THE MISSING FISHER EFFECT:

A THEORY WITH SOME TESTS

USING UK DATA

WILLIAM COLEMAN

Abstract

This paper expounds and tests an explanation the lack of relationship between the nominal rate of interest and the expected rate of inflation in a small open economy. It is hypothesized that what explains an economy’s interest rate is the expectation by the rest of the world of the inflation rate in the economy in question, not the expectation prevailing in the economy itself. Consequently, there will be no Fisher relation when domestic expectations are not shared by the rest of the world. The hypothesis is tested with UK data from 1959 to 1988 with fairly favorable results.

I. Introduction

Since the close of the 1960s it has become widely accepted that an increase in the expected

rate of inflation will increase the nominal rate of interest, as Fisher originally hypothesized.

However, notwithstanding the widespread acceptance of the hypothesis, not all empirical

enquiry has been favorable to it. Mishkin, for example, in a multinational study of interest

rates in the 1970’s concluded, “A fascinating finding of the present study is that the Fisher

Direct all correspondence to: William Coleman, Economics Department, University of Tasmania, GPO Box 252C, Hobart, TAS 7001, AUSTRALIA.

International Review of Economics and Finance, 2(3) 267-285 Copyright Q 1993 by JAI Press, Inc.

ISSN: 1059-0560 All rights of reproduction in any form reserved.

267

WILLIAM COLEMAN

effect does not generalize to all countries . . . the Fisher effect is weak for France, Germany,

the Netherlands, and Switzerland.” (Mishkin, 1984)

Several explanations have been advanced why this may be so. The general equilibrium

assumptions on which the hypothesis is founded may be at fault. Consider a model which

allows for disequilibrium; an IS-LM model, for example. An increase in expected inflation

of x percent will shift up the IS curve by x percent, but as long as the LM curve is not vertical

the nominal interest rate will rise by less than x percent. (The increase in real income, by

increasing saving, has increased investment and therefore reduced the real rate of interest.‘)

Taxation or imperfections in capital markets could also invalidate Fisher’s one-for-one

relation between nominal interest rates and expected inflation.

This paper advances a different theory of the absence, or near absence, of the Fisher

relation. The critical ingredient of our theory is the existence of disagreement over the future

rate of inflation. Specifically, our theoretical model predicts that there will be no Fisher type

relation in a small open economy when the country’s residents disagree with the rest of the

world over the rate of inflation they will experience. Effectively, the expectations of the rest

of the world are overwhelming the impact of those of a small country.

Our theory has an easy intuition. It is fairly obvious that the interest rate at which an

individual borrows and lends in a closed economy depends on an average of the expectations

of all the individuals in the economy; it does not depend on his/her own expectation of

inflation. This paper just extends that idea to the world economy, so that the interest rate at

which country n borrows depends on an average of what all countries expect inflation in n

to be.

Our model is tested against the facts of UK experience since 1959, with largely favorable

results.

Section II outlines some empirical results which motivate the model. Section III outlines

the model which explains the results. Section IV tests the model against UK experience since

1959 and examines some alternative explanations of the results. Section V draws some

conclusions.

II. An Empirical “Anomaly”

The Fisher thesis is usually interpreted as meaning that the nominal interest rate in country

k is determined by the expectation of inflation in country k. The expectation of inflation in

country n should be entirely irrelevant to the nominal interest rate in k. However, for two

economies with highly developed financial sectors, the United States and the United

Kingdom, this appears not to be the case. Regression analysis strongly suggests that US

expectations of US inflation are important for UK nominal interest rates, and are more

important than UK expectations of UK in.ation. That is in a regression of the form,

iUK = 4 + p us inflation expectation + y UK inflation expectation,

p is significant and y is insignificant. Consider Table 1. It reports a regression of the UK

short interest rate on UK and US expectations of consumer price inflation over the period

The Missing Fisher Effect

1959-1988. The UK interest rate used was the three month prime bill rate. The US interest

rate was the three month Treasury Bill rate. Since expectations of inflation are difficult to

measure, Table 1 reports the regression using three different measures (or proxies) for

expected inflation:

1. “Expert” forecasts of inflation. The Livingston Survey provides a good measure of

expert opinion for the United States. There is no equivalent over any significant

sample periods for the UK. We will use the forecasts of the National Institute of

Economic and Social Research, which have been appearing regularly since 1959. It

might be worth stressing at this point that the quality of the National Institute’s

forecasts has no relevance for the wisdom of our choice here; neither does their

supposed Keynesian accent have any relevance. The only thing which is of relevance

here is how representative they are of UK expert opinion. We venture that the National

Institute’s forecasts are the forecast series nearest the median of UK expert opinion.

2. Current Actual Inflation; this will be an accurate measure if people forecast inflation

in a “static” fashion. This will also be the rational expectation when inflation is a

random walk.

3. Future Actual Inflation; This will be an accurate measure if there is perfect foresight

of inflation.

