over
the time horizon relevant for the study of business cycles, the short run. For
this horizon, the ﬁxed capacity output level, Y c , provides the effective “lid” on
the economy. Cyclical ﬂuctuations in output correspond to deviations of actual
output, Y, from the constant full-employment output level, Y ∗ . Resources are
less than fully employed when Y < Y ∗ , while they are more than fully employed
when Y > Y ∗ —that is, people and capital work overtime. The cyclical ﬂuctuations in output are driven by shifts in the aggregate demand curve, stemming
from changes in consumption, investment and government expenditures or in
the stock of money. The ﬁgure abstracts from economic growth, a long-run
phenomenon, which can be imagined as increasing Y ∗ and Y c gradually over
time and also shifting the demand and supply curves outward slowly over time.
Such growth is due to improvements in technology and increases in the stock
of capital and the work force.
In this theory, the aggregate supply curve is nonlinear. This nonlinearity
implies that the relationship between the price and output responses to demand
Figure 1 A Keynesian Theory

P
D2

S

D1

P

2

P1

+

Y1

Y*

Y2

Yc

Y

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

3

shifts depends on the level of real output or, alternatively, the level of overall
resource use in the economy. At low levels of output, say Y = Y1 , resources
such as labor and capital are underemployed. Firms can obtain as much of these
resources as they wish at constant wages and rents. Therefore, ﬁrms are willing
to supply whatever amount of ﬁnal goods is demanded at the existing price
level, P1 . When output is high, say Y = Y2 , the story is different. Now resources
are more than fully employed—capital and labor are working overtime. If ﬁrms
want to increase resource usage, they must offer higher wages and rents. Consequently, ﬁrms are willing to accommodate demand expansions only if they
can pass along the resource price increases to consumers in the form of higher
ﬁnal goods prices. Moreover, these price increases occur at an increasing rate
as Y c is approached and resources become increasingly overworked. In short,
at low output levels, a rise in demand causes a rise in output with little or no
accompanying price inﬂation. When output is high, demand expansions cause
both output increases and rising inﬂation.1
The Federal Reserve’s capacity utilization rate for total U.S. industry is the
ratio of actual to capacity industrial production. The capacity industrial production index attempts “to capture the concept of sustainable practical capacity,
which is deﬁned as the greatest level of output that a plant can maintain within
the framework of a realistic work schedule, . . . assuming sufﬁcient availability
of inputs to operate the machinery and equipment in place” (Federal Reserve
Statistical Release G.17, March 15, 1994, p. 18). Thus, the capacity utilization
rate intends to measure Y/Y c . The above theory explains the attention devoted
to tracking the capacity utilization rate in discussions of inﬂation.
The isolation of the central features of the theory immediately invites some
criticism and questions. First, the concept of capacity output relies on the impossibility of quickly expanding the work force, capital stock and technological
knowledge. But the work force is elastic even in the short run—for example,
retired workers and young adults acquiring education can be induced to enter the work force. New investments and bringing back “on line” previously
obsolete capital can rapidly increase the stock of capital. Improvements in
technology, leading to more efﬁcient production techniques and labor-saving
capital equipment, can quickly occur as well. These points are emphasized
in recent discussions in The Wall Street Journal (Harper and Myers, June 6,
1994) about “companies forced to buy equipment to keep up with technological
improvements” (p. A6) and in The New York Times (Uchitelle, April 24, 1994):
“Labor-saving machinery permits the extra production with fewer workers”
(p. 24).
1 Notice

that there is an ambiguity in the theory. It is not clear how the theory develops the
relationship between the level of output and the rate of change of prices. A strict interpretation
of the theory’s underlying arguments points to a relationship between the levels of output and
prices.

4

Federal Reserve Bank of Richmond Economic Quarterly

Second, the short-run changes in the work force, capital stock and technology will impact on deviations of Y from Y ∗ , since they shift the aggregate
supply curve. Therefore, cyclical output ﬂuctuations may not always be demand
driven. Indeed, quantitative real business cycle theory shows that between 54
and 70 percent of the postwar output ﬂuctuations in the United States can be
explained by short-run changes in technology (see Kydland and Prescott [1991]
and Aiyagari [1994]).
Third, the possible variation in the relationship between price and output
responses to demand shocks rests on the presumed existence of a critical level,
or levels, of output at which underemployment of resources sets in and prices
no longer adjust so as to clear the goods and factor markets. It is difﬁcult to
see why such critical levels should exist. The theory is silent on this issue.
Thus, there are some good reasons for wondering whether the theory
connecting high capacity utilization with inﬂation provides a useful guide in
explaining the real world. Therefore, the questions arise: What are the facts
on the relationship between capacity utilization and inﬂation? Is the Keynesian
theory consistent with these facts?

2.

THE EVIDENCE

The Empirical Data
The evidence examined here involves seasonally adjusted, quarterly data for
the United States over the period 1953:1–1994:1. A complete description of
the data is in the appendix. The individual series will be gradually introduced
into the discussion.
Regarding the above theory, two theoretical variables measure the overall
degree of resource utilization: Y/Y c and Y/Y ∗ . Trend output growth does not
affect these variables since it affects the numerators and denominators to the
same extent. Consequently, movements in these variables are purely cyclical in
nature. Furthermore, they bear a perfect positive relationship to one another—
stemming from the common cyclical variation in Y. Also notice that the average
values of Y/Y c and Y/Y ∗ , when taken over the long run, are Y ∗ /Y c and Y ∗ /Y ∗
respectively—since in the long run the temporary cyclical deviations of Y from
Y ∗ do not, by deﬁnition, occur.
The closest empirical counterparts to Y/Y c and Y/Y ∗ are the capacity utilization rate and cyclical per-capita GDP (henceforth referred to as cyclical
GDP), respectively. The former measure was described earlier. Cyclical GDP
is the percentage deviation of per-capita GDP from its smoothly evolving time
trend.2 Both empirical variables exhibit purely cyclical variations. The mean
2 This trend is derived by using the Hodrick-Prescott ﬁltering method (with the “smoothing”
parameter value set equal to 1600). See Kydland and Prescott (1990) for a description of the
method.

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

5

value of utilization provides an estimate of Y ∗ /Y c ; the mean value of cyclical
GDP is zero (i.e., it is an estimate of the logarithm of Y ∗ /Y ∗ ). In contrast to the
perfect sychronization between movements in the theoretical variables, Y/Y c
and Y/Y ∗ , movements in the utilization rate and cyclical GDP may diverge for
at least two reasons.3,4 First, the underlying output measures are different—the
utilization rate uses industrial production, while cyclical GDP uses the more
comprehensive production measure, GDP. Second, cyclical GDP captures output relative to its smoothly evolving trend, while capacity utilization is output
relative to its capacity level. Even though the theory assumes that the trend
growth in Y ∗ and Y c is the same, there is no reason why this has to hold in the
data.
Figure 2 shows the capacity utilization rate and cyclical GDP. The two
series move together closely but not perfectly. The contemporaneous correlation between the series is 0.74.5 Movements in the two variables are sufﬁciently
different that both merit attention as alternative measures of resource utilization
from the point of view of the Keynesian theory.
The capacity utilization rate is shown with the inﬂation rate in Figure 3.
Inﬂation is measured by the quarter-to-quarter annualized percentage change
in the CPI. Notice that there are time periods when the utilization rate is
both high (in excess of 85 percent) and rising and inﬂation is also rising—
for example, during 1964 and 1972. This is consistent with the theory. But,
there are also periods of high utilization when utilization and inﬂation move
in opposite directions. For example, early in 1955 and 1965, rising utilization
simultaneously occurs with falling inﬂation, and during most of 1973–74 and
1979, falling utilization coincides with rising inﬂation. Furthermore, during
much of the time when utilization is low and variable, inﬂation exhibits substantial variation. These episodes are inconsistent with the theory. A similar
story emerges regarding cyclical GDP and inﬂation. It is depicted in Figure 4.
The upshot is that the linkage between high utilization or high cyclical GDP
and inﬂation is not immutable; neither is the linkage between low utilization
or low cyclical GDP and inﬂation.

that cyclical GDP measures log(Y/Y ∗ ). The exponential of cyclical GDP gives a
measure of Y/Y ∗ , which is in comparable units to the utilization rate. This exponential transformation makes no substantive difference to the quantitative analysis, a reﬂection of the transformation
being close to a linear one (at least for the underlying range of variation in cyclical GDP). It is
not undertaken for the quantitative analysis discussed in the article since it is common practice
to measure cyclical GDP as a percentage deviation, not a ratio (again see Kydland and Prescott
[1990]).
4 See Shapiro (1989) for a detailed description and discussion of the capacity utilization rate.
5 The correlation between capacity utilization and cyclical per-capita industrial production is
higher. It is 0.82. This shows that the second reason explaining the difference between utilization
and cyclical per-capita GDP is the more important of the two reasons.
3 Notice

6

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Capacity Utilization Rate and Cyclical Per-Capita GDP

6

95

Capacity Utilization Rate (left scale)
Cyclical Per-Capita GDP (right scale)
4
90

2

0

82.58
80

Percent

Percent

85

-2

75

-4

70
1953:1 57:1

-6
61:1

65:1

69:1

73:1

77:1

81:1

85:1

89:1

93:1

+
Notes: (1) Cyclical per-capita GDP is the percentage deviation of per-capita GDP from its HodrickPrescott trend component. (2) The horizontal line is drawn at the mean of both variables.

Regression Analysis
Here regression analysis quantiﬁes the average historical relationships between
inﬂation and each of the two real economic activity variables. The analysis also
tests the signiﬁcance and possible asymmetries in those relationships.
The regression equation with capacity utilization is
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βh uh + βl ult + t .
t

(1)

Including cyclical GDP, the regression equation is
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γh y h + γl y lt + vt .
t

(2)

πt is the time t inﬂation rate, uh (ult ) is the time t utilization rate series pert
taining to high (low) utilization rates and y h (y lt ) is the time t cyclical GDP
t
series corresponding to high (low) cyclical GDP. These high and low series
are explained more fully below. t and vt are disturbance terms at time t. The
αi (i = 0, 1, 2, 3), βi (i = h, l) and γi (i = h, l) are parameters.

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

7

Figure 3 Capacity Utilization Rate and CPI Inﬂation

105

100

18

Capacity Utilization Rate (left scale)
CPI Inflation (right scale)

16
14
12
10

90

8
85

Percent

Percent

95

6

82.58
4

80

2
75
0
70
1953:1 57:1

-2
61:1

65:1

69:1

73:1

77:1

81:1

85:1

89:1

93:1

+
Notes: (1) CPI inﬂation is measured quarter to quarter at an annualized rate. (2) The horizontal
lines are drawn at the mean of the utilization rate and zero CPI inﬂation.

The equations include three lagged values of the inﬂation series. Their
inclusion is essential to ensure an adequate speciﬁcation—omitting lagged inﬂation results in equations with almost no explanatory power. The choice of
lag length is predicated on a sequence of F-tests, which establish that additional
lagged values of the inﬂation series are statistically insigniﬁcant. Allowing inﬂation to depend on its lagged values constitutes a generalization of the simple,
static Keynesian theory to admit persistence in the inﬂation process.
In line with the theory, it is the contemporaneous values of the real economic activity variables that enter into the regression equations. Their coefﬁcients indicate the magnitude of the average, contemporaneous relationship
obtaining between them and inﬂation, given the past path of inﬂation. A sequence of F-tests establishes that lagged values of the real economic activity
series are insigniﬁcant once the contemporaneous values are accounted for in
the regression equations.
To test for possible asymmetry in the relationship between utilization
and inﬂation, one must ﬁrst specify a threshold value deﬁning high and low

8

Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Cyclical Per-Capita GDP and CPI Inﬂation
18

6

Cyclical Per-Capita GDP
(left scale)

16

4

14

CPI Inflation
(right scale)

12

2

8

0

Percent

Percent

10

6
-2

4
2

-4

0
-6
1953:1

-2
57:1

61:1

65:1

69:1

73:1

77:1

81:1

85:1

89:1

93:1

+
Notes: (1) Cyclical per-capita GDP is the percentage deviation of per-capita GDP from its HodrickPrescott trend component. CPI inﬂation is measured quarter to quarter at an annualized rate.
(2) The horizontal lines are drawn at the mean of cyclical per-capita GDP and zero CPI inﬂation.

utilization rates and then create corresponding high and low utilization rate
series. This exercise is accomplished as follows. Initially the utilization rate
is expressed in terms of deviations from its mean. If rt denotes the time t
¯
utilization rate and r its mean (equaling 82.58 percent), then ut = rt − r is the
¯
time t deviation of utilization from its mean. The threshold value is set at 85
percent since that value is the one most often used in media discussions. Two
new variables, uh and ult , are then derived as follows:
t
uh = ut if rt ≥ 85 percent and zero otherwise;
t
ult = ut if rt < 85 percent and zero otherwise.
Should a difference exist between the coefﬁcients of uh and ult , it signiﬁes a
t
difference in the relationship between inﬂation and each of high and low rates
of utilization.6
6 It

is important to express the utilization rate, rt , as a deviation from its mean before deriving
and ul series. Not doing so results in series, say uh and ul , that have a correlation equaling
¯t
¯t
the
t
−0.99, causing extreme multicollinearity problems in the regression equation. By contrast, uh and
t
ul are mildly correlated (with correlation 0.32), so multicollinearity problems do not arise.
t
uh
t

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

9

A similar procedure is followed regarding cyclical GDP, the time t value of
which is denoted by yt . By construction, yt has a zero mean. Using a threshold
value equal to one standard deviation above the mean of yt (equaling 0.0175),
one can derive the two variables, y h and y lt :
t
y h = yt if yt ≥ 0.0175 and zero otherwise;
t
y lt = yt if yt < 0.0175 and zero otherwise.
A difference between the coefﬁcients of y h and y lt would mean a difference in
t
the relationship between inﬂation and each of high and low cyclical GDP.7
The regression results for equation (1) are presented in the top panel of
Table 1. The coefﬁcients of uh and ult are βh = 0.10 and βl = 0.19. They
t
are individually signiﬁcant (that is, signiﬁcant at the 5 percent level). An Ftest of the hypothesis βh = βl strongly indicates nonrejection at signiﬁcance
level 0.28. In other words, the evidence suggests that the relationship between
inﬂation and utilization is the same regardless of whether or not the utilization
rate is high.
Setting βh = βl = β improves the precision of the estimation and therefore
leads to the preferred regression equation:
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut + t .

(3)

The ﬁndings for this equation are displayed in the lower panel of Table 1. The
coefﬁcient of ut is β = 0.15, which is signiﬁcant. The relationship between
capacity utilization and inﬂation is a signiﬁcantly positive one—with a one
percentage point increase in utilization being associated, on average, with a
0.15 percentage point increase in inﬂation.
The top panel of Table 2 gives the regression results for equation (2). The
coefﬁcients of y h and y lt are found to be γh = 0.58 and γl = 0.36, respectively.
t
Both are individually signiﬁcant. The hypothesis γh = γl cannot be rejected,
using an F-test, at a fairly high signiﬁcance level of 0.38. Once again, the
evidence reveals no important asymmetry.
Setting γh = γl = γ yields a more precise estimation, leading to the
preferred speciﬁcation:
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γyt + vt .

(4)

The lower panel of Table 2 shows the ﬁndings for this equation. The coefﬁcient of yt is γ = 0.45. It is signiﬁcant. A one percentage point increase in the
deviation of per-capita GDP from its trend path has been linked with a 0.45
percentage point increase in inﬂation, on average.
7 The

ﬁndings, to be discussed below, are robust to alternative choices of thresholds. More
exactly, for utilization threshold values equaling 82.58 percent and 80 percent, and for cyclical
GDP threshold values equal to zero and one standard deviation below the mean of yt (−0.0175),
similar ﬁndings obtain.

10

Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Regression Results for Inﬂation-Utilization Relationship,
1953:1–1994:1
Regression Equation (with utilization threshold value = 85%)
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βh uh + βl ul +
t
t
α0 = 0.38
(1.70)
2

R = 0.80

α1 = 0.58
(8.20)

α2 = −0.05
(−0.61)

α3 = 0.41
(5.77)

t

βh = 0.10
(2.07)

βl = 0.19
(4.04)

DW = 1.94

For test of hypothesis βh = βl , F(1, 159) = 1.19 (0.28)
Regression Equation (imposing βh = βl )
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut +
α0 = 0.25
(1.32)
2

R = 0.80

α1 = 0.58
(8.25)

α2 = −0.05
(−0.62)

α3 = 0.41
(5.73)

t

β = 0.15
(5.12)

DW = 1.94

Notes: (1) The number of observations is 165. (2) t-statistics are in parentheses below the correspond2
ing coefﬁcient values. (3) R is the regression goodness-of-ﬁt statistic. DW is the Durbin-Watson
statistic. (4) The F-statistic with the relevant degrees of freedom is denoted by F (. , .); its signiﬁcance
level follows in parentheses.

Table 2 Regression Results for Inﬂation-Cyclical GDP Relationship,
1953:1–1994:1
Regression Equation
(with cyclical GDP threshold value = one standard deviation above its mean)
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γh y h + γl y l + vt
t
t
α0 = 0.26
(1.21)
2

R = 0.81

α1 = 0.55
(7.94)

α2 = −0.05
(−0.58)

α3 = 0.40
(5.88)

γh = 0.58
(3.59)

DW = 1.99

For test of hypothesis γh = γl , F(1, 159) = 0.78 (0.38)
Regression Equation (imposing γh = γl )
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γyt + vt
α0 = 0.36
(1.94)
2

R = 0.81

α1 = 0.55
(8.02)
DW = 1.99

Notes: See notes for Table 1.

α2 = −0.04
(−0.54)

α3 = 0.41
(5.94)

γ = 0.45
(6.23)

γl = 0.36
(2.78)

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

11

Forecasting Analysis
Another way of gauging the linkages between inﬂation and each of the two
real economic activity variables is to assess the marginal predictive content of
the latter variables for inﬂation. This assessment is done as follows. First, ut−1
and yt−1 replace ut and yt , respectively, in equations (3) and (4) to give the
forecasting equations:
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βu t−1 +

t

(5)

πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γyt−1 + vt .

(6)

and

Omitting the real variables gives the univariate forecasting equation:
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + dt ,

(7)

where dt is the time t disturbance term. All right-hand-side variables in these
equations are in the time t − 1 information set. Consequently, the disturbance
terms t , vt and dt are the one-period-ahead forecasting errors.
The estimation results for equations (5)–(7), over the period 1953:1–1994:1,
are presented in Table 3. The ﬁndings for equations (5) and (6) are similar to
those for equations (3) and (4), respectively—a reﬂection of ut and yt being
highly autocorrelated.
By sequentially estimating (5), (6) and (7) over the 17 sample periods that
start in 1953:1–1990:1, increasing by one quarter at a time and ending with
1953:1–1994:1, one can generate a sequence of 17 one-period-ahead forecasting
errors for each equation.8 Computing and then comparing the mean squared
forecast errors (MSE) of the equations allows one to assess the marginal predictive content of utilization and cyclical GDP for inﬂation. Ireland (1995)
introduces this measure of predictive contribution.
Table 4 lists the MSE for each forecasting equation. The ratio of the MSE
for the utilization-inﬂation equation to the MSE for the univariate equation
is 0.87. In the case of the cyclical GDP-inﬂation equation, the ratio of MSE
is 0.82. These ﬁndings highlight that both utilization and cyclical GDP have
substantial predictive power for inﬂation, over and above that stemming from
the accounting for past inﬂation behavior. Also cyclical GDP has the edge on
the utilization rate in this respect.
The forecasting results naturally lead to the question: Does utilization have
predictive content for inﬂation once account is taken of past inﬂation and
cyclical GDP? To answer this, Table 3 gives the estimation ﬁndings (1953:1–
1994:1) for the forecasting equation that includes both utilization and cyclical
GDP:
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut−1 + γyt−1 + bt ,
8 Each

forecasting equation is stable over various forecasting periods.

