Are expectations of inflation rational? or Is variation of the expected real interest rate unpredictable?

Journal of Monetary Econr~ks

8 (1981) 59-84. North-Holland

ARE EXPECTATIONS

OF INFLATION

Publishing Company

RATIONAL?

OR IS VARIATION

OF THE EXPECTED REAL INTEREST UNPREDKTABLE?

RATE

Gerald P. DWYER, Jr.* %tcrsA&.%#UniWrsity, Colkge

Station, TX

77843, iJSA

Emory Uniwrsitb: Atlanta, GA 30322, USA

The Joint hypothesis devel and tested in this paper is that the nominal interest rate is a cZrtimd expectation of the r &rest rate plus the inflation rate and that variation of the expected rcrtl interest rate is unprc&table on the basis of information US& m the test. This test is applied to quarterly data on three-month United States Tremor Sills . 1954 to 1973. The information U in the tests includes, besiies past interest rates _mti :nflaiadn rates, past growth rates of the sou~c(: base, the money supply, and real GNP. Smx I the tests allow for a positive marginal tax rate. which changes the results little. The hypothesis is generally consistent with the data, which provides support for the proposition that predictable chr.nges of the money supply do not +.fkct expected real interest rates over perk& as short as a quarter.

1.

Introduction

The hypothesis that expectations of inflation are unbiased is most appealing as a long-run equilibrium proposition. The most general statement that might be made is that a long-run equilibrium will not exist if people’s tations are systematically incorrect and lower their wealth. This hesis can be used to enerate the ‘natural-rate hypothesis’ that rn~~~t~y policy cannot lx u to permanently change the unemployment rate. As succinctly stat by Gordon and Wjnes ( 1970, p. 387): ‘

” . * the

monetary authorities cannot choose between alternative quil~brium rates of inflation and equilibrium levels of unemployed isian units will cve~~a’!y discover -- or, more estimates of _-__the equilibrium rate of m that monetary authorities on average ScfVC Bwnkof Ckic

ero?lsIy supported this research. konetheless. no rify be attributed to any employee or part of the ssions 0 ’ the paperwith Robert E. Lucas, Jr.. obert D. Lau ‘ent were very helpful. Other members ap at the Universit:, of Chicago RXI of a seminar at the helpful comma nts, as did pa! kipants in a seminar at si~nif~cant~~ilnprove the pa+;.

csn

0304-3932/8 l/

$02.50 @ North-Hol:land

60 G. Dwyer, Jr., Are ir#7ationexpectations rational? Expected

real interest rate unpredictrrble?

could reduce the unemployment rate below the private equilibritum rate] by continually increasing tire rate of monetary expansion, thus making sellers’ estimates of the “true” or equilibrium rate of inflation continually . biased. However, if we invoke the assumption that economic units will eventually learn of any stable situation, they will become aware of such a stable policy consistently affecting the acceleration of the money supply (,and thus of prices) and will incorporate this information into their estimating functions . . .‘. There is no necessar)’ suggestion that people will learn of changed circumstances quickly or that their expectations will be unbiased for a period not characterized by long-run equilibrium. As Milton Friedman (1969, p. 99) pointed out in his original statement of the natural-rate hypothesis, ‘essentially the same theoretical analysis’ applies to the real interest rate. Any attempt to permanently lower the real interest rate through monetary policy will ultimately fail because people will learn from their past errors. The hypothesis that expectations are rational in Muth’s (1961) sense is a restrictive cousin of this equilibrium unbiased-expectations hypothesis. The hypothesis of rational expectations is that expectations of variables are based on the same distributions as those implied by the posited model. This hypothesis about expectations, like the long-run version used by some to generate the natural-rate hypothesis, can be used to generate a hypothesis that monetary policy cannot peg the unemployment rate or the real interest rate [Lucas (1972, 1975), Sargent and Wallace (1975)J Because it is a stronger hypothesis, it also has stronger negative implications about the efficacy of counter-cyclical monetary policy and the optimality of ‘optimal control applied to the economy [Kydland and Pre.scott (197711. The restrictiveness of the rational expectations hypothesis, which generates the stronger conclusions, also generates restrictions on the data which are less ambiguous than empirical restrictions from the long-run version. The hypothesis of rational expectations in the sense of Muth is one part of the hypothesis tested in this paper. The behavior of interest rates, which reflect expected occurrences over time only, is an obvious context in which to examine the rationality of expectations. Because a test which does not impose restrictians on the real interest rate is vacuous, it is necessary to impose restrictions on the function relating the real interest rate to information available. Fama (1970) assumes that the expected real interest rate is a constant. The test in this paper is more general because the expected real interest rate need be only unpredictable by the variables used to predict the expected inflation rate. A consequence of imposing this restriction on the expected real interest rate is that a joint hypothesis of rational expect~tior~s and an ~n~r~~icta

G. Dwl>er. Jr., sirs i&tinn

eqwctations

rariorzd? Expected real interest rate unpredictable? 61

expected real interest rate is tested. Therefore, rejection of the hypothesis could be rejection of rational expectations, an unpredictable expected real interest rate, or both: onlv failure to reject the h:!jpothesis provides any clearcut evidence on the rationality of expectations in Muth’s sense.” 2. Fisher’s hypothesis with rational expectations There are three components of the operational hypothesis to be tested. The first is Fisher’s hypothesis that the nornina: interest rate is the same as the expected return from commodities including the expected increase of the prices of commodities. ?‘he second component is the hypothesis that expectntiots of inflation are rational. The third Il:ornpJnent is the hypothesis that the expected reltl interest rate sannoi be predicted by the variables used to predict the expected inflatlon rate 2.1. Fishtr’s

hypothesis

Irving Fisher’s (1896) proposition in Ape irttlvti ~~ncl Interest is that the expected return on a Ioan is invkirianr eC. :he stP[ldard of the deferred payments. If loan contracts in terms of gold and loan contracts in terms r>f silver are alternatives in a market, then the diflerence between the gold rate of interest and the silver rate of interest will reflect the expected change is! the price of gold relative to the pric-: of silver. This proposition can hold for any asset. A loan contract could b;: specified un terms of wheat instead or money >,vith the delivery of the ecjuivalent amount of money specified. The hypothesis that the interest rate in terms of money is the expected return in terms of P commodity from holding, the commodity plus the expected percentage change of the price is a particular application of this proposition. F’isher’s proposition is a result of the equilibrium condition that expected marginal nominal returns on assets with the s;,me initial nominal value arc equal if adjustment and storage cosls and non-pecuniary returns are zero. If costs of adjustin a portfolio of n~sets exist and ‘$re a function of the speed of adjuutmcnt, then an the mar. in expected tromi~ai returns net ot‘ adjustment cosl~i are equal but rns need not be equal. Similarly, storaln,e costs and u~~ap~uni~~ry returns can driv L’ a wedge Ix~wct~ the nominal mtcrest If these costs rate ai3 a lawn and the gross retilr n i’rc9m holdin~;r ;I a~rt3modity.

