INFLATIONARY EXPECTATIONS
AND THE
TERM STRUCTURE OF INTEREST RATES
BY
JAMES MCCALL TIPTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
1981
INFLATIONARY EXPECTATIONS
AND THE
TERM STRUCTURE OF INTEREST RATES
BY
JAMES MCCALL TIPTON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN
PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
Copyright 1981
by
James McCall Tipton
ACKNOWLEDGEMENTS
To my patient and confident adviser, Professor
William A. Bomberger, I would like to express my deepest
gratitude and sincerely say thank you very much for all
that you have done for me.
I also wish to thank Professors Leonall C. Andersen,
David A Denslow, H. Russell Fogler, and James T. McClave
for their reading and comments on this manuscript.
I am also grateful to Michael Conlon for his invalu
able computer programming assistance and to Barbara Long
for her superb typing of the manuscript.
I wish to thank Patti Dunett of Action Graphics,
Waco, Texas for her excellent graphics work.
Finally, I would like to express my sincere apprecia
tion for the understanding and support of three special
people: my wife, Barbara, my mother, and my father.
in
TABLE OF CONTENTS
CHAPTER PAGE
ACKNOWLEDGEMENTS
ABSTRACT V
I INTRODUCTION 1
Background 2
The Term Structure of Interest Rates 4
Implications of Both Questions 6
II THEORETICAL MODEL 7
Macroeconomic Model 7
Other Variables 14
III TESTING AND RESULTS 19
Preliminary Step to the Test 20
The Main Test 3 6
The Subperiod Test 42
Notes 43
IV ANALYSIS OF RESULTS 4 9
The Term Structure of Inflationary Expec
tations 4 9
Interest Rates and Inflationary Expec
tations 52
Summary Statistics and Graphs 52
Maturity Matching 56
Problems and Extensions 60
V CONCLUSIONS 64
BIBLIOGRAPHY 66
BIOGRAPHICAL SKETCH 7 5
IV
Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
INFLATIONARY EXPECTATIONS
AND THE
TERM STRUCTURE OF INTEREST RATES
By
James McCall Tipton
March 1981
Chairman: William A. Bomberger
Major Department: Economics ,
Individuals find it to their advantage to gather
information about the sources of inflation. In this
study I assume they use this information by forming
rational forecasts based on the structure of the process
which precedes inflation. Individuals then are assumed
to use the most important variables of that process in
order to form more accurate forecasts of the future course
of inflation.
The stochastic processes which generate those patterns
of inflationary expectations for the different monthly
horizons were then matched with interest rates of different
monthly horizons. This was to test if the horizon of
inflationary expectations matched the horizon of the same
interest rate as the best explanatory variable for the
v
Fisher equation
r
t
e
Pt
*t + Ptir
(1)
The rfc represents the nominal rate of interest at time
period t, pt represents the real rate of interest deter
mined by the intersection of the supply and demand curves
(roughly speaking the net savings and net investment
curves) from the real section of the economy at time t,
tt^ represents the expected future price appreciation at
time t, and ptir is the crossproduct which is of such
small magnitude that it is dropped.
Using this matching approach the paper investigates
the single idea that the expected real returns from hold
ing debts to maturity of different terms are more similar
than the nominal returns. Therefore, as one examines the
structure of interest rates at a point in time, part or
all of that structure is a reflection of the underlying
structure of expected inflation.
This study shows, both in a main period, January 1959
to December 1978, and in an inflationary subperiod,
January 1965 to December 1978, the horizon of inflationary
expectations does not hold as the best explanatory vari
able for the nominal interest rate of the same horizon.
Thus the underlying structure of expected inflation is
not completely reflected in the term structure of
interest rates.
vi
CHAPTER I
INTRODUCTION
Perhaps as never before, today's financial economist
faces tremendous uncertainty in our present economy.
Each day seems to bring big surprises as interest rates
soar and fall and inflation spirals upward to previously
unheard of rates. Indeed, the uncertainty of the present
age seems to be the only thing we can be certain of.
One of the more interesting questions in the theory
of interest rates then becomes, "How does inflation affect
interest rates?" More specifically, "How does expected
inflation affect the nominal (market) rate of interest?"
It is rightly noted that expected inflation is not the
only argument in interest rate determination, but these
types of questions are important to economic theory
because it is through their assessment that we evaluate
the effect of inflation on the real cost of capital and
the real return to investors. These questions are quite
broad in scope and currently involve much disagreement
over the manner in which "expectations" of inflation are
formed as well as the actual number of determinants to be
used in their formulation.
1
2
The purpose of this study is to test a single idea:
the expected real returns from holding debts to maturity
of different terms are more similar than the nominal
returns. Therefore, as one examines the structure of
interest rates at a point in time, part or all of that
structure is a reflection of the underlying structure of
expected inflation.
An approach which would shed some light on these and
related issues of expectations theories would be to answer
the question, "Do different patterns which provide the
best predictors of inflation over a given horizon also
provide the best explanation of the movements of market
interest rates of comparable duration?" This then will
indicate the underlying expectations structure and its
relationship to the nominal interest rate structure.
Because we are studying the effects of expected
inflation on interest rates of different durations, we
are in effect studying the effects of expected inflation
on the term structure of interest rates. Alternatively
stated, the study becomes a study of the effect that
expectation of inflation has on the relationship among
different maturities of debt.
Background
Interest Rates and Inflation. Irving Fisher (1930)
was the first to formulate a systematic theory of the
business cycle. This theory involved money stock changes,
3
commodity price changes, and a disparity between money
interest rates and real interest rates induced by the
commodity price changes which then caused the cycle to
evolve.
It is from this work of Fisher, which drew upon the
earlier work of Henry Thornton (1802), that we obtain the
hypothesis that the nominal interest rate moves in the
opposite direction to changes in the value of money
(therefore, the same direction as price changes). This
proposition is generally presented in some form similar to
rt = pt + ut + pt7Tt (1)
where rfc represents the nominal rate of interest at time
period t, p represents the real rate of interest deter
mined by the intersection of the supply and demand curves
(roughly speaking the net savings and net investment
curves) from the real section of the economy at time t, tt
represents the expected future price appreciation at time
t, and is the crossproduct which is of such small
magnitude that it is dropped.
The hypothesis is then generally tested in some form
of a distributed lag equation as
rt = Pt + (2)
Then the heroic assumption is made under the economic
theory of a longrun classical equilibrium that the real
rt (D
4
rate is equal to some constant plus an error term, with
zero mean and finite variance,
5t = o + t (3)
With statistical independence of the terms on the
righthand side in equation (2), the test equation
generally has become
rt = a0 + + Â£t (4)
where
m
i=o
aipti
(5)
That is, expectations of inflation are dependent on some
past series of inflation, and (Sargent 1973C) the weights
relating that influence sum to one:
T
E ai = 1 (6)
i=o
The Term Structure of Interest Rates
The theoretical relationship between interest rates
of different maturities likewise was being developed dur
ing this and subsequent periods. This question of the
relationship between interest rates and maturity is com
monly called the yield curve. Lutz (1940) restated this
5
classical relationship of the yield curve in terms of five
propositions from which we develop our modern theories.
He hypothesized that given three assumptions: that
everyone knows the future shortterm rate, there are no
transactions cost, and there did not exist any constraints
on shifting investments for either the borrower or the
lender, that the relationship between the shortterm and
longterm rates:
1. could be conceived of as the longterm rate
being an average of the future shortterm rates,
2. that the longterm rate could never fluctuate as
widely as the shortterm rate,
3. that it is possible that the longterm rate may
move temporarily opposite to the short rate,
4. that the current yield to redemption of a long
term bond will be above the current shortterm
rates up to the maturity date of the bond is
above the current shortterm rate (and vice
versa) and
5. that the return on an investment for a given
time is the same no matter in what form the
investment is made:
nominal interest rate + capital gains (or capital loss)
purchasing price
The excitement then comes when the assumption of
certainty about future shortterm rates is dropped.
Allowing for uncertainty then creates theories based on
expected future shortterm rates being related to long
term rates.
6
Implications of Both Questions
Until the recent article by Modigliani and Shiller
(1973) both bodies of economic theory, i.e., the question
of inflation and the interest rate and the question of
the term structure of interest rates, were being developed
in isolation. The Modigliani and Shiller article appears
to be the first attempt to combine the two questions into
one. Their article, as well as others, views only one
stochastic process as generating expectations for the debt
maturities studied.
If one believes the Fisher hypothesis to be true, the
fact that we observe different expected future shortterm
rates and negative real rates, ex_ post, may be compatible
with the notion of more than one stochastic process
generating the expectation equation during periods of
transition from one longrun equilibrium to another long
run equilibrium.
CHAPTER II
THEORETICAL MODEL
Macroeconomic Model
Sargent (1972b) has pointed out that if we wish to
study the relationship between inflationary expectations
and the nominal rate of interest it is most appropriate
to do so in the context of a macroeconomic system.
Of course, what is rational, or consistent, for the
individual to use in formulating his expectations is a
function of what one places within the model. The follow
ing simplified ISLM model is presented in order to talk
about possible different stochastic processes generating
inflationary expectations for different horizons.
Let the production sector be described by the follow
ing IS curve:
Yt = + aQ + ct]_ (rt7r) + vt (7)
where aQ > 0 and a]_ < 0, Yt is the log of real output, Y
is the log of real fullemployment output, and vt is the
disturbance term.
The monetary sector is represented as the following
LM curve:
7
r+hi
8
m = Y. + 8
t t o
Blrt + Â£t
(8)
where 8 ]_ < 0 and mt is defined as the log of real money
balances, i.e., log MSt log PLt, where MSfc is the
nominal money supply and PLt is the price level and et
represents the disturbance term.
The model's price adjustment equation represents an
intermediate position between the Keynesians and
Monetarists. Friedman (1970) described the price adjust
ment equation as the crucial difference in the way they
close ISLM models. Keynesians regard the money wage
rate as being exogenously determined and assume that firms
set prices equal to or proportional to marginal costs.
Assuming that shortrun marginal cost are constant this
means constant priced and thus implies that PLt is exoge
nous and that we need to solve for Yt and rfc. Monetarists,
on the other hand, assume output, Yt, equals the exoge
F
nously given fullemployment level, Yfc, and treat r^ and
PLt as endogenous variables.
This model assumes that the percentage change in
money wages can be represented by an expectational version
of the Phillips curve,
wt = a Ut + btir.1 (9)
where a < 0 and b = 1. In this equation, a defines the
shortrun tradeoff between the unemployment rate and the
rate of wage inflation, w, for a given
, and b
9
measures the extent to which inflationary expectations are
reflected in current wage changes. The shortrun capital
stock is fixed, KQ: thus real output is a function of the
other input factor, i.e., employment
Yfc = f(Nt) (10)
where f" (Nt) > 0.
Assuming the unemployment rate is a linear function
about the fullemployment level, N, then
N.
U = f = k(YtY^) (11)
Nt
where k < 0.
With the assumptions that prices are a constant
markup on unit labor costs, no technological change, and
constant labor productivity, then we may define the unit
of output to be such that
Pt = wfc (12)
Substituting (12) and (11) into (9) and rearranging terms,
the Phillips Curve becomes
Yt = + nt (13)
where y > 0 and r), is the disturbance term. If in the
long run one held expectations based on some constant
10
level of past inflation, then (P^it^) would equal zero
and
(13a)
The three equations of the system
(7)
(8), and
(13) ,
have four unknowns if tt is treated for the moment as
exogenous: mt, Yr^r and Pt. Thus the system is under
determined. However, mfc and Pfc are related by the
identity
(14)
where represents the exponential growth rate for the
money supply i.e., log MSfc log MSt_^, and P repre
sents the exponential growth rate for inflation, i.e.,
log PL^_ PL^_^. With these four equations then the
system can be solved in the sense that the four endogenous
variables may be expressed as linear functions of the pre
determined variables and the disturbance terms. Let the
exponential growth rate for capacity output, log 
log he defined as Gt. Continue to substitute for
the lagged endogenous variables in the solution for P^
11
until the solution may be written as an infinite sum of
weighted past exogenous variables. The solution for Pt is
Pt =
CO
z
i=o
ki
mti +
oo oo
Z 1 Gt.i + Z
1=0 1=0
n
i ti^tli
+
00
z
i=o
9ivti
+
z
i=o
wiÂ£ti
+ z
i=o
sipti
(15)
where j k^, 1^, n^, q.j_, w, and s^ are simply removed
coefficients that are functions of aQ, aj_, 3]_, and y
which can in principle be solved as well.
Up until now we have been treating ttÂ£ as a constant,
more specifically, t^tl* This is justified on the basis
that the inflationary variable has been determined in the
prior period. We now wish to extend our model to future
periods. Thus with the longerrun extensions of the
model it becomes necessary to treat tt^ as an endogenous
variable. Assume that the objective of the individual is
to forecast this variable minimizing the sum of errors in
the forecasting process, i.e., the individual is rational
if he minimizes the expected value of mean square forecast
error,
min E[e^+i] = E[Pfc+i t+irJ]2 i=l,2, ...
(16)
This implies that if the forecast of inflation is to be
rational then
12
t+i/t E (Pt+i} 1 1,2
(17)
For example, from (15) we can derive the E(I?t+]_) and thus
the function for t+l^t'
0 00 CO
. tt, = j_ + Z k.m, ,, + Z
t+1 t Jo j_=^ i t+li Â¡_=
LiGt+li +
1 niTrt+ii + ,.n ^ivt+ii + iEi wiÂ£t+ii
i=l
i=l
+
^ si nt+ii + ko E
noE(ut+l) + ^o E(vt+1} + wo E(Â£t+1> + soE(T1t+l>*
(18)
To estimate equation (18) directly it is necessary
to have observations on all variables, including the
expected rates of inflation. Unfortunately, direct
observations on expectations for the various horizons are
not widely available. Therefore, a proxy variable needs
to be substituted for expected inflation. In the litera
ture, the typical solution to this problem is to assume
that expectations are generated by a distributed lag on
past values of that variable which are observable. This
will be the procedure followed in this case. However,
the number of lagged variables will be restricted by use
of max x method for autoregressive schemes. The lag
structure generating expectations will follow an
13
extrapolation hypothesis that expected inflation equals a
weighted average of past rates of inflation together with