(8)

12

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Regression Results for the Forecasting Equations,
1953:1–1994:1
Forecasting Equation with Utilization
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut−1 +
α0 = 0.34
(1.73)
2

R = 0.79

α1 = 0.58
(8.02)

α2 = −0.06
(−0.66)

α3 = 0.40
(5.45)

t

β = 0.13
(4.47)

DW = 1.88
Forecasting Equation with Cyclical GDP
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γyt−1 + vt

α0 = 0.45
(2.37)
2

R = 0.80

α1 = 0.54
(7.36)

α2 = −0.05
(−0.62)

α3 = 0.41
(5.78)

γ = 0.41
(5.34)

DW = 1.90
Univariate Forecasting Equation
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + dt

α0 = 0.36
(1.73)
2

R = 0.77

α1 = 0.66
(8.89)

α2 = −0.06
(−0.64)

α3 = 0.31
(4.21)

DW = 1.88
Forecasting Equation with Utilization and Cyclical GDP
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut−1 + γyt−1 + bt

α0 = 0.43
(2.23)
2

R = 0.80

α1 = 0.54
(7.36)

α2 = −0.05
(−0.63)

α3 = 0.42
(5.81)

β = 0.03
(0.69)

γ = 0.34
(2.83)

DW = 1.90

Notes: See notes for Table 1.

where bt is the one-period-ahead forecast error. The coefﬁcient of ut−1 , β =
0.03, is insigniﬁcant, while that of yt−1 , γ = 0.34, is signiﬁcant.
This ﬁnding suggests that utilization does not have independent predictive
content for inﬂation. But this conclusion, based on the relative signiﬁcance of β
and γ, must be tempered by the recognition that the high collinearity between
ut−1 and yt−1 , noted earlier, makes it difﬁcult to disentangle the relative predictive contributions of ut−1 and yt−1 . What is revealing is that the predictive
relationship between inﬂation and cyclical output, speciﬁed by equation (6),
is sufﬁciently stronger than that between inﬂation and utilization, given by
equation (5), so that yt−1 is still signiﬁcant even when ut−1 is included in the
forecasting equation for inﬂation.

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

13

Table 4 Forecasting Results for Inﬂation, 1990:1–1994:1
Forecasting Equation with Utilization, Equation (5)
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + βut−1
MSE5 = 1.4599
Forecasting Equation with Cyclical GDP, Equation (6)
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3 + γyt−1
MSE6 = 1.3723
Univariate Forecasting Equation, Equation (7)
πt = α0 + α1 πt−1 + α2 πt−2 + α3 πt−3
MSE7 = 1.6818
Mean Squared Error Ratios
MSE5 /MSE7 = 0.8681

MSE6 /MSE7 = 0.8159

Notes: (1) Forecasts are one-period-ahead forecasts. The number of forecasts is 17. (2) MSEi is
the mean squared forecasting error for forecasting equation i (i = 5, 6, 7).

3.

ASSESSING THE THEORY

The empirical evidence, presented above, on the relationship between inﬂation
and either capacity utilization or cyclical GDP provides a basis for evaluating the Keynesian theory that connects high resource usage with inﬂationary
conditions.
Analysis of the time proﬁles of the data establishes that utilization or cyclical GDP does not always move together with the inﬂation rate. There are several
episodes during which inﬂation and utilization or cyclical GDP move in the
opposite direction. For example, during 1973–74 and 1979, falling utilization
rates and cyclical GDP coincided with rising inﬂation rates. The theory cannot account for episodes such as these. Thus, the theory’s assumption that
the aggregate supply curve and associated capacity output level are constant
over the business cycle is called into question. For if the supply curve were
ﬁxed, inﬂation and utilization or cyclical GDP would always move in the same
direction.
The empirical regression analysis conﬁrms, on average, the theory’s prediction of a positive relationship between utilization or cyclical GDP and inﬂation.
A one percentage point increase in the utilization rate (cyclical GDP) is associated, on average, with a 0.15 (0.45) percentage point increase in the rate of
inﬂation. But, the regressions also indicate that the asymmetries predicted by the

14

Federal Reserve Bank of Richmond Economic Quarterly

theory are not present in the data. That is, the utilization-inﬂation relationship is
the same regardless of whether the utilization rate is high or low, and the stage
of the cycle is immaterial to the cyclical GDP-inﬂation relationship. This casts
doubt on the theory’s joint assumption of a ﬁxed and nonlinear supply curve.
Alternatively expressed, there is no evidence of the existence of immutable
threshold levels of utilization or cyclical GDP, levels at which underemployment of resources sets in and prices become relatively unresponsive to market
conditions, as asserted by the theory.
The forecasting exercise lends support to the Keynesian theory by showing
that both capacity utilization and cyclical GDP have substantial marginal predictive content for inﬂation. The alternative inclusion of utilization and cyclical
GDP in an otherwise univariate forecasting equation for inﬂation reduces the
mean squared one-period-ahead forecast error by 13 and 18 percentage points,
respectively. Of the two measures of real economic activity, cyclical GDP exhibits the stronger predictive relationship with inﬂation, presumably because it
is the broader measure of real economic activity. This ﬁnding, together with the
high degree of correlation between utilization and cyclical GDP, suggests that
there is nothing special about the capacity utilization rate relative to cyclical
GDP in predicting price inﬂation. In other words, the empirical linkage between
utilization and inﬂation evidently obtains only in so far as utilization is highly
correlated with cyclical GDP.

4.

CONCLUSION

Media discussions of inﬂation inextricably link high capacity utilization with
inﬂationary conditions. The present study outlines the Keynesian theory underlying this viewpoint and then evaluates it in terms of its ability to explain the
facts about the U.S. economy over the period 1953:1–1994:1. The outcome is
summarized as follows.
First, capacity utilization has often moved in the opposite direction to that
of inﬂation. Of particular note are the oil-price shock periods, 1973–1974 and
1979, when capacity utilization plummeted while inﬂation soared. The theory
cannot explain such occurrences. Second, the relationship between utilization
and inﬂation is, on average, a positive one as asserted by the theory; however,
it is the same regardless of whether utilization is high or low, which conﬂicts
with the theory. Third, the theory derives support from the facts that utilization
helps predict future price inﬂation and that the broader measure of economic
activity, cyclical GDP, works better than utilization as an inﬂation predictor.
The upshot of this evaluation is that problems for the Keynesian theory
include not only misspeciﬁcation of the channels through which shocks impact
on the economy but also its ignoring of shocks to the supply side of the economy. More exactly, the evidence is not supportive of the theory’s assumption
of a nonlinear aggregate supply curve that remains constant over the business

M. G. Finn: Is “High” Capacity Utilization Inﬂationary?

15

cycle. An alternative theory that drops the assumption of a nonlinear relationship between inﬂation and real economic activity and that incorporates both
demand and supply shocks is called for. In particular, by including supply
shocks the alternative theory has the potential to explain why inﬂation and real
economic activity sometimes move in opposite directions.
One such alternative theory is that of Greenwood and Huffman (1987) and
Coleman (1994). This theory emphasizes technology shocks as the main source
of cyclical output ﬂuctuations. It also stresses the endogenous responsiveness
of the money supply to those shocks. Other alternative theories are advanced
by Lucas (1975), Cho and Cooley (1992) and Ireland (1994). These theories
feature signiﬁcant sources of monetary nonneutralities, thereby allowing not
only technology shocks but also money supply shocks to play an important role
in driving cyclical output ﬂuctuations. All of these theories offer explanations
of the empirical relationship between inﬂation and cyclical output. They all
show a relationship that is positive, but only on average, since opposite movements sometimes occur. Further research will reveal which of these theories or
alternative theories provides the best explanation.

DATA APPENDIX
The data are quarterly, seasonally adjusted (unless otherwise indicated) and for
the United States over the period 1952:1–1994:1. The 1952 data are used only
in forming lagged variables for the regression analysis. The source for all data
is the FAME database. A description of the individual series follows.
Inﬂation Rate:

CPI inﬂation measured quarter to quarter at an
annualized rate. The average value of the underlying CPI index is 100 over the period 1982–1984.

Utilization Rate:

Total industry (consisting of manufacturing, mining and utilities) utilization rate for the period
1967:1–1994:1. Manufacturing industry utilization
rate for the period 1952:1–1966:4.

Industrial Production:

Total industry output index, 1987 = 100.

Gross Domestic Product:

Gross domestic product measured at constant 1987
prices and in billions of dollars.

Population:

Civilian, noninstitutionalized, aged 16 and above,
measured in thousands. This series is not seasonally
adjusted.

16

Federal Reserve Bank of Richmond Economic Quarterly

REFERENCES
Aiyagari, S. Rao. “On the Contribution of Technology Shocks to Business
Cycles,” Federal Reserve Bank of Minneapolis Quarterly Review, vol. 18
(Winter 1994), pp. 22–34.
Cho, Jang O., and Thomas F. Cooley. “The Business Cycle with Nominal
Contracts.” Manuscript. University of Rochester, 1992.
Citicorp. “Capacity Utilization: The Rate Looks Much Too High,” Economic
Week, January 18, 1994.
Coleman, Wilbur J., II. “Prices, Interest, and Capital in an Economy with Cash,
Credit and Endogenously Produced Money.” Manuscript. Duke University,
May 1994.
Davies, Stephen A. “Economists See an Inﬂation Signal in Rise of Factory
Operating Rate,” American Banker, January 12, 1994.
Epstein, Gene. “Yes, Determining Capacity Utilization Is Crucial, But How
Much Capacity Do Our Factories Have?” Barron’s, June 20, 1994, p. 48.
Greenwood, Jeremy, and Gregory W. Huffman. “A Dynamic Equilibrium
Model of Inﬂation and Unemployment,” Journal of Monetary Economics,
vol. 19 (March 1987), pp. 203–28.
Harper, Lucinda, and Henry F. Myers. “Perking Along: Economy’s Expansion
Gets a Boost as Firms Move to Build Capacity,” The Wall Street Journal,
June 6, 1994, pp. A1 and A6.
Ireland, Peter N. “Using the Permanent Income Hypothesis for Forecasting,”
Federal Reserve Bank of Richmond Economic Quarterly, vol. 81 (Winter
1995), pp. 49–63.
. “Monetary Policy with Nominal Price Rigidity.” Manuscript.
Federal Reserve Bank of Richmond, March 1994.
Kydland, Finn E., and Edward C. Prescott. “Hours and Employment Variation
in Business Cycle Theory,” Economic Theory, vol. 1 (1991), pp. 63–81.
. “Business Cycles: Real Facts and a Monetary Myth,” Federal
Reserve Bank of Minneapolis Quarterly Review, vol. 14 (Spring 1990),
pp. 3–18.
Lucas, Robert E., Jr. “An Equilibrium Model of the Business Cycle,” Journal
of Political Economy, vol. 83 (December 1975), pp. 1113–44.
Shapiro, Matthew D. “Assessing the Federal Reserve’s Measures of Capacity
and Utilization,” Brookings Papers on Economic Activity, 1:1989, pp.
181–241.
Uchitelle, Louis. “Growth of Jobs May be Causality in Inﬂation Fight,” The
New York Times, April 24, 1994, pp. 1 and 24.

An Empirical Measure of
the Real Rate of Interest
Robert Darin and Robert L. Hetzel

T

he interest rate adjusted for expected inﬂation, the real rate of interest, is
a key variable in macroeconomics. It is the price one pays for currently
available resources expressed in terms of future resources. How does
the real rate of interest behave? Despite the importance of this question, there
is no generally available measure of the real rate of interest one can use to
answer it. Economists who have studied the real rate have had to create their
own series. The purpose of this article is to construct and make available a
number of alternative empirical measures of the real rate of interest.
As noted above, the real rate of interest is the difference between the
observed market rate of interest and the inﬂation rate expected by the public.
Expected inﬂation, however, is not directly observable.1 In order to construct a
real rate series, one must select a proxy for expected inﬂation. We examine two
possibilities—inﬂation forecasts made by Data Resources Incorporated (DRI)
and by the staff of the Board of Governors of the Federal Reserve System.
The DRI forecasts are made monthly. Through 1978, the Board staff produced
monthly forecasts. Thereafter, it produced them eight times per year. These
forecast series, therefore, allow construction of real rate series that are observed
frequently enough to study cyclical timing relationships.
As an illustration of the usefulness of having a real rate series, we ﬁrst
review recent public debate over the typical level of the real rate. The main
part of the article provides a defense of the plausibility of the real rate series
constructed here and listed in the appendix. We compare forecasts of inﬂation
from four different sources: the staff of the Board of Governors, DRI, the
Michigan Survey of Consumers, and the Livingston Survey. We argue that the
The views expressed are those of the authors and do not necessarily represent those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
1 Hetzel (1992) makes a proposal for indexed bonds that would render expected inﬂation
directly observable. Expected inﬂation would be the difference in yields between nonindexed and
indexed Treasury securities of the same maturity.

Federal Reserve Bank of Richmond Economic Quarterly Volume 81 Winter 1995

17

18

Federal Reserve Bank of Richmond Economic Quarterly

broad agreement exhibited among all these series is evidence that the series
used here (Board of Governors staff and DRI) have been representative of the
expectations of inﬂation affecting ﬁnancial markets.
We then discuss other approaches to estimating the real rate. In this context,
we examine the predictive ability of the four forecast series. We point out the
persistent underprediction of inﬂation by survey forecasts in the 1970s. We
argue that this underprediction does not reﬂect a basic defect in the survey
data, but rather the special difﬁculty in predicting inﬂation during the ﬁnal
transition from a commodity to a ﬁat money standard.

1.

CONTROVERSY OVER THE BEHAVIOR OF
THE REAL RATE

Recently, the behavior of the real rate of interest has become an issue in debates
over monetary policy. For example, during Humphrey-Hawkins testimony in
July 1993, the chairman of the Federal Reserve System, Alan Greenspan (1993),
drew attention to the unsustainably low value of the then current short-term
real rate:
Currently, short-term real rates, most directly affected by the Federal Reserve, are not far from zero; long-term rates, set primarily by the market, are
appreciably higher, judging from the steep slope of the yield curve and reasonable suppositions about inﬂation expectations. This conﬁguration indicates
that market participants anticipate that short-term real rates will have to rise
as the head winds diminish, if substantial inﬂationary imbalances are to be
avoided.
(P. 853)

In spring 1994, after the Federal Reserve began to raise the funds rate,
controversy arose over what constitutes typical behavior of the real rate of
interest. This controversy is illustrated by the following excerpts from The
Wall Street Journal.
[W]ith the economy now growing at a robust pace . . . the Fed has concluded
that it is time to take the foot off the accelerator and put monetary policy into a
“neutral” stance. . . . Robert Reischauer, director of the Congressional Budget
Ofﬁce, said neutral probably means inﬂation-adjusted rates of somewhere between 3⁄4% and 11⁄2%. But chief White House economist Laura Tyson has said
that—excluding the anomalous 1980s—inﬂation-adjusted interest rates “have
always been below 1%.”
(Wessel, 4/19/94, p. A2)
The federal funds rate . . . now stands at 3.5%. And inﬂation is running at
roughly 3%. That means the “real” interest rate . . . is only .5%. That is well
below historical experience, says Barry Bosworth of the Brookings Institution,
who adds that the norm is “1.5% to 2% real short-term rates in the middle of
an economic expansion.”
(Murray, 4/11/94, p. A1)

R. Darin and R. L. Hetzel: Real Rate of Interest

19

In 1992 and 1993 real interest rates had been stuck around zero because of
a weak world economy. Rates have since increased with global economic
prospects, but the recent level of real rates, 1.9% to 2%, is not high by historical standards; it is just about the average since 1961. Real rates remain well
below the average of 3% that prevailed during the period of high growth and
robust investment from 1984 to 1989.
(Barro, 8/19/94, p. A10)

2.

REAL RATE SERIES

Figure 1 shows two real rate series for Treasury bills, one using inﬂation forecasts from the staff of the Board of Governors and one using forecasts from
DRI. (The data appendix provides a detailed discussion of data sources and the

Percent

Figure 1 Greenbook’s and DRI’s One- to Two-Quarter
Real Treasury Bill Rates
5
4
3
2
1
0
-1
-2
-3

••• •
•

•
••

•• • •
•

•

•
• •• • •
••
•• •• ••
• •
•

Jan
1966

•

•
••
•
• •
• •

Greenbook
DRI

••
•
• • ••
••
••
•• •• •
• ••• • • • • •• •
• • • • ••
•••
••
• •• • • ••
• ••
••
•

•

67

68

69

70

71

72

73

74

75

76

77

78

67

68

69

70

71

72

73

74

75

76

77

78

10

Greenbook
DRI

Percent

8
6
4
2
0
-2
1979 80

+

81

82

83

84

85

86

87

88

89

90

91

92

93

94

Notes: The Greenbook real rate is calculated for dates on which Greenbooks were published. It is
the difference between the yield on those dates on a Treasury bill maturing at the end of the subsequent quarter and a weighted average of Greenbook inﬂation forecasts for the contemporaneous
and subsequent quarter. Inﬂation is measured by the GNP (GDP from 1992 on) implicit price
deﬂator. The DRI real rate is calculated using the same Treasury bill yield and DRI predictions
of inﬂation from the DRI publication immediately preceding publication of the Greenbook. Observations in the top panel are monthly. Observations in the bottom panel correspond to FOMC
meetings, which have been held eight times a year since 1981, and tick marks indicate the ﬁrst
FOMC meeting of the year.

20

Federal Reserve Bank of Richmond Economic Quarterly

construction of the real rate series.) The Board staff forecasts are contained in a
document referred to as the Greenbook, which is prepared prior to Federal Open
Market Committee (FOMC) meetings. Because Greenbooks remain conﬁdential for ﬁve full calendar years after the year in which they are published, the
Greenbook real rate series ends in 1989. The DRI forecasts are from the table
“Quarterly Summary for the U.S. Economy—Control” in the DRI/McGrawHill monthly publication Review of the U.S. Economy. Observations in the top
part of Figure 1 are monthly. In the bottom part of the graph, they correspond
to FOMC meetings, which have occurred eight times a year since 1981.
We calculate the Greenbook real rate series for dates on which the Board
staff issued Greenbooks. The real rate is the difference between the yield
(recorded on the Greenbook issue date) on a Treasury bill that matures on
the last working day of the subsequent quarter and a weighted average of the
Greenbook inﬂation forecasts for the contemporaneous and subsequent quarter.
Inﬂation is measured by the GNP (GDP from 1992 on) implicit price deﬂator.
We calculate the DRI real rate series using the same Treasury bill yield and DRI
predictions of inﬂation from the most recent monthly DRI Review of the U.S.
Economy available as of the issue of the Greenbook. The Greenbook and DRI
real rate series generally move together. Some discrepancies in the two series
arise because the dates on which the Greenbook and DRI inﬂation forecasts
are made can differ by as much as a month.

3.

HOW SIMILAR ARE THE INFLATION FORECASTS?