62 G. Dwyer, Jr., Are inflation expectations rational? Expected real interest rate unpredictable?

and returns are not zero, the variation of the nominal interest rate can be linked directly to variation of the real return from commodities only if the variation of these costs and returns is zero. Given Fisher’s hypothesis, the nominal interest rate, i,, on a one-period security at the beginning of period E can be represented by

where p,. is the constant part of the interest rate in terms of commodities or the real rate of interest, LIP:+I is the anticipated change of the logarithm of the price level from period t to t + 1, and rff is a variable with a mean of zero \I;hich reflects variation in the real interest. rate and error in measuring tht: nominal interest rate. AS Darby (1975) has pointed out, eq. (1) is not necessarily strictly correct w!ien income taxes affect the borrower or the lender. If a lender pays income taxes on the nominal interest payment, then the after-tax proceed is (1 - r)i, times the amount of the loan, where z is the tax rate. Furthermore, if a borrower is allowed to deduct interest payments from gross income, then the net payment by the borrower is also (1 -z)i, times the amount of the loan. Given the reciprocal nature of t.he tax, the after-tax nominal rate will adjust to expected inflation. If the appreciation from holding a commodity is not taxed and p(r+ P$is the after-tax real interest rate, then

or

and the nominal interest rate will adjust by more than the anticipated inflation rate. If taxes are paid on the appreciation of the real commodities, however, then the increase of the nominal interest rate is less than l/(1 _St) because the after-tax appreciation is less than the full appr The size of this marginal tax rate will be a matter of some irnp~rt~n~e in the empirical analysis. For simplicity, I will suppress the tax rate in the discussion of expectations. No important property of the test is ehan this omission. 2.2. Expectutiorzs of hjlation Fisher’s hypothesis has no testable content without a hypothesis concerning expectations of inflation or a data series measuring the anticipated inflation sate. I introduce a hypothesis. co:icerni;;G cxpectntions of inflation.

G. Dqvr,

Jr., Arc inflction expectations rational?

Expected real interest rate unpredictable? 63

The hypothesis that the securitie market is ‘efficient’ [Fama (197O)J in reflecting a rational e**pectation of inflation in the sense of Muth (1961) is the hypothesis that the anticipated rate of change of prices contained in the nominal interest rate is the same as E[dp,, 1 1i2J, the ‘actual’ or ‘objective’ expected value of the rate of change of prices conditional on the information set, Ll,. available to market participants. Substitution of this term into (1) yields

(2) If the variation of the real interest known, then the derivation of the deftnition, the actual rate of inflation inflation plus the prediction error, r$+ ,,

rate and the measurement error is appropriate test is immediate. By next period is the expected rate of i.e.,

If all p.W prediction errors, #, 9:. , . . . , are inci ded in a,, then gp is a serially independent random variable. Su:: ,titu .R of (3) into (2) and rearrangement yields *4~,+, I= --In,+!&--rlf)+qP+,.

(4)

This regression of hfp, + , on i, - iii satisfies the assumptions of the general linear model necessary for unbiased estimation, *md, under the hypothesis, the coefficient of i, ---vi shc~uld be one. Fama’s (1375) test is a particular case of (5) with qf assumed to be identtcalty equal to zro. If sf is unknown, however, there is no restrrction on the estimated coefftcient from a regression of the rate of inflation on the interest rate. [Malinvaud (1970, pp. 374-383).] The regression of LIP,.+I on i, has the whrch is correlated with the right-hand sidi: varirble, i,. If residual FIN+ 1 -t#, 81: is a serially uueorrelatcd variable with constant variance and the ~orre~~ti~n of Efd;p, + I 1ii!,] and Y$is zero, then estimation by ordinary least sq~~~res will praduce the usual result: a bias toward zero. More generally, if the ~~rr~J~tion of E[dp,, 1G?J and qf is not constrained to zero, then the ~rd~~~ry least squares co :fficierrt estimator of the coefficient of i, can be :he correct coefficient is one. ative or crofter than on@, cw n thou ~~~~~t~e~~~s~even if rjf is Ano~~n and used by market participants but is ~~l~k~~w~ to an investigator, a test can h;: constructed by imposing restrictions on vi. ectetF

In the test used in

this

p

real irircrest

me

64 G. Dwyer, Jr., Are it#lation expectations rational?

Expected red interest rate unpredictable?

does not help to predict P$ even though it helps to predict it and E[dp,, 1 1Sz,]. Let Y, denote an information set which is a subset of $ that excludes k (which will help to predict q:’ because it contains qf) and possibly other variables in a,. Then the expectation of i, conditional on Y’r yields, under the hypothesis (2),

The expectation of f[dp, + 1 I L!,] conditional on Yy, yields, as long as Ylvf includes only elements in a,, E[Ap, + i I YJ If the elements of Yy,do not help predict the variation in the real rate or the measurement error in i,, then E[qf I YJ =O. Therefore, (4) becomes

That is, given an information set which does not include the current interest rate or other variables that help to predict the variation of the real interest rate or the measurement error in the nominal interest rate, the predicted value of the interest rate at r differs from the predicted rate of change of the price level from t to t + 1 by a constant. Implementation of the test only requires the specification of an information set that is a subset of $2, and testing the constraints across the interest rate and inflation expectations. In general, only serially independent (or random with respect to time) variation of the expected real rate aud measurement error is consistent with the condition that the variation not be predictable by the information set. For example, if only past n-krlal interest rates are used to predict the infiation rate and interest rate, +erial dependence of the expected real interest rate will result in part of the c.trrent variation of the expected real rate being predictable by the past movements of the real rate included in past nominal interest rates. The hypothesis of an unpredictable expected real interest rate is admittedly ad hoc, but anj’ other hypothesis would require the specification of the parameter ualues which characterize the predictability. Ikcause there is no independent measure of the real rate which is adequate for present purposes, any calculation would have to be based on the dilIerence between the nominal interest rate and the rate of change of prices. The exact ~~r~~~~t~r~ determining the expected value of this ex post red rwtc, IIOWOVW, cm determined from the parameters det,ermining the cxpectcd values of the inflation rate and nominal interest rate.2 Therefore, without a restriction on ‘This follows because

EEr,1Kl = Eli, I Kl where r, is the real interest rate.

fT.&

+ l 1%I.

G. Dwym. Jr.. Are i&tinn

expecrarions rational?

Expem~d real inter.*st rat@ unpredictable?

the expected real interest rate, there are no testable restrictions relationship between the nominal inter-st rate and the inflation rate. 2.4. CdCtikJtiUFt

of

65

on the

the rrstrictiorrs

In order to have a tractable lest, linear regressions instead of mathematical expectations wilt be used in the actual test. 3 To emphasize this distinction, E,_ ,X, will be used where expectations are taken and will denote the regression of A’, on :,ome subset of the information available at t - 1. Consider 8~ information set that includes i,_ i and ilp, -j, j = 1,. . ., k. Suppose Ap, and i, form a bivarlate covariance stationary stochastic process which ‘possesses the autoregressive representation

where the errors (E’S)are serially uncorrelahzu, jointly covariance stationary random variables with zero mean, Finite variances, IY&and ~r,2~, a~,d a firrite Constants are suppressed, as they will be in succeeding covatiance, CC,~.~~. equations; variables can be interpreted as deviations from their means. plus the The hypothesis that expectati,>nS of inflation are rational hypothr;sis that the expected real interest rate and measurement error are not predictable by lagged inflatian rates and interest rates implies that, except for the suppressed constant, the one-step-ahead forecast of the inflation rate from (7) and (8) eqrals the forecasted interest rate from (8). Let E,_ , Ap, den&e the ratisnal expectation of Ap, ven knowledge of i, .-j and AP,-jj= I,..., k, and similarly for i,. Updating ey. (7) one period and calculating the expectation given information dated t - 1 and earlier,

By hypothesis, E,.; 1 &I, and E,... 1i, “The @alculationaf the

r~st~i~ti~ns

wn

be

obtained from (‘7) an3 (8). Making

is similar to th, t in Sarp2nt (1978,

1979) except that the

66 G. Dwyer, Jr., Are iqflation expectations rational? Expected real interest rate unpredictable?

this substitution

and collecting terms results in

. + 1 ~0. From (6), I?,_, dp,,, equals E,- 1 i, (except for the where crp+I = elf, suppressed constant part of the real rate of interest); therefore tile coefkients of (9) are equal to the coefftcients of J3). This Implies that, after rearrangement, g&?-(-I$? /jjp=

