Full text of DOI:10.1016/S0304-3932(99)00040-9

Journal of Monetary Economics 45 (2000) 3}35
In#ation and the great ratios:
Long term evidence from the U.S.ଝ
Shaghil Ahmed*, John H. Rogers
Board of Governors of the Federal Reserve System, Division of International Finance, Washington,
DC 20551, USA
Received 6 October 1997; received in revised form 1 October 1998; accepted 26 January 1999
Abstract
Using over 100 years of U.S. data, we "nd that the long-run e!ects of in#ation on
consumption, investment, and output are positive. Also, great ratios like the consump-
tion and investment rates are not independent of in#ation, which we interpret in terms of
the Fisher e!ect. However, the variability of the stochastic in#ation trend is small relative
to the variability of the productivity and "scal trends. Thus, models generating long-term
negative e!ects of in#ation on output and consumption seem to be at odds with data
from the moderate in#ation rate environment we consider. ( 2000 Elsevier Science
B.V. All rights reserved.
JEL classixcation: E31 and E32
Keywords: In#ation; Investment; Great ratios; Tobin e!ect; Fisher e!ect
ଝThe opinions expressed in this paper are those of the authors and do not necessarily re#ect those
of the Board of Governors of the Federal Reserve System or other members of its sta!. Without
implicating them in any remaining errors, we would like to thank an anonymous referee, Allan
Brunner, Tom Cooley, Joe Gagnon, Dale Henderson, Doug Joines, Prakash Loungani, Jim Nason,
Eric Rasmusen, Chris Waller, as well as workshop participants at the Federal Reserve Board,
Indiana, UC Irvine, UC Santa Barbara, USC, and the Mid-West Macro Conference for very helpful
comments.
* Corresponding author: Tel.: 202-452-3209; fax: 202-452-6424.
E-mail addresses: shaghil.ahmed@frb.gov (S. Ahmed), john.h.rogers@frb.gov (J.H. Rogers)
0304-3932/00/$- see front matter ( 2000 Elsevier Science B.V. All rights reserved.
PII: S 0 3 0 4 - 3 9 3 2 ( 9 9 ) 0 0 0 4 0 - 9

4
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
1. Introduction
Consider a situation in which, with the economy in a low-in#ation steady
state, the rate of in#ation falls permanently, say by 2 percentage points. What
would be the long-run e!ects on real economic variables such as output,
consumption, the real interest rate, investment, and the capital stock? Would the
long-run path of the so-called &great ratios', such as the investment rate and the
consumption-output ratio get altered? Economic theory provides no clear-cut
prediction. On the one hand, there is the famous superneutrality result due to
Sidrauski (1967). Yet, Sidrauski's result emerges from a very speci"c theoretical
set-up, requiring in particular the strong assumption that consumption and
leisure are separable in utility. In several theoretical models, the superneutrality
result breaks down as in#ation can have either positive or negative e!ects on
real variables such as output and investment, depending on the exact assump-
tions concerning preferences and how money is introduced into the economy.
Additionally, in these models the real interest rate, and, therefore, implicitly the
great ratios, may or may not be independent of in#ation in the long run. (see
Orphanides and Solow (1990) for a survey).
Therefore, whether the long-run e!ects of in#ation on real economic aggreg-
ates are positive or negative, and whether the real interest rate is independent of
in#ation in the long run are empirical issues. Recently, there has been consider-
able interest in the existence and nature of a long-run trade-o! between in#ation
and unemployment (or output) (e.g. King and Watson, 1994, 1997; Akerlof
et al., 1996) as well as in the e!ects of in#ation on economic growth. Under-
standing these e!ects is crucial for evaluating monetary policy, among other
things, especially in light of the debate about moving from the current low
in#ation rate to price stability.
Existing empirical results are mixed. Cross-country growth regressions sug-
gest that the e!ects of in#ation on output or investment are negative. However,
these results may be driven largely by the presence of high in#ation countries
and, for reasons discussed by Levine and Renelt (1992), lack robustness. On the
other hand, King and Watson (1994, 1997) "nd that results on neutrality and
superneutrality are sensitive to the short-run identi"cation assumptions made,
although for the assumptions they consider plausible, departures from neutral-
ity tend to be insigni"cant. It should be noted that the King}Watson "ndings
are based on bivariate systems and do not use a multivariate structural model.
With these considerations in mind, we re-examine the empirical evidence on
the long-term interactions between in#ation and the real economy. Our goal is
to sort out which of several theoretical channels characterizing these interac-
tions are empirically more relevant. Using long-term U.S. data, we ask whether
a once-and-for-all permanent increase in in#ation leads to an upward or
a downward jump in the balanced-growth paths of output and investment. We
use the label &Tobin-e!ect' to denote an upward jump and &reverse-Tobin e!ect'

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
5
to indicate a downward jump. Although we do not test for the precise transmis-
sion channel that Tobin (1965) originally had in mind, we use these labels
because they are convenient and used often. Additionally, we present indirect
evidence on the &Fisher e!ect' (Fisher, 1930): the hypothesis that in#ation rate
has a one-to-one positive e!ect on the nominal interest rate and, consequently,
ꢀ
does not a!ect the real interest rate. Since we use U.S. data, our results should
not be used to infer what would happen in high in#ation environments.
Our empirical "ndings are organized in three parts. First, the univariate
properties of the data are described and cointegration (CI) vectors are estimated.
Hypothesis testing on these CI vectors reveals whether the data are consistent
with a long-run Fisher e!ect. Our test relies on the direct long-run correspond-
ence between the investment}output ratio and the real rate of interest; this
implies that, if the real interest rate is independent of in#ation, the latter should
have equal e!ects on investment and output over a long enough horizon. This is
testable using cointegration analysis. Our approach to examining the Fisher
e!ect is thus di!erent from previous studies in that we avoid explicitly modelling
inyationary expectations. However, our method does not shed light on the
short-to-medium run validity of the Fisher e!ect.
Second, under certain restrictions, we are able to identify and estimate
additional structural parameters, which allow us to compute the e!ects of
exogenous changes in the long-run component of in#ation on the levels of
consumption, investment, and output (as opposed to the e!ects on ratios to
GDP obtained from the cointegration analysis). We do this by estimating a fully
identi"ed structural vector error correction model (VECM), similar in spirit to
King et al. (1991), but with the major di!erence that we allow and test for
long-run nonneutralities.
Finally, we examine the robustness of our "ndings over di!erent sub-periods
of the data. This allows for the possibility that there may have been structural
breaks in the interactions between in#ation and real variables over the long
period covered in the full sample.
Our estimates indicate the presence of a Tobin-type e!ect and also indicate
that the Fisher e!ect does not hold in annual U.S. data from 1889 to 1995.
However, the full-sample variance decompositions also show that the stochastic
trend in in#ation is not particularly important for explaining real economic
#uctuations. We use these results to assess the potential usefulness of di!erent
theoretical models for understanding the long-run real e!ects of in#ation, and
compare our results to others in the literature.
The remainder of our paper is organized as follows: In Section 2, we set up
a general framework that nests the di!erent types of e!ects of in#ation on the
ꢀ As will become clear, the absence of a Tobin or reverse-Tobin e!ect and the Fisher e!ect holding
are not one and the same thing when leisure is endogenous.