The regressions were conducted in both levels and differences. The data is annual averages

of monthly or quarterly observations.2

Table 1 strongly suggests that UK short interest rates are completely dominated by the

expectation of inflation in the United States! This is true for both “Expert” “Static” and

“Perfect Foresight” measures. It is true for levels and difference regressions. The levels

regression are marred by substantial autocorrelation errors, but Table 1 also shows that the

addition of a lagged dependent variable eliminates that autocorrelation, without affecting

the predominance of US expectations.

A scoffer might dismiss the dominance in Table 1 of US Expert forecasts over UK Expert

forecasts as merely the result of the (supposedly) low (and therefore uninfluential) standard

of forecasting by the National Institute. A doubter might suggest that the National Institute

was so incompetent that US Expert forecasts of US inflation were a better forecaster of UK

inflation than the Institute’s forecasts of UK inflation! This however is not the case; Table

3 shows how the National Institute’s forecasts dominate the Livingston series as a predictor

of UK inflation.3

Thus the domination of US Expert forecasts over UK Expert forecasts as an explanation

of UK nominal interest rates cannot be explained away by supposing US forecasts are

incidentally better forecasters of UK inflation than UK Expert opinion (as we have measured

them). Some other explanation must be found.

With regard to the problem of measuring inflation forecasts, it is worth noting that US

Expert forecasts of US inflation also dominate survey based measures of UK expectations

270 WILLIAM COLEMAN

Table 1. Regression of the UK Interest Rate on UK and US Expected Inflation,

1959-1988

c neUK l-hi I’UK-1 R2 SE DW h

Expert

forecasts

static expectations

Perfect

foresight

Differences

Expert

fotiXasts

Static

expectations

Perfect

4.1 0.003 1.11 0.81 1.4 1.24

(11.7) (0.1) (9.0)

2.8 0.01 0.81 0.28 0.85 1.3 0.75

(5.1) (0.2) (5.4) (2.6)

5.0 0.10 0.63 0.62 2.0 0.65

(7.2) (1.2) (4.1)

1.9 4.04 0.5 0.56 0.82 1.4 0.08

(3.1) (0.6) (4.3) (6.0)

5.0 0.15 0.55 0.58 2.2 0.50

(7.1) (1.7) (3.0)

1.4 -0.004 0.47 0.59 0.84 1.3 0.41

(2.6) (0.05) (3.8) (7.8)

0.16 0.81

(1.1) (2.8)

0.07 0.48

(0.8) (3.4)

0.20 0.41

(1.8) (3.1)

Heteroskedastic consistent ?-statistics in brackets

0.42 1.5 2.34

0.37 1.6 2.35

0.44 1.5 2.21

as an explanation of UK nominal interest rates. Table 3 shows the results of repeating Table

l’s regressions but with UK expectations measured by the Carlson and Parkin’s series (1975)

based on Gallup surveys, and a series provided by Papadia (1984) based on a EEC survey.4

The predominance of US expectations with “Expert” measures does have some sensitivity

to the sample period; tests for structural instability suggests a structural shift in favor of US

Table 2. Regression of UK Inflation on UK and US “Expert” Forecasts of Domestic Inflation 1965-1988

c National Institute Livingston R= SE DW

a.20 1.29

(0.2) (7.3)

1.31 1.57

(0.9) (5.3)

Hetezoskedastic consistent t-statistics in brackets

0.74 3.0 1.09

0.40 4.5 0.58

The Missing Fisher Effect 271

Table 3. Interest Rate Regressions Using Alternative Survey Measures of UK Expectations

Sample

C kJK his Rz SE DW Period

Carlson- (?:, -0.20 1.57 0.76 1.0 1.73 1961-1973

Parkin (0.6) (2.8)

Papadia 4.1 0.07 0.97 0.60 1.22 1.22 1974-1982

(1.6) (0.6) (4.5)

Note: IIeus is the Livingston Series. Heteroskedastic consistent t-statistics in brackets

expectations in the late 70’s. The stability of Table l’s levels regression using Expert forecasts

was assessed using the following procedure. The levels regression using Expert expectations

was repeated with a slope dummy on the US expectations variable. The slope dummy

assumed a value of zero before some “critical year” and the actual value thereafter. This type of

regression was estimated for 10 critical years; 1970 to 1979 inclusive. The slope dummy was

significant on a two-tailed test in only the year, 1979. The results are reported in Table 4.

It is significant that the “anomaly” which Table 1 reports does not occur if US interest

rates are substituted for UK interest rates as the dependent variable. If the regression of Table

1 is repeated with the US interest rate as a dependent variable, a larger coefficient is estimated

on the US expectation variable than on the corresponding UK expectation variable in every

regression. In some regressions the UK variables are significant, but only when they are

estimated to have a negative impact. No UK expectations variable is estimated to have a

statistically significant and positive impact on US interest rates.