1;

a’l+‘,

-

1

j=l,...,k. ai=

(10)

ap ai.+ f$. 1 ;_a,+1, 1

Thus. under the null hypothesis, the coefficients of the interest-rate regressions CP~I 3e derived from those in the inflation-rate regression. There are 2k non-linear restrictions across (7) and (8), and instead of the 4k seemingly independent parameters in these two equations, there are only 2k independent parameters.4 This test can easily be generalized to one or more additional variables which help to predict either the interest rat<;, the inflation rate, or under the hypothesis both. In the case of one additional variable, X, a regression to predict X, is added to (7) and (8) and lagged values of X are added to (7) and (8). Restricted values of the coefficients in the interest rate regression can then be calculated ,which are a function of the coefkients in the re 4An alternative test could be based on estimating a rcgrcssion of the real interest rute an p&t inflation rates and interest rates. Then the restrictions that all of the coefhcients in the rcggwiinn are zero could be tested. Consistency with the formulation in the ext and the evitlcncc prcsentbd by Hess tind Bicksler (5975) would require that both the unrestrtcted and the restricted regressions have a firsl-order moving-average parameter. The error term is first-order moving overage because JI, is assum to be unknown at the beginning of period r; themfore, E: is unknown and is not inchtded in the information cet. As a consequence, the errors for rr md r, ,. , contain a cammon element, c4p, which results in the moving-average error. Because the error of the real-interest-rate: regression follows a first-order moving process, two problems in non-linear Icast syuarcs would hava to be solved: one with the 2k coefficients and the error term; and one with only t,he!em% tmm. The procedure 1 use requires two linear least-squares regressions with 4k estimated coeflScients and solution of one non-linear least squares probletn with 2k estimated coe:flc$nts. Besides being somewhat simpler, the later procedure also highlights the ability of variable; to hdp predict the inflation rate and the nominal interest rate,

C. DWJW. Jr., Ate i@ation expectations rational? Expected real ir?terest rate unpredicta&?

67

for X and the inflation rate. Tn order to keep the computations manageitble, onfy one series besides the interest rate and inflation rate will be considered at a time.

Esti-,nating the sions (7) and (8) and testing the restrictior:s (IO) is a relativrsly straight exercise in non-linear least squares. For simplicity and to most close low Wilson, (1972), the exposition of the econometrics unde-Aging the p re is based on the assumption that the errors are normally distributed. This is not strictly necessary for the estimated coefGcients to be consistent estimates of the parameters or for the test to be valid. [Malinvaud (1970, pp. 326X8).] If the error terms in (7) and (8) are normally distributed and fiilutually serially uncorrelated, the natural logarithm 6f the likelihood function L, is In L- -Tln(2n)-g

lnIrl--$

i

I=1

~;19-‘~~,

ill)

where 6, is a two-by-one vector of errors of the regression equations for each of the 7’ observations and f is the contemporaneous covariance matrix of the errors.5 Besides the error covariance matrix, this likehhood function also depends upon the coefficients of the inflation-rate and the interest-rate ressions, which arr denoted by the vectors a and /_Irespectively. Evaluated at the maximum with respect to I’, the concentrated log likolihood is lnL=

--Tln(2k)-~TlnlPI-+?V.

Max~mi~ati~~n of the concentrated log likelihood with respect to a and /? is equivalent to minimization of lnlPl with respect to the same quantities. The unrestricled estimates of the regressions can be obtained by ordinary least squares. ause the right-hand side variables in the regressions are the same, the minimum In IP! is obt;tine$ by ordinary least squares [Zellner (1862)J; the ordinary lcast squirms estimator is also the maximum likelihooo rrors are normalIF distributed and serially uncorrelated. ~stirn~t~s of the ru ssions arc obtained by minimization of the non~l~n~~~rrcstr- ’ ms. Utadcr the null hypotvlesis that the rcstr~ct~ons arc true, ordinary least squares provides consistent estimates of t!~ elcmcets of x; thercfor~, ordinary least squares estimates are an adequate point for minimization of in lpj with respect to B with fi derived fro*n

68 G. Dwyer, Jr., Are inj7ation expectations rational? Expccte& real interest rate unpredictable?

ai. Under the null hypothesis, with serially uncorrelated errors, the fin:_1 restricted coefficient estimator is the maximum likelihood estimator. The k x k variance-ccvariance matrix of the estimators is the information matrix -E [

d21nL -’ i3ai3a’

1

’

and can be esltimated by the inverse of the Hessian matrix from the nonlinear minimization of In Irl [Wilson (1972, pp. 82--83)J. The square roots of the diagorlal elements are estimated standard errors which can be used to obtain asymptotic confidence intervals for the coefficients using the asymptotic normaiity of the esrimators. The variance-cov- ,,,ance F: matrix of the restricted coen”cients in fi can bi: calculated by a simple procedure pointed out by Malinvaud (1970, pp. 623 628). The san e conditions resultmg in the asymptotic normality of a? resuh irl the asymr 3tic normality of 6; therefore, confidence intervals can b2 constructed once the variances of / are obtained. Let da =&-a and dg = b - j?. The variance of /?$‘,for example, is calculated from d@‘, which is equal tv

Squaring this expression and using the fact that, e.g., dc$ is the variance rjf c$’provides an estimator for the variance of &‘. The test of the restrictions is based on the Skelihood ratio of the rcstricteJ and unrestricted values of the likelihood function (L’ and L” respectively!. The null hypothesis is that the restrictions in (10) are true and the alternativi” hVpo:hesis is that the equalities do not hold. The likelihood ratio is

where li’rl and lP”l are the restricted and unrestricted determinant of r. The null hypothesis is rejected if i is specified value, i.e., the 1iMihood of the sample under the hypothesis is substantially less than under the unrestricted hypothesis. As ii well known, -2 lnfi has an asymptotic chi-square d:lstribution with de!grecs of freedom equal to ti:s r.umber of restrictions under the null hypothesis. 7 Therefore, an appropriate rest is to calculate %f course under the null hj >&hcsis. there is no large-sample mascn far non-linear itcrstian to obtain a detter estimate of CLand fi. Experience with this apnlir~aticn sqggests, however, that nmerZal!ly smallchanges in Q can have substantial effects on 1P 1, “For a pertinent exposition, see Malinvaud (1970, pp. 3.58-360).

G. Dwyer, Jr.,

Are ~~~tio~ expectations ratiord?

Expected reui interest rate unpredictable? 69

and comparti this to the value of a chi-square indicating relation of the null hypothesis.8

distribution

with large values

4. Empirical results The test is applied to quarterly interest rates on United States Government three-month Treasury bills since the Treasury- Federal Reserve Accord. Results based on only the past history of interest rates ana ehe inflation rate by results using times series that help to predict the interest rate and the inflation rate: the rate of change of the source base of the money supply, the rate of change of the money supply, and real gross national product (GNP). A brief description of the data precedes the analysis of the tests. 8.1. The data The interest rates on three-month Treasury bills are from Salomon Brothers ( 1976, Part IV). The interest rates in the source are on a discount basis at bid and are converted to ?I-day returns. The interest rates are for the first month of each quarter. They are measured ‘mid-month’ through 1959 and ‘first of month’ thereafter. The price level used is the Consumer Price Index (Cal) calculated by the Bureau of Labor Statistics. The rate of change of p;~ces is the first difference of the logarithm, as are other rates of change. The immediate source of the below is the data tape compiled by the Federal Reserve Bank price data u of San Francisco. from the uses side is currency held by the non-bank The source ba public and commercial banks plus member bank deposits in the Federal RLSXV~.‘The money supply is measured by M2, currency held by the nonbank public plus demand, time. and savings deposits in commercial banks otiable certificates of deposit. The source base and money res arc from the Federal Reserve Hank of St. Louis and are ures fur the first month of each quarter. the Slrrvey o/’ Currsnt Business (1976) through 1974; ures are from more recent issues of the same publication. ‘Alt~o~~~h there is no evidence of ~ubstantk.1 fdrial correlation that serial c~~f~~~tion of the errors will nt:t bias sh~~~~ be not strl~ctu~~ of the serial correlation changes. These either way. There was no su stion of such c~~~~~s in the cases that rcjcctril.

in the empirical results. it the test of the restrictions chrnl vx cold bias the test the ~11 hypothesis was not

70 G. Dwyer, Jr., Are inJation expectations rational?

Expected real interest r&e unpredictable?