6
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
real economy found in the theoretical literature. In this framework, the long-run
paths of the variables are driven by three stochastic trends: a productivity (or
output) trend, a "scal trend, and an in#ation trend. Section 3 links the theoret-
ical framework to our empirical estimation. In this Section, we also discuss the
identi"cation assumptions and present and interpret our empirical results.
Section 4 concludes.
2. Theoretical predictions on the long-term real e4ects of in6ation
Consider a simple, deterministic, optimization problem of an in"nitely lived,
ꢁ
integrated household-"rm unit. While this is a standard framework that does
not yield major new theoretical insights, it does help to serve two purposes.
First, the problem is set up in a general enough way that most of the theoretical
results on the real e!ects of in#ation emerge as special cases. Second, it gives us
an opportunity to introduce a "scal trend (which the typical theoretical litera-
ture on in#ation and growth does not have) in addition to the usual in#ation
and productivity trends. Incorporating our "scal variable is necessary in order
to adequately characterize the data in our empirical work.
The representative agent's optimization problem is
$
ꢃ
MRR B
R
max
R>ꢀ
! ꢂ) ꢂ+
+ b u C ,1!N ,
R
R
A
P
$
+
,
R
R>ꢀ
Rꢄꢅ
$
ꢃ
R
,
max
! ꢂ) ꢂ+
+ b lnC #ꢀ ln(1!N )#ꢀ ln
(1)
R
*
R
+
G
A
P
MRR BH
$
R>ꢀ
+
,
R
R>ꢀ
Rꢄꢅ
subject to the sequence of constraints (2) and (3) below:
$
M #Q
M
R
R
R>ꢀ
F
Z F(N , K )#
![K !(1!d)K ]!C !
"0
(2)
(3)
R
R
R
R>ꢀ
R
R
P
P
R
R
$
M #Q
R
R
!a C !a [K !(1!d)K ]50
A R
)
R>ꢀ
R
P
R
where b is a subjective discount rate, C is consumption, N the fraction of time
spent working, M (M) the desired (initial) holdings of money, Q the lump-sum
$
transfer of money from the government, P the price level, ꢀ , ꢀ '0 are
*
+
preference parameters, K the capital stock, d the depreciation rate (0(d(1), h,
ꢁ Sticky prices and/or imperfect information (e.g., Ball et al., 1988) models can also generate
nonneutralities and highly persistent real e!ects of in#ation, as can monetary business cycle models
of the sort analyzed by Cooley and Quadrini (1998). However, in such models, these e!ects do not
last forever and, strictly speaking, the long-run neutrality propositions apply.

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
7
a , a are parameters satisfying 0(h(1, 04a , a 41, F represents techno-
!
)
!
)
logy, changes in Z represent shifts to the production function, and where
investment " K !(1!d)K .
R>ꢀ
R
In Eq. (1), utility is log-linear. Eq. (2) is the representative agent's budget
constraint implying an equality of the sources of funds } which are private sector
output (the "rst term), initial real money holdings, and the real value of
lump-sum transfers from the government } to the uses of funds } which are
ꢆ
consumption, investment, and holdings of real balances. It is assumed that
agents internalize the government's budget constraint and treat the transfers of
money from the government as lump-sum, although monetary policy sets these
ꢇ
transfers proportional to existing money holdings. Eq. (3) is a cash-in-advance
(CIA) constraint, with a fraction a of consumption and a fraction a of
!
)
investment required to be "nanced by cash holdings. It seems rather awkward to
have money in the utility function as well as a CIA constraint. However, this is
for convenience only: in the special cases we consider below either money
provides utility (ꢀ '0) or the CIA constraint is relevant (a and/or a '0)
+
!
)
but never both.
The output available to private agents, >!G, is given by
F
F F ꢀ\F
(1!g )> "(1!g )exp+ꢀg ,A F(N , K ),Z N K
,
(4)
R
R
R
R
R
R
R
R
R
R
where g"G/> represents the size of the government with G being aggregate
government purchases of goods and services, A is the technology shift variable,
ꢀꢉF
F ꢀ\F
Z"[exp+ꢀg,(1!g)] A, and F(N, K)"N K . The above speci"cation of
production is standard Cobb}Douglas, except that we focus on private output
and also allow government size to directly a!ect private output, with no
ꢈ
particular stance taken on the sign of this e!ect (ꢀ4 or 50).
The equations describing steady-state paths for the above model are of the
standard form and are relegated to Appendix A for the sake of brevity. We
highlight here the main properties of three well-known models of the long-term
e!ects of in#ation on real variables that emerge as special cases of the above
framework:
ꢆ Writing (2) as an equality, rather than an inequality, recognizes that we have imposed standard
conditions on preferences and technology that lead to the option of free disposal never being
exercised.
ꢇ While we have abstracted from government debt, with Ricardian equivalence holding our
long-run properties would be robust to the introduction of such debt.
ꢈ Barro and Sala-i-Martin (1995), p.158) argue that it is appropriate to have private production
depend on government size (rather than the level of government purchases) if public goods use is
subject to congestion e!ects.

8
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Model 1: Sidrauski Model. This arises if money enters the utility function
(ꢀ '0), but there is no CIA constraint (a "0"a ). In this case constraint (3)
+
!
)
ꢊ
is not binding, and we get Sidrauski's well-known superneutrality result.
Model 2: CIA-for-consumption model. In this case, money provides no direct
utility (ꢀ "0), but cash is needed in advance to "nance consumption expendi-
+
tures (a "1, a "0). The model resembles that of Cooley and Hansen (1989):
!
)
in#ation acts as a tax on market activities and induces households to switch
from market to non-market activity (leisure). As a result, consumption and work
e!ort fall in response to a permanent rise in in#ation. However, in the long run
the real interest rate is still independent of in#ation. With constant returns to
scale, the real rate depends only on the productivity-adjusted-capital-labor ratio
(K/ZN), which is still constant in the steady state if the CIA constraint applies
only to consumption. The model also generates a reverse-Tobin e!ect: with
N falling and K/ZN constant, the productivity-adjusted capital stock (K/Z), and
hence productivity-adjusted investment (I/Z), must fall in the long run. The
intuition is that the fall in work e!ort decreases the marginal productivity of
capital. But, the long-run negative e!ects of in#ation on investment and output
are equal, leaving the ratio, and hence the real rate unchanged.
Model 3: CIA-for-consumption-and-investment model. As in model 2, money is
not allowed to enter the utility function (ꢀ "0), but the CIA constraint now
+
applies to both consumption and investment (a "1"a ). In this case, the
!
)
negative e!ects on consumption, investment, and work e!ort noted above still
apply. But, since in#ation now represents an additional cost to investment, K/Z
falls by more than >/Z in response to a rise in in#ation, and consequently the
real interest rate rises and the investment rate falls in the steady-state. Thus,
the Fisher e!ect does not apply, even in the long run. Stockman (1981) examines
the implications of a CIA constraint applying to investment. Abel (1985)
compares (abstracting from the labor/leisure choice) the dynamic accumulation
of capital in models in which the CIA constraint applies only to consumption
with those in which it applies to both consumption and investment.
Before turning to models in which in#ation has positive e!ects on capital
accumulation, two other points are worth emphasizing. First, as the above
results illustrate, when labor supply is endogenous, having the Fisher e!ect
holding and in#ation not a!ecting investment and output are not one and the
same thing. Second, there is also the more modern genre of endogenous growth
models with money, which we have not discussed. These models also generate
a reverse-Tobin e!ect of the type discussed above. But they display the impor-
tant additional feature that a once-and-for all rise in in#ation has a negative
ꢊ Alternatively, superneutrality arises within a cash-in-advance framework if the CIA constraint
applies only to consumption and labor supply is exogenous. However, in fairness to Sidrauski (1967),
which is the classic reference in this area, we have chosen to present the result in this way.

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
9
e!ect on the steady-state growth rate of the economy as well (see, for example,
Gomme, 1993).
Model IV: Tobin model. The "nal model we discuss generates a positive e!ect
of in#ation on the steady-state capital stock and investment, and is exempli"ed
by the portfolio adjustment model of Tobin (1965). The intuition behind the
original Tobin e!ect is that, with a "xed savings rate, higher in#ation increases
the opportunity cost of holding money, inducing savers to shift from holding
real balances to holding physical capital. This permanently lowers the marginal
product of capital and thus the real interest rate falls; the Fisher e!ect does not
hold, but the prediction goes in the opposite direction from the CIA-for-
consumption-and-investment model.
The Tobin e!ect, as originally formulated, was criticized on the grounds that
ꢋ
it assumed an exogenous savings rate. This criticism led to a literature that has
shown that a &Tobin-type e!ect' can arise even in optimizing models, although
not in the type of framework used above. For example, it can arise in two-period
OLG models, in in"nite horizon models with individual heterogeneity and
family disconnectedness due to uncertain lifetimes, and in models with con-
sumption and money entering utility in a nonseparable way with particular
assumptions about how the marginal utility of consumption is a!ected by
ꢌ
money. A positive relationship between in#ation and investment can also arise
if there are distortions in the tax system. Speci"cally, Bayoumi and Gagnon
(1996) argue that if it is nominal, rather than real, capital income, that is taxed,
as in Feldstein (1976), higher in#ation countries should have greater investment.
They "nd this to be consistent with the data. Ireland (1994) shows that a Tobin-
type e!ect can arise even in a representative-agent, in"nite-horizon model. This
happens in an environment where not only can sustained in#ation a!ect equilib-
rium steady-state paths of real variables, as in traditional money-growth
models, but capital accumulation in turn a!ects money's role in an evolving
payments system.
The widely di!erent predictions on the real e!ects of in#ation generated by
the four models discussed above are summarized in Table 1. This table encapsu-
lates just how varied is the theoretical literature on the link between in#ation
and real variables (including the so-called &great ratios' such as the investment
ꢍ
rate, consumption}output ratio, and the capital}labor ratio). The empirical
ꢋ Moreover, the mechanism in Tobin's original formulation cannot possibly lead to a very large
e!ect of in#ation on capital given plausible values of the interest elasticity of money demand and the
ratio of non-interest-bearing money to capital. We thank Joe Gagnon for pointing this out to us.
ꢌ See Orphanides and Solow (1990) for a survey. See also Wang and Yip (1992) for the role of
nonseparability in utility.
ꢍ There are also models in which in#ation has an ambiguous e!ect on the steady-state capital stock
(e.g. Brock, 1974 and Fischer, 1983).