We now examine the correspondence among four inﬂation forecasts: the Greenbook, the DRI Review, the Livingston Survey, and the Michigan Survey. Different groups make the four forecasts. The staff of the Board of Governors makes
the Greenbook forecasts. The 19 members of the FOMC critically examine
the Greenbook forecasts at their meetings. Professional forecasters trained as
economists make the DRI forecasts and sell them to a variety of corporations
and state governments. Economists working for banks, corporations, and in
ﬁnancial markets make the forecasts in the Livingston Survey. The Survey Research Center of the University of Michigan randomly selects respondents from
the public for the inﬂation forecasts in its Survey of Consumers. A straightforward explanation for the similar behavior among these different measures of
expected inﬂation is that they do in fact capture movements in the public’s expectation of inﬂation. This similarity suggests that the real rate series proposed
here capture, at least broadly, the real rate as perceived by the public.
The Livingston Survey is available starting in June 1946. Joseph Livingston
was a ﬁnancial columnist from Philadelphia who surveyed business economists
twice yearly on their expectations of CPI inﬂation. Among others, Carlson
(1977), Caskey (1985), Hafer and Resler (1980), Jacobs and Jones (1980), and

R. Darin and R. L. Hetzel: Real Rate of Interest

21

Mullineaux (1978) examine the properties of this series. A series for the real
rate of interest constructed from Livingston Survey data on expected inﬂation
consists of only two observations per year.
The Survey Research Center of the University of Michigan has collected
data on expected inﬂation quarterly since 1966 and monthly since 1978. (Before 1966, it asked respondents only whether they expected prices to go up or
down.) Starting in 1978, the median, as well as the mean, of the individual respondents’ forecasts from the Michigan Survey becomes available. The survey
median has been lower than the survey mean in 95 percent of the observations.
For all observations, the median prediction is lower than the mean prediction
by an average of 1.0 percentage points. (This fact indicates that a small number
of respondents regularly expected inﬂation to be unusually high relative to the
group forecast.)
Figures 2 and 3, which compare Greenbook forecasts with DRI and Livingston forecasts, respectively, reveal a great deal of similarity between the
paired forecasts. The standard deviation of the difference between the Greenbook and DRI forecasts from 1970Q3 to 1989Q4 is about 1 percent. The
standard deviation of the difference between the Livingston and matching (May
and November) Board staff forecasts from 1968 to 1989 is 0.60 percent.
Figure 4 plots Livingston Survey forecasts of four-quarter CPI inﬂation. It
also plots the Michigan four-quarter mean forecasts of CPI inﬂation made in the
same month and, beginning in 1978, the median forecasts as well. Although the
Livingston and Michigan series move together, the Livingston series regularly
lies below the Michigan series until 1980. From 1982 through the middle of
1988, the Livingston and Michigan mean forecasts are close. Thereafter, the
mean of the Michigan forecast lies above the Livingston forecast. Over the
period starting with the November survey of 1967 and ending with the May
survey of 1994, the standard deviation of the difference in the Michigan (mean
value) predictions and the Livingston predictions is 1.1 percent.
We maintain that the broad underlying similarity among the series examined above indicates that they capture movements in the public’s expectations
of inﬂation. Of course, as indicated by the discrepancies in inﬂation forecasts
among the series, individual observations from a particular series are only rough
estimates of the consensus view of expected inﬂation that shapes the behavior of
market rates. Nevertheless, we believe the real rate series contained in Table 1,
which are constructed from Greenbook and DRI inﬂation forecasts, do capture
the general behavior of the short-term real rate of interest.

4.

COMPARING PREDICTED AND ACTUAL INFLATION

Two characteristics of the various inﬂation forecasts examined above warrant
close scrutiny. First, through the 1970s, the inﬂation forecasts generally fall
short of subsequently realized inﬂation. These persistent forecast errors could

22

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Greenbook’s and DRI’s Inﬂation Predictions
12

Greenbook
DRI

10

Percent

8

6

4

2

0

+

1971:1

73

75

77

79

81

83

85

87

89

Notes: Observations of predicted inﬂation are from Greenbooks for February, May, August, and
November FOMC meetings and are of the annualized quarterly percentage change in the GNP
price deﬂator for the subsequent quarter. DRI predictions are from DRI publications with the
same monthly date as the Greenbook.

indicate a defect in the survey data on expected inﬂation. Second, the forecasts
do have some predictive power. That is, they perform more accurately than
naive forecasts that simply employ past observations of inﬂation to predict
future inﬂation. This latter characteristic, however, is not a necessary property
of forecasts. The process generating inﬂation could be such that predicting
inﬂation is simply very hard.
Figure 5 compares quarterly predictions of CPI inﬂation over future fourquarter periods from the Michigan Survey with the subsequently realized CPI
inﬂation. It illustrates the persistent underprediction of inﬂation over much of
the period shown. The Michigan Survey respondents underpredict inﬂation except during the early 1970s, the mid-1970s, the mid-1980s, and the early 1990s.
They underpredict inﬂation whenever inﬂation rises. From 1973 through 1981,
the average underprediction is 1.6 percentage points. (The standard deviation
of the prediction errors is 2.3 percent.) This pattern of errors in predicting inﬂation is similar for the other sources—Greenbook, DRI, and Livingston. From
1966 to 1981, the Livingston Survey underestimates inﬂation by 1.8 percentage
points on average. (The standard error of the forecast errors is 2.1 percent.)
In evaluating the Greenbook forecasts, we use forecasts of one-quarterahead (nominal output deﬂator) inﬂation made for FOMC meetings held in

R. Darin and R. L. Hetzel: Real Rate of Interest

23

Figure 3 Greenbook’s and Livingston’s Two-Quarter
Inﬂation Predictions

14

12

Greenbook
Livingston

Percent

10

8

6

4

2

+

1969

71

73

75

77

79

81

83

85

87

89

Notes: Greenbook predictions of inﬂation are for the GNP implicit price deﬂator before 1980
and for the CPI thereafter. Livingston predictions are for the CPI. The tall tick marks correspond
to May Greenbook predictions of the annualized inﬂation rate for the last two quarters of the
year. Tall tick marks also correspond to Livingston predictions of the annualized inﬂation rate
for the eight-month period ending December and are from the June release. The short tick marks
correspond to the December Greenbook predictions of the annualized inﬂation rate for the ﬁrst
two quarters of the following year. Short tick marks also correspond to Livingston predictions of
the annualized inﬂation rate for the eight-month period ending in June and are from the December
release.

February, May, August, and November (Figure 2). In general, from 1966
through 1981, subsequently realized inﬂation exceeds predicted inﬂation. During this period, the Greenbook underpredicts inﬂation by 1.1 percentage points
on average. (The standard deviation of the one-quarter-ahead prediction errors
is 1.6 percent.) In 1982, actual inﬂation falls below predicted inﬂation. The predictions are then fairly accurate from 1983 through 1989. For the corresponding
DRI forecasts, from 1970Q3 through 1981Q4, the average underprediction is
1.2 percentage points and the standard deviation of the one-quarter-ahead prediction errors is 2.4 percent.2
One way to assess whether the forecasts shown in Figure 2 have predictive
value is to compare them with naive forecasts made by simply extrapolating
2 The actual inﬂation series changes over time as nominal and real output are rebenchmarked
and as seasonal factors change. The original forecasts, therefore, were for a somewhat different
inﬂation series than the one to which they are compared here.

24

Federal Reserve Bank of Richmond Economic Quarterly

Figure 4 Michigan’s and Livingston’s Inﬂation Predictions
12

Livingston

Percent

10

Michigan
(survey mean)
Michigan
(survey median)

8

6

4

2
1968

70

72

74

76

78

80

82

84

86

88

90

92

94

+
Notes: Observations of predicted CPI inﬂation are for the subsequent four-quarter period. The
Livingston Survey was conducted in May and November. Its forecasts are matched with Michigan
Survey of Consumers forecasts also made in the months of May and November. Michigan forecasts
are the mean (black line) and the median (grey line) of respondents’ forecasts. The median is
available only starting in 1978. Tall tick marks indicate ﬁrst observation of the year.

past inﬂation. Accordingly, we use the inﬂation rate from the quarter prior to
the quarter in which the inﬂation forecast was made as a simple benchmark
forecast. For the period 1966 through 1981, if the Greenbook’s forecast of
(GNP deﬂator) inﬂation for the subsequent quarter is replaced with the past
quarter’s actual inﬂation rate, the correlation with subsequently realized inﬂation is 0.63. For this period, the correlation between the Greenbook predictions
of inﬂation and subsequently realized inﬂation is 0.79. This latter correlation
represents an improvement of 25 percent over the naive prediction made using
the prior quarter’s actual inﬂation ﬁgure. For the period 1982Q1 to 1994Q2,
the correlation between the prior quarter’s inﬂation rate and the subsequently
realized inﬂation rate is 0.42, while the correlation between the predicted and
subsequently realized inﬂation rate is 0.62, an improvement of 48 percent.
In evaluating the DRI predictions, as with the Greenbook, we use forecasts
of one-quarter-ahead (nominal output deﬂator) inﬂation dated as of February,
May, August, and November. (The forecasts were made at the end of the preceding month.) For the period 1973Q1 through 1981Q4, the correlation between
the naive prediction (using the actual inﬂation rate two quarters in the past) and

R. Darin and R. L. Hetzel: Real Rate of Interest

25

Figure 5 Michigan’s Inﬂation Predictions and Realized Inﬂation

16
14

Predicted
Realized

12

Percent

10
8
6
4
2

+

0
1969:1 71

73

75

77

79

81

83

85

87

89

91

93

Notes: Observations of predicted inﬂation are from the Survey of Consumers conducted by the
Survey Research Center of the University of Michigan. Before 1978, predicted inﬂation consists
of quarterly observations of the mean inﬂation rate predicted by respondents. From 1978 on,
observations are quarterly averages of monthly observations of the median inﬂation rate predicted
by respondents. Observations of actual inﬂation are the subsequently realized annual percentage
changes in the CPI (all urban consumers).

subsequently realized inﬂation is 0.49, while the correlation between predicted
and subsequently realized inﬂation is 0.60, an improvement of 22 percent.3

5.

IS THERE A BETTER WAY TO ESTIMATE
THE REAL RATE?

Empirical work on the real rate of interest divides two groups. In one group,
researchers use survey data to measure expected inﬂation. They regress the
observed market rate of interest on a proxy for expected inﬂation derived
from survey data (almost invariably the Livingston Survey) and on a collection
of variables believed to be determinants of the real rate (government deﬁcit,
price of oil, etc.). Makin (1983) and Mehra (1985) represent examples of this
methodology. Researchers in the other group assume that expected inﬂation
3 For the same period, the Greenbook’s average underprediction is 1.0 percentage points and
the standard deviation of the one-quarter-ahead prediction errors is 1.9 percent. The correlation
between predicted and subsequently realized inﬂation is 0.72.

26

Federal Reserve Bank of Richmond Economic Quarterly

equals subsequently realized inﬂation plus a white-noise error term. They use
subsequently realized inﬂation over the relevant forecast period as a proxy for
expected inﬂation (see Fama [1975]).
Researchers in the latter group use the ex-post real rate of interest (the
market rate minus subsequently realized inﬂation) as a noisy measure of the
ex-ante real rate. Using either a time-series or a structural model of the real
rate, they then often ﬁt a regression explaining this ex-post real rate. Then
they use the ﬁtted parameters of the model to generate a less noisy, smoother
series for the real rate. For example, Antoncic (1986) generates estimates of
the real rate by assuming the real rate is a random walk. (See also Garbade
and Wachtel [1978] and Fama and Gibbons [1982].) Huizinga and Mishkin
(1986) generate estimates of the real rate by assuming it can be represented as
a linear combination of variables, that is, as a distributed lag of ex-post real
rates, inﬂation rates, and the price of energy. (See also Bonser-Neal [1990].)
The approaches used by each group yield quite different measures of real
rates over the earlier and latter parts of the 1970s.4 Figure 6 plots a oneyear real rate calculated as the difference between the one-year Treasury bill
rate and predictions of four-quarter CPI inﬂation from DRI. It also plots the
realized real rate for the corresponding four-quarter period, that is, the oneyear bill rate minus the subsequently realized inﬂation rate. The increases in
the rate of inﬂation that began in 1973, 1977, and, to a lesser extent, 1989 are
associated with a realized real rate signiﬁcantly less than the real rate calculated
using inﬂation forecasts. Conversely, when inﬂation falls starting in 1981, the
realized real rate lies well above the predicted real rate.
Researchers in the second group discussed above justify their use of
realized inﬂation as an unbiased measure of expected inﬂation through the
assumption of rational expectations. Speciﬁcally, they assume that participants
in ﬁnancial markets understand the nature of the monetary regime that generates
inﬂation. The assumption of rational expectations, together with the assumption
that individuals make efﬁcient use of information, implies that forecast errors,
apart from special cases, will not exhibit persistent bias. Because measures of
expected inﬂation derived from survey data persistently underpredict inﬂation
through the end of the 1970s, they fail to meet the requirements set by this
second group.
A variant of the rational expectations approach is to assume that the public
understands the time-series behavior of inﬂation. One can then use past observations of inﬂation to recreate the public’s predictions of inﬂation. (For an
interesting application, see Choi [1994].) Under the assumption that inﬂation is
an autoregressive process, we regress inﬂation on its lagged values to generate
4 The issue of which approach generates better measures of the real rate will be settled only
when a consensus develops over the validity of a structural model of the real rate. The predictions
of that model can then be compared with the alternative empirical measures of the real rate.

R. Darin and R. L. Hetzel: Real Rate of Interest

27

Figure 6 DRI’s Ex-ante Real Rate and the Ex-post Real Rate
14
12

Ex-ante Real Rate
Ex-post Real Rate

10
8

Percent

6
4
2
0
-2
-4
-6
1971

+

73

75

77

79

81

83

85

87

89

91

93

Notes: The one-year real ex-ante rate is the Salomon Brothers one-year government bond yield
read from a yield curve minus four-quarter predicted CPI inﬂation from DRI publications. The
bond yield is for the last working day of the month. Observations are dated as of the subsequent
month. If that month is the ﬁrst or second month of a quarter, the quarter in which that month
falls is the ﬁrst quarter used in the four-quarter inﬂation forecast. If that month is the third month
of a quarter, the subsequent quarter is the ﬁrst quarter used in the four-quarter inﬂation forecast.
No observation is plotted in cases in which the DRI forecast was unavailable. The ex-post oneyear real rate is the bond yield minus the subsequent four-quarter CPI inﬂation rate. Tick marks
indicate ﬁrst observation of the year.

predictions of inﬂation. Equation (1) is a regression of contemporaneous (implicit GNP deﬂator) inﬂation on its three lagged values for the period 1966Q1
to 1979Q4.
πt = 0.45πt−1 + 0.31πt−2 + 0.24πt−3 + u
ˆ
(1)
2

R = 0.32

SEE = 1.96

DW = 2.0

Degrees of Freedom = 50

We employ regressions like (1) to generate inﬂation forecasts whose predictive
accuracy can be compared to the Greenbook and DRI predictions displayed in
Figure 2.
The forecasts of Figure 2 were made close to the middle of a quarter
(February, May, August, and November) and were for the succeeding quarter.
For example, the prediction of 1970Q4 inﬂation shown in Figure 2 was made
by the Board staff in the August 12, 1970, Greenbook and by DRI at the end of
July. At the time these predictions were made, the forecasters would have just
received GNP data for the preceding quarter, 1970Q2. We therefore conduct
the comparison as follows.

28

Federal Reserve Bank of Richmond Economic Quarterly

To begin, we regress inﬂation on its three lagged values over the period 1966Q1 to 1970Q2.5 We then use this regression to forecast inﬂation
for 1970Q3. Next, we substitute the resulting prediction for 1970Q3 and the
realized inﬂation rates for 1970Q2 and 1970Q1 into the regression equation to
obtain a prediction of inﬂation for 1970Q4. This predicted value is comparable
to the Greenbook and DRI predictions of one-quarter-ahead inﬂation made in
1970Q3 for 1970Q4 and shown in Figure 2: all three predictions use data for
the period predating 1970Q3. We repeat this procedure for each quarter through
1980Q4. That is, we run a series of rolling regressions, each of which starts in
1966Q1, with each successive regression containing one additional quarter.
The resulting comparison of forecast errors highlights the Board staff’s and
DRI’s persistent underprediction of inﬂation through the 1970s (see Cullison
[1988]). From 1970Q3 through 1980Q4, Greenbook and DRI forecasts underestimate inﬂation by 1.3 percent and 1.6 percent on average, respectively, while
the time-series forecasts slightly overestimate inﬂation by −0.2 percent. The
time-series predictions, however, are not superior on all dimensions. The sum
of the squared errors of the predictions from 1970Q3 to 1980Q4 is lower for
the Greenbook than for the autoregressive predictions, 171 compared to 204
(267 for DRI). Also, the autocorrelation in the Greenbook and DRI forecast
errors is negligible, while the autocorrelation in the autoregressive predictions
is 0.4.
We conduct one ﬁnal test in the spirit of the rational expectations literature
to see whether Greenbook forecasts made efﬁcient use of information. We calculate the correlation between the forecast errors of one-quarter-ahead inﬂation
(derived again from the series shown in Figure 2) and the ﬁgure for the most
recently available rate of growth of GNP as of the date of the forecast. (The
latter ﬁgure is taken from the Greenbook.) It seems likely that when the rate
of growth of GNP was high, the Board staff would underestimate inﬂation,
and vice versa. In this event, the correlation between forecast errors and GNP
growth would be positive. However, the correlation is in fact negligible (−0.03).
In this case, the Board staff was making efﬁcient use of available information.

6.

WHY THE PERSISTENT FORECAST ERRORS?

We maintain that the underprediction of inﬂation exhibited by survey data reﬂected the long period of time required for the public to realize that the process
generating inﬂation under the prior commodity and Bretton Woods monetary
standards had disappeared irrevocably. (See Caskey [1985] for a thorough
5 We choose 1966 as a starting date under the assumption that as of this date the FOMC no
longer conducted monetary policy subject to constraints imposed by the Bretton Woods system.
We choose the end date on the basis of when DRI predictions become available.

R. Darin and R. L. Hetzel: Real Rate of Interest

29

exposition of this view.) Before World War II, the quantity of money had been
determined through ﬁxing its value in terms of gold or silver. After World War
II, the United States was part of the Bretton Woods system, which mimicked
the international gold standard. Under the Bretton Woods system, the Federal
Reserve maintained a dollar price of gold. In order to maintain the reserves
necessary to peg the price of gold, the Fed had to respond to reserve outﬂows
by raising rates, just as central banks had responded to gold outﬂows under the
gold standard.
After the mid-1960s, the monetary regime changed to a pure ﬁat money
regime. In 1968, Congress eliminated the gold cover on Federal Reserve notes.
With the closing of the gold window in August 1971, the last vestigial, institutional relationship tying the value of the dollar to the value of a commodity
disappeared. Under the new ﬁat money regime, there were no institutional
arrangements to tie down the inﬂation rate. Moreover, monetary policy from
the mid-1960s until the end of the 1970s was unique in the history of the United
States through the emphasis placed on controlling growth of real output and
unemployment. As a consequence, the character of the process generating inﬂation changed. The level of the inﬂation rate began to move randomly instead
of reverting to a low average value.
Given that prior to World War II the United States had been on a commodity standard for all of its history apart from wars and that the Bretton Woods
system replaced the gold standard after the War, it is no surprise that the public required some time in order to understand that “high and rising” inﬂation
would not necessarily entail subsequent reductions in inﬂation. Furthermore,
because of the particular historical circumstances surrounding the appearance
of inﬂation, the public was slow to develop an understanding of the new,
nonstationary character of inﬂation. Inﬂation surged ﬁrst in conjunction with
the Vietnam War. Inﬂation in wartime had been the historical norm, however.
Inﬂation then surged after two oil-price shocks, one in 1973 and one in 1978.
Given the association of inﬂation with these real shocks, the public required
considerable time to realize that changes in the inﬂation rate were likely to be
persistent rather than transitory.

7.

WHAT IS THE NORMAL LEVEL OF
THE REAL RATE?