All data from the sources are not seasonally adjusted, except real gross national product Iwhich is only available seasonally adjusted. The serious questions raised by the Advisory Committee on Monetary Statistics (1976) suggest that the rmon-seasonally-adjusted data on the base and the money supply are to be preferred. For present purposes, the most serious problem probably is that the final seasonal adjustment of a month’s data depermds upon future values of the series.’ Nonetheless, systematic seasonal variation of ihe real interest rate is not a topic that I want to analyze. It is doubtful if seasonality in the construction of the CPI, which has been analyzed by Fama and Schwert (1977), can be distinguished from any seasonality in the ‘true’ real interest rate on consumers’ goods and services. 1 have dealt with this by subtractin quarterly means from the interest rates and inflation rates; this procedure is eq,uivalent to regressing them on an intercept and three quarterly dummy ,variables and using the residuals a>f the regression. In general, this is not suflicient seasonal adjustment, but the results suggest that further correction for seasonality would serve little pklrpose. The means of the rates of change of the source base, the money supply, and real GNP are also allowed to vary by quarter. Quarterly means of the variables are presented in table 1 for the period from January 1953 to July 1976. Table 1 Quarterly means of variables (I-1953 to III-1976).

Quarter

rate

Interest rate

Source base growth rate

Money supply growth rate

Kcal GYP growth rate

I I1 III IY

0.0055 0.0082 0.0106 0.0076

0.0104 0.0099 0.0100 0.0104

0.0141 - 0.0057 0.0196 0.0070 _- Dsss

0.0247 0.0094 0.0105 0.0158 --

Q.0080 B.oOR6 0.0085 O.11060

Inflation

*”

.~_-

4.2. inflation and irzterest rates Table 2 presents restricted and unrestricted regressions of the inflation rate 2nd the interest rate on two lags of both series. The data on the left-hand side of the regressions begin in January 1994 and end in July 1976. The ‘The seasonal adjustment of ,x%1 GNP’, as well as the, t”\:blished source base and rnonrIf supply, is ba& VII LL EYWW cai the Census X-11 m&xl, a ratio-to-maviny_avel~L~~ technique that uses future values to estimate the final seasonal adjustment af the series, This final series is not part of the kkrmation actually available when holders deterniine the rate an Treasury bills.

In genera!. aspmetric seasonal adjustment of series in a regression which hold&filr the nonSeaSOnally-adJuSted data

results in systematic changes of the coeflicients. Seasonal adjustment with dummy variables, or Ehe same linear filter for a!! series, w&l not result in such changes. For a useful discussion, see Sims (I 974).

Uwmfrktad

~@~~~~sj~~~s

Inflation rate

0.1556 (0.1~)~

0.2688 (0.0950)

0.8497 (0.2634)

-- 0.0388 (0.2829 j

0.719 0.4J5 x 10 2

Interest rate

O.O*bJ1 (0.0396)

- Q.Oi80 (0.0373)

0.9610 (0.1034)

-0.1149 (0.1112)

0.879 0.159 x 10-z

NrstrirtL”d wgrcssivns Inflation rate

0.1940 $Mw4) *

- 0.0278 (0.0112)

0.58 15 (0.0510~

- 0.0583 (!).O?W,

0.649 0.452 x I(!- z

Infcte~st rat ,’

0 0830 (0.0277 I

- 0.0455 (0.0172)

0.9508 (0.0761 )

-0fW55 0 ’ 26)

0.878 0.159 x IO_’

_--_---_-

Likelihood ratio test statist IC Marginal significance level _.-. .II._.s-_l_-_--_l_.. __.

“Estimated

20.19 0.459 x !O_’

standard errors of the coefficrcnl: estimates are in parentheses.

unrestricted regressions are simply two ordinary-least-squares regressions; the restricted regressions arc estimated regressions subject to the non-hear restrictions implied by the hypothesis that expectations are rational and the expected real interest rate is unpredictable. The nekt to the last row of tht table contains the likelihood ratio test statistic for testing the restrictinns; the last row cantains the marginal signikancc level at which the restrictions can be rejected. Only if one were to use an extraordinarily low significznce level less than about 0.05 cf a percent would one not reject the hypothesis. Therefore, loa the period as a whole, the hypothesis is rejected at any ucual signi~cance level. It is plausible, however, that rejection of the hypothesis is largely due to events in the last part of the sample. Fama’s (1 even subsequent amendments [Farnil ct al. (1977)], suggest the hypothesis tested paper is more plausible Fama’s hypothesis that the before price controls in Au ust expected rate

72 G. Dwyer, Jr., Are in/lotion expectations rational? Expected real interest rate unpredictable?

price of crude oil by thz Organization of Petroleum Exporting Countries significantly reduced real rates of return on capital and consequently on other assets such as Treasury bills. Berndt and Wood find that energy is a complement with non-human capital in the production of commodities. If this is correct, the increase of the price of oil may have significantly reduced real rates of return; the rejection of the hypothesis may only reflect the gradual adjustment of asset holdings after this severe exogenous shock. This suggests that data ending July 1973 - the beginning of the quarter in which oil prices were quadrupled - would provide a more appropriate test. Estimated regressions for the two possible ending dates of July 1971 and July 1973 are presented in tables 3 and 4. For neither period can the hypothesis be rejected at usual significance levels. The data for both periods produce no evidence inconsistent with the joint hypothesis that nominal interest rates include rational expectations of inflation and that variation of the expected real interest rate cannot be predicted by interest rates and inflation rates a quarter or more past. Table 3 Estimated regressions with the inflation rate and the interest rate (l-1954 to 111-1971). Explanatory variables’ coefficients Inflatlcn rate lags Dependent variable

(1)

(2)

-

~-_

Interest rate lags -, (1)

(2)

~-.

P/s,

Unrestricted regressions inflation rate

0.0623 (0.1165)”

0.1465 (0.1080)

1.1397 (0.3020)

- 0.3505 (0.3380)

0.557 0.337 x 1o-z

Interest rate

0.0286 (0.0465)

0.0573 (0.0431)

1.1035 (0.1206)

-0.2577 (0.1350)

0.886 0.134 x 10’ z

Restricted reg vessions Inflation rate

-0.0173 (0.0376)

-0.0004 (0.0040)

1.1193 (O.OS29)

- 0.0937 (0.0591)

0.542 0.342 x 10‘ a

Interest rate

0.0012 (0.0315)

-0.0001 (0.0006)

0.9474 (0.0467)

- 0.0136 (0.0240)

0.8BO 0.138 X IQ-”

Likelihao 1 ratio ml statistic Marginal siqificanca level

6.12 0.191 WV_._--P. ‘Estimilted standard errors of the coefficient cstimatcs are in prtrcnthescs.