10
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Table 1
Summary of theoretical results on the long-term real e!ects of in#ation!
Model
E!ect of a once-and-for-all change in n on the
steady-state value of
c"
k"
i"
y"
Super- Tobin Reverse Fisher
neutral e!ect Tobin e!ect
C/ZN K/ZN I/ZN Y/ZN
N
I/Z
Sidrauski
CIA (C)
CIA(C,I)
Tobin
0
0
!
?
0
0
!
#
0
0
!
#
0
0
!
#
0
Yes
No
No
No
No
No
No
Yes
No
Yes
Yes
No
Yes
Yes
No
No
!
!
0"
!
!
#
!#indicates positive e!ect, !indicates negative e!ect, 0 indicates no e!ect, and ? indicates
ambiguous e!ect.
"Models generating a Tobin e!ect typically focus on capital accumulation. In general, a negative
e!ect of in#ation on work e!ort could still be present in these models.
work that follows should be helpful in distinguishing between the di!erent types
of models.
3. The empirical framework and results
3.1. The data
We use annual U.S. data from 1889 to 1995. Output, consumption, invest-
ment, and government spending on goods and services are expressed in per
capita billions of 1987 dollars. In#ation is the rate of change of the GDP
de#ator. Total resident population is used to obtain per capita values. The
notation used in reporting the results is as follows: y, c, and i are the log-levels of
per capita real values of output, consumption, and investment, respectively; G/>
is the ratio of real government spending to output; and n is in#ation. Appendix
B gives sources of the data.
Table 2 reports summary statistics. Over the full sample period 1889}1995,
in#ation averages about 3 percent, while the shares in total GDP of government
spending, consumption, and investment are 19, 64, and 17 percent, respectively.
As is well-known (and can be seen from the data plots in Fig. 1), there have been
dramatic shifts in in#ation and the real variables during sub-periods of our
sample. For example, while the Great Depression era was a time of de#ation and
very low investment, the pre-WWI period was characterized by low in#ation (an
annual average rate of under 1 percent) and high investment (23 percent of
GDP). The post-WWII period was just the opposite from the pre-WWI period,
with a relatively high in#ation rate (greater than 4 percent) and a relatively low

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
11
Table 2
Summary statistics } means and standard deviations!
Full sample
(1889}1995)
Pre-WWI
(1889}1914)
Inter-war
(1918}41)
Post-war
(1950}95)
Post-19}
(1973}95)
G/>
p
19.4
(8.56)
3.05
11.6
(0.92)
0.81
18.3
(5.66)
0.36
21.7
(3.19)
4.30
19.0
(0.69)
5.56
(5.54)
63.8
(5.40)
16.6
(5.41)
3.13
(2.72)
64.1
(2.10)
22.9
(3.13)
3.49
(7.03)
67.8
(4.52)
13.7
(4.98)
2.92
(2.42)
63.3
(3.27)
15.8
(1.25)
3.08
(2.48)
66.1
(1.52)
16.1
(1.32)
2.39
C/>
I/>
*y
(5.92)
(5.82)
(7.29)
(2.40)
(2.15)
!G/>, C/>, and I/> denote, respectively, the ratios of real government purchases, real consump-
tion, and real investment to real GDP; n denotes the annual percentage change in the GDP de#ator,
and *y denotes real GDP growth. Reported above is the mean and standard deviation (in
parenthesis) of each series, in percent.
investment share (16 percent). Of course, these summary statistics are simple
correlations and should not be given any deep structural interpretation. Note
also from Table 2 that the share of consumption in total output has varied much
less than that of investment over the di!erent sub-periods, although the WWII
years 1942}45 were marked by a large negative spike in the consump-
tion}output ratio (see Fig. 1).
3.2. Univariate properties
The use of per capita data for output, consumption, and investment amounts
to de#ating the aggregate quantities of these variables by the deterministic
component of the trend in work e!ort driven by population growth. The
representative agent theoretical framework laid out earlier presumes that the
variables y, c, i, G/> and p have stochastic trends embedded in them (i.e. have
unit roots). We conduct three types of univariate analysis to evaluate this:
looking at plots of the data, examining the autocorrelations, and conducting
two formal tests for unit roots.
Examining the plots in Fig. 1, the deterministically detrended logs of per
capita values of output, consumption, and investment appear to be nonstation-
ary. The "gure also plots government size, in#ation, and the great ratios (c!y)
and (i!y). The question of the nonstationarity of government size and in#ation
is not so clear-cut from the plots of the data. Plots of the "rst di!erences (not
reported) give a strong indication that the di!erences are stationary.

12
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Fig. 1. Data in levels.
The autocorrelations of the variables are plotted in Fig. 2. The autocorrela-
tions of the detrended per capita levels of consumption, investment, output, and
government size do not die away quickly, indicating nonstationarity. The
in#ation autocorrelations decay at a rate quicker than the autocorrelations of
the detrended levels of output, consumption, and investment, but slower than

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
13
Fig. 2. Autocorrelations levels.
the autocorrelations of the "rst di!erences (not shown). This reinforces that the
issue of the nonstationarity of the in#ation rate is not clear-cut.
Table 3 reports the results of two tests for unit roots: the augmented Dickey
Fuller (ADF) test, which has the unit root as the null hypothesis (Dickey and

14
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Table 3
Unit roots tests!
ADF
(level)
ADF
(1st di!.)
KPSS
(level)
KPSS
(1901}95)
ADF- level
(war dynamics)
Variable
c
i
y
n
!2.14
!2.24
!2.95
!3.51"
!2.28
!4.24"
!4.80#
!4.57#
!6.00#
!5.98#
0.24#
0.28#
0.08
0.05
0.65"
2.12#
0.47#
2.02#
0.25#
0.65#
!2.05
!2.24
!2.82
!2.62
!0.49
G/>
!c, i, and y denote, respectively, the logs of real per capita consumption, investment, and GDP;
p denotes the annual percentage change in the CPI, and G/> is the ratio of government purchases to
GDP. ADF denotes the Augmented Dickey}Fuller test statistic for the unit root null hypothesis.
KPSS denotes the Kwiatkowski, Phillips, Schmidt, and Shin test of the null of stationarity. ADF
(war dynamics) refers to the ADF tests that allow the short-run dynamics for the world war years to
be di!erent. The sample period is 1889}1995, unless otherwise noted.
"ꢂ#indicate rejection of the null at 5%, and 1%, respectively. A time trend is included in all tests for
all variables except (G/>). A lag length of 5 is used in all tests.
Fuller, 1979), and the KPSS test, for which the null is trend-stationarity or
stationarity (Kwiatowski et al., 1992). (The trend is not included in the case of
the government size variable, since this ratio is bounded between 0 and 1.) The
results are presented in the "rst four columns. There is substantial evidence for
unit roots in per capita values of output, consumption, investment, and govern-
ment size, but the results on in#ation are very borderline and sensitive to the
exact sample period used. Because some of the series display unusual dynamics
during the wars, we also conducted ADF tests that allowed war-time dynamics
to be di!erent (by including variables that interact the lagged "rst di!erences
with a war-time dummy in the Dickey}Fuller regressions). For this speci"cation,
the results (reported in the "nal column) indicate the presence of unit roots in all
the variables.
We thus proceed with the maintained hypothesis of unit roots in y, c, i, and n.
In the case of G/>, we alternatively report results both under the assumption of
a unit root and stationarity. Two factors in#uenced our decision in this respect.
First, many RBC models that incorporate "scal policy assume that G/> is
mean-reverting, although deviations from the mean are modelled as being very
persistent (e.g. Baxter and King, 1993). Second, although our univariate tests do
support the unit root in G/>, these standard tests are based on an assumption of
a linear process, whereas this variable}being a ratio bounded between 0 and
1 } cannot be a restriction-free linear unit root process.
We also realize from our own results and those of others that the question of
a unit root in in#ation is controversial. However, there is a vast theoretical
literature analyzing the real e!ects of once-and-for-all unanticipated changes in