past rates of growth of the two other variables indicated
in equation (18). Then just what information becomes
important in the formation of rational expectations depends
upon the stochastic processes which generate the R.H.S.
variables, the assumptions we make about the error terms.
For example, suppose that after first differencing all of
the time series on the R.H.S. of (18) we approximate
"white noise." Then the best estimate for the future
value of the expected variable would be its present value
plus some error term. This then would have some weight
attached to it in the overall formulation of expecta
tions. Likewise, if the time series did not convert to
"white noise," but instead showed some pattern which con
tained information on the future value of its series, then
those observations would be included by their weight in
the formation of expectations.
Of course, the disturbance terms in (18) will be
proxied, as well as "full capacity" output. Expectations
themselves are unmeasurable and will require some proxy.
It is proposed that in terms of equation (18) expecta
tions may be represented as a function of the past series
of monthly inflation rates, Consumer Price Index, monthly
growth rate of the money supply, MIA, and the growth rate
of monthly personal income, i.e.,
14
(19)
where the interest rate series serves as a proxy for
E.
Let the exponential growth rate for expected infla
tion one period (one month) forward, log PLt+^ log
. e
PLt, be defined as t+l^t* Likewise, let the exponential
growth rate for expected inflation two periods (two
months) forward, log PLt+2 log PLt, be defined as
0
t+27rt* Continue this definitional pattern for expected
inflation forward for 60 periods. Thus the variable
represents the 5 year expected inflation rate as
of period t.
Other Variables
The theoretical model described above is not too
interesting if the only conclusion is a mathematical
manipulation, since virtually any result could be obtained
depending on what is included in the model. What makes
the "other variables" interesting is that they carry an
implication for different stochastic processes over time.
The economic interpretation of different processes for
the shortterm with several variables versus the longterm
with only one variable has a very interesting explanation
in terms of informationprocessing by the market place.
15
It is suggested that additional variables are
necessary in the shortterm equations because information
sources (data) on which investors form shortterm infla
tionary expectations are embedded with a high degree of
unreliability and possibly conflict. Shortterm forecasts
of inflation are influenced by credit market conditions,
while medium and longterm forecasts are not, and investors
behave as though changes in fundamental economic relation
ships in the shortterm are highly unpredictable and use
several sources (different variables) of information to
reduce their uncertainty. Also, within a shorter time
period expectational effects occasioned by excluded vari
ables, e.g., changes in the money stock, act to reduce
(increase) the real rate of interest temporarily. That
is, money is nonneutral in the shortterm. This was
tested in the above mentioned model by adding additional
variables, e.g., money growth, etc., in the interest rate
determination equation as the variable Zt, i.e.,
rt = ao + 3 lt+n71? + 3 iZt + et 1=2'3'4
(20)
This new formulation for shortterm expectations
does not imply that the markets are not efficient. Rather,
the markets are efficient, i.e., all the currently avail
able information is in the market place, but it implies
that our method of extracting the current information
16
about expected prices is inefficient if we restrict our
selves to only one time series. Thus, the information of
how that series to be predicted will tend to move in the
future is a function not only of its own past, but also
of the indirect effects of other series, which act on
the future course of the price series itself.
Roughly speaking this implies that a chnge in cur
rent personal income carries some information about the
future disturbance term. If income is rising toward full
employment income, it is valid to assume that this imparts
expectations about prices rising in the future. The
individual may even be said to expect prices to start
rising prior to Y^, rather than have some clean break
assumption for price increase or decrease for Yt ^ Y^..
Rapid changes in price levels may signal fear of escalat
ing prices in the immediate future.
Likewise, current money supply changes may signal
the path of future prices over the next few months. This
information may or may not be reflected in the past price
series. Therefore, changes in the money supply were
included in the equation for the formation of expecta
tions.
Finally, the last theoretical variable included in
the category "other variables," i.e., variables other
than the past series of prices, is the change in the
17
shortterm interest rate, rt. This is included on the
grounds that it affects expectations of individuals
regarding the future monetary policy of the government.
This analysis provides a useful approach to the two
bodies of literature which analyze the role of inflation
ary expectations. The first of these deals with the
question of the determinates in the formulation of infla
tionary expectations. The second of these deals with the
question of testing Fisher's hypothesis that rt fully
adjusts to inflationary expectations, thus the sum of the
coefficients of lagged prices equals one. The majority of
the empirical studies yield estimates well below unity and
these are taken to reject the hypothesis (Yohe and
Karnosky 1969 and Gibson 1970). It is possible that the
reason the coefficients sum to less than unity is because
the specification of the formulation of expectations in
these studies is incorrect for the shortterm interest
rates.
All of these "other variables" would have a positive
effect on expectations of changes in prices at least in
the immediate term. Thus they would have an indirect
positive effect on the shortterm interest rate.
With respect to the longterm interest rates, all
information about inflationary expectations would be con
tained in a single variable time series, the price series.
That is, as the time horizon lengthens then the basis for
18
the formulation of expectations about inflation should be
reduced from an informational set of four series to one
series because all relevant information has passed from
the other series to the price series.
These different stochastic processes then answer
questions about the determination of the term structure
of interest rates. They imply that as the informational
set is unrestricted that the empirical conflict in the
literature will be reduced, as R. Craine and J. L. Pierce
(1978) claim. It almost seems counterintuitive to imply
that the agent is more certain about a fiveyear fore
cast of inflation than about a twoyear or threeyear
forecast. But perhaps the individual knows the eventual
outcome of fundamental economic relationships, though
highly uncertain of the shortterm timing for those events.
CHAPTER III
TESTING AND RESULTS
The hypothesis that the market rate of interest on
debt of different durations can best be explained by
inflationary expectations formed over different horizons
that correspond to those debt horizons is a joint hypoth
esis. The first part, a test for the formulation of
expectations, is described below as the preliminary step.
The preliminary step is necessary since inflationary
expectations are constructed by statistical modeling.
The second part, a test matching the different maturities
of interest rates and inflationary expectations, is
designed to answer the question, "How much of the struc
ture of interest rates is caused by the structure of
expected inflation?" The second part is designated below
as the main test.
Finally, a control is conducted for the main test
results by repeating the main test again for a subperiod
which has a different inflation rate. The control, des
ignated below as the subperiod test, allows comparison
and contrast of two different inflation periods.
19
20
Preliminary Step to the Test
The preliminary step to the test is based on the
additional explanatory power yield by the additional
variables in the expected inflation equation. The
traditional linear autoregressive structure:
= a + 3
t1
+ 3
t2
+
+ Â£
(21)
is derived from a generalized polynomial function with
power terms for the various P's including time and its
powers as variables:
P
t
00 j CO
Â£ a. t + Â£
i=o 1 k=l
Â£
j = l
3 .P
D tj
+ Â£
(22)
under the assumption that the linear specification is
correct and will thus lead to an unbiased and efficient
forecast. The best prediction made at the end of the
current month for the next month t+1 then becomes
t+i^t = a + bq Pfc + b2 PÂ£l + +0 (23)
where the forecast error is
Â£t+l = (pt+l ~ t+l71?) (24)
The purpose of including additional autonomous vari
ables is to reduce the size of the error term. For each
duration of actual inflation there exist several possible
autoregressive schemes. The best two, one a single
21
series model and the other a multiple series model, are
pitted against one another to select the better model in
terms of the minimum size error term.
Thus, consider the equations for expected inflation
one month hence tested for fit with the following two
autoregressive schemes:
(A) t+l77? (ML)pti i=0, 1
(25)
versus
(B) t+l *t =
(26)
where (L) 9 (L) 'l (L) and 2 (L) represent the coeff i
w
cients for each lagged polynomial, and the other vari
ables are as described before.
The preliminary step should be considered as a set
of screening steps which, after BoxJenkins (1976) identi
fication for differencing requirements, divides the set
of possible autoregressive orders into those of an
efficient set and those of an inefficient set for both
the single series case and the multiple series case. This
division into subsets is accomplished for the single
series case by performing the max x operation on the
differenced price series. The optimal lag list is then
used to construct the single series model. The multiple
22
series model requires that Pt be regressed onto Mt, Yt,
and rt
Pt = 3i + B2Mt + 8 3 Yt + rt + et (27)
by OLS regression and the residuals from the regression,
p
Â£(./ be operated on by the max x procedure. Selection
is made then from the efficient set of that alternative
ARI order which, according to the max x method, mini
mizes the residual variance for each case, i.e. the
single series case and the multiple series case. Finally,
one selects between alternatives, the single series
equation and the multiple series equation, the better
method of describing inflationary expectations for each
duration.
For example, the choice of the equation to represent
inflationary expectations one month hence, n=l, was
between the single series model:
t+i^t = *31 P.J + .36 P_i + .28 Pj 5 (28)
(.06) (.06) (.06)
with summary statistics:
R2 .78 DW 2.15
MSE 4.8E6 Prob > F .0001
and the multiple series model:
23
. 10 M +
(.05)
.06 Mt1 +
(.05)
.01 Mt_4 .00 Yt .01 Yt1 +
(.05) (. 03) (. 03).
.06 Y + .00 r + .00 r + .00 r
(. 00) (. 00) t (. 00) (. 00)
(2 9)
with summary statistics:
R2 .80
MSE 4.6E6
DW 2.15
Prob > F 0001
The values in parenthesis represent the standard
0
errors. The symbol t+l^t represents the expected infla
tion rate, log PLt+i PLt. Likewise, Pt represents the
actual difference in the past log levels of the monthly
Consumer Price Index, log PLt log PLt_]_, Pt_]_ repre
sents log PL^_2_ log PL^_2, and Pt5 represents log
PLt_5 PLt_g. The actual difference in the past log
levels of the monthly money supply (Ml, currency plus
demand deposits) is represented by where i=0, 1,
2, ... In equation 29, Mt represents log MSt log
represents log MSt_^ log MSt_2, and
represents log MSt_4 log MSt_5 The symbol Y,
where i=0, 1, 2, ... represents changes in the past
levels of monthly Personal Income. For example,
represents log PIfc log PIt_]_/ Yti
represents log
 log
PIt_^ log PIt_2' an<^ Yt4 rePresents log PIt ^
PIt_5 Finally r where i=0, 1, 2, ... represents
the difference in the end of the month annualized bond
equivalent yield of one month seasoned U.S. Government
Treasury Bills. In this case rt represents rt rt_j_,
rt_]_ represents rt_j_ rt_2, and rt_ represents rt_^ 
r 1
rt5
In the above manner then one is able to select the
multiple series model as the better fit for expected
inflation one month forward, t+l77?' during the period
January 1959 to December 1978.
To extend this process forward for t+n periods, we
make use of the general equation for single series fore
casting:
t+n77? = ^1 (t+nl11?) + ^2 (t+n2 ^t) + + ^nl (tn+l17?)
Vtn + ^n+l Ptnl + * (30)
which gives an optimal forecast in the sense of the
minimum extrapolation error (Mincer 1969).
0
For example where n=2, i.e. t+27Tt' substitute the
as yet unknown magnitude into the autoregression by its
extrapolated value. Thus the traditional formulation:
t+lut = a + ^i^t + ()2ptl + * (31)
is substituted into
25
t+27Tt a + ^lPt+1 + ^2pt + ^3Ptl +
A
for Pt+i, giving
(32)
t+2nt a + ^l^t+l^t + Â£t+l^ + ct>2pt + ct)3ptl + +
Â£t+2 (33)
Thus the mean squared error of extrapolation in predic
ting  + 27Tt' the two month expected inflation at time
period t, for the single series forecast is the variance
of (Gget+l + Â£t+2)
Likewise, given that all the R.H.S. series of the
multivariate model have been whitened with constant
variance, the alternative formulation for the multi
variate scheme is
t+2nt ~ a + ^l^t+l^t + et+l) + ^2 pt + ^3ptl + +
0l(t+lMt + et+l^ + ^2 Mt + 3Mtl + +
h + Vt + Vti +
+
^1 (t+l^t + Â£t+l) + ^ 2^t + ^ 3^tl + + ^t+2
(34)
Therefore, the mean squared error in predicting t+2Trt with
P
the multivariate model is the variance of ($1^+1 +
m v v
0lÂ£t+l + 'i'lÂ£t+l + ^lÂ£t+l + Â£t+2 )
26
Renaming the coefficients to eliminate rewriting the
constants and bracketed terms, the choice is between
t+2t 1=0, 1,
(35)
and
t+2Trt = (i)*(L)pti + 0*(DMt_i + 'F*(L)Yt_i +
i2*(L)rt_i i=Q/ lf
(36)
for the two period forecast.
For the horizon n=2, the best single series model
was:
t+27Tt = .60 Pt + 57 Pti + 32 Pt2 + 42 pt5
(.11) (.10) (.10) (.10)
(37)
with summary statistics:
R2 .85 DW 1.14
MSE 1.2E5 Prob > F .0001
The best of the multiple series models, and the better of
the single series versus multiple series models, was:
Tre
t+27rt
.44 P^_ +
(.10)
. 45 P^.i + .22 P, +
(.10) (.10)
. 35
(.10)
pt5
+
.17 Ml. +
(.07)
11 M
(.08) t
+
. 07
(.07)
M
t2
. 00 M
(.07)
t4
+
.01 Yt .00 Yj__i
(. 04) (. 05)
.00 Y^_9 + .12 Y+._a +
(. 05) (. 05)
27
.00 r, .00 r, .00 r. 7 + 0008 r. 4
(.00) C (. 00) (.00) (. 0004)
(38)
with summary statistics:
R2 .88 DW 1.12
MSE 1.1E5 Prob > F .0001
In summary, the technique used to forecast the for
ward one period rate of inflation for periods where n > 1
is that after the usual BoxJenkins identification for
2
stationary.ty and the initial preliminary max x screen for
the optimal lag list, set up a two phase least squares
procedure. The lag list for the future values of the vari
ables for time periods less than n from the first OLS
regression are then used as R.H.S. variables in the
estimation procedure of the second OLS regression. In
this manner, the estimation procedure steps the forecast
forward one step at a time giving the one period forward
rate of expected inflation for that duration and assures
approximately the same variance for each forward rate.
Thus by operating on the autoregressive schemes for
ward, applying Wold's "chain principle of forecasting,"
replacing stochastic elements on the R.H.S. with their
prior lag last, expectations of inflation were derived
for different durations. Table 1 then shows the best
autoregressive scheme using the max x2 approach, and has
several interesting points for later discussion.
Table 1
The Term Structure of Inflationary Expectations, t+nir
1/59 to 12/1978
Monthly
2
Horizon, t*n
Equation
R
MSE
Prob > F
t.i"f
.206 Pt +
(.064) (
.301 Pt_1 4 .
.062) (.
243 Pt_5 3
059) (
. 100 Mt 4 .060 Mt_1 4
.045) (.047)
.007 Mt_4 
(.046)
.001 Yt .
(.029) (.
0U Yt1 +
030) (
.057 Yt_4 4 .0004 rt 4
.029) (.0003)
.0001 rfc1
(.0003)
4 .0003 rt_4
(.0003)
.00
4.6E6
.0001
t*2wt
.444 P 4
(1.04) (
.450 P 4
.099) t"1 (.
.220 P 4
.102) t"2 (
.354 P 4 .171 M +
.095) (.075) t
. 100 4
.067 Mt_2 ~
.001 Mt_4 4
.008 Yt .002 Yt_i

