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The International Fisher Effect: theory and application

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This paper uses an asset pricing based approach to derive an international version of the Fisher effect, denoted the "International Fisher Effect", and tests it for the US and the UK interest rates and inflation rates differentials. We apply the casewise bootstrap technique that is robust to heteroscedasticity and non-normality, which usually characterize financial data. We also allow for a structural break in October 1987 in our estimations. The results show that the international Fisher effect is slightly less than unity. This means that nominal interest rates differential responds less than point-for-point to the changes in the inflation rates differential. The implication of this empirical finding is explained in the main text.
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Investment Management and Financial Innovations, Volume 6, Issue 1, 2009
Abdulnasser Hatemi-J (UAE)
The International Fisher Effect: theory and application
Abstract
This paper uses an asset pricing based approach to derive an international version of the Fisher effect, denoted the
“International Fisher Effect”, and tests it for the US and the UK interest rates and inflation rates differentials. We apply
the casewise bootstrap technique that is robust to heteroscedasticity and non-normality, which usually characterize
financial data. We also allow for a structural break in October 1987 in our estimations. The results show that the
international Fisher effect is slightly less than unity. This means that nominal interest rates differential responds less
than point-for-point to the changes in the inflation rates differential. The implication of this empirical finding is
explained in the main text.
Keywords: interest rates, inflation rates, international Fisher effect, casewise bootstrap, structural break.
JEL Classification: G15, E40, E43, C22.
Introduction
pected inflation rate becomes remarkably high. The
money illusion phenomenon is expressed as another
The relationship between the nominal interest rates
explanation for not finding a full Fisher effect
and the expected inflation is of fundamental impor-
(Modigliani and Cohn, 1979; Tanzi, 1980; Sum-
tance in financial markets. In his seminal book mers, 1983). The existence of peso problems in the
Fisher (1930) establishes the foundation of the un-
market for nominal debt is additional argument put
derlying relationship between the nominal interest forward by Evans and Lewis (1995)1. The fourth
rate and the purchasing power of money measured reason, suggested by Fried and Howitt (1983), is the
by the inflation rate. The response of the nominal existence of a liquidity premium included in finan-
interest rate to the inflation rate is known as the cial assets that increases when expected inflation
Fisher effect in the literature and it is of paramount
rate increases. It can also be argued that one of the
importance pertinent to the efficiency of the finan-
reasons for not finding a full Fisher effect might be
cial markets and the performance of the monetary due to parameter instability. Previous studies take
policy. The Fisher effect predicts that the real inter-
usually for granted that the estimated parameter
est rate is not affected by the changes in the ex-
values remain constant across the time. However,
pected inflation rate because it results in equal there are many reasons to expect that structural
changes in the nominal interest rate. This implies breaks can take place. Changes in peoples’ prefer-
the nominal interest rate responds one-for-one to the
ences and their behavior, major technological ad-
expected inflation rate, which in turn implies the vancements, financial crises, policy alteration, insti-
long-run real interest rate is established in the real tutional and organizational development can result
sector of the market by means of “technology and in structural breaks. In addition, both the interest
preferences”.
rates and the inflation rates are usually non-normal
Despite its sound theoretical foundation, the full and heteroscedastic, which calls into question the
Fisher effect has not been strongly supported em-
application of standard methods.
pirically (Hatemi-J and Irandoust, 2008). The esti-
The aim of this paper is to derive an international
mated slope coefficients in regressions of nominal version of the Fisher effect using an asset pricing
interest rates on different measures of expected in-
based approach. We call this phenomenon the inter-
flation rates are significantly different than the theo-
national Fisher effect. To our best knowledge, this is
retical value of unity. Fama and Gibbon (1982), a notation that has been introduced in this paper. It
Huizinga and Mishkin (1986) and Kandel et al., should be pointed out that all previously mentioned
(1996) found that real interest rates were negatively
arguments for not finding a full Fisher effect might
related to the expected inflation rates. Crowder and
be valid for an international Fisher effect also. The
Wohar (1999) showed the Fisher effect is similar for
exchange rate risk and the existence of transaction
taxable and non-taxable interest rates in the US and
costs, especially for trading across international
the Fisher effect was found to usually be less than markets, might be additional factors for not finding
unity. According to the literature, there are several a full international Fisher effect. We also test
reasons for not finding a full Fisher effect. In a whether this effect is empirically supported between
seminal paper Tobin (1969) argues that investors
shift their portfolios towards real assets if the ex-

1 The phrase "peso problem" was launched by options trader Nassim

Taleb to represent a scenario in which a financial asset or trading strat-
egy that has demonstrated high stability and produced outstanding
© Abdulnasser Hatemi-J, 2009.
returns across a long period of time swiftly and surprisingly falls down.
117