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
15
in#ation and whether a long-run in#ation-unemployment tradeo! exists or not.
This literature seems to put a strong prior on the presence of a unit root in
in#ation. Moreover, there are empirical results that are sympathetic to unit
roots in in#ation (e.g. Ball and Cecchetti, 1990; Mishkin, 1992). Barsky (1987)
has also argued that since 1914, shocks to in#ation have become more persist-
ent, particularly in the post-war period. However, more recently, Culver and
Papell (1997) "nd that, while in individual country time-series data in#ation
appears to have a unit root, in a panel setting in#ation appears to be stationary.
We proceed with the assumption of a unit root in in#ation, but recognize that
the empirical evidence is mixed.
3.3. The general model in a stochastic environment
The univariate analysis suggests the presence of stochastic trends. Our general
model allows for three stochastic trends: a productivity (output) trend (the
long-run component of the technology shift variable A in the production
function), a "scal trend (the long-run component of government size) and an
in#ation trend. It is convenient to separate out the trend and cyclical compo-
nents:
ln A "ln A #a , g "g #g , n "n #n ,
(5)
R
R
R
R
R
R
R
R
R
where an overbar represents the trend component and tilde denotes the station-
ary component. The theoretical results summarized in Table 1 suggest that
steady-state paths (denoted by asterisks) depend on the stochastic trend in
in#ation. In particular,
H
H
C
I
R
R
H
H
cR ,AZ NR B "c(n ), i ,AZRNR B "i(n ),
R
R
R
R
R
H
> (1!g )
R
H
H
yR ,C Z N D "y(n ),
N "N(n ),
(6)
R
R
R
R
R
H
where y is long-run e!ective per capita private (not total) output. In (6),
invoking certainty equivalence, we have replaced the previously constant
steady-state in#ation rate by the expected value of its permanent component,
which from the random walk property of stochastic trends is just the current
long-run component. In general, the theory discussed earlier implies that the
relationships in (6) will be nonlinear. However, in our empirical work we
postulate linear relationships between the logs of the variables on the left
hand side of (6) and the permanent component of in#ation. This can be
viewed as a linear approximation to the underlying nonlinear processes for
the purposes of estimation. Given this, (5), (6), and the de"nition of Z

16
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
[which implies ln Z"(ꢀ/h)g#(1/h) ln (1!g)#ln A+ln A![(1!ꢀ)/h]g],
yield
ln C "ln A !a g #(b #b )n #+a #c #n !a g ,,
(7)
(8)
R
R
! R
,
!
R
R
R
R
! R
ln I "ln A !a g #(b #b )n #+a #i #n !a g ,,
R
R
' R
,
'
R
R
R
R
' R
ln > "ln A #(1!a )g #(b #b )n
R
R
7 R
,
7
R
#+a #y #n #(1!a ) g ,,
(9)
R
R
R
7
R
where b , b , b , b represent the long-run e!ects of the in#ation on the
,
!
'
7
H
H H
H
logarithms of N , c , i , y , respectively, and a , a , a represent the long-run
!
'
7
H H
H
ꢀꢅ
e!ects of government size on the logs of c , i , y , respectively. Our particular
theoretical set-up of Section 2 implies that a "a "a "(1!ꢀ)/h. However,
!
'
7
in a more general model (e.g. when government spending enters utility in
a nonseparable way), these restrictions will not necessarily hold, and hence we
test them. The terms in curly brackets in (7)}(9) represent stationary components
that are constant along steady-state paths.
Eqs. (7)}(9) imply the two independent structural cointegrating CI relation-
ships given below:
g
R
n
R
[a #(1!a )] [!(b !b )] !1 1 0
!
7
!
7
aꢂX ,
ln> &I(0). (10)
R
R
A
B
[a #(1!a )] [!(b !b )] !1 0 1
'
7
'
7
lnC
R
A B
lnI
R
These long-term relationships can be estimated and we can do hypothesis
testing on them. We estimate the following CI relationships using the max-
imum-likelihood Johansen method (see Johansen and Juselius, 1990):
b
b
g #b n #b ln > #ln C "e ,
ꢀꢀ R
ꢁꢀ R
ꢀꢁ R
ꢀꢆ
R
R
ꢀR
(11)
g #b n #b ln > #ln I "e ,
ꢁꢁ R
ꢁꢆ
R
R
ꢁR
where e , i"1,2, represent stationary deviations from long-run paths. These
estimates are conditional on a cointegrating rank of 2, which we test
for. Comparing Eq. (10) with Eq. (11), our structural model implies
G
b
"!1"b . In addition, if a "a "a , we have b "1"b . When
ꢀꢆ
ꢁꢆ
!
7
'
ꢀꢀ
ꢁꢀ
the Fisher e!ect holds, we also have b ,b !b "0"b ,b !b (i.e.
in#ation has no e!ect on the shares of investment and consumption in GDP).
These are the restrictions we test.
ꢁꢁ
'
7
ꢀꢁ
!
7
ꢀꢅWe treat N, K as unobserved variables due to lack of long-term data on them and thus do not
report the corresponding equations for these variables.

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
17
Table 4
Tests of cointegrating rank!
Results for 3-trend model
Max
eigenvalue
statistic
Max
eigenvalue
statistic (df) value
95%
critical
Trace
statistic
(df)
H : CI
ꢅ
Trace
statistic
95% critical
value
rank"p
p"0
p41
p42
p43
p44
44.84"
28.08"
19.85
13.26
0.12
38.31#
24.0#
16.96
11.33
0.10
33.5
27.1
21.0
14.1
3.8
106.1"
61.3"
33.23#
13.38
0.12
90.69"
52.38#
28.39
11.43
0.11
68.5
47.2
29.7
15.4
3.8
System: G/>, n, y, c, i; Sample"1893}1995; Lag length"3. Constant included in the deterministic
component.
Results for 2-trend model
p"0
p41
p42
p43
42.3"
27.4"
12.7
37.4"
24.2"
11.2
27.1
21.0
14.1
3.8
82.6"
40.3"
12.8
73.0"
35.6"
11.4
47.2
29.7
15.4
3.8
0.22
0.20
0.22
0.20
System: n, y, c, i; Sample"1893}1995; Lag length"3. Constant, G/> and 3 lags of G/> are
included in the deterministic component.
!The maximum eigenvalue statistic (df) and trace statistic (df) apply a simple small-sample
correction to Johansen's statistics (replacing ¹ by ¹!nm, where ¹"number observations,
n"number of variables, m"number of lags), as recommended by Reimers (1992).
"ꢂ#indicates signi"cance at the 1% and 5% level
3.4. Cointegration test results: Evidence on the relationship between inyation
and great ratios
We "rst examine the cointegrating rank for both the 3-trend and 2-trend
speci"cations of the model. In the former, all "ve variables are treated as having
unit roots. The 2-trend speci"cation also contains all "ve variables, but treats
G/> as a stationary, deterministic variable so there are four variables with unit
roots. In each case we should have two CI vectors, since the CI rank will be the
number of variables with unit roots less the number of stochastic trends.
The results are in Table 4. The lag length in the VAR was selected by starting
with a lag length of "ve and sequentially eliminating lags with F-tests used to
check the validity of each reduction. The null hypothesis of p cointegrating
vectors (CI rank"p) is tested against the alternative of p#1 cointegrating
vectors using the maximum eigenvalue test statistic, and the more general
alternative of at least p#1 cointegrating vectors using the trace statistic. Test

