This note demonstrates that Bennett McCallum's recent critique of low frequency estimates of macro-economic relationships is of little empirical significance. It also demonstrates that readily available and frequently used techniques can be used to diagnose the problem McCallum raises. Finally, it shows that the standard critique of expectational distributed lags is not warranted once the role of learning by economic agents is recognized.

Lawrence H. Summers

English

Research Papers in Economics

NBER WORKING PAPER SERIES
ESTIMATING THELONG—RUNRELATIONSHIP
BE1MEENINTEREST RATES AND INFLATION:
ARESPONSE TO McCALLUM
Lawrence H. Summers
WorkingPaper No. i48
NATIONALBUREAU OF ECONOMIC RESEARCH
1050 Massachusetts Avenue
Cambridge, MA 02138
September 1984
I am indebted to Robert Barsky for research assistance and helpful
discussions, to Brad DeLong for helpful discussions and to the NSF
for financial support. The research reported here is part of the
NBER's research programs in Economic Fluctuations and Financial
Markets and Monetary Economics. Any opinions expressed are those
of the author and not those of the National Bureau of Economic
Research.NBER Working Paper #i8
September 198)4
Estimating the Long—Run Relationship Between
Interest Rates and Inflation:
A Response to McCallum
ABSTRACT
This note demonstrates that Bennett McCallum's recent critique of
low frequency estimates of macro—economic relationships is of little empirical
significance. It also demonstrates that readily available and frequently
used techniques can be used to diagnose the problem McCallum raises.
Finally, it shows that the standard critique of expectational distributed
lags is not warranted once the role of learning by economic agents is
recognized.
Lawrence H. Summers
Departmentof Economics
Littauer Center 229
HarvardUniversity
Cambridge, MA 02138In a recent contribution to this journal Bennett McCallum (198I) sharply
criticizes econometric tests of long run economic relationships, which make use
of frequency domain time series techniques. Summers (1982) is held out as an
offending example. This note demonstrates that McCallum's criticisms are not
valid either theoretically or empirically. It nkes three points. First, as
asserted in ntr original paper, the issue of nasuring inflation expectations,
which is the heart of McCallum's critique, is of limited empirical significance
in low frequency studies of the Fisher effect. Second, very simple econometric
procedures implemented in itrjr original paper and in another paper criticized by
McCallum make it possible to bound the bias arising from the problems McCallum
raises. The potential bias is small. Third, once the role of learning is
recognized, the Sargent—Lucus critique of expectational distributed lags which
provides the basis for McCallum's comment is incorrect.
I. Measuring Expected Inflation
McCallum summarizes his comment by noting that "the low frequency measures
in question are simply not designed to reflect the distinction between antici-
pated and unanticipated fluctuations that is crucial for accurately charac-
terizing inter variable relationships in dynamic nDdels." The central issue of
measuring inflation expectations was recognized in rir original paper. Indeed
the Appendix contains an exposition of Sargent's (19T1) point which underlies
McCallum's comment. The text also addresses the issue of measuring inflationary
expectations and makes the empirical assertion that "low frequency variations in
the rate of inflation are almost completely forecastable so that the assumption
that expected inflation can be proxied by actual inflation is warranted.—2—
Indeed, when the equations reported here were re—estimated with various proxies
for expected inflation the results were not significantly affected."
Below this asssertion is documented using asures of expected inflation
which are free from the Lucas—Sargent—McCallum taint. Before turning to
this empirical evidence it should be made clear that the ntivation for per-
forming low frequency econometric tests is the conviction that history should
not be viewed as a sequence of realizations of a single stationary stochastic
process. The data can only illuminate long—mn issues if regime changes of the
type envisaged in comparative steady state economic theory have actually taken
place. The possibility that such changes may take place andthatif they do
low frequency econometric techniques may provide a good way to study their
effects is not disputed by McCallum. If regime changes have taken place, they
will acccount for nest of the variance in both expected and actual inflation, so
