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"A Comparative Analysis of Expected Inflation
Estimates and the Resulting Implication's for
Financial Assets Returns"

presented by

Robert T. Kleiman

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A COMPARATIVE ANALYSIS OF EXPECTED INFLATION
ESTIMATES AND THE RESULTING IMPLICATIONS

FOR FINANCIAL ASSET RETURNS

BY

Robert T. Kleiman

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of

DOCTOR OF PHILOSOPHY

Department of Finance and Insurance

1986

HJW'QVbb

ABSTRACT
A COMPARATIVE ANALYSIS OF EXPECTED INFLATION
ESTIMATES AND THE RESULTING IMPLICATIONS
FOR FINANCIAL ASSET RETURNS
by
Robert T. Kleiman

The purpose of this study is to re-examine the
relationship between nominal asset returns and
inflation. Since previous empirical findings may be a
function of the proxy employed for expected inflation,
this study provides a comprehensive analysis of several
alternative proxies for expected inflation. In addition,
this analysis encompasses the post—1971 period of high
and volatile inflation. Furthermore, this study examines
the relationship between inflation and the returns on
some previously unstudied asset categories.

The empirical findings of this analysis indicate that
none of the inflation proxies fully satisfies
rationality. Of the proxies considered, the Survey
Research Center measure has the smallest dispersion of
forecast errors.

The findings of this study indicate that financial
assets are poor hedges against expected inflation, which

is contrary to the Generalized Fisher Hypothesis. The

results also indicate that the returns on both debt and
equity securities are negatively related to unexpected
inflation and the changes in inflationary expectations.
Although the results are generally consistent for
different inflation proxies, the magnitude of the
relationship is, in some cases. dependent on the
inflation measure that is employed.

This analysis also finds that the Geske-Roll reversed
causality model holds for long-term debt securities and
"small" stocks as well as for common stocks.
Furthermore, this study provides an alternative test of
the Darby-Feldstein Hypothesis by comparing the
coefficient for expected inflation for non-taxable and
taxable debt instruments. Although the results do not
support the model in the short-run. the long-run results

are consistent with the Darby-Feldstein Hypothesis.

CHAPTER ONE

INTRODUCTION

In the past decade. the United States has experienced
high and volatile rates of inflation. With the
variability in the general price level has come an
appreciation of the importance of inflationary
expectations in nominal interest rate determination and
an increased interest in the relationship between
inflation and the returns on various financial assets.

The Generalized Fisher Hypothesis expresses the
nominal return on a financial asset as the sum of the
expected real return on the asset and the expected
inflation rate over the life of the instrument. Although
asset payoffs are in nominal terms, rational investors
are concerned with the real value of their wealth
measured in terms of final goods and services.
Therefore. investors should seek investments that are
efficient in real terms.

The seminal study of the association between
inflation and financial asset returns was undertaken by
Fama and Schwert (1977). Fama and Schwert's analysis.
which encompassed the 1953-1971 period, utilized the
nominal Treasury Bill rate minus a constant for the real
interest rate as an estimate of the expected inflation

1

2
rate. The years 1953-1971 are unique in American history
for their record of price stability. Subsequent studies,
such as Hess and Bicksler. have indicated that the
expected real return is not constant during periods of
high inflation. Consequently. Fama and Schwert's
methodology utilizing short-term interest rates as
predictors of inflation may not be accurate in periods of
rapid inflation. In turn. this leads to the possibility
that Fama and Schwert's findings were period dependent
and not representative of the post-1971 period.

In another comprehensive study. Huizinga and Mishkin
(1984) analyzed the monthly data from 1959 to 1981 on the
real returns for seven financial assets. They found that
the ex—ante real rates on all seven assets were
negatively correlated with actual inflation rates. and
longer maturity assets were the worst inflation hedges.
However. Huizinga and Mishkin's study utilized actual
ex-post data for inflation. The relationship between
asset returns and inflation is more properly investigated
using expectations data.

This thesis re-examines the relationship between the
returns on alternative asset categories and inflation.
Since Gultekin (1983) suggested that the asset return-
inflation relationship was dependent on the proxy used to
measure inflation. it is instructive to consider other

inflation proxies than those employed by Fama and Schwert

3
and Huizinga and Mishkin. This thesis provides a
comprehensive analysis of several alternative measures of
expected inflation which have been identified in the
literature. This should provide insight regarding the
consistency of results for different inflation proxies.

This disSertation also employs a single comprehensive
period. 1959-1983. which encompasses periods of both
relatively high and low inflation. This will provide
evidence as to whether Fama and Schwert's findings were
period dependent. In addition. this study investigates
the relationship between inflation and the returns for
previously unstudied asset categories.

Chapter Two traces the historical development of the
relationship between nominal interest rates and expected
inflation and cites some associated empirical studies.

In addition to the Fisher Effect, three alternative
models of nominal interest rate determination are
considered--the Mundell-Tobin Effect. the Darby-Feldstein
Effect, and the Inverted Fisher Effect. The
Mundell-Tobin Effect argues that the relationship between
nominal interest rates and inflation is less than unity
as a result of a decline in the real interest rate. 0n
the other hand. the Darby-Feldstein Effect contends that
the taxation of interest income (and/or tax deductibility
of interest payments) implies a more than complete

adjustment of nominal interest rates to anticipated

4
inflation. The Inverted Fisher Effect suggests that
nominal interest rates are constant and that real
interest rates move inversely with the expected inflation
rate.

Chapter Three reviews several alternative measures of
expected inflation which have been utilized in previous
empirical work. These include:

1. distributed lag model

2. extrapolative model

3. adaptive model

4. monetary information model

5. Livingston survey data

6 Survey Research Center data

7. nominal T-Bill rate model

8. time series (ARIMA) model

The distributed lag model argues that price
expectations are formed autoregressively. The
extrapolative model expresses the expected inflation rate
as a linear combination of last period's actual inflation
rate and a term reflecting the change in the actual
inflation rate. The adaptive model expresses the
expected inflation rate as a linear combination of the
previous period's expected inflation rate and the
previous period's actual inflation rate. The monetary
information model hypothesizes that inflationary

expectations reflects past rates of change in the money

5
supply in addition to past rates of inflation. The
Livingston survey data represent the consensus inflation
forecasts of academic and professional economists. The
Survey Research Center data are an alternative price
expectations series which represent the inflation
forecasts of consumers. The T-Bill rate model contends
that the nominal T-Bill rate minus a constant reflecting
the real interest rate is the best predictor of future
inflation. The ARIMA model uses Box—Jenkins univariate
time series techniques to construct a model to forecast
the expected inflation rate.

Chapter Four considers the forecasting accuracy of
the alternative inflation proxies. Each measure is
statistically evaluated in terms of Muthian rationality
which holds that expectations about future inflation are
formed in a manner that fully incorporates all currently
available and relevant information. Following the
methodology suggested by Mullineaux (1977). this study
tests the unbiasedness and efficiency of the alternative
inflation measures. In addition. this chapter also
considers three alternative measures of forecast
dispersion--mean square error. mean absolute error. and
Theil's U coefficient.

Chapter Five presents a literature review of the
inflation hedging capability of alternative asset

categories. The first section of the chapter defines the

6
concept of an inflation hedge. The next section presents
the major theoretical arguments regarding the inflation
hedging capability of several assets. For equity
securities. four alternative theories are considered--the
Classical Hypothesis. the Price-Cost Sensitivity
Hypothesis. the Tax Effects Hypothesis. and the External
Financing Hypothesis. All but the Classical Hypothesis
suggest that the returns on equities should be negatively
impacted by an increase in inflation. This section also
indicates that the returns on fixed—income securities
should be negatively related to unanticipated inflation.
The final section of Chapter Five reviews previous
empirical studies of the relationship between inflation
and the returns on alternative investments.

Chapter Six provides an empirical investigation of
the relationship between inflation and asset returns.
This chapter examines the extent to which a variety of
assets were hedges against anticipated and unanticipated
inflation for the 1959-1983 period. In addition to the
financial assets studied by Fama and Schwert. this
analysis includes municipal bonds. "small" common stocks.
preferred stocks, and commodities. Chapter Six also
tests the Geske—Roll reverse causality model which argues
that the nominal return on an asset signals a change in
inflationary expectations of the opposite sign. In

addition. this chapter provides an alternative test of

7
the Darby-Feldstein Hypothesis. This is accomplished by
comparing the coefficients of inflation expectations for
regressions involving nominal yields on taxable and
tax-exempt debt instruments.

Finally. Chapter Seven summarizes the results of this
study and makes some concluding remarks. In addition,
this chapter presents a discussion of C.P.I. inflation
futures. which should enable investors to lock in real

rates of return.

CHAPTER TWO

INFLATION. INTEREST RATES. AND THE FISHER HYPOTHESIS

Irving Fisher's Theory of Interest (1930) has proven
to be a durable and influential contribution to financial
theory. A central element of Fisher's contribution is
the "Fisher effect" which holds that the nominal rate of
interest is equal to the sum of the real interest rate
and the expected rate of inflation. A great amount of
literature. both theoretical and empirical. has developed
around the Fisherian analysis. This chapter provides a
review of the significant studies following Fisher's
pioneering work. The first section provides a discussion
of the real rate of interest. Next. nominal interest
rates are discussed. The third section covers the Fisher
Effect. Next. theoretical issues concerning the
relationship between inflation and nominal interest rates
are summarized. The final section of this chapter
reviews previous empirical findings concerning this

relationship.

I. THE REAL RATE OF INTEREST

The real rate of interest is the equilibrium rate at
which claims to future income are traded for current
income. The real interest rate thus reflects the premium
that individuals place on providing themselves with

9
present consumption goods (or income) versus future
consumption goods (or income). Thus, it may be regarded
as the expected positive return in real purchasing power
that is sufficient to induce potential lenders to part
with their funds. thereby deferring their own spending.
The real rate is expressed without reference to prices
and represents the real growth in goods and services.

Investors expect the real rate of interest to be
positive because: (1) resources have productive uses
that should increase their value over time and (2)
consumers have a positive time preference and thus must
be paid to forego the present use of their resources.

The real interest rate is the equilibrium rate which
equates the supply and demand for capital. The supply
function depends on willingness of economic units to
save, that is. to postpone consumption whereas the demand
function depends on the opportunities for productive
investment.

Each firm in the economy faces a downward sloping
menu of investment opportunities which is depicted in the
investment opportunity schedule. As the firm invests
more. it exhausts opportunities for high rates of return
and is forced to accept projects with gradually
diminishing returns. Eventually, it will discontinue

investing when there are no more projects offering rates

10
of return greater than the opportunity cost of capital.
The total demand for capital in the economy is the
aggregation of all firms' investment opportunity
schedules.

The total supply of capital depends on the amount
economic units are willing to save at various interest
rates. The supply of capital is an increasing function
of the interest rate. The equilibrium real interest
rate. r*, is achieved when savings (the supply of
capital) equals investment (the demand for capital) for
the economy as a whole. This occurs at the point of
intersection of the investment and savings curves in
Figure 2.1.

The real interest rate changes when one or both of
these curves change. Hence, movements in the real rate
of interest result either from changes in the underlying
preferences for consumption goods vis-a-vis capital goods
(which are the sources of future consumption goods) or

from an increase in the productivity of capital.

II. NOMINAL INTEREST RAT§§

Because interest and principal payments are expressed
in monetary terms and because the monetary standard
changes over time. the real rate of return on a financial
asset can differ from its nominal return. The expected

return from a security is the discount rate which equates

11

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[INERNWflUIJWOFTHEFEfiLBHERESTRAHE

Demand tor capital
(amount all tirms
are willing to invest)

   
     
 
 
 
  
  

Supply of capital
(amount all individuals
are willing to save)

Equilibrium interest rate

 

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_ investment
Equilibrium all firms
Investment

 

The expected real rate of interest, r*, is the rate
at which the amount that savers are willing to lend

is exactly equal to the amount that-investors find
worthwhile to borrow.

12
the present value of the stream of expected cash inflows
with the purchase price of the asset. For a fixed-income
security. the cash inflows consist of interest payments
and the sales price. Likewise. for an equity security,
the inflows include dividends and the sales price. The
expected return from holding these financial assets is
expressed in nominal terms. that is. the cash inflows are
denominated in current dollars at the time of receipt.
Thus. the interest rates observed in financial markets
are nominal returns. The real rate of return may be
viewed as the return realized when the future cash flows
are placed on the same purchasing power basis as the
security's initial purchase price.

Nominal interest rates represent the premium that a
borrower must pay in a financial transaction involving-
the exchange of dollars now in return for a promise to
repay dollars at some future date. Consequently. the
parties involved in financial contracts are interested in
the expected future value of goods and services that can
be purchased when the loan is repaid. Because the (real)
value of money varies inversely with movements in the
general price level. the expected inflation rate over the
period of the financial contract is one component of the_
nominal interest rate.

When inflation is expected to accelerate. lenders

expect the real value of the principal and interest

13
payments to depreciate and borrowers expect to repay
loans with money having less real value. Thus. the
impact of an increase in inflationary expectations is to
increase the quantity of loans demanded and to decrease
the quantity of loans supplied. both of which increase

equilibrium nominal interest rates.

III. THE FISHER EFFECT

Fisher (1930) expressed the nominal rate of interest
on a security as the sum of the expected real return on
the security and the expected rate of inflation over the
life of the financial instrument. More formally. we can
express the nominal rate. 1. as:

(2.1) 1 + i = (1 + r)(1 + n) or

i = r + n + rn
where: r is the expected real rate of return and
n is the annual rate of inflation expected to
prevail over the life of the security.

When inflation is moderate. the cross product term. rn.
is small and therefore can be ignored in the
formulation. Thus. we have:

(2.2) i = r + n

Traditionally. this formulation is known as the
"Fisher Effect."1 Essentially. it states that the

difference between the nominal rate of interest and the

1‘
expected real rate of interest is the expected rate of
inflation.

As the expected rate of inflation changes. the spread
between the nominal interest rate and the expected real
rate changes by the same amount. The change in the
spread might result from two different events. At one
extreme. the increase in expected inflation might result
in an identical increase in the nominal interest rate
with no change in the expected real rate. At the other
extreme. the expected real interest rate might decrease
by the amount of the increase in expected inflation. with
no change in the nominal interest rate.

Fisher's theory emphasized the case where the real
rate remains unchanged with an increase in inflation
(i.e.. %% s 0). He believed that the real and monetary
sectors of the economy were largely independent. Fisher
argued that the expected real rate was determined by real
factors such as productivity of capital. investor time
preferences. and tastes for risk. Moreover. he argued
that inflation had no lasting effect on real magnitudes.
Thus. the only long run effect of an increase in expected
inflation was to increase the nominal interest rate
leaving the real rate unchanged.

Assuming a constant real rate. the Fisher effect
suggests that the relationship of changes in nominal

interest rates to changes in expected inflation is unity.

15
(i.e.. %% e 1). Holding risk constant. economic units
would be indifferent between investing in real assets or
financial assets since financial markets equilibrate in
terms of expected real rates of return. In other words
both real and financial assets would provide the same

expected return after adjusting for inflation.

IV. NOMINAL INTEREST RATES AND INFLATION:

THEORETICAL ISSUES

The question of whether the relationship between
changes in nominal interest rates and changes in expected
inflation is one—to-one is a subject of considerable
controversy. On a theoretical level. there are arguments
justifying less than a one-to-one as well as more than a
one-to-one relationship. Mundell (1963) presented a
theory where changes in the anticipated rate of inflation
increase or decrease the nominal interest rate by less
than the change in the expected inflation (i.e..«%% <
1). In the Mundell model. the expected real return is
inversely related to expected inflation. Thus. in the
case of an increase in expected inflation. there is both
an increase in the nominal rate of interest and a
decrease in the real rate.2 Fluctuations in the rate
of inflation affect real economic variables and not only
the nominal rate of interest.

Mundell contends that real interest rates fluctuate

over time because of portfolio adjustments that accompany

16
changes in expected inflation. An increase in the price
level reduces the value of real money balances since
money depreciates in real terms. Thus. expected
inflation raises the opportunity cost of holding money.
This factor. combined with the resulting decline in
wealth. stimulates increased savings. In turn the
increased supply of savings depresses the real return on
financial assets (i.e.. %% < 0). Finallyu the decline in
the real rate of interest stimulates an increase in
capital expenditures and real activity.3

Tobin (1965) obtained a result similar to
Mundell--the response of nominal interest rates to
anticipated inflation is only partial. In Tobin's model
the partial response is a result of an inflation induced
decrease in the demand for real money balances and an
increase in capital intensity as wealth owners shift
funds from money balances to capital.

Steindl (1973). however. contends that the
Mundell-Tobin hypothesis. is valid only if the reduced
demand for real money balances that results from an
increase in expected inflation is in turn reflected in an
increased real demand for bonds. A priori. it is not
apparent that the decreased real demand for money will be
reflected primarily in the bond market as opposed to the
commodities market. Therefore. Steindl argues that it is

not possible to predict the exact impact of changes in

17
the expected rate of inflation on the nominal interest
rate.

On the other hand. Darby (1975) and Feldstein (1976)
posited a model whereby the response of the nominal rate
of interest to the change in expected inflation would be
greater than one-to-one. Their principal argument for
the response being greater than unity has to do with tax
effects.

A significant contribution to the Fisher equation
literature was made by the introduction of taxation on
interest income by Darby (1975). He argued that
investors base their decisions not on expected real
returns. but on expected real after tax returns:

(2.3) r = 1(1 - t) — r
where: t is the proportional personal income tax rate
and the other variables are as previously defined.
Therefore. the nominal interest rate is given by:

(2.4) 1 .H
Thus. if the marginal tax rate is positive. the nominal
interest rate must increase by more than the increase in
expected inflation in order for the after tax real return
to remain unchanged (i.e.. %%. > 1).

Feldstein (l976) arrived at a result similar.to
Darby's. The Feldstein effect differs from the Darby

effect since it uses the corporate tax rate as compared

to Darby's use of the personal income tax rates. and it

18
arises in the context of allowing for the tax
deductibility of interest payments in calculating the
real cost of capital. However. the implications of the
two models are identical-~a greater than unity adjustment
of nominal interest rates to expected inflation.

Darby and Feldstein's models were modified by Nielsen
(1981) and Gandolfi (1982). These authors introduced the
taxation of capital gains. and found that the response of
nominal interest rates to anticipated inflation should be
greater than the complete response suggested by Fisher.
but less than the result suggested by Darby and
Feldstein.‘

(2.5) 1 < 31 5 1

“5'1? l-t
The introduction of a capital gains tax by Nielsen and

 

Grandolfi increases the real cost of acquiring capital
and thus reduces the magnitude of 3% below Té'T
Carmichael and Stebbing (1983) suggest that the
Fisher Effect was intended to apply to returns on capital
rather than financial assets. Provided a minimum degree
of regulation of interest rates (that being the
non-payment of interest on money balances) and providing
a relatively high degree of substitutability between
money and financial assets. the Fisher hypothesis as
applied to financial assets may give inverted results.

In this model. referred to as the Inverted Fisher Effect.

real interest rates move inversely with inflationary

19
expectations and nominal interest rates are constant

(i.e.. %%»= -1 and %% s 0). Thus. interest rates on
financial assets behave in the exact opposite manner to

that predicted by Fisher's hypothesis.

V. PREVIOUS EMPIRICAL FINDINGS

Despite the problems involved in empirical testing.
there have been numerous direct and indirect tests of the
Fisher Hypothesis. These studies point to a positive
relationship between changes in inflation and changes in
nominal interest rates during the post-World War II
period. However. a number of econometric studies
indicate that this relationship is less than one-to-one.
In addition. the evidence of the 19608 and 19703 suggests
that lenders' after-tax real returns decline with
unanticipated increases in the inflation rate.. Based on
the empirical studies. it is possible that the real rate
of interest is not constant. Furthermore. it is also
possible that the statistical association between
interest rates and expected inflation might reflect other
factors affecting both inflation and interest rates.
Hence. while expected inflation is of primary importance
in explaining the movement of nominal interest rates. the
relationship may not be as direct as implied by the
Fisher effect. A review of some of the representative

tests of the Fisher Hypothesis follows:

20

The standard Fisher equation may be rearranged to
obtain

(2'5) “t t it - rt
Although investors can not predict actual inflation
perfectly. we would expect the average forecasting error
to be zero in an efficient market. If. on average.
inflation estimates by market participants are realized.
the actual rate of inflation (Pt) could be used as an
estimate of the expected rate of inflation (fit).