18
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Table 5
Estimates of structural cointegrating vectors!
Coe$cient
3-trend model
Vector 1:
2-trend model"
Vector 2:
Vector 1:
Vector 2:
Variable
consumption
investment
consumption
investment
c
i
1.00ꢀ
0.00ꢀ
0.00ꢀ
1.00ꢀ
1.00ꢀ
0.00ꢀ
0.00ꢀ
1.00ꢀ
y
n
!1.00ꢀ
4.02
(0.72)
!0.63
(0.39)
3
!1.00ꢀ
!5.58
(1.17)
3.84
(0.63)
!1.00ꢀ
11.0
(1.95)
*
!1.00ꢀ
!6.52
(1.26)
*
G/>
Lag length
3
vꢁ(2) [p-value]#
16.5 [0.00]
17.7 [0.00]
Coe$cient on y
(unrestricted)$
!1.075
(0.024)
!0.99
(0.09)
!1.08
(0.03)
!0.97
(0.10)
!Standard errors are in parentheses. A &ꢀ' indicates that the coe$cient was constrained to the value
shown.
"In the four-variable system, G/> is treated as stationary and exogenous; its contemporaneous
value and three lagged values are included as deterministic components in the system.
#This is the Chi-squared statistic associated with the likelihood ratio test of the null hypothesis that
the restrictions imposed on the output variable in the two vectors are jointly satis"ed. `p-valuea
refers to the marginal signi"cance level of the sꢁ statistic.
$The coe$cient on y from a separate estimate in which the unit coe$cient restriction is relaxed.
statistics are reported both with and without a small sample correction due to
Reimers (1992). Consistent with the theoretical set-up, results fairly strongly
support the null of two CI vectors for both speci"cations. The exception is
a borderline rejection when the trace statistic is used without the degrees-of-
freedom correction.
Table 5 displays our estimates of the cointegrating vectors for the 3-trend and
2-trend speci"cations. Recall that, according to the theoretical model of Section
2, in both vectors the coe$cients on output and government size should be !1
and 1, respectively. We begin by testing these overidentifying restrictions. For
the 3-trend model, the restrictions on the government size variable are rejected
at conventional signi"cance levels. (In the 2-trend model G/Y does not enter the
CI vector.)
The last row of the table shows that the restrictions on output are not rejected
in the investment vector, but rejected in the consumption vector for both
speci"cations. The joint hypothesis that the output coe$cient is !1 in both
vectors is also rejected (see the second last row of the table). Although the

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
19
estimate of the output coe$cient in the consumption vector for both speci"ca-
tions (about !1.08) is not far from the predicted value of !1, the standard
errors are very small, making the deviation from the predicted value statistically
signi"cant at conventional levels. Based on economic signi"cance, however, it
could reasonably be claimed that the output restrictions are satis"ed. Moreover,
economic theory considerations put a very strong prior on the !1 coe$cient
on output: it arises from permanent productivity innovations leading to bal-
anced growth. By contrast, it is possible for a more general speci"cation of our
theoretical model in which the coe$cient on G/> is not unity (e.g. if government
spending enters utility in a nonseparable manner). In light of this, we proceed to
estimate the cointegrating vectors with the output variable restricted and G/>
unrestricted.
Next turn to the in#ation coe$cients in the estimated CI vectors. For the
3-trend model, keeping "xed the e!ects of government size, a permanent in-
crease in in#ation is associated with a drop in the consumption}output ratio
and a rise in the investment}output ratio. The coe$cient estimates shown in
Table 5 are statistically signi"cant at customary levels. The estimates of 4.02 in
the consumption vector and !5.58 in the investment vector indicate that
a permanent one percentage point increase in in#ation is associated with
ꢀ
a long-run drop in the consumption}output ratio of about 2 percentage points
ꢁ
and rise in the investment share of about 1 percentage point, using as initial
shares the full-sample means reported in Table 2. These estimates are large, as
they imply that a permanent, one standard deviation change in the rate of
in#ation (5.5 percent according to Table 2) is associated with approximately
a one standard deviation change in I/> and considerably more than a one
standard deviation change in C/>.
The estimates from the 2-trend speci"cation also show that the relationship
between in#ation and the investment}output ratio is positive and that between
in#ation and the consumption}output ratio is negative. The point estimates are
again statistically signi"cant and the coe$cient on consumption is noticeably
larger than in the 3-trend system.
The results in Table 5 also imply that the Fisher ewect does not hold in the long
run: as in#ation rises permanently, the long-run investment}output ratio goes
ꢀꢀ
up and, therefore, the capital-output ratio goes up and the real interest falls.
ꢀꢀSuppose we parameterize h, d, k, which represent the labor share of output, the depreciation
rate, and the long-term growth rate of the economy. Then, we can use our estimated e!ect of n on
I/>, the steady-state relationship I/>"(k#d)K/>, and the equality of the real interest rate to the
net marginal product of capital to determine the e!ect of in#ation on the steady-state real rate of
interest. For plausible parameter values, the computed e!ect on the real interest rate is implausibly
large. However, introducing quadratic adjustment costs to changing the capital stock can yield more
reasonably sized e!ects. We do not report these numbers because there is quite a wide range,
depending on the size of the adjustment costs, which are di$cult to pin down. Also, our estimates for
the post-WWII sub-period (discussed later) yield e!ects on the real rate of more plausible magnitudes.

20
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Since, typically, models in which the real rate falls with higher in#ation are ones
in which a Tobin-type e!ect holds, indirectly this provides some evidence in
favor of this e!ect.
3.5. Modelling the stochastic trends: Evidence on the ewects of inyation
on Y, I, and C
The analysis so far, based on estimated CI vectors, does not tell us about the
direction of causation between in#ation and the great ratios, nor about the
relationship between in#ation and the levels of consumption and investment.
Therefore, in order to interpret the results more cleanly in light of economic
theory, we need to identify exogenous shocks to the in#ation trend, which (as
always) comes at the expense of further restrictions. To this end, assume that the
stochastic trends behave according to
g "g #e ,
(12)
(13)
(14)
R
R\ꢀ
ER
ln A "k#ln A #e ,
R
R\ꢀ
ꢀR
n "n #b g ; n "k #n
#e ,
LR
R
ꢀR
% R
ꢀR
L
ꢀꢂR\ꢀ
where k is the long-term growth rate of the economy, b , k are "xed para-
%
L
meters, and the e's are zero-mean, serially uncorrelated, independent distur-
bance terms. Eqs. (13) and (14) imply that the stochastic parts of the in#ation
and productivity trends are independent. Also, we have assumed that the "scal
trend is exogenous and allowed to a!ect steady-state in#ation (b O0). The sign
%
of b tells us whether the in#ation tax is complementary to (b '0) or substitut-
%
%
able with (b (0) general income taxation. King et al. (1991) also estimate
%
a 3-trend model, including an in#ation trend, but they impose long-run neutral-
ity as an identi"cation assumption. Later, we provide a comparison of our
results to theirs and also examine robustness to changing some of the causality
assumptions on the stochastic trends.
A key issue before we proceed is whether the identi"cation restrictions
embedded in (12)}(14) are consistent with economic theory or not. In particular,
is it plausible that the stochastic trends in in#ation and productivity are
independent? First, economic theory clearly suggests that it is reasonable to
assume that the stochastic part of the productivity trend does not a!ect in#ation
in the long run. Once-and-for-all permanent shocks to the supply of output
a!ect the long-run price level but not its long-run rate of change.
Second, and more controversially, is it reasonable to assume that a permanent
change in in#ation can have a long-run e!ect on the investment rate } and hence
the real interest rate } but no long-run e!ect on productivity growth } and hence
the growth rate of the economy? At "rst pass, this would seem inconsistent with
the standard intertemporal e$ciency condition for consumption: with log-
utility, this is given by b(1#r)"uꢂ(C )/uꢂ(C )"(1#k). From this equation,
R>ꢀ
R

