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MONETARY AND NON MONETARY
S O U R C E S O F INFLATION: A N E R R O R
C O R R E C T I O N ANALYSIS
Kenneth N. Kuttner
Working Paper Series
Macro Economic Issues
Research Department
Federal Reserve Bank of Chicago
August, 1989 (WP-89-15)
M o n e t a r y a n d N o n - M o n e t a r y S o u r c e s o f I n flation:
A n E r r o r Correction Analysis
Kenneth N. Kuttner
The economic events of the past two decades have done a great deal to discredit
the old monetarist dictum, that inflation is “always and everywhere a monetary
phenomenon.” First, the oil shocks of 1973 and 1979 reminded economists that
major shifts in relative prices can be associated with large, iftransitory, inflation
shocks. Second, in spite of the rapid growth of many monetary aggregates, the
early 1980s saw an abrupt reduction in the rate of inflationin conjunction with the
deepest recession since the 1930s.
These episodes have underscored the importance of three distinct, ifinterrelated,
sources of inflation: first, the monetary effect; second, the demand side effect
represented by the Phillips Curve; and third, the price level adjustment resulting
from exogenous supply shocks, or shifts in the terms of trade.
This isnot to say that the three mechanisms operate independently of one another,
however. A given increase in the stock ofmoney may raise theprice level directly
(for example, in the context of a rational expectations model in which agents di
rectly observe the money supply), and indirectly through a Phillips Curve mecha
nism, as in textbook IS-LM models, or a rational expectations model with a Lucas
supply curve. Furthermore, while these mechanisms may be conceptually dis
tinct, in practice itmay be difficultto identify their structural coefficients, owing
tocontemporaneous feedback into the monetary aggregates through money supply
rules.1
An additional important relationship is the interaction between real output and the
money supply implicit in the Quantity Equation, M V = PX, where P is the price
level, X red output, M the money supply, and V the velocity of circulation. On
one hand, the Phillips Curve relation implies that an increase in real output above
the full employment level is inflationary. On the other hand, an increase in real
output, ceteris paribus,is deflationary from a monetary perspective; with M and
V held constant, an increase inX implies a lower P.
In fact, ifthis deflationary effect exactly equals the inflationary effect implied by
the Phillips Curve, then the net effect of that change in real output will be zero;
the level of output willbe irrelevant to the determination of inflation and theprice
level. The money stock becomes the sole determinant (apart from any supply
shocks) of the price level.
Itisthis idea which motivated therecentpaper by Jeffrey Hallman, Richard Porter
and David Small (1989).2 Their paper (referredtosubsequently as the HPS study)
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proposes and estimates an error-correction model of inflation in which the infla
tionary and deflationary effects of output changes exactly cancel, leaving a vari
able labelledp* as the sole determinant of inflation. Because p* depends only on
M 2 and potential GN P (adjusted by the mean velocity of M2), this specification
reasserts the purely monetary nature of inflation.
This paper uses a comparable error-correction framework to conduct a more thor
ough analysis of the dynamics of monetary-based inflation, comparing them with
those stemming from Phillips Curve effects. In particular, itscrutinizes both the
empirical and theoretical basis for thep* specification in the HPS study.
Our main empirical finding is to reject the p* specification in favor of a similar,
but lessrestrictive, errorcorrection model, embodying distinctdynamic responses
to money supply and real output changes. By allowing the real and monetary re
sponses todiffer, the alternative model does not allow the two inflationary sources
to be combined into a single variable, as they were under thep* restrictions. The
practical significance of this finding is to uphold the importance of real GN P (or
itsdeviation from itspotential) as an important short-run determinant of inflation.
Specifying a n error correction inflation m o d e l
The attractiveness of the error correction models advanced by Davidson et al.
(1978) is their ability to integrate a certain amount of theoretical content with a
flexibledynamic adjustment process, which can be freely estimated. Suppose the
ory suggests that two variables, y and y*, obey an equilibrium relationshipy = y\
Ifan error correction mechansim exists, then the deviation of y from its equilib
rium y* willpredictsubsequent movements iny. Such a mechanism can be written
as:
Afeyr= A(L)o£_1 -;y(-i).
where L is the lag operator, A is the difference operator (1 - L), and A(L) is a
polynomial in the lag operator representing the adjustment dynamics.