Fama (1975) fitted the following equation to data on
U.S. Treasury Bills for the 1953-1971 period:

<2-7) P. = a + b(it) + e.
If the Fisher equation is correct. the coefficient b
should be close to 1.0 and the constant term a should be
equal to minus the real interest rate. Fama found b =
.98. consistent with the hypothesis that changes in
short-term interest rates are entirely due to changes in
inflationary expectations. In addition. the results
supported a constant real interest rate. Yohe and
Karnosky (l969). Feldstein and Eckstein (1970). and
Gibson (1972) also found evidence of complete adjustment
of nominal interest rates to expected inflation.
However. these studies have been criticized for being
period specific since they are all confined to the

1953-1971 sample period.

21

On the other hand. a number of empirical
investigations. beginning with Fisher himself. have found
the relationship less than one-to-one. Although Fisher's
theoretical analysis predicted a complete adjustment of
nominal interest rates to expected inflation. his
empirical results indicated only a partial adjustment of
nominal interest rates to expected inflation. Likewise.
using one hundred. twenty years of data. Summers (1983)
found evidence of less than complete adjustment. For
every subperiod of the 1860-1979 sample period. the
coefficient for expected inflation was consistently below
unity. Fisher and Summers rationalized their empirical
findings as reflecting "money illusion" on the part of
money market participants. that is. market participants
confused the distinction between nominal and real
interest rates.

A number of other studies. such as Tanzi (1980) and
Friedman (1982) also found evidence for only partial
adjustment. Although "money illusion" may be a possible
explanation. these results also suggest that additional
macro-economic factors. such as the level of economic
activity or the stage of the business cycle. play a
significant (although secondary) role in the
determination of nominal interest rates.

Evidence on the Darby-Feldstein hypothesis of more

than complete adjustment has been mixed. Several

22
studies. including Carr. Pesando. and Smith (1976).
Cargill (1977). and Tanzi (1980) did not find compelling
evidence for the hypothesis. However. Cargill and Meyer
(1980) and Peek (1982) found support for more than
complete adjustment of nominal interest rates to expected
inflation.

There have also been a number of studies
investigating the notion of a constant real rate of
interest. Hafer and Hein (1982) rejected the hypothesis
that the expected real rate on short term investments was
constant over the 1955-79 period. Likewise. Hess and
Bicksler (1975). Carlson (1977). and Nelson and Schwert
(1977) found the real rate of interest to vary over
time. Nelson and Schwert and Hess and Bicksler pointed
out that the real interest was unusually stable during
the period examined by Fama (1975). and hence the results
may be period dependent. Fama and Gibbons (1982)
discovered that the expected real rate varied inversely
with the expected inflation rate. Rather than attribute
the results to the Mundell-Tobin Hypothesis. they
concluded that real returns varied with measures of real
activity. On the other hand. Santoni and Stone (1981)
utilized real as well as financial variables as proxies
for the real rate and concluded that the real rate was

constant over the 1954-80 period.

23
FOOTNOTES
1It should be noted that several 18th and 19th
century economists--including Douglas. Thornton. and
Marshall--expressed the proposition that equilibrium
nominal rate adjustments entail no real effects.
However. Fisher gave this concept its classic exposition.
21n the case of a decrease in the expected
inflation. the opposite occurs. In this instance. the
real rate rises and the nominal rate falls by less than
the change in inflation.
3Fama and Gibbons (1982) found the relationship
between the expected real return and measures of real
activity to be positively rather than negatively related.
‘Gandolfi also suggested that a relationship close
to unity could occur even in the presence of taxes. This
relationship would occur if the capital gains and‘
ordinary income tax rates were equal and if the
elasticity of investment with respect to after tax real
rates was significantly greater than the elasticity of
savings.
5rne idea that the nominal yield on financial

assets is constant and thus does not respond to inflation

dates back to Keynes (1936).

CHAPTER THREE

THE MEASUREMENT OF EXPECTED INFLATION

This chapter provides a discussion of the measurement
of changes in the general price level. The first section
distinguishes between the actual rate of inflation and
the expected rate of inflation. The remainder of the
chapter provides a discussion of several alternative
proxies for estimating expected inflation which have been
employed in previous empirical investigations of the

Fisher Effect.

1. ACTUAL~g§RSUS EXPECTED INFLATION

Inflation occurs when there is a general rise in the
price level of goods and services over a period of time.
This movement in the price level is typically measured by
changes in various indices such as the Consumers Price
Index. the G.N.P. Implicit Price Deflator. or the
Producer Price Index and represents the "actual" rate of
inflation.

As with most empirical studies. the Consumer Price
Index (C.P.I.) is used to measure the actual inflation
rate in this analysis.1 The C.P.I.. which is computed
monthly by the U.S. Bureau of Labor Statistics. is the
nominal or money value of a "market basket" of
consumption goods and services where the weights given to

24

25
individual items in the bundle are based on the
proportions of family budgets allocated to these items by
a large sample of urban wage earners.

Although the observed relationship between interest
rates and actual inflation is interesting. it cannot be
utilized to test the effect of expected inflation. The
nominal interest rate in the Fisher Hypothesis is agreed
to by borrowers and lenders based on ex-ante inflationary
expectations rather than the ex-post realized rates of
inflation. Since financial market participants do not
have perfect foresight. actual inflation is not an
appropriate measure of expected inflation.

Expected (or anticipated) inflation is that price
level change currently recognized by financial market
participants and embodied in expected asset returns. The
anticipated rate of inflation is defined in terms of the
expected annual rate of change in one of the
aforementioned price indices. If the realized rate of
inflation that actually occurs over the life of the
instrument is exactly that which was anticipated. neither
borrowers nor lenders gain (or lose) with respect to
inflation. On the other hand. unanticipated (or

unexpected) inflation represents a change in the rate of

26
expected inflation that was unanticipated in advance by
market participants. and is measured by the difference
between actual and expected inflation. When
unanticipated inflation occurs. borrowers gain and
lenders lose since the unanticipated inflation has not

been reflected in expected asset returns.

II. PROXIES FOR EXPECTED INFLATION
A problem in empirical testing of the Fisher
hypothesis is that the expected rate of inflation is not
directly observable and hence must be estimated.
Although inflation expectations are invariably regarded
as the fundamental determinants of nominal interest
rates. no consensus exists regarding the appropriate
determination of expected inflation. This analysis will
examine several alternative methods of deriving expected
inflation estimates which have been utilized in previous
empirical studies. These proxies include:
1. Treasury bill rates
2. Survey estimates
a. Livingston data
b. Survey Research Center data
3. Expectations measures
a. distributed lag
b. adaptive

c. extrapolative

27
d. monetary information

4. Time series (ARIMA) model
In order to permit a systematic comparison of the
proxies. each model will utilize a six-month (bi-yearly)
time horizon.

A. Treasury Bill Rates

One proxy often used for measuring the expected
inflation rate is the nominal interest rate on U.S.
Treasury Bills (minus a constant) under the assumption
that the real interest rate is constant. This measure.
which was suggested by Fama (1975). holds that short-term
interest rates fully reflect expectations of changes in
the future price level.2 Fama found that short-term
interest rates contained significant information beyond
that embodied in past inflation rates. and hence
concluded that nominal interest rates (on T-Bills) were
the single best predictor of the future inflation rate.
This proxy has the advantage of being’based on observed
economic behavior and of capturing the latest information
available in the financial markets. This model may be
represented as follows:

(3-1) "t a 1t - r
where: “t is the expected inflation rate for a

six-month period beginning at time t;

1t is the nominal interest rate on six-month

Treasury bills at time t: and

28
r is the real rate of interest which is assumed
to be constant.

p. Survey Estimates

Another method of measuring expected inflation is to
use direct inflation estimates from survey data. The
direct survey estimates most employed in empirical work
are the Livingston data and the University of Michigan
Survey Research Center data.3

1. Livingston Data

Twice a year since 1947. Joseph Livingston. a
financial columnist for the Philadelphia Inquirer. has
compiled the inflation forecasts of approximately sixty
business. government. and academic economists. These
forecasts have the advantage of incorporating additional
information beyond that contained in past inflation
rates. Also. the economists selected tend to have a
fairly significant influence in the financial markets.
However. the sample surveyed is small and specialized so
generalization to all market participants is difficult.

In order to accurately utilize the results of the
Livingston surveys. researchers must be careful in
specifying (1) the time horizon over which the
forecasters are predicting the C.P.I. and (2) the most
recent information available on the actual level of the

C.P.I. when the forecast is made. In each survey.

29
Livingston requests forecasts of the level of the C.P.I.
for the following six and twelve months.‘ Livingston
mails the survey participants the questionnaire one month
prior to the publication of the survey data. The
questionnaire includes the most current data available on
the C.P.I.. which is the value two months prior to
publication. Thus. the actual forecasting horizon is
eight rather than six months.

Livingston also sometimes adjusted the mean forecasts
in an attempt to incorporate new information that was
unavailable when the forecasts were made. Since the
adjustments were not consistently applied. the original
forecasts published by Livingston may contain measurement
errors. In an effort to correct this deficiency. Carlson
(1977) employed the unadjusted mean values to create a
consistent series using an eight-month forecasting
horizon. The adjusted Livingston measure. converted to

an annual rate of change. is:

(8.2). “6.t+2 = (Ft/Ptllz/8 ’ 1

where: "6.t+2 is the expected rate of inflation;
Ft is the consensus expected level of the
C.P.I.; and
Pt is the current level of the C.P.I.
(October for December surveys and April

for June surveys)

30
2. Survey Research Center_2§£§

The Survey Research Center (SRC) data are an
alternative series to the Livingston data of directly
measured expectations about changes in the price level.
This quarterly series. which consists of approximately
1.000 randomly selected households who are asked the
prices of things "they buy." may be viewed as a measure
of popular inflationary expectations. Rather than
representing the expectations of professional economists
(as with the Livingston data). the Survey Research Center
series represents the expectations of consumers and
businessmen.

Given their different perspectives and levels of
knowledge of economic variables. we would expect that the
process used by the two groups to forecast inflation
would differ substantially. Juster and Comment (1975)
have compared the behavior of the SRC expected price
change series with the Livingston data. In general. they
found that the SRC data exhibited greater mean values and
substantially large standard deviations.5 In addition.
the distribution of the SRC data appears to be skewed to
the right.

The form in which the SRC data has been obtained has
gone through three major alterations since the series was
first compiled. Until 1966. the survey only obtained

qualitative information about the direction of expected

31
price changes. Beginning in 1966. respondents expecting
price increases were asked how much--1-2 percent. 5
percent. or 10 percent. In 1977. the questionnaire was
improved to enable respondents to provide an open-ended
response concerning the amount of expected inflation.
Juster and Comment produced a consistent quantitative
series by establishing a relationship between the survey
answers during the years (1966-1977) in which the
qualitative and quantitative data overlapped and

extrapolating backwards. The SRC measure is given by:

Q Q
Ti 4" 1T
8 t t+1
(3.3) Tit 2
where: “t is the expected semi-annual inflation
rate;

n3 is the SRC quarterly expected inflation
rate at time t (1 quarter ahead);
“2+1 is the SRC quarterly expected inflation
rate at time t+1 (2 quarters ahead).
C.4_§xpectations Measures
Lahiri (1976) recognized that the Livingston price
expectations may contain errors that might make them
differ from the true. unobservable price expectations.

Consequently. he combined the information from the

32
Livingston price expectations series with that from past
rates of inflation to derive new estimated price
expectations variables. In doing so. Lahiri employed
three expectation hypotheses previously proposed by
Turnovsky (1970)--distributed lag. adaptive. and
extrapolative. Subsequently. other researchers suggested
an expectations measure which included past monetary
growth rates in addition to past inflation rates.
1. Distributed Lag

The distributed lag hypothesis utilizes some weighted
average of past inflation rates as a proxy for expected
inflation. This hypothesis assumes that price
expectations are formed autoregressively. that is. the
subset of available information used in forming price
expectations is restricted to past rates of inflation.
The distributed lag model may be represented as follows:

(3") It = i wi Pt-i
where: “t is as before
W1 are the weights assigned to past inflation

rates and
t-i the actual rate of inflation for a period

ending at the time price expectations are

formed.
In this study. 1 is assumed to have 3 lags.

In order to utilize the distributed lag model. the

researcher must determine (1) how many past inflation

33
rates are relevant and (2) the associated weights. In
the case where the weights add to unity. the model is
referred to as weighted expectations.

Gibson (1970) noted that prior to the mid-19605.
there were long lags in the formation of price
expectations. Consequently. bond yields adjusted very
slowly to past changes in the price level. However.
studies using the distributed lag model with data
obtained since the mid-19605 have indicated a marked
acceleration in the formation of price expectations. In
addition. when inflation is rapidly changing and
volatile. past rates of inflation may not be a good proxy
for future expectations. Hence. a model which employs a
fixed number of past inflation rates may be inappropriate
when there is a change in economic policy or a structural
change that affects the inflation generating process.

2. Adaptive_§xpect§tions

The original form of the adaptive expectations model
expressed the changes in price expectations as some
fraction. A. of the last period's forecast error.
Formally. this version of the adaptive expectations model
may be written as:6

(3.5) (nt _ n

t—i) = AlPt-l ’ "t-l’

where: fit and Pt are as before; and "t-i is the

34
expected inflation rate for the six-month period
beginning six months ago.
One problem with this formulation is that it leads to
systematic underestimation of a trend in price changes.
Therefore equation (3.5) was modified to:

(3.6) fit a WIPt-l + W2Wt—2
where: ”1 and W2 are the weights and the other

variables are as previously defined.

If W1 + W2 > 1. the forecaster will increase his
expectations above last period's in order to reflect the
trend.

3. Extrapolative Model

The extrapolative model may be expressed as:

(3.7) wt : Wo + w1Pt_1 + WzlPt-i - Pt-2)

where: W0 is a constant.

Pt-1 is the actual inflation rate for the

six-month period beginning six months ago.
and the other variables are as previously
defined.
In the case where we = 0 and W1 = 1. the hypothesis
asserts that the expected price change for the next six
months equals the price change for the last six months
plus a correction to reflect the trend in changes in the

actual price level over the past six months. The case

where W0 = W; = 0 and W1 = 1 corresponds to static

35
expectations. where next period's expected inflation rate
equals the previous period's actual inflation rate.
4. Monetary Information Model

The aforementioned expectations hypotheses fail to
consider other relevant macro-economic variables that may
affect inflationary expectations. Therefore. these
proxies may result in biased and inconsistent parameter
estimates. Consequently. Rutledge (1976). Maital (1979).
and Mullineaux (1980) have hypothesized that inflationary
expectations should reflect past rates of change in the
money stock in addition to past rates of inflation.

As Friedman (1969) has noted: "To the best of my
knowledge there is no instance in which a substantial
change in the stock of money per unit of output has
occurred without a substantial change in the level of
prices in the same direction . . . ." The monetary
information model may be represented as:

‘3'“) 1't = E biPt-i + g Ci"t-i
where: “t and Pt are defined as previously.

P1 are the coefficients for the past inflation

rates.

03 are the coefficients for the monetary

growth rates and

Mt is the six-month growth rate in the money

BuPP1Y (i.e.. M1) for the period ending at
time t.

36

In this analysis. it is assumed that both i and j have 2
lags. It is possible that additional macro-economic
variables besides the growth rates in the money stock may
influence inflationary expectations. However. Maital
(1979) found inflationary expectations to be monetarist
rather than fiscalist in nature. Consequently. this
study will only consider monetary as opposed to fiscal
variables (e.g.. changes in the level of government
expenditures).

9. Time Series Model

The final method used to derive estimates of expected
inflation is the Box-Jenkins (1976) technique. A
univariate time-series model. which is based on a single
time series. describes the present value of the series as
a function of the past values of the same series and a
random error. In contrast to the assumption of
statistical independence. the time series methodology
presumes that the observations in a time series may be
correlated. The appropriate time series model is used to
explain the correlation pattern of the observations.

Box-Jenkins univariate time series modeling involves
three steps: (1) model identification; (2) parameter
estimation; and (3) diagnostic checking. These steps are
done iteratively in order to derive an appropriate time
series model. Appendix A contains a more detailed

discussion of time series methodology.

37

The time series estimates in this analysis will be
calculated as follows: First. Box-Jenkins time series
techniques will be utilized to model the monthly Consumer
Price Index for a base period. January 1953 to April
1959. The estimated model will then be used to forecast
the C.P.I. for December 1959. Next. six more months of
data will be added. the model will be re-estimated. and
another forecast will be computed. This procedure will
be followed until the last forecast date. April 1983. is
reached.7

Analysis of the base period C.P.I. series indicates
that the second difference of the C.P.I. can be modeled
as a first order moving average process (ARIMA 0.2.1):

Pt = 2Pt-l " Pt-2 + at ’ eat-l

where: Pt is the C.P.I. in month t and

t is the random error term which is assumed

to be normally and independently distributed
with mean zero and constant variance.
All estimates of the moving average parameter fall within
the range of .76 to .79 and there is no evidence that the
form of the stochastic process changed over the 1959-1983
period.
Chapter Three has reviewed the primary models for

estimating expected inflation that have been used in

38
previous empirical tests of the Fisher hypothesis. In
the next chapter. the accuracy of these alternative
inflation forecasts will be evaluated with the objective

of determining a "preferred" inflation proxy.

39

FOOTNOTES

1The use of the C.P.I. can be questioned on several
grounds. First. the C.P.I. largely excludes the prices
of long-lived goods and existing capital assets. Second.
the C.P.I. overstates inflation during the 1970s due to
inappropriate treatment of residential hdusing costs.
Third. improvements in the quality of goods are seldom
incorporated in the Index. Finally. the substitution of
relatively less expensive goods for those that have
become relatively more expensive is not considered.
However. any attempt to measure cost of living changes is
likely to have drawbacks.

2Prior to Fama. the majority of empirical research

indicated that there was no statistically reliable
relationship between the rates observed in the market at
a point in time and the rates of inflation subsequently
observed. For a summary. see Roll (1972).

3There are two additional survey expectation series

which could be used. Since 1970 the Bureau of Economic
Analysis has surveyed businesses on their estimates of
the rates of price change for goods and services sold and
capital goods purchased. Also. since 1978. the Decision
Makers Poll of institutional portfolio managers provides
estimates of long-run inflation expectations over the

next five- and ten-year periods. These proxies are not

40
employed in this analysis due to their comparatively
short existence.

4Each December. Livingston publishes expected values

of the level of the C.P.I. for June and December of the
following year. In June. he presents forecasts for
December and the following June.

5The SRC standard deviations are larger primarily

because they are measured over a more informationally
heterogeneous group than the Livingston standard
deviations.

6This version of the adaptive expectations is the

error-learning model proposed by Cagan (1956).

7To enhance comparability with the other proxies for

expected inflation. the timing and horizons match those

of the adjusted Livingston series.

CHAPTER FOUR

AN EVALUATION OF THE ACCURACY OF THE INFLATION PROXIES

Chapter Three discussed a number of alternative
proxies for expected inflation. This chapter will
examine the forecasting accuracy of these proxies. that
is. how well the proxies track the corresponding actual
inflation series. A number of alternative measures of
forecasting accuracy including the coefficient of
correlation. unbiasedness. efficiency. mean square error.
mean absolute error. and Theil's U coefficient will be
considered.

The means and standard deviations of the alternative
inflation models for the 1959-1983 period are given in
Table 4.1.1 With the exception of the time series
model. the means of the various models are less than the
mean of the actual inflation rate. indicating that the
models generally underestimate the actual inflation
rate. Also. the standard deviation of the actual
inflation rate is greater in all cases than the standard

deviations of the alternative models.

I. CORRELATION COEFFICIENT

 

One measure of the forecasting accuracy is the
coefficient of correlation between the alternative
proxies for expected inflation and the actual inflation
rate as measured by the C.P.I. The correlation

Cl

‘2

TABLE 4.!
MEANS AND STANDARD DEVIATIONS OF ALTERNATIVE
INFLATION MODELS

 

 

Inflation Standard
Measure Mean Deviation
ACTINF .05199 .03715
TSINF .05243 .03704
FAMAINF ' .05953 .02951
WEXPINF . .03938 .02743
SRCINF .05121 .02585
EXTRINF .03961 .02732
ADAPINF .04055 .02781
MDNINF .04058 .0268?