that actual inflation will, at low frequencies, be a satisfactory proxy for
expected inflation.
Ultimately then, the validity of the low frequency econometric techniques
used in rrr paper is an empirical question depending on the properties of the
inflation process. I tried to resolve it in preliminary work on the paper by
seeing if substantively different empirical results regarding the Fisher
effect were obtained when proxies for expected inflation were used. A rolling
ARMAprocedureof the type first used in Feldstein andSummers(19T8) was
employed. In this procedure each period a measure of expected inflation is
by forecasting inflation based on an ARMA estimated over the previously
available data. This method produces direct estimates of 11e which depend on the—3—
stochastic process actually followed by inflation, and so is free of the
problemsraised by McCallum. Note that McCallum raisesno objection to the low
frequency estimation of a structural relation betweeninterest rates and a
direct measure of inflationary expectations. His only difficulties involve the
use of distributed lags in measuring inflationary expectations.
Some of the results in nr original paper are reproduced along with results
using proxies for expected inflation in Tables 1 and2.The alternative results
suggest larger estimates of the effect of inflation on interest rates than do
thosein nrearlier paper. However, its principal conclusions are confirmed.
Forthe186O—l9IO period, thehypothesisthat the Fisher coefficient is one is
rejected. For the Post—War interval, the hypothesis that the Fisher coefficient
equals its tax adjusted value is also rejected. However, in some sub—samples,
theuse of the alternative inflation variablesdoes have dramatic effects. This
is particularly the case for the l951_197l interval, where the Fisher coef-
ficient rises from .79 to 1.39, at a frequency of over 3 years. The observation
that over short intervals the choice of an expectations measure has significant
effects should not be too surprising since they are less likely to contain
regime changes of the type discussed above.
II. Bounding Possible Errors'
The analysis so far establishes the empirical irrelevance of McCallum's
criticism of evidence pointing to the non—adjustment of nominal interest rates.
This section reconsiders McCallum's econometric argument and the next section re—
examines its theoretical basis.
Consider McCallum's example:itP+EtIIt+i+vt
(i)
(2)
where I have changed his notation 'but nothing else. McCallum's analysis focuses
on low frequency estimates of the relationship:
=+
+ i 11t + Vt
He notes that a researcher estimating this relationship in either the time or
frequency domain would conclude that the Fisher relationship failed to hold
even though it was built into the econonr by assumption in (1) as long as iii, < 1.
Equation (3) is temporally misaligned relative to the actual estimation
presented in Summers (1982),
=(p+ +
11t+l + Ut
where the actual realization of inflation is used as a proxy for its expec-
tation.2 McCallum in a footnote asserts that the distinction between (3) arid
(14)is immaterial. To see that this is in general incorrect, consider the spe-
cial case where e1 is perfectly forcastable using information available to
agents but not available to econometricians. Then clearly, 1Ttl =E(fl+1)
and
so (14) provides a legitimate test for the Fisher effect even though (3) does
not.
As McCallum (19T6)himselfwas the first to point out, (14) can be thought
of as having the classical errors in variables problem. Under the assumption of
rational expectations, is a noisy measure of E(JTt+i). This means that
will be 'biased towards 0. As Delong (1983) demonstrates, a parallel result to
be classical errors in variables formula holds in the frequency domain as well.—5—
Thereis no deep identification problem here, only one of errors in variables.
Aprimitive solution to the errors in variables problemis to bound the bias
by-running theregression equation in the opposite direction. Infact this Is
notnecessary in the two variable case since the reverse regression coefficient
can be calculated as the ratio of the to the slope coefficient in the origi-
nal regression. This primitive solution to the errors in variables (and
simultaneity) problem wasdiscussedand implemented in my original paper.
Implicitly, it wasalsoimplemented by Lucas (1980) who exhibits his data
graphically without explicitly estimating regressions. The results in my work
suggested that in almost all cases, the point estimates bounded the effect of