Overall, we believe the results of Tables 1 and 3 constitute strong evidence for UK interest

rates being set by the inflation expectations of Americans. At the same time there is little

evidence that US interest rates are affected by the inflation expectations of Britons.

Table 4. Regressions of UK Interest Rates Including a Slope Dummy for US

Expectations

Beus C neUK film (dummy) R2 DW SE

Levels

Expea forecasts 4.6 0.18 0.54 0.34 0.85 1.42 1.3 (11.0) (1.9) (2.2) (2.4)

Heteroskedsstic consistent t-statistics in brackets

272 WILLIAM COLEMAN

I I I. A Theoretical Explanation

How can the empirical results of Section II be explained? This section presents a very simple

economic model which relies on heterogeneity in expectations, and in the size of national

economies, and which goes some way to explain the results.

A. The Structural Equations

The model is an “opened” version of the flex-price IS-LM model developed by Metzler

(195 1) and Patinkin (1965). The model has three types of commodities; money, goods and

bonds. Expectations are predetermined. All markets clear.

We assume a world in which there is one good, N countries, and N currencies (one currency

for each country). There are no impediments or costs in trade so an international parity in

purchasing power is always observed. There are also N bonds, one for each national currency.

The bonds promise to pay one unit of currency in the next period and nothing thereafter. The

bonds are perfect substitutes.

The supply of goods of each country is exogenous. There is no endogeneity in output

resulting from sticky wages; wages are completely flexible.

The demand for goods by each country is proportional to the wealth of the country. So,

Dk=Zwk all k. (1)

Each country has a positive demand for its own national money, and a zero demand for

all foreign money. Country k’s demand for its own money is unit elastic to the wealth of

country k, but the proportion between wealth and money is a negative function of the

domestic interest rate.

Hk = h(ik)wk all k. (2)

The wealth of each country equals its current output of goods plus the present value of its

future output plus its endowment of money plus the value of the bonds it bought in the

previous period.

Yk = the CUrmt output of country k. RkYk = present value of country k’s future output, under static income expectations.

Rk = the preSent Vahe of an infinitely long stream of one unit of real income per

period.

I&/?+ = the real money endowment of country k. X&-I/& = is the real value of bonds purchased by k in the preceding period.

wk = Yk + Rk(ik - nt?,) Yk + i%!t#k + x&-,/Pk. (3)

Notice that Rk, the present value factor, is assumed to be a negative function of the current

expected real interest rate, ik - &.M, where Hek is k’s expectation of inflation in k.

The Missing Fisher Effect 273

B. Equilibrium

A general world-wide equilibrium requires that the national excess demands for goods

sum to zero. A general world-wide equilibrium also requires that the total excess demand

for each money is zero.

The third condition of equilibrium is the equilibrium condition for bonds. Since the N

bonds are perfect substitutes, asset equilibrium requires that the real rate of return offered

by all bonds must be equal. Thus the ex ante real rate of interest is the same the world over.

i, - He, = ik - He, all n,k. (4)

It is crucial to note there is an ambiguity about the expected inflation terms in equation

(4) as they stand: who do these expectations “belong” to; n or k ? The answer is that they

“belong” to both n and k: equation (4) must be satisfied for both n’s and k’s expectations if

there is to be an asset equilibrium. So, asset equilibrium requires,

ifi - He& = ik - ne,& (5)

He& = k’s expectation of inflation in n, etc.

These equations must be satisfied for any pair of n and k one cares to select. It might seem

that such an asset equilibrium condition would be difficult to achieve, since equations (5)

and (6) imply that n and k have the same expectation of the inflation differential between n and k. Why should it be true that, for any pair of countries you care to select, their expectations

of the inflation differential between the two countries be the same? In fact there are (at least)

three hypotheses about the interrelationships of expectations which will ensure that countries

agree about the differentials.5 We will call these three hypotheses “Homogeneous Expecta-

tions”, “Non-Discriminating Expectations” and “Mixed Expectations”.

C. Expectations

1. Homogeneous Expectations. Perhaps the simplest hypothesis about expectations

which produces identical expectations of differentials is the hypothesis of “Homogeneous

Expectations”; tbe expectations of complete agreement. Homogeneous Expectations sup-

poses every country has the same expectation of inflation in a given country. For example,

Britain’s expectation of inflation in the US is the same as Germany’s expectation of inflation

in the US, which is the same as the US’s expectation of inflation in the US. Effectively, each

country can be seen as deciding its own forecast of inflation and then forecasting foreign

inflation by adopting foreigners’ own forecasts about their own inflation. Consequently all

country’s agree on the inflation differential between two countries.

In algebraic terms the hypothesis (which we denote by HO) states;

HO: lie, = lie, all n,k. (7)

274 WILLIAM COLEMAN

That is, k’s forecast of inflation in n is the same as n’s forecast of inflation in n.