LIVINF .04143 .02795

 

l3

coefficient. r. is given by
Exin

lzxi i/ZY:

 

(4.1) r =

where: y is the dependent variable (the actual inflation
rate) and
x is the independent variable (the inflation
proxy).
The closer r is to +1. the stronger the degree of
association between the inflation proxy and the actual
rate of inflation.

The matrix of correlation coefficients for the
alternative inflation measures is given in Table 4.2.
The Survey Research Center data proxy is the model which
is the most highly correlated with the C.P.I.. whereas
Fama's T-Bill measure is the least correlated with the
actual inflation rate. The primary disadvantage
associated with the correlation coefficient is that it

does not penalize a model for exhibiting systematic

linear bias.

II. RATIDNALITY

There has been a growing emphasis in the theoretical
literature that expectations formation should conform to
the notion of rationality pioneered by John Muth (1961).

The rational expectations hypothesis holds that

44

TABLE 4.2

MATRIX 0F CORRELATION COEFFICIENTS FOR ALTERNATIVE

 

 

 

 

INFLATION MEASURES: ORDINARY LEAST SQUARES
1959-1983
ACTINF TSINF FAMAINF WEXPINF SRCINF
ACTINF 1.000 0.815 0.646 0.817 0.944
TSINF 0.815 1.000 0.712 0.962 0.840
FAMAINF 0.646 0.712 1.000 0.789 0.672
WEXPINF 0.817 0.962 0.769 1.000 0.865
SRCINF 0.944 0.840 0.672 0.865 1.000
EXTRINF 0.748 0.886 0.785 0.953 0.804
ADAPINF 0.784 0.938 0.802 ;0.976 0.846
MDNINF 0.827 0.953 0.786 0.988 0.874
LIVINF 0.832 0.914 0.775 0.947 0.884
EXTRINF ADAPINF MONINF LIVINF
ACTINF 0.748 0.784 0.827 0.832
TSINF 0.886 0.938 0.953 0.914
FAMAINF 0.785 0.802 0.786 0.775
WEXPINF 0.953 0.976 0.988 0.947
SRCINF 0.804 0.846 0.874 0.884
EXTRINF 1.000 0.951 0.952 0.929
ADAPINF 0.951 1.000 0.963 0.943
MDNINF 0.952 0.963 1.000 0.950
LIVINF 0.929 0.943 0.950 1.000

 

45

expectations about future inflation are formed in a
manner that fully incorporates all of the relevant
information contained in a specified information set that
is available at the time the forecast is made.2 In the
context of this analysis. rationality encompasses two
requirements--u§biasedness and efficiency. The
unbiasedness criterion suggests that inflationary
expectations are unbiased estimates of the actual
inflation rate. Thus. the observed (actual) rate of
inflation differs from the expected rate of inflation
only by some random error term. The unbiasedness
criterion may be stated algebraically as:

(4.2) Pt 3 “t + at
where: Pt is the actual rate of inflation experienced

during period t:

"t is the anticipated rate of inflation for
period t; and

5t is a random error term with mean 0 and
constant variance 02.

The unbiasedness criterion may be tested empirically
by regressing the actual inflation rate on the predicted
inflation rate. i.e.. running the regression

(4.3) Pt 3 a + Bflt + ct
where the variables are as previously defined.

The predictor "t is termed unbiased if the sample
estimates of a and 8 do not differ significantly from 0

46
and 1. respectively. Moreover. ct should exhibit no
evidence of autocorrelation.

An additional criterion for rationality requires that
inflation forecasts be efficient. In other words. the
process by which inflation expectations are formed should
be identical to the process that generates actual
inflation. Therefore. any evidence suggesting that the
specified information set is not being fully (i.e..
efficiently) utilized would indicate rejection of
rationality. Pesando (1975) tested this concept of
rationality by assuming that both the expectations of
inflation and actual inflation itself were described by
the history of past rates of actual inflation (i.e.. a
distributed lag model). This interpretation of rational

expectations may be expressed mathematically as:

(4.4a) Pt 3 i 81 Pt-i + "t and

(4.4b) nt . i: a; pt,1 + vt
Efficiency requires that 81 g B; for all 1.....n.
Mullineaux (1978) demonstrated that the error
variances of equations (4.4a) and (4.4b) were not
identically distributed and therefore not homogeneous.
Consequently. the test for rationality expressed in
equations (4.4a) and (4.4b) was inappropriate.
When forecasting is done over several periods. a

series of forecast errors is obtained. Mullineaux

proposed an alternative efficiency test which alleviated

47
the difficulty associated with the previous test by
employing the forecast errors series:

(4-5) FEt = (Pt - "ti = f(information set

used to generate the forecast)
Thus. assuming that the relevant information set is the
past history of realized inflation rates. the concept of
efficiency may be expressed as:

(4.6) FEt g a + i 81 Pt-i + Et
where: at = “t.” Vt and the other variables are

as previously defined.

Efficiency requires that FEt be unrelated to any
information known at the time the forecast was made
(i.e.. the series of past inflation rates). The null
hypothesis requires that the coefficients of all the
information variables equal 0. Stated formally. this
requires that Ho: a = 0 and 31 3 0 for all i.....n. A
non-zero coefficient indicates that information was
available at the time the forecasts were made which could
have reduced forecast errors. but was not properly
utilized in forming expectations. In addition.
efficiency requires that the error term is serially
uncorrelated. i.e.. Cov (ct. £1) = 0 for t g 1.

Pesando (1975) and Figlewski and Wachtel (1981)
examined the Livingston expectations series for
unbiasedness. Pesando was unable to reject unbiasedness

using the consensus Livingston forecasts for the

48
1959-1969 periods. On the other hand. Figlewski and
Wachtel were able to reject unbiasedness using a pooled
time series/cross-section analysis of individual
forecasts for the 1947-1975 period. Hafer and Resler
(1980) also found the Livingston data to be biased for
the 1959-1978 period.

Pesando. Carlson. and Mullineaux tested the
efficiency of the Livingston inflation forecasts.
Pesando was not able to reject the efficiency criterion
for the 1959-1969 period. Using the adjusted version of
the Livingston data. Carlson found that the inflation
forecasts did not satisfy efficiency for the 1959-1969
period. Mullineaux employed the alternative test for
rationality given in equation (4.6). Using Carlson's
version of the Livingston data. he was unable to reject
the efficiency hypothesis for the 1959-1969 period.
However. for the data set used by Pesando (the original
Livingston data). Mullineaux was able to reject
efficiency.

Gramlich (1983) investigated the rationality of the
Livingston and Survey Research Center forecasts of
expected inflation. For both the Livingston and Survey
Research Center data. he rejected the rationality
hypothesis. In both instances. the expected inflation
series appeared to be biased and inefficient. The

results also indicated that the inflation forecasts of

49
economists were more biased and inefficient than the
forecasts of households.3 This suggests that.
households are somewhat better in predicting inflation
than economists.

To test for bias in the inflation forecasts. equation
(4.3) was estimated and t-tests were conducted for the
joint hypothesis that a c 0 and 5 c 1 for each of the
alternative inflation proxies.‘ The results of the
unbiasedness test are shown in Table 4.3. The results
indicate that the Fama T-Bill proxy. the SRC proxy. and
the extrapolative and adaptive models are biased
estimates of the actual inflation rate. On the other
hand. the weighted expectations. monetary information.
and Livingston measures appear to satisfy the
unbiasedness criterion.

It is possible for inflation forecasts to show
evidence of bias and yet still be "weakly" rational in
the sense that forecasters efficiently utilize the
information set. To test for efficiency. FEt 1;
calculated for each measure of expected inflation and
used to estimate equation (4.5). Since the adaptive.
extrapolative. and monetary information models utilize
additional information other than past rates of inflation
in generating inflationary expectations. the relevant
information set is expanded to include the additional

information.

50

TABLE 4.3

UNBIASEDNESS TEST OF EXPECTED INFLATION MEASURES

 

 

Pt 8 a + Bnt + et
a B c

Proxy (T-Ratio)a (T-Ratio)b R2 D.W.d
FAMAINF £3152) (:2823) .1586 2.27
WEXPINF (33:3) ::8222) .4683 2.16
SRCINF (Lgfggl 2593;; .8972 1.95
EXTRINF £2233) (:1???) .1921 2.25
MDNINF i392?) #:2322) .5383 2.10
LIVINF if???) 2:232?) .4555 2.11

 

aT-ratio is for o = 0.

bT-ratio is for 8 - 1.

cAdjusted for degrees of freedom.

d

Obtained from

the maximum likelihood iterative

technique to adjust for first order serial correlation
of the error.

51

The results for the efficiency test are shown in
Tables 4.4 and 4.5. These results indicate that only the
Fama inflation proxy satisfies the efficiency test. The
weighted expectations. Survey Research Center.
Livingston. extrapolative. adaptive. and monetary
information proxies do not appear to have efficiently
utilized the appropriate information set. Taken
together. the unbiasedness and efficiency tests indicate
that none of the proxies for expected inflation fully

satisfies the rationality criterion.

II. DISPERSION M ASURES

_ v-L—ij

 

Granger and Newbold (1973) argue that any measure
that looks at only the relationship between the predictor
and actual series and not the magnitude and behavior of
the forecast error series gives a misleading impression
about the accuracy of the forecasts. Therefore. this
anlaysis considers three alternative statistics which are
commonly used to measure the accuracy of forecasts--mean
absolute error (MABE). mean square error (MSE). and
Theil's U coefficient.

While both the mean absolute error and the mean
square error provide evidence on the dispersion of the
forecast error. MABE gives equal weighting to all

forecast errors whereas MSE gives greatest weight to

52

TABLE 4.4

EFFICIENCY TEST OF EXPECTED INFLATION MEASURES

 

 

= + + +
FEt “ BlPt-l BZPt-Z B3Pt-3 + Et
0.
e1 82 63 a

Proxy (T-Ratio) (T-Ratio) (T-Ratio) (T-Ratio) a? b.w.b
FAMAINF .0110 .0343 .1490 -.1471

(-.6673) (.2142) (1.10) (-.9221) -.0191 2'30
WEXPINF .0067 .9233 -.4254 -.2944

(2.19) (5.96) (-1.92) (-2.16) '5727 2'02
sacrnr -.0105 .3204 .1935 -.2933

(4.79) (4.07) (1.72) (-3.85) '5957 2'08
LIVINF .0053 .6441 .0353 -.5926 6‘30 2 14

(1.88) (5.79) (.2264) (-5.41) “" '

 

aAdjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique

to correct for first order serial correlation of the
error.

53

TABLE 4.

(fl

EFFICIENCY TEST OF EXPECTED INFLATION PROXIES
1959-1983

EXTRAPOLATIVE PROXY

 

 

 

rat = “ElPt-l + EzlPt_1 - Pt-Z) + at
0 81 82 a
(z-Ratic) (T-Ratio) (T-Ratio) 32 b.w.b
a: -
($33., fizz.) .Léiii. 4205 ...o
ADAPTIVE PROXY
FEt = ° + B1Pt-l + Ezrt-z + 8t.
0 8l 82 2a
(T-Ratio) (T-Ratic) (T-Ratio) a 0.w.b
. . . -
(3775) (:édg) (;§?i§) "205 1'50
.ONETARY INFORMATION pnoxv
FEt = a + 81Pt_1 + BZPt-Z + BZMt-l + B4Mt-2 + Et
“ e1 82 B3 84 2a
(r-natio) (T-Ratio) (T-Ratio)4(T-Ratio) (T-Ratio) a n.w.b
(91?53) (;§?Z:) (géii) (:2???) (86:70) '0509 2'00

3Adjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique

to correct for first order serial correlation of the
error.

54
large forecast errors. These two measures can be

represented as follows:

 

 

N r - A
(4.5) MABE =%]- i: Jih—t
t=l t
' 2
N r -— A
(4.6) use =-;-' )3 J—K—t
t=l t

where: Ft is the forecast value for time t:

At is the actual value for time t: and

N is the number of forecasts.

Theil's U statistic is defined as:5

(4.7) u= ”:3
JZAt/N

The coefficient always falls between zero and one.6

 

It is equal to zero when there are perfect forecasts. 0n
the other hand. U assumes the value of one when the
predictive performance of the forecasting model is as bad
as it possibly could be.

The results for the MABE. MSE. and Theil's U
coefficient measures are reported in Table 4.6. These
results indicate that the Survey Research Center measure
is the most accurate inflation proxy since it has the
lowest values for each of the measures of forecasting

accuracy.

55

 

 

TABLE 4.6
FORECAST ERROR MEASURES OF INFLATION PROXIES
1959-1983
MABE MSE THEIL'S U

SRC .01146 .00023 .2388
TSINF .01665 .00049 .3509
LIVINF .01692 .00054 .3639
WEXPINF .01789 .00062 .3902
MONINF .01798 .00059 .3827
ADAPINF .01834 .00067 .4080
EXTRINF .01841 .00068 .4080
FAMAINF .02329 .00088 .4673

 

56

It is interesting that the forecasts of households
are better than those of economists. Presumably. the
respondents to the Livingston surveys should be able to
apply greater knowledge and expertise in weighing the
various factors affecting the formation of inflation
expectations. In addition. they may also have access to
relevant information not available to the general
public. Therefore. the Livingston forecasts would be
expected to exhibit greater forecast accuracy than the
SRC forecasts.

The SRC forecasts also are superior to those of the
ARIMA model. Since the ARIMA model produces MSE
inflation forecasts that fully incorporate all the
information contained in the past series of inflation
rates. this suggests that the respondents to the SRC.
surveys utilize additional information beyond that
contained in the past history of inflation rates in
forming price expectations.

Chapter Four has investigated the forecasting
accuracy of the alternative inflation proxies. The SRC
measures was the most highly correlated with the actual
inflation rate. This chapter also found that none of the
alternative inflation measures fully satisfied the
rationality criterion. Finally. the SRC measure had the
lowest mean absolute error. mean square error. and

Theil's U coefficient.

57

FOOTNOTES

1The time period considered begins with 1959 to

permit comparison among all the alternative inflation

proxies.

2Rationality may be contrasted with completeness.

Rationality implies that all information is utilized in
an Optimal manner. whereas completeness implies only that
all information is utilized. Since such use of
information need not be optimal. completeness is a

necessary but not sufficient condition for rationality.

Gramlich. however. failed to correct for first
order serial correlation. The Livingston data exhibit

significantly more serial correlation than the SRC data.

4The time series ARIMA model is not evaluated since

by definition the ARIMA proxy provides rational inflation
forecasts that fully incorporate all the information

contained in the past series of inflation rates.

5Theil first defined the U coefficient as:

JM—ss

JZAE/N + / IPE/N

Us:

However. Granger and Newbold (1983) argue that this

measure does not provide a useful ranking of forecasts.

58

6Often. researchers decompose the MSE and Theil's U

coefficient into bias. variance. and covariance

proportions where:

——2
u” c izfigfiél— = bias proportion

(sf— 6 )2

Us a MSE a variance proportion
2(1 - r)S S

0C a f a

 

MSE c covariance proportion

However. Granger and Newbold (1983) argue that it is

difficult to give any meaningful interpretation to Us

and UC. Therefore. this study does not report the

decomposition.

CHAPTER FIVE

INVESTMENT ASSETS AS INFLATION HEDGES

The impact of inflation on the rates of return of
various investment assets are complex and difficult to
disentangle. This chapter provides a discussion of the
inflation hedging capability of various financial assets.
The theoretical arguments regarding the ability of various
asset categories to protect against price level changes is
presented. In addition. previous empirical studies of the
relationship between price level changes and the returns on

different asset categories are summarized.

I. THE CONCEPT OF AN INFLATION HEDGE

As indicated earlier. investors are typically concerned
with the real value of their income and wealth as measured
in terms of final consumption goods and services.

Therefore. investors will be concerned with the real or
inflation-adjusted rate of return rather than the nominal
rate of return on their investments.

Chapter Two introduced the "Fisher Effect." which holds
that the nominal rate of interest is equal to the sum of the
real interest rate and the expected rate of inflation. The
Fisher equation can be generalized to the rates of return on
common stock and other financial assets. The generalized
Fisher equation asserts that an efficient market will set

59

60
the prices of financial assets such that the expected
nominal return is the sum of the equilibrium real return and
the market's assessment of expected inflation over the life
of the asset. This can be expressed formally as:

(5.1) it e r + "t
where: 1t is the expected nominal rate of return on the

financial asset as time t:
r is the expected real rate of return: and
ht is the expected rate of inflation at time t.
The generalized Fisher equation indicates that an increase
in the expected inflation rate will increase the nominal
rate of return. thereby leaving the real return on the asset
unchanged.

In addition to expected inflation. the returns on
securities could also be affected by unexpected inflation
and changes in inflation expectations. Given efficient
capital markets. investors can be expected to react to
unanticipated inflation by revising their expectations of
future returns. which in turn may lead to changes in the
portfolio composition of assets. Furthermore. if forecasts
of future inflation rates are influenced by economic factors
other than past forecast errors. the change in inflationary
expectations may reflect more information than that
incorporated in the unexpected inflation rate. Thus. in

evaluating the inflation hedging characteristics of various

61
asset categories. it is useful to consider three aspects of
inflation: (1) current expectations of inflation. (2)
unexpected inflation. and (3) changes in inflation
expectations.

An asset can be thought of as a hedge against expected
inflation if the real return on the security is unaffected
by expected inflation rates.1 Likewise. an asset is a
hedge against unexpected inflation if its real return is
unaffected by the unanticipated inflation rate. Finally. an
asset is a hedge against changes in inflationary
expectations if the real return on the security is
unaffected by changes in the expected rate of inflation.

Reilly. Johnson. and Smith (1976) have provided an
alternative definition of an inflation hedge. They defined
an asset to be a complete inflation hedge if the real rate
_of return in inflationary periods is at least as great as
the real rate of return in non-inflationary periods.2 In
addition. they defined an asset to be a partial inflation
hedge if the nominal rate of return in inflationary periods
was greater than the nominal rate in non-inflationary
periods.

Reilly. Johnson. and Smith's definition suffers from two
major deficiencies. First. according to this
interpretation. securities must be free of downside risk
originating from all sources. not only from inflation. It

is quite possible that factors other than inflation depress

62
returns in inflationary periods. Second. Reilly et al. fail

to distinguish between the three components of inflation.

II. INFLATION AND THE VALUATION DECIQUITIES

This section reviews some alternative theories about the
relationship between inflation and common stock prices.
Although classical economic theory suggests that there
should be a positive relationship between the returns on
common stocks and inflation. three alternative hypotheses--
price cost sensitivity. tax effects. and external
financing--suggest a negative relationship. In addition.
this section will also discuss the association between
inflation and small capitalization stocks.

A. ClassicaI:Iheory

Traditionally. financial theorists have viewed common
stocks as inflation hedges whose real returns should be
independent of the rate of inflation. The classical
position suggests that real returns accruing to the
ownership of capital goods should be invariant to changes in
the general price level. since these returns depend upon
production functions and input-output relationships which
are independent of the inflation rate. Furthermore. this
position also held that the real capitalization rate should
be invariant to the general price level. since this rate
should reflect the marginal real product of capital goods

and the marginal rate of time preferences. neither of which

63
depend on the price level. Since both the flows of real
returns and the real capitalization rate are invariant to
inflation. the real present value of these cash flows should
be unaffected by inflation. The invariance of real values
implies that a change in the inflation rate should be
accompanied by an equal change in the nominal rate of return
on equity.

Classical economic theory also suggests that
unanticipated inflation should result in a transfer of real
wealth from net creditors to net debtors because the value
of fixed monetary claims declines.3 A net creditor is
defined as an economic unit whose monetary assets exceed its
monetary liabilities. whereas the opposite holds true for a
net debtor. Given the fixed nature of debt obligations. net
debtor firms should enjoy lower real capital costs during
periods of unanticipated inflation. Therefore. the value of
the common equity of net debtor firms should increase when
inflation rates increase above their expected levels. Since
the consolidated balance sheet of U.S. nonfinancial
corporations has consistently been in a net debtor position.
the classical theory would predict that the returns on
common stock. in the aggregate. should increase when there
is unanticipated inflation.4

p. Price-Cost SensItivity

Van Horne and Glassmire (1972) contend that the costs of

a firm's inputs and the prices and quantities of a firm's

64
outputs may also be responsive to changes in the price
level. The sensitivity of these factors to the inflation
rate can result in significant changes in a firm's value.
If the prices of the firm's output increase more rapidly
than the costs of the firm's inputs. the Operating earnings
and the value of the firm should rise. Similarly. if they
rise more slowly. the value of the firm should fall.