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
21
anything that a!ects the real interest rate in the long run, including in#ation,
should also a!ect the long-term growth rate of the economy. The puzzle can be
solved by realizing that the exact form of the above intertemporal e$ciency
condition holds only in those cases in which the CIA constraint does not apply
to investment, which are precisely the cases in which the real interest rate is
independent of in#ation. In general, if the CIA constraint applies to investment
also (a "1), the appropriate intertemporal e$ciency condition includes the
)
ꢀꢁ
Lagrange multipliers associated with constraints (2) and (3). Also, as an
empirical matter, if output growth is stationary (as the data strongly seem to
suggest), shocks to the random walk component of in#ation cannot have any
signi"cant permanent e!ect on growth.
Considering the fundamental shocks, one natural interpretation of the
permanent shock to in#ation (e ) is that it represents changes in the
L
monetary authority's target in#ation rate. However, following most of
the theoretical literature on in#ation and growth, we will, for the most part,
refer to it as an in#ation shock, rather than explicitly as a money growth
shock.
3.5.1. The estimated VECM
The vector of our "ve observed variables, X"(g, n, ln>, lnC, lnI)ꢂ is deter-
mined by the three permanent innovations to the stochastic trends, e , e ,
ER LR
2
2
e
and two transitory disturbances, which we label e , e . In moving average
ꢀR
ꢀR ꢁR
form, the structural model is:
*X "h(¸)e
(15)
R
R
2 2
where e"(e e e e e )ꢂ. The CI relationships given in (10) imply that the
E L ꢀ ꢀ ꢁ
matrix of long-run multipliers, h(1) } obtained by setting ¸"1 in (15) } is
1
0
1
0
0
1
1
1
0
0
0
0
0
0
0
b
%
h(1)" b (b #b )#(1!a ) b #b
0 ,[h$0],[h
0
I
C$0],
%
,
7
7
,
7
!
b (b #b )!a
b #b
b #b
%
,
!
!
'
,
A
B
b (b #b )!a
0
%
,
'
,
'
(16)
ꢀꢁThe general version of the consumption intertemporal e$ciency condition in the CIA model
can be shown to be: b(1#r)"(1#k)#[1#k!b(1!d)](c/j)a , where j,c are the Lagrange
)
multipliers associated with (2) and (3), respectively. This reduces to the familiar b(1#r)"(1#k)
only when a "0. There are also other models with a unit root in the investment rate, yet with
)
suitable parameterization, the growth rate e!ects of this unit root are small (e.g. Mendoza et al.,
1995).

22
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
where 0 is the 5;2 null matrix and
1
0
1
0
1
0
0
0
h"
1 ,
1
!(a !a ) (b !b )
!
7
!
7
A
B
!(a !a ) (b !b )
1
'
7
'
7
(17)
1
0
1
0
C"
b
0 .
%
b (b #b )!a
b #b
1
A %
B
,
7
7
,
7
Eqs. (16) and (17) imply that the matrix h is the product of a matrix consisting
of known coe$cients (since b !b , b !b , a !a , and a !a can be
!
7
'
7
7
!
'
7
obtained from the estimates of the CI vectors) and a lower triangular matrix.
Given lower triangularity and the independence of the permanent innovations,
it can be shown that the parameters b , (b #b ), (b #b ), (b #b ),
%
,
7
,
!
,
'
a , a , a are identi"ed and can be retrieved from the reduced-form VECM.
7
!
'
Also, under the assumption that the transitory disturbances are independent of
the permanent disturbances, variance decompositions and impulse responses
with respect to the permanent shocks can be computed.
The formal proof of identi"cation is very similar to that in King et al. (1991)
and is given in Appendix C. The intuition for why these parameters are
identi"ed lies in the recursive nature of the long-run model. Speci"cally, since
the "scal trend is causally prior to the in#ation trend and independent of the
productivity trend, the long-term behavior of g will identify this trend. Ac-
counting for the e!ect of this trend on in#ation, the long-term behavior of
in#ation then identi"es the in#ation trend. Similarly, accounting for the long-
run e!ects of in#ation and government size on consumption, investment, and
output, the long-run behavior of any one of these three variables identi"es the
productivity trend. (Note that the long-run responses of these three variables to
the productivity trend have been constrained to be equal from the CI vectors
imposed, which just re#ects the property that the productivity trend by itself
leads to balanced growth.)
3.5.2. Coezcient estimates
The point estimates and standard errors of the b parameters are reported in
Table 6. Estimates from both the 3-trend and 2-trend models indicate that
a Tobin-type e!ect is present: a permanently higher in#ation rate increases
output, investment, and consumption (b #b , b #b , b #b '0). With
7
,
!
,
'
,

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
23
Table 6
Estimates of structural parameters!
Model
Coe$cient
b #b
b #b
b #b
b
,
7
,
!
,
'
%
3-trend model
2-trend model
7.47
(2.14)
20.7
3.45
(1.77)
9.74
13.1
(2.44)
27.3
0.48
(0.18)
*
(5.84)
(5.09)
(5.98)
*
!Standard errors, shown in parenthesis, were computed by Monte Carlo simulation using 1,000
replications.
the exception of b #b , which is borderline, these are all statistically signi"-
!
,
cant, using two standard-deviation con"dence bands. These results do not
represent e!ects on lifetime utility and, therefore, should not be given any welfare
connotations. The issue of the optimal rate of in#ation is beyond the scope of our
paper, but it should be noted that there are models, e.g. Ireland (1994), in which
the welfare costs of in#ation are substantial even though higher rates of in#ation
are associated with a larger capital stock.
The results in Table 6 also indicate that b 'b 'b , so that a rise in
'
7
!
in#ation leads to an increase in the investment-output ratio and a drop in the
consumption}output ratio. The e!ects on these ratios are of exactly the same
magnitude as those reported in Table 5, since the CI vectors estimated there
have been imposed in estimating the VECM. For the 3-trend model, we also "nd
that b '0, suggesting that the revenue creation function of in#ation is used in
%
a complementary fashion to other taxes. (b is not identi"ed from the long-run
restrictions in the 2-trend model, since G/> is treated as stationary in that
%
model.)
3.5.3. Variance decompositions
Table 7 displays the fraction of the forecast error variance of each variable
that is attributable to the three permanent shocks in the 3-trend model. The
in#ation shock (panel B) accounts for under 20 percent of the forecast error
variance of either output or investment, and no more than 6 percent of the error
variance of consumption, at any horizon. Most of these point estimates are
less than twice the standard error. Fiscal shocks (panel A) account for most of
the variance of G/>, and a sizable amount of the variance of in#ation and
output. The error variance of consumption is almost entirely accounted for by
the permanent output (productivity) shock (panel C), while investment is
explained by a combination of the permanent and transitory shocks. The lack
of importance of transitory shocks in explaining consumption is consistent

24
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Table 7
Variance decompositions: 3-trend model!
Horizon: G/>
(A) Fraction of the forecast error variance attributed to the xscal shock
n
y
c
i
1
77.8
(17.6)
71.1
42.9
(14.9)
37.9
49.7
(16.3)
53.2
0.17
(7.35)
1.86
0.18
(8.43)
1.45
2
(17.4)
62.3
(13.5)
32.8
(14.6)
51.1
(6.96)
1.82
(8.05)
1.94
5
(15.4)
60.8
(11.4)
32.4
(13.3)
51.0
(6.52)
1.92
(7.23)
2.87
20
(14.9)
(11.1)
(13.2)
(6.47)
(6.97)
(B) Fraction of the forecast error variance attributed to the inyation shock
1
1.57
(4.33)
9.37
2.68
(6.64)
8.89
17.5
(12.0)
18.5
5.13
(8.31)
5.35
1.94
(7.01)
3.31
2
(6.30)
18.3
(10.6)
14.8
(6.56)
17.8
(7.58)
5.63
(6.48)
14.1
5
(6.95)
19.0
(7.01)
15.3
(9.54)
17.9
(6.64)
5.77
(6.27)
15.7
20
(7.03)
(7.04)
(9.43)
(6.66)
(6.68)
(C) Fraction of the forecast error variance attributed to the output shock
1
1.93
(6.54)
3.71
4.06
(6.83)
7.27
28.8
(15.8)
24.2
90.9
(14.1)
83.9
25.1
(13.0)
28.6
2
(7.08)
3.61
(7.79)
6.81
(12.4)
23.9
(12.7)
81.4
(11.8)
24.7
5
(6.26)
3.90
(6.87)
6.74
(11.2)
23.9
(11.7)
81.0
(9.95)
24.3
20
(6.16)
(6.75)
(11.1)
(11.6)
(9.46)
!Notes: Standard errors, shown in parenthesis, were computed by Monte Carlo simulation using
1000 replications.
with predictions of the life-cycle permanent-income hypotheses of consumption
ꢀꢆ
behavior.
Table 8 reports the variance decompositions from 2-trend speci"cation.
Generally, the results are the same as above. In particular, in#ation shocks
ꢀꢆTo examine robustness, we also estimated a system in which in#ation is placed after the
consumption, investment, and output block in the long-run causal ordering. In the 3-trend speci"ca-
tion, the contribution of in#ation shocks to the variability of output is slightly less and to the
variability of consumption is slightly more under the new ordering than the original ordering. The
shock's contribution to investment is approximately the same under either ordering.