What is the average level of the real rate of interest? Before examining this
question, we would like to know to what extent generalizations about the shortterm rate of interest carry over to the long-term rate of interest. Figure 7 displays
a ten-year real rate constructed using forecasts from two surveys of ten-year
expected inﬂation. The initial forecasts are from a survey conducted by Richard
Hoey. The ﬁrst observation in this series is for September 1978. Hoey conducted

30

Federal Reserve Bank of Richmond Economic Quarterly

Figure 7 Long-Term and Short-Term Real Rates

10
•
8

•• •

••
•

•

6
Percent

+

•

•
•

4

•

•• •
••

•

Long-Term
(Professional Forecasters)

•••
••

Short-Term
•
••

•

• •
• •• ••
• •
• •
••
• •
• • ••
•
•• • • •
•

••
•
• ••

•
2 •

Long-Term (Hoey)

••• •
• •••

•
•

+
+
+

+++ +
+ +
++++
•••••

0

-2

+

1979 80
J-79

81

82

83

84

85

86

87

88

89

90

91

92

93

94

Notes: The long-term real rate is the ten-year bond yield minus the predicted ten-year inﬂation
rate from the “Decision Makers Poll” conducted in the 1980s by Richard B. Hoey (for Warburg, Paribus, Becker; Drexel, Burnham, Lambert; and Barclay’s de Zoete Wedd). The Hoey
Survey was discontinued in March 1991, but was reinstated by Cowens Investment Strategy
for ﬁve months beginning in March 1993. Starting in October 1991, the Survey of Professional
Forecasters conducted by the Federal Reserve Bank of Philadelphia (formerly conducted by the
American Statistical Association and the National Bureau of Economic Research) began to collect
data in its quarterly survey on expected CPI inﬂation for a ten-year horizon. The long-term real
rate is calculated using both series whenever possible. Observations of the long-term real rate
are matched with monthly observations on the short-term real rate calculated as the difference
between Salomon Brothers one-year government bond yields and DRI predictions of four-quarter
CPI inﬂation. Tick marks indicate ﬁrst observation of the year.

his survey only intermittently before 1981. He conducted it more frequently
starting in 1983 and discontinued it in March 1991. Cowens Investment Strategy conducted the survey again for ﬁve months in 1993. Toward the end of
1991, in its quarterly Survey of Professional Forecasters, the Federal Reserve
Bank of Philadelphia began to collect predictions of CPI inﬂation over future
ten-year intervals. This latter series ﬁlls in most of the missing observations
from the Hoey Survey, although the observations are less frequent.
Figure 7 also plots the DRI one-year real rate from Figure 6. For the
period 1978 to 1991, short- and long-term rates are quite close.6 From July
1980 through March 1991, the long-term real rate averages 4.25 percent, while
6 For

the period shown in Figure 7, the standard deviation of the one-year real rate (2.2) is
slightly higher than that for the Hoey long-term real rate series (1.5).

R. Darin and R. L. Hetzel: Real Rate of Interest

31

the short-term real rate averages 4.3 percent. With the sharp fall in short-term
market rates in 1991, however, long-term and short-term real rates diverge.
From November 1991 through August 1994, the long-term real rate averages
3.0 percent, while the short-term real rate averages only 0.63 percent. This
divergence suggests that statements about the behavior of the short-term real
rate of interest do not necessarily carry over to the behavior of the long-term
rate of interest.
From November 1965 through the end of 1993, the mean of the Treasury
bill real rate calculated using Greenbook inﬂation forecasts (the series shown in
Figure 1) is 2.3 percent.7 As a check on this ﬁgure, we calculate a semiannual,
one-year Treasury bill real rate using inﬂation forecasts from the Livingston
Survey for the period June 1951 through June 1965. The mean of this series
is 2.1 percent, which lies close to the ﬁrst estimate.8 Trehan (1995) calculates
the realized real rate on one-year Treasury bonds as 1.8 percent over the period
1954 to 1993.
The real rate, however, is variable over time. From November 1965 to
June 1974, the Greenbook Treasury bill real rate is 1.6 percent. It falls to
−0.38 from July 1974 to September 1978. It then begins to rise and is 1.1
percent from October 1978 to October 1979. From November 1979 to October
1990, the real rate averages 4.3 percent. It reaches its maximum value of 9.7
percent in May 1981. The DRI one-year Treasury bill real rate, whose monthly
observations are shown in Figure 6, falls in the 1990s, to 2.1 percent over the
interval November 1990 to June 1992 and then to 0.5 percent over the interval
July 1992 to the end of 1993.

8.

CONCLUDING COMMENTS

We have examined four sources of short-term inﬂation forecasts: the Greenbook
issued by the staff of the Board of Governors before FOMC meetings, the DRI
monthly publication Review of the U.S. Economy, the semiannual Livingston
Survey, and the Survey of Consumers conducted by the Survey Research Center
of the University of Michigan. The inﬂation forecasts in these series can diverge
signiﬁcantly for individual observations and moderately over extended periods.
7 The standard deviation of the real rate is somewhat lower than for the nominal rate. From
November 1965 to July 1979, the standard deviation of the one- to two-quarter Greenbook real
Treasury bill rate shown in Figure 1 is 1.3, while the standard deviation of the nominal bill rate
is 1.6. The corresponding ﬁgures for the period August 1979 through July 1994 are 2.1 and 3.4.
8 We use the one-year Treasury bill rate from Salomon Brothers’ “Analytical Record of
Yields and Yield Spreads.” From 1951 through 1958, the bill rate is for mid-month. For this
period, in matching the Livingston inﬂation forecasts, we use the Treasury bill rate from May
and November. From 1959 through June 1965, the bill rate is for the ﬁrst of the month. For this
period, we use an average for May and June and also for November and December.

32

Federal Reserve Bank of Richmond Economic Quarterly

Nevertheless, they display the same broad patterns. We conclude that these
series can be used to construct measures of the real rate of interest. The average
short-term real rate on Treasury bills is about 2 percent. The real rate exhibits
considerable variation, however, and at times has remained considerably above
or below the 2 percent norm.

DATA APPENDIX
Sources of Inﬂation Forecasts
1. The Greenbook, formally titled “Current Economic and Financial Conditions,” is prepared by the staff of the Board of Governors of the Federal Reserve
System and is circulated prior to FOMC meetings. Part 1 of the Greenbook,
“Summary and Outlook,” has made forecasts for nominal and real output and
the implicit output deﬂator since November 1965. Since 1980, the Greenbook
has also made predictions for CPI inﬂation. Greenbooks remain conﬁdential
for ﬁve full calendar years after the year in which they were published.
Initially, Greenbook forecasts were entirely judgmental. The Board staff
ﬁrst made a forecast using a large-scale econometric model in May 1969,
although model forecasts did not inﬂuence the Greenbook forecasts until the
early 1970s. Since the early 1970s, Greenbook forecasts have made use of a
judgmental forecast and a model forecast. Senior staff decide how to weight
these two kinds of forecasts in the combined forecast that appears in the Greenbook. Once a combined forecast for nominal and real GNP is arrived at, the
equations in the staff’s econometric model are adjusted to produce the combined
forecast. This adjusted model is then estimated to provide consistent forecasts
of the various components of the National Income and Product Accounts.
2. The DRI/McGraw-Hill monthly publication Review of the U.S. Economy
publishes quarterly forecasts of CPI and implicit GNP deﬂator inﬂation. Forecasts are taken from the table “Quarterly Summary for the U.S. Economy—
Control.” We have issues of the DRI Review starting in March 1973. (We are
indebted to John Caskey for these issues. We are indebted to Steve McNees
and Delia Sawhney of the Boston Fed for the earlier observations.)
3. Begun in 1947 by Joseph Livingston, the Livingston Survey is currently
conducted by the Federal Reserve Bank of Philadelphia. Twice annually (in
June and December) the Philadelphia Fed asks about 50 business economists
for their forecasts of the level of the CPI at six- and twelve-month horizons.
The forecasts of inﬂation in the article follow Carlson (1977). Carlson notes
that the December survey is mailed early in November when respondents have
available the October CPI. The respondents forecast the level of the CPI for
the following June. The forecast of inﬂation, therefore, is assumed to be the

R. Darin and R. L. Hetzel: Real Rate of Interest

33

annualized rate of growth of the CPI over the eight-month period from October
to June. Similarly, the inﬂation forecast based on the forecasted December level
of the CPI for the following year is assumed to be the annualized rate of growth
of the CPI over the 14-month period ending in December of the following year.
4. The Survey of Consumers conducted by the Survey Research Center of
the University of Michigan includes questions on expected price changes in
the following 12 months. The survey consists of a random telephone sample
of 500 or more individuals and asks the questions “During the next twelve
months, do you think prices in general will go up, or go down, or stay where
they are now?” and “By about what percent do you expect prices to go up, on
the average, during the next twelve months?” The survey begins in 1946, but
quantitative estimates of the predicted inﬂation rate are continuously available
only since May 1968. Before 1978 the survey is quarterly; thereafter, it is
monthly. The mean of the individual survey responses is available from 1966
to the present. The mean and median are available from 1978 to the present.
5. Richard B. Hoey in “Decision Makers Poll” conducted irregularly timed
surveys of inﬂation expectations when he worked for Bache, Halsey, Stuart
& Shields; Warburg, Paribus, & Becker; Drexel, Burnham, Lambert; and Barclays de Zoete Wedd Research, respectively. The ﬁrst ten-year inﬂation forecast
is from September 1978. The survey begins collecting shorter-term (approximately one-year) forecasts in October 1980. The number of respondents varies
between 175 and 500 and includes chief investment ofﬁcers, corporate ﬁnancial
ofﬁcers, bond and stock portfolio managers, industry analysts, and economists.
The survey dates are the dates on which the polls were mailed to Hoey. The
survey was discontinued in March 1991, resumed in March 1993, and ended
again deﬁnitively in August 1993.
6. The Survey of Professional Forecasters was ﬁrst conducted by the American
Statistical Association and National Bureau of Economic Research in 1968Q4.
It is currently conducted quarterly by the Federal Reserve Bank of Philadelphia.
In 1981Q3, the survey begins collecting forecasts of four-quarter rates of CPI
inﬂation. In 1991Q4, it begins to collect forecasts of CPI inﬂation over the next
ten years.
Constructing the Real Rate Series
Greenbook Real Rate Series
a) Real rate series of one- to two-quarter maturity calculated as the difference between the Treasury bill rate and expected inﬂation measured by the
implicit output deﬂator—Table 1, column (4)
This series is shown in Figure 1. It is calculated as the difference between
the Treasury bill yield and predicted inﬂation from the Greenbook. Inﬂation is

34

Federal Reserve Bank of Richmond Economic Quarterly

for changes in the implicit GNP (GDP from 1992 on) deﬂator. A weightedaverage inﬂation rate for the period from the Greenbook date to the end of the
succeeding quarter is calculated from the Greenbook’s inﬂation forecasts for
the current and succeeding quarters. The weight given to the current quarter’s
inﬂation rate is the ratio of the number of days left in the current quarter to
the number of days from the Greenbook date until the end of the succeeding
quarter. The weight given to the succeeding quarter’s inﬂation rate is the ratio
of the number of days in that quarter to the number of days from the Greenbook
date until the end of the succeeding quarter. This weighted-average expected
inﬂation rate is subtracted from the Treasury bill yield. The Treasury bill yield
is for the date the Greenbook appeared and is for the bill maturing on the last
working day of the succeeding quarter. It is copied from the Federal Reserve
Bank of New York’s daily release “Composite Quotations for U.S. Government
Securities.” (For August 1972, the Treasury bill yield is for January 4, 1973,
instead of December 31, 1972.)
In the 1960s, the FOMC usually met more than 12 times per year. For
example, it met 15 times in 1965. In order to make the real rate series monthly
through 1978, we record an observation for only the ﬁrst FOMC meeting of
the month for those months in which there was more than one meeting. The
FOMC met only nine times in 1979. (Because the October 6, 1979, meeting
was unscheduled, there was no Greenbook and no real rate is calculated for
this date.) It met 11 times in 1980. Starting in 1981, it has met eight times a
year. For this reason, starting in 1979, the observations of the Greenbook real
rate series are less frequent than monthly.
The real rate series begins in November 1965 because the Greenbook ﬁrst
began to report predictions of inﬂation for the November 1965 meeting. Until
November 1968, for FOMC meetings in the ﬁrst two months of a quarter, the
Greenbook often reported a forecast of inﬂation for only the contemporaneous
quarter. For this reason, for the following FOMC meeting dates, the real rate
calculated is for the period only to the end of the contemporaneous quarter, not
to the end of the succeeding quarter: 11/23/65, 1/11/66, 2/8/66, 4/12/66, 5/10/66,
6/7/66, 7/26/66, 11/1/66, 12/13/66, 1/10/67, 7/18/67, 10/24/67, 11/14/67, 1/9/68,
2/6/68, 4/30/68, 5/28/68, 7/16/68, 10/8/68, 10/17/72, and 11/21/72. For these
dates, the maturity of the Treasury bill used to calculate the real rate varies
between one and three months. For other dates, the maturity varies between
three and six months. For this reason, some of the variation in real rates reﬂects
term-structure considerations. This variation is a consequence of the fact that
the FOMC meets at different times within a quarter and the Greenbook inﬂation
forecasts are for the quarters of the year.
b) Real rate series of one- to two-quarter maturity calculated as the difference between the commercial paper rate and expected inﬂation measured
by the implicit output deﬂator—Table 1, column (5)

R. Darin and R. L. Hetzel: Real Rate of Interest

35

This series is calculated like the one above except that the interest rate is
the commercial paper rate for prime paper placed through dealers. Observations
are matched with the publication dates of the Greenbook. From 1965 through
1969, rate data are from the New York Fed release “Commercial Paper.” Subsequently, they are from the Board’s FAME database. From 1965 through April
1971, the paper rate is for four- to six-month paper. Thereafter, if there are
fewer than 135 days from the Greenbook date to the end of the subsequent
quarter, the three-month paper rate is used; otherwise, the six-month paper rate
is used.
DRI Real Rate Series
a) Real rate series of one- to two-quarter maturity calculated as the difference between the Treasury bill rate and expected inﬂation measured by the
implicit output deﬂator
This series is shown in Figure 1. It is calculated like the Greenbook series
discussed above except for the substitution of predictions of (implicit GDP deﬂator, GNP before 1992) inﬂation from the most recent DRI Review of the U.S.
Economy available as of the publication of the Greenbook. In order to keep the
Greenbook and DRI real rate forecasts as closely comparable as possible, we
keep the interest rate the same. Consequently, unlike the Greenbook forecasts,
the matching between the date on which the interest rate is recorded and the
date of the inﬂation forecast is not exact.
b) Real rate series of one-year maturity calculated as the difference between
the Treasury bill rate and expected inﬂation measured by the consumer price
index—Table 1, column (3)
This series is the difference between the one-year Treasury bill rate and the
four-quarter inﬂation rate predicted by DRI. The one-year Treasury bill rate is
from Salomon Brothers “Analytical Record of Yields and Yield Spreads” and
is read from a yield curve. The yield for each month is for the last business day
for the preceding month. Because the DRI forecasts for a particular “control”
month are made at the end of the preceding month, the date of the interest rate
and forecast are fairly closely matched.
Four-quarter predicted inﬂation is a geometric average of the quarterly DRI
predictions of CPI inﬂation. When the control date on the DRI forecasts is the
ﬁrst or second month of the quarter, the initial quarterly inﬂation forecast is the
one reported for the contemporaneous quarter. For example, if the control date
is January or February, then the initial quarter used in constructing the inﬂation
forecast is the ﬁrst quarter of the year. If the control date is the third month of
the quarter, the initial quarter used in constructing the four-quarter forecast is the
inﬂation forecast for the subsequent quarter. For example, if the control date is
March, then the initial quarter of the four-quarter forecast is the second quarter.

36

Federal Reserve Bank of Richmond Economic Quarterly

c) Real rate series of two-quarter maturity calculated as the difference
between the Treasury bill rate and expected inﬂation measured by the consumer price index—Table 1, column (1)
The calculations for this series are like those for the preceding series with
two changes. First, the interest rate is the six-month Treasury bill yield from
Salomon Brothers. Second, the geometric average of the quarterly predictions
of inﬂation is for two quarters.
d) Real rate series of two-quarter maturity calculated as the difference
between the commercial paper rate and expected inﬂation measured by the
consumer price index—Table 1, column (2)
The calculations for this series are like those for the preceding series with
two changes. First, the interest rate is the 180-day commercial paper rate for the
last working day of the month preceding the control date on the DRI forecast
of inﬂation. Second, the geometric average of the quarterly DRI predictions of
CPI inﬂation is for two quarters.
Hoey and Survey of Professional Forecasters Ten-Year Real Rate Series
These series are shown in Figure 7. The ten-year market rate is the ten-year
Treasury constant maturity yield taken from the Federal Reserve’s Statistical
Release G.13, “Selected Interest Rates.”
Real Rate Series
Table 1 presents ﬁve series for the real rate of interest. The ﬁrst three are constructed using CPI inﬂation predictions from DRI. The ﬁrst two are for interest
rates of two-quarter maturity and the third is for one-year maturity. The ﬁrst
and third use the Treasury bill real rate, while the second uses the six-month
commercial paper rate.9 The last two real rate series use inﬂation forecasts from
the Greenbook. Depending upon when the Greenbook was published within the
quarter, the maturity of the real rate varies from slightly more than three months
to almost six months. One series uses the Treasury bill rate and the other the
commercial paper rate.

9 In periods such as 1969–1970 and 1973–1974, when market rates were high relative to
Regulation Q ceilings, disintermediation out of bank deposits apparently drove down the bill rate
relative to the paper rate. Consequently, in these periods the two real rate series differ.

R. Darin and R. L. Hetzel: Real Rate of Interest

37

Table 1 Real Rate of Interest
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1965
1966

1967

1968

1969

11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3

(DRI data are not available
until August 1970.)