(075)
(.073)
(.073)
(.045) (.046)
.005 Yt_2 4
. ns vt.4 +
.0003 rt 
.0003 rt_1 
(.047)
(.045)
(.0005)
(.0005)
.0006 rt2
4 .0008 rt_4
(.000S)
(.0004)
.88
1.IE5
.0001
t + 3wt
.597/ Pfc +
(.144) (
.559 Pt_!
.141) (
.359 Pt_2
.144) (
.331 Pt_5 4 367 Pt_?
.141) (.145)
4
. 204 M 4
.138 M
.106) (
.153 M 4
.030 M .4 .204 M ?
4
(.102) (
.102) t_2 (
.102) (.107)
.019 Yt 
.014 Yt_x 4
.015 Y._2 +
.105 Yt_4 .081 Yt_7
4
(.063) (
.065) (
.066) (
.064) (.064)
.0002 rt 
(.0007)
.0004 ft_x 4
(.0007)
.0001 rfc2
(.0007)
4 .0001 r. 4
(.0006)
K>
00
.0009 r _
(.0006)
90
2.0E5
0001
98
Table 1 Cont'd
Monthly
Horizon t+n
t*4
e
t
t+5 t
Equation
.500 Pt_2 f .536 Pt_5 t
(.179) (.175) (.179) (. 178)
(.120) (.132) (
186 Mt_2 .041 Mt_4
120) (.128)
.033 .026 Yt1 .066 Y^_, + .060 Y
t2
(.079) (.081) (.082) (.079)
.0002 rt .0003 rt_1 +
(.0009) (.0008)
.0004 rt_2 .002 rt
(.0008) (.0007)
.001 1,
(.0000)
t7
(.025) (.202) (.212) (.203)
.563 P 4 .466 M
(.208) (.148) (.
200 M + 238 M 4
151) 1 (.149)
(.151) (.158)
.074 Yt .075 Yt_1
(.093 (.093)
.004 Yt_4 .079 Yt_5 
(.091) (.096)
.081 Yt_? t .0006 rt 4
(.092) (.001)
(.001 ) (.001)
.001 r.
.003 rf4 003 i
(.0009) (.0009)
(.0009)
t7
.426 Pt
(.181)
.228 Mt
(.134)
.058 Yt
(.080)
~4 *
.597 Pfc
(.211)
.200 M
(.157)
.073 Yt
(.094)
t5 *
MSE
Prob > F
DW
.91 3.1E5 .0001 .81
K)
.93
4.0E5
0001
74
Table 1 Cont'd
Monthly
Horizon t*n
Equation
t*6"t
.939 P 4
967 P 6
.48 3 P 4
657 Pt4 4 '
661 P
(.735)
t.
234)
(.245)
(.234) (.
.243) 1
M3 Pt.7
4
.573 Mt 4
.208 Mt_j
.351 Mt_2 4 .
197 Mt
(.241)
(.170)
(.174)
(.171) (.
, 181)
.5 Mt.s
4
101 "t.7
4 .125 Y 4
.052 y .
,005 Yt
(.175)
(.182)
(.107)
(.106) (,
. 108)
.0002 Yt .
1
 049 Yt_5
( 4 .OOfl Yt_?
.001 it ^ .
.0004 i
< 10S)
(.111)
(.106)
(.001)
(.001)
.001
4
.004 r
4 .004 rL r
* .002 *
(.001)
(.001)
(.001) J
(.001)
e
t*7"t
1.020 Pt 4
.954 Pt_1 4
.616 Pj._2 *
.600 Pt_3 4
.634 P
(.201)
(.
. 20H)
(. 272)
(.267)
(.259)
.749 Pt.s
4
.685 Pt_7
4 .725 Mt 4
.484 M. .4
.637 Mt
(. 276)
(.266)
(.196)
(.196) (
. 203)
112 Mt_3
4
.038 Ht.4
 .117 Ht.5
4 .151 Mt_7 4
.117
(.202)
(.206)
(.202)
(.200)
(.119)
035 Yt.t