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
25
Table 8
Variance decompositions: 2-trend model!
Horizon
n
y
c
i
A. Fraction of the forecast error variance attributed to the inyation shock
1
2.80
(7.70)
8.04
34.1
(18.6)
34.7
5.23
(8.98)
7.80
1.89
(7.63)
1.86
2
(7.22)
10.8
(18.1)
38.8
(8.44)
7.92
(7.12)
4.78
5
(7.91)
11.0
(14.5)
39.6
(7.44)
8.16
(6.74)
6.31
20
(8.01)
(14.2)
(7.44)
(6.65)
(B) Fraction of the forecast error variance attributed to the output shock
1
0.11
(3.89)
2.89
56.6
(19.1)
55.5
94.4
(12.0)
86.2
37.1
(15.8)
39.0
2
(5.24)
3.24
(18.0)
50.0
(11.0)
83.5
(14.7)
36.6
5
(5.13)
3.21
(14.6)
49.0
(10.2)
83.2
(13.3)
36.5
20
(5.13)
(14.4)
(10.3)
(13.0)
!Notes: Standard errors, shown in parenthesis, were computed by Monte Carlo simulation using
1000 replications.
explain a very small percentage of the forecast error variance of consumption or
investment. However, the contribution of in#ation shocks to output variability
(above 30 percent) is higher than in the 3-trend speci"cation, but standard errors
are large.
The point estimates of the structural parameters and the variance decomposi-
tions are suggestive of the following interpretation: Over the entire 1889}1995
period, permanent shocks to in#ation (when they do occur), appear to have
fairly large positive long-run e!ects on output, investment, and consumption.
However, signi"cant shocks to in#ation appear not to occur very frequently
during much of the sample. In other words, the unit root in in#ation is small
compared to the unit roots in the productivity and "scal trends. This suggests
that superneutrality may not be too bad an approximation when analyzing
historical U.S. economic data on real variables. However, this should not be
taken to imply that when we are discussing hypothetical or proposed signi"cant
changes in in#ation, we can ignore the long-run e!ects of in#ation on the real
economy; this is because our results also suggest that when in#ation shocks do
occur, they have signi"cant real e!ects and that models that generate the
reverse-Tobin e!ect (including endogenous growth models and RBC models
that incorporate money) appear to be at odds with the data.

26
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
3.6. Comparison with other results
One popular source of existing evidence on the output e!ects of in#ation is
cross-country growth regressions. Examples are Kormendi and Meguire (1985),
Barro (1991), and De Gregorio (1993). Typically, research in this area "nds
a negative impact of in#ation on output growth, although in some cases (e.g.
Barro, 1991), the e!ects are quite small. When large negative e!ects of in#ation
on growth are found, the sample generally includes high in#ation countries. For
example, De Gregorio (1993) focuses on Latin American countries. Bruno and
Easterly (1998) also "nd that it is during &discrete' high in#ation crises that
growth falls sharply. Thus, one way to reconcile our results with these cross-
country studies is to argue that the relationship between in#ation and output is
markedly di!erent in low to moderate in#ation environments than in high
in#ation environments. Moreover, Levine and Renelt (1992) have found the
results of these cross-country growth regressions to be &fragile', in the sense that
estimates are very sensitive to the set of conditioning variables used.
Another type of evidence on the real e!ects of in#ation comes from estimates
of dynamic Phillips curves. This work is exempli"ed by King and Watson (1994).
Using bivariate systems, both neutrality and superneutrality propositions are
examined, depending on the assumed order of integratedness of prices (or
money). The identi"cation method is to make an assumption about the magni-
tude of some impact elasticity. King and Watson "nd that the results on the
superneutrality proposition are sensitive to the assumptions made about the
magnitude of the impact elasticities, although for the range of short-run e!ects
they consider plausible (based on Keynesian, monetarist, and RBC types of
ꢀꢇ
models), the departures from superneutrality are not big. They also test for the
Fisher e!ect in a two-variable system (nominal interest rate and in#ation) and
"nd mixed evidence.
King and Watson's "ndings are based on bivariate systems, using (deliberate-
ly) only minimal structural information. Our results, based on a multivariate
model whose structure is derived from equilibrium growth theory, appear to
lead to di!erent conclusions about the long-run real e!ects of in#ation; however,
if the full range of the King/Watson estimates is considered, there is some
overlap.
King et al. (1991) (after which we have closely patterned the empirical
methodology used in this paper) do use a structural multivariate model and
include a stochastic in#ation trend. In contrast to our paper, King et al. impose
long-run neutrality as part of an identi"cation scheme, rather than test for it.
ꢀꢇThere are also papers that have used the assumption of zero impact elasticities to test long-run
neutrality (e.g. Geweke, 1986; Stock and Watson, 1989; Fisher and Seater, 1993).

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
27
Nevertheless, in the short run, shocks to the in#ation trend contribute almost
nothing to output #uctuations, which is roughly consistent with our full-sample
variance decomposition results. But, their impulse responses also indicate vir-
tually no short-to-medium-run response of consumption, output, and invest-
ment to the in#ation shock. Although their impulse responses do not represent
long-run properties, they appear to be inconsistent with our full-sample results
showing signi"cant real e!ects of in#ation. However, as discussed in the next
section, the King et al. results are more consistent with our results from the
post-war sample.
King et al. do "nd that their &real interest rate' trend is very important
in the variance decompositions and impulse responses. However, it is di$cult
to give a fundamental interpretation to an exogenous real interest rate trend
(in particular, their real interest rate trend is independent of their in#ation
trend). Our identi"cation scheme allows for in#ation to a!ect the investment
rate (and thus implicitly, the real interest rate) and allows for long-run depar-
tures from the Fisher e!ect, in line with at least some of the growth theory
literature.
3.7. Analysis of sub-periods
Are our results driven by special sub-periods such as wars or the Great
Depression, which was a period of de#ation and low investment? As noted
above, Table 1 and Fig. 1 indicate that the processes for in#ation and real
variables have been quite di!erent across the pre-WWI, interwar, and post-1949
periods. Therefore, to examine robustness, we also estimate the models over two
subsamples: the post-war period (1950}1995) and the inter-war period
ꢀꢈ
(1918}1941).
These results are presented in Table 9. The "rst row of results (reporting the
estimated CI vectors) shows that both sub-periods are characterized by a posit-
ive relationship between in#ation and the investment}output ratio and a nega-
tive one between in#ation and the consumption}output ratio. The long-run
relationship between in#ation and the investment rate is much stronger for the
inter-war period than the full sample: a permanent 1 percentage point drop in
in#ation leads to a long-run increase of nearly 30 percent in the investment rate
(which translates into a 4 percentage point increase taking as initial shares the
sample mean). This 30 percent change should be contrasted with a 5 1/2 percent
change obtained from the full sample. This large e!ect probably is the conse-
quence of the dramatic movements in in#ation and investment during the Great
ꢀꢈThe results from the sub-periods should be interpreted with some caution, however, since CI
techniques are more appropriate for longer spans of data.