4.3. Injlation

W*VI__

information

The series considered additional information for predicting inflation rate and the interest rate are the growth rotas of the

G. Dryer,

Jr., Are injlatio,! expectations rational? Expected real interest rate unprcdicrable? 73

Table 4 Estimated regressions Hith the inflation rate and the interest rate (I-1954 to III-1973). P-W--Explan&oryvariables’ coefIicients ddent variable Usrestricted

Innation rate laps ~,.,~.~ ~~~,~~~~. __

Interest rate lags

(1)

(1)

(2)

R2!s,

(2)

regressions

Inflation rate

0.0494 (0.11233”

0.1498 (0.1135)

0.9999 (0.2752)

-0.1807 (0.2943)

0.561 0.359x 1U 2

Interest rate

0.0677 (QMSQ)

0.0493 (0.0464)

1.0991 (0.1125)

- 0.2730 (0.1204)

0.871 0.147x 10 a

-0.0136

0.550 0.363 x 10 2

Rsrstricrud rPgrussions Inflation rate

0.0145 (0.0X5&

-0.0002 (0.00; 7)

i O.mJ)

Interest rate

0.0681 (0.0324)

- 0.0026 (0.0104)

1.0979 (0.08:

0.9991

(OAW 1)

- 8 2297 ,829)

0.869 0.148~10~~

Likelihood ratio test s:;.Y ,;#c 2.66 Marginal significance levrl 0.617 -_.. ~~-.-~~VZstirnated standard errors of the coeflicient estimates are in parentheses.

-

source base, the money supply, and real GN,P. Besides helping to predict inflation, the source base and the money supply are likely to capture any loanable funds or liquidity effects on real interest rates [Friedman (1068)] if they persist for periods as long as a quarter. If there are any ‘business cycles’ in real interest rates on three-month Treasury bills, it is plausible that the rate of growth of real GNP will heIp to predict this component; in fact, some forecasters define recessions in terms of real GNP growth. In order to adequately characterize the autoregressive representation of the trivariate stochastic process, four lags af the inflation rate, the interest rate, and the additional series are included in each regression. Each of tbesc time series provides a statistically significant increase of the e~pl~n~ti~n of the joint inflation-rate, interest-r,ate process. A likelihood ratio to test if the ~~~ffi~i~nts of the additional series are non-zero.” test is us ‘@The calculrrtion of the test statistic is the same !IS presented for testing the restrictions. The restricts determinant of the covariance matr:x of rhe re!,iduh:s is based on ordinary leas1 cssiwns of the inflation rate and the interest rate on four lags of both series. The Qtwminant is has& an ordinary least squarcsl regressions of the inflation rate and rate on roar lags of these variables plus four lags of the new variable. Despite the and an estimation of eight ~~~ition~~l eneficicnts, under the hypothesis of rationalty od real intmcst me, thcrc are actually only four B klitional coefficients. It :L unpr~ict~ble ex most appropriate to test if the new variab!es help to predict the interr i rate and inflation rate under this hypothesis; ?~~r~fo~~‘the test is based on chi-square values with four degrees of freedom.

74 G. Dwyes, Jr., Are inflation expectations rational? Expected real interest me unpredictable?

The likelihood rati. test statistics for testing if the additional series help to predict the joint process of inllation and interest rates are 9.83 for the source tease, 25.51 for the money supply, and 20.36 for real GNP. Each of these likelihood ratio test statistics is greater than the five percent significane value of 9.49 for the chi-square distribution with four degrees of freedom, although the source base’s test statistic is only marginafly greater. In fact, the likelihood ratio test statistics for the money supply and real GNP are substantially greater than the one-half of one percent significance value of 12.8 Unrestricted and restricted regressions estimated with data through July 1971 are presented in tables 5 through 7. All of the results are quite consistent with the hypothesis that expectations of inflation ate rational and that movements of expected real interest rates cannot be predicted by movements of these series one quarter or more past. It is interesting that the restricted coefficients of both the source base and GNP are numerically close to zero in the inflation regressions. ii This is not, however, true for the money supply, which also is the series that provides the largest increase of the explanation of the interest-rate and inflation-rate process. Results obtained for the period through July 1973 largely replicate the results for the shorter period. These results are presented in tables 8, 9, and 10. At no usual significance level can the hypothesis be rejected. In fact, judging by marginal significance levels, the restrictions are less binding for this longer period than for the shorter period. One further problem still remains: the effect of taxes. All the results above are based on the assumption that the marginal tax rate is zero, or equivalently, that the marginal tax rate on the interest payments from Treasury bills is the same as the marginal tax rate cm the appreciation of consumption commodities. This is not a particularly plausible condition to apply to the United States in recent years. 4.4, A positive tax rate Testing the restrictions with a positive marginal tax rate requires an estimate of the tax rate. Estimating the m nal tax rates on sh~rt-&~~~ government securities and consumables wo itself. For the present, I have estimated a crude ran assumption that appreciation of the commodities in the all and that the marginal. tax rate on government securities is the same 8%the marginal rate on corporalte bonds compared to tax-free municipal ~CYI “As a referee notes, the coefficients of real WP in tbe Mation-rrmte and int~~ar~~~ regressions in table 7 (and table LObelow) tend to hsve stron$otilktary Gompant%r% ‘IX8 seasonalityof the coeficientsmay he due to the seasonal adjustment of suggests rhat the asymmetric seasonal adjustment may explain why real GNP ap predict inflation and interest rates.

“d stimat

Interest rate

errors of the coefkient

estimates

0.028 I (O10233)

0.0307 (’ 0216)

0.0443 (0.0835) 0.0310 (0.0174)

0.4486 (0.0785) 0.0339 (0.0170)

0.0312 (0.0188)

0.0056 (0.0883,

- O&J49 iO.0062)

Likelihood ratio test sta:istic Marginal significance level

-0.0065 (0.0098)

0.0011 (O.GiW)

0.0298 (0.0239)

0.0057 (0.1247)

0.0374 (O.T,ZiY:

0.4403 (0.1129)

are in parentheses.

- 0.0323 (0.0190)

0.0021 (0.0337)

0.032 1 (0.0340)

standard

0.2917 (0.1674)

-0.3126 (0.1623)

- 0.0132

0.1940 (0.2140)

-0.0118 (0.1914)