.076 Y*.,
 033 Yt_,
.026 Yt_4 
(. 120)
(.120)
(.120)
(.117)
074 Yt.s

.021 Yt.7
* .001 rt +
.001 ft1 4
.003 rt
(. 122)
(.117)
(.001)
(.001) (
.001)
. 005 r
4
.006 r
4 .005 t
4 .002 t
(.001) t_3
(.001) t_4
(.001) t5
(.001) t'7
2
R MSB Prob > F DW
.93 5.3E5 .0001 .72
U)
O
94
6.3E5
0001
67
Table 1 Cont'd
Monthly
Horizon t*n
t*8"t
Equation
701 Pt *
926 P+
858 Pt_2 *
752 Pt_,
650 Pt.4
4
266)
(
286)
(
287)
(.284)
(.283)
661 Pt_5
4
.687 Pt_6
4
.773 Pt.7
+ .856 M. +
.840 M. ,
4
2lJ2)
(.265)
(.286)
(.210)
(.213)
725 Mt.2
f
. 262 Mt ,
4
197 Mt.4
.084 Mt 5

216)
(.221)
(.221)
(.218)
202 Mt.6

.128 Mt_?
4
.073 Yt 
077 Yt_j 
.147 yt_2
221)
(. 223)
(.127)
(.127) i
1.126)
osi yt.3

.051 Yt_4

.047 Yt.s
 .065 yt.6
+
127)
(.128)
( 130)
(.124)
0J1 Vt7
4
.002 t
.003 rt_j
.004 t_2 *
.005 rt.3
4
125)
(.001)
(
.001)
(.001)
(.001)
007 rt_4
4
.008 rt_5
4
.006 it.6
.004 Pt_7
001)
(.001)
(.001)
(.001)
796 p. *
.996 Pt *
.883 P. t
.925 P +
.664 P
4
t
t1
t2
t3
t4
113)
(
.312)
(
.312)
(.309)
(.309)
.674 Pt.5
(.318)
4
049 Pt_6
(. 311)
4
.857 P,
(312)
.803 Mc_2
(.239)
4
.221 Mt.3
(.241)
4
. 325 M,
(.241)
21 "t6
(.241) 6
4
.226 H
(.244)
4
.052 Y
(.138)
.060 Yt_3
(.139)
4
037 t_4
(. 139)
.091 Y,
(.141)
4 .975 Mt 4 .930 Mt_1 4
(.234) (.234)
4 .073 Mt_5 
(.239)
098 y
(.139)
. 116 Y
(.130)
t2
 .019 Y_ Â£ 4 .070 
t O XL~ I
(.136) (.136)
.003 lt 4 .004 ft_1 4 .005 tt_2 4 .006 *t_3 4 .000 *t_4
(.001) (.001) (.002) (.001) (.002)
.009 rt 5 4 .007 r 4 .004
(.002) (.002) (.002)
MSE
Prob > F
DW
.95 6.9E5 .0001 .44
.95
B.2E5
0001
39
Table 1 Cont'd
Monthly
Horizon tn
t*iowt
Equation
.826 Pt
1.014 P
1
. 074P. j 1
975 Pt3 +
626 P
4
(. 343)
(.341)
1
[.341)
(
330)
(. 338)
.878 Pt_5
* 971 Pt6
4
.787 Pt_7
4
1.141 Mt +
(. 349)
(. 341)
(. 342)
(.257)
1.048
.748 Mt_2
4
.351 Mt.3
4
336 Mt_4
4
(256)
(.263)
(.264)
(.264)
.070 Mt_5
 3S1 Mt.fe