2. Non-Discriminating Expectations. There are other expectations hypotheses apart

from HO which ensure uniformity of expectations about inflation differentials between

countries.

Consider the situation where each country expects the rest of the world to share its own

inflation rate. This would imply that each country has a zero expectation of the inflation

differential between every pair of countries. Consequently, it makes possible the existence

of asset market equilibrium. We will call this hypothesis the “non- discriminating hypothesis

of expectations”, ND, since, under this hypothesis, each country does not discriminate

between countries in making forecasts of inflation.

In algebraic terms ND states,

ND: ne,,k = IIeM all n,k. (8)

Equation (8) says k’s expectation of inflation in n equals k’s expectation of inflation ink.

Notice that ND allows expectations to be “heterogeneous” since countries have, or may

have, different expectations about inflation of a given country. But at the same time ND does

not necessitate heterogeneous expectations: suppose all countries agree that inflation in all

countries will be p percent; then expectations are both “non-discriminating” and “homoge-

neous”.

But ND expectations does not always seem very plausible; especially when countries do

have different inflation trends.

3. Homogeneous Expectation of Differentials and Heterogeneous Expectations of Lev-

els. There is a third expectations hypothesis which ensures countries agree on inflation

differentials, but does not require everyone totally agree, nor that everyone think everyone

will be the same as themselves. We will call this mixed expectations, M.

M states,

(9)

where rr,, is a parameter which is invariant across k.

Mixed expectations supposes every country agrees on the differentials between inflation

in various countries, but they disagree on the average level of inflation experienced in the

world, with some countries taking a high estimate of the average level (a positive 6) and

some countries taking a low level (a negative 6).

It is worth noting that the hypothesis can be rewritten in a way which makes it comparable

to HO and ND. Equation (9) implies He,,” = rr, + 6,. Substituting this expression back into

equation (9) yields IIe,,k = He,,, + Sk - 6,. It is also worth noting that if n, is invariant across

n then expectations are “non-discriminating”. And if Sk is invariant across k they are “homogeneous.”

M appears to be more plausible than HO or ND, owing to its greater generality. Unlike

ND, it allows, for example, Americans to expect that Italy will have a higher rate of inflation

The Missing Fisher Effect 275

than themselves. But, unlike HO, it does not require Americans and Italians to agree on the

level of inflation in Italy or America.

How does the hypothesis of rational expectations bear on the three hypotheses? Do rational

expectations favor any of them, and disallow any others?

Of the three hypotheses we have advanced, the one with the greatest presumption of the

favor of rational expectations is homogenous expectations, But the inference of homogeneity

from rationality cannot be an immediate one. It is true that rational expectations would ensure

that forecasters’ expectations are the same (homogenous) on average over time, since the

expected value of any rational expectation of x is always the expected value of x. But with

different information sets at a given point in time, it is perfectly possible for two persons

with rational expectations to have different inflation forecasts at that point in time. However,

the mere difference in private information sets need not lead to heterogeneity in forecasts

since the less informed may use market prices as indicators of the information of the more

informed. Suppose, for example, a small minority of persons are well informed about the

prospects of future inflation, while the majority are not. The competition of the well-in-

formed for funds will push up interest rates, until nominal returns on bonds equal the nominal

returns on real assets which the well informed anticipate. The less informed, appreciating

they are not well informed, can use the interest rate as their predictor of inflation us long as

they share with the better informed the same expectation of the real rate of interest. In

this circumstance their expectation of inflation will coincide with that of the better informed.

However, the incorporation of rationality into our model requires a far more substantial

revision than merely reducing our set of admissible expectations hypotheses to homogeneous

expectations. This is because rationality requires all but the omniscient to be uncertain. And

this means that, for all but the risk neural, our equilibrium conditions would be invalidated.

This is because, with uncertainty and risk aversion, equilibrium does not imply the equality

of ex ante rate of returns. Bonds with higher volatility in real returns may have a risk

premium.

Therefore our model lies outside the domain of rational expectations. We believe the

validity of this property should be judged on its empirical performance rather than on a priori

terms.

D. Equilibrium Solutions

Annex 3 demonstrates that the equilibrium value of i, responds to changes in inflation

expectations according to the total differential,

di,, = x sk dnenk (10)

Sk = k’s share in world outpu@.

Equation (10) essentially says that country n’s interest rate varies on a one-to-one basis with

a weighted average of each country’s expectation of inflation in n.7 This is the central result

of the paper.

276 WILLIAM COLEMAN

But the import of equation (10) depends sensitively on how expectations of n’s inflation

are determined.

Suppose that expectations of inflation are formed according to HO; KIenk = lIenn. That is,

each country forms its forecast of a foreign country’s inflation by adopting the foreigners

own forecast. Then equation (10) becomes,

di,, = dlIenn. (11)

When expectations are homogeneous the national rate of interest varies with the national expectation of inflation on a one-to-one basis.

This conclusion must be completely reversed if we suppose expectations are formed

according to ND; lIe,,k = l-k,.