1.66
2.1
2.65
2.22
2.85
2.8
2.35
0.9
1.5
1.49
1.32
1.27
1.48
2.59
2.71
2.49
2.31
1.08
0.98
0.87
0.55
0.73
0.69
0.32
0.14
1.63
1.41
1.66
1.16
1.27
1.27
1.85
1.27
1.32
1.32
1.47
2.23
2.13
2.95
2.81
2

One- to TwoQuarter
Commercial
Paper
2.27
2.34
2.82
2.32
3.38
3.59
3.07
2.16
2.36
2.38
2.1
2.67
3.1
3.23
3.84
3.24
3.25
1.95
1.9
1.89
1.48
1.5
1.45
1.19
1.18
2.24
2.08
2.04
1.73
1.81
2.1
2.29
2.28
2.14
1.98
2.09
2.49
2.25
3.1
3.27
2.96

38

Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1969

1970

1971

1972

4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8

3.75
3.7

5
4.5

3.34
3.17

3.24
2.3

3.625
2.715

2.66
2.3

1.15
−1.5

1.525
−0.84

1.09
0.41

1.1
−0.39

1.8
0.21

1.65
0.93

1.25
1.58

1.11
2.325

2.02
1.22

0.58
−0.42

1.3
−0.065

0.91
0.64

0
−1.14

0.125
−1.065

0.45
0.23

−0.45
−0.69

−0.05
−0.39

0.23
0.54

−0.17

0.2

0.54

1.79
1.53
1.61
2.63
2.56
2.89
3.82
3.72
3.81
4.27
3.39
2.86
2.01
2.54
2.68
2.97
2.7
2.62
2.25
1.3
0.14
0.63
−0.04
−0.71
−0.7
−0.46
−0.64
0.5
1.4
2.25
1.49
0.8
0.43
−0.46
−0.53
0.06
0.97
0.39
0.45
1.06
1.05

One- to TwoQuarter
Commercial
Paper
2.71
2.85
3.43
4.41
4.12
4.19
5.23
4.96
4.79
5.34
4.47
4.45
3.66
3.83
4.19
4.53
4.37
3.7
2.83
2.18
0.96
1.41
0.52
0.26
0.22
0.08
0.26
0.81
2.62
3.32
2.59
1.42
1.05
0.03
0.14
0.49
1.31
0.92
1.05
1.66
1.63

R. Darin and R. L. Hetzel: Real Rate of Interest

39

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1972

1973

1974

1975

1976

9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1

0.04

0.16

1.44

1.04
0.5

1.05
0.46

1.53
1.48

0.61
1

0.825
1.2

1.33
1.73

1.33
2.42
3.34
1.1
2.53

1.68
2.7
3.9
2.18
3.93

1.86
2.73
3.71
2.27
2.9

1.31
0.91
0.46
−1.7
−0.82
0.57
−0.08
−0.06
0.08
0.9
0.17
−2.82
−1.13
0.24
−1.16
−4.17
−1.4
−0.3
0.41
−0.85
0.67
0.72
−2.21
−1.45
−2.51
−0.63
−0.5

1.91
1.93
1.75
−0.95
−0.56
1.21
1.46
1.63
3.53
3.15
2.13
−0.02
−0.37
1.5
0.7
−3.66
−0.9
−0.07
0.475
−0.95
0.73
0.38
−2.65
−1.82
−2.35
−0.97
−0.4

1.69
1.71
0.99
−0.87
−0.04
1.49
0.88
1.4
1.68
1.39
1.17
−1.28
−0.76
−0.28
−0.38
−3.08
0.04
0.36
1.02
−0.11
1.03
1.05
−0.92
−0.33
−1.78
−0.2
0.06

1.58
1.85
1.58
1.17
1.5
1.29
2.17
2.25
1.72
2.44
3.6
2.71
2.48
1.63
2.74
0.69
0.91
0.05
0.64
2.05
0.77
0.78
−0.14
0.2
0.18
−1.19
−1.85
−1.5
−0.67
−0.84
−0.96
0.27
−0.66
−1.33
−0.15
−0.88
−2.54
0.26
−0.06
0.39
−0.05

One- to TwoQuarter
Commercial
Paper
2.09
2.57
2.1
1.44
1.75
1.84
2.73
2.79
2.46
3.16
4.46
4.01
3.94
3.33
3.37
2.43
1.51
0.55
1.08
2.87
3.32
3.36
4.05
2.78
2.92
0.65
−0.71
0.13
0.2
−0.2
−0.33
0.39
−0.17
−0.87
−0.18
−0.91
−2.32
0.41
0.24
0.65
0.05

40

Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1976

1977

1978

1979

2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6

−1.31
0.45
0.95
0.39
0.58
0.7
0.55
0.64
0.91
−0.17
−0.19
−0.85
−2.39
−1.15
−1.22
−0.85
0.17
−0.04
0.57
0.65
0.96
1.27
0.25
0.5
1.1
0.87
0.99
0.56
1.5
1.33
1.47
1.63
2.02
2.39
2.42
1.73
1.38
1.69
1.35
0.59
0.33

−1.27
0.05
0.65
0.15
0.33
0.5
0.35
0.48
0.83
−0.27
−0.02
−0.87
−2.71
−1.5
−1.27
−1.07
0.38
0.05
0.15
0.55
0.8
1.05
0.18
0.49
0.92
0.64
0.75
0.34
1.28
1.36
1.63
1.61
1.94
2.05
2.92
2.26
1.55
1.64
1.06
0.48
0.25

−0.79
0.62
1.16
0.81
1.32
1.01
1.04
0.9
0.85
0.06
−0.24
−0.78
−0.96
−0.47
−0.52
−0.16
0.24
0.1
0.6
0.58
0.77
1.23
0.78
1.08
1.52
1.32
1.5
1.35
1.8
2.2
2.14
1.76
1.96
2.68
2.97
2.75
2.05
2.3
1.89
1.65
1.15

One- to TwoQuarter
Commercial
Paper

−0.24
−0.08
−0.09
−0.01
0.13
−0.02
−0.34
−0.72
−0.66
−0.92
−1.57
−0.4
−0.67
−0.94
−0.85
−0.67
−1.2
−0.94
−0.52
−0.25
0.37
0.09
0.01
0.79
0.5
0.16
0.1
−0.08
−0.04
0.87
−0.23
0.86
1.55
1.41
1.38
1.79

−0.06
0.01
−0.09
−0.2
0.42
0.06
−0.21
−0.55
−0.67
−0.91
−1.26
−0.54
−0.73
−0.96
−0.85
−0.74
−0.97
−1.03
−0.55
−0.13
0.09
0.21
0.36
0.47
0.53
0.41
0.03
0.07
0.67
1.11
0.62
1.37
1.73
3
2.45
2.17

−0.23
2.28
1.88

0.05
2.2
2.01

R. Darin and R. L. Hetzel: Real Rate of Interest

41

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1979

1980

1981

1982

7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11

−0.23
−0.38
0.92
0.5
1.76
2.42
1.58
1.39
2.77
−1.21
−4.7
−0.67
0.3
0.11
1.06
1.76
2.13
4.51
2.37
2.35
4.14
3.78
6.6
6.34
6.96
7.47
9.75
8.85
5.58
4.79
6.28
7.53
8.52
9.07
9.8
6.57
7.17
4.02
4.6
3.92
4.7

−0.08
−0.24
1.46
1.34
2.98
2.54
1.5
1.36
2.44
−0.24
−3.91
−0.51
0.2
−0.52
1.02
2.23
1.89
4.36
2.47
2.74
3.68
3.73
6.06
6.86
7.13
7.55
8.74
8.5
5.84
4.07
6.13
7.29
7.82
8.85
9.67
6.87
7.33
4.75
5.36
5.06
4.72

0.62
0.33
1.43
1.61
3.04
2.89
1.49
1.67
4.31
1.63
−1.46
−0.59
0.05
0.26
1.34
2.24
3
4.24
2.43
2.45
4.04
3.66
6.41
5.71
6.69
7.8
9.38
9.12
6.21
4.73
6.16
7.52
7.87
8.2
8.6
6.38
7.25
5.1
5.35
4.77
4.22

One- to TwoQuarter
Commercial
Paper

−0.12
0.36
1.04

0.12
0.7
2.11

3.08

3.98

3.86
4.48
6.81
3.51
−1.29

3.67
4.53
7.21
5.05
−0.5

−1.08
−0.45
−0.46
1.33
3.3
6.62

−0.96
0.69
0
1.78
3.6
9.04

5.67
4.68

5.36
4.67

9.71

9.39

7.62
7.95

7.48
7.97

6.49
4.33
4.22

7.13
4.51
5.2

6.95
6.7

6.97
7.23

7.54

7.49

7.86
3.4

8.82
3.95

2.5
2.97

4.67
3.45

42

Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1982
1983

1984

1985

1986

12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4

3.65
3.25
3.58
4.53
4.6
4.52
4.43
4.68
5.62
5.15
3.91
4.08
4.51
5.62
5.25
4.94
5.32
5.8
5.9
6.3
7.32
7.39
7.14
5.98
5.59
5.04
5.28
5.86
5.43
4.95
4.15
3.81
4.53
4.28
4.37
4.61
3.98
4.21
4.53
5.8
5.24

3.34
3.33
3.3
4.15
4.32
4.28
3.94
4.39
5.06
4.7
3.63
3.77
4.02
5.34
4.87
4.5
5.04
5.54
5.88
6.5
7.16
7.14
6.76
5.8
5.24
4.72
4.83
5.71
5.26
4.78
4.12
4.04
4.63
4.24
4.6
4.65
4.09
4.41
4.71
5.88
5.69

4.06
3.39
3.67
4.16
4.44
4.33
4.59
5.03
5.97
5.65
4.47
4.64
5.11
5.77
5.36
5.38
5.61
6.01
6.84
7.44
7.58
7.74
7.49
6.55
6.03
5.59
5.75
6.12
5.81
5.22
4.25
3.98
4.54
4.57
4.61
4.74
4.34
4.3
4.38
5.09
4.66

One- to TwoQuarter
Commercial
Paper

3.31

3.9

4.66
4.36

4.62
4.33

5.36

5.25

6.04
6.12

6.07
5.98

5.03
4.65
4.61

4.84
4.5
4.91

4.95
6.02

4.66
5.98

6.15

6.61

6.75
7.05

7.17
7.5

6.41
5.39
4.06

6.54
5.39
4.3

4.71
5.66

4.6
5.89

5

5.17

4.37
4.36

4.48
4.77

4.36
4.07
3.4

5.02
4.28
3.92

3.72

3.97

3.7

4.26

R. Darin and R. L. Hetzel: Real Rate of Interest

43

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1986

1987

1988

1989

5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9

5.66
2.75
2.29
2.8
0.55
1.04
1.24
1.42
1.34
1.24
0.96
1.74
1.38
1.86
1.52
1.72
2
2.39
1.86
2.05
2.87
2.52
1.55
2.26
2.34
2.86
2.37
2.83
2.79
2.9
3.13
3.55
4.08
4.41
4.55
4.57
4
5.06
3.96
3.94
4.17

5.72
2.73
2.55
2.95
0.61
1.18
1.31
1.51
1.51
1.2
1.32
1.83
2.17
2.54
2.34
2.12
2.41
3.15
3.04
3.11
3.16
2.87
2.13
2.66
2.78
3.28
2.96
3.47
3.35
3.32
3.52
4.08
4.24
4.53
5.14
5.08
4.52
5.27
4.63
4.23
4.45

4.64
3.61
3.2
2.72
1.3
1.61
1.75
1.45
1.55
1.57
1.38
1.88
1.91
2.26
2.14
2.16
2.42
3.08
2.41
2.6
2.9
2.48
1.98
2.48
2.62
3.05
2.63
2.99
3.21
3.16
3.4
3.84
4.36
4.5
4.7
4.72
4.42
4.76
3.89
3.63
4.14

One- to TwoQuarter
Commercial
Paper

4.16

4.4

4.02
3.49
3.7

4.31
3.71
4.17

3.38
3.21

3.63
3.49

2.89
2.71

3
3.24

2.62

3.6

3.09
3.02
3.78

3.95
3.55
4.81

2.43
1.93

3.94
4.05

2.59
2.83

3.14
3.52

2.81
2.57

3.51
3.35

3.33
3.09

4.22
3.82

3.55
5.08

4.02
5.82

4.64
4.77

4.98
5.57

4.6

5.21

4.77
4.56

5.56
4.81

44

Federal Reserve Bank of Richmond Economic Quarterly

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1989

1990

1991

1992

1993

10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12
1

4.31
4.11
3.59
3.63
3.63
4.8
5.49
5.31
4.42
4.27
3.36
2.49
0.33
1.04
3.36
3.48
4.33
4.21
3.75
3.37
2.92
2.88
2.96
2.25
1.54
1.43
0.74
0.29
0.79
0.72
0.86
0.51
0.51
0.32
0.08
−0.01
−0.57
0.15
0.88
0.89

4.65
4.21
3.72
3.66
3.55
4.71
5.53
5.35
4.41
4.32
3.33
2.54
0.82
1.27
3.89
4.07
4.73
4.37
3.96
3.48
2.98
3.16
3.11
2.41
1.71
1.54
1.04
0.5
0.86
0.84
0.92
0.59
0.57
0.5
0.19
0.11
−0.26
0.3
1.17
1.01

4.38
3.83
3.45
3.53
3.83
4.3
4.86
4.77
4.15
3.93
3.39
3.42
2.25
2.42
3.84
3.43
4.01
3.65
3.25
2.85
2.7
2.83
2.84
2.25
1.65
1.51
0.97
0.36
0.73
0.86
1
0.82
0.74
0.63
0.32
0.13
−0.27
0.5
0.92
0.81

5.02
4.11
3.87

One- to TwoQuarter
Commercial
Paper
5.71
4.3
4.48

R. Darin and R. L. Hetzel: Real Rate of Interest

45

Table 1 Real Rate of Interest (Continued)
(1)

(2)
DRI

(3)

(4)

(5)
Greenbook

Two-Quarter
One- to TwoTwo-Quarter Commercial One-Year
Quarter
Year Month
T-bill
Paper
T-bill
T-bill
1993

1994

2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
10
11
12

0.51
0.4
0.34
0.27
0.17
−0.24
0.13
−0.35
−0.68
−0.36
0.14
0.14
−0.06
0.2
0.61
0.99
1.35
1.12

0.63
0.49
0.49
0.35
0.18
−0.05
0.23
−0.26
−0.52
−0.33
0.21
0.18
−0.03
0.39
0.84
1.15
1.45
1.32
1.84
1.96
2.48
2.63
3.07

0.61
0.38
0.28
0.15
0.36
0.08
0.31
0.15
0
0.12
0.46
0.44
0.26
0.62
1.12
1.66
1.96
2.07

One- to TwoQuarter
Commercial
Paper

46

Federal Reserve Bank of Richmond Economic Quarterly

REFERENCES
Antoncic, Madelyn. “High and Volatile Real Interest Rates,” Journal of Money,
Credit, and Banking, vol. 18 (February 1986), pp. 18–27.
Barro, Robert J. “What the Fed Can’t Do,” The Wall Street Journal, August
19, 1994, p. A10.
Bonser-Neal, C. “Monetary Regime Changes and the Behavior of Ex Ante
Real Interest Rates,” Journal of Monetary Economics, vol. 26 (1990), pp.
329–59.
Carlson, John A. “A Study of Price Forecasts,” Annals of Economic and Social
Measurement, vol. 6 (Winter 1977), pp. 27–56.
Carlson, John B. “Assessing Real Interest Rates,” Federal Reserve Bank of
Cleveland Economic Commentary, October 1993.
Caskey, John. “Modeling the Formation of Price Expectations: A Bayesian
Approach,” The American Economic Review, vol. 75 (September 1985),
pp. 768–76.
Choi, Woon Gyu. “Inﬂation Forecastability and the Fisher Relationship: A
Reexamination Using a Threshold Model,” Working Paper. Los Angeles:
UCLA Department of Economics, October 15, 1994.
Cullison, William E. “On Recognizing Inﬂation,” Federal Reserve Bank of
Richmond Economic Review, vol. 74 (July/August 1988), pp. 4–12.
Fama, Eugene F. “Short-Term Interest Rates as Predictors of Inﬂation,” The
American Economic Review, vol. 65 (June 1975), pp. 269–82.
, and M. Gibbons. “Inﬂation, Real Returns, and Capital Investment,”
Journal of Monetary Economics, vol. 9 (1982), pp. 297–323.
Garbade, Kenneth D., and Paul Wachtel. “Time Variation in the Relationship
between Inﬂation and Interest Rates,” Journal of Monetary Economics,
vol. 4 (November 1978), pp. 755–65.
Greenspan, Alan. “Statement before the Subcommittee on Economic Growth
and Credit Formation of the Committee on Banking, Finance and Urban
Affairs, U.S. House of Representatives, July 20, 1993,” Federal Reserve
Bulletin, vol. 79 (September 1993), pp. 849–55.
Hafer, R. W., and David H. Resler. “The ‘Rationality’ of Survey-Based
Inﬂation Forecasts,” Federal Reserve Bank of St. Louis Review, vol. 62
(November 1980), pp. 3–11.
Hetzel, Robert L. “Indexed Bonds as an Aid to Monetary Policy,” Federal
Reserve Bank of Richmond Economic Review, vol. 78 (January/February
1992), pp. 13–23.

R. Darin and R. L. Hetzel: Real Rate of Interest

47

Huizinga, John, and Frederic S. Mishkin. “Monetary Policy Regime Shifts
and the Unusual Behavior of Real Interest Rates,” Carnegie-Rochester
Conference Series on Public Policy, vol. 24 (Spring 1986), pp. 231–74.
Jacobs, Rodney L., and Robert A. Jones. “Price Expectations in the United
States: 1947–75,” The American Economic Review, vol. 70 (June 1980),
pp. 269–77.
Makin, John H. “Real Interest, Money Surprises, Anticipated Inﬂation and
Fiscal Deﬁcits,” Review of Economics and Statistics, vol. 65 (August
1983), pp. 374–84.
Mehra, Yash P. “Inﬂationary Expectations, Money Growth, and the Vanishing
Liquidity Effect of Money on Interest: A Further Investigation,” Federal
Reserve Bank of Richmond Economic Review, vol. 71 (March/April 1985),
pp. 23–35.
Mullineaux, Donald J. “On Testing for Rationality: Another Look at the
Livingston Price Expectations Data,” Journal of Political Economy, vol.
86 (April 1978), pp. 329–36.
Murray, Alan. “Fed Moved too Slow on Increasing Rates,” The Wall Street
Journal, April 11, 1994, p. A1.
Trehan, Bharat. “The Credibility of Inﬂation Targets,” Federal Reserve Bank
of San Francisco Weekly Letter, January 6, 1995.
Wessel, David. “Fed Boosts Short-term Rates,” The Wall Street Journal, April
19, 1994, p. A2.

Using the Permanent Income
Hypothesis for Forecasting
Peter N. Ireland

P

ersonal consumption expenditures grew by almost 2 percent during 1993
in real, per-capita terms. Real disposable income per capita, meanwhile,
actually fell slightly. By deﬁnition, households draw down their savings
when consumption grows faster than income. In fact, the ﬁgures for consumption and income just mentioned underlie a decline in the personal savings rate
from over 6 percent in the fourth quarter of 1992 to only about 4 percent in
the fourth quarter of 1993.1
One popular interpretation of these data starts with the idea that reductions
in the savings rate cannot be permanently sustained. Eventually, households
must rebuild their savings by cutting back on consumption; to the extent that
lower consumption leads to lower income, income must fall as well. Thus,
in U.S. News & World Report, David Hage (1993–94) used the behavior
of consumption, income, and savings to forecast that the economy would
slow in 1994: “[A] slowdown in consumer spending is likely, and it could
trim an additional 0.6 percentage point off growth” (p. 43). Around the same
time, Gene Epstein (1993) of Barron’s quoted economist Philip Braverman
as saying that “consumers don’t have the wherewithal to keep up the current
spending pace. The prevailing euphoria will get knocked for a loop” (p. 37).
Similarly, in DRI/McGraw-Hill’s Review of the U.S. Economy, professional
forecaster Jill Thompson (1993) wrote: “All is not rosy, of course. Consumers
went out on a limb to give the economy a needed jump-start. . . . They have
pushed the saving rate very low and incurred more debt. . . . Consumption
must slow” (pp. 16 and 18).
In light of this conventional wisdom, which suggests that a decline in
savings presages a slowdown in economic growth, the continued strength of
The author would like to thank Mike Dotsey, Mary Finn, Bob Hetzel, Tom Humphrey, and
Tony Kuprianov for helpful comments and suggestions. 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.
1 Appendix

B describes all data and their sources.

Federal Reserve Bank of Richmond Economic Quarterly Volume 81/1 Winter 1995

49

50

Federal Reserve Bank of Richmond Economic Quarterly

the U.S. economy in 1994 came as a surprise, raising the question of whether
an alternative framework can better reconcile the recent behavior of consumption, income, and savings. This article considers one such alternative: Milton
Friedman’s (1957) permanent income hypothesis. This hypothesis implies that
households save less when they expect future income to rise. Thus, according
to the permanent income hypothesis, a decline in savings like that experienced
during 1993 signals that faster, not slower, income growth lies ahead.
Although developed in detail by Friedman in his 1957 monograph, the
permanent income hypothesis has its origins in Irving Fisher’s (1907) theory
of interest. Thus, the article begins by reviewing Fisher’s graphical analysis
and by indicating how this analysis extends to a full statement of the permanent income hypothesis. The article goes on to show how Robert Hall (1978)
derives the permanent income hypothesis from a mathematical theory that has
very speciﬁc implications for the joint behavior of consumption, income, and
savings. Following John Campbell (1987), it draws on Hall’s version of the hypothesis to formulate a simple econometric model that exploits data on savings
to forecast future income growth. Estimates from this model show that the U.S.
data conform not to conventional wisdom, but to the intuition provided by the
permanent income hypothesis: historically, declines in savings have preceded
periods of faster, not slower, income growth. Finally, the article uses the model
to generate forecasts for the U.S. economy.