28
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
29
ꢀꢊ
Depression. In the post-WWII period, the long-run relationship between
in#ation and investment, although still positive and signi"cant, is considerably
weaker than in the full sample: the estimate implies that a permanent 1 percent-
age point drop in in#ation is associated with an increase in the investment rate
of only 1 percent (which translates into only a two-tenths of a percentage point
increase). This is more consistent with the "ndings of King et al. (1991) discussed
above.
The variance decomposition results are reported in the "nal row of Table 9.
In contrast to the full-sample results, the in#ation shock accounts for a
large percentage of the forecast error variances of output, consumption,
and investment in the inter-war and especially the post-WWII periods. This
suggests that the in#ation trend became increasingly important relative to the
other permanent trends in the latter part of the sample, a result that probably
re#ects the increased persistence of in#ation itself, as noted by Barsky (1987) and
others.
We conclude from the above that the post-WWII period is noticeably di!er-
ent than other sub-periods. Although signi"cant &permanent' shocks to in#ation
are a more regular feature of the data, the long-run e!ects of a given size shock
are much smaller. For instance, the decade-average in#ation rate was about
5 percentage points lower in the 1980s than in the 1970s. Despite this large
permanent drop in the in#ation rate, neither investment nor the investment rate
fell by as much as would be implied by the full sample estimates.
4. Concluding remarks
Understanding the long-run real e!ects of changes in in#ation is essential to
debates concerning the channels of monetary policy transmission and the goal
of price stability. Long-term U.S. data indicate that a permanent unanticipated
rise in in#ation is associated with a rise in the share of investment in GDP in the
long run and also has positive long-run e!ects on the levels of output, consump-
tion, and investment. These results are consistent with the real interest rate
falling in response to a permanent rise in in#ation and the existence of a &Tobin-
type e!ect'. However, the results should not be given a welfare connotation, as
we have not examined their implications for utility.
Our empirical approach does not tell us the exact mechanism that generates
a Tobin-e!ect and we leave this as an open question. Possible factors are "nite
ꢀꢊEichengreen (1992) argues persuasively that the direction of causation in this interwar relation-
ship likely goes from monetary contraction (and hence a fall in in#ation) to investment and output.
This is consistent with our variance decompositions for the inter-war period (reported later), which
indicate that in#ation shocks are important over this period in explaining the forecast error variance
of real variables.

30
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
lifetimes, individual heterogeneity, and uncertain lifetimes highlighted in the
literature cited in the Orphanides and Solow (1990) survey paper, tax distortions
of the type highlighted by Feldstein (1976) and more recently by Bayoumi and
Gagnon (1996), or downward rigidity of nominal wages with individual "rms
experiencing stochastic shocks to the demand for their output, as emphasized by
Akerlof et al. (1996).
Our results are inconsistent with a variety of theoretical models that generate
a reverse-Tobin e!ect, models that at the present time seem to be slightly
favored in the in#ation and growth literature. However, our full-sample variance
decompositions indicate that, although signi"cant non-neutralities are found,
the role of in#ation in explaining #uctuations in real variables is very limited,
compared to the role played by productivity and "scal trends. This suggests that
real-business-cycle models and endogenous growth models without money
might be useful approximations when analyzing historical U.S. data on real
variables, but only because the unit root in inyation is small (i.e., permanent shocks
to inyation have small variance) and not because long-run superneutrality applies.
The results from the post-war (1950}1995) and inter-war (1918}1941) sub-
samples con"rm the existence of a &Tobin-type e!ect', but di!er from the
full-sample estimates in two ways. First, the estimated long-run e!ects on
output, investment, and consumption are much larger in the inter-war period
and much smaller in the post-war period than in the full sample. Second, as
measured by the variance decompositions, the in#ation trend is quite important
in the inter-war and post-war periods, unlike the full-sample results. The results
suggests that in those periods when permanent changes in in#ation are esti-
mated to have large long-run real e!ects such as pre-WWII, such shocks did not
occur often; by contrast, in the post-WWII period when permanent shocks to
in#ation are a more regular feature of the data, such shocks are estimated to
have smaller long-run e!ects.
Three avenues of further research in this area seem to us to be particularly
worthy of pursuit: First, it would be useful to try to identify the exact mechanism
by which an exogenous increase in in#ation leads to a rise in consumption,
investment, and output. Second, what accounts for the "nding that the observed
response of real variables to a given size in#ation shock is smaller in the
post-WWII period? Finally, it would be interesting to estimate these types of
structural VECMs using panel data and look for di!erences in the real e!ects of
in#ation across high in#ation and low in#ation countries.
Appendix A. Steady-state paths
$
The optimization problem is to choose the sequence +C , K , M , to
R
R>ꢀ
R>ꢀ
maximize (1) subject to the sequence of constraints (2) and (3) in the text. If, in the
long run, g is constant and technology, A grows at rate k, then the "rst order

S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
31
conditions and market equilibrium imply that the steady-state paths satisfy the
following conditions:
1
"j #a c ,
(A.1)
(A.2)
(A.3)
R
! R
C
R
ꢀ
*
F F\ꢀ ꢀ\F
"j Z N
K
,
R R
R
R
1!N
R
P
R>ꢀ
bꢀ
#bc "(1#n)j ,
A
M
B#bj
+
R>ꢀ
R>ꢀ
R
B
R>ꢀ
F
F
\F
K
b[(1!h)Z
N
#(1!d)]j #bc a (1!d)"j #a c ,
R>ꢀ R>ꢀ R>ꢀ
R>ꢀ
R>ꢀ )
R
) R
(A.4)
F
Z F(N , K )!(k#d)K !C "0,
(A.5)
(A.6)
R
R
R
R
R
M (1#k )
R
+
c
!a C !a (k #d)K "0,
R
A R
)
ꢀ
R
G
R
P
H
R
where j , c represent the Lagrange multipliers associated with (2) and (3)
R
respectively, k is the long-run money growth rate, and p represents the
+
constant steady-state in#ation rate, which can be shown to be k !k.
+
Appendix B. Data sources
(1) >"real gross domestic product in billions of 1987 dollars. The sources
are Kendrick (1961) table A-IIa from 1889 to 1928, and the National Income
and Product Accounts (NIPA) from 1929 to 1995 (U.S. Department of Com-
merce (1993) and various issues of U.S. Department of Commerce, Survey of
Current Business).
(2) C"real consumption expenditures in billions of 1987 dollars. Sources are
the same as for >.
(3) I"real gross private domestic investment in billions of 1987 dollars.
Sources are the same as for >.
(4) G"real federal government expenditures on goods and services in billions
of 1987 dollars. Sources are the same as for >.
(5) P"GDP de#ator, taken as the ratio of nominal GDP to real GDP
(1987)"1.00). Nominal GDP data are taken from Kendrick (1961) table A-IIb
from 1889 to 1928, and NIPA from 1929 to 1995.
(6) POP"total resident population of the United States, taken from U.S.
Bureau of the Census (1976,1992) and updates.

32
S. Ahmed, J.H. Rogers / Journal of Monetary Economics 45 (2000) 3}35
Appendix C. Identi5cation and estimation strategy
The structural model in MA form is (15) in the text and reproduced below for
convenience:
S
S
h 0
ꢀ
ꢀꢀ ꢀꢁ
S
*X "h(¸)e ,
,
h(1), h 0 ,
,
(C.1)
var(e )"S,CS D C D Ch 0D
R
R
R
ꢀꢁ ꢁꢁ
ꢁ
2 2
where recall that e"(e e e e e ) consists of the three permanent and two
E L ꢀ ꢀ ꢁ
transitory shocks and X"(g, n, ln >, ln C ln I). S (3;3), S (2;2) are the
ꢀꢀ
ꢁꢁ
diagnol covariance matrices of the structural permanent and transitory distur-
bances respectively, with S "S "0, implying the independence of the
ꢀꢁ
ꢁꢀ
permanent and transitory disturbances. The matrix h (5;3) is the product of the
two matrices given in (17) in the text; we have partitioned h further into h (3;3)
ꢀ
and h (2;3), with h lower triangular.
ꢁ
ꢀ
The reduced-from <ECM can be used to obtain the following reduced-form
MA representation:
C
ꢀ
*X "C(¸)e , var(e )"<,
,
(C.2)
C(1)"CCꢁD
R
R
R
where C(1) has been partitioned into its "rst three rows, C (3;5), and its last
two rows, C (2;5). Next, express the structural disturbances as linear combi-
ꢀ
ꢁ
nations of reduced-form disturbances:
P
ꢀ
\ꢀ
e "P e ,
\ꢀ
P
,
(C.3)
"CPꢁD
R
R
\ꢀ
where P
ꢀ
has been partitioned for convenience into its "rst three rows, P
(3;5), and its last two rows, P (2;5). To show that our model identi"es the
ꢁ
permanent structural disturbances, we have to demonstrate that, under the
assumptions given above (that S , S are diagnol matrices, S , S are null
ꢀꢀ ꢁꢁ
ꢀꢁ ꢁꢀ
matrices, and h is lower triangular), P is determined uniquely.
ꢀ
ꢀ
1
\
A.1. Obtaining the xrst three rows of P (P1)
From (C1)}(C3),
h(1)e "C(1)e ,
R
R
Nh(1)Sh(1)ꢂ"C(1)<C(1)ꢂ,=,
ꢂ
Nh S h "= ,
(C.4)
ꢀ ꢀꢀ ꢀ ꢀꢀ