Then,

di, = Csk dlIe,. (12)

Equation (12) says i, is the weighted average of each country’s expectation of its own

national inflation. Since the weights are the shares in world output, this implies that n’s

expectation of its own inflation will only affect its interest rate by its share of world output.

With “non-discriminating” expectations a small country’s expectation of inflation will have very little impact on its interest rate.

Evidently, the relation Fisher hypothesized need not be observed in this model.

What if expectations are Mixed? We have already noted that in this case,

IIenk = IIenn + & - 8,. (13)

Therefore

di, = dIIenn + d @s&k - a,,). (14)

Here we see a determinant of home interest rates in addition to the home expectation of

inflation, d(&& - a,,), which can be interpreted as the difference between the home

country’s expectation of the average level of world inflation and the average expectation of

that average level.

IV. Testing The Model

This section tests the model outlined in Section III against the facts of UK history since 1959.

To recapitulate, the model predicts interest rates will obey, as an approximation, the

following equation:

+ = exogenous real interest rate.

That is, the interest rate in country n is the weighted average of each country’s expectation

of inflation in country n. To test this for UK interest rates we could attempt to directly

The Missing Fisher Effect 277

measure the expectations which other countries have of the UK’s inflation rate. We will not

do this. Rather, we will begin by advancing two hypotheses about each country’s expectation

of the UK’s inflation rate, and then conduct joint tests of the expectations hypotheses and

the model.

A. Testing Under Homogeneous and Non-Discriminating Expectations

The first hypothesis we will consider is HO; country k’s expectation of inflation inn equals

country n’s expectation of inflation in n. We have noted that the model predicts an

(approximately) one-for-one relation between the UK interest rate and the UK’s expectation

of inflation when HO prevails.

The second hypothesis we will consider is ND; that is, country k expects country n to have

exactly the same inflation as it expects to have itself. This would imply the UK interest rate

is a weighted average of all the other countries’ expectation of their own inflation. This

implication would be best tested by gathering the expectation of domestic inflation of every

country in the world. However, over the period we are studying inflation expectations data

is readily available only for the United States. So our test will consist of a regression of the

UK interest rate on UK and US expectations of their own inflation. Under ND such a

regression should yield (i) a coefficient on US expectations equal to the US’s share of output,

and (ii) a coefficient on the UK’s expectation equal to the UK’s share of world output,

assuming the excluded expectations are uncorrelated with the included expectations.

Table 5 summarizes the predicted coefficients in a regression of the UK interest rate on

the UK’s expectation of UK inflation, and the US’s expectation of US inflation, according

to HO and ND. Quite clearly, the results of Table 1 reject the joint hypothesis of the model and homoge-

nous expectations. They also seem to sit nicely with the prediction of the joint hypothesis of

the model and ND that US expectations should appear with a large coeffkient.s

In fact, the regressions suggests US expectations maybe fooa dominant because, according

to ND, the coefficient should equal the US share of world output; much less than near unit

coefficients (under “expert forecasts”) of Table 1. Such coefficients are only consistent with

ND if US expectations are also proxying for other foreign expectations.

Table 5. The Predicted Coefficients of Regressors in the Interest Rate Regression Under Two Expectations

Hypotheses.

CoefJicient

Expectations Hypothesis c kJK neus

HO positive 1 0

ND positive UK’s share in US’s share in

world outmt world outuut

278 WILLIAM COLEMAN

Further, we can extract a testable implication of the joint hypothesis of the model and ND

which appears to be rejected by the data for all of our measures of expected inflation. It will

be recalled that the model was built on this asset equilibrium condition,

i”=ik+ne,-ne, (16)

The interest rate in n equals the interest rate in k plus the expected inflation differential.

Both HO and ND make predictions about the differential; HO says it equals IIenn - IIeu

and ND says it equals zero. In terms of the US and the UK, HO predicts the differential will

equal lIeuk - IIe,, while ND predicts it will equal zero. So one test of the model would be

to regress UK interest rates on US interest rates and IIe,,k - IIe,. Homogeneous expectations

predicts the coefficient on the lIeuk - IIe, term will be one, non-discriminating expectations

predicts it will be zero. The results of such a regression are reported in Table 6.

For Expert Expectations, Static Expectations and Perfect Foresight, the results of Table 6

are inconsistent with both HO and ND; for all three expectational measures the coefftcient

on the differential is positive (which is inconsistent with ND) but less than one (which is

inconsistent with HO). The failure of ND is not surprising: if US investors really did assume UK inflation was

the same as US inflation, then they would have perceived substantially higher real interest

rates in UK securities over the sample period, owing to their substantially higher nominal

interest rates over the sample period. But such a perceived discrepancy in real interest rates

can hardly be consistent with equilibrium. But by the same token, nominal returns on UK

securities over the sample period were not as high as they should have been according to the

orthodox interpretation of the Fisher thesis which implicitly assumes expectations are

homogeneous. And this is reflected in the rejection of HO by Table 6.