.153 Mt_7
4
.029 Yt 
.066 Yt1

(.261)
(.264)
(.267)
(.151)
(.152)
.117 Vt_2
* .024 Yt_,

01) Yt.4

051 Yt_5
4
( 151)
(.152)
(.52)
(.155)
019 Vt_6
* .058 Yt_7
4
. 004 rt 4
005 +
.005 rt_2
4
(.149)
(. 149)
(.002)
(
002)
(.002)
.007 rt_3
* .009 rt.4
4
.010 rt_5
4
.OO7 rt.6
4 .004 rt_
7
(.002)
(.002)
(.002)
(.002)
(.001)
.838 Pfc +
1.193 P. +
1.
162 Pt2 +
.996 P. +
*780 Pt4
4
(.376)
(.374)
(
374) 1 2
(
.370) t_3
(.371) 11 4
.996 Pt_s
907 pt6
4
.781 Pt_7
4
1.265 Mt 4
1.045 M(;_1
4
(.383)
(. 373)
(. 374)
(. 282)
(.282)
.935 Mt.2
f 3T2
4
374 Mt.4

.026 Mt_5

(. 288)
(.290)
(.289)
(.286)
.270 Mt_6
 .247 Mt.7
4
.087 Yfc 
041 
.035 Yt_2

(. 289)
(.293)
(.166)
(
.167)
(.165)
.022 Yt_3
.042 Yt_4

.035 Yt_5
4
.005 Yt.6
+
(.167)
(.167)
(.169)
(.163)
.040 Yt_7
.004 ft
.005 rt. *
.006 rt_2 +
.008 rt_3
4
(.163)
(.002)
(
.002)
(
.002)
(.002)
.010 rt_4
+ .010 r._.
4
.008 ;
4
.005 rt7
(.002)
(.002)
(.002)
(.002)
2
R MSE Prob > F DW
.96 9.8E5 .0001 .38
OJ
K>
96
1.2E4
0001
35
Table 1 Cont'd
Monthly
Horizon t4n
Equation
t*12n
e
1.038 Pt 4
(.414)
1.319 Pti + 1.209 Pt_2 1.144 Pt_3 4
(.411) (.412) (.408)
.860 Pt_4 4
(.408)
.961 P._5 4 .957 P 4 .592 P ? 4 1.328 M 4 1.217 M 4
(.421) (.410) (.413) (.311) (.310)
.933 Mt_2
(.317)
. 399 Mt_i 4 291 M_4
(.318) (.318)
.079 Mt_5 
(.315)
.408 Mt_6
(.320)
.139 Mt_?
(. 322)
.113 Y4 .045 Yt_1 .077 Yt_2 4
(.184) (.184) (.182)
t4l3"t
.047 Yt_3
* .072 Yt_4

.046 Y,.,
 0i6 Yt6
4
(.183)
(. 184)
(.186)
(. 179)
104 Vt_7
4 .004 rt 4
.005 rt_3 4
.007 rt_2 +
.008 rt_3
4
(.180)
(.002)
(
.002)
(.002)
(.002)
.oio t_4
4 .010 rt_,j
4
.008 rt_6 4
.005 rt_7
(.002)
(.002)
(.002)
(.002)
1.131 Pt +
1.532 Pt_x 4
1
429 Pt_2 +
1. 246 Pt_3 *
742 Pt.4
4
(. 1006)
(1.001)
(
.984)
(. 970)
(.970)
1.369 P
4 1.602 P,. ,
4
1.160 P
t 2.579 M +
2.085
4
(1.009)
(.994) t6
(1.006)
(.732)
(731)
1.877 Mt_2
.894 Mt_3
4
1.017 Mt_4
+ .671 H. c
+
(. 747)
(.755)
(.752)
(.743)
34i Mt6
+ .921 Mt_7
4
.522 Yt 4
.310 Yj.,3 +
.310 Yt.j
4
(.751)
(.760)
(.429)
(.430)
(.428)
.384 yt_3
+ .408 Yt.4
4
.277 Yt.5
* 154 Yt_6
+
(.431)
(.431)
(.438)
(.419)
2J9 *t7
O
o
rr
+
.007 r. 4
.008 ft_2 +
.008 ft_3
4
(.420)
(.004)
(
.005)
(.005)
(.005)
.011 f. .
4 .010 r. c
4
.006 r
+ .004 r ,
, t4
t5
t6
t7
(.005)
(.005)
(.005)
(.004)
MSE
Prob > F
DW
.95 1.4E4 .0001 .31
OJ
U)
.94
7.6E4
0001
15
Table 1 Cont'd
Monthly
Horizon t+n
t+36nt
t*48"t
Equation
.423 Pt 4 1.044 Pt_4 1.009 Pt_2 1 082 Pfc_3 4 1.071 Pfc_4 4
(1.448) (.1429) (1.380) (1.371) (1.377)
2.205 Pt_5 4 3.0B6 Pt_6 4 2.004 Pt? 4 3.112 Mfc 4 2.240 Mtl 4
(1.4)9) (1.439) (1.47B) (1.044) (1.055)
2.337 Mt_2 1.1607 Mt_j 2.160 M^_4 1.019 Mt_s +
(1.078) (1.095) (1.081) (1.079)
1.255 Mt.6 2.203 Mt_7 .859 Yt + .763 Yt1 + .930 Yt.2
(1.099) (1.123) (.604) (.605) (.602)
.913 Yt_j + .818 Yt_4 .592 Yt_j .272 Yt_6 +
(.0608) (.607) (.625) (.594)
.387 Yt_7
(.607)
.002 rt .002 rt_i .001 rt2 .001 rt_3 +
(.006) (.007) (.007) (.007)
.003 rt_4 4 .001 rt_5 .003 rt_6 .001 rt_7
(.007) (.007) (.007) (.006)
.394 Pt 4 .802 Pfc1 4 1.296 Pt_2 4 1.634 Pt_3 42.310 Pfc_4 4
(1.893) (1.890) (1.809) (1.789) (1.820)
3. 393 P
+
4655 Pt6
4 3.609 P
4
2.961 H +
(1.858)
(1.872) t
(1.899)
(1.371)
2.087 Mt1
4
2.498 Mt_2
2.069 Mt_3
4
2.450 Mt_4
4
(1.37?)
(1.392)
(1.415)
y.393)
2.056 Mt_5
4
1.685 Mt_6
2.851 Mt_7
4
1.287 Yt +
1.359 Yt_x 4
(1.390)
(1.407)
(1.402)
(.774)
(.775)
1.540 Yt2
4
1.636 Yt_3
4 1.354 Yt_4
4
1375 Yt_5
4
(.777)
(.776)
(.775)
(.789)
Yt.6
4
.851 Yt_,
 .011 rt 
.012 ft.3 
.011 rt_2 
(.755)
(.759)
(.008)
(
.009)
(.009)
.013 rt_3