Many firms are unable to raise prices sufficiently to
recapture increased production costs. The more rapid the
inflation. the greater the likelihood of governmental action
such as fiscal and monetary restraint or price guidelines or
controls. To the extent that these policies succeed.
operating income may be reduced. In addition. competition
from substitutes with more stable cost structures limits
firms' abilities to raise their prices. The inability to
pass on price increases because of governmental pressure or
competition from substitutes has an adverse impact on common
stock returns.

C. Tax Effects Hypothesis

Feldstein (1980) argued that biases in the United States
tax laws impair equity values during inflationary periods.
These biases. chiefly the use of historic cost depreciation
and first-in. first-out (FIFO) inventory accounting in the
computation of the corporate profits tax base. erode
after-tax earnings of firms by raising the effective tax

rate on corporate profits.5

65

Depreciation charges based on the historical cost of
assets do not change with inflation. However. if inflation
results in large cash inflows. an increasing portion of
these cash inflows is subject to taxation. Because the
corporation is able to deduct only historical cost
depreciation rather than replacement cost depreciation. its
real rate of return is lower. This. in turn. depresses the
value of common equities.

The same phenomenon also applies to FIFO inventory
valuation. If a corporation uses the first-in. first-out
method of inventory valuation. recorded inventory costs will
be lower than replacement costs if inflation is positive.
When the FIFO cost is used in the cost of goods sold
calculation. accounting profits are overstated resulting in
greater taxes than would be paid if replacement costs were
used. Again. the real return on capital decreases which. in
turn. depresses share values.

Q4 ExternalygInanchg Hypothesis

Lintner (1975) argues that even if a corporation is able
to maintain its real profit margin by increasing prices in
proportion to costs. the companies relative dependence upon
external financing will be higher. the greater the inflation
rate. whether anticipated or unanticipated.6 The greater
relative dependence on outside financing reduces the value
of outstanding equity and the real rate of return on

equity. These results hold whether additional debt or new

66
equity is issued to meet the additional financial
requirement. If the added financing is obtained by debt.
the after-tax cost of the new debt will reduce the real
returns to equity owners even though the firm's real profits
are maintained. Likewise. if the financing is obtained with
a new equity issue. the owners of the previously issued
shares will own a smaller portion of the firm's equity.
thereby reducing their real returns.

However. in the long run. stocks must provide a positive
expected real return in order for firms to attract capital.
Since investors have little incentive to invest at a loss.
firms must be able to offer positive expected returns. If
new investments offer positive expected returns. the share
of firms owning equivalent existing capacity should be bid
up to comparable values. Therefore. after a period of
adjustment to higher rates of inflation. the real rates of
return on common equities should be just as high as before
the increase in inflation rates.7

By reversing the arguments given above for an increase
in inflation. we see that subsequent reductions in either
expected or unexpected inflation will reduce firms' relative
dependence on external financing. During the transitional
period to lower rates of inflation. there will be unusually
large capital gains on equities. Hence. the holding period

returns on common stocks will be larger than expected.

67

E. Small CapItaIIzatIon Stocks angIIanatIgg

Ban: (1981) and Reinganum (1981) examined the
relationship between firm size and risk adjusted rates of
return. The total market value of the firms was used as a
proxy for size. Banz ranked all the firms on the New York
Stock Exchange on the market value of their stock.
Likewise. Reinganum ranked all the stocks on the American
Stock Exchange on the basis of the market value of their
stock. In each instance. the ranked samples were divided
into ten portfolios assuming equal weighting of the stocks
in the portfolio. Banz and Reinganum derived risk adjusted
abnormal returns for the ten portfolios. Both found that
the portfolios composed of small firms consistently
experienced significantly greater risk adjusted abnormal
returns than the portfolios consisting of large firms.

It is possible that inflation may assist in explaining
the superior performance of small capitalization stocks.
The prices and the returns of financial assets are affected
by changes in the future expected cash flows and discount
rates for those assets. If the effects of inflation are not
perfectly incorporated in the cash flows and discount rates,
the prices of financial assets will change. It is possible
that the impact of inflation on the cash flows and discount
rates of small capitalization stocks is different than for

the aggregate stock market. If this is the case. there may

68

be a differential relationship between the returns on the

, two categories of equities and inflation.

III. INFLATION AND THE VALUATION 0f
FIXED INCOME SECURITIES

In this section. theoretical arguments regarding the
relationship between the returns on fixed income securities
and inflation are presented. The analysis distinguishes
between a number of alternative instruments offering fixed
payments at periodic intervals. including conventional debt
securities. municipal bonds. utility bonds. and preferred
stock.

A. ConventIonaI Debt Securities

As inflation becomes more uncertain. conventional debt
securities offering a fixed nominal return (such as
long-term government and corporate bonds) become less
attractive to investors. To investigate the association
between inflation and bond returns. it is instructive to
review the general bond valuation model:8

(5.2) P0 .. c + c

+ . . . +
(1 + kl) (I + 1:1) (1 + k?

c+r
Tl+kpll +k3). . .(l+knl

where: P0 is the current price of a bond that

matures in n years:
0 is the annual coupon interest;

F is the face value at maturity: and

69

k1.k2.....kn are the discount rates expected

to prevail during each of n years.

Since the coupon interest payments incorporate
anticipated inflation at the time of issue. we would
anticipate that all bonds would provide a hedge against
expected inflation. However. to the extent that actual
inflation exceeds anticipated inflation..all bonds.
regardless of maturity. should suffer from unanticipated
inflation.

Fama (1975) and Roll (1972) have found that inflation
rates are positively serially correlated suggesting that the
expected rates of inflation in future periods are related to
the amount of unanticipated inflation in the present
period. Since an increase in the inflation rates expected
for future periods should raise the term structure of
interest rates. changes in inflationary expectations should
have a negative impact on debt securities. Moreover. the
price drop in long-term bonds should be greater than the
price drop in short-term instruments for a given upward
shift in the term structure.

Although both long-term and short-term interest rates
contain an inflation premium. long-term bonds lock investors
into the current interest rate for the life of the bond.
Since the coupon payments are fixed in nominal terms
(i.e.. %% c 0). the value of bonds decline if the discount

factors subsequent to the initial period (i.e..

70

k2.....kn) increase to reflect a revision in
inflationary expectaions (i.e.. %% > 0). Since short-term

bonds are better able to modify the coupon interest payments
to incorporate increased inflationary expectations. we would
expect the shorter the maturity of the bond. the better the
hedge against changes in inflationary expectations.

B. MunipraIIBonds

The interest payments received from holding municipal
bonds issued by state and local governments are exempt from
federal income taxes. If we assume that the total return on
taxable and tax exempt bonds is equal to its coupon yield
and the two bonds are identical in all respects except for
the taxation of the coupon payments. the respective yields
for the two instruments are linked by the following
equilibrium condition:

(5.3) RT(1 - tm) a 35
where: RT is the nominal return on the taxable bond:

RE is the nominal return on the municipal bond;
and
tm is the marginal tax rate of the marginal
purchaser of municipal bonds.

Equation (5.3) indicates that there is some break-even
tax bracket above which investors prefer tax-exempts to
taxables and below which investors prefer taxables to
tax-exempts. The equilibrium relationship among the two

financial instruments should also be reflected in their

71
respective responses to anticipated inflation in order to
equalize the expected after-tax real returns. This can be
expressed as:

(5.4) R5 - n = RT(1 - t - n

m)
If the Darby-Feldstein hypothesis is an appropriate
description of financial market behavior. the nominal
tax-exempt rate of interest should adjust on a one-to-one
basis with anticipated inflation. whereas the nominal
taxable interest rate should adjust at a rate greater than
one-to-one with changes in expected inflation.

Within the context of this analysis. it is also possible
to infer the marginal tax bracket applicable to municipal
bond pricing. The marginal tax bracket of the marginal
investor in municipal bonds can be computed by comparing the
yields on tax-exempt and taxable bonds. i.e.:9

(5.5) tm g 1 - RE/RT

The equilibrium marginal tax bracket changes in response
to supply and demand conditions in the municipal bond
market. Hendershott and Koch (1977) suggest that the supply
of municipal bonds is interest inelastic. Therefore. the
equilibrium marginal tax rate is principally determined by
demand considerations. The major purchasers of municipal
bonds are commercial banks and individuals. These two
groups are segmented according to their preferred habitat.

Commercial banks prefer to concentrate their holdings in

shorter maturities. Consequently. the pricing of short-term

72
municipals is determined primarily by the marginal tax rate
of commercial banks. On the other hand. longer maturity
tax-exempts are held primarily by individuals who have lower
marginal tax rates than the commercial banks.10 As a
result. the yields on long-term tax-exempts must be higher
relative to taxable bonds to induce individuals to hold
them.

Mussa and Kormendi (1979) compared the actual tax-exempt
and taxable yields in which the bonds of similar maturities
and bond ratings were paired. They found that the relative
long-term yield ratios were always higher than the relative
short-term ratios. Moreover. they found that the applicable
tax rate for short-term municipal yields was the corporate
(commercial bank) rats. whereas the applicable tax rate for
long-term municipal bond yields was the personal tax rate.
Therefore. the differential response of short-term taxable
and tax-exempt yields to expected inflation should reflect
the corporate tax rate. whereas the long-term differential
response should reflect the individual tax rate.

C. Utility Bonds

Expectations of increases in the general price level
might cause interest rates on utility bonds to increase more
than rates on industrial bonds. The logic behind this
argument relates to the differences in pricing flexibility
for the two segments of the economy. Utilities. which must

obtain regulatory approval of rate increases. may find that

73

their costs are increasing faster than their revenues.
This. in turn. reduces the operating earnings of utilities.
and increases the difficulty of repaying the principal and
interest on bonds outstanding. On the other hand.
industrial companies often have grater flexibility in
raising their prices in response to increased costs.
Therefore. industrial firms should be better able to
maintain profit levels and their ability to meet debt
obligations.

9. Preferred Stock

Preferred stock is a hybrid security having some of the
characteristics of both debt and equity. Preferred
dividends are a contractual obligation like the interest
payments on debt. However. if the preferred dividends are
not earned. the firm can forego paying them without danger
of bankruptcy. In this regard. preferred stock is similar
to common stock. Although preferred stock has rights and
claims ahead of common stock. preferred shareholders do not
generally benefit from increased earnings. Thus. to the
investor. preferred stock is less risky than common stock
but riskier than bonds.

Although preferred stock issues may be callable and may
be retired. most are perpetuities. The yield on preferred

stock is calculated as:

0
(5.6) 7? - F:

74

where: Yp = the yield on preferred stock:

DP c the fixed preferred dividends; and

Pp = the price of preferred stock.

If the economy experiences unanticipated inflation.
preferred stock with fixed dividend rates becomes less
desirable. With unanticipated inflation. the prices of
existing preferred stock decline in order to make the
dividend yields on existing issues competitive with the
yields on newly issued preferred stock.11

Moody's classifies preferred stock into three
homogeneous default-risk categories:

High grade (11)

Medium grade (20)

Speculative grade (10)
The number of preferreds used in each category is shown in
parentheses following each group. Bildersee (1971) has
shown that high grade preferred stock behaves more like
bonds than common stock. whereas low quality preferreds are
more similar to common stock. The returns of medium grade
preferred stock were correlated with the returns on both

bonds and common stock.

IV. INFLATION AND THE VALUE OF COMMODITIES
Robichek. Cohn. and Pringle (1972) and Bodie (1983)
found that movements between various types of real and

financial assets were less positively correlated than those

75
for financial assets alone. The lack of a significant
positive correlation of the returns between commodities and
other investments suggest that commodities may provide
opportunities for portfolio diversification. Since
commodities represent ownership of real assets which reflect
the price behavior of the economy as a whole. they should be
a hedge against anticipated inflation. Similarly. since
commodity prices and consumer prices tend to move together.
commodities should be positively related to unanticipated

inflation.

V. PREVIOUS EMPIRICAL STUDIES

There have been numerous tests of the Generalized Fisher
Effect. although none considers all the asset categories
mentioned above. This section reviews the major empirical
studies investigating the relationship between the returns
on alternative investments and inflation. This section is
divided into three parts: (1) comprehensive empirical
studies; (2) studies of the common stock/inflation
relationship; and (3) studies of the relationship between
other assets and inflation.

A. ComprehensIveyEmpirical Tests

Fama and Schwert (1977) examined the relationship
between realized rates of return on various assets and the
inflation rate. Their study concerned both expected and

unexpected inflation and involved the return to stock

76
portfolios. U.S. Treasury Bills. U.S. government bonds. and
private residential real estate for the 1953-1971 period.
Of all the assets. only private residential real estate was
a complete hedge against both expected and unexpected
inflation during 1953-1971. Government debt instruments
(i.e.. bonds and bills) were a complete hedge against
expected inflation. but not against unexpected inflation.
Finally. common stock returns were negatively related to
both anticipated and unanticipated inflation.

Huizinga and Mishkin (1984) analyzed the monthly data
from 1959 to 1981 on real returns for seven securities: (1)
3-month Treasury bills; (2) 6-month Treasury bills: (3)
12-month Treasury bills; (4) intermediate term (5- 10-year)
Treasury bonds: (5) long-term Treasury bonds; (6) long-term
corporate bonds; and (7) common stocks. They found that the
ex-ante real rates on all seven assets were negatively
correlated with actual inflation rates. Moreover. as the
maturity length of the asset increased and the asset became
closer in its risk characteristics to equity. an increase in
inflation was associated with an even larger decrease in the
ex-ante real return. Mishkin and Huizinga concluded that
all seven assets have been imperfect hedges against
inflation. and the longer maturity assets have been the
worst hedges. Hence. Fama and Schwert's concluson that U.S.

Treasury bills and Treasury bonds of five years maturity or

77
less were reasonably good hedges against inflation was not
supported.

8. Tests of Common Stock/Inflation ReIatIonshIp

A number of studies have found a negative relationship
between inflation and the returns on common stocks. Reilly.
Johnson. and Smith (1970) found that the average real return
on several well-known stock price series were negative or
below the indices' long-term average return for five rapid
inflation periods during 1937-1973. A study by Oudet (1973)
indicated that. during the total period 1953-1970. the
returns on common stock were highest during the periods of
least inflation and lowest during the periods of high
inflation. Jaffe and Mandelker (1976) regressed both
monthly and yearly stock returns on inflation rates. The
results using monthly data for the 1953-1971 period
indicated a significant negative relationship for
contemporaneous data or using data with various leads or
lags. In a similar study. Nelson (1976) found a significant
negative relationship between the returns on common stock
and inflation.

Bodie (1976) examined the extent to which common stocks
could be used to reduce the uncertainty of real returns
resulting from uncertainty about the future price level.
Using the proportionate reduction in the variance of the
real return of a nominal bond attainable by combining it

with an equity portfolio as a measure of hedging

78
effectiveness. he found that the real return on equities was
negatively related to both anticipated and unanticipated -
inflation. Bodie concluded that only a short position in
common stock would have produced a hedge against inflation.

A study by Cagan (1974). which dealt with the long-run
returns from common stocks. found that U.S. stock prices
increased by 3 percent more annually than wholesale prices
over 1871-1971. However. a careful analysis of the results
indicated that stocks did not do well during periods of high
inflation. but made up for the loss when inflation
subsided. In contrast to earlier studies. Gultekin (1983)
used the Livingston data to relate expected stock returns to
expected inflation. His results indicated a strong positive
relationship between expected stock market returns and
expected inflation.

Following the publication of empirical studies
indicating a negative association between the returns on
common stock and inflation. a number of theoretical
arguments were advanced to explain the negative
relationship. Modigliani and Cohn (1979) argued that
investors systematically confused real and nominal discount
rates when valuing common equity. Malkiel (1979) attributed
the decline in share values to an increase in the perceived
riskiness of common stock investment as compared to bonds.
Fama (1981) hypothesized that the negative relationship

between stock returns and inflation was a result of a

79
negative relationship between inflation and changes in real
variables that reduced the return on capital. This
proposition was supported and further elaborated by Geske
and Roll (1983). Pindyck (1984) argued that the variance of
firms' gross marginal returns on capital increased which. in
turn. increased the relative riskiness of real returns from
common stock.

C. The ReIationshIp Between Other Assets_§nd;Inflation
Using data up to 1972. Gibson (1972) found that bond
yields were positively related to expected inflation. Jaffe

and Mandelker (1979) investigated the relationship between
the returns on fixed income securities and inflation for the
1953-1971 period. Their empirical findings suggested a
positive relationship between the returns to bondholders and
anticipated inflation. In addition. their research
indicated that the returns on fixed-income securities were
negatively related to unanticipated inflation. On the other
hand. Eddy and Seifert (1984) found that the returns on both
long-term corporate and government bonds were negatively
related to both expected and unexpected inflation for the
1968-1981 period.

Bodie and Rosanski (1980) reported that for 1950-1976
commodity futures prices tended to move inversely with stock
prices. and to do best during periods of rapid inflation.
Bodie (1983) found that the real returns on commodity

futures were positively correlated with the actual inflation

80
rate and negatively correlated with the real rates of return
on other major asset categories.

Chapter Five has reviewed the concept of an inflation
hedge. Next. it summarized theoretical arguments regarding
the association between inflation and the returns on
equities. fixed-income securities. and commodities.
Finally. the chapter discussed previous empirical tests of
the inflation hedging capability of alternative assets.
With the exception of commodities. previous studies have
indicated a negative relationship between asset returns and
inflation. Chapter Six will present another comprehensive

study of the relationship between asset returns and

inflation.

81

FOOTNOTES

1Alternatively. an asset is said to be a hedge against
a particular aspect of inflation if the nominal return on
the asset varies in one-to-one correspondence with the
particular inflation component.

2Reilly. et al.. referred to the rate of return in

non-inflationary periods as the "normal" return. The normal
rate of return should equal the risk-free interest rate plus
a risk premium commensurate with the business and financial
risks involved.

3There have been a number of tests of debtor-creditor

wealth transfers with inflation. Early studies by Hessel
(1956) and Alchian and Kessel (1959) suggested that there is
a wealth transfer from net creditors to net debtors with
inflation. However. subsequent tests by Hong (1977) and
French. Ruback. and Schwert (1983) did not find support for
the debtor-creditor hypothesis.

‘Since the total real value of the firm is invariant to

changes in the price level. the losses incurred by debt
holders accrue to the owners of equity when there is
unanticipated inflation.

5Since depreciation and inventory tax shields are based

on historical costs. their real value declines with
inflation. This. in turn. depresses the real value of the

firm.

82

6It is important to note that the dependence on

external financing increases when inflation rates increase.
If the price level rises at a constant rate. the dependence
on external funds will remain constant. Also. if inflation
rates decline. the dependence on external funds will

decrease.

7The real return on common equity will be maintained at
the level it otherwise would have been. assuming the same
level of business risk and degree of financial leverage.

8Because the federal government has the power to tax

and print money. securities issued by the U.S. Treasury are
free of default risk. On the other hand. fixed income
securities issued by corporations carry the risk that the
issuer will default on the contractually promised cash
flows. In this case. the expected cash flows can be
calculated by multiplying each possible cash flow in the
period by its probability of occurrence and summing the
products.

9Skelton (1982). however. argues that it is

inappropriate to infer marginal tax brackets directly from
yield ratios since long-term relative yields are sensitive
to a measure he derived which forecasts future short-term

yields.

10Miller (1977) and Fama (1975) assert that the

marginal bondholder's tax rate must be equal to the

83
corporate (i.e.. commercial bank) rate. Fama suggests that
arbitrage across the tax-exempt and taxable bond markets
ensures that the relative pricing of two securities reflects
the marginal tax rate of the banks. Likewise. Miller
suggests that the choice of debt or equity financing by
firms in the aggregate ensure that the relevant tax bracket
is the corporate rate.

11In order to make preferred issues more attractive to

investors. many companies. particularly utilities. have
begun to issue adjustable rate preferred stock with the
dividends tied to rates on U.S. government obligations.

With adjustable rate preferred stock. the price fluctuations
resulting from unanticipated inflation are significantly

reduced.

CHAPTER SIX
EMPIRICAL TESTS OF THE RELATIONSHIP BETWEEN

INFLATION AND ASSET RETURNS

This chapter re—examines the empirical relationship
between the components of inflation and the returns on a
variety of assets. The first section of this chapter
discusses the deficiencies associated with previous
empirical investigations of the Generalized Fisher
Hypothesis. The next section presents the total return
and yield data. The third section discusses the
methodology that is used in empirical tests of the
inflation hedging capability of alternative assets. The
fourth section presents the results of a series of
regression equations. The fifth section tests the
Geske-Roll reverse causality model. Finally. the last
section of this chapter provides an alternative test of

the Darby-Feldstein Hypothesis.