The case of money and the price level is an obvious candidate for such a longrun equilibrium relationship. No economist would argue with the proposition that
over a sufficientlylong horizon, the price level willbe proportional to some appro
priately measured monetary aggregate. (The debate instead tends to focus on the
direction of causality between money and nominal income, or the “measurability”
and stabilityof the appropriate money supply concept in the presence of financial
system innovation.)
This observation suggests that an error correction mechanism may be operating
between either between money and nominal income, or between money and the
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price level, or both.3 The obvious starting point is the logarithmic form of the
Quantity Equation, where lower case letters denote logarithms:
p t + x t = m t + Vt.
Unlike the other monetary aggregates, the M 2 velocity has (since 1955, at least)
tended toreverttoitshistoricalmean, v.4 This factsuggests using M 2 toconstruct
an “equilibrium” price level, p, implied by a given stock of M 2 and real output,
corresponding to the steady-state velocity of M2, v:
pt = mt -X t + v.
Ifp indeed tends to return to p over time, then an appropriate error correction
specification would be in terms of the deviation of p from p. Thus, a plausible
error correction model of price adjustment would be:
= A(L)(p,-i
While this addresses the monetary side of the story, one might also want to in
clude other factors contributing to inflation. One obvious additional factor is the
effect of demand pressure on prices, falling under the Phillips Curve rubric. The
natural specification for this demand pressure is in terms of the gap between the
unemployment rate and the “natural” rate of unemployment Alternatively, be
cause most measures of potential output rely on some form of Okun’s Law, one
can recast the unemployment gap in terms of the difference between (the log of)
actual output x ,and potential output x.
As the employment or output gap is not a function of the price level, it would
be considered exogenous in the context of this single equation model. With its
inclusion, the error correction specification becomes:
(1)
A kp t = A(L)(p t-1 -p t-\ ) + B(L)(xt-1 - x t-0
where B{L) is another polynomial in the lag operator, reflecting the adjustment
dynamics specific to the exogenous output gap term.
With the general form of an error correction equation in hand, one part of the
search for an appropriate inflation model requires finding the right k, the degree
of differencing ofp ,which determines the model’s steady state properties and its
potentialforoscillatorybehavior. These issues willbe discussed laterin the paper;
the next section is concerned with the search for the appropriate lag polynomials
A(L) and B(L).
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Deriving the p specification
Before considering theproperties ofgeneral errorcorrection model in Equation 1,
we turn to a discussion of the restrictions required to deliver thep* model.
One startsby noting thatbecausep isa function ofrealoutput,x implicitlyappears
intheerrorcorrectionterm, (/?M - p t-\ ),as wellas theoutputgap term (xt- i - * m ).
This means that the model, as itstands, is not closed; in order to forecast future
inflation, one must alsoknow the trajectory ofx. Alternatively, from the monetary
authority’s point of view, controlling m is not sufficient to achieve a given target
price level, or targetrate of inflation.
But what ifit were true thatA(L) and B{L) were equal, implying identical price
level responses to both ( p t-\ ~ P t -i ) and (xt- i - x m )? If so, then the two terms
could be combined into one:
Akp t = A(L)[(p*_i -/7m )+ (*m
~ * m )].
Recalling thatp t = mt-X t + v, the output terms cancel, leaving:
Akp t = A(L)[m m - Xt-x + v -/7 m ], or
= A(L)(/7;_1-/7 m ),
wherep* s mt- x t+v. Thus, M 2 per unitofpotential G NP becomes the sole anchor
forthepricelevel. Because ofthiscancellation, actualGN P (oritsdivergence from
potential GNP) is completely irrelevant to inflation.
ImposingA(L) = B(L) isonly the firststeptowards thep* specification; the second
is to choose k = 2 ,
A2pt=A{L){pU-P^).
so that the change in inflation is a function of the gap between p and p *. The mo
tivation for choosing k - 2 comes from the familiar inflation-augmented Phillips
Curve story,
Apt = 0( * M - * m ) + (A/?,)*,
where (Apt)e isthe expected rate of inflation, and 0 isa positive constant. A naive,
but tractable, version of this equation can be obtained by assuming (Apt)e = A/?m ;
thatis,settingtomorrow’sexpected inflationequal totoday’srateofinflation. Con
sistent with elementary textbooks’discussion of the “Non Inflation Accelerating
Rate ