This paper reassesses the long-run relation between nominal interest rates and inflation using German data. It shows that the empirical rejection of the strict Fisher effect in previous studies, i.e., the finding of interest rates not fully adjusting to changes in inflation, can be attributed to the particular time series behavior of inflation and interest rates which cannot be accounted for by standard non- stationary models. It is argued that the stochastic process governing the bivariate system of inflation and interest rates depends on the level of the variables and should be modeled as a thresh-old cointegration (TC) model. Contrary to the unit root hypothesis this model can be given an economic interpretation in terms of the opportunistic approach to disinfla-tion. The full Fisher effect, even in its tax-adjusted form, cannot be rejected when a threshold cointegration model is estimated. The TC model not only explains the downward bias of the coefficient estimates, but also the sample and country sensitivity observed in previous studies. The TC model may prove useful in testing other long-run relations such as uncovered interest rate parity or purchasing power parity.Inflation, interest rates, unit-roots, cointegration, bootstrap, Monte Carlo, threshold cointegration, SETAR-models

Jens Weidmann

English

Research Papers in Economics

Revised version of
University of Bonn, Sonderforschungsbereich 303 Discussion Paper B-385
NEW HOPE FOR THE FISHER EFFECT?
A Reexamination Using Threshold Cointegration
JENS WEIDMANN
*
University of Bonn
Institut für Internationale Wirtschaftspolitik
Lennéstraße 37, D-53113 Bonn
E-Mail: weidmann@iiw.uni-bonn.de
January  1997
Abstract: This paper reassesses the long-run relation between nominal interest rates
and inflation using German data. It shows that the empirical rejection of the strict
Fisher effect in previous studies, i.e., the finding of interest rates not fully adjusting to
changes in inflation, can be attributed to the particular time series behavior of inf la-
tion and interest rates which cannot be accounted for by standard non-stationary
models. It is argued that the stochastic process governing the bivariate system of inf la-
tion and interest rates depends on the level of the variables and should be modeled as
a threshold cointegration (TC) model. Contrary to the unit root hypothesis this model
can be given an economic interpretation in terms of the opportunistic approach to
disinflation. The full Fisher effect, even in its tax-adjusted form, cannot be rejected
when a threshold cointegration model is estimated. The TC model not only explains
the downward bias of the coefficient estimates, but also the sample and country sens  i-
tivity observed in previous studies. The TC model may prove useful in testing other
long-run relations such as uncovered interest rate parity or purchasing power parity.
Keywords: Inflation, interest rates, unit-roots, cointegration, bootstrap, Monte Carlo,
threshold cointegration, SETAR-models
JEL classification: E43, E31
                                           
* I am indebted to Axel Cron, Charles Goodhart , Huston McCulloch, Manfred J.M.
Neumann, Pete r Pedroni, Dan Thornton, and seminar participants at the Ohio State
University, Indiana University, and the Federal Reserve Bank of Cleveland for helpful
comments. Partial financial support from the German Marshall Fund of the United
States is gratefully acknowledged.2
1 Introduction
Ever since Irving Fisher’s “The Theory of Interest,” the conjecture that nominal
interest rates vary, ceteris paribus, point-for-point with expected inflation has
become one of the most studied topics in economics.  The Fisher effect is  a
cornerstone of many theoretical models that generate monetary neutrality and
is important for understanding movements in nominal interest rates.
Because  changes in the value of money redistribute purchasing power be-
tween debtors and creditors, a unity response of nominal interest rates to
changes in expected inflation is required to avoid such re-distributions and in-
sulate the real rate of interest.  This “full” or “strict” (point-for-point) Fisher ef-
fect, however, applies only to economies without taxes. Darby (1975), Feldstein
(1976), and Tanzi (1976) have shown that because of taxation, nominal rates
must change by more than the change in expected inflation if the real after-tax
rate of interest is to be invariant to anticipated changes in the value of money.
This effect produces an “augmented” Fisher effect. Darby (1975) suggests that
the nominal rate should change by 1.3 to 1.5 times the change in expected in-
flation.
 Generally empirical investigations, e.g., McDonald and Murphy (1989),
Wallace and Warner (1993), Mishkin (1992), Phylaktis and Blake (1993), and
Evans and Lewis (1995), tend to support the notion that movements in nominal
interest rates primarily reflect fluctuations in expected inflation.  However,
support for the full Fisher effect has been scant. In line with Fisher’s (1930) own
results, changes in inflation generally seem to have less than a point-for-point
effect on nominal rates, suggesting that expected inflation is non-neutral. Fur-
thermore, the strength of this effect depends heavily on the period and country
considered.
There have been several rationalizations of the apparent failure of the strict
Fisher hypothesis. Fisher himself explained it by some form of money illusion,3
while Tobin (1969) stressed the negative effect of inflation rates on money de-
mand. Mishkin (1984) argues that the failure of the full Fisher effect hypothesis
is due to the negative correlation between inflation and the real interest rate.
This paper takes a completely different approach and extends the literature
on the Fisher effect in a new direction. It argues that the rejection of the full
Fisher effect is due to a downward bias of the coefficient estimates that results
from a failure to adequately model the stochastic features of the data generat-
ing process. It suggests that this failure also accounts for the sample and coun-
try sensitivity of the results reported in previous studies. To model the particu-
lar time series behavior of inflation and interest rates, the threshold cointegra-
tion (TC) model is introduced and estimated. Contrary to the finding with con-
ventional cointegration techniques, the conjecture that interest rates respond to
inflation in the way the full Fisher effect suggests is confirmed. The contribu-
tion of the paper may be of relevance not only for the Fisher effect, but also for
other long-run relations involving inflation and/or interest rates, e.g. uncovered
interest rate parity or purchasing power parity.
 The remainder of the paper consists of five sections. The next section pres-
ents the Fisher hypothesis more formally and reexamines standard findings in
the literature that suggest the full Fisher effect does not hold. Section 3 pro-
vides a critical assessment of the variables’ time series properties and questions
the appropriateness of the unit root hypothesis. Then, section 4 introduces the
threshold cointegration model and shows how the presence of threshold coin-
tegration leads to a bias in coefficient estimates of standard cointegration re-
gressions. In section 5 the Fisher effect is reexamined using a TC-model. Fi-
nally, section 6 concludes the paper.
2 Testing for the Fisher effect using cointegration techniques
Starting with Rose (1988), several empirical studies recently recognized the im-
portance of accounting for non-stationarity when testing for the Fisher effect
and pointed to the danger of spurious regressions when not taking the non-6
effect depends heavily on the period considered, with the Fisher effect being
most apparent in periods when there is strong evidence for stochastic trends.
Working on postwar US data as well, both Wallace and Warner (1993), and
Crowder and Hoffman (1996) apply the fully-efficient Johansen (1988) cointe-
gration estimator which has several advantages over the Engle and Granger
two-step technique. Unlike the Engle-Granger specification, the long-run coef-
ficient estimates do not depend on the essentially arbitrary choice of normali-
zation, i.e., on the choice of the left-hand-side variable. Moreover, this ap-
proach does not suffer from the small sample bias of the Engle-Granger static
regression. Generally, their results tend to confirm a one-to-one relationship
between inflation and interest rates, but are also sensitive to the time period,
country, and data frequency.
Evans and Lewis (1995) also corroborate a long-run (cointegration) relation
between nominal interest rates and inflation. They show that nominal rates
move less than one-for-one with inflation and attribute this finding to struc-
tural shifts in the inflation process as these may induce small sample serial cor-
relation in forecast errors. Finally, Phylaktis and Blake (1993) observe drastic
differences between low and high inflation economies. For high-inflation
countries they find strong evidence of a full Fisher effect.  The results are at
best mixed for low-inflation countries, however.
In the next paragraphs, the empirical evidence of previous studies is reex-
amined, using conventional univariate unit-roots tests and the Johansen (1988)
test for cointegration. The data are monthly observations of the German 12-
month interest rate on T-bills and the consumer price index (CPI) from January
1967 to June 1996. The 12-month maturity was chosen because of its wide-
spread use in the empirical literature. The inflation series is calculated as the
annual percentage change in CPI shifted one year forward. Both series are
taken from the German Bundesbank’s datatape. Figure 1 plots the two series.
[FIGURE 1 ABOUT HERE]8
tives with roots close to but different from unity. Recently, Pippenger and Go-
ering (1993) have shown that the power of these tests falls dramatically under
threshold processes. Loosely speaking, this means that the researcher is likely
to accept the null of a unit root even if the true process is a threshold process
and therefore these tests should be interpreted with caution.
Second and more importantly, the unit root hypothesis is hard to reconcile
with economic intuition and “stylized” facts. Saying that inflation and the
nominal interest rate are non-stationary variables which are tied together in the
long-run by a cointegration relation means that random shocks have perma-
nent effects and the variables do not tend to return to their means. Thus, per-
sistently high and negative realizations are entirely consistent with this repre-
sentation. While theoretically under a fiat standard nothing assures monetary
stability, a Central Bank committed to price stability will not allow inflation
rates to become negative or persistently high. Generally, when inflation is “too”
high there is likely to be a public pressure on the Central Bank to conduct a
more restrictive monetary policy and to bring inflation down to “tolerable” re-
gions. If this pressure prevails, like in Germany and the post-war United States,
the inflation rate may become mean reverting for high levels of the series.
3 Be-
sides, for nominal interest rates there is a clear lower bound since they cannot
fall below zero. But also inflation rates seldom realize below zero, i.e., deflation
is clearly the exception.
                                           
3 Sometimes this public pressure does not overcome, e.g. in high inflation countries and
post World War I Germany,  and inflation and interest rates are driven by a stochastic trend.
Therefore, the upper trigger value is country specific and depends on the institutional frame-
work as well as the society’s preferences. While for some countries these factors may have
changed over the last 30 years, for Germany they can be assumed constant. See Barsky (1987)
for details on the dramatic differences in inflation persistence under different monetary re-
gimes.18
post real interest rates. However, I attribute this rejection of the Fisher effect to
the failure of conventional non-stationary cointegrated models to fully describe
the time series behavior of inflation and interest rates.
Instead, I propose a threshold cointegration model which can account for
the stylized fact that, at least in most industrialized countries, inflation and in-
terest rates seldom occur outside some narrow band of “normal” values.
Threshold cointegration not only explains the serious downward bias in pa-
rameter estimates, but also the sample sensitivity observed in previous studies.
When a bivariate threshold cointegration model is estimated to reexamine the
long-run relationship between interest rate and inflation one cannot reject that
nominal interest rates vary one-for-one with inflation. Despite the imperfec-
tions of the present approach, it seems evident that the threshold model is a
better approximation of the true underlying DGP than the conventional linear
model. Finally, note that undetected threshold cointegration may not only af-
fect estimates of the Fisher equation, but also of other long-run relations, e.g.,
uncovered interest rate parity or purchasing power parity. Obviously further
research is needed.19
GRAPHS AND FIGURES
Figure 1. German inflation and 12-month interest rate
1967 1970 1973 1976 1979 1982 1985 1988 1991 1994
-2
0
2
4
6
8
10
12
14
CPI-inflation (t+12)
12-month interest rate (t)
TL
TU
Figure 2a. Decision rules of the conventional and opportunistic policymaker
TU
TL
p*
Activist
pursuit of
inflation target Opportunistic pursuit
of i.t.
Activist
pursuit of
i.t.
p
Intended y-gap
Note: The optimal decision rule of the conventional policymaker is drawn as a
dashed line, whereas that of the opportunistic policymaker is drawn in bold.20
Figure 2b. Response function under the opportunistic approach
TH
TL p(t-1)
p*
mean-
reversion
(p=rp-1) unit root (Dp=e)
mean
reversion
(p=r’p-1)
Dp
Figure 3. Distribution of LR coefficient with Engle-Granger
0.480 0.528 0.576 0.624 0.672 0.720 0.768 0.816
0
12
24
36
48
60
72
Distribution of b-estimates
with Engle-Granger (true value: 1.52)21
Figure 4. Distribution of LR coefficient with equation (3)
0.752 0.848 0.944 1.040 1.136 1.232 1.328
0
16
32
48
64
80
96
112
Figure 5. Distribution of LR coefficient with equation (4)
1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00
0
25
50
75
100
Distribution of b-estimates
in THC-model (true value: 1.5)22
Figure 6. Bias of LR coefficient for various values of kL
Bias of LR coefficient
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
-0,5 -0,4 -0,3 -0,2 -0,1 0 0,1 0,2 0,3 0,4
lower threshold coefficient
b
i
a
s
Mean
Fract05
Fract95
Figure 7. Bias of LR coefficient for various values of kH
Bias of LR coefficient
-0,6
-0,5
-0,4
-0,3
-0,2
-0,1
0
0 -0,1 -0,2 -0,3 -0,4 -0,5 -0,6 -0,7 -0,8 -0,9 -1
upper threshold coefficient
b
i
a
s
Mean
Fract05
Fract9523
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New Hope for the Fisher Effect? A Re-Examination Using Threshold Cointegration

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New Hope for the Fisher Effect? A Re-Examination Using Threshold Cointegration

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