I. DEFICIENCIES 0F PREVIOUS STUDIES

As mentioned in Chapter Five. Fama and Schwert (1977)
and Huizinga and Mishkin (1984) previously conducted
comprehensive studies of the relationship between asset
returns and inflation. Fama and Schwert's (1977) study.
which employed the nominal T-Bill rate as the measure of
expected inflation. assumed that the real interest rate

84

85
was constant. However. a number of previous studies--
such as Hess and Bicksler (1975). Nelson and Schwert
(1977) and Levi and Makin (1979)--have indicated that the
expected real interest rate is not constant during
periods of high inflation. Consequently. Fama and
Schwert's methodology utilizing short-term T-Bill rates
as predictors of inflation may not be accurate in periods
of rapid inflation.

Small changes in the real rate of interest can cause
significant and opposite percentage change in the prices
of financial assets. This effect can be demonstrated
utilizing a simple valuation formula:

ZCFt

(6.1) v,c =._____
(1 + k)t

where: Vt is the price of the financial asset at time
t;
CFt is the (perpetual) cash flow at time t; and
k equals the constant discount factor.
The real interest rate. r. is an integral subset of
the discount factor since k = r + n. Since«%¥ = 2%E. a

change in the real rate of interest induces an opposite

change in financial asset values. Thus. to the extent

86
that changes in the Treasury Bill rate--Fama and
Schwert's inflation proxy--are due to changes in the real
interest rate rather than to changes in expected
inflation. we would expect a contemporaneous asset return
of the opposite sign.

Also. as Nelson and Schwert (1977) and Fama and
Gibbons (1982) have indicated. the 1953-1971 period
examined by Fama and Schwert was one of little variation
in the rate of inflation. On the other hand. the
post-1971 period was characterized by high and variable
rates of inflation. The heterogenous characteristics of
the inflation rate for the two periods are shown in
Figure 6-1. The differences between the two periods lead
to the possibility that Fama and Schwert's findings were
period dependent and not representative of the post-1971
experience.1

Huizinga and Mishkin (1984) utilized actual ex-post
data as their measure of inflation. However. the
relationship between asset returns and inflation is more
properly investigated using expectations data. Also.
Huizinga and Mishkin failed to distinguish between the
three components of inflation. Furthermore. Huizinga and
Mishkin used real returns rather than nominal returns in
their analysis. Estimates of the real rate of return on

an asset are dependent on the proxy used to measure

87

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88
inflation. whereas nominal returns have the advantage of
being directly observable.

This thesis will re-examine the relationship between
the returns on financial assets and the components of
inflation. Since Gultekin (1983) found that the effect of
inflation on asset returns may be largely a function of
the proxy used to measure inflation. this analysis will
include three alternative inflation proxies--the ARIMA
measure. Fama and Schwert's T-Bill measure. and the
Survey Research Center measure. This should provide
evidence regarding the consistency of results across
different measures for expected inflation. The thesis
will use a comprehensive single period. 1959 through
1983. which encompasses periods of both relatively high
and low inflation. In addition. this study will
investigate the relationship between inflation and the

returns for previously unstudied asset categories.

II. THEIQATA

This study will analyze the relationship between
inflation and the semi-annual nominal returns on the
following asset categories for the 1959-1983 period:2

(1) Treasury Bills

(2) Long-term government bonds

(3) Long-term corporate bonds

(4) Commodities

89

(5) Common stock and

(6) Small common stock.
In addition. the thesis will explore the association
between inflation and the six-month yields for the
following fixed-income securities:

(1) Ass corporate bonds

(2) Baa corporate bonds

(3) Composite industrial bonds

(4) Composite utility bonds

(5) Long-term municipal bonds and

(6) High grade preferred stock.

A. TotaI Return Series

For the U.S. Treasury Bill index. this analysis uses
the data from the CRSP Government Bond File. The index
measures one-month holding period returns for Treasury
Bills of not less than one month in maturity. Since
T-Bills do not pay coupon interest and are sold at a
discount from maturity value. their returns may be

represented by:

(6.2) 1 = capitalygain or_Ioss
pur0hase price

Then. the average of all the Treasury Bills is calculated
to obtain the returns for the index.

To measure the total returns on long-term U.S.
governmental bonds. this analysis uses the U.S.

Governmental Bond File prepared by the Center for

90
Research in Security Prices (CRSP). This index attempts
to maintain a twenty-year bond portfolio whose returns do
not reflect the potential tax benefits. impaired
negotiability. or special redemption or call privileges
that may characterize government bonds. The return for

each bond in the portfolio is calculated as follows:

(5.3) 1 = Efipitalggain or loss-tcoupon interest
purchase price at_beginfiing of period

Then. the returns of all the bonds were averaged to
calculate the index's return.

This index measures the total return on corporate
bonds with approximately twenty years to maturity. The
index is based upon Salomon Brothers' Long-Term Corporate
Bond Index from its beginning in 1969 through 1983. For
the 1959-1968 period. Salomon Brothers' index is
backdated using Salomon Brothers' monthly yield data.

The return on each bond in the portfolio is calculated in
accordance with equation (6.3)

The returns on commodities are based on Moody's Daily
Price Commodity Index. which is a weighted average price
of fifteen leading commodities in which there is an
active business and speculative interest. The index is
based on daily closing spot prices for the commodities.
Each quotation is weighted and the weighted total is
expressed as a relative number.3 The return for the

index is computed as:

91

(6 4) i g end of period number-beginning of period number
° beginfiing of pefiod numBer

The Standard & Poor's (SEP) 500 Composite Index of
diversified common stocks with all cash dividends
reinvested is used as the index for common stock
returns. Each common stock's rate of return is computed
as follows:

(5.5) 1 : Q§§I£§I_gain or loss + cash divIdend
Purc see price atFEEginnIng of—period

Adjustments are made to compensate for any stock that has
a stock split or a stock dividend. so that these changes
do not distort the returns. Then the rates of return on
all 500 stocks are averaged together using their relative
market values as weights. Therefore. stocks of the
larger companies are given larger weight in this index.
To represent the equities of smaller companies. this
analysis uses a historical series developed by Banz.
This index is comprised of the set of stocks making up
the fifth smallest quintile of New York Stock Exchange
stocks. where the stocks are ranked by market
capitalization (price times number of shares
outstanding). The returns on the portfolio of small
common stocks are calculated in the same manner as the

returns from the overall common stock index above.

92

BI, YieId SerIgs

For the yields on debt securities. the composite data
from Moody's Bond Yield Averages are used. The yield-
to-maturity is calculated for each bond in a given
category. Then. the yields of the individual bonds are
averaged on a simple arithmetic basis to obtain the
averages for each rating group.

The yield-to-maturity is the interest rate that
equates the future interest payments and the payment at
maturity to the current market price of the bond. The
yield-to-maturity is approximated by:

I+PV;;MV
(6.6) YTM = TMV_:_PVT7§
where: I is the periodic interest payment;
PV is the par value of the bond ($1.000):
MV is the market value of the bond: and
N is the number of periods to maturity.

This study considers the yields-to-maturity for both
Aaa and Baa bonds. Bonds which are rated Aaa are judged
to be of the highest quality and carry the smallest
degree of investment risk. The capacity to pay interest
and repay principal on these bonds is extremely strong.
Bonds which are rated Baa are considered medium grade
obligations. which are neither highly protected nor
poorly secured. These bonds lack outstanding investment

characteristics and have some speculative characteristics

93
as well. The use of two indices should provide insight
into the differential impact of inflation on the returns
of bonds of different risk classes.

The composite yields for industrial bonds are the
average yields of forty long-term industrial bonds
including 10 Ass. 10 As. 10 A. and 10 Baa bonds. The
composite yields for utility bonds are also composed of
the average yields of forty long-term bonds. ten in each
of four rating groups. The composite yields for
municipals are based on a broad sample of long-term
municipal bonds composed of bonds in each of the four
rating categories.

The yields on preferred stocks are based on the
yields of Moody's High-Grade Preferred Stock Index. This
group of preferred stocks has the least risk of default.
The yields were calculated in accordance with equation
(5.6).

The means and the standard deviations of the total
return series and the yield series for the 1959-1983
period are shown in Tables 6.1 and 6.2. respectively. In
general. there is a positive relationship between risk
and return which is in accordance with capital market
theory. However. the mean return for T-Bills is higher
than for long-term government and corporate bonds even
though the standard deviation of T-Bills is lower. This

suggests that holders of long-term debt instruments

94

suffered capital losses during this period as a result of
unanticipated inflation. In addition. Tables 6.1 and 6.2
also suggest the importance of distinguishing between the
total return and the yield of fixed—income securities.
As these tables indicate. the two series have
significantly different characteristics.
_II. MEIHODOLOGY

The generalized Fisher equation can be tested with
the following simple regression model:

(6.7) it = o + B"t + 5t
where all terms are as previously defined.
As in previous empirical studies investigating the Fisher
hypothesis. this analysis assumes that the expected real
return is constant or independent of the expected
inflation rate.4 Thus. equation (6.7) represents a
test of the joint hypothesis that the market is efficient
in pricing expected inflation and that the expected real
return and the expected inflation rate are independent.

An estimate of the a coefficient that is
indistinguishable from unity is consistent with the
hypothesis that the expected return on financial assets
moves in a one-to-one correspondence with the expected
inflation rate. Furthermore. an estimate of B which is
indistinguishable from unity is consistent with the
hypothesis that the expected real return on the financial

asset and the expected inflation rate are unrelated.

95

TABLE 6.1

MEANS AND STANDARD DEVIATIONS 0F

TBILL

GVTBND
CRPBND
COMMOD

COMMSTK

SMLLSTK

TOTAL RETURN SERIES

 

 

1959-1983
Standard

Mean QevIation
.0610 .0315
.0485 .1388
.0544 .1545
.0760 .2215
.1149 .2695
.2453 .4667

TABLE 6.2

MEANS AND STANDARD DEVIATIONS OF

AAABND
BAABND
INDBND
UTILBND
MUNIBND

PDFSTK

YIELD SERIES

1959-1983

Standard
Heap Deviation
.0762 .0297
.0863 .0241
.0786 .0306
.0820 .0337
.0576 .0236

.0745 .0258

96

Equation (6.7) fails to differentiate between
expected and unexpected inflation. It is possible that a
security might serve as a hedge against expected
inflation. but not against unexpected inflation.
Therefore. it is preferable to treat the two aspects
separately. To incorporate the effect of unanticipated
inflation. we modify equation (6.7) to obtain:

(6.8) it e a + Bln

t I 82”): ' "t’ * Et

where: Pt is the actual inflation rate for time
t;
(Pt - 1it) is the unanticipated component of
the inflation rate; and
5t is the random error term.
An estimate of the regression coefficient 32 in
equation (6.8) which is statistically indistinguishable
from 1.0 is consistent with the proposition that the
nominal return on the financial asset varies in
one-to-one correspondence with unexpected inflation.

If the empirical results of equation (3) suggests
that 81 = 1. the asset is said to be a complete hedge
against expected inflation. When 82 equals 1.0, the
asset is a complete hedge against unexpected inflation.
Finally. when the results suggest that 81 g 82 g 1.0,
the asset is said to be a complete hedge against both
expected and unexpected inflation. In this instance. the

nominal return on the asset varies in one-to-one

97
correspondence with both expected and unexpected
inflation. and the ex-post real return on the asset is
uncorrelated with the ex-post inflation rate.

In addition. it is also useful to test a model which
incorporates a term representing the revision in
inflationary expectations. To test the impact of changes
in inflationary expectations. equation (6.8) is modified

as follows:

(6.9) it = a + Blnt + 82(Pt - nt) + 83hrt - flt-l) + at

where:' (“t — "t-i) represents the change in

inflationary expectations and all other terms are
as previously defined.
If the asset is a hedge against changes in inflation
expectations. the estimated value of B3 in equation
(6.9) should be close to 1.0. When the 31 = 82 =
83==1.0. the asset is said to be a complete hedge
against all three aspects of inflation.

It is also possible to modify equations (6.7). (6.8).
and (6.9) to obtain the results of regressing real
financial returns on inflation. For example. the ability
of the real returns on a financial asset to provide a
hedge against anticipated inflation is tested with the
following model:5

(6.10) r = a + (81 _ 1)nt 1 et
If the expected real returns on the asset are independent

of inflationary expectations. then (81 - 1) should be

98
close to zero. Similarly. one can test the ability of
the real returns on a security to provide a hedge against
unanticipated inflation and against the changes in

inflationary expectations. This is accomplished by

examining whether (82 - 1) = 0 for unanticipated

inflation and (33 — 1) = 0 for changes in inflationary

expectations.

Iv. assuprs AND:INI§RPRETATION

Initially. this study investigates the extent to
which various assets were hedges against anticipated and
unanticipated inflation. This is done by estimating
equation (6.8) for the various asset categories using
three alternative measures for expected inflation--Fama
and Schwert's T-Bill proxy. the ARIMA proxy. and the
Survey Research Center proxy. The results using Fama and
Schwert's inflation proxy are given in Table 6.3 for the
total return data and Table 6.4 for the yield data.

Using Fama and Schwert's inflation proxy. the
following results are obtained:

1. T-Bills are a complete hedge against expected
inflation. Although the T-Eill coefficient for
unexpected inflation is positive and more than two
standard errors from zero. the estimate is reliably less
than unity. This suggests that T-Bills are a partial

hedge against unexpected inflation.

HEDGES AGAINST

99

TABLE 6.3

EXPECTED AND UNEXPECTED INFLATION--

 

 

rang AND scstar INFLATION encxv: 1959-1923
it I a + 811:t + 82(1’t - It) + 6t .
B1 82 i a
Asset o (T-Ratio) (T-Ratio) r a- o.w.b
TBZLL (I:3§§§) 15:12:) (532;? 284.57 .9267 1.14
“”5“” .2222; (£23.? 2522? 5'“ ...... “2
CRPBND (282:: (:3322) ’iéfzg? 7.17 .2031 1.90
.291:
COMMSTK (ligi) 1:321g5) (3113:? 1.55 .0245 2.02

 

.Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique
to correct the first order serial correlation of the error.

100

TABLE 6.‘

HEDGES AGAINST EXPECTED AND UNEXPECIED INFLATION--
FAMA AND SCHWERT INFLATION PROXY: 1959-1983

it ‘ ° + B1"1: + 82(Pt ' “0) + ‘t

 

 

(31 02 ‘
Asset a- (T-Ratio) (T-Ratio) r a? 0.w.b
AAABND .0721 .1641 .0202
(2 50) (3.20) (.5134) 3.40 .0900 1.29
BAAEND .0737 .1007 -.0160
(2.73) (3.15) (-.3527) ‘°50 136° 1'03
IhDBND .0720 .1022 .0004 50“ .1626 1.27
(2.74) (3.60) (.0090)
UTILBND .0766 .1006 .0202
(2.62) (3"4) (.4602) 4.11 .1199 1.10
MUNI .0565 .0035 - 0350 1.‘6 .0209 1.27
(3.10) (1.45) (-.7606)
PFD .0607 .1604 -.3110 , ,
(3.96) (“.0975) 0.42 .2445 1.5-

(3.41)

 

aAdjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique
to correct the first order serial correlation of the error.

101

2. The coefficient for expected inflation for the
total return on both government and corporate bonds is
positive but not significantly different from zero. This
indicates that neither government nor corporate bonds are
a hedge against expected inflation.6

3. Although the coefficients for both expected and
unexpected inflation for commodities are positive.
neither is significantly different from zero. This
suggests that the return on commodities are not a hedge
against expected and unexpected inflation.

4. The coefficient for expected inflation for the
S&P 500 common stock index is negative, whereas this
coefficient is positive for small common stocks.

However, the estimates are not significantly different
from zero in either case indicating that neither asset is
a hedge against expected inflation. Likewise, neither
asset is a hedge against unexpected inflation since the
coefficient for unexpected inflation is negative in both
instances.

5. Yields on both Aaa and Baa bonds are partial
hedges against expected inflation since the estimated
coefficients are positive and more than two standard
errors from zero. However, the coefficients are less
than one in both cases. The yields on both Aaa and Baa

bonds are ineffective hedges against unexpected inflation

102
since the coefficients for unexpected inflation for both
assets are insignificantly different from zero.

6. Similarly. the yields on both industrial and
utility bonds are positive and significantly different
from zero. However. both coefficients are significantly
less than one. Thus. the yields on industrial and
utility bonds are a partial hedge against expected
inflation.

7. The dividend yield on preferred stocks is a
partial hedge against expected inflation. However. the
yield on municipal bonds is not an effective hedge
against expected inflation since the coefficient for
expected inflation is not reliably different from zero.
Neither the dividend yield on preferred stocks nor the
yield on municipal bonds is a hedge against unexpected
inflation as the coefficient for unexpected inflation for
both assets is negative.

The results for the T-Bills and common stocks for the
1959—1983 period are in agreement with Fama and Schwert's
(1977) earlier study. However. in contrast to Fama and
Schwert's prior results. government bonds are not a hedge
against expected inflation.

The results for equation (6.8) for the ARIMA
inflation proxy are shown in Tables 6.5 and 6.6. In
general. the results for the total return data are quite

similar to those obtained with the Fama and Schwert

103

TABLE 6.5

HEDGES AGAINST EXPECTED AND UNEXPECTED INFLATION--

 

 

ARIMA inrnarxon pnoxv: 1959—1903
1t - a + 81": + 82(Pt - at) + st
81 B2 a

Asset a (T—Ratio) (T-Ratio) r 82 D-"ob
TBILL igfgg) igfgg) i§?;;) 7.60 .2156 1.99
GVTBND £2922) IL?§§34) (l?:3§0) .55 .1910 2.11
CRPBND (g?§:) (£3333) (3i?§§? 1.46 .0190 2.03
COHMOD (?:330) (77:23) :57321 .73 .0111 1.96
coxnsrx (2?:2) I;?;;:O) Iféfii? 7.00 .2002 2.05
SHLLSTK i§?32) (T3330) :35?;;: 4.94 .1400 1.99

 

‘Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique

to correct the first order serial correlation of the error.

104

 

 

TABLE 6.6
BEDGES AGAINST :xrecrro AND unexpecrzo INFLATION--
ARIMA INFLATION PROXY: 1959-1903
it - a + 81 t + 82(Pt - 't) + at
01 02 I.
Asset a (T-Ratio) (T-Ratio) r 02 0.5:.b
AAABND .0773 .1521 .0710
(2 70) (2.37) (1.91) .99 .0033 1.05
BAABND .0061 .1603 .0557
(3.03) (2.12) (1.24) '79 '°°91 1°°2
INDBND .0700 .1365 .0642
(2.90) (2.00) (1.50) .30 .0162 1.14
UTILBND .0026 .1055 .0027 1
(2.67) (2.65) (1.90) 1.66 .0260 .9.
HUNT .0599 .0735 .0026
(3.43) (1.02) (.1926) '33 '1‘2‘ '97
PFD -.0752 .0954 .0522
.(3.63) (1.60) (1.55) '85 '°°55 1'32

 

‘Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique
to correct the first order serial correlation of the error.

105
inflation proxy. However. using the ARIMA inflation.
measure. T-Bills are shown to be only a partial hedge
against expected inflation since the coefficient for
expected inflation is reliably different from one. In
addition. the signs for the expected inflation
coefficients for both government and corporate bonds are
negative (although insignificantly different from zero).
The signs for unexpected inflation for the ARIMA proxy
are negative and significantly different from zero for
both common stocks and small stocks. On the other hand.
although the coefficients for unexpected inflation for
corporate and government bonds are negative. they are
insignificantly different from zero.

The results using the yield data for the ARIMA proxy
are also somewhat different than for the T-Bill measure.
The yields for Aaa bonds and utility bonds are positive
and significantly different from zero. although reliably
different from one. This suggests that the yields on Aaa
bonds and utility bonds are partial hedges against
unanticipated inflation. Also. the signs for the
unexpected inflation component are positive for each of
the yield series.

