# The International Fisher Effect: theory and application

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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:

*B*ˆ

.

*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

*B*ˆ . 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

*B*ˆ 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

*v*1

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.

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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

(

*a*1)

intercept (

*a*2)

(

*b*1)

slope (

*b*2)

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-*

nomics9: 297-324.

nomics

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.

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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”,

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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.

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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.),

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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