1.

FISHER’S THEORY OF INTEREST AND
THE PERMANENT INCOME HYPOTHESIS

In presenting his theory of interest, Irving Fisher (1907) uses a graph like that
shown in Figure 1 to illustrate how a household makes its consumption and
savings decisions. To simplify his graphical analysis, Fisher considers only two
periods. His horizontal axis measures goods at time 0, and his vertical axis
measures goods at time 1. Fisher’s representative household receives income
y0 during time 0 and y1 during time 1.
The representative household faces the ﬁxed interest rate r, which serves
as an intertemporal price. It measures the rate at which the market allows the
household to exchange goods at time 1 for goods at time 0. In particular, if the
household lends one unit of the good at time 0, it gets repaid (1 + r) units of
the good at time 1. Similarly, if the household borrows one unit of the good
at time 0, it must repay (1 + r) units of the good at time 1. Thus, in Figure 1,
the household’s budget constraint AA , which passes through the income point
(y0 , y1 ), has slope −(1 + r).
The household’s preferences over consumption at the two dates are represented by the indifference curves U0 and U1 , each of which traces out a set of
consumption pairs that yield a given level of utility. Utility increases with

P. N. Ireland: Permanent Income Hypothesis

51

Figure 1 Fisher’s Diagram
Time 1
Consumption
B

A

y '
1
c '
1

c1
y1"
U

1

slope = - (1+r)
y1
slope
= - (1+r)
A'
y0'

c0

c0'

y0

U0
B'

Time 0
Consumption

s0' = y0 - c0'

+

s =y -c
0
0
0

consumption in both periods; hence, U1 > U0 . The slope of each indifference
curve is determined by the household’s marginal rate of intertemporal substitution, the ratio of its marginal utilities of consumption at dates 0 and 1, or the
rate at which it is willing to exchange goods at time 1 for goods at time 0.

52

Federal Reserve Bank of Richmond Economic Quarterly

To maximize its utility, the household chooses the consumption pair (c0 , c1 ),
where the indifference curve U0 is tangent to the budget constraint AA . At
(c0 , c1 ), the household’s marginal rate of intertemporal substitution equals the
gross rate of interest (1 + r). The household saves amount s0 = y0 − c0 .
Now suppose that the household’s income pair changes to (y0 , y1 ). Since
this new income point lies on the same budget constraint as (y0 , y1 ), the household continues to select (c0 , c1 ) as its optimal consumption combination. In
fact, the household chooses (c0 , c1 ) starting from any income point along AA .
Since all income points along AA have the same present value, equal to
PV = y0 +

y1
,
(1 + r)

(1)

this example illustrates the ﬁrst implication of Fisher’s theory: the household’s
consumption choice depends only on the present value of its income pair
(y0 , y1 ), not on y0 and y1 separately.
Next, hold y0 constant and suppose that the household’s income at time
1 increases to y1 . This change increases the present value of the household’s
income pair. It shifts the budget constraint out from AA to BB and leads the
household to choose the preferred consumption pair (c0 , c1 ). Since c0 > c0 , the
increase in time 1 income allows the household to reduce its time 0 savings
from s0 = y0 − c0 to s0 = y0 − c0 . This example illustrates the second implication of Fisher’s theory: the household saves less when it expects future
income to be high. Conversely, the household saves more when it expects
future income to be low.
This second implication makes Fisher’s model useful for forecasting. It
suggests, in particular, that data on household savings help forecast future
income. A low level of savings today indicates that households expect higher
income in the future. A high level of savings today signals that households
expect lower income in the future. Note that both of these predictions contradict the conventional wisdom, which indicates that low savings predate lower
income.
Milton Friedman’s (1957) permanent income hypothesis generalizes
Fisher’s analysis to a model in which there are more than two periods and the
representative household may be uncertain about its future income prospects.
Thus, Friedman also derives the result that a representative household’s consumption depends not on its current income but on the present value of its
future income. With an inﬁnite number of periods and uncertainty, this present
value can be written
∞

PV =
t=0

Eyt
,
(1 + r)t

where Eyt denotes the household’s expected income at time t.

(2)

P. N. Ireland: Permanent Income Hypothesis

53

Friedman deﬁnes the household’s permanent income yp as the constant
income level that, if received with certainty in each period t, has the same
present value as the household’s actual income path. That is, yp satisﬁes
∞
t=0

yp
= PV =
(1 + r)t

∞
t=0

Eyt
.
(1 + r)t

(3)

In light of the formula
∞
t=0

1+r
1
=
,
(1 + r)t
r

(4)

equation (3) simpliﬁes to
yp =

r
PV.
1+r

(5)

Thus, the ﬁrst implication of the permanent income hypothesis is that the household’s consumption at date 0 can be written as a function of its permanent
income:
c0 = f (y p ).

(6)

Similarly, Friedman generalizes the second implication of Fisher’s analysis.
Friedman’s representative household borrows to increase consumption today
when it anticipates higher income in the future. In other words, it saves less
when it expects future income to be high. Conversely, the household uses additional savings to buffer its consumption against expected declines in income;
it saves more when it expects future income to be low. Thus, like Fisher’s
theory of interest, Friedman’s permanent income hypothesis suggests that data
on savings help forecast future income.

2.

HALL’S VERSION OF THE PERMANENT
INCOME HYPOTHESIS

Robert Hall (1978) develops a mathematical version of the permanent income
hypothesis that makes the relationship between savings and expected future
income identiﬁed by Fisher and Friedman more precise. In fact, the details
of Hall’s model indicate exactly how data on savings can be used to forecast
future changes in income.

54

Federal Reserve Bank of Richmond Economic Quarterly

Hall, like Friedman, assumes that there are many time periods t = 0, 1, 2,
. . . and that the representative household is uncertain about its future income
prospects.2 Hall’s inﬁnitely lived representative household has expected utility
∞

β t u(ct ),

E

(7)

t=0

where E once again denotes the household’s expectation, u(ct ) measures its
utility from consuming amount ct at time t, and the discount factor β lies
between zero and one.
The household begins period t with assets of value At . It earns interest on
these assets at the constant rate r; its capital income during period t is therefore
ykt = rAt . The household also receives labor income ylt during period t.
At the end of period t, the household divides its total income yt = ykt + ylt
between consumption ct and savings st = yt − ct . It then carries assets of value
At+1 = At + st = (1 + r)At + ylt − ct

(8)

into period t + 1.
The household is allowed to borrow against its future labor income at the
interest rate r; because of borrowing and the associated accumulation of debt,
its assets At may become negative. Its borrowing is constrained in the long run,
however, by the requirement that
lim

t→∞

At
= 0.
(1 + r)t

(9)

To see how equation (9) limits the household’s borrowing, note that equation
(8) is a difference equation in the variable At . Using equation (9) as a terminal
condition, one can solve equation (8) forward to obtain
∞

At =
j=0

ct+j − ylt+j
.
(1 + r)j+1

(10)

Equation (10) shows that the household must repay any debt owed today (−At ) by setting future consumption ct+j below future labor income ylt+j .
Equation (10) also implies that
∞

At =
j=0

Et ct+j
−
(1 + r)j+1

∞
j=0

Et ylt+j
,
(1 + r)j+1

(11)

where Et denotes the representative household’s expectation at time t. This
condition states that the household’s current level of assets At must be sufﬁcient to cover any discrepancy between the present value of expected future
consumption and the present value of expected future labor income.
2 This

presentation of Hall’s model follows Sargent (1987, Ch. 12) quite closely.

P. N. Ireland: Permanent Income Hypothesis

55

The representative household chooses consumption ct and asset holdings
At+1 for all t = 0, 1, 2, . . . to maximize the utility function (7) subject to the
constraints (8) and (9). The solution to this problem dictates that
u (ct ) = β(1 + r)Et u (ct+1 ).

(12)

Equation (12) simply generalizes the optimality condition shown in Figure 1 as
the tangency between the household’s indifference curve and its budget constraint. It indicates that the household sets its expected marginal rate of intertemporal substitution, the ratio of its marginal utility of consumption at time t
to its expected marginal utility of consumption at time t + 1, u (ct )/βEt u (ct+1 ),
equal to the gross rate of interest (1 + r).
Assume that the interest rate r is related to the household’s discount factor
β via β = 1/(1 + r). Assume also that the household’s utility is quadratic, with
u(c) = u0 + u1 c − (u2 /2)c2 for some positive constants u0 , u1 , and u2 .3 Under
these additional assumptions, equation (12) reduces to
ct = Et ct+1 .

(13)

Equation (13) states Hall’s famous result that under the permanent income
hypothesis, consumption follows a random walk.
Equation (13) implies that Et ct+j = ct for all j = 0, 1, 2, . . . . Substituting
this result into equation (11) yields
ct = rAt +

r
1+r

∞
j=0

Et ylt+j
.
(1 + r)j

(14)

The right-hand side of equation (14), equal to current capital income plus
r/(1 + r) times the present value of expected future labor income, deﬁnes the
representative household’s permanent income in Hall’s model. Equation (14),
like equation (6), states the ﬁrst main implication of the permanent income
hypothesis: consumption is determined by permanent income.
Using ykt = rAt and st = ykt + ylt − ct and denoting the change in labor
income by ∆ylt = ylt − ylt−1 , one can rearrange equation (14) to reveal the
second main implication of the permanent income hypothesis:
∞

st = −
j=1

Et ∆ylt+j
.
(1 + r)j

(15)

According to equation (15), the household’s current savings st equals the present
value of expected future declines in its labor income. Thus, equation (15) states
that the household saves less when it expects future gains in income, that is,
3 These assumptions, while restrictive, greatly simplify the analysis. Hansen and Singleton
(1982) derive and test the implications of Hall’s (1978) model under more general assumptions
about r and u(c).

56

Federal Reserve Bank of Richmond Economic Quarterly

positive values for ∆ylt+j . Conversely, the household saves more when it anticipates future declines in income, that is, negative values for ∆ylt+j . Once
again, this second implication of the permanent income hypothesis suggests
that data on savings help forecast future changes in income.

3.

THE PERMANENT INCOME FORECASTING MODEL

John Campbell (1987) shows exactly how Hall’s version of the permanent
income hypothesis can be used to formulate an econometric forecasting model
for the U.S. economy. Since the permanent income hypothesis implies that data
on savings will help forecast future changes in labor income, Campbell starts
with a bivariate vector autoregression (VAR) for ∆ylt and st of the form
∆ylt
a(L) b(L)
=
st
c(L) d(L)

∆ylt−1
u1t
+
,
st−1
u2t

(16)

where, for example, a(L) = a1 +a2 L+a3 L2 +· · ·+ap Lp−1 , L is the lag operator,
and u1t and u2t are serially uncorrelated errors.4 Campbell then shows how the
relationship (15) between savings and future labor income identiﬁed by Hall’s
model translates into a set of parameter restrictions on the VAR (16).
Campbell works through the series of linear algebraic manipulations outlined in Appendix A. First, he uses the VAR (16) to compute the expected
future declines in labor income −Et ∆ylt+j that appear on the right-hand side
of equation (15). Next, he demonstrates that these expected future declines
depend on the coefﬁcients of the lag polynomials a(L), b(L), c(L), and d(L).
In particular, if the present value of the expected future declines in income
are to equal the current value of savings st , as required by equation (15), the
following parameter restrictions must hold:
a1 = c1 , . . . , ap = cp , d1 − b1 = (1 + r), b2 = d2 , . . . , bp = dp .

(17)

Equation (17) gives the restrictions imposed by Hall’s version of the permanent
income hypothesis on the VAR (16).

4.

PERFORMANCE OF THE PERMANENT INCOME
FORECASTING MODEL

Quarterly data, 1959:1–1994:3, are used to estimate the VAR in equation (16)
both with and without the permanent income restrictions (17). The speciﬁcation (16) assumes that ∆ylt and st have zero mean; in practice, adding constant
terms to the VAR removes each variable’s sample mean. The estimated models
include six lags of each variable on the right-hand side.
4 See

Sargent (1987, Ch. 9) for details about the lag operator.

P. N. Ireland: Permanent Income Hypothesis

57

Panel (a) of Table 1 shows the unconstrained equation for labor income
growth. The negative sum of the coefﬁcients on lagged savings indicates that a
decrease in savings translates into a forecast of faster income growth, exactly
as implied by the permanent income hypothesis. Moreover, an F-test easily rejects the hypothesis that the savings data do not help to forecast future income
growth; again as predicted by the permanent income hypothesis, the coefﬁcients
on the lags of st are jointly signiﬁcant at the 0.00037 level.
Panel (b) of Table 1 displays the equation for labor income when the
permanent income constraints (17) are imposed on the VAR. The estimates
assume that r = 0.01, which corresponds to an annual real interest rate of 4
percent. The coefﬁcients of the constrained equation closely resemble those of
the unconstrained equation, indicating once again that the data are consistent
with the permanent income hypothesis.

Table 1 Estimated Labor Income Equation from the
Permanent Income Forecasting Model
(a) Unconstrained Model
+ 0.107∆ylt−1
(0.118)

− 0.0286∆ylt−2
(0.121)

+ 0.236∆ylt−3
(0.118)

− 0.111∆ylt−4
(0.121)

+ 0.0261∆ylt−5
(0.120)

− 0.0636∆ylt−6
(0.0876)

− 0.379st−1
(0.102)

+ 0.218st−2
(0.143)

− 0.140st−3
(0.144)

+ 0.172st−4
(0.142)

− 0.109st−5
(0.145)

+ 0.170st−6
(0.110)

+ 0.0448∆ylt−1
(0.109)

− 0.0734∆ylt−2
(0.111)

+ 0.168∆ylt−3
(0.109)

− 0.0417∆ylt−4
(0.111)

+ 0.0115∆ylt−5
(0.111)

− 0.0742∆ylt−6
(0.0806)

− 0.375st−1
(0.0942)

+ 0.235st−2
(0.131)

− 0.109st−3
(0.132)

∆ylt = 88.9

+ 0.0899st−4
(0.131)

− 0.0840st−5
(0.133)

+ 0.159st−6
(0.101)

(b) Constrained Model
∆ylt = 106

Note: Standard errors are in parentheses.

A statistical test rejects the constraints (17) at the 99 percent conﬁdence
level. As noted by King (1995), however, formal hypothesis tests seldom fail
to reject the implications of detailed mathematical models such as Hall’s.5
5 Moreover, as indicated in footnote 3, Hall’s model makes very restrictive assumptions about
the interest rate and the household’s utility function. The statistical rejection of the constraints
(17) may therefore reﬂect the failure of one of these additional assumptions to hold in the data,

58

Federal Reserve Bank of Richmond Economic Quarterly

Ultimately, the permanent income hypothesis must be judged on its ability to
forecast the data better than alternative models.
Thus, Table 2 reports on the forecasting performance of the permanent
income model. First, the constrained VAR is estimated with data from 1959:1
through 1970:4 and is used to generate out-of-sample forecasts for the total
change in labor income one, two, four, and eight quarters ahead. Next, the
sample period is extended by one quarter, and additional out-of-sample forecasts are obtained. Continuing in this manner yields out-of-sample forecasts for
1971:1 through 1994:3.
The table computes the permanent income model’s mean squared error at
each forecast horizon. It expresses each mean squared error as a fraction of
the mean squared error from a univariate model for labor income growth with
six lags. Thus, ﬁgures less than unity in Table 2 indicate that the VAR’s mean
squared forecast error is smaller than the univariate model’s.
The table shows that the permanent income forecasts improve on the univariate forecasts at all horizons. The gain in forecast accuracy exceeds 10
percent at horizons longer than one quarter. The permanent income model is
especially valuable for forecasting at the annual horizon, where it reduces the
univariate forecast errors by 25 percent.

Table 2 Performance of the Permanent Income Forecasting Model
Horizon
(Quarters Ahead)

Improvement Over
Univariate Model

Improvement Over
Unconstrained VAR

1
2
4
8

0.95
0.87
0.75
0.89

1.00
0.97
0.92
0.78

Note: Performance is measured by the mean squared forecast error from the permanent income
model expressed as a fraction of the mean squared forecast error from two alternative models:
a univariate model for labor income and an unconstrained vector autoregression for savings and
labor income.

Table 2 also compares the forecasting performance of the constrained VAR
to the performance of the VAR when the permanent income constraints (17)
are not imposed. Once again, the ﬁgures less than unity indicate that the permanent income forecasts have lower mean squared error than the unconstrained
forecasts. The improvement is most dramatic at longer horizons. Thus, Table 2
shows that the permanent income constraints help improve the model’s out-ofsample forecasting ability relative to both a univariate model for labor income
growth and an unconstrained VAR.
rather than a more general failure of the permanent income hypothesis itself.

P. N. Ireland: Permanent Income Hypothesis

59

Figure 2 plots the data for real personal savings per capita. It shows that
savings increased from 1987 through the end of 1992, but have fallen since
then. According to the permanent income hypothesis, this recent decline in
savings indicates that households expect future gains in income.
Indeed, forecasts from the permanent income model reﬂect these expectations. When estimated with data through 1994:3, the constrained VAR predicts
growth in real disposable labor income per capita of $181 for 1995. Since real
disposable labor income now stands at about $12,300 and the population is
growing at an annual rate of about 1 percent, this ﬁgure translates into a gain
in aggregate real labor income of 2.5 percent. Thus, the permanent income
model predicts that the U.S. economy will continue to expand in 1995.
Figure 2 Real Personal Savings
1987 Dollars per Capita

1300
1200
1100
1000

Dollars

900
800
700
600
500
400
300
1959:1

64

69

74

79

84

89

94

+

5.

CONCLUSION

Conventional wisdom suggests that the recent decline in personal savings cannot be sustained. Eventually, households will have to reduce their consumption,
causing economic growth to slow. The permanent income hypothesis, however,
contradicts this conventional wisdom. According to this hypothesis, households
reduce their savings when they expect future income to be high; a low level of
savings indicates that faster, not slower, income growth lies ahead.

60

Federal Reserve Bank of Richmond Economic Quarterly

This article uses a mathematical version of the permanent income hypothesis to formulate a simple econometric forecasting model for the U.S. economy.
Estimates from the model reveal that the data are broadly consistent with the
hypothesis’ implications. Most important, the data indicate that declines in savings typically precede periods of faster, rather than slower, growth in income.
The results show that the permanent income model improves on univariate forecasts for annual labor income growth by 25 percent. The model also
improves on the forecasting ability of an unconstrained vector autoregression
for savings and labor income. In light of the recent decline in savings, the
permanent income model forecasts continuing growth in personal income for
1995.

APPENDIX A:

DERIVATION OF THE PERMANENT
INCOME RESTRICTIONS

Campbell (1987) rewrites the vector autoregression (16) as
zt = Azt−1 + vt ,

(18)

zt = [∆ylt . . . ∆ylt−p+1 st . . . st−p+1 ] ,

(19)

where



a1
1







A=
 c1







·
·
·

·

·

ap

b1

·

·

·

1
·

cp

d1
1

·

·

·

·
·

·

·

bp











,
dp 







(20)

1
and
vt = [u1t 0 . . . 0 u2t 0 . . . 0] .

(21)

Equation (18), along with the conditions Et vt+j = 0 for all j = 1, 2, 3, . . . ,
makes it easy to write the time t expectation of the vector zt+j as
Et zt+j = Aj zt .