The results for equation (6.8) for the Survey
Research Center inflation measure are shown in Tables 6.7
and 6.8. For the total return data. the following

results are obtained:

HEEGES

AGAINST EXPECTED

106

TABLE 6.7

AND UNEKPECTED INFLATION--

 

 

SURVEY RESEARCH CENTER INFLATION pnoxr: 1959-1903
it I a + Blflt + 82(Pt - wt) + ct
81 82 a
Asset o_ (T-Ratio) (T-Ratio) r a? 0.w.b
TEILL (gfig) E3???) (ifif, 6.13 .1761 .00
GVTBND iOSSEC) ti?2;? zféfizfi 2.42 .0550 .99
SMLLSTK 1.03:2?) ?§?3:§ Ii§::§?8 3.34 .0000 .03

.Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique
to correct the first order serial correlation of the error.

.107

TABLE 6.8

HEDGES AGAINST EXPECTED AND UNEXPECTED INFLATION--
SURVEY RESEARCH CENTER INFLATION PROXY: 1959-1983

it I a + Slut + 82(Pt In t) + at

 

(31 B

 

Asset o ‘ (T-Ratio) (T-Raiio) r 32. 0.w.b
AAABND {2?33, (?::;9) QS?2§, 1'23 '0109 '77
BAABND (5013) (93057) (9§8§) .55 —.0109 .69
INDBND (g?22) (?:ig9) (g?gg) '88 -.0413 '83
”Tm-23;,
PFD ”£3?§3) (2?;3) (?§3$1 1.11 .0050 .91

 

‘Adjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique

to correct the first order serial correlation of the error.

108

l. T-Bills are a partial hedge against both expected
and unexpected inflation.

2. The coefficients for expected inflation for both
government and corporate bonds are positive. but
insignificantly different from zero. Thus. neither debt
instrument is a hedge against anticipated inflation. The
coefficients for unanticipated inflation are negative and
significantly different from zero. This indicates that
the returns for both government and corporate bonds are
negatively related to unanticipated inflation.

3. The coefficients for both anticipated and
unanticipated inflation for commodities are positive. but
not reliably different from zero. Thus. commodities are
not a hedge against either expected or unexpected
inflation.

4. The coefficient for expected inflation for common
stocks is positive but insignificantly different from
zero. On the other hand. the coefficient for expected
inflation for small stocks is positive and significantly
different from zero. These results suggest that small
stocks are a hedge against expected inflation. whereas
common stocks. in general. are not. The coefficients for
unexpected inflation for both common stocks and small
stocks are negative suggesting that neither asset

category is a hedge against unexpected inflation.

109

The results using the yield data indicate that the
nominal yields for each of the fixed-income securities
are unrelated to either expected or unexpected
'inflation. Thus. the nominal yields on these financial
assets appear to be constant. and the real rates are
inversely related to the expected inflation rate as
suggested by the Inverted Fisher Hypothesis.

Next. this study estimates equation (6.9) for each of
three alternative inflation proxies (see Appendix B for
other inflation proxies). This provides evidence
regarding the ability of the various asset categories to
hedge against changes in expected inflation. The results
for the T-Bill inflation proxy are shown in Tables 6.9
and 6.10 for the total return and yield data.
respectively. The coefficients for the change in
expected inflation are negative with the exception of
government bonds and commodities for the total return
data. Although the coefficients are positive for
government bonds and commodities. they are
insignificantly different from zero. For the yield data.
all the coefficients are negative and significantly
different from zero. These results suggest that none of
the asset categories provide a hedge against changes in
inflationary expectations.

The results for the ARIMA model are given in Tables
6.11 and 6.12. For the total return data. all the

coefficients for the change in inflationary expectations

110

 

 

TABLE 6.9
srrscrs or CHANGES IN EXPECTATIONS or INFLATION:
FAMA AND scnwsar INFLATION PROXY: 1959-1903
' + + - -
1t ‘ ° Blnt B2‘91: 1‘t.’ + B3("t "t-l) + Et
61 82 83 3
Asset a (T-Ratio) (T-Ratio) (T-Ratio) 02 .w.
TBILL -.0009 1.064 .1657 -.2460
(-.2325) (10.79) (3.35) (-3.04) 116°33 '8853 99
6v70N0 —.0091 .7546 -1.0740 .1656 q
(-.2063) (1.10) (-2.04) (.1407) “'79 '1511 '99
CRPBND .0020 .5693 -2.4660 -.1197
(.0409) (.7609) (—3.41) (- 9046) "68 '1903 '92
common .0093 -.0009 1.3175 1.7191 . .
(1.01) (—.0509) (1.05) (.9501) '6‘ 0‘1‘ '96
cxxsrx .0013 .5170 -1.5040 —5.57621
(1.14) (.4650) (-1.33) (—2.33) 2'91 1157 '08
SMLLSTK .1533 1.5106 -1.6064 -5.6044
(1.00) (.6066) (-.7276) (-1.34) '97 0116 ’99

 

aAdjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

111

TABLE 6 . 10

EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:

 

 

FAMA ANO scxwznT INFLATION PROXY: 1959-1903

it ' ° I Bi't I aZ‘Pt ' “0’ I 83(“t ' “tel, * ‘t

51 B2 B3 29

Asset o (T-Ratio) (T-Ratio) (T-Ratio) F R D.“
AAABND Eg?f:) (2?:Z) (Igégg) 1;;?§0) 6'83 '2788 1'57
BAABND £2922) ngéi) (;?;f:0) ;;;?Z§) 0.00 .3226 1.52
IfiDBND {2?33, (2?:i) 1;?g027) :3132) 8'67 '3‘03 1's‘
“2222222222, .2222, 22222.. 1:22.22,
.2222, .2222, 2:222, 2:222.
"D ($33, (2‘???) 3:10. $2332. we -‘°°° 24°

 

.Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

112

TABLE 6.11

EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:

 

 

AFINA INFLATION PROXY: 1959-1906
it - a + 01st + 02(Pt - at) * 83(1)t - “t-l) + at
81 32 53 a
Asset a (T-Ratio) (T-Ratio) (T-Ratio) F R? .w.
TBILL {2?32, (3?0:? (5133? Eli???) 5'7° '2272 '89
GVTEND (213:) 1;?8330) 1;?3235) (321:0: 2'81 ‘1015 '15
.2222, 22222., 1222222 2.2222
C°””°” z?§3§a. (:2223, i;?28? 21$333) ... -°2‘17 -9‘
°°””S’” z§?§é, I;?§2§., Ifs?i§? ;;?§§33. .... -125’ -°5
SMLLSTK .1924 .9714 (32122? (31:32:) 3.32 .1259 .99

(1.90) (.5945)

 

‘Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

113

TABLE (.12

EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:
ARIMA INFLATION PROXY: 1959-1983

it = ° + 81"t + 82(pt ’ 1't) + B3‘“t ' 1't-—1’ + ‘t

 

5 1 3 2 8*3 a
Asset a (T-Ratio) (T-Ratio) (T-Ratio) F F? 0.14.
AAABND .0730 .2416 .0536 -.1121 . a
(2.90) (3.39) (1.44) (-2.43) "71 .1532 1‘25
0AA0N0 .0009 .2994 .2537 -.1749
(3.20) (3.70) (.612) (—3.41) "52 '1802 1'32
INDBND .0746 .2440 .0400 —.1357 o ,
(3.15) (3.30) (1.05) (-2.63) 2'95 '10'3 1'“8
UTILBND .0704 .2955 .0509 -.1357 . ..
(2.03) (3.57) (1.40) (-2.79) 3'87 ‘15‘3 1"“
NUNI .0564 .1636 -.0121 -.1140
(3.24) (2.03) (-.2065) (-2.20) 1'8‘ '0‘93 1'18
FFO .0714 .1936 -.03000 -.1234 , 99 1573 1 55
-(3.09) (3.10) (.9735) (—3.14)

 

.Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

114
different from zero. Again. this indicates that none of
the asset categories provides a hedge against the change
in inflationary expectations.

The results for equation (6.9) for the Survey
Research Center proxy are given in Tables 6.13 and 6.14.
For the return data. all the coefficients for the change
in inflationary expectations are negative suggesting that
none of the asset categories provides a hedge against the
change in expected inflation. Similarly. for the yield
data. all the coefficients are negative for the change in
expected inflation.

The coefficients for the changes in inflationary
expectation for fixed-income securities merit further
analysis. As indicated in Chapter Five. short-term debt
instruments should provide a better hedge against changes
in inflationary expectations since they are better able
to incorporate increased inflationary expectations. The
results using the ARIMA proxy are consistent with this
hypothesis since the coefficient is more strongly
negative for government and corporate bonds than for
T-Bills. On the other hand. the results using Fama and
Schwert's inflation proxy and the Survey Research Center
proxy are not consistent with the hypothesized
relationship. For Fama and Schwert's proxy. the
coefficient for the change in expected inflation for

government bonds is positive. Moreover. the coefficient

115

TABLE 6.13

EFFECTS IN CHANGES IN EXPECTATIONS 0F INFLATION:

 

 

suavzy RESEARCH CENTER INFLATION PROXY: 1959-1903
it ‘ “ I blrt I 62(Pt ' rt) + 63(“t ' "t-l) + Et

81' B2 . 13 . 20 b

Asset O (T-Ratio) (T-Ratio) (T-Ratio) F R D.W.
22222. (22222 ..2222 7:222.
““3"” 1:?2327. 1.233? 135???? H32? “4 -°“’ 1°99
.2222, 222222 7123222 7323222
COMMOD (?gg§9) 11311:) (Z0323) 1:13:14) '22 '0143 1'92
CMMSTK 102321) 11113? 132131? }}:Z$Z?9 0.77 .3270 2.06
22222.. 2.2222 22.22223 722:222’4-23

 

aAdjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique to
correct the first order serial correlation of the error.

116

TABLE 6.14

EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:

 

 

SURVEY RESEARCH CENTER INFLATION PROXY: 1959-1983

it I 0 + elflt + 82(Pt - 7t) + 83(11t - "t-l) + Ct

81 B2 B3 20
Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R 0.w.
...... 22222, .2222, .2222 153222..
...... .2222, 22222. .2222. 22222.
22222. 22322. .2222. 1:22:27,
”TILBND}2?Z§) (1???) (i913? 1:333) 1-26 -°199 "‘6
”m” 22%?) 291332) 7933350) (23%;?) -°2'~°“3 -9°'
22322. 22722. .2222, 1.22722.

 

aAdjusted for degrees of freedom.

b

correct the first order serial correlation of the error.

Obtained from the maximum likelihood iterative technique to

117
for T—Bills is more strongly negative than for corporate
bonds. For the SRC proxy. the coefficient for T-Bills is
more strongly negative than for government bonds.

In sum. the results of this chapter suggest that Fama
and Schwert's results may have been period dependent as
well as proxy dependent. In general. this study reaches
a negative assessment regarding the inflation hedging
capabilities of the various asset categories. The
results using the yield data indicate that nominal yields
on financial assets adjust partially to inflation
expectations when the T—Bill and ARIMA inflation proxies
are used. However. nominal yields do not appear to be
influenced by inflation expectations when the SRC

inflation proxy is used.

V. A TEST OF THE GESKE-ROLL‘REYERSE CAUSALITY MODEL

This chapter also presents a test of the Geske-Roll
reverse causality model. Osaka and Roll (1983) argue
that a change in the nominal return on common stocks
signals a change in inflationary expectations through a
chain of macro-economic events. First. a change in stock
returns predicts a change in government tax revenues.
Second. since government expenditures are largely fixed.
fluctuations in government revenues lead to budget
deficits. Third. budget deficits lead to an increase in

Treasury borrowing and debt monetization. Finally. debt

118
monetization results in an increase in the rate of
inflation. Thus. the negative association between
inflation and stock market returns is a spurious
relationship which is induced by a combination of a
reversed adaptive inflation expectations model and a
reversed money growth/stock returns model.

Although this model was originally developed for
stock market returns. it should be applicable to bond
market returns as well. Since the findings of the
earlier empirical tests indicate an inflation
relationship for bond returns which is similar to stock
returns. bond returns should also signal changes in the
inflationary process. The Geske-Roll model is tested by
estimating the following regression model:7

(6'11) (“t ' "t-l) ‘ a + Blit + B2"t—1 + 6t

where all the terms are as previously defined. Osaka and
Roll hypothesize that both 81 and 32 should be

negative.

The model is tested for (1) T-Bills; (2) government
bonds; (3) corporate bonds; (4) common stocks; and (5)
small stocks for the 1959-1983 period. The results for
the ARIMA proxy. Fama and Schwert T-Bill proxy. and the
Survey Research Center proxy are given in Tables 6.15.
6.16. and 6.17. respectively.

Geske and Roll hypothesized that both the

relationship between (1) the return on the asset and

3119

TABLE 6.15

TESTS OF THE GESEE-ROLL MODEL: 1959-198°
FAMA AND SCHWERT PROXY

 

 

 

(It ' "t-l) = a + slit + B2”t-1 + 6t
81 82 "a
Asset a (T-Ratio) (T-Ratio) F R‘ D w
TBILL £2023) £§§?§7) [:53135, 90.45 .7357 2.43
OVTENO 2002:) {:2323) z;?:§fe) 4.00 .1191 2.29
CRPBND (1?:Z) ;;33:§) 1;?5231) 0.04 .1000 2.33
COMMSTK .00;;) {:3236) Zl?;i;0) 5.15 .1535 2.10
SMLLSTK £0022) ELE?:§) 1:303?) 1.96 .0410 2.00
a

Adjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique

to correct the first order serial correlation of the error.

1120

TABLE 6.16

TESTS OF THE GESKE-ROLL MODEL:
ARIMA PROXY

(fl

1959-1383

t ' 't-l) ' “ + slit + 82"t-1 I ‘t

 

 

8 8
1 2 .-

Asset a (T-Ratio) (T-Ratio) F R‘ D.W.b
TBILL .0003 .3009 -.3519

(.0512) (2.17) (-3.04) "55 '132‘ 1'99
CVTENO .0070 —.0461 -.0909

(1.97) (_2.65) (_1.57) 5.41 .1551 1.96
CRPBND .0000 -.0503 -.0911 7.67 .2176 1.97

(2.19) (-3.39) (-1.60)
COMMSTK .0077 -.0000 —.1393

(1.66) (_.0031) (_1.92) 1.07 .0351 1.99
SMLLSTx .0070 .0017 -J1501

(1.64) (.3052) (-2.00) 2.01 .0403 2.00

 

‘Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique

to correct the first order serial correlation of the error.

12].

TABLE 6.17

TESTS OF THE GESXE-ROLL MODEL: 1959-1983
SURVEY RESEARCH CENTER PROXY

(“t ’ “t-1’ ' ° I “lit I B2"t-1 I ‘t

 

 

B1 82
.3
Asset a (T-Ratio) (T-Ratio) F R‘ 0.w.t
TEILL .0050 .1144 —.2303 . 07 0.0- . I.
(1.40) (1.20) (-2.23) ‘ “ ' "‘ ‘ "
GVTBNO .0055 -.0154 -.1001 . 35 0537 . .3
(2.01) (-1.21) (-1.70) ‘ “" ‘ '
CRFEN0 .0071 -.0194 -.1066 , .9 2943 . 09
(2.22: (-1.76) (-1.93)
COMMSTK .0033 -.020: -.0324 .. $1 ..61 . A-
(2.72; (-3.37) (-1.60) “ “‘ ‘ “‘
SMLLSTK .0077 -.0142 —.0600 “.26' . a.
2.37) _ (-3.99: (-1.17) '5C91 '“"' 7"“

 

aAdjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique

to correct the first order serial correlati a of the errcr.

122
changes in inflationary expectations and (2) the previous
period's expected inflation rate and changes in
inflationary expectations should be negative. For
T-Bills. the coefficient for the return on the asset is
positive in all three cases. However. for both
government and corporate bonds. both coefficients are
negative for all three inflation proxies. With the
exception of the ARIMA model. the two coefficients are
also negative for both common stock and small stock
returns. Thus. the results indicate that the Ceske-Roll
reversed causality model is a reasonable explanation for
the relationship between inflation and the returns on

both debt and equity securities.

VI. AN ALTERNATIVE_T§ST OF THE DARBY-FELDSTEIN
NYFOTHESIE
As discussed in Chapter Two. Darby (1975) and
Feldstein (1976) modified Fisher's analysis to allow for
the taxation of nominal interest income and/or the tax
deductibility of nominal interest payments. These
authors argued that investors base their decisions not on
expected real returns. but on expected real after-tax
returns. Assuming that the real after-tax interest rate
remains constant. this implies that the response of
nominal interest to anticipated inflation is given by:
dR l

a? I 1 - t

123
As discussed in Chapter Five. if the Darby-Feldstein
hypothesis is an appropriate characterization of
financial market behavior. the nominal tax-exempt
interest rate should adjust on a one-to-one basis with

expected inflation. Similarly. the nominal taxable
interest rate should adjust at a rate greater than

one-to-one with changes in expected inflation.
Most of the prior empirical studies have attempted to

examine the effects of taxes on nominal interest rates by

distinguishing between the Fisher and Darby-Feldstein
hypotheses. This. in turn. was usually done by looking
at the coefficient of inflation expectations in a reduced
form equation for the nominal interest rate. According
to this approach. a coefficient of expected inflation
greater than one provides support for the Darby-Feldstein
hypothesis.

Using a random walk intercept model. Yun (1984)
estimated the relative responses of taxable and
tax-exempt yields to expected inflation. The results
indicated that the nominal yields on Treasury Bills rise
at a rate greater than one-for-one with expected
inflation. while the nominal yields on default-free
municipal bonds rise approximately one-for-one with
expected inflation. However. Yun fails to relate the
results to empirical evidence of pricing in the municipal
bond market by distinguishing between short run and long

run results.

124

A. Methodology

This study examines the effects of taxes on the
response of the nominal interest rate to expected
inflation by comparing the coefficient of inflation
expectations in the reduced-form equations for nominal
interest rates on tax-exempt and taxable debt instruments
for both the short run and the long run. For this
purpose. we estimate the following regression equations:

E

(6.12) R = a + BEflt + pt

I a + BRW

fl'D-Ifl'

(6.13) R + V

t t

Since the interest income on municipal bonds is
tax-exempt. the coefficient of inflation expectations in
the municipal bond rate regressions (BE) provides an
estimate of the response of nominal interest rates to
expected inflation in the absence of any tax effects. In
order for this coefficient to be consistent with the
Fisher and Darby-Feldstein effects. one might hypothesize
that this coefficient should not be significantly
different from unity.

The coefficient of expected inflation (ET) in the
taxable rate regression provides the magnitude of the
response of nominal interest rates which includes the
Darby—Feldstein tax effect. In accordance with the
Darby-Feldstein hypothesis. this coefficient should be

greater than one. its level being determined by the

prevailing marginal tax rate.

125

In our approach. the difference between the
coefficients of inflation expectations in the tax—exempt
and taxable rate regressions provides a measure of the
tax effect from which the implied marginal tax rate
across the tax-exempt and taxable debt markets can be
calculated. Assuming that the equilibrium condition
between tax-exempt and taxable debt holds. the marginal
tax rate can be calculated from:

(6.14) tm'.= 1 - 133/0T

a. ;§mpirical Resultg

Initially we estimate equations (6.12) and (6.13) for
short-term debt instruments. One-year Treasury bills and
one-year prime housing notes are used as our proxies for
default-free taxable bonds and tax-exempt bonds.
respectively. The yields on one-year Treasury bills and
one-year prime housing notes were obtained from Salomon
Brothers' An Analytiggl_8ecggd of Yields and Yigld
Spreads. As our measure of short—run inflation
expectations. we use the twelve-month direct survey
estimates from the Livingston series.8 Our sample
period covers 1961 through 1984.

Using short-term data. we obtain the following
estimates for equations (6.12) and (6.13). (t statistics

are given in parentheses.)

126

(6.12') R5 = 2.010 + .455“
(3.06) (4.63)

t

R2 e .1397 0.w. . 2.070 5.0.(85) e .0933

(6.13') RE . 3.166 + .096n
(3.71) (5.57)

t

R2 = .2001 D.W. c 2.103 S.E.(BT) g .1507

The equations indicate that estimate of both BE
and 8T are less than one. The coefficient of expected
inflation in the municipal bond rate regression is
significantly different from one thus contradicting the
Darby-Feldstein hypothesis. Although the coefficient of
expected inflation in the Treasury bill rate regressions
is not significantly different from one. the coefficient
is less than unity which is inconsistent with the
Darby-Feldstein hypothesis. In both cases the adjusted
R2 is quite low suggesting that factors other than
expected inflation influence nominal yields. Using
equation (6.14). the implied short-run marginal tax rate
applicable to municipal bonds is 49.2 percent. This is
consistent with the hypothesis that the relevant
short-run tax rate is that of commercial banks.