Investment Management and Financial Innovations, Volume 6, Issue 1, 2009
the US and the UK economies. The international
E
,


(3)
Fisher effect has implications for market integration
t m
t
t m
t m
and market efficiency. A full international Fisher
effect would imply that arbitrage possibilities across
E
*
t m
t m
t m ,

(4)
t
economies do not exist. A casewise bootstrap ap-
proach that is insensitive to the presence of hetero-
where
and
are white noise error terms.
t m
t m
scedasticity and non-normality is utilized to obtain Assuming an insignificant risk premium in each
more precise estimates. The impact of a potential market and applying equations (3) and (4) we can
structural break due to October 1987 is also taken express equations (1) and (2) as:
into account in the estimations.
m
m
The rest of this paper consists of four sections. Sec-
i
r
,


(5)
t
t
t m
t m
tion 1 derives an international version of the Fisher
m
m
equation. Section 2 describes the data and the i t
r t
t m
t m .


(6)
econometric methodology. Section 3 presents the If the steady state value of the real interest rate is
empirical findings, and the last section offers con-
constant, then the nominal interest rate responds
clusions.
one-for-one to the expected inflation rate according
1. The International Fisher Effect
to the Fisher effect formulated in equations (5) and
(6). In an international version of the Fisher effect
According to the literature, a standard asset pricing
the interest rate differential between the two
model in which both nominal and real bonds are countries should be equal to their expected inflation
traded in the domestic market or the foreign market
differential. This means that we can combine the
provides the following conditions:
equations (5) and (6) to obtain the following interna-
tional version of the Fisher equation:
m
m
1
i
r
E
Var
t
t
t m
t
t
t m
2
m
m
m
m
i t
i t
r t
r t
t m
t m
Cov
,d
,



(1)
t
t m
t m
t m
.



(7)
t m
m
m
1
i t
r t
E
t m
Var
t m
t
t
2
It should be pointed out that the international Fisher
effect can be higher than one if the interest income
Cov
t m , d t m ,


(2)
t
is imposed to taxation. This point is shown in the
Appendix.
where
m
it is the nominal return on the m-period
2. Data and methodology
bond, m
r represents the return in real terms, m
t
t
The source of the data used in this study is the In-
signifies the inflation rate between the periods t and
ternational Financial Statistics (CD-ROM). The
t
m . Both nominal and real returns are assumed data frequency is monthly and it covers observations
to be continuously compounded. The notation m
d
on short-term nominal interest rates and CPI infla-
t
tion for the US and the UK economies. The sample
stands for the real discounting factor that is a func-
period is 1964:M1-2007:M1.
tion of the consumption growth in the consumption
based capital asset pricing model. The expectation By putting m
1 we can represent equation (7) in
operator based on information set available in pe-
the form of the following regression relationship:
riod t, i.e.
, is denoted by E . The denotations
t
i
a
b
e ,


(8)
t
t 1
t
Var and Cov represent the variance and covari-
t
t
where
represents the difference between the do-
ance measures, which are also based on the informa-
mestic and the foreign variable, and a and b are
tion available in period t. A star above a variable parametric coefficients to be estimated. The error
indicates that variable being a foreign variable. term is denoted by e
Equation (1) states that the nominal interest rate is a
t and it is assumed to be a white
noise process. To explore the international Fisher
linear function of the real interest rate, the inflation
effect when there is a potential structural break, we
rate and a risk premium that is measured by the extend equation (8) as:
second movements of m
d
t m
and
m
t m . A similar
i
a
a I
b
b I
v ,
(9)
t
1
2 t
1
t 1
2 t
t 1
t
result holds for the foreign market according to
equation (2). By applying the rational expectations where a1, a2, b1 and b2 are parametric constants to be
hypothesis we have
estimated. It is a dummy indicator that is equal to
118