kO873 (~.~~)

~~~~ess~~~

Source base growth rate

raie

IIlfllltiO~

Restrlcted

Interest rate

3,

- 0.0240 (0.0641)

0.0345 (0.0580)

0.0192 (0.0627)

11.39 0.521

0.9771 (0.0961)

c. lY47 (0.5I 89)

C.3197 (0.3397) O.C@05 (0.0179\

1.1737 (6) 1964)

0.9695 (0.1227)

O.E9ll @WllO)

3.1313 (0.32%)

-- O.0050 (WO66)

0.0030 (0.0230)

0.3233

-0. (0.0617)

- 0.2645 (0.1518)

0.1895 (0.9423)

- 0.2724 IFi XII I___ ._I7)

-0.1853 (0.1873)

0.345 1 (0.9786)

- 0.6772 (O.xu2)

-0.0158 (0.1181)

- 0.9465 (0.8757)

0.0695 (0.0849)

- 0.2339 (0.1896)

0366 (0.5094)

growth rate. and the inter sate (I-1954 to HI-l971). ---_.-----P.-P-

Table 5 idlation rate, the smme

- 0.0039 (0.0427)

0.7774 (0.5927)

- 0.0023 (0.0214)

0.902 0.125 x IO- 2

0.539 (i.Y’5 x 10-Z

0.575 0.342 x lo- 2

0.908 0.121 x I@‘ z

0.541 0.634 x 10‘ z

I OR77 (0.6936) 0.2070 (0.1327)

c).614 0.326 x 10’. 2

-0.1487 (0.3566)

^ “,.....--

0.0317 (0.0414)

-0.6649 (0.2150)

%stimated

standard errors of the co&c&t

-0.1053 (0.0100)

0.0238 (0.0234)

- 0.0059 (0.1215)

-0.1064 (0.0645)

(2)

0.0392 (0.0568)

-0.0187 (0.0652)

0.?067 (0.3360)

0.0416 (0.0074)

-0.0118 (0.0217)

0.0442 (0.1125)

0.0062 (0.0597)

(3)

Liielihood ratio test statistic Margmd si$YiYiM;e k:je:

0.0355 (0.0743)

0.4385 - 0.1327 (0.045 1) (0.0380)

0.1697 (0.0154)

0.0421 (0.0221)

0.3002 (0.1146)

0.1561 (0.0608)

(1)

Money supply growth rate lags

estimates are in parentheses.

- 0.09w (0.1004)

0.1162 :o.K2Wj

II&rest rate

0.0161 (0.1050)

0.1170 (0.280)

0.5569 (0.0776)

-0.5518 (O-0738)

Money suppKy -0.6860 growth rate (0.0992)

0.0350 (0.1200)

- 0.1087 (0.0149)

-0.1092 (0.0413)

0.3457 (0.2144)

0.4718 (0.2271) 0.0224 (0.0437)

-0.1263 ;0.1138)

0.2242 (0.1205)

(4)

0.0833 0.0784 {O.O16~j rao147j

0.1119 (0.0437)

(0.2267)

-0.4404

0.2014 (0.1203)

(2)

0.1819 (0.0302)

Inflation rate

Restricted regressbas

Interest rate

Mosey suppIy mowth rate

0.058 1 (OJi41)”

regressions

Unrestricted

Inflation rate

(1)

Dependent variable (3)

variables’ coefficients

Inflation rate lags

Explanatory

16.42 0.173

0.0592 (0.0341)

0.0454 (0.0107)

- 0.0426 (0.0068)

(0.1028)

0.?146

-0.0448 (0.0546)

(4)

- 0.0776 (0.1815)

0.9541 (Z9426)

-0.4818 (0.5002)

(2)

0.9706 (0.3639)

-1.6640 (0.2393)

-0.0840 (0.4102)

1.3599 (0.2819)

0.0996 (E%) (0.0565)

0.9955 (0.1131)

- 2.0061 (0.5873)

1.3121 (0.3117)

(1)

Interest fate lags

-0J778 (0.4417)

1.1489 (0.17373

- 0.2489 (0.0598)

--(X859 (0.1845)

1.5389 (0.9580)

O.Oi48 (0.5084)

(3)

Estimated regressions with the inflation rate, the growth rate of money, and the interest rate (I-1954 to III-!971).

Table 6

0.0815 (0.4093)

(0.0432)

0.1599

-0.1496 (0.0372)

0.0755 (0.1410)

0.2067 (0.7323)

-0.2183 (0.3886)

14)

x

io- i

0.919 0.114 x 10-z

G.576 0.624 x lo- !

0.330

0.604

0.319 0.113 x 10-z

0.623 0.X8 x 1O-2

O.&l5 0.312 x 10 ’

lz=/s,

- 0.34Qf 10.264?]

a.1096 (O.Q429j

0.0769 (O.Q428)

- 0.0160 (0.0410)

- u.3 170 (0.2*)21)

O.QW (O.Q3?7)

0.0801 Real GNP growth rate (0.22c\?’

0.9776 (0.0388) CO.0405)

- MQQ9

- 0.5841 (0.2O.zO)

(0.0302)

- o.om4

Q-0195 ~Q.~8~

0.181S iO.1227)

43)

ents

- 0.0016 (0.0079)

0.1499 (0.1277)

0.0012 (0.~0058)

- O.Q478 (0.0427)

0.0589 (0.2632)

(0.1168)

0.0300

t4!

(1)

0.052 1 (0.0154)

- 0.0556 (0.095 1)

0.0426 (0.0192)

0.0046 (0.0185)

o.QQO2 (0.0855)

0.0066 (0.0205)

Likelihood ratio test statistic Marginal significance level

0.0305 (0.0149)

0.4300 (0.0723)

0.0359 (O.Ol76)

16.82 0.156

0.0011 (O.QO94)

0.0285 (0.0192)

(0.1044)

0.9Q45

- 1.3054 (0.4868)

1.1165 (0.3775)

0.8728 (0.1239)

- 1.3678 (0.76413

(4)

- 0.074 1 0.1432 (0.1186) (oo:%) (E%, - (0.1183)

0.0020 (0.0562)

(3)

1.2838 (0.3391)

0.0304 (O.OS26,

(2)

-0.1990 (0.1451)

2.5543 (0.9789)

-0.1977 (0.4607)

-0.1349 (0.1907)

2.6799 (1.17S8)

- 1.0313 (0.5219)

(2)

Imxest rate lags

0.0428 (0.0525)

-0.0818 (0.048 1)

(1)

Real GNP growth rate lags

- aEstinbated standard errcrs of thy coeflicient estimates are in parentheses.

Interest rate

Inflation rate

0.0834 (0.1134)

Restricted regressions

Interest rate

Inmion rate

0.0787 Real GNP growth rate (0.2641)

(aj

0.1099 (Q.l l7S)

regressium

(1)

Inllation rate lags

0.u (Q.1172p

Unrestrictd

Dependent variable

Explanatory rariabW

0.0963 (0.1763)

- 3.4692 (1.45913

0.0381 (0.1368)

-0.1559 (0.1991)

- 3.9652 (1.2274)

0.4554 (0.5448)

(31

Estimated regrcssiom with the inflation rate, the real GNP growth rate, and the interest rate (I-1954 to I&1971).

Table 7

-0.0048 (0.1414)

2.6499 (1.0160)

-0.0123 (0.0478)

Q.2223 (0.1394)

3.0975 (0.8595)

-0.0811 (0.38151

(4)

-

0.903 0.121 x 10 -i

0.453 0.718 x 1O-2

0.574 0.342 x IO- 2

&id 0.116x lO-2

0.458 0.715 x 10-Z

0.6’34 0.317 x !O- 2

R2/s

(19

0.0855 (0.0441)

Interest rate

o.O941 (O.OB30)

..

0.116 (0.1535)

0.0021 (O-0326)

0.0542 {O&464)

0.1366 (0.1167)

(2)

rate lags

0.1017 (3.2252)

-C3118 (0.2347)

(0.0351)

-0.0230

- 0.3632 (0.1546)

0.0070

(0.0192)

0.0282 (O.Ol58)

-0.0165 (0.0215)

oB08O (0.0166)

0.0066 (0.0826)

0.0069 (0.0188)

0.0140 (0.0232)

0.0188 (0.1172)

o&l67 (0.0583)

(2)

0.0375 (0.0170)

0.0172 (0.0847)

o.O005 (0.0032)

0.0257 (0.0246)

- 0.0107 (0.1242)

- 0.0392 (0.0619)

(3)

estimates are in parentheses.

Likelihood ratio test statistic .Marginal signikance level

0.3924 (0.0777)

0.O063 (0.0185)

0.0203 (0.0218)

0.3740 (0.1100)

(0.0549)

- 0.0352

(1)

Source base growth rate lags

0.2646 (O.1540)

(0.0151)

- 0.0061

-0.O810 (0.0446)

(0.1123)

oBO14 (0.0455)

- o.ooo7

(4)

0.2048 (0.1170)

(3)

“Estimated standard errors Oz the c&Gent

rate

Interest

Source base

0.0062 growth rate (0.1534)

rate

Inflation

Restricted t~gtesions

OAJ540 (0.1110)’

Mation rate

Unrestticted regtmions

Dependent variable

Mation

-.

10.541 0.569

-0.O089 (0.0136)

0.2654 (: 9845)

0.0052 (0.0137)

0.0166 (0.0233)

0.3297 (0.1176)

(2%)

(4)

0.9951 (0.1042)

- 0.0945 (0.5 196)

0.8639 (0.3575)

0.9982 (0.1159)

(0.5847)

-0.0914

0.9586 ((J.2915)

(11

- 0.2555 (0.1317)

1.1442 (0.6668)

0.0536 (0.4170)

- 0.2776 (0.1640)

1.1085 (0.8279)

- 0.3645 (0.4128)

(2)

Interest rate lags

0.0319 (0.1122)

- 1.3411 (O-6468)

- 0.0325 (0.0978)

0.0200 (0.1690)

- 1.3844 (0.8528)

0.2278 (0.4252)

(3)

Estimated regressions with the inflation rate, the source base growth rate, and the interest rate (I-1954 to III-1973).

Explanatory variables’ coeflkients

-..--

Table 8

0.0347 (0.0436)

0.880 0.141 x lo-’

0.7301 0.485 (0.4529) 0.700~ 1O-z