We might expect the long-run response of nominal
yields to anticipated inflation to be larger than the
short-run response. It is possible that nominal rates
adjust to expected inflation with a lag. Consequently.

we re-estimate equations (6.12) and (6.13) using

127

long—term interest rates. In this instance. we use the
yields on long-term corporate bonds and long-term
municipal bonds from Moody's Industrial Manual as our
proxies for taxable and tax-exempt nominal rates. In
order to equate the risk of taxable and non-taxable
obligations. we use Aaa rated debts for both the
financial instruments.9 It is. however. inappropriate
to use short-run expectations for regressions involving
long-term interest rates. Following Summers (1963). we
derive a series of long-term inflation expectations by
removing cyclical effects. -However. unlike Summers we do
not use the actual inflation rates as a proxy for
expected inflation. Instead. we use a twenty-period
moving average of six—month inflation forecasts from the
Livingston series.10

We re-estimate equations (6.12) and (6.13) using the
long-term interest rates and our measure of long-run
inflation expectation (NLR). The results of these
regressions are (t statistics in parentheses):

(6.12") RE 2 2.434 + .0091:LR
(3.512) (6.055)

R2 . .2467 0.w. . 1.940 5.0.(82) . .1453

(6.13") RT = 3.701 + 1.14611LR
(7.327) (10.537)

52 - .4905 0.w. = 2.146 5.0.(81) - .1067

128

The long-run coefficients on expected inflation are
greater in both instances than the short-run
coefficients. For both taxable and tax-exempt
obligations. the coefficients on expected inflation are
insignificantly different from one. For municipal bonds.
this suggests that long-term nominal yields fully
incorporate expected inflation. which is consistent with
the Fisher and Darby-Feldstein hypotheses. The inflation
coefficient in the corporate bond rate regressions.
although insignificantly different from one. is greater
than 1.0 suggesting that long-term corporate yields
incorporate tax effects.

The implied marginal tax bracket applicable to
long-term municipal bonds is calculated from equation
(6.14) to be 22.4 percent. Seater (1985) has calculated
a series of marginal federal income tax rates for private
individuals in the United States based on data reported
in the United States Treasury Department's Statistics of
Income. For the 1961-1980 subperiod. this rate was
calculated to be in the 14 to 18 percent range. However.
Seater‘s estimates are based on all incomes. not just
interest income. One might expect that the marginal
investor in municipal bonds would have a rate somewhat
greater than those reported by Seater. In fact.
Feldstein and Summers (1979) and Gordon and Malkiel

(1979) have provided estimates of average marginal tax

129
rates for interest income which are much higher than
Seater's. Feldstein and Summers estimated this rate to
be 42 percent. and Gordon and Malkiel's study suggested a
tax rate of 33 percent.

Even if we use Gordon and Malkiel's intermediate rate
of 33 percent. to be consistent with the Darby-Feldstein
hypothesis. the estimate of 8.1. should be 1,327, We
found that the value of 8T so calculated is not
significantly different from our estimate of 1.146 (at 5
percent level of significance) thus supporting the
Darby-Feldstein hypothesis. However. as our estimate of
quies between 1.000. required to be consistent with
the Fisher hypothesis. and 1.327. to be consistent with
the Darby-Feldstein hypotehsis. one may conjecture that
the value Of 8T of 1.146 is also consistent with the
Nielsen-Gandolfi hypothesis. Moreover. the implied
marginal tax bracket applicable to long—term municipal
bonds is consistent with individuals being the primary
holders of long-term municipal debt. Thus. our evidence
is consistent with the preferred habitat model of
municipal pricing.

It may be noted that while estimates of the
coefficient of inflation expectations in the tax-exempt
long-term municipal rate and taxable long-term corporate
rate regressions are not significantly different from

one. the absolute difference between them is not

130

trivial. In fact. we found that the difference between
the two coefficients is significantly different from zero
at the 99 percent level of confidence. This evidence is
consistent with Darby-Feldstein and Nielsen-Candolfi
hypotheses.

Chapter Six has conducted a number of empirical tests
investigating the relationship between asset returns and
inflation. The next chapter summarizes the results of

9

this study and presents some concluding remarks.

131

FOOTNOTES

1Fama and Schwert excluded observations after 1971
because queues. side payments. and increases in the
various forms of non-price rationing due to price
controls prevented stated prices from accurately
measuring the costs of acquiring goods.

2The time period considered begins with 1959 in

order to permit comparison with the ARIMA model. The
price data used to estimate the ARIMA model began with
1953. Data prior to 1953 would not be particularly
meaningful in estimating the ARIMA model since there was
a substantial upgrading of the C.P.I. at the beginning of
1953. The number of items in the C.P.I. increased
substantially and the major items were sampled more
frequently beginning in 1953. Also. we used 1959 because
there were no U.S. Treasury Bills with more than three
months of maturity until April of that year.

3The commodities included in Moody's Daily Commodity

Price Index. with approximate weights are: cocoa--2;
coffee-~4; copper--5; corn--4: cotton--13; hides-~5;
hogs-~13; lead-~3; rubber--4; silk--4; silver--3; steel
scrap--10; sugar-—10; wheat-~13; and wool-~7. This index
may not completely reflect the returns obtainable from
commodities futures contracts. When investors take a

long position in a futures contract. they do not buy it

132
in the sense that they would buy a stock or bond or the
physical commodity itself. Instead. they agree to '
purchase the commodity for a specified price at a
specified time in the future. Since traders must deposit
only 5 to 15 percent of a contract's value. price
fluctuations are magnified greatly.

4As indicated earlier. there is a great deal of

controversy regarding this point.

5The standard error of the coefficient is unaffected
by algebraic manipulation.

6It can also be noted that the coefficients for

expected inflation for long-term government and corporate
bonds are also insignificantly different from 1.0 due to
the large standard errors of the estimate. Thus. the
power °f the test. 3°: 81 I 1. is weak. Following Eddy
and Seifert (1985). this study adopts the convention that
if a coefficient is insignificantly different from zero.
the asset is not an inflation hedge (even though the

coefficient may also be insignificantly different from

one).

7Although the original Geske-Roll model was a

non-linear one. they found that the ordinary least

squares estimates were very close to the ones using the

non-linear method.

133

8The twelve-month Livingston data used in this
section is the original unadjusted series published by
Livingston.

9Trzcinka (1982) argues that the rating agencies do

not attempt to make ratings comparable between corporate
and municipal bonds. He suggests that municipals are

somewhat more risky than corporate bonds of the same

rating.

10This measure of long-run inflation expectations may

be questioned. However. in the absence of the
availability of direct survey data on long-term inflation
expectations. any empirical study employing long-run

expectations is likely to suffer from this drawback.

CHAPTER SEVEN

SUMMARY AND CONCLUDING REMARKS

This chapter consists of three sections. In the
first section, the major empirical results of this study
are reviewed and summarized. The second section cites
some limitations of this study. The third section
discusses C.P.I. Inflation Futures Contracts which can
assist the investor in protecting against price level

changes.

I. SUMMARY

The importance of inflationary expectations in
nominal interest rate determination and the association
between inflation and the returns on alternative asset
categories have attracted substantial attention in the

literature. However. theoretical and empirical studies

on these issues have led to conflicting results.

Although the Fisher Hypothesis suggests that the
nominal rate of return on a financial asset is equal to
the real rate of return and the expected inflation rate
over the life of the asset. three alternative hypotheses
have been proposed. The Mundell-Tobin Effect argues that
the real rate is inversely related to the expected
inflation rate and therefore the nominal interest rate
changes by less than the changes in expected inflation.

134

135
The Feldstein-Darby Effect argues that the response of
nominal interest rates to expected inflation is greater
than unity due to tax effects. Finally. the Inverted
Fisher Effect suggests that the nominal interest rate is
constant and the real interest rate is inversely related
to the expected inflation rate.

Since the expected inflation rate is not directly
observable. several alternative measures have been
utilized in previous empirical work. These include:

1. distributed lag model

2. extrapolative model

(0

adaptive model

4. monetary information model

5. Livingston survey data

6. Survey Research Center data

7. nominal T-Bill rate measure

8. time series (ARIMA) model.

This study provides a comprehensive analysis of the
forecasting accuracy of these inflation proxies for the
1959-1983 period. The Survey Research Center measure was
the most highly correlated with the actual inflation
rate. This study also found that none of the alternative
inflation proxies fully satisfied rationality.

Therefore. expectations about future inflation do not
appear to have been formed in a manner that fully
incorporates all currently available and relevant

information.

136
Finally. this analysis found that the Survey Research
Center measure had the lowest forecast dispersion as
indicated by the mean absolute error. mean square error.
and Theil's U coefficient.

After reviewing the theoretical arguments regarding
the inflation hedging capabilities of several alternative
asset categories. the dissertation re-examines the
relationship between the returns on the alternative asset
categories and the components of inflation. This study
augments the previous work on the Fisher Hypothesis as it
applies to the returns on alternative asset categories in
three important ways. First. this analysis covers a more
recent time period than earlier studies which used data
prior to the 1970s. Rates of inflation have been
substantially higher and more volatile since 1971.
Second. the conflicting results of prior studies suggest
that the effects of inflation on asset returns may be a
function of the proxy used to measure expected
inflation. Therefore. this thesis uses three different
proxies--including the ARIMA model. the Survey Research
Center model. and the nominal Treasury Bill rate model to
measure the effect of inflation on asset returns. Third.
this study includes previously unstudied asset categories
such as "small" common stocks. municipal bonds. and
commodities.

The empirical results of this study suggest that. in

general. financial assets are poor hedges against

137
expected inflation. In contrast to Fama and Schwert's
previous study. g0vernment bonds do not appear to be a
hedge against expected inflation. This suggests that
Fama and Schwert's previous results may have been period
dependent. Although the coefficients for expected
inflation for small stocks and commodities are positive
for both the ARIMA and Treasury Bill proxies. the
estimates are not significantly different from zero. On
the other hand. the coefficient for expected inflation
for small stocks is significantly greater than zero.
suggesting that small stocks are a partial hedge against
expected inflation. This suggests that the results of
empirical investigations of the relationship between
asset returns and inflation may be proxy dependent. Of
the assets considered. only Treasury Bills appear to
offer a partial hedge against expected inflation. This
suggests that the real returns on debt and equity
securities are negatively related to the expected
inflation rate.

The results indicate that the returns on both debt
and equity securities are negatively related to both
unanticipated inflation and the changes in inflation
expectations. However. the magnitude of the coefficients
for the changes in inflation expectations for Treasury
Bills. government bonds. and corporate bonds differs

depending on which proxy is employed.

 

138

The coefficients for expected inflation for
fixed-income securities are. in general. significantly.
greater than zero for both the ARIMA and Treasury Bill
inflation proxies. However. these coefficients are also
significantly different from one. suggesting partial
adjustment of nominal interest rates to inflation. On
the other hand. the coefficients for expected inflation
for fixed—income securities are insignificantly different
from zero when the Survey Research Center proxy is
employed. Therefore. the results for the Survey Research
Center measure are consistent with the Inverted Fisher
Hypothesis.

This study also tests the Geske-Roll reverse
causality model which argues that a change in the nominal
return on a financial asset signals a change in
inflationary expectations of the opposite sign. The
results indicate that the Geske-Roll reverse causality
model holds for long-term government and corporate bonds
as well as for equity securities.

The dissertation also provides an alternative test of
the Darby-Feldstein hypothesis by comparing the
coefficient of inflation expectations for two regression
equations involving nominal yields for taxable and
tax-exempt debt. Using short-run inflation expectations.
the Darby-Feldstein hypothesis is not empirically

supported. However. using long-run inflation

139
expectations. the nominal yields on corporate bonds
adjust at a rate greater than one—for-one with expected
inflation. whereas the nominal yields on municipal bonds
rise one—for-one with expected inflation. This evidence
is consistent with the Darby-Feldstein hypothesis in the
long run. In addition. these results also support a

preferred habitat theory of municipal bond pricing.

:1. gLIMITATIQNS

Although this study has provided additional insight
into the relationship between asset returns and
inflation. several limitations may be noted. First. it
is apparent that the relationship between inflation and
the returns on financial assets does not lend itself to
simple explanations. nor does it appear to be stable over
time. Therefore. it would be useful to examine the
association between inflation and asset returns over
heterogeneous subperiods. In particular. it would be
interesting to determine if there was a change in the
relationship following the shift in Federal Reserve Board
operating procedures in October 1979. In addition. the
generally low coefficients of determination in this
analysis indicate that other macro-economic variables
affect nominal asset returns. Further research should be
done to identify these variables.

In addition. it would be useful to examine the

relationship between inflation and asset returns over a

140
longer time period. As Chapter Six indicated. it is
possible that nominal interest rates adjust to inflation
expectations with a lag. Therefore. researchers must be
careful in specifying the holding period of the
security. The influence of six-month inflation
expectations should diminish as the holding period of the
asset extends beyond six months. Consequently. the
adjustment of long-run nominal interest rates to
inflation expectations should be investigated using some
measure of long—run inflation expectations. It is likely
that interest rates adjust more completely to long—run
inflation expectations. Although it is difficult to
predict expected inflation when one is dealing with long
time periods. future research efforts should be devoted
to determining a measure of long-run inflation
expectations.

Finally. it would be instructive to examine whether
the relationship between inflation and asset returns is
consistent across different industries. It is likely
that there are cross-sectional differences since
inflation should have a heterogeneous impact on different

industries.

II. THE C.P.I. INFORMATION FUTURES MARKET

The preceding analysis has indicated that

conventional financial assets do not provide sufficient

141

protection against price level changes. However. the
C.P.I.-W futures market should enable investors to lock
in real rats of return through the creation of inflation-
proof derivative investments.

A. Contract Characteristics

The C.P.I. Inflation Futures Contract was developed
by the Coffee. Sugar. and Cocoa Exchange in 1985 in order
to provide a mechanism for hedging against purchasing
power risk. The basis of the futures contract is the
C.P.I.-W. the Consumer Price Index for Urban Wage Earners
and Clerical Workers.1

The futures contracts use 1967 as the base year and
one hundred as the C.P.I.-W index for that year. The
contracts are traded for quarterly settlement (in
January. April. July. and October) during the nearest
twelve months and semi-annual settlement (in January and
July) over the next 24 months. The face value of the
futures contract is $1.000 multiplied by the actual index
number released each month by the Bureau of Labor
Statistics of the U.S. Department of Labor. The exchange
requires the trader to put down only $1,500 in cash for
each contract.

Cash settlement is an integral feature of the C.P.I.
futures contract. Unlike most futures contracts on
commodities and government securities. no intent to

deliver are required. Settlement involves a cash payment

142
or collection on any contract not offset prior to the
expiration of trading. Settlement takes place on the day
the C.P.I.-W is announced by the Bureau of Labor
Statistics. Trading in the futures contract ends at the
close of trading two business days prior to the
settlement date.

A position in the C.P.I. futures contract can be
liquidated prior to the expiration of trading through the
execution of an offsetting transaction. A long (buy)
position in a futures contract can be liquidated through
the sale of a futures contract with the same expiration
date. Likewise a short (sell) position can be liquidated
by an offsetting purchase. In either case. the gain or
loss on the futures position is determined by the
difference between the purchase price and the sales
price.

Futures prices are quoted and traded in two deCimal
points. The minimum price fluctuation is .01 or $10 per
contract. The maximum daily price fluctuation is 3.00
points. or $3,000 per contract.

The C.P.I.-W is not seasonally adjusted and has only
been revised twice since February 1971. Although the
C.P.I.-W is only compiled and released on a monthly
basis. the underlying data for the index are constantly
changing. The fact that the index is compiled on a

monthly basis is no more an impediment to trading than

143
the fact that securities which are actively traded report
earnings on a quarterly basis.

Trading in the C.P.I. futures contracts provides an
ongoing market derived consensus of inflation
expectations. A major difficulty in previous empirical
studies involving inflation has been the problem of
measuring the ex-ante inflation rate. Economists have
often failed to correctly forecast future price level
changes. The C.P.I. futures market should provide
valuable data concerning future inflation trends. thus
facilitating empirical investigation of inflation

expectations.

B; Uses 9;;the Futures Contracts

In the C.P.I. futures market. investors can buy or
sell claims to future market baskets of goods services by
trading claims to dollars. The current price of the
futures contract represents an equilibrium price in the
sense that those who believe that inflation will be more
rapid than that level implied by futures price will buy
futures contracts. whereas those who believe inflation
will be less rapid will sell the contracts. Thus. buyers
of the futures contracts can achieve protection against a
higher than expected rate of inflation. Likewise.
sellers of the future contract can achieve protection

against a lower than expected rate of inflation.

144

Investors and money managers can buy the C.P.I.
futures contracts to hedge against inflation risk and
lock in real rates of return. For example. an investor.
could look in the real rate of return by combining fixed
income contracts with C.P.I. futures contracts. Except
for transaction costs. the buyer's financial position
after liquidating his C.P.I. future position would be the
same as if inflation had advanced at the consensus rate
forecasted by C.P.I. futures traders. If inflation
pushes the index to a level above that predicted by
C.P.I. futures traders. the investor would lose more
purchasing power to inflation than anticipated. However.
that loss would be offset by gains on the C.P.I.
contract. Similarly. if future inflation is less than
the consensus foreCasted value. the futures buyer's
contract will lose some of its value. but the underlying
asset that is being hedged will retain more of its value.

Alternatively. participants in the inflation futures
market may utilize speculative strategies in the pursuit
of profit. Buyers of C.P.I.-W futures profit from an
increase in the rate of inflation and lose from a
decrease in the rate of inflation. On the other hand.
sellers of C.P.I.-W futures contracts gain from a
decrease in the inflation rate and lose from an increase

in the inflation rate.

145

Most participants in the C.P.I. futures market are
expected to be buyers rather than sellers. Therefore.
the market might be expected to have a forward price
which is greater than the spot price. This is often
cited as the major potential difficulty associated with
this market.

C. Concluding Comments

Inflation changes have not been well correlated with
the changes in returns of financial assets. In addition.
economists have encountered difficulty in forecasting
price level changes. These concerns may be alleviated by
the C.P.I. futures market. This market enables investors
to hedge against unforeseeable variability in rates of
inflation. and therefore lock in real rates of return.
Also. the inflation futures market provides an ongoing
measure of consensus price expectations which should he

useful in future empirical studies.

146

FOOTNOTES

1The Bureau of Labor Statistics also publishes the
C.P.I.-W. the Consumer Price Index for all urban
consumers. The price movements of the two indices are
highly correlated. The futures contract is based on the
C.P.I.-W because it is the index used in conjunction with

wage and Social Security cost of living adjustments.

APPENDIX A

TIME SERIES METHODOLOGY

147

APPENDIX A
TIME SERIES METHODOLOGY

This dissertation uses the Box-Jenkins modeling
techniques to construct a forecasting model for the C.P.I.
inflation rate. This methodology has been used in several
previous studies of the Fisher hypothesis including Hess and
Bicksler (1975) and Pearce (1979). A discussion of the
classes of time series models follows. The second section of
the appendix presents a discussion of the ARIMA model
building process.
I. CLASSES 0F ARIMA MODELS

ARIMA models can be used to forecast future values of a
time series using the past values and/or the past error terms
(shocks) of a time series. These models are used to
determine the process generating an economic variable using a
set of autoregressive. integrative. and moving average terms.

A fundamental assumption of the time series technique is
that the time series is stationary. A stationary series is
in a state of statistical equilibrium. This occurs when the
joint probability distribution of the series is invariant
with respect to where one looks in the series. Thus. the
mean. variance. and higher moments are the same throughout
the series. If the underlying series is not stationary.
various transformations are available in order to obtain a
stationary series. Differencing is one of the most common

transformations used to induce stationarity.

148

149

If differencing is used. one must sum the differences to
a starting value in order to return to the original data.

The process of summing the differences to a starting value is
referred to as "integrating" in time series nomenclature.

If the current observation of the time series can be
expressed as a linear function of the previous values of the
series and a random shock. the analysis is referred to as an
"autoregressive" (AR) process. If the current observation is
a linear function of the current and previous disturbance
terms or shocks. the analysis is referred to as a "moving
average" (MA) process. A combination of autoregressive and
moving average components can often be used to explain the
sequence of observations. A model using both autoregressive
and moving average terms is called an ARMA process. In the
case where differencing is used along with autoregressive and
moving average components. the process is referred to as an
"autoregressive. integrative. moving average" or ARIMA model.