.006 rt_4
 .009 rt_5

.oio it_6
 .007 rt_7
(.009)
(.009)
(.010)
(.010)
(.009)
MSE
Prob > F
DW
OJ
4^
.95 1.5E3 .0001 .13
95
2.2E3
0001
17
Table 1 Cont'd
Mon th1y
Horizon t+n
t+60nt
Equation
1.028 Pt
2.380 Pt_L 4
3.006 Pt_2
4.202 Pt_3
4.681 P,
(1.97b)
(1.986)
(1.980)
(1.914)
(2.005)
5.947 Pt_5
6.502 Pt_6
6.946 P. 7
2.578 Mt
2.321 M(
(2.250)
(2.286)
(2.255)
(1.382)
(1.378)
2.703 M _
1.963 M
2.406 M_ .
1.810 M _
+
(1.403)
(1.421)
(1.309) k'4
(1.388)
1.630 Mt_6
> 2.665 M^_7
1.303 Yt
1.596 Yt1
1.995 Y,
(1.403)
(1412)
(797)
(.792)
(.793)
2.197 Yt3
1.943 Yt_4
1.821 Yt_5
1.036 Yt_6
+
(.704)
(.786)
(.807)
(.782)
.497 Yt_7
 .018 rt 
.018 ht.1 
.018 rt_2 
.019 rt.
(. 775)
(.009)
(.010)
(.010)
(.010)
.014 ft4
 on rt_5
 .008 rt_6
.006 rt_7
(.010)
(.010)
(.011)
(.010)
U>
U1
2.1E3 .0001 .27
36
The Main Test
The best scheme for each horizon of expected infla
tion having been formulated in the preliminary step,
interest rates of different maturities were then regressed
on the various annualized horizons of Table 1 using a
first order autoregressive scheme
rt =
+ 31 t+n^t +
(39)
where
et = yt al Â£tl (40)
and yt is normally and independently distributed with
mean 0 and variance a We should expect the better fit
for inflation expectations to cause the estimated value
beta, 3 to approach in value the true value of beta.
If we expect the true value of beta to be one, then we
should observe the values of beta increasing towards the
value one as we substitute the better expectation fits
into the equation. The true value of beta is unknown.
It should be noted that it is possible for movement to
be away from the true value. Table 2 is the first order
autoregression of interest rates on the annualized infla
tion schemes from Table 1.
Variables which were placed in the expectations
formulation are then placed into the interest rate equa
tion as additional variables, zt,
control variables. Thus
37
following the same procedure as described above,
rt = ao + 31 t+lut + et
(41)
is compared with
(42)
to conclude if any additional explanatory power can be
gained with the added determinates. The test is then
repeated for each interest rate. Table 3 gives the
results of equation (42) for a first order autoregression.
Thus the horizons are mixed according to the follow
ing pattern.
Pattern A
e
etc.
If the regressions show that the expected inflation for
mulated for a given maturity best explains the interest
rate of that maturity, then the pattern for the highest
R observed would be that of the following,
Table 2
The Fisher Equation, 1/59 to 12/78
Beta/R Square Values
Inflationary Expectations t+nir
1 mo.
2 mo.
3 mo.
4 mo.
5 mo.
6 mo.
7 mo.
8 mo.
9 mo.
I
N
1 mo.
.35/.37
.34/.31
.35/.34
.37/.35
.40/.41
.42/.44
.43/.45
.44/.41
.45/.41
T
E
2 mo.
.32/.35
.31/.29
.32/.32
.33/.33
.36/.37
.38/.40
.38/.40
.40/.35
.40/.35
R
T?
3 mo.
.31/.34
.30/.28
.31/.31
.33/.32
.35/.36
.37/.38
.37/.38
.38/.32
.38/.32
s
T
4 mo.
.32/.35
.31/.30
.32/.32
.34/.34
.37/.38
.38/.41
.38/.40
.39/.34
.39/.34
R
5 mo.
.33/.36
.32/.30
.33/.33
.34/.35
.37/.39
.39/.41
.38/.40
.39/.33
.39/.33
A
T
6 mo.
.33/.36
.31/.30
.33/.33
.34/.34
.37/.39
.39/.41
.38/.40
.39/.33
.39/.33
E
S.
12 mo.
.41/.37
.43/.36
.43/.37
.45/.39
.46/.42
.47/.43
.46/.44
.46/.42
.47/.42
24 mo.
.30/.36
.29/.31
.31/.36
.31/.36
.32/.37
.32/.37
.29/.31
.27/.20
.27/.19
t
36 mo.
.29/.35
.28/.30
.30/.35
.30/.35
.30/.35
.29/.35
.26/.27
.24/.16
.23/.15
48 mo.
.27/.33
.27/.29
.29/.34
.29/.34
.28/.33
.28/.33
.24/.24
.22/.14
.21/.13
60 mo.
.26/.32
.25/.28
.28/.34
.28/.33
.27/.32
.26/.31
.22/.22
. 19/.12
. 18/. 11
u>
oo
Table 2 Cont'd
Inflationary Expectations, t+rnr
10 mo. 11 mo. 12 mo. 24 mo.
I
N
1 mo.
.45/.41
T
E
2 mo.
.41/.35
R
E
3 mo.
.39/.32
S
T
4 mo.
.40/.33
R
5 mo.
.39/.33
A
T
6 mo.
.40/.33
E
s,
12 mo.
.43/.42
r
4
24 mo.
.28/.20
U
36 mo.
.24/.16
48 mo.
.22/.13
60 mo.
.19/. 11
.46/.41
.46/.39
.41/.19
.41/.34
.41/.33
.36/.16
.39/.31
.39/.30
.33/.14
.40/.32
. 40/. 31
.34/.15
.39/.32
.40/.31
.34/.15
.40/.32
.40/.30
.35/.16
.48/.42
.49/.42
.53/.33
.28/.19
.29/.20
.29/.14
.24/.15
.25/.16
.27/.13
. 22/.13
.23/.13
.25/.12
. 19/.11
.20/.11
.23/.11
36 mo.
48 mo.
60 mo.
.38/.13
.40/.14
.44/.20
.34/.12
.34/.12
.38/.18
.30/.09
.31/.10
.36/.16
.32/.10
.33/.11
.38/.18
.31/.10
.32/.11
.36/.17
.32/.11
.32/.11
.37/.17
.51/.23
.50/.23
.46/.30
.28/.11
.30/.13
.34/.20
.28/.11
.31/.14
.34/.23
.27/.11
.30/.15
.34/.23
.26/.11
.29/.15
.33/.24
Table 3
Multiple Regression, 1/59 to 12/78
Beta^/R Square Values
Inflationary Expectations t+nfr
1 mo.
2 mo.
3 mo.
4 mo.
5 mo.
6 mo.
7 mo.
8 mo.
9 mo.
I
N
1 mo.
.39/.43
.40/.41
.42/.45
.44/.48
.46/.52
.47/.54
.47/.54
.49/.53
.49/.53
T
E
2 mo.
.36/.41
.38/.40
.39/.44
.41/.46
.43/.50
.44/.52
.44/.51
.46/.49
.46/.48
R
E
3 mo.
.36/.41
.37/.40
.39/.44
.41/.46
.43/.49
.44/.51
.43/.50
.45/.47
.45/.46
S
T
4 mo.
.37/.42
.38/.41
.40/.45
.42/.48
.44/.52
.45/.53
.44/.52
.46/.48
.46/.47
R
5 mo.
.38/.43
.39/.42
.41/.46
.43/. 49
.45/.53
.45/.54
.44/.52
.45/.47
.45/.47
A
T
6 mo.
.38/.43
.39/.42
.41/.46
.43/.49
.45/.53
.46/.54
. 45/.52
.45/.47
.45/.46
E
12 mo.
.41/.33
.43/.37
.43/.39
.45/.41
.46/.43
.47/.44
.46/.44
.46/.42
.47/.42
/
24 mo.
.32/.39
.33/.37
.35/.42
.35/.42
.35/.43
.35/.42
.32/.34
.30/.24
.30/.23
rt
36 mo.
.29/.36
.30/.34
.32/.39
.32/.38
.31/.38
.31/.37
.27/.28
.26/.18
.25/.17
48 mo.
.28/.34
.28/.32
.30/.36
.30/.36
.29/.34
.28/.33
.24/.24
.23/.15
.22/.14
60 mo.
.26/.33
.27/.30
.28/.35
.28/.34
.27/.32
.26/.31
.22/.21
.20/.12
.19/.11
4^
o
2 Eh
Table 3 Cont'd
Inflationary Expectations, t+niT
E
R
E
S
T
R
A
T
E
S,
10 mo.
11 mo.
12 mo.
24 mo.
1 mo.
.50/.52
.51/.52
.51/.51
.52/.36
2 mo.
.47/.48
.47/.48
. 48/.46
.49/.34
3 mo.
.46/.46
.46/.46
.47/.45
.47/.32
4 mo.
.47/.47
. 47/. 47
.48/.46
.49/.34
5 mo.
.46/.47
.47/.46
.47/.45
.48/.33
6 mo.
.46/.46
.47/.46
.47/.45
.49/.34
12 mo.
.48/.42
.48/.42
.49/.43
.53/.35
24 mo.
.31/.24
.32/.24
.33/.24
.38/.24
36 mo.
.26/.17
.27/.17
.28/.18
.34/.19
48 mo.
.23/.14
.23/.14
.24/.14
.31/.17
60 mo.
.20/.12
.20/.11
.22/.12
.28/.15
36 mo.
48 mo.
60 mo.
.49/.26
.50/.26
.51/.36
.46/.24
.45/.25
.47/.33
.44/.23
.44/.23
.46/.32
.45/.24
.45/.25
.47/.34
.45/.24
.45/.25
.47/.34
.46/.25
.46/.25
.48/.35
.53/.26
.52/.26
.48/.34
.39/.22
.42/.25
.43/. 35
.36/.20
.40/.24
.41/.33
.34/.18
.38/.23
.39/.31
.33/.13
.37/.23
.38/.30
42
Pattern B
r
1
r
2
r
3
etc.
rather than Pattern A, or some scheme similar to that of
Pattern A. Thus if Pattern B is observed we can con
clude that the market rate of interest on debt of dif
ferent durations could best be explained by inflationary
expectations over different horizons that correspond to
those debt horizons.
The Subperiod Test
subsequent test was performed for the time period
January 1965 to December 1978 using the autoregressive
schemes developed from the main period, January 1959 to
December 1978. The subperiod testing allowed the inclu
sion of seven to eleven month Tbills in the output
results in addition to the purpose of studying the
results for consistency during a period of rapid infla
tion. The yearly mean inflation rate was larger in the
subsequent period, changing from an arithmetic mean of 4.3
in the whole period to an arithmetic mean of 5.7 in the
subperiod.
43
Tables 4 and 5 contain the results for the sub
period January 1965 to December 1978 using the infla
tionary expectations formulation indicated on Table 1 for
Tables 2 and 3.
From initial data screens on the subperiod it was
obvious that this was a period where formulation of
inflationary expectations had changed. Thus some simi
larity of results in a period with different auto
regressive orders and rapid inflation would be taken as
additional confidence in the original results.
Notes
1. Data for Gross Personal Income and Money Supply, Ml,
are seasonally adjusted monthly observations from the
November 1979 CITIBASE tape compiled and updated by
Citicorp National Bank, New York, and supplied
courtesy of the Bureau of Business and Economic
Research, University of Florida, Gainesville. Data
for Consumer Price Index was supplied courtesy of the
Bureau of Labor Statistics, Department of Commerce,
Washington, D.C. Data for Tbills are discount yields
(bid yields) as of the end of the preceding month
from Salomon Brothers quote sheets. The bid yields
are converted to bond equivalents by the following
calculation
r(semiannual) = 2[( 1 )365/2t ^
1dxt
360
where the discount rate on a 360day Tbill is con
verted to a semiannual compounded rate based on a
365day year, r represents the semiannual interest
rate, d represents the annual discount rate, and t
represents the time in days til maturity. Bond data
are beginning of the month series of yields of United
Table 4
The Fisher Equation, 1/65 to
Beta/R Square Values
Inflationary Expectations, t+r
I
N
T
E
R
E
S
T
R
A
T
E
S,
1 mo.
2 mo.
3 mo.
4 mo.
5 mo.
6 mo.
1 mo.
.22/.18
.23/.15
.22/.14
. 23/.14
.26/.17
.27/.19
2 mo.
.21/.17
.22/.16
.21/.15
.21/.14
.23/.16
.24/.17
3 mo.
.20/.16
.22/.16
.20/.14
.21/. 13
.23/.15
.23/.16
4 mo.
.21/.18
.22/.17
.21/.15
.23/.16
.25/.18
.26/.19
5 mo.
.22/.19
.23/.17
.22/.16
.23/.16
.25/.19
.26/.20
6 mo.
.22/.20
.24/.18
.22/.16
.23/.16
.26/.19
.26/.20
7 mo.
.23/.21
. 24/. 19
.22/.16
.23/.16
.26/.19
.26/.20
8 mo.
.23/.21
.24/.18
.23/.16
.24/.16
.26/.20
.27/.20
9 mo.
.24/.23
.24/.20
.22/.18
.23/.17
.25/.20
.26/.21
10 mo.
.25/.24
.25/.21
.22/.18
.23/.18
.25/.21
.26/.22
11 mo.
.25/.24
.24/.21
. 22/.19
.23/.18
.25/.21
.26/.22
12 mo.
.31/.19
.32/.17
.30/.15
.32/.17
.35/.20
.36/.21
24 mo.
.17/. 16
. 18/. 14
.18/.15
.19/.14
.20/.17
.20/.17
36 mo.
.15/.13
.16/.13
.17/.14
.17/.13
.18/.15
.10/.15
48 mo.
.14/.11
.15/.11
.15/.12
.16/. 12
.16/.13
.16/.13
60 mo.
.12/.10
.13/.10
.14/.11
.14/.11
.14/.11
.14/.11
12/78
7 mo.
8 mo.
9 mo.
,32/.24
.34/.23
.35/.23
29/.23
.30/.19
.30/.19
, 28/.20
.29/.17
.29/.17
30/.24
.31/.19
.31/.19
, 30/.24
.31/.20
.31/.20
,31/.25
.31/.20
.31/.19
.31/.25
.30/.19
.30/.19
.30/.24
. 30/.18
.30/.18
.30/.26
.29/.18
.29/.17
.30/.27
.29/.17
.28/.16
.30/.27
.28/.17
.27/.16
,35/.23
.36/.23
.36/.23
.21/.16
.20/.11
.19/.10
.18/.13
.10/.09
.16/.07
.16/.10
.16/.07
.14/.06
.13/.08
.14/.06
.12/.04
Table 4 Cont'd
Inflationary Expectations tnir
10 mo.
11 mo.
12 mo.
24 mo.
36 mo.
48 mo.
1 mo.
.35/.23
.35/.22
.36/.21
.28/.07
.28/.06
.31/.07
2 mo.
.32/.20
.31/.18
.32/.IB
.26/.07
.26/.06
.27/.06
3 mo.
.30/.17
.29/.16
.30/.16
.22/.05
.20/.03
.23/.04
4 mo.
.32/.19
.31/.18
.32/.18
.25/.06
. 24/.05
.26/.06
5 mo.
.32/.20
.31/.19
.32/.18
.24/.06
.22/.04
.23/.05
6 mo.
.32/.20
.32/.19
.33/.19
.26/.07
.24/.05
.24/.05
7 mo.
.32/.19
.31/.18
.33/.18
.26/.07
.24/.05
.24/.05
8 mo.
.31/.19
.31/.18
.32/.17
.26/.07
.24/.05
.24/.05
9 mo.
.32/.19
.31/.17
.33/.18
.26/.07
.23/.05
.21/.04
10 mo.
.31/.IB
.30/.17
.32/.17
.25/.07
.22/.04
.21/.04
11 mo.
.31/.10
.29/.16
.32/.17
.26/.07
.22/.04
.20/.04
12 mo.
.37/. 23
.37/.23
.38/.23
.39/.12
.30/.04
.25/.03
24 mo.
.21/.11
.21/.10
.23/.11
.22/.07
.19/.04
.20/.04
36 mo.
.18/.08
.17/.08
.20/.09
.20/.06
.19/.04
.21/.05
48 mo.
.16/.07
.15/.06
.18/.07
.19/.06
.19/.05
.21/.06
60 mo.
.14/.06
.13/.05
.15/.06
.17/;05
.10/.05
.20/.06
60 mo.
.37/.12
.32/.09
.30/.00
.34/.10
.30/.00
.31/.09 ^
Ul
.29/.08
.29/.08
.25/.07
.26/.07
.25/.07
.29/.07
.24/.00
.25/.09
.24/.09
.23/.09
Table 5
Multiple Regression, 1/65 to 12/78
Beta^/R Square Values
Inflationary Expectations, t+nn
1 mo.
2 mo.
3 mo.
4 mo.
5 mo.
6 mo.
7 mo.
8 mo.
9 mo.
I mo.
.29/.29
.31/.29
.31/.29
.34/.31
.36/.36
.38/.37
.40/.41
.43/.43
.43/.43
2 mo.
.27/.28
.30/.20
.30/. 29
.32/.30
.34/.34
.35/.35
.38/.39
.40/.39
.40/.38
3 mo.
.27/.27
.30/.29
.30/.28
.32/.29
.34/.33
.35/.34
. 37/. 37*
.39/.38
.39/.37
4 mo.
.28/.30
.31/.30
.31/.30
.33/.32
.36/.36
.36/.37
.39/.40
.40/.40
.40/.39
5 mo.
.29/.31
.31/.30
.31/.31
.33/.32
.36/.37
.37/.38
.39/.41
.40/.39
.40/.38
6 mo.
.29/.31
.32/.31
.32/.31
.34/.33
.36/.37
.37/.38
.39/.41
.40/.39
.40/.38
7 mo.
.29/.32
.32/.31
.32/.31
.33/.32
.36/.37
.36/.37
.38/.40
.39/.37
.39/.37
8 mo.
.30/.32
.32/.31
.32/.31
.34/.33
. 36/.37
.36/.38
.38/.40
.38/.36
.38/.35
9 mo.
.30/.34
.32/.33
.30/.31
.32/.33
.35/.37
.35/.37
.37/.40
.37/.34
.37/.33
10 mo.
.30/.34
.32/.33
.30/.32
.32/.33
.35/.37
.35/.37
.37/.39
.37/.33
.36/.31
11 mo.
31/.36
.32/.35
.30/.33
.32/.34
.34/.38
.35/.38
.36/.40
.36/.33
.35/.31
12 mo.
.31/.21
.31/.20
.30/.18
.32/.20
.35/.23
.35/.24
.35/.25
.35/.25
.35/.25
24 mo.
.21/.22
.23/.22
.23/.23
.24/.23
.25/.25
.24/.25
.24/.23
.25/.18
.23/.16
36 mo.
.17/.17
.19/.17
.20/.19
.20/.19
.21/.19
.20/.19
.20/.16
.20/.12
.18/.11
48 mo.
.15/.13
.17/.14
.17/.16
.17/.15
.18/.16
.17/.15
. 16/.12
. 17/. 09
.15/.08
60 mo.
.13/.11
.15/.12
.15/.13
.16/.13
.15/.13
.15/.13
.14/.09
.15/.07
.13/.06
Table 5 Cont'd
Inflationary Expectations, t*nnÂ£
\
\
10 mo.
11 mo.
12 mo.
24 mo.
I
1 mo.
.44/.42
.44/.41
.45/.40
.41/.20
N
T
2 mo.
.41/.38
.41/.37
.42/.36
.39/.19
E
R
3 mo.
.40/.37
.41/.36
.41/.35
.37/.17
E
S
4 mo.
.41/.39
.42/.38
.42/.37
.38/.19
T
5 mo.
.41/.38
.41/.37
.42/.36
.38/.18
R
A
6 mo.
.41/.38
.41/.37
.42/.36
.39/.19
T
E
7 mo.
.40/.37
.40/.36
.41/. 35
.38/.18
s,
8 mo.
.40/.35
.40/.34
.41/.34
.38/.18
rt
9 mo.
.39/.34
.39/.32
.40/.32
.34/.15
10 mo.
.39/.32
.39/.31
.40/.31
.34/.15
11 mo.
.38/.32
.38/.31
.39/.31
.34/.16
12 mo.
.36/.25
.37/. 25
.37/.25
.38/.14
24 mo.
.26/.18
.26/.17
.27/.18
.28/.13
36 mo.
.21/.12
.21/.11
.22/.12
.24/.09
48 mo.
.18/.09
.17/.08
. 19/. 09
.22/.08
60 mo.
.15/.07
.15/. 06
.17/.07
.19/.07
36 mo.
48 mo.
60 mo.
.34/.13
.36/.14
.43/.21
.32/.12
.32/.12
.39/.19
.28/.10
.29/.10
.38/.18
.30/.12
.31/.12
.40/.20
.30/.11
.31/.11
.39/.19
,31/.12
.32/.12
.40/.20
.32/!11
.32/.12
.40/.20
,32/.11
.33/.12
.40/.20
.25/.09
.25/.09
.36/.17
. 26/.09
.26/.09
.37/.17
.25/. 10
.24/.10
.35/.18
.29/.07
.27/.06
.31/.11
.25/.09
.27/.10
.33/.17
.23/.08
.26/.10
.31/.16
.22/.07
.26/.09
.30/.15
.21/.07
.24/.09
.28/.13
48
States securities read from monthly yield curves pre
pared by Saloman Brothers. Whenever there is a choice
of coupons, the curves follow the yields of higher
coupon issues in the longer maturities.
2. The annualized interest rates were given, and the
inflationary patterns of Table 1 were annualized by
compounding each to one year as in the following
example of the monthly expectation
t+iut^A^ n (i + t+i^t^
k=l
likewise to annualize t+21Tt
e JL_ e
t+2Trt(A) = n (1 + t+2V
k=l
1
Thus Table 1 was annualized and regressed against
annualized interest rates. Missing observations, due
to differencing the data for equation fit in Table 1,
were generated by the expectation equation for the
respective horizon. Thus expectations values for
longer horizons were "filled" in for the later years
of the data set under the assumption that expectation
were formulated in the same manner. Thus sample sizes
are equal.
CHAPTER IV
ANALYSIS OF RESULTS
The Term Structure of Inflationary Expectations
Table 1 indicates a limited number of different
inflationary patterns. The actual number of different
lag patterns to be matched is six. The spread between
the patterns and the change in loading patterns in Table 1
implies that individuals formulate, at a minimum, two
distinct sets of inflationary expectations.
Two sets are isolated primarily by the statistical
significance of the lagged inflation variables at a .10
level. In the first set, lagged values of price changes
(inflation) plus various combinations of additional vari
ables generate expectations. The second set of expecta
tions, horizons twentyfour to fifty months, shall be
called longterm expectations. In the second set, lagged
values of money supply growth plus various combinations
of additional variables generate expectations.
The term structure of inflationary expectations has
an upward drift and indicates a four year cycle. Thus
the lag pattern is unique in most of the first eight
periods and tries to correct itself with additional
49
50