expected inflation on interest rates below that predicted by the Fisher theory.
An alternative procedure for calculating the possible bias in band spectral
estimates of (1) is to calculate explicitly the noise to signal ratio for as
of E(ll). This approach is taken by Delong (1983) who concludes that,
"there is no Fisher effect before World War II and after World War I it is not
possible to believe both in the distortionary effects of the tax system and the
Fisher effect."
There is both a methodological and a substantive point in this section.
The latter confirms in a different way the empirical analysis of the first sec-
tion suggesting that problems of measuring inflation expectations are not an
important explanation for negative results regarding the Fisher effect. The
methodological point is that it is easy to determine whether expectational
errors are an important problem, and to estimate the nEgnitude of any biases
they engender.
a measure—6—
III.The Validity of Expectational Distributed Lags
As the previous section n.de clear, the use of actual inflation as a proxy
for expected inflation involves errors in variables problems, but not deeper
issues of the type raised first by Sargent (1971). Here, I demonstrate that
even if lagged rather than actual values of inflation had been used in the ana-
lysis, there would be no serious problem. The now traditional critique of iden-
tification in distributed lag irodels is incorrect once the need for economic
agents to learn the true economic model is recognized. The point made here is
spelled out in xrore detail, and its implications for rational expectations eco-
nometrics are spelled out in more detail in Summers (19814).
Consider again McCallum's example given in (1) and (2). McCallum and
Sargent (1971), (1973) implicitly argue that a regression of i on any distri-
buted lag of past JI would yield a coefficient of on and 0 on all lagged
ITs. This argument implicitly assumes that agents know and divinely and
need not try to infer them from the data, unlike underprivileged ecoriometri—
cians. Friedman (1980) discusses the implausibility of this assumption.
Assume on the other hand that agents always use some finite amount of past
data to estimate the parameter c in (2). If their estimates are unbiased,
they must have the property that:
+l
' =
orequivalently that:
=
(1—p1)11 (6)
whereITrefersto the mean value of fl in the sample period over which the para——7—
meter P0 is being estimated. Suppose for example that agents always use only
the nxst recent N observations in estimating P0. Then an estimte of' (3) in
which additional laggedvalueswere included would yield:
=++ (i_)pt—i +e (7)
Note that the weights on the inflation terms in (7) sum to unity. This result
generalizes easilytothe case where follows a irore complicated process than
(2). It also can be generalized to allow for alternative learning procedures in
which the weight given to past data declines gradually.
A more significant generalization of this result proceeds as follows.
Consider any method by which agents trytodiscern structure and forecast a sta-
tionary time series. Estimation and forecasting using time series models is one
example of such a method. Any explicitly described method will give rise to a
functional relationship between forecasts and past data of the form:4
=F(llt,JItl,.....lltNl)
This function can be approximated by using its Taylor expansion about the sample
mean value of II.This yields:
=F(tL...tNl)+y. [ll •
— (8)
where y. = ). Ifwe impose the minimal rationality requirement on the
1
t—i
functionF that F(Tt,L...T!Nl) =ii,we obtain:
N—l N—i
=+.
—. i=0 i=O—8—
1
N-i
Substituting for II using the definition11 = yields
1=0
=ili+ - —r] 11ti (10)
N
where r= 1.•Thesum of the bracketed terms in (10) is unity.This
1=01
establishesthat in general learning procedures applied to finite bodies of data
which satisfy a minimal rationality requirement will have the property that they
arebest approximated as weighted averages of past data with weights summing to
unity.5
Thus,once agents need to learn the mean inflation rate is recognized, tra-
ditional expectational distributed lags and their maintained hypothesis that the
sum of the lag weights used in forming inflation expectations is unity are vin-
dicated. McCallum's equation (9) shows that when the sum of the weights on
lagged inflation used in forecasting inflation is unity, band spectral
regressions will provide a legitimate test of the Fisher effect. Of course,
depending upon how agents process information, the lag length may be quite long.