Investment Management and Financial Innovations, Volume 6, Issue 1, 2009
zero for the period before October 1987 and it is 1. Create Y and Z by resampling with replacement
equal to one for each observation during the period
and denote them Y and Z , i.e. generate:
after the break. The denotation vt is a stochastic
error term, which does not have necessarily to be
Y
Y ,Y ,
,Y
,
1
2
n
homoscedastic or non-normally distributed. The
i
,
1
, n
break period is selected to be at 1987:M9 due to the
Y
Y
i , where
.
i
Black Monday stock market crash, which took place
Z
Z , Z ,
, Z
,
on October 19, 1987. By the end of this month,
1
2
n
stock markets in the UK and the US had fallen
Z
Z
i , where i
,
1
, .
n The denotation n
i
26.4% and 22.68%, respectively.
represents the bootstrap sample size.
It is widely accepted that the probability of extreme
2. Calculate the parameter vector by using Y* and
events in the financial markets is much higher than
~
what a normal distribution would suggest. To take
Z* and denote it B , that is, estimate
this issue into account in our estimations we apply a
1
~
casewise bootstrap approach, which has been devel-
B
Z Z
Z Y .
oped recently by Hatemi-J and Hacker (2005). This
method is robust to heteroscedastic and non-
3. Repeat steps one and two N times, where N is
normally distributed error term in the regression and
the number of bootstrap iterations, which is
it performs well in the presence of a structural
10000 in this study.
break. This method is used to estimate the coeffi-
cients and it is also used to test the statistical signifi-
4. Calculate the casewise bootstrap coefficient
cance of these estimated coefficients. In order to
N ~
make the presentation more compact we represent
B j
j 1
equation (9) in matrix format as the following:
vector ( B
ˆ ) via: Bˆ
.
N
Y
ZB
v ,



(10)
The casewise bootstrap method is also used to ob-
where
tain the p-values for all elements in the parameter
i
vector Bˆ . For example, let us concentrate on the
1
i
situation in which the null hypothesis that is tested
Y
2
a (T
)
1 vector,
is a
0 . We obtain the p-value for this hypothe-
1
sis by ranking the calculated values for Bˆ as the
iT
first step. If the estimated value of the median is a
1
I
I
2
1
1
2
positive value for a , then the p-value is the per-
1
1
I
I
3
2
2
3
Z
a (T
)
4 matrix,
centage of elements in the bootstrap distribution
for a that are negative added to those that are
1
1
I
I
T 1
T
T
T 1
greater than twice the median. If the estimated
B
a
a
b
b a (4
)
1 vector,
median for a is negative, the p-value is the per-
1
2
1
2
1
centage of elements in the bootstrap distribution
v1
for a that are positive plus the percentage of
v
1
and v
2
a (T
)
1 vector.
elements in a that are less than twice the median.
1
v
The cut-off point of twice the median of a is
T
1
equivalent to p-values that are similar to those
The ordinary least squares estimator for the parame-
symmetric two-sided tests in a traditional hypothe-
ter vector is obtained by calculating the following:
sis testing approach as shown by Hatemi-J and
Hacker (2005). The p-values for the other parame-
B
Z Z 1
ˆ
Z Y .



(11)
ters are estimated in a similar way. All the boot-
By using these denotations, we describe the strap simulations in this paper are conducted by
casewise bootstrap technique to be performed via using a GAUSS program code, which is accessible
the following steps:
upon request.
119