However, even these inconsistencies of predictions with the data could be attributable to

failings in HO or ND, rather than the model itself. Is it not possible that expectations are

“mixed” rather than homogeneous or non-discriminating? In that case the model does not

make the predictions of Table 5, and therefore “rejections” of Table 6 are invalid. Regretta-

bly, the joint hypothesis of the model and A4 does not yield any testable implications for the

interest rate differential. To see this, recall, that M implies,

IIenk = IIenn + & - 6,. (17)

Substituting into the asset market equilibrium implies,

(18)

The Missing Fisher Effect 279

Table 6. The Impact of Expected Inflation Differentials on the UK Interest Rate, 1959-88

c ‘us HeuK - &JS R2 SE DW

Levels Expert 2.3 0.89

(3.0) (6.4) static 2.0 0.96

(2.9) (7.6) Perfect 2.0 0.94

(2.8) (7.1) Differences Expert 0.58

(2.3) static 0.65

(2.4) Perfect 0.63

(2.4) Heteroskedastic consistent f-statistics in brackets

0.29 0.81 1.4 1.75

(3.9) 0.25 0.81 1.5 1.77

(3.6) 0.29 0.82 1.4 1.74

(4.2)

0.10 0.26 1.7 2.31

(0.6) 0.05 0.25 1.8 2.33

(0.5) 0.13 0.29 1.7 2.47

(1.2)

We have no direct measures of & or 6,. So we cannot estimate this. The only way to test the theory (without jointly assuming HO or ND) would be to obtain a survey of data on US expectations of UK inflation, something for which there is no direct measure.9

C. Alternative Explanations

The model outlined in Section IV has some success in predicting the dominance of US inflation in explaining UK interest rates. Yet it relies on very strong assumptions: general equilibrium, perfect mobility of goods, and a completely integrated global capital market. The last assumption might seem especially questionable for the period exchange controls in the UK, which extend up to 1979. But what other explanations are there for the success of US inflation variables in the explanation of UK interest rates? One alternative explanation would rely on the equality of nominal rates of return on UK and US securities, but dispense with the equality of real interest rates this model has been built around. lo It could be that US nominal interest rates are the direct and fundamental determinant of UK interest rates, and that US inflation only succeeds in explaining UK interest rates insofar as it proxies for US nominal interest rates. Such an interpretation retains the huge size of the US economy as a part of the explanation of the data, but it dispenses with the elaborate Fisherine general equilibrium approach.

V. Conclusions

This paper has used a very simple, but fairly orthodox, macroeconomic model to argue that it is the expectation by the rest of the world of inflation in a given country which is relevant

280 WILLIAM COLEMAN

for that country’s interest rate, not the expectation prevailing in the country itself. So, for

example, it is the expectation which non-Britons have of British inflation which is the

fundamental determinant of British interest rates, not British expectations.

It is true that, insofar as British expectations are correlated with non-British ones, the UK

interest rate will be correlated with British expectations. But any deviations of British

expectations from those of non-Britons will have very little impact on nominal interest rates.

The conclusion that an increase in expected inflation, which is not shared by the rest of

the world, will have little impact on the home interest rate has important implications for

other questions. Consider all those decisions which are influenced by the real interest rate.

Our model implies that any increase in expected inflation, which is not shared by the rest of

the world, will have a substantially negative impact on the real interest rate the home country

expects. Consequently, the model predicts an increase in expected inflation, not shared by

the rest of the world, will have a positive impact in home consumption, investment and a

negative impact on home net lending.

Empirical scrutiny of this model using UK data since 1959 indicates that American expecta-

tions of US inflation were better predictors of UK interest rates than UK expectations of UK

inflation. This empirical result is surely something of a puzzle and calls for an explanation.

Appendix 1

Inflation Expectations Series Sources

UK interest rate: 1959-1982, “Prime Bill Rate”, Capie and Webber (1985), Table III( 10).

1983-1988, “3 month bill rate”, Annual Abstract of Statistics, (UK Central Statistical

Office), various issues, Table 17.9.

US interest rate: 1959-1988,3 month US Treasury bill rate, Main Economic Indicators:

Historical Statistics (OECD), various issues.

UK inflation: 1959-1988, Retail Price Index, Table 25, Economic Trends: Annual

Supplement, 1990 (UK Central Statistical Office).

US inflation: 1959-1988, Consumer Price Index, Main Economic Indicators: Historical

Statistics, (OECD), various issues.

UK “Expert Forecasts” 1965-1988: The percentage forecast of inflation between the

current and next quarter, derived from the “Personal Income and Expenditure” table in

National Institute Economic Review. Between 1959 and 1964 the NIESR published

forecasts for inflation “over the next four quarters”. Linear interpolation of these have

produced proxies for quarterly forecasts over the period 1959-1964. Our series is reproduced

in Appendix 2.