0.551 (0.0165) 0.363 x lO-Z

- 0.011

-0.0018 0.886 (0.1248) 0.138 x 1O-2

0.8282 0.491 (0.6297) 0.695 x 10-I

0.590 0.347 x 10-2

--’ R=/s,

- 0.2970 (0.3140)

(4)

0.2283 (0.1198) 0.4462 (0.2287) 0.0322 (0.0458)

K2382 (0.1195)

- 0.2872 (0.228!1

0.1127 ~0.0457~

-0.4621 (0.25?4)

0.0587 (0.0416)

0.0306 (Omsdy

-

-_I_

+tjtimated

0.0189 (0.0439)

-- 0.1384 (0.0448)

-0.1336 (0.0164)

0.0156 (0.0245)

- 0.0439 ;0.1224)

- 0.1394 (0.0641)

0.0456 (0.0412)

-0.0189 (0.0429)

Marginal

significance lev:!

13.83 0.312

(0.0118)

0.03 ifI

‘?‘wH)

-00532

0.0394 (0.0200)

0.2801 (0.0996)

- 0.0252 (0.0522)

0.2215 (0.0353)

0.0480 (0.13113j

-0.0153 (0.0225)

0.0823 (0 1125)

0.0343 (0.0589)

Likelihood ratio XSI statistic

0.0594 (0.0475)

0.4956 (0.0498 j

0.0856 (0.0346) - 0.1283 (O.OS53)

0.2093 (0.0193)

0.0635 (0.0225)

0.3639 (0.1124i

0.2Kv (0.05S9)

-0.1246 (0.02 15)

-0.1335 !0.043;1

0.2320 (0.2158)

-0.13% (0.1130)

I .oO53 (0.1737)

- 1.3001 (0.2467)

1.0011 (0.0331)

-0.1133 (0.2610)

1.2675 (0.3023)

0.1282 (0.0709)

-0.1235 (0.1549)

1.1938 (0.7729)

- fs6S6 (0.5244) 1.0241 (0.1051)

-0.1336 (0.40+8)

i.1574 (0.2746)

4.2)

Interest rate fags

- 0.0537 11.2716)

0.7678 (0.2470)

.- 0.2834 $0.0734)

- 0.0457 fO.1601)

0.5548 (0.7990)

- 0.2347 (O.(rlSq

(311

with the inflation rate, the growth rate of money, and the interest rate (f-1954 to 111-1973).

standard errors of the coefficient estimates are in parentheses.

0.0222 (0.0878)

0.1081 (0.0957)

0.0662 (O.OSS1)

Interest rate

0.5393 (0.0910)

- 0.4546 (0.1016)

-0.5231 (0.0922)

Money supply growth rate

0.0823 (0.0224)

0.0891 $.O22 i j

0.1426 (0.0326!

Inilation rate

Restricted regressions

1 ate

It&rest

?.%?aey siipply growth rate

rate

IniIation

Cinrestricted regresstons

Estimated ~&OGS

Table 9

- 0.0638 (0.2146)

O.OSZ& (0.0427)

-0.1202 (0.0578)

(0.1303)

- 0.0800

0.4284 /0.6501)

-0.2430 (0.3405)

(41

______-

0.906 0.125 x 10-I

0.654 x 10-Z

0.550

0.605 0.340 x lQ- *

0.906 0.125 x 10-2

0 590 0.624 A IO-”

0.637 0.327 x iO-2

R2/S,

fl)

0.1208 (0.0381)

-0-4149 (OA77O)

0.0116 (0.0255 )

0.0899 (OB436)

-0.0178 (0.0148)

0.2400 (0.1836)

-03480 (0.1733) o.O019 (0.0339)

- 0.0028 (0.0059)

!O.tMM)

-0.Q4a4

0.1023 (0.2560)

0.1011 (0.1198)

0.0052 ~0.0111)

0.0188 (0.0453)

- 0.3492 (0.2670)

0.2372 (0.1249)

(4)

o-0655 (0.0151)

- 0.0239 (0.0666)

o.OO75 (0.0126)

0.0635 (0.0191)

- 0.0432 (0.1127)

0.0523 (0.0527)

(2)

-0.0102 (0.0171)

0.0389 (0.0853)

o.OO67 (0.0103)

- 0.0120 (0.0201)

(k%,

- 0.0365 (0.0554)

(3)

Likelihood ratio test statistic Marginal significance level

0.0319 (0.0128)

0.4574 (0.0666)

- 0.0077 (0.0119)

0.0337 (0.0171)

0.4684 (O.lOOS)

-0.0251 (0.0472)

(1)

Real GNP growth rate iags

12.08 0.439

0.0X18 (0.0082)

-0.1836 (0.0748 J

-0.0013 (0.0022)

0.0279 (0.0188)

-tJ.l540 (0.1107)

0.1039 (0.0518)

(4)

0.8928 (0.0862)

- 1.6128 (0.5651)

0.8878 (0.1834)

0.8907 (0.1129)

- 1.6379 (0.6654)

0.9527 (0.3112)

(1)

-0.2401 (0.1143)

2.2867 (0.6384)

0.0024 (0.1447)

- 0.2572 (0. I 530)

2.3780 (0.9021)

- 0.5366 (0.4219)

(21

Interest rate lags

0.2339 (0.1063)

- 2.5236 (0.6170)

-0.0116 (0.02971

0.1439 (0.1559)

- 2.9280 (0.9190)

0.2%X ft r.4299)

(3)

rate, the real GNP growth rate, and the interest rate (I-1954 to 111-1973).

“Estimated standard errors of the coeffkient estimates are in parenthesff.

Interest rate

Real GNP 0.2294 grow*h rate (0.1624)

inflation rate

Restricted regressions

0.1230 (O.O404)

-0.4171 (0.2572)

Real GNP 0.2430 growth rate (0.2379)

I lterest rate

0.1460 (0.1203)

0.0396 (0.1113)s

Inflation rate

(22)

(3)

variables’ coefficients

Znnatioo rate Iags

Unrestricted regressions

Dcmderlt v&able

Explanatory

Estimated regressions with thi mfktion

Table IO

-0.1235 (0.0610)

1.5783 (0.4413)

0.0077 (0.0178)

- 0.0088 (0.1153)

2.3523 (0.6795)

- 0.2220 (0.3178)

(4)

0.905 0.126 x IO- 2

0.420 0.734 x 10-2

0.551 0.363 x lo- 2

0.908 0.124x 1O-2

0.426 0.730 x 1O-2

0.603 0.341 x lo- 2

R21s,

C. I)Nyer, Jr., Are injIation expectations

rational?

!ixp*cted real interest rate ut.predtctable?

Sl

The differentials between annual average yields of corporate

and ruugicipal range of marginal tax

bonds from 1954 through 1970 suggt*t a plausible rates from about 15 percent to 33 percent.” (Board of Governors 1976.) For

Aaa bonds, the range of estimated tax rates is from 20 to 32 percent with a me&u of 27 percent. For Aa bonds, the range is from 17 to 32 percent with a mean of 26 percent. For A bonds, the range is from 14 to 31 percent, with a mean of 23 percent. These values are only suggestive at best, especially since I will suppose that the margiual tax rate is constant, but the upper Lnd lower ends of the range can be used as approximations.‘3 Table 11 presents summary statistics for the restricted and unrestricted regressions based on the various information sets through July 1971. Results Table

11

Tests of restrictions with positive tax rates (I-1954 to 111-1971). IInterest and inflation rates plus: No addit ionai series

Growth rate of source base ---.

Grc b~tlr rate or a,loney -_

Growth rate of real GNP

Marginal tax rate equal to 15 percent

Likelihood ratio test StatisticE

7.34

Marginal significance level

0.119

Marginal

12.11 0.437

IS.18 0.110

17.94 0.118

tax rate equal to 33 percent

Likelihood ratio test statistic@

13.33

Marginal signkficance level

0.978x IO-’

lo.83 0.156

23.6 8 0.023

22.48 0.03 1

“If no additional series,the test statistic is &i-square null hypothesis.

With an additional

kedom under the null

with four degrees of freedom under the series, the test statistic is chi-square with twelve degrees of

hypothesis.

are presented both for the lower-bound marginal tax pate of 15 percent and the upper bound rate cf 33 percent. With a marginal tax rate of 15 percent, the results are quite consistent with the hypothesis. With a marginal tax rate of 33 percent, the results are marginally in favor of or opposed to the hypothesis depending upon one’s pi*ior. At the five-percent significance level but not the o~e~~ercent level, the restrictions can be rejected for the money supply and real GNP; at the one-percent significance level, the restrictions “Friedman and Schwartz (1976) cstimatc a m:uginal tax rate of about 33 percent. ISAn alternative would be to estimate the marginal tax rate as part of the non-linear optimization. It is clear, however, that the estimated rate would tend tc go to the boundary value of zero.