Investment Management and Financial Innovations, Volume 6, Issue 1, 2009
3. The results
Conclusions
Table 1 presents the estimation results applying the This paper derives the international Fisher effect
casewise bootstrap method1. Based on these results we
analytically using an asset pricing approach and
can observe that the intercept as a measure of risk tests it empirically using the casewise bootstrap
premium is positively significant. There is also a sig-
method, which performs accurately when the error
nificant break in this intercept, which suggests a statis-
term in the regression is heteroscedastic and non-
tically significant increase in the intercept after Octo-
normally distributed. We also allow for a break due
ber 1987 period. The slope parameter that represents to the October 1987 stock market crash. The sample
the international Fisher effect is statistically signifi-
covers the period of 1964:M1-2007:M1. Monthly
cant. There is no statistically significant break in this
data for the US and the UK markets are used. The
slope. Thus, the October 1987 event has resulted in a
estimated results reveal that the October 1987 stock
break in the risk premium but not in the international
market crash has resulted in a significant break in
Fisher relation between the US and the UK. However,
the risk premium but has not resulted in any break in
the estimated value of the slope is slightly less than the international Fisher relation between the two
one. Finding an international Fisher effect less than economies. The interest rates differential between
unity might be explained by the arguments expressed
these countries does positively and significantly
in the introduction of this paper.
respond to their inflation rates differential by the
Table 1. The estimation results using the casewise
same amount for the pre-break and post-break peri-
bootstrap method
ods. Nevertheless, this response is slightly less than
unity. However, the parameter value that we ob-
Intercept
Change in
Slope
Change in
tained is close to unity. Hence, taking into account
(a1)
intercept (a2)
(b1)
slope (b2)
the transaction costs and the existence of an ex-
Estimated
change rate risk premium, earning arbitrage profits
1.899 0.449 0.772
-0.291
value
may still not be possible. Thus, the markets may still
p-value <0.0001 0.0151 0.0005 0.4598 be considered as efficient.
References 1
1.
Crowder, W.J. and Wohar, M.E. (1999), Are Tax Effects Important in the Long-Run Fisher Relationship? Evi-
dence from the Municipal Bond Market, Journal of Finance, 54 (1), 307-17.
2.
Darby, M.R. (1975), “The Financial and Tax Effects of Monetary Policy on Interest Rates”, Economic Inquiry 13:
266-276.
3.
Evans, M. and Lewis, K. (1995), “Do Expected Shifts in Inflation Affect Estimates of the Long-Run Fisher Rela-
tion?”, Journal of Finance 50: 225-253.
4.
Evans, L.T., Keef, S.P. and Okunev, J. (1994), “Modelling Real Interest Rates”, Journal of Banking and Finance
18: 153-165.
5.
Fama, E. and Gibbons, M.R. (1982), “Inflation, Real Returns, and Capital Investment”, Journal of Monetary Eco-
nomics
9: 297-324.
6.
Fisher, I. (1930), The Theory of Interest. New York: Macmillan.
7.
Fried, J. and Howitt, P. (1983), “The Effects of Inflation on Real Interest Rates”, American Economic Review 73:
968-979.
8.
Haliassos, M. and Tobin, J. (1990), “The Macroeconomics of Government Finance”, in Friedman, B.M. and Hahn,
F.H. (Eds.), Handbook of Monetary Economics, Vol. II: 889-959. North-Holland.
9.
Hatemi-J, A. and Hacker, R.S. (2005), An Alternative Method to Test for Contagion with an Application to the
Asian Financial Crisis, Applied Financial Economics Letters, 1 (6), 343-347.
10. Hatemi-J, A. and Irandoust M. (2008), The Fisher Effect: A Kalman Filter Approach to Detecting Structural
Change, Applied Economics Letters, 15 (8), 619-624.
11. Huizinga, J. and Mishkin, F.S. (1986), “Monetary Policy Regime Shifts and the Unusual Behavior of Real Interest
Rates”, Carnegie-Rochester Conference Series on Public Policy 24: 231-274.
12. Kandel, S., Ofer, A., and Sarig, O. (1996), “Real Interest Rates and Inflation: An Ex-ante Empirical Analysis”,
Journal of Finance 51: 205-225.
13. Mishkin, F.S. (1992), “Is the Fisher Effect for Real? A Reexamination of the Relationship between Inflation and
Interest Rates”, Journal of Monetary Economics, 30, 195-215.
14. Modigliani, F. and Cohn, R. (1979), “Inflation, Rational Valuation, and the Market”, Financial Analysts Journal
35: 24-44.

1 It should be pointed out that we tested each differential variable for unit roots. The results, not reported, were supporting the stationary property of
each variable.
120

Investment Management and Financial Innovations, Volume 6, Issue 1, 2009
15. Summers, L. (1983), “The Non-adjustment of Nominal Interest Rates: A study of the Fisher Effect”, in J. Tobin
(Ed.), Symposium in Memory of Arthur Okun: 201-244. Washington, D.C.: Brookings Institution.
16. Tanzi, V. (1980), “Inflationary Expectations, Economic Activity, Taxes and Interest Rates”, American Economic
Review 70: 12-21.
17. Tobin, J. (1969),“A General Equilibrium Approach to Monetary Theory”, Journal of Money, Credit, and Banking
1: 15-29.
18. Wallace, M.S. and Warner, J.T. (1993), “The Fisher Effect and the Term Structure of Interest Rates: Tests of Coin-
tegration”, Review of Economics and Statistics 75: 320-324.
Appendix. The tax-adjusted International Fisher Effect
According to Darby (1975), if the interest income is imposed to taxation then the Fisher effect might be higher than
one. To see this point analytically, we assume that the interest income is taxed by
decimal points in the domestic
market and by
decimal points in the foreign market. In this case the nominal returns are equal to
m
1
i and
t
m
i*
1
t , respectively. By substituting these values into equations (5) and (6) we can obtain the following tax-
adjusted Fisher equations:
m
1
m
1
1
i
r
,







(A1)
t
t
t m
t m
1
1
1
m
1
m
1
1
i
*
*
t
r t
t m
t m .






(A2)
1
1
1
Equations (A1) and (A2) show that the Fisher effect is higher than one in the domestic market if
0 and it is higher
*
than one in the foreign market if
0 . By taking the interest rate differential using equations (A1) and (A2) we
obtain the following tax-adjusted international Fisher equation:
m
m
1
m
1
m
1
1
1
1
i t
i t
r t
r t
t m
t m
. (A3)
t m
t m
1
1
1
1
1
1
Assuming that the tax rates are equal to each other in the two markets we can express equation (A3) as
m
m
1
m
m
1
1
i t
i t
r
r t
t m
t m .
(A4)
t
t m
t m
1
1
1
Thus, the tax-adjusted international Fisher effect can be higher than one.
121

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