information from past inflation. It adjusts itself for
the remainder of the term structure to the eight month lag
pattern.
The relatively short pattern of lagged values is
taken to indicate that individuals anticipate the longer
run impacts of changing economic conditions much quicker
than in previous periods of stable monetary growth. The
use of additional information from the other time series
and the more active inflation rates after 1965 are offered
as explanations for the change from past formations of
long lag patterns.
Table 1 indicates the question, "Do changes in the
inflation rate cause changes in interest rates?" is
justified only as a reflection of the primary determinate
of expected inflation. In the study of shortterm
interest rates there exists more than the one determinate,
past and current price changes, for the formation of
expected inflation. The study of longterm interest rates,
however, is the study of the movement of both interest
rates and inflation responding primarily to changes in the
growth rate of the money supply.
This observation from Table 1 of the different load
ing patterns of shortterm and longterm interest rates
described above is weakened when the business cycle repeats
itself. The various alternating upswings and downswings of
varied length and intensity occur at slightly less than
51
fortyeight months. The observation also is weakened at
turning points, directional changes, within the business
cycle. This marked difference in loading on interest
rates at turning points suggests a new way to forecast
phases of the cycle.
For example, the twentyfour month inflationary
expectations are influenced not only by the growth rate
of the money supply, but are influenced also by the past
change in one month interest rates. Expectations for the
fourth year repeat the business cycle and are influenced
by income changes, as well as, money supply growth.
Determinates for the five year forecast are a similar
combination of the four year determinates and the one year
determinates. This is taken to indicate that the indi
vidual is forecasting that five years from the present
time of forecast the business cycle will be one year into
repeating another second cycle.
Table 1 shows the assumption made in Chapter 2 con
cerning the convergence of information to a single series,
the price series, is unjustified.
The following conclusions concerning expectations
formulation were reached. There exist two distinctly
different sets of inflationary expectations. Each set of
expectations is generated by a different stochastic process.
Expectations of inflation are influenced by the use of
information other than the price series. The market place
52
takes into consideration not only the current and past
values of inflation to predict the future course of infla
tion, but also uses information concerning the rate of
growth of personal income, the growth rate of the money
supply, and past changes in the onemonth interest rate.
Interest Rates and Inflationary Expectations
To answer the question, "How much of the structure of
interest rates is caused by the structure of expected
inflation?" two types of evidence are presented. First,
an examination of summary statistics and graphs is pre
sented to determine whether there is a structure to
expected inflation which is in some way comparable to the
structure of interest rates. Second, the major results
of matching interest rate maturities and expectations
horizons are presented for the different time periods and
different interest rate equations to determine the match
ing with the highest explanatory power.
Summary Statistics and Graphs
The information presented in Table 6 and Figure 1 is
designed to answer the following two questions: "Do the
0
tt 's vary as much across the horizons as the r's vary
across maturities?" and "Does the expectation "yield
curve" bend up and down when the interest rate curve does?"
Going vertically down Table 6 the mean r's rise and
the variances of the r's fall. This is as expected. The
53
same is true of the Tre's at least for the variances; the
means behave slightly different by having only a general
upward trend. However, the behavior of the mean Tre's does
not damage the theory. Only if the variances indicated a
different trend pattern would the theory be damaged.
Table 6
Summary Statistics for
Interest Rates and Inflationary Expectations
1/59 to 12/78
Standard
Standard
Horizon
r mean
Deviation
irf mean
Deviation
in months
yrt
/a 2
rt
Jn
/ure
1
4. 61
1.78
4. 50
2.73
2
4.70
1. 68
4. 54
2.66
3
4.90
1. 67
4.56
2.61
4
4.90
1. 67
4.56
2.61
5
4.98
1. 67
4. 56
2.64
6
5. 04
1. 66
4.57
2. 60
12
5.21
1.75
4.53
2.56
24
5. 44
1. 63
4.56
2.13
36
5.56
1. 58
4. 56
2.01
48
5. 64
1. 55
4. 58
1. 97
60
5. 69
1. 55
4. 59
1. 95
Figure
1 helps
to highlight
the results
of Table 6.
It adds evidence that the two structures are comparable.
The TTe,s do vary as much across horizons as the r's vary
across maturities. As the expectations "yield curve"
bends up the interest rate curve bends up, and as the
expectations "yield curve" bends down the interest rate
curve does also. Interesting is the observation of
increased frequency after 1968 of the times the portion
54
Figure 1 A. The Term Structure of Interest Rates
T9S9 616365 67 69 71 73 75 77 79
Figure 1 B. The Term Structure of Inflationary
Expectations
55
i
4
Figure 1 C. Inflationary Expectation and the Term
Structure of Interest Rates
56
of shortterm expectations curve lies above the interest
rate curve. This is interpreted as the result of the
increased inflation activity and more rapid information
processing by the market place.
The above results from Table 6 and Figure 1 shows
the structure of expected inflation is comparable to the
structure of interest rates.
Maturity Matching
Tables 2 and 4 represent the various interest rates
regressed on the different inflation horizons with an
adjustment for first order serial correlation for the
Fisher equation. Tables 3 and 5 show the same results
for the multivariate regression model. Therefore the
result of the existence of at least two distinct horizon
was not model sensitive. However, the results of the
loading patterns were sensitive to the rate of inflation,
i.e., the main period or the subperiod, and the specifica
tion of a single variable or multiple variable interest
rate equation. The results below show the highest load
ings during the main period. Thus if horizons do matter,
the pattern of loadings exhibited in the main period
imply that the horizons of the interest rates does not
match that of the expected inflation rate as a unique
set.
57
Interest Rates
(monthly maturities)
Inflationary Expectations
Table 2 Table 3
(annualized horizon)
1
2
3
4
5
6
12
24
36
48
60
7 6,7
6,7 6
6,7 6
6 6
6 6
6 6
7 6
6,6 5
1,3,4,5,6 3
3,4 3,4
3 3
Likewise the loading patterns for the subperiod of
more rapid inflation indicate
Interest Rates
(monthly maturities)
Inflationary Expectations
Table 2 Table 3
(annualized horizons)
1
2
3
7 8,9
7 7,8
7
8
4
5
7
7
7,3
7
6
7
7
7
7
7
8
7
7
9
10
11
7
7
7
7
7
7
12
24
36
48
7,8,9,10,11,12
5,6
5,6
5,6
7,8,9,10,11,12
5,6
3,4,5,6
3,5
60
3,4,5,6
3,4,5,6
Again if horizons do matter, i.e., the small changes
in the values are thought to imply some significance,
then the pattern of loadings does not support the Fisher
hypothesis.
53
Information processing by the market place concerning
inflationary expectations shows little if any real dif
ference between matches in shortterm rates or the long
term rates. The major difference in loading patterns
appears to be between the two groups short and long rates.
It appears that individuals forecast the 6 month and
7 month inflation rate very well, and that they use these
two durations as the basis for most of the interest rate
equations.
At no time with the use of any of the different models
did the nominal interest rate fully adjust to expected
inflation. The sign of the price expectations influence
was always positive and slightly sensitive to the use of a
single series or a multiple series expectation formulation.
This is not taken as a rejection of the Fisher hypothesis.
It is taken as an indication that the Fisher equation will
hold only during periods that occasion conditions of long
run classical equilibrium. Therefore, if the data are dur
ing periods of transition, the Fisher hypothesis is not
expected to empirically hold for the duration of debt instru
ments used. The more rapid the rate of price increase, the
weaker will be the Fisher relationship between expected
inflation and nominal interest rates. This is easily seen
in the comparison of Table 2 of the main period, 1/59 to
12/78, to Table 4 of the subperiod, 1/65 to 12/78.
59
Likewise, the comparison of Table 3 to Table 5, the
more rapid the rate of price increase the more valuable
the change in the money supply became in explaining the
nominal interest rate. The negative sign and strength of
money indicates the strong influence the "liquidity effect"
(Friedman 1968) had on nominal interest rates during the
period of rapid inflation for interest rates with a 11
month or less horizon.
The statistical loss of significance for the "real
output" proxy, personal income, from the intermediate and
longterm interest rates, from Table 3 to Table 5 is
interpreted as a result of the disturbance from monetary
sources, Mt.
The increase in parameter values of t+n^t from
Tables 2 and 4 to Tables 3 and 5, respectively, and the
2 ,
increase m R value is taken to imply the correct speci
fication of interest rate determinates in Tables 2 and 4
had been relegated to the disturbance term. That is, a
multivariate model is more desirable in explaining nominal
interest rates than a single variable model. This speci
fication was sensitive to the duration of the rates. Thus
both the determinates of the interest rate and the results
of the Fisher equation "test" are sensitive to sample
period choice and duration of rates chosen.
It is interesting that the general matching pattern
increases in maturity as the subperiod loadings are
60
compared to the main period. Thus individuals would need
to extend their expectations horizon during periods of
rapid inflation in order to better explain interest rate
movements. This maturity extension, due to the increased
uncertainty of the holding (opportunity) cost of money,
implies expected inflation will underestimate actual
inflation. The six month Tbill should have reflected
within its "price" the seven month expected inflation
instead of the six month expectations. Interest rate fore
cast with matched horizons will also tend to be nonoptimal.
The final point to be noted is the reduction in MSE of
the interest rate with the use of the multivariable model.
If the MSE is considered as a proxy for the variance, and
the variance is the appropriate measure of risk for the
purchase of an investment (or a portfolio of investments),
then the reduction in risk, and the measure of performance
by a risk manager, can be improved with the additional
variables in the interest rate equation, especially during
periods of rapid inflation.
Problems and Extensions
Any empirical study within the field of economics can
be said to have problems, either with theory or measurement.
This study is certainly no exception.
The two major problems concern the develooment of
Table 1. First, serial correlation in the forward rates
61
makes use of summary statistics typically used for OLS
comparison very suspect, i.e., the t statistic, the F
statistic, and the MSE. Likewise, the lagged values of
the L.H.S. variable on the R.H.S. invalidates the D.W.
statistic except as a measure of direction for the increase
or decrease of correlation. Second, the decrease in the
sample size in order to fit the various expectations
horizons with the actual differenced price series makes
the stability of the parameters and comparison across
sample sizes suspect. Generating values for missing data
was justified with the assumption that for the observa
tions lost at the end of the dataset the formation of the
expectations did not change.
Timing of the data is suspect when using the C.P.I.
and endofmonth interest rates to generate Fisher equa
tions. The regressions were adjusted forward and lagged
back by one month to test sensitivity. The results showed
that timing did not statistically matter.
The concern with developing and analyzing an inte
grated macromodel, rather than with any detailed discussion
of particular expenditure functions, caused the omission
of devotion to detailed consumption, investment, govern
ment, or money demand functions. Thus the theoretical
model is simple and permits inflationary expectations to
enter into the system only through the investment function
and the price adjustment process and overlooks the
62
elasticity with respect to the demand for money (Friedman
1956) and the proportion of government expenditure financed
by increasing the monetary base (Christ 1968) and taxes.
It is suggested that if the variable on the R.H.S.
meant to capture future government anticipations in the
credit market were altered to using a listing of future
debt sales of the government that the model could be
improved.
A very interesting observation is the movement of the
real rate ot0. The real rates for Tables 2, 3, 4 and 5,
with the exception of the twelve month rates, increased
slightly in value as the maturity of the debt increased.
Because the equation was not controlled for the various
effects on the real rate (Mundell 1963 and Phillips 1958
and Darby 1975 and Friedman 1977 and Feldstein and Summers
1978), the various movements in the real rate from Table 2
to Table 3, and along the horizon on the same Table 2 or
3, and the comparison with the subperiod Tables 4 and 5,
must have offered as an explanation of occurrence being
due to statistical modeling. However, several interesting
questions arise. Does the real rate remain constant, or
increase with the horizon? Does orice uncertainty,
measured by the variance of the forecast, cause the real
rate to rise during periods of rapid inflation? Does the
real rate approach a constant for the longer maturity
horizons, e.g., 5 year bonds? What are the determinates
63
for the real rate of interest, i.e., should it be con
sidered as a variable in the Fisher hypothesis when test
ing during transition periods,
rt = ^ opt + ^l^t + Â£t (43)
Did the increase in the savings to disposable income ratio
during the subperiod, from 6.2% to 6.5% shift the net sav
ings curve to the right by a smaller dynamic growth than
the shift to the right in the government deficit training,
thus causing the real rate to rise? There are many pos
sible economic and statistical answers that could be
offered for the movement of the real rate. It is hoped
that additional study will answer these questions.
CHAPTER V
CONCLUSIONS
The results of this study show that for the main
period, 1/59 to 12/73, that the use of a multivariate
model to forecast inflation both reduces the MSE of the
2
forecast and gives a better R (explanatory fit) for the
interest rate. Moreover, the different stochastic pro
cesses underliying the term structure of inflationary
expectations, as indicated by both the different lag
patterns and the different loading patterns, suggest
segmentation of the expectations curve.
" The analysis of the structure of inflationary
expectations and the structure of interest rates indicates
the two structures are comparable. A slight difference
was observed between the shortterm and longterm relation
ships. This difference in relationships was much more
pronounced during periods of rapid inflation.
It was shown that in both the main period, 1/59 to