(22)

P. N. Ireland: Permanent Income Hypothesis

61

Equation (22) implies, in particular, that
Et ∆ylt+j = e1 Aj zt ,

(23)

where e1 is a row vector consisting of a one followed by 2p − 1 zeros.
Equation (23) shows that expected changes in future labor income depend
on the coefﬁcients of the lag polynomials a(L), b(L), c(L), and d(L), which are
contained in the matrix A. The current value of savings, meanwhile, is just
st = ep+1 zt ,

(24)

where ep+1 is a row vector consisting of p zeros followed by a one and p − 1
zeros. Equation (23) can be substituted into the right-hand side of equation
(15), and equation (24) can be substituted into the left-hand side of equation
(15), so that Hall’s (1978) relationship between savings and income becomes
∞

ep+1 zt = −

(1 + r)−j e1 Aj zt .

(25)

j=1

Equation (25) must hold if, as required by equation (15), the current value of
savings is to equal the discounted value of expected future declines in labor
income.
Campbell uses the matrix analog to equation (4), which implies
∞

(1 + r)−j Aj = (1 + r)−1 A[I − (1 + r)−1 A]−1 ,

(26)

j=1

to rewrite (25) as
ep+1 zt = −e1 (1 + r)−1 A[I − (1 + r)−1 A]−1 zt ,

(27)

ep+1 [I − (1 + r)−1 A] = −e1 (1 + r)−1 A,

(28)

or, more simply,

where I is an identity matrix of the same size as A. The deﬁnition of A given
by equation (20) implies that this last condition is equivalent to equation (17).
Thus, equation (17) restates Hall’s permanent income condition (15) as a set
of parameter constraints that can be imposed on the VAR (16).

62

Federal Reserve Bank of Richmond Economic Quarterly

APPENDIX B:

DATA SOURCES

All data used in this article come from the National Income and Product
Accounts, as reported in the DRI/McGraw-Hill Data Base. The underlying
quarterly series, 1959:1–1994:3, are the following:
WSD = Wage and Salary Disbursements
YOL = Other Labor Income
YENTADJ = Proprietors’ Income
YRENTADJ = Rental Income
DIV@PER = Personal Dividend Income
YINTPER = Personal Interest Income
V = Transfer Payments
TWPER = Personal Contributions for Social Insurance
TP = Personal Tax and Nontax Payments
YD = Disposable Personal Income
C = Personal Consumption Expenditures
INTPER = Interest Paid by Consumers to Business
VFORPER = Personal Transfer Payments to Foreigners
YD87 = Disposable Personal Income, 1987 Dollars
NNIA = Population
From these series, disposable personal labor income (YDL) and disposable
personal capital income (YKL) are constructed as
YDL = WSD + YOL + (2/3) ∗ YENTADJ + V − TWPER
−(2/3) ∗ TP − VFORPER
and
YDK = (1/3) ∗ YENTADJ + YRENTADJ + DIV@PER + YINTPER
−(1/3) ∗ TP − INTPER.
When converted into real, per-capita terms using the deﬂator for disposable
personal income (YD/YD87) and the population NNIA, the series YDL, YKL,
and C correspond to ylt , ykt , and ct in the text. Real savings per capita is deﬁned
by st = ylt + ykt − ct .

P. N. Ireland: Permanent Income Hypothesis

63

REFERENCES
Campbell, John Y. “Does Saving Anticipate Declining Labor Income? An
Alternative Test of the Permanent Income Hypothesis,” Econometrica,
vol. 55 (November 1987), pp. 1249–73.
Epstein, Gene. “Oh, Say Can You Foresee the GDP?” Barron’s, December 27,
1993, p. 37.
Fisher, Irving. The Rate of Interest. New York: MacMillan Company, 1907.
Friedman, Milton. A Theory of the Consumption Function. Princeton: Princeton
University Press, 1957.
Hage, David. “Will the Economy Pick Up Steam?” U.S. News & World Report,
December 27, 1993–January 3, 1994, pp. 42–43.
Hall, Robert E. “Stochastic Implications of the Life Cycle-Permanent Income
Hypothesis: Theory and Evidence,” Journal of Political Economy, vol. 86
(December 1978), pp. 971–87.
Hansen, Lars Peter, and Kenneth J. Singleton. “Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models,”
Econometrica, vol. 50 (September 1982), pp. 1269–86.
King, Robert G. “Quantitative Theory and Econometrics.” Manuscript.
University of Virginia, 1995.
Sargent, Thomas J. Macroeconomic Theory, 2d ed. Orlando: Academic Press,
1987.
Thompson, Jill. “Risks to the Forecast,” Review of the U.S. Economy,
November 1993, pp. 16–23.

The Rational Expectations
Hypothesis of the Term
Structure, Monetary Policy,
and Time-Varying
Term Premia
Michael Dotsey and Christopher Otrok

M

ost empirical studies of the rational expectations hypothesis of the
term structure (REHTS) generally ﬁnd that the data offer little support for the theory.1 In many cases this large body of empirical
work indicates that the theory does not even provide a close approximation of
market behavior. This feature has led some investigators to search for alternative “irrational” theories of behavior in order to explain the data. We, on the
other hand, believe that the rejections are so striking that the large amount of
irrationality implied by the data is too implausible for this avenue to be treated
seriously. Since the rejection of rational expectations in these studies generally
involves the rejection of more complicated joint hypotheses, we choose to focus
our energies on exploring a broader class of models that are consistent with
REHTS.
In particular, we examine a model that incorporates Federal Reserve behavior along with a reasonable parameterization of term premia to revise the
theory. The consideration of Fed behavior was ﬁrst suggested by Mankiw and
Miron (1986), who found that REHTS was more consistent with the data prior
to the founding of the Fed. Even stronger evidence is presented in Choi and

The authors wish to thank Peter Ireland for many useful and technically helpful suggestions.
The comments of Tim Cook, Douglas Diamond, Tony Kuprianov, and John Weinberg are
also greatly appreciated. Sam Tutterow provided excellent research assistance. The views
expressed in this article are those of the authors and do not necessarily reﬂect those of the
Federal Reserve Bank of Richmond or the Federal Reserve System.
1 For

an extensive set of results, see Campbell and Shiller (1991). Cook and Hahn (1990)
and Rudebusch (1993) also give excellent surveys.

Federal Reserve Bank of Richmond Economic Quarterly Volume 81/1 Winter 1995

65

66

Federal Reserve Bank of Richmond Economic Quarterly

Wohar (1991), who cannot reject REHTS over the sample period of 1910–
14. Cook and Hahn (1990) and Goodfriend (1991) argue persuasively that the
Federal Reserve’s use of a funds rate instrument, and, in particular, the way in
which that instrument is employed, is partly responsible for the apparent failure
of REHTS.
Recently Rudebusch (1994), in a study very much in the spirit of ours, provides some empirical support for the Cook and Hahn (1990) and Goodfriend
(1991) hypothesis. Further, McCallum (1994) shows the theoretical linkage
between the Fed’s policy rule and the regression estimates in various tests of
REHTS when the Fed responds to the behavior of longer-term interest rates.
While Fed behavior represents a potentially important component for explaining the empirical results of tests of REHTS, any explanation of these
results that also maintains rational expectations must include time-varying term
premia. Without time-varying term premia, tests of REHTS will not be rejected.
This fact is pointed out in Mankiw and Miron (1986) and Campbell and Shiller
(1991). Further, Campbell and Shiller indicate that white-noise term premia are
insufﬁcient to reconcile theory with data. We ﬁnd this to be the case as well.
Thus, we examine a more elaborate model of term premia coupled with Fed
behavior in an attempt to explain some of the empirical results on REHTS.
Before developing a theory of Fed behavior and linking it to empirical
work on REHTS, we present, in Section 1, a brief overview of the rational
expectations hypothesis of the term structure. Then, in Section 2, we construct
a model of Fed behavior that emodies the key elements described in Goodfriend
(1991). We use the resulting model along with REHTS to generate returns on
bonds of maturities ranging from one to six months. In Section 3 we focus,
in essence, on the empirical regularities documented by Roberds, Runkle, and
Whiteman (1993). We show that Fed behavior is not enough to reproduce their
ﬁndings. Then, in Section 4, we turn our attention to incorporating a more
realistic behavior of term premia. Combining these term premia with rational
investor and Fed behavior generates data that is roughly consistent with the
Roberds, Runkle, and Whiteman results. Section 5 concludes.

1.

THE RATIONAL EXPECTATIONS THEORY OF
THE TERM STRUCTURE

Tests and descriptions of the rational expectations theory of the term structure
constitute a voluminous literature. An excellent survey can be found in Cook
and Hahn (1990), and an exhaustive treatment is contained in Campbell and
Shiller (1991). The basic idea is that with the exception of a term premium,
there should be no expected difference in the returns from holding a long-term
bond or rolling over a sequence of short-term bonds. As a result, the longterm interest rate should be an average of future expected short-term interest

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

67

rates plus a term premium. Speciﬁcally, the interest rate on a long-term bond
of maturity n, rt (n), will obey
rt (n) =

1
k

k−1

Et rt+mi (m) + φt (n, m),

(1)

i=0

where rt+mi (m) is the m period bond rate at date t + mi, Et is the conditonal
expectations operator over time t information, and φt (n, m) is the term premia
between the n and m period bonds.2 In equation (1), k = n/m and is restricted
to be an integer.
The rational expectations hypothesis implies that rt+mi (m) = Et rt+mi (m) +
et+mi (m), where et+mi (m) has mean zero and is uncorrelated with time t information. Using this implication, one can rearrange equation (1) to yield the
following relationship:
k−1

1
k

[rt+mi (m) − rt (m)] = α + rt (n) − rt (m) + vt (n, m),

(2)

i=1

where vt (n, m) =

1
k

k−1

et+mi (m) − [φt (n, m) − α] and α is the non-time-varying

i=1

part of the term premium. Thus, future interest rate differentials on the shorterterm bond are related to the current interest rate spread between the long- and
short-term bond.
Equation (2) forms the basis of the tests of the term structure that we focus
on in this article. This involves running the regression
1
k

k−1

[rt+mi (m) − rt (m)] = α + β[rt (n) − rt (m)] + vt (n, m)

(3)

i=1

and testing if β = 1. We shall focus our attention on n = 2, 3, 4, 5, and 6
months and m = 1 and 3 months. For m = 3 and n = 6 (implying k = 2), the
appropriate regression would be
1

/2 [rt+3 (3) − rt (3)] = α + β[rt (6) − rt (3)] + vt (6, 3).

(3 )

That is, the change in the three-month interest rate three months from now
should be reﬂected in the difference between the current six-month and threemonth rates because the pricing of the six-month bill should reﬂect any expected
future changes in the rate paid on the three-month bill.
In the absence of time-varying term premia, the coefﬁcient β should equal
one. In practice, however, that has not been the case. For example, Table 1
2 Term

premia arise naturally in consumption-based asset pricing models and involve the
covariance of terms containing the ratio of future price-deﬂated expected marginal utilities of
consumption to the current price-deﬂated marginal utility of consumption, the price of the longterm bond, and future prices of the short-term bond. See Labadie (1994).

68

Federal Reserve Bank of Richmond Economic Quarterly

reports some estimates obtained by Roberds, Runkle, and Whiteman (1993)
and Campbell and Shiller (1991). Not only is β < 1, but the degree to which
β deviates from one increases as k increases. Also, the coefﬁcient in the regression when n = 6 and m = 3 is of the wrong sign and insigniﬁcantly different
from zero.
This latter result is in stark contrast to estimates obtained by Mankiw and
Miron (1986) and Choi and Wohar (1991), who ﬁnd that prior to the advent
of the Fed, the theory fared much better. These two sets of results, which
primarily involve r(6) − r(3), imply a number of possibilities among which
are the following: (1) REHTS once held but no longer does (perhaps because
investors have become irrational), (2) the nature of term premia has changed, or
(3) Federal Reserve policy has in some way affected the nature of the empirical
tests.
In analyzing these possibilities, we ﬁrst note that the term premia must be
ˆ
time-varying for plim β = 1 (i.e., the predicted value of β to be something
ˆ
other than one). To show this, we report the probability limit of β in (3 ), which
is adopted from the derivation in Mankiw and Miron (1986):
ˆ
plim β =

σ 2 [Et ∆rt+1 (3)] + 2ρσ[Et ∆rt+1 (3)]σ[φt (6, 3)]
, (4)
σ 2 [Et ∆rt+1 (3)] + 4σ 2 [φt (6, 3)] + 4ρσ[Et ∆rt+1 (3)]σ[φt (6, 3)]

where σ 2 [Et ∆rt+1 (3)] is the variance of the expected change in the threemonth interest rate, ρ is the correlation between Et ∆rt+1 (3) and φt (6, 3), and
σ 2 [φt (6, 3)] is the variance of the term premium.3
Expression (4) is informative for our purposes. Notice that for nonstochasˆ
tic term premia, plim β = 1. Hence stochastic term premia are required for
ˆ
ˆ
plim β = 1. Also observe that as σ 2 [φt (6, 3)] increases, plim β decreases. Furˆ
ther, note that plim β is a complicated function of σ 2 [Et ∆rt+1 (3)], but as this
ˆ
ˆ
term gets fairly large, plim β goes to one. More generally, β’s deviation from
a value of one will depend on the ratio of the variance of the term premium to
the variance of the expected change in interest rates.
It is this latter variance that Fed behavior may inﬂuence. In this regard,
Mankiw and Miron (1986) document the variation over time in this variable
and show that σ 2 [Et ∆rt+1 (3)] was much larger prior to the creation of the
Federal Reserve System. Mankiw and Miron attribute this ﬁnding to the Fed’s
concern for interest rate smoothing.
As Cook and Hahn (1990) point out, however, rate smoothing cannot be the
total story since the regression coefﬁcient on [rt (2) − rt (1)] is highly signiﬁcant
and close to one; moreover, for longer-term bonds the term structure does
help predict future changes in interest rates. Regarding the short end of the
yield curve, Cook and Hahn postulate that one must consider the discontinuous

3 Rudebusch

(1993) derives a similar expression with ρ = 0.

Table 1 Coefﬁcient Estimates from Literature

Source

Short (m)
Long (n)

1 period
2 period

1 period
3 period

1 period
4 period

1 period
6 period

2 period
4 period

3 period
6 period

Campbell & Shiller
Table 2, T-bills,
1952–87

coefﬁcient
standard error

0.5010
0.1190

0.4460
0.1990

0.4360
0.2380

0.2370
0.1670

0.1950
0.2810

−0.1470
0.2000

Roberds, Runkle &
Whiteman, Table 6
F Fund, 1984–91

coefﬁcient
standard error

0.5925
0.0983

0.3935
0.1437

na
na

0.2121
0.2822

na
na

−0.1411
0.6079

Roberds, Runkle &
Whiteman, Table 9
F Fund, 1984–91, SW*

coefﬁcient
standard error

0.7596
0.1359

0.2953
0.1399

na
na

0.1557
0.1861

na
na

−0.2971
0.3675

Roberds, Runkle &
Whiteman, Table 11
F Fund, 1984–91, FOMC†

coefﬁcient
standard error

0.7119
0.1720

0.4104
0.1688

na
na

0.0869
0.1878

na
na

−0.3149
0.4553

* Settlement Wednesday.
† FOMC meeting date.
Note: Roberds, Runkle, and Whiteman use daily data in their regressions.

70

Federal Reserve Bank of Richmond Economic Quarterly

and infrequent changes in policy. Therefore, economic information that will
affect future policy is often known prior to actual policy reactions. This factor
implies that movements in the short end of the term structure will anticipate
policy and hence have predictive content. In terms of equation (4), the variance
of ∆Et rt+1 (1) is likely to be greater than the variance of ∆Et rt+1 (3).
Additional arguments supporting the relevance of monetary policy for tests
of REHTS can be found in Goodfriend (1991) and McCallum (1994). McCallum shows that if the Fed reacts to movements in the term structure, then the
strength of that reaction will inﬂuence estimates of β in tests of REHTS.
Taken together, these papers indicate that capturing Fed behavior is potentially important for understanding the term structure. We now attempt such an
exercise.

2.

A MODEL OF FED BEHAVIOR

Our model of Fed behavior is designed to capture the basic characteristics described by Goodfriend’s (1991) analysis of Federal Reserve policy. In particular,
we model the Federal Reserve’s adjustment of its funds rate target as occurring
at intervals and only in relatively small steps. Also, funds rate changes are often
followed by changes in the same direction so that the Fed does not “whipsaw”
ﬁnancial markets. While the Fed is generally viewed as adjusting the funds
rate to achieve various economic goals, for our purposes it is sufﬁcient to let
the Fed’s best guess of an unconstrained optimal interest rate target follow an
exogenous process. For simplicity, let
∗
∆rt∗ = ρ∆rt−1 + ut ,

(5)

where rt∗ is the unconstrained optimal interest rate. That is, it is the interest
rate the Fed would choose before the arrival of new information if it were not
constrained to move the funds rate discretely. One could think of rt∗ as arising
from a reaction function, but equation (5), along with additional behavioral
constraints, is sufﬁcient for the purpose of our investigation. To capture Fed
behavior, we model changes in the funds rate according to the following criteria:
r ft = r ft−1 + 1 /2 if rt∗ − r ft−1 ≥

1

/2 ,

r ft = r ft−1 + 1 /4 if 1 /4 ≤ rt∗ − r ft−1 <
r ft = r ft−1 if − 1 /4 < rt∗ − r ft−1 <

1

1

/2 ,

/4 ,

(6)

r ft = r ft−1 − 1 /4 if − 1 /2 < rt∗ − r ft−1 ≤ − 1 /4 ,
r ft = r ft−1 − 1 /2 if − 1 /2 > rt∗ − r ft−1 .
The behavior described by equation (6) implies that at each decision point
the Fed is guided by its overall macroeconomic goals as depicted by the

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

71

behavior of rt∗ . It adjusts its instrument r ft incrementally and discretely. Thus,
for a big positive shock to rt∗ , the Fed would be expected to raise the funds
rate at a number of decision points until r ft approximated rt∗ . There would also
be only a small probability that the Fed would ever reverse itself (i.e., raise the
funds rate one period and lower it the next).
We parameterize the variance of ut and the parameter ρ so that the behavior
of the funds rate target, r ft , is consistent with actual behavior over the period
1985:1 to 1993:12. The parameter ρ is set at 0.15, and ut has a variance of
0.09. In particular, ut is distributed uniformly on the interval [−0.525, 0.525].
A uniform distribution is used to facilitate the pricing of multiperiod bonds in
the next section.
The behavior generated by a typical draw from our stochastic process
and by the actual funds rate target are reasonably similar. These are depicted
in Figures 1 and 2. Table 2 provides some additional methods of comparison.
Figure 1 Federal Funds Rate Target

12

Percent

10

8

6

4

2 J
an Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul Jan Jul
85 85
86 86
87 87
88 88
89 89
90 90
91 91
92 92
93 93

+

The low p-values of Fisher’s exact test indicate that both actual and model data
are consistent with targeted interest rate changes not being independent of the
sign of previous changes.4 However, a somewhat smaller percentage of interest
rate changes are of the same sign in the model. The Fed, as modeled here, is
4 Model

data are from 250 draws of a series with 300 observations.

72

Federal Reserve Bank of Richmond Economic Quarterly

Figure 2 Representative Draws of Funds Rate Target

14
12

10

Percent

Draw 1
8
6

4

Draw 2

2

0
1

+

25

49

73

97

121

145

169

193

217

241

265

289

Months

more likely to reverse itself than the Fed actually did over this period. Also,
the Fed of our model is more likely to leave the funds rate unchanged. Finally,
the correlation coefﬁcient between funds rate changes in the model is not signiﬁcantly different from the actual correlation coefﬁcient displayed by the data.
We thus feel that equations (5) and (6) jointly represent a reasonable and
tractable model of Federal Reserve behavior, especially if rt∗ is thought of as
depending upon underlying economic behavior.

3.