But this corresponds to the available empirical evidence on inflation and
interest rates, particularly for the l860—l94O period, and also confirms the
desirability of using low frequency techniques.
IV. Conclusions
The point MeCallummakeswhile technically correct is of negligible
substantiveimportance for studies of the Fisher effect. Nor should his paper—9—
deter future investigations from using low frequency econometric techniques in
appropriate settings. There is no econometric "wonder technique" that will give
right answers in all settings. Empirical workers have at their disposal a
variety of techniques which will work well in some situations and poorly in
others. Successful empirical work requires an appropriate nRtch betweeen sta-
tistical technique and the data and question being examined. In order to be
telling, criticism of econometric techniques must go far beyond denonstrating
that conceivable settings exist where they will not yield meaningful results.
While McCallum shows that such settings exist for band spectral techniques, this
does not call into questions their utility in the setting where they have been
applied.If economists are going to test the "long run" implications of their
theories, some technique likeband spectral regression will be necessary.—10—
Footnotes
1. This section draws heavily on DeLong (1983).
2. Because of problems of data alignment, the equation estimated r.y not be
exactly equivalent to (4).
3. We assume implausibly that is known. This assumption is relaxed below.
4. Oneexample of a learning procedure would be the rolling ARMA method
describedin Section I. By nRking N arbitrarily large, any procedure can be
approximated. Note that there is no requirement that observations be used sym-
metrically. Givingnre weight to recent data is consistent with (7).
5. Note that it is possible that some of the I will be negative. This will
occur if plausible learning procedures are applied to non—stationary series.—11—
References
DeLong, B., "Observations on the Non—Adjustment of Nominal Interest Rates,"
Mimeo (1983).
Engle, Robert F., "Band Spectrum Regression," International Economic Review
15 (February 197k), 1—11.
Feldstein, Martin, andLawrenceSummers, "Inflation Tax Rules, and the Long
Term Interest Rate,"BPEA1 (1978), 61—109.
Friedman,Benjamin M., "Rational Expectations and the Extreme Information
Assumptions of Rational Expectations Macromodels," Journal of Monetary
Economics, 1979.
Lucas,Robert E., Jr., "Econometric Testing ofthe Natural Rate Hypothesis," in
0.Eckstein, ed., The Econometrics of Price Determination Conference
(Washington: Boardof Governors ofthe Federal Reserve System, 1972) (a).
__________'TwoIllustrations of the Quantity Theory of Money," American
Economic Review TO (December 1980), 1OO5lOl1.
McCallum, Bennett, "Rational Expectations and the Natural Rate Hypothesis,"
Econometrica (1976), 43—52.
________"OnLow Frequency Estimates of "Long Run" Relationships in
Macroeconomics,"Journal of Monetary Economics, June l984.
Sargent,Thomas J., "A Note on the Accelerationist Controversy," Journal of
Money, Credit, and Banking 3 (August 1971), 50—60.
__________"RationalExpectations, the Real Rate of Interest, and the Natural
Rate of Unemployment," Brookings Papers on Economic Activity (No. 2, 1973)
129_372.—12—
__________MacroeconomicTheory (New York: Academic Press, 1979).
Suinnrs, Lawrence H., "The Non—Adjustment of Nominal Interest Rates: A Study
ofthe Fisher Effect," in J.Tobin, ed., Essays in Honor of Arthur Okun
(1982).
_______"RationalLearning Mechanisms Implying Irrational Expectations,"
in preparation, 1981k.

A Note on the Accelerationist Controversy,&quot;

andLawrenceSummers, &quot;Inflation Tax Rules, and the Long Term Interest Rate,&quot;BPEA1

Econometric Testing ofthe Natural Rate Hypothesis,&quot;

Observations on the Non—Adjustment of Nominal Interest Rates,&quot; Mimeo

Rational Expectations and the Extreme Information Assumptions of Rational Expectations Macromodels,&quot;

Rational Expectations and the Natural Rate Hypothesis,&quot; Econometrica

RationalExpectations, the Real Rate of Interest, and the Natural Rate of Unemployment,&quot;

RationalLearning Mechanisms Implying Irrational Expectations,&quot;

The Non—Adjustment of Nominal Interest Rates: A Study ofthe Fisher Effect,&quot;

TwoIllustrations of the Quantity Theory of Money,&quot;

Estimating the Long-Run Relationship Between Interest Rates and Inflation: A Response to McCallum

Estimating the Long-Run Relationship Between Interest Rates and Inflation: A Response to McCallum

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