82 G. Dwyer, Jr., Are inflation expectations rational? Expected real interest rate unpredie~ablee?

can barely be rejected for inflation and interest rates without additional information. The results for the extended period through July 1973 are presented in +able 12. With a marginal tax rate of fifteen percent, the hypothesis cannot be rejected at the usual five- or one-percent significance levels for any information set. With a marginal tax rate of 33 percent, the hypothesis can be rejected at the k-percent significance level but not at the on~-~r~ent level for the regressions including the money supply. The hypothesis can be rejected at any usual significance level when only interest rate and prices are included in the information set. For this period, consistency of the data with the hypothesis is sensitive to the marginal tax rate assumed. Table 12

-

Tests of restrictions with positive tax rates (l-1954 lo Ill-1?73’. - - -.__I~interest and inflation rates plus: -_. _p. Growth rate Growth rate No additional Growth rate of real GNP sesies of source base of money -___ -

Marginal tax rate equal to f-5 percent

L.ikelihood ratio test statistica

9.01

hIarginal significance level

O.Ct’

12.68 0.393

16.41 0.173

12.70 0.391

Marginal tax rate equal to 33 perttwt

Likelihood ratio test statistic

16.15

18.38

22.59

20.18

Marginal slgnilicance level

0..282x lo- 2 0.105 0.03 1 0.064 --_.__eVS “If no additional series, the test statistic is chi-square: with Tour degrees of frcQdom under the null hypothesis. With an additional series, the test statistic is &i-square with twelve degrees of freedom under the null hypothesis.

5. Conelusiom

Given its restrictiveness, the joint hypothesis that ~x~~t~ti~ns are ratmniri and that the expected real interest rate on three-month Treasury bills is unpredktable does surprisingly well. Variables that economic theory sug will help to predict inflation and interest rates do help to predict t Nonetheless, the results from 1954 to 1371 or 1973 largely indicate that these vatiables do not help to predict expected real interest rates. The consistency of the data for this part of tile post-war period su that, at least over a period as long as a quarter, predictable than

G. Dwyer. Jr.. Are ~~~~fi~~g~_~~~~~f~~~l~ rational? Expected rrnl interest rate unpredictable? 83

money supply do not have any effec: on the expected real interest rate on three-month U.S. Treasury bills. This result is consistent with the implication of some rational-expectations models that predictable changes of the moriey supply win have no effect on expected real interest rates. The results indicate that the growth rate of real GNP does not help to predict rn~vern~nt~ of the expected rest interest rate. Because business ~uctuations are often defined in terms of real GNP growth, these results st that there are no si Cant business fluctuations in the expected real interat rate on three-month Treesury bilk

Advisory Committee on Monetary Statistics. 1976, Improving the monetary aggregates (Board of aovernors sr the Federal Reserve System, Washington, DC‘). fkrndt. Ernst R. :tnd David 0. Wet d, 1975, Tmhnology, prices, and the derived demand for energy, Revtew of Ecsnomics and Statistics 57. Aug. 259- 268 Darby, Michael R., 1975, The financial and tax effects oi mot:;:: y policy on interest rates, Economic lnqutry 13. June. 266-276. ~~ar~~rlt of Commerce, Survey of current business Fama. Eugene F.. 1970. Eflicient capital market\: ’ cview ot theory and empirical work, Journal of Finance 25, May 38s-417. Fams. ene F., 1995. Short-term interest rates as predictors of inT.*tion. American Economic Rc 65. June, 269 282. Fama, Eugene F. et al.. 1977. Intcrcst rtitcs and inflation: Commeilts and a reply, American Econixmic Review 63, June, 469496. Fwmu. Eugene F. cmd G. William Schwert, 1977, The behzior of relative and money prices of consnm n gnods, Unpublishtut paper. Feb. (University of ChIcago, Chicago, IL). 1896. Appr&atlon and tntcrcst (Au~dst M Kelley, Bookseller, New York; 1965). Fisher, lrvi l~rierlnren. ilton, 1968, Factors aff’ectmg !he level of interest rates, in: 1968 conference proceedings. Snvingq and residential financing conference (IJ.S. Savings & Loan League. Chicago. IL). Friedman. Milton, 1969. The role al’ monetary policy, in: The optimum quantity of money (Aldine. Chicago, IL). Friedman, Milton and Anna .I. SC nrtn 1976, Money and interest rate-s, Unpublished chapter States Etnd the United Kingdam, S:pt. of Monetary Trends in the Unit Gordon, Dan&d F. and Allan Hynes, 1670. On the theory of price dynamics, in: The micrmonomic foundattions of em~loyt~nt and inflation theory (W.W. Norton, New York). Has, Patrick J. and James t. #ickslcr. 1995, Capital asset prices versus time series models as predictors of inflation, Journal of Financial Economics 2, Dec.. 341 350 Ky~l~nd, Finn E. and f$+JwurJ C. Prescott, 1999, Rules rather than discretion: The inconsistency of optimal plays. Journal of Political Economy 85, June, 473-492. trtcns. Jr., R&xx? E.. 1972, Expectations and the nlutrality of money. Journal of Economic Theory 4, hpri;, 103 124.

M~tb, John F , t961, R~tio~~~l ~x~~t~~t~o~s and the theory of price movements, Econometrica 29, :u1y, 315,-33>. on Brothers. 1976, An analytical record of yictds and bond sprh ,jds (Salomon Zrothers.

34 G. Dwyer, Jr., Are ir@ation expectations

rational? Expected red interest rate unpr&ctable?

Sargent, Thomas J., 1978, Rational expectations, econometric exogeneity and consumption, Journal of Political Economy 86, Aug., 673-700. Sargent, Thomas J., 1979, A note of maximum likelihood estimation of the rational expectations model of the term structure, Journal of Monetary Economics 5, 133-144. Sargent, Thomas J. and Neil Wallace, 1975, Rational’ expectations, the op:imal monetary instrument, and the optimal money supply rule, Journal of Political Econom]. 83, April 241254. Sims, Christopher A., 1974, Distributed lags, in: M.D. Intriligator and D.A. Kendrick, eds., Frontiers of quantitative economics, Vol. II (North~Holland, Amsterdam). Wilson, G. Tunnicliffe Wilson, 1972, The estimation of parameters in multivariate time seriw models, Journal of the Royal Statistical Society, Series B 35, 76-85. Zellner, Arnold, 1962, An efficient method of estimating seemingly unrelated regressions and tests for aggregation bias, Journal of the American Statistical Association 57, June, 348-368.