MONETARY POLICY AND THE TERM STRUCTURE

As described by equations (5) and (6), the Federal Reserve determines the behavior of the one-period nominal interest rate. The FOMC meets formally eight
times per year and informally via conference calls. Also, the chairman may act
between FOMC meetings so that in actuality the term of the one-period rate is
less than one month. Further, the timing between decision periods is stochastic
and can be as little as one week or as long as an intermeeting period.5 For
simplicity, we model the decision period as monthly. Thus, the pricing of a
5 For

a more detailed modeling of behavior along these lines, see Rudebusch (1994).

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

73

Table 2 Actual and Model Data Comparison
Actual

Model

Percent of Changes of Same Sign

0.833

0.648

Fisher’s Exact Text
p-value (standard error)

2.6E-06

0.003600
(0.011)

Corr(∆r t , ∆r t−1 )
(standard error)

0.2661

0.3242
0.04948

Std(Et rt+1 − rt )
(standard error)

0.2409

0.1234
(0.0034)

Std[Et rt+1 (2) − rt ]
(standard error)

0.2562

0.1457
(0.0039)

Std[Et rt+1 (3) − rt ]
(standard error)

0.2687

0.1565
(0.0040)

Std[Et rt+3 (3) − rt (3)]∗

0.1858

0.1306
(0.0052)

f

f

* Goldsmith-Nagan yields 0.2011.
Notes: Model data are from 250 draws of a series with 300 observations. The null of Fisher’s
exact test is that the sign of the change in the funds rate is independent of the sign of the previous
change.

two-month bond or, more accurately, a two-month federal funds contract will
obey
∗
2rt (2) = r ft + 1 /2 (Prob[rt+1 − r ft ≥ 1 /2 ])
∗
+ 1 /4 (Prob[ 1 /4 ≤ rt+1 − r ft <

1

/2 ])

∗
− 1 /4 (Prob[− 1 /2 < rt+1 − r ft ≤ − 1 /4 ])
∗
− 1 /2 (Prob[rt+1 − r ft ≤ − 1 /2 ]) + 2φt (2, 1).

(7)

In calculating the expectation of interest rates further than one period ahead,
say, for example, two periods ahead, one needs to form time t expectations of
∗
terms such as Prob[rt+2 − r ft+1 > 1 /2 ]. Assuming that ut is uniformly distrib∗
uted, the expressions we obtain for the various probabilities are linear in rt+j
f
and r t+j−1 . Thus, one can pass the expectations operator through the respective
cumulative distribution functions.
For pricing three-, four-, ﬁve-, and six-month term federal funds, we use
expressions analogous to equation (7). In order to examine the effect that our
model of monetary policy has on tests of the rational expectations hypothesis of
the term structure, we generate 250 simulations, each containing 300 values of
each rate. The results are presented in Table 3, where standard errors have been
corrected using the Newey-West (1987) procedure. We report results when

74

Federal Reserve Bank of Richmond Economic Quarterly

Table 3 Coefﬁcient Estimates Using Model-Generated Data
Independent
Variable

r(2) − r r(3) − r r(4) − r r(5) − r r(6) − r r(4) − r(2) r(6) − r(3)
(a) σ(φ) = 0

Coefﬁcient
Standard Error

1.03
(.005)

1.00
(.118)

1.00
(.109)

.99
(.158)

.99
(.179)

.97
(.22)

.95
(.42)

.64
(.138)

.35
(.083)

.31
(.114)

(b) σ(φ) = .10
Coefﬁcient
Standard Error

.28
(.055)

.49
(.084)

.57
(.105)

.62
(.122)

there is no term premium (row 1) and when there is a white-noise term premium
with standard deviation 0.10 (row 2).
The results indicate that in the absence of time-varying term premia, there
is no departure of estimates of β from one. This essentially serves as a check
on our calculations, since all interest rates are calculated using REHTS. With
a time-varying term premia, REHTS is rejected. However, the rejection of
the model’s data is not in keeping with the result on actual data. The estimates of β are increasing in k = n/m rather than decreasing. Also, the
results for k = 2 and m = one month, two months, and three months, respectively, are almost identical for the model, while they are strikingly different
for the data. Looking at Table 2 and equation (4) shows why. Table 2 indicates that σ 2 [Et ∆rt+1 (m)] is approximately the same for m = 1 and m = 3.
(When m = 2, its value is 0.151.) With σ 2 [φ(n, m)] equivalent by construction,
the estimate of β will not vary much across experiments. For a model with
white-noise term premia to replicate actual empirical results, it must generate
σ 2 (Et ∆rt+1 ) > σ 2 [Et ∆rt+2 (2)] > σ 2 [Et ∆rt+3 (3)], which does not happen in
our particular model.
Interestingly enough, as shown in Table 2, the required behavior of
σ 2 [Et rt+1 (m) − rt (1)] does not occur in the data either. We are therefore forced
to conclude that our description of monetary policy, along with white-noise
term premia, is insufﬁcient to explain the empirical results in Roberds, Runkle,
and Whiteman (1993) as well as in Campbell and Shiller (1991). Our failure
could be due primarily to an insufﬁcient model of policy or to an inadequate
model of term premia. In the next section we modify our model of term premia
and reexamine REHTS on data generated by our modiﬁed model.

4.

A DESCRIPTION OF TERM PREMIA

To generate term premia that potentially resemble the stochastic processes of
actual term premia, we need some way of estimating term premia. For this we

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

75

turn to the multivariate ARCH-M methodology described in Bollerslev (1990).
We use a multivariate model since the term premia generated from a univariate
model are highly correlated. In essence, we estimate a multivariate ARCHM model of excess holding period yields then use the estimated process to
simulate time-varying term premia. The simulated processes, along with the
model in Section 2, are used to generate data on interest rates. This simulated
data is then used to estimate regressions like (3 ).
In estimating term premia (for the case in which k = 2), ﬁrst deﬁne the
excess holding period yield, yt (n, m), as
2rt (n) − rt+m (m) − rt (m).
From equation (1) we see that this is merely Et rt+m (m) − rt+m (m) + 2φt (n, m),
which is the sum of an expectational error and twice the actual term premium
as deﬁned in (1).
The multivariate ARCH-M speciﬁcation that we estimate over the sample
period 1983:1–1993:12 is given by
yt = β + δ log ht + εt ,

(8)

where εt conditioned on past information is a normal random vector with
variance-covariance matrix Ht . The elements of Ht are given by
12

h2 = γj + αj
jj,t

wi ε 2
j,t−i
i=1

hij,t = ρij hii,t hjj,t ,

(9)

where yt is a 3 by 1 vector of the ex-post excess holding period yields on
Treasury bills that includes the two-month versus one-month bill, the threemonth versus one-month bill, and the six-month versus three-month bill.6 The
wi are ﬁxed weights given by (13 − i)/78. In this speciﬁcation of the model, the
covariances hij are allowed to vary but the correlation coefﬁcients, ρij , between
the errors are constant. The coefﬁcient estimates are reported in Table 4. Almost
all the coefﬁcients are highly signiﬁcant.
The term premia derived from this model are depicted in Figure 3 and are
labeled T2, T3, and T6. Recall that T2 and T6 are twice φ(2, 1) and φ(6, 3),
respectively, while T3 is three times φ(3, 1). One notices the term premia spike
upward in 1984, in late 1987, and in early 1991. The term premium on twomonth bonds also spikes in late 1988 and early 1989. The 1987 episode is
associated with the October stock market crash. Interestingly, the 1984 and
6 We

use T-bills rather than term federal funds because coefﬁcient estimates using the federal
funds rate are insigniﬁcant. One possible explanation for this result is that the excess holding
period yield on federal funds involves both a term premia derived from a consumption-based
asset pricing model as well as default risk that may be uncorrelated with the term premia. This
default risk may add sufﬁcient noise that it is difﬁcult to estimate the term premia using ARCH-M
type regressions.

76

Federal Reserve Bank of Richmond Economic Quarterly

Table 4 Coefﬁcient Estimates for ARCH-M Model
Log likelihood = 149.72
Coefﬁcient

Estimate

Standard Error

Signiﬁcance Level

β2
δ2
γ2
α2
β3
δ3
γ3
α3
β6
δ6
γ6
α6
ρ23
ρ26
ρ36

.98
.46
.058
.92
1.46
1.25
.52
.44
.94
.79
.25
.23
.92
.48
.67

.13
.13
.018
.20
.13
.47
.089
.13
.45
.82
.08
.17
.016
.082
.060

.0000
.0005
.0009
.0000
.0000
.0076
.0000
.0010
.0389
.331
.0011
.0601
.0000
.0000
.0000

1988–89 episodes correspond to the inﬂation-scare episodes documented in
Goodfriend (1993). The last spike in the term premia occurs around the time
of the Gulf War and a recession.
Statistical data for the in-sample residuals, the estimated term premia, and
the ex-post holding period yields are depicted in Table 5. In attempting to
ascertain the joint importance of Fed behavior and time-varying term premia
in explaining the regression results of Campbell and Shiller (1991) as well as
Roberds, Runkle, and Whiteman (1993), we perform three experiments. First
we generate a funds rate that is stationary and thus does not display the interest
rate smoothing or discrete interest rate changes that are embodied in our model
of Fed behavior. Longer-term interest rates are then derived using equation (1)
and the rational expectations hypothesis. We do this to see if our model of term
premia by itself can account for the actual regression results. Next we examine
an interest rate process that includes a greater degree of smoothing but does
not require discrete changes in the funds rate. Finally, we combine our model
of term premia with our depiction of Fed behavior and investigate whether this
model of interest rate determination can explain the regression results obtained
using actual data. The results we analyze involve the cases in which n = 2,
m = 1, and n = 6, m = 3 (i.e., the term spread between the two-month and
one-month bills and the six-month and three-month bills).
∗
To begin, we model the one-period interest rate as rt∗ = 0.75rt−1 + ut .
As in our actual model of Fed behavior, ut is distributed uniformly on the
interval [−0.525, 0.525]. Combining this behavior with term premia generated
from our estimated ARCH-M model, we generate data on longer-term interest

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

77

Figure 3 Interest Rate Term Premia

T2 and T3

2.5

2.0

Percent

T3
1.5

1.0

T2

0.5

0.0
Jan
Jan 1984
1984

85

86

87

88

89

90

91

92

93

91

92

93

T2 and T6

1.2

Percent

1.0

0.8

T2
0.6

T6

0.4

+

0.2
Jan 1984
Jan
1984

85

86

87

88

89

90

Notes: T2 is the term premium between the two-month and one-month bonds. T3 is the term
premium between the three-month and one-month bonds. T6 is the term premium between the
six-month and three-month bonds.

78

Federal Reserve Bank of Richmond Economic Quarterly

Table 5 Statistical Data from ARCH-M Model
Residuals

Standard Error

ε2
ε3
ε6

.45
.99
.64

Estimated Term Premia

Mean

Standard Error

T2
T3
T6

.60
1.35
.53

.15
.25
.13

Actual Ex-post Yields

Mean

Standard Error

y2
y3
y6

.65
1.51
.62

.47
1.04
.68

Correlation Matrix
1.0
.92
.50

1.0
.70

1.0

Correlation Matrix
1.0
.90
.77

1.0
.85

1.0

Correlation Matrix
1.0
.93
.55

1.0
.74

1.0

rates using equation (1). The regression results based on 500 simulations of
125 observations are
rt+1 (1) − rt (1) = a0 + 0.43 [rt (2) − rt (1)] + ε1t ,
(0.16)
rt+3 (3) − rt (3) = α0 + 1.00 [rt (6) − rt (3)] + ε3t ,
(0.18)
where standard errors are in parentheses. REHTS is not rejected by the second
regression, and the results of this regression are consistent with those documented in Mankiw and Miron (1986) and Choi and Wohar (1991) for the
period prior to the founding of the Fed. One must therefore conclude that our
model of term premia is not sufﬁcient for generating data that are capable of
replicating regression results using actual post-Fed data.
∗
Next we model the short-term interest rates as ∆rt∗ = 0.15∆rt−1 +ut , which
is consistent with our modeling of rt∗ in equation (5). Thus, the only element
lacking from our complete model of Fed behavior is the discrete nature of funds
rate behavior given by equation (6). Generating data using this nonstationary
model of rt∗ , along with our model of term premia, we obtain the following
regression results:
rt+1 (1) − rt (1) = b0 + 0.10 [rt (2) − rt (1)] + e1t ,
(0.19)
rt+3 (3) − rt (3) = β0 + 0.12 [rt (6) − rt (3)] + e3t .
(1.24)

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

79

Here both coefﬁcients are insigniﬁcantly different from zero. Thus, this experiment does not generate the statistically signiﬁcant coefﬁcient commonly found
when using actual data on two- and one-month interest rates.
Finally, we combine the joint modeling of term premia using the ARCHM process and Fed behavior given by equations (5) and (6). These regression
results are the following:
rt+1 (1) − rt (1) = c0 + 0.46 [rt (2) − rt (1)] + w1t ,
(0.10)
rt+1 (3) − rt (3) = γ0 + 0.64 [rt (6) − rt (3)] + w3t .
(0.59)
Here the joint modeling of term premia and Fed behavior is capable of
explaining a statistically signiﬁcant coefﬁcient that is less than one in the
shorter-maturity regression, whereas the coefﬁcient in the regression involving longer maturities is insigniﬁcantly different from zero. An explanation for
the increased signiﬁcance of the coefﬁcient in the ﬁrst regression from that
estimated in the previous experiment goes as follows. Due to Fed behavior, the
standard deviation of the expected change in the one-month rate has risen from
a value of 0.095 to 0.123, while the standard deviation of the term premia
has remained unchanged. However, there are only 35 episodes in which the
coefﬁcient in the ﬁrst regression is greater than 0.5 while the coefﬁcient in the
second regression is also less than zero. Thus, the coefﬁcient estimates that are
consistent with the results presented in Roberds, Runkle, and Whiteman (1993)
occur in approximately 7 percent of the trials.
The results presented above are not entirely satisfactory because the generated term premia do not exactly match the ﬁtted term premia of the model
(perhaps because the correlation coefﬁcients are constrained to be time invariant). The standard deviations of the generated term premia are somewhat
less than those depicted in Table 5, whereas the correlation coefﬁcients are
appreciably less. With generated data, σT2 = 0.17, σT3 = 0.14, and σT6 = 0.06
while ρ23 = 0.66, ρ26 = 0.20, and ρ36 = 0.41.
To remedy this situation, we generate data by also allowing the correlation
coefﬁcients, ρij , to vary intertemporally. We do this by allowing them to depend
on the hjj,t s in equation (9), producing standard deviations of σT2 = 0.14,
σT3 = 0.44, and σT6 = 0.12 and correlation coefﬁcients of ρ23 = 0.88,
ρ26 = 0.73, and ρ36 = 0.83. In a regression using data generated by this
mechanism, the coefﬁcient on the 2,1 term is 0.93(0.16) and on the 6,3 term
is 0.79(0.63), where standard errors are in parentheses. Also, in 10 percent
of the cases the 6,3 coefﬁcient is less than zero, while the 2,1 coefﬁcient is
greater than 0.5. When there is no discretization of movements in the funds
rate, these coefﬁcients are 0.54(0.47) and 0.11(1.95). Both coefﬁcients differ
insigniﬁcantly from zero.

80

Federal Reserve Bank of Richmond Economic Quarterly

While the term premia in the last simulation do not come from any estimated model, the experiment at least shows that regression results that are in
accord with those obtained in practice can be generated by the combination
of (1) Fed behavior that both smooths the movements in interest rates and
only moves interest rates discretely and (2) time-varying term premia that are
calibrated to match data moments.

5.

CONCLUSION

This article explores the linkage between Federal Reserve behavior and timevarying term premia and analyzes what effect these two economic phenomena
have on tests of the rational expectations hypothesis of the term structure.
Adding both these elements to a model of interest rate formation produces
simulated regression results that are reasonably close to those reported using
actual data. We thus feel that a deeper understanding of interest rate behavior
will be produced by jointly taking into account the behavior of the monetary
authority along with a more detailed understanding of what determines term
premia. Reconciling theory with empirical results probably does not require
abandonment of the rational expectations paradigm.

REFERENCES
Bollerslev, Tim. “Modelling the Coherence in Short-Run Nominal Exchange
Rates: A Multivariate Generalized ARCH Model,” The Review of Economics and Statistics, vol. 72 (August 1990), pp. 498–505.
Campbell, John Y., and Robert J. Shiller. “Yield Spreads and Interest Rate
Movements: A Bird’s Eye View,” Review of Economic Studies, vol. 58
(May 1991), pp. 495–514.
Choi, Seungmook, and Mark E. Wohar. “New Evidence Concerning the
Expectation Theory for the Short End of the Maturity Spectrum,” The
Journal of Financial Research, vol. 14 (Spring 1991), pp. 83–92.
Cook, Timothy, and Thomas Hahn. “Interest Rate Expectations and the Slope
of the Money Market Yield Curve,” Federal Reserve Bank of Richmond
Economic Review, vol. 76 (September/October 1990), pp. 3–26.
Engle, Robert F., David M. Lilien, and Russell P. Robins. “Estimating Time
Varying Risk Premia in the Term Structure: The ARCH-M Model,”
Econometrica, vol. 55 (March 1987), pp. 391–407.
Fama, Eugene F. “The Information in the Term Structure,” Journal of Financial
Economics, vol. 13 (December 1984a), pp. 509–28.

M. Dotsey and C. Otrok: Rational Expectations Hypothesis

81

. “Term Premiums in Bond Returns,” Journal of Financial
Economics, vol. 13 (December 1984b), pp. 529–46.
Goodfriend, Marvin. “Interest Rates and the Conduct of Monetary Policy,”
in Carnegie-Rochester Conference Series on Public Policy, Vol. 34.
Amsterdam: North Holland, 1991, pp. 7–30.
. “Interest Rate Policy and the Inﬂation Scare Problem: 1979–
1992,” Federal Reserve Bank of Richmond Economic Quarterly, vol. 79
(Winter 1993), pp. 1–24.
Labadie, Pamela. “The Term Structure of Interest Rates over the Business
Cycle,” Journal of Economic Dynamics and Control, vol. 18 (May/July
1994), pp. 671–97.
Mankiw, Gregory N., and Jeffrey A. Miron. “The Changing Behavior of the
Term Structure of Interest Rates,” Quarterly Journal of Economics, vol.
101 (May 1986), pp. 211–28.
McCallum, Bennett T. “Monetary Policy and the Term Structure of Interest
Rates.” Manuscript. Carnegie-Mellon University, June 1994.
Mishkin, Frederick S. “The Information in the Term Structure: Some Further
Results,” Journal of Applied Econometrics, vol. 3 (October–December
1988), pp. 307–14.
Newey, Whitney K., and Kenneth D. West. “A Simple Positive Semi-Deﬁnite
Heteroskedasticity and Autocorrelation Consistent Covariance Matrix,”
Econometrica, vol. 55 (May 1987), pp. 703–8.
Roberds, William, David Runkle, and Charles H. Whiteman. “Another Hole in
the Ozone Layer: Changes in FOMC Operating Procedure and the Term
Structure,” in Marvin Goodfriend and David H. Small, eds., Operating
Procedures and the Conduct of Monetary Policy: Conference Proceedings.
Washington: Board of Governors of the Federal Reserve System, 1993.
Rudebusch, Glenn D. “Federal Reserve Interest Rate Targeting and the Term
Structure.” Manuscript. Board of Governors of the Federal Reserve
System, May 1994.
. “Propagation of Federal Funds Rate Changes to Longer-Term
Rates Under Alternative Policy Regimes,” in Marvin Goodfriend and
David H. Small, eds., Operating Procedures and the Conduct of Monetary
Policy: Conference Proceedings. Washington: Board of Governors of the
Federal Reserve System, 1993.
Full text of Economic Quarterly (Federal Reserve Bank of Richmond) : Winter 1995, Vol. 81, No. 1

FRASER