12/73, and the subperiod, 1/65 to 12/78, the horizons of
the same length of interest rates and inflationary expec
tations did not yield a unique match. It was suggested
that the Fisher hypothesis applies only to equilibrium
periods thus the study was not taken to be inconsistent
with the longrun theory.
64
65
The main conclusion of the study has to be that if
horizons are thought to be important, then the Fisher
hypothesis stated with the duration of the interest rate
matching the duration of expected inflation does not hold
for the different durations along the yield curve.
In addition to using a multivariable expectation
forecast, the results show an interest rate which includes
additional variables as determinates provides a better fit
and a smaller MSE than those equations without the addi
tional variables. Finally, the real rate of interest is
noted as not being constant across different horizons,
but the explanation is left to question because the
equation was not adequately controlled to discuss move
ments in the real rate.
BIBLIOGRAPHY
Bomberger, W. A. and W. J. Frazer, Jr., "Interest Rates,
Uncertainty and the Livingston Data," Working Paper,
May 1980.
Box, G.E.P. and G. M. Jenkins, Time Series Analysis,
San Francisco, CA: HoldenDay, Inc., 1976.
Brainard, W. C. and J. Tobin, "Pitfalls in Financial
Model Building," American Economic Review, 58, May
1968.
Branson, W. H., Macroeconomic Theory and Policy, New York,
NY: Harper & Row Publishers, Inc., 1972.
Brick, J. R. and H. E. Thompson, "TimeSeries Analysis of
Interest Rates: Some Additional Evidence," Journal
of Finance, 33, March 1978.
Cargill, T.F., "Estimating Term Structure Phenomena
From Data Aggregated Over Time," Journal of Money,
Credit and Banking, 6, November 1974.
Cargill, T. F., "Notes: Anticipated Price Changes and
Nominal Interest Rates in the 1950's," Review of
Economic and Statistics, 58, August 19761
Cargill, T. F., "The Term Structure of Inflationary
Expectations and Market Efficiency," Journal of
Finance, 35, March 1980.
Cargill, T. F. and R. A. Meyer, "Interest Rates and Prices
Since 1950," International Economic Review, 15, June
1.97 4.
Cargill, T. F. and R. A. Meyer, "Intertemporal Stability
of the Relationship Between Interest Rates and Price
Changes," Journal of Finance, 32, September 1977.
Carlson, J. A., "Systematic Errors in Inflation Forecasts,"
Conference on Models of Expectations, Gainesville,
Florida, May 1978.
66
67
Christ, C. F., "A Simple Macroeconomic Model With a
Government Budget Restraint," Journal of Political
Economy, 76, February 1968. '
Cornwell, B., "Monetary Policy, Inflation Forecasting and
the Term Structure of Interest Rates," Journal of
Finance, 33, March 1978.
Craine, R. and J. L. Pierce, "The Time Structure of
Interest Rates: A New Approach to an Old Problem,"
Federal Reserve Bank of San Francisco, West Coast
Academic/Federal Reserve Economic Research Seminar,
1978.
Cramer, H., "On Some Classes of Nonstationary Stochastic
Processes," Proceedings on the 4th Berkeley Symposium
on Mathematical Statistics and Probability" 2~, 1961.
Cukierman, A., "The Relationship Between Relative Prices
and the General Price Level: A Suggested Interpre
tation," American Economic Review, 69, June 1979.
Darby, M. R., "The Financial and Tax Effects of Monetary
Policy on Interest Rates," Economic Inquiry, 13,
June 1975.
Dhrymes, P. J. "A Note on an Efficient TwoStep Estimator,"
Journal of Econometrics, 2, 1974.
Dornbusch, R. and S. Fischer, Macroeconomics, Highstown,
NJ: McGrawHill, 1978.
Eckstein, 0. and M. Feldstein, "The Fundamental Deter
minants of Interest Rates," Review of Economic and
Statistics, 52, November 1970.
Fama, E. F., "Efficient Capital Markets: A Review of
Theory and Empirical Work," Journal of Finance, 25,
May 1970.
Fama, E. F., "Shortterm Interest Rates as Predictors of
Inflation," American Economic Review, 65, June 1975.
Fama, E. F., "Inflation Uncertainty and the Expected
Returns on Treasury Bills," Journal of Political
Economy, 84, June 1976a.
Fama, E. F., "Forward Rates as Predictors of Future Spot
Rates," Journal of Financial Economics, 3, October
1976b.
68
Feige, E. L. and D. K. Pearce, "The Casual Causal Rela
tionship Between Money and Income: Some Caveats
for Time Series Analysis," Review of Economics and
Statistics, 59, February 1977.
Feldstein, M. and G. Chamberlain, "Multimarket Expecta
tions and the Rate of Interest," Journal of Money,
Credit and Banking, 5, November 1973.
Feldstein, M. and L. Summers, "Inflation, Tax Rules, and
the LongTerm Interest Rate," Brookings Papers on
Economic Activity, 1, 1978.
Fisher, D., Monetary Theory and the Demand for Money,
New York, NY: John Wiley & Sons, Inc., 1978.
Fisher, Irving, The Theory of Interest, New York, NY:
Macmillan Co. 1930.
Fisher, S., "The Demand for Indexed Bonds," Journal of
Political Economy, 83, June 1975.
Frenkel, J. A., "Inflation and the Formation of Expecta
tions," Journal of Monetary Economics, 1, October
1975.
Friedman, B. M., "Price Inflation, Portfolio Choice, and
Nominal Interest Rates," American Economic Review,
70, March 1980.
Friedman, Milton, "The Quantity Theory of Money, a
Restatment," M. Friedman, ed., Studies in the
Quantity Theory of Money, Chicago, IL: University
of Chicago Press, 1956.
Friedman, Milton, "The Role of Monetary Policy," American
Economic Review, 58, March 1968.
Friedman, Milton, "A Theoretical Framework for Monetary
Analysis," Journal of Finance, 78, 1970.
Friedman, Milton, "Noble Lecture; Inflation and Unemploy
ment." Journal of Political Economy, 85, June 1977.
Gibson, W. E., "The Lag in the Effect of Monetary Policy
on Income and Interest Rates," Quarterly Journal of
Economics, 83, March 1969.
Gibson, W. E., "Price Expectations Effects on Interest
Rates," Journal of Finance, 25, 1970.
69
Gibson, W. E., "Interest Rates and Inflationary Expecta
tions," American Economic Review, 62, December 1972.
Granger, C.W.J. and P. Newbold, Forecasting Economics
Time Series, New York, NY: Academic Press, Inc.,
'1977.
Granger, C.W.J., Forecasting in Business and Economics,
New York, NY: "Academic Press, Inc. IT8a7
Granger, C.W.J., "Some Properties of Time Series Data
and Their Use in Economic Model Specification,"
Working Paper 804, March 1980b.
Hamburger, M. J. and C. M. Latta, "The Term Structure of
Interest Rates," Journal of Money, Credit and
Banking, 1, February 1969.
Hamburger, M. J. and E. N. Platt, "The Expectations
Hypothesis and the Efficiency of the Treasury Bill
Market," Review of Economics and Statistics, 57,
May 1975.
Haugh, L. D. and G.E.P. Box, "Identification of Dynamic
Regression (Distributed Lag) Models Connecting Two
Time Series," Journal of the American Statistical
Association, 72, March 1977^
Hess, P. J. and J. L. Bicksler, "Capital Asset Prices
Versus Time Series Models as Predictors of Inflation,
Journal of Financial Economics, 2, June 1975.
Johnston, J., Econometric Methods, Highstown, NJ:
McGrawHill, T9 72~
Kane, E. J. and B. G. Malkiel, "Autoregressive and Non
autoregressive Elements in Crosssection Forecasts
of Inflation," Econometrica, 44, January 1976.
Kennedy, C., "Inflation and the Bond Rate," Oxford
Economic Papers, 12, April 1960.
Keynes, J. M., The General Theory of Employment, Interest
and Money, New York, NY: Macmillan Co., 1936.
Kmenta, J., Elements of Econometrics, New York, NY:
Macmillan Co., 1971.
Lahiri, K., "Inflationary Expectations: Their Formation
and Interest Rate Effects," American Economic Review,
76, March 1976.
70
Langetieg, T. G., "A Multivariate Model of the Term
Structure," Journal of Finance,. 35, March 1980.
Levi, M. D. and J. H. Makin, "Fisher, Phillips, Friedman
and the Measured Impact of Inflation on Interest,"
Journal of Finance, 34, March 1979.
Luckett, D. G. and A. R. Soltow, "Determinates of Interest
Rate Expectations," Journal of Money, Credit, and
Banking, 3, September 1971.
Lutz, F., "The Structure of Interest Rates," Quarterly
Journal of Economics, 54, November 1940.~
Maddala, G. S., Econometrics, Highstown, NJ: McGrawHill,
1977.
Maital, S., "Inflation Expectations, in the Monetarist
Black Box," American Economic Review, 69, June 1979.
Makinen, G. E., Money, the Price Level and Interest Rates,
Englewood Cliffs, NJ: PrenticeHall, Inc., 1977.
McClave, J. T., "Subset Autoregression," Techonometrics,
17, May 1975.
McClave, J. T., "Estimating the Order of Autoregressive'
Models: The Max Method," Journal of the American
Statistical Association, 73, March 1978.
Merril, G. and S. S. Bhalla, "Direct Measurement of
Popular Price Expectations," American Economic
Review, 65, March 1975.
Mincer, J., "Models of Adoptive Forecasting," in Economic
Forecasts and Expectations, ed. by J. Mincer, New
York, NY: National Bureau of Economic Research, 1969.
Mishkin, F. S., "Efficient Markets Theory: Implications
for Monetary Policy," Brookings Paper on Economic
Activity, 3, 1978.
Modigliani, F. and R. J. Schiller, "Inflation, Rational
Expectations, and the Term Structure of Interest
Rates," Econometrica, 40, February 1973.
Modigliani, F. and R. Sutch, "Innovations in Interest
Rate Policy," American Economic Review, 56, May 1966.
71
Modigliani F. and R. Sutch, "Debt Management and the Term
Structure of Interest Rates," Journal of Political
Economy, 75, August 1967.
Mullineaux, D. J., "Inflation Expectations and Money
Growth in the United States," American Economic
Review, 70, March 1980.
Mundell, R. A., "Inflation and the Real Interest Rate,"
Journal of Political Economy, 71, June 1963.
Mussa, M., "Adoptive and Regressive Expectations in a
Rational Model of the Inflationary Process," Journal
of Monetary Economics, 1, 1975.
Muth, J. F., "Rational Expectations and the Theory of
Price Movements," Econometrica, 29, 1961.
Nelson, C. R., The Term Structure of Interest Rates,
New York, NY: Basic Books, 1972.
Nelson, C. R., Applied Time Series Analysis for Managerial
Forecasting^ San Franc i s c o, CA: HoldenDay, Inc. ,
1973.
Nelson, C. R., "Recursive Structure in U.S. Income,
Prices, and Output," Journal of Political Economy,
87, 1979.
Nelson, C. R. and G. W. Schwert, "On Testing the Hypothesis
That the Real Rate of Interest is Constant," American
Economic Review, 67, June 1977.
Pearce, D. K., "Comparing Survey and Rational Measures of
Expected Inflation," Journal of Money, Credit and
Banking, 11, November 1979.
Pesaudo, J. E., "Alternative Models of the Determination
of Nominal Interest Rates," Journal of Money, Credit
and Banking, 8, May 1976.
Pesaudo, J. E., "On the Efficiency of the Bond Market:
Some Canadian Evidence," Journal of Political
Economy, 86, December 1978. 
Phillips, A. W., "The Relationship Between Unemployment
and the Rate of Change of Money Wage Rates in the
United Kingdom, 18611957, Economica, 25, 1958.
72
Phillips, L. and J. Pippenger, "Preferred Habitat vs.
Efficient Market: A Test of Alternative Hypothesis,"
Review, Federal Reserve Bank of St. Louis, 58, May
1976.
Phillips, L. and J. Pippenger, "The Term Structure of
Interest Rates in the MITPENNSSRC Model," Journal
of Money, Credit and Banking, 11, May 1979.
Poole, W. "Rational Expectations in the Macro Model,"
Brookings Paper on Economic Activity, 2, 1976.
Priestley, M. B., "Evolutionary Spectra and Nonstationary
Proccesses," Journal of Royal Statistical Society,
27 1965.
Pyle, D. H., "Observed Price Expectations and Interest
Rates," Review of Economics and Statistics, 54,
August 1972.
Resler, D. H., "The Formation of Inflation Expectations,"
Review, Federal Reserve Bank of St. Louis, 62,
April 1980.
Roll, R., "Assets, Money and Commodity Price Inflation
Under Uncertainty," Journal of Money, Credit and
Banking, 4, May 1973^
Rutledge, J., A Monetarist Model of Inflationary Expecta
tions, Lexington, MA: D. C. Heath and Co., Inc.,
1974.
Sargent, T. J., "Commodity Price Expectations and the
Interest Rate," Quarterly Journal of Economics, 83,
February 1969.
Sargent, T. J., "A Note on the 'Accelerationist' Contro
versy," Journal of Money, Credit and Banking, 3,
August 1971.
Sargent, T. J., "Rational Expectations and the Term
Structure of Interest Rates," Journal of Money,
Credit and Banking, 4, February 1972a.
Sargent, T. J., "Anticipated Inflation and the Nominal
Rate of Interest," Quarterly Journal of Economics,
86, May 1972b.
Sargent, T. J., "what Do Regressions of Interest on
Inflation Show?" Annals of Economic and Social
Measurement, 2/3 1973a"!
73
Sargent, T. J., "Interest Rates and Prices in the Long
Run," Journal of Money, Credit and Banking, 5,
May 1973b. ~ ~
Sargent, T. J., "Rational Expectations, the Real Rate of
Interest, and the Natural Rate of Unemployment,"
Brookings Papers on Economic Activity, 2, 1973c.
Sargent, T. J. "Interest Rates and Expected Inflation: A
Selective Summary of Recent Research," Explorations
in Economic Research, 3, Summer 1974.
Sargent, T. J., "A Classical Macroeconomic Model of the
United States," Journal of Political Economy, 84,
April 1976.
Sargent, T. J., Macroeconomic Theory, New York, NY:
Academic Press, Inc. 1979".
Shiller, R. J., "Rational Expectations and the Term Struc
ture of Interest Rates," Journal of Money, Credit and
Banking, 5, September 1973.
Sims, C. A., "Money, Income and Causality," American
Economic Review, 62, September 1972.
Tanzi, V., "Inflationary Expectations, Economic Activity,
Taxes, and Interest Rates," American Economic Review,
70, March 1980.
Theil, H., Principles of Econometrics, New York, NY:
John Wiley & Sons, Inc., 1971.
Thornton, Henry, The Paper Credit of Great Britain, 1802,
New York, NY: Augustus Kelly, 1965.
Tipton, J. and J. T. McClave, "Time Series Modeling: A
Comparison of the Max Approach and the BoxJenkins
Approach," Working Paper in progress.
Tobin, J., "A General Equilibrium Approach to Monetary
Theory," Journal of Money, Credit and Banking, 1,
February 1969.
Turnovsky, S. J., "Empirical Evidence on the Formulation
of Price Expectations," Journal of the American
Statistical Association, 65, December 1970.
Turnovsky, S. J. Macroeconomic Analysis and Stabiliza
tion Policies, New York, NY: Cambridge University
Press, 1977.
Wood, J. H., "The Expectations Hypothesis, the Yield
Curve and Monetary Policy," Quarterly Journal of
Economics, 77, August 1964.
Wood, J. H., "Metzler on Classical Interest Theory,"
American Economic Review, 70, March 1980.
Yohe, W. P. and D. S. Karnosky, "Interest Rates and Price
Level Changes, 195269," Review, Federal Reserve
Bank of St. Louis, 51, 19691
BIOGRAPHICAL SKETCH
I was born July 20, 1948, to James Reed and Lois
McCall Tipton in Knox County, Tennessee. I grew up and
attended public schools in Chattanooga, Tennessee,
graduating from Brainerd Senior High School in 1966.
I graduated from the University of Tennessee,
Knoxville, with a Bachelor of Science in Economics in
1970. Then after spending three years as a regular Army
officer, I served an internship with the Chicago North
western Transportation Company as a Treasury Analyst.
Entering the University of Florida in September 1974, I
graduated with a Master of Business Administration, con
centrating in Finance in 1976. I married Barbara Ann
Miller in.June 1976. I then received a Master of Arts
in Economics in 1978. I have accepted a joint appoint
ment in both Finance and Economics from Baylor University,
Waco, Texas beginning August 1980.
75
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
c< C
William A. Bomberger, Chairman
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
David A. Denslow
Professor of Economics
I certify that I have read this study and that in my
opinion it conforms to acceptable standards of scholarly
presentation and is fully adequate, in scope and quality,
as a dissertation for the degree of Doctor of Philosophy.
V a
Jam'te T. McClave
Professor of Statistics