US “Expert Forecasts” 1959-1988: Livingston Survey report of forecast of six month

inflation in consumer prices. supplied by Federal Reserve Bank of Philadelphia.

The Missing Fisher Effect

Appendix 2

Inflation Expectations Series

National Institute Livingston

1959 0.5265 0.79

1960 1.6875 0.33

1961 1.8125 1.04

1962 1.6875 0.99

1963 2.0 0.94

1964 2.225 1.189

1965 3.72 1.252

1966 3.2 2.33

1967 4.2 2.35

1968 7.2 2.9

1969 3.76 3.38

1970 6.0 3.56

1971 5.2 3.51

1972 4.8 3.41

1973 11.2 4.59

1974 16.4 7.41

1975 18.4 5.73

1976 15.4 5.26

1977 10.0 5.96

1978 10.8 6.69

1979 16.3 9.22

1980 12.8 10.59

1981 13.6 7.91

1982 6.0 4.99

1983 5.6 4.9

1984 5.6 4.68

1985 4.08 3.97

1986 3.62 3.16

1987 5.20 3.89

1988 5.20 4.75

Appendix 3

The Model

The N national excess demands for goods are:

z( Yk( 1 + &(ik - rk,)) + x&-I/Pk + kfk/Pk) - Yk. W)

281

282 WILLIAM COLEMAN

The N national excess demands for currencies are:

h(ik)( Yk( 1 + Rk(ik - I&TM)) + XBkl/Pk + MJPk) - Mk/Pk. (24

For analytic convenience we suppose the existence of a World Central Bank. When

exchange rates float the World Central Bank’s demand for each currency is zero. When

exchange rates are fixed the Bank’s excess demand for each money is whatever the national

excess supply of money is. Because of this accommodating nature, the Bank’s excess money

demands under fixed exchange rates do not have an independent functional form. But they

must still obey the Bank’s budget constraint. Assuming the World Central Bank does not

deal in bonds the World Central Bank’s budget constraint is,

c XcBMk -=t)

Pk

(All summations are from k = 1 to k = N).

Having specified the excess demands, we now turn to the conditions for equilibrium.

In equilibrium the aggregate excess demand for goods is zero.

I: Yk=zc(Yk(l +&)+xBk-,/Pk+b’fk/Pk)

= Z .c( Yk (1 + &) + lt’fk/Pk)

(3A)

(4A)

(The XBpI’s cancel out because all debts are inside debts.)

In equilibrium the aggregate demand for each money is zero. So, for k = 1 to N,

i&/Pk = h(ik) (Yk( 1 + Rk) + XBk_, /Pk-1 + Mk/Pk) + XcBMk/Pk (54

The final equilibrium condition is the equality of the anticipated real rates of return on

different bonds. That is, for k = 1 to N,

i,, = ik + n&k - &?fi. (W

(Our results will be exactly the same if we use the alternative condition for asset equilibrium:

Ill ’ = ik + Penn - ne&

Finally, for the sake of analytical convenience, we will assume that at the initial equilib-

rium, (i) all countries expect their own rate of inflation to be x (i.e., IIee = x for all k), and

(ii) all countries expect inflation abroad to be x (i.e., n&k = n for all 12 and k). These assumed

equalities are not functional equalities; they are just assumed to be satisfied at the initial

equilibrium. Some assumption is needed about the values of expectations at the initial

equilibrium, and this assumption is (i) simple, and (ii) consistent with HO, ND and M. It

will be noticed that in conjunction with equation (6a) these assumptions imply in = ik for all

n and k at the initial equilibrium.

To derive the interest rate comparative-statics we first rewrite the N money market

equilibrium conditions in the following manner;

The Missing Fisher Effect 283

Mk/Pk = h(ik)( Yk( 1 + Rk) + x&/p,) + xcBik&/Pk

1 - h(ik)

Summing these N money market conditions yields,

c Mk,pk = c h(ik)Yk(l + Rk) . 1 - h(ik)

(74

@A)

Notice that the summation has eliminated theXCBMk terms. The sum of the World Central Bank’s money demands is obviously zero in the floating case. But it is also necessarily equal to zero in the fixed exchange rate case owing to the Bank’s budget constraint. In both the fixed and floating exchange rate cases summation also eliminates the XBk_1 terms because all debts are “inside” debts.

Substituting equation (8A) into the goods market equilibrium condition, equation (4A), yields,

z Yk=t): yk(l + Rk) (9A) 1 - h(ik) ’

Eliminating all the iks from equation (9A) using equation (6A) yields,

c yk = a tyk + y,Rtin - n&k>) 1 - h(in - ne& + neti) ’

(lOA)

Equation (1OA) is an equation in one variable, i,,, and implicitly determines i,,. Notice that this equation describes the determination of the interest rate in both floating and fixed exchange rates. The equation has been derived only with elements common to both types of economy, and no assumptions have been made about the specification of the World Central Bank’s demand for money, which is what distinguishes the fixed from floating arrangement. Neither has any assumption been made about what expectation hypotheses are held.