Box and Jenkins (1976) provided a set of symbolic
operators for describing ARIMA models in a concise manner.

If we let 2t be the deviation of an observation from

its mean value. then the AR model may be represented as
'7

t = ¢12t_1 + 022t_2 + . . . + Opzt_p + at (A01)

Where the 0's are the AR parameters. and the at is the

random shock term. In this process it can be seen that the

series Zt is "regressed" against past values of itself.

150

Equation (A.1) is referred to as an autoregressive process of
order "p."

Alternatively. AR models can be represented using the
backshift operator notation. The backshift operator. 8, is

defined as:
k
B 2t = zt_k. (A.2)
Note that B shifts the time subscript backward by the
exponent of B.
Equation (A.1) may now be represented as
Z=(¢B+¢52+...+ 3p) (A3)
t 1 2 4p .
and letting
4(0) = 1 - 4B - 4202 - - . - - ¢psp (A.4)
Likewise. equation (A.1) can be represented symbolically by:
(.1002. = at. (AS)
The MA class of models may be represented symbolically

as:

zt = at - e

1at_1 - 623t_2 - . . . ' eqat-q (A.6)
Where 2t is the deviation of the observation from its

mean. at is the ramdom shock term at time "t." and the 's

are the MA parameters. Equation (4.9) is referred to as a
moving-average process of order "q."
Employing the backshift operator. equation (A.6) can be

represented as:

2 9
Z = - - - -
Letting
(B) = l -613 _ 9232 - . . . - Bqu (A.8)

151

then Equation A.9 may be represented compactly as

2t = 6(a) at . ’ (A.9)

An important reason for employing a moving average
process is that the number of parameters can be greatly
reduced. Parsimony is the concept of adequately modeling the
process with the fewest number of parameters. In the
interest of parsimony. a mixed model incorporating both
autoregressive and moving average components (i.e.. an ARMA
process) may lead to a model having fewer parameters than
either an AR or a MA process used alone.

In the case of an ARMA model. the current observation is

a function of past values and past disturbances. This model

may be written as:
zt==¢lzt_1 + ... + ¢pzt_P+at--81at_1 - ... - eqat-q (A.10)
Combining the compact representation of the AR model in
equation (A.5) with the compact representation of the MA
process in equation (A.9). the mixed ARMA model in equation
(A.11) may be written as

¢(B)Zt = 4(0) at (A.11)

Often a time series has no affinity for a mean value. If
this is the case. it is said to be nonstationary and certain
transformations are necessary to render the series
stationary. A stationary process is a process in which the
joint distribution (21, , , ,, 2t and zt+1) 13 .
invariant with regard to displacements in time. Thus. a time

series process is stationary if the mean and variance are

152
constant over time. and if the autocorrelation between values
of the process depends only on the distance between the two
time points and not on the time period itself.

To construct a time series model such as that in equation
(A.11) it is first necessary to render the series stationary.
using transformations such as taking the natural logarithm or
dividing by a scale variable. Often. first and higher order
order differencing can be used to induce stationarity. For
example. the process may be nonstationary. but the
differences may be stationary. This can be represented as:

¢(B)(1 - 8) Zt = 4(0) at (A.12)
or in the general form

¢(B)(1 - 8) dZt s (0) at (A.13)
where "d" is the number of consecutive differences.

In sum. the general class of Box-Jenkins autoregressive.
integrated. moving average (ARIMA) models can be expressed
as:

(1- 410-...-¢p0p) (1-0)d zt=(1-3113-...eating“;t (11.14)
The On. m I 1.2....p. are the autoregressive parameters

whereas 9m. m I 1.2.....q are the moving average

parameters. The model is based on the assumption that the

series being modeled is stationary. or that stationarity is
induced by differencing the series an appropriate number of
times. d. In simplified notation. we may denote the general

model as ARIMA (p.d.q).

153

II. MODEL BUILDING

The Box-Jenkins iterative approach for constructing
linear time series models follows a three stage process.
These stages are identification. estimation. and diFTnostic
checking. The following paragraphs describe these steps in
the model building process.

A. Identification

In the identification stage. one selects several possible
models from the general class of ARIMA models presented
earlier. There are two basic tools available for the
identification of time series models. the autocorrelation
function and the partial autocorrelation function. Several
models are usually identified because the sample
autocorrelation (ACF) and partial autocorrelations (PACF) are
subject to sampling error and may be difficult to match with
a specific ARIMA process.

The first step in the identification process is to
examine the series in order to determine the transformations
necessary to convert the original series into a stationary
series. This is accomplished by plotting the data and
calculating the sample autocorrelation function. If the
sample autocorrelation function dies out slowly.
nonstationarity is suspected and further differencing is
suggested. When the spikes in the ACF die out rapidly. there

is no need for additional differencing.

154

Next. the ACF and the PACF of the differenced series are
calculated. Each form of the ARIMA model has a theoretical
ACF and PACF. The orders of p and q are determined by
comparing the sample ACF and PACF with the theoretical
functions for a particular form of the model. An AR model of
order p is characterized by an ACF that tails off after lag p
and a PACF that cuts off abruptly at lag p. Similarly. a MA
model of order q exhibits an ACF that cuts bff abruptly at
lag q and a PACF that tails off after lag q. Finally. if
both the ACF and PACF die off slowly it is an indication of a
mixed ARMA process.

B. .Estimatggp

For the preliminary models identified previously. the
values for the autoregressive and moving average parameters
are then determined. Preliminary estimates of the parameter
values are obtained by comparing the values of the sample
autocorrelations with the parameter representation of the
population autocorrelations and then solving for each
individual parameter. An iterative process is then used to
obtain the maximum likelihood estimates of the parameters.

§;__Qiggnostic Checking

After the parameter estimates for the identified models
are generated. several checks need to be made on the adequacy
of the models. One check is the significance of the

coefficients of the estimated models. Another check on the

estimated models is whether their residuals are random

155
noise. If a posited model adequately describes the
underlying process then the residuals should exhibit no
. significant pattern in their autocorrelation function.
Another factor used in determining the appropriateness of a
model is parsimony. As mentioned earlier. parsimony is the
concept of adequately representing the model with the
smallest number of parameters.

A plot of the residuals and their autocorrelation
function is helpful in checking for patterns in the
residuals. From the plot of the residuals. one can inspect
for the stationarity of the residuals and for constant
variance.

Two statistics have been developed to test if the
residuals are serially uncorrelated. The first of these is
the Box-Pierce 0 statistic which is compared to the
chi-square distribution. Ljung and Box (1978) reported that
the statistic was biased downward. They proposed a statistic
which eliminated the bias found in the Box-Pierce statistic.

The previous sections of this appendix discussed the
categories of time series models. The symbolic notation used
to describe ARIMA models was presented. In addition. the
three steps of the model building process were summarized.
Those desiring a more comprehensive discussion of time series

models should refer to Box and Jenkins (1976).

ACF

@Ih‘4@\nthfll”

AUTOCORRELATION FUNCTION FOR THE ARIMA MODEL

-100

-.366.

-elut
.002
-0312
-0067
.100
‘0131
.034
.519
.166
-0091

-.095

.050

‘ -0573

.13h
-0029
‘0066

.052
-0581

.067

.003

-.S

156

FIGURE A.1

’05

‘0“

.Z .0

.----.--..§----Q----.---09----.----.---o,----f----'

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XXXXX

XXX

X
XXX

XXX

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XX

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XX!
XXX

XX

XXX

XXXX

XX
XXX

xx
xxx .
xxx

X

.Z

.5

.3

1.0

WDVO‘WJ‘HNP

FACF

157

FIGURE A.2
PARTIAL AUTOCORRELATION FUNCTION

FOR THE ARIMA MODEL

‘100 '05 '05 ’0‘. ’02 .0 .2 0'0 0;) 08 100

-0386
-0331
‘0162
-0119
.0155
-0019
-a172
-0125
-a125

.156

.115
-9013
-0318
-.180

.098

.008

.016

e31“
-0191
’OOQT
-0110

.----.- ..- .----.---- O---.Q----.----.---o.----§--o-p

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APPENDIX B
EMPIRICAL TESTS OF THE RELATIONSHIP BETWEEN
INFLATION AND ASSET RETURNS:

OTHER INFLATION PROXIES

158

159

TABLE B.1

EFFECTS OF CHANGES IN EXPECTATIONS

DISTRIBUTED LAG

INFLATION PROXY:

OF INFLATION:
1959-1983

 

 

. _ + _
it I u + Blflt + 82(Pt Ht) 83(1)t "t-I) + ct
81 82 83 2a
Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R 0.w.
TEILL .0326 .7101 .2553 -.1500
(2.91) (3.02) (3.33) (-.0729) 5'23 25°8 1‘90
GVTBND .0511 .6242 —1.9500 -2.4252 ,
(1.69) (1.02) (-2.06) (—1.15) 3'19 1'02 2'°°
CRPBND .0600 .4434 -2.2947 -4.3410 6 9
(2.00) (.6695) (-2.25) (—1.92) 5'15 2195 "°“
COMMOD .0474 .1371 1.4254 .2740 9
(.7010) (.0990) (1.25) (.0017) '51 0'54 1 96
CMMSTK .1204 1.3550 -5.1974 -.0731 , a
(2.21) (1.15) (-2.96) (-.2245) -.00 150° 2'0“
SMLLsTx .1907 3.7101 -6.9262 -4.3011 4 o
(1.06) (1.79) (-2.21) (-.6214) "'16 1284 1"8

 

aAdjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

160

TABLE B.2

EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:

 

 

DISTRIBUTED LAG INFLATION PROXY: 1959-1903
1t I ° + B1“t + B2(Pt ' “t’ + 83(“t ' "t-l) + Et
81 B2 83 a2
Asset o (T-Ratio) (T-Ratio) (T-Ratio) F R D.W.
AAABND .0713 .4000 .0654 - 1567 4 .
(2.77) (3.73) (1.40) (-1.02) “ 1‘ ‘1265 1'27
BAABND .0779 .5372 .0190 - -073 ,
(3.01) (4.61) (.4094) (-3.25 5'85 "46‘ 1°‘°
INDBND .0716 .4302 .0452 - 1012
(3.04) (3.06) (1.17) (—2.02) 3'48 '1445 1'39
UTILEND .07575 .5023 .0403 - 2020 . ,
(2.62) (4.50) (1.25) (-3.15) 5'32 22”9 1"“
MUNI .0530 .3229 -.0139 - 1914 .
(3.27) (2.67) (-.3256) (-1.95) 2“° .0760 1‘23
PFD .0604 .4051 .0319 - 2051
(3.60) (4.69) (1.06) (—2.96) 6'65 '2757 1'56

 

8Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

EFFECTS OF

CHANGES IN
ADAPTIVE INFLATION

161

TABLE B.3

PROXY:

EXPECTATIONS OF INFLATION:
1959-1983

 

 

1t ' ° ‘ “I“t * 8zu’t ' “t’ I 83‘“t ' “tei’ I ‘t
81 82 33 2a
Asset a (T-Ratio) (T-Ratio) (T-Ratio) F R D.w.b
TBILL (31:6) (216:) (316:) 11111;) “'73 ‘3258 1'83
GvanD (31:9? i}?§§, Ii§?§:, (31131? 4.05 .1940 2.03
.2222, 22222., 2:22.22 2.2222
.2222... .2222... 222.22., 2.2222
.2322, 2222:. 3.22:. 222222
ENLLsTR .1792 3.1937 -4.7295 -0.9941 ..90 .106‘ 1.99
(1.75) (1.55) (-1.71) (- .30)

 

.Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct for the first order serial correlation of the error.

162

TABLE B.4
EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:
ADAPTIVE INFLATION PROXY: 1959-1983

it I o + Slut + 82(Pt - Ht) + 83(1)t - “t-I) + t

 

 

t
81 32 B3 20

Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R D.
AAABND .0672 .5004 .0634 -.2176

(2.67) (4.21) (1.76) (-2.69) "57 ‘192‘ 1'
0AA0ND .0711 .6914 .3194 -.3030

(3.22) (5.56) (.0460) (-4.52) 9"‘ '3‘5‘ 1'
INDBND .0667 .5555 .0499 -.2779 5 03 1,17 1

(3.11) (4.60) (1.36) (—3.30) ' 7 °‘“ °
DTILEND .0697 .6400 .0654 -.3140 ,

(2.02) (5.36) (1.70) (-3.02) 8'15 '7089 1'
MUNI .0467 .4550 -.0029 -.2305

(3.60) (3.40) (-.0699) (-2.61) “'10 '1527 1‘
FFD .0639 .4010 .0479 -.2063

(3.90) (4.93) (1.61) (-3.09) 7'53 '2897 1'

 

aAdjusted for degrees of freedom.

bObtained from the maximum likelihood iterative technique to

correct for the first order serial correlation of the error.

163

 

 

TABLE 0.5
EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION.
EXTRAFOLATIVE INFLATION PROXY: 1959-1903
. I + + - -
1t ° 81"t B2‘91: "t’ + B3“"t 1't-I’ * ‘t
B] 82 83 2!
Asset a (T-Ratio) (T-Rat16) (T-Ratio) F R .w.b
AAABND .0690 .4251 .0700 - 0906 .
(2.00) (3.07) (1.97) (-1 17) ‘°35 '1732 ‘7
0AAEND .0745 .5592 .0371 -.2343
(3 11) (4.46) (.9141) (-2.66) 5'39 '2155 '37
INDBND .0609 .4657 .5701 -.1360
(3.00) (4.06) (1.55) (-1.60) "50 '1800 '41
UTILEND .0730 .4056 .0650 -.1900 4
(2.77) (3.94) (1.65) (-2.19) “'91 '1539 '32
NUNI .0494 .3742 .0010 -.1316 4 . .
(3.61) (3.00) (.2500) (-1.40) “"1 1‘11 '22
PFD .0653 .4411 .0373 -.2003
(3.00) (4.90) (1.20) (-3 29) 7'29 '2555 '77

 

aAdjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to
correct for the first order serial correlation of the error.

164

 

 

TABLE B.6
EFFECTS OF CHANGES IN EXPECTATIONS 0F INFLATION:
EXTRAPOLATIVE INFLATION PROXY: 1959-1903
' + + - -
It I “. B1“t B2‘91; 1't) I B3("t "t-l) I et
81 B2 B3 20
Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R D.W.
TBILL .0306 .7310 .2010 -.0105
(2.07) (3.91) (3.03) (-.122 '01 '3046 1'96
GVTBND .0560 .7750 -2.5760 —5.0299 4
(1.39) (1.64) (-2.09) (-1.63) '79 '“27‘ 2'05
CRPBND .0560 .7750 -2.5760 -5 0299
(1.96) (1.31) (-3.36) (-2.62) '79 '327‘ 2'05
COMMOD .0423 .4356 1.4140 -1.1302
(.6014) (.3024) (.9974) (-.3697) '40 '0262 1'95
CMMSTK .1103 .9464 —3.1379 -5.3991 n
(2.00) (.0105) (-2.15) (-1.50) '12 '1159 "°‘
SMLLSTx .1731 3.3609 -4.730 -0.4073
(1.67) (1.50) (-1.02) (-1.34) '71 '0967 3'99

 

8Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct for the first order serial correlation of the error.

EFFECTS

165

TABLE 8.?

OF CHANGES IN EXPECTATIONS

OF INFLATION:

 

 

MONETARY INFORMATION INFLATION PROXY: 1959-1933
i I + + — -
t ° Blnt B2(Pt 1't’ I 83("t "t-l’ I Et
8 B. 8 a
1 2 3 2
Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R .W.
AAAABND .0714 .3460 .0543 -.1400 q
(2.72) (3.00) (1.40) (-1.63) 2'05 '061‘ “7
0AABND .0779 .4416 .0205 —.2553 a 05 11,5 ,5
(2.95) (3.46) (.4649) (-2.60) “' ' ' '
INDBND .0724 .3523 .0420 -.1640 ,
(2.07) (2.97) (1 05) (-1.01) 1'87 '0515 “7
UTILBND .0750 .4205 .0401 -.2403 ,9
(2.64) (3.49) (1.16) (-2.69) 3'17 '1195 ““
MUNI .0542 .2443 -.O140 -.1599
(3.49) (1.92) (-.3152) (—1.62) 1'18 1133 '21
9 ‘3 - ."
PFD .0601 .6416 .0274 .2015 4.31 .1712 .64
(3.57) (3.66) (.0532) (-2.01)

 

8Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

166

 

 

TABLE 0.0
EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:
MONETARY INFORMATION INFLATION PROXY: 1959-1903
‘ + + - -
It I ° B1“t B2(Pt “t’ I B3‘“t "t-i) I Et
81 82 B3 20
Asset 0 (T-Ratio) (T-Ratio) (T-Ratio) F R D.W.
TBILL .0266 .0492 .2363 -.2639 O,
(2.64) (4.01) (3.10) (-1.65) 9'0“ '33‘5 1"“
GYTBND .0400 .5102 -1.6204 -3.3910
(1.61) (.0139 (-1.761) (-1.74) 2'91 '106‘ 2'02
CRPBND .0607 .3475 -2.2615 -4.4792 , a
(2.03) (.4914) (—2.20) (-2.13) 4°“3 '17‘4 "°‘
COMMOD .0357 .6709 .9675 1.695 .4 0.10 1 93
(.5033) (.4501) (.6200) (.5261) “‘ ' ‘ ' “
CMMSTK .1190 1.4760 -5.6206 .0209
(2.07) (1.22) (-3.31) (.2307) 3'90 '2064 2'04
SMLLSTX .1741 4.002 -0.1035 -.9515 0 ,
(1.75) (1.95) (~2.70) (—.1492) 3"“ 1‘74 1'97

 

8Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to
correct the first order serial

correlation of the error.

EFFECTS OF CHANGES IN
LIVINGSTON INFLATION PROXY:

167

TABLE 8.9

1959-1983

EXPECTATIONS OF INFLATION:

 

 

. = + I '1' - -
1t 0 8ft Bz‘pt 1't.) I B3("1: "t-l’ I et
81 32 33 2a
Asset o (T-Ratio) (T-Ratio) (T-Ratio) F R D.W.b
TBILL .0233 .9252 .2164 -.3635 ,
(2.35) (5.33) (2.99) (-2.60) 1°'55 376‘ 1'95
CYTEND 0462 .3501 -.6576 —7.1311 g . a
(1.67) (.6490) (—.7797) (-3.01) 6'7“ 26‘s "01
CRPBND 0671 .1419 -1.299 -7.6609 ,
(2 11) (.2230) (—1.30) (-3.76) 7"5 '287“ 2'01
COMMOD .041- .5196 1.0601 1.0370
( 5976) (.3772) (.6306) 39 '0253 1'94
CMMSTH .1322 .5276 -3.3603 -4.3636 9 _
(2.20) (.4411) (-1 05) (-1.10) 2'“5 '0778 2'04
SMLLSTX .1924 2.0050 -5.0655 -4.9237 n
(1.06) (1 40) (-1.07) (-.7101) "26 '0731 2'00

 

8Adjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

168

 

 

TABLE 0.10
EFFECTS OF CHANGES IN EXPECTATIONS OF INFLATION:
- LIVINGSTON INFLATION PROXY: 1959-1903
. g + _ _ .
It a Slut + 52(Pt wt) + 83(71t rt_1) + It
81 B2 83 20
Asset o (T-Ratio) (T—Ratio) (T-Ratio) F R .W.
AAEND .0640 .6314 .4791 -.2746
(2.09) (5.03) (1.40) (-3.74) 5'84 '267‘ '23
0AABND .0726 .5551 .0300 -.3102 .
(3.25) (4.30) (.7196) (-3.54) 5'04 "01‘ ‘9
INDBND .0650 .5231 .0427 -.2099 a ,
(3.11) (4.60) (1.17) (-3.67) 5'7“ '2282 “2
OTILEND .0704 .5527 .4907 -.2557 4 2.
(2.02) (4.50) (1.20) (-3.05) “ 57 2“1 '09
NONI .0463 344 -.1371 -.2614 4
(3.64) (3 52) (-.3363) (-3.01) ‘°“° '1709 '11
PFD .0650 .-920 .0420 —.2306
(3.90) (4.06) (1.35) (-3.55) 5"5 '2177 '57

 

aAdjusted for degrees of freedom.

b

Obtained from the maximum likelihood iterative technique to

correct the first order serial correlation of the error.

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