Equation (1OA) plainly implies i,, does depend on expectations of inflation. To calculate the dependence of i,, on expectations of inflation we calculate the total differential of equation (lOA), yielding,

di,,=~Skdk’,&-x.sk (1 + R)h’

(1 -h)R’+(l+R)h’dneM. (1 14

where Sk is the kth country’s share of world output, and R’ is the derivative of R with respect to the current ex ante real interest rate.

To pursue the implications of equation (11A) we will consider these two hypotheses about I&?,&: HO and N. Under HO (1 IA) becomes,

di,, = dllenn - z Sk (1 +R)h’ me

(1 -h)R’+(l +R)h’ &’ (124

Under ND equation (12A) becomes,

284

din = c Sk (1 -h)R’+(l +R)h’ kk.

WILLIAM COLEMAN

If one assumes h’ = 0, then i, varies on a one-to-one basis with ne,,,, under HO. But under

ND Ilen” will only impact on i, by n’s share of world output.

Acknowledgments

I would like to express my thanks for the helpful criticism of two anonymous referees, Ian Harper and the participants of the University of Melbourne Economics Department Seminar.

Notes

1. In the extreme case of horizontal LM curve all of the impact of an increase in expected inflation will be felt on the real rate of interest and none on the nominal rate. In a similar vein, Carmichael and Stebbing (1983) have pointed out that if the implicit rate of return on real balances is exogenous, the nominal rate of interest is determined prior to expected inflation. An increase in expected inflation would not increase the nominal interest rate; it would only necessitate an equal fall in the ex ante real interest rate.

2. The interest rate series are annual averages of monthly observations of three monthly interest rates. The “static expectation” series is an annual average of quarterly observations of consumer price inflation. The “perfect foresight” series is au annual average of quarterly observations of consumer price inflation, shifted ahead one quarter. The “Expert” series for the UK are annual averages of quarterly forecasts of quarterly inflation. The Livingston series does not provide forecasts of quarterly inflation. It provides only biannual forecasts of inflation six months ahead. We use an annual average of this. This average is only an imperfect proxy for forecasts of inflation over a three month horizon. However, this imperfect measurement does not prevent US forecasts dominating UK forecasts.

3. Table 2 does not compare quarterly forecasts: the Livingston series does not include quarterly forecasts. Table 2 compares the accuracy of forecasts of inflation over the next 12 months.

4. The Carlson-Parkin data is an annual average of monthly observations of expected inflation over the next six months. The Papadia data is an annual average of thrice yearly observations of expected inflation over the next 12 months.

5. What if two countries did not share the same expectation of the inflation differential? Then an equilibrium would not exist in this model. Consider the two possibilities. Possibility I: Both countries agree on which currency is offering the higher real rate of return. Then both would want to do all their lending in the currency with the higher rate and all their borrowing in that currency with the lower rate. Possibility 2: The two countries disagree on which currency offers the higher rate. Then both countries will perceive an indefinitely large arbitrage gain in borrowing in one currency from the other country and lending in the second currency to the other country. Since there is no limit to these perceived arbitrage gains equilibrium would seem unattainable. The possibility that inconsistent expectations could be au obstacle to equilibrium was noted long ago by Hicks. ‘There is no reason why expectations of different individuals should be consistent; one of the main causes of disequilibrium in the economic system is the lack of consistency in expectations and plans”. Hicks (1939, p. 177).

6. Since income expectations are static these shares of world output will also conform closely to shares of world wealth.

The Missing Fisher Effect 285

7. Equation (11) is actually an approximation of the exact result, secured by neglecting the expectation terms in the money demand function. Complications in the analysis of the impact of expectations on interest rates due to money demand effects were first noticed by Mundell(1963), but have usually been ignored on the grounds of small quantitative significance.

8. The only minor relevance of UK expectations for US interest rates which we noted in Section I also sits well with the joint hypothesis of the model and ND. The UK being “small”, UK expectations should only have a small impact on US interest rates.

9. However there is one proxy for US expectations of UK inflation. Recall the asset equilibrium condition equation (5). We can arrange this to read,

Consequently we can derive a measure of IIenk from in, k i and our proxies for lieu. In terms of the

US and the UK, our new measure of the US’s expectation of inflation in the UK equals the US’s expectation of inflation in the US plus the excess of the UK’s interest rate over that of the US. According to HO this will equal the British forecast; according to ND it will equal the American forecast. As it happens, it is rather better correlated with the US forecast of inflation than with the British forecast; 0.79 compared with 0.50.

10. In this context it is worth noting that according to the Bank of International Settlements Annual Report of 199 1 issuance by UK private sector entities in the international securities markets constituted 84.5 percent of total UK private securities outstanding in domestic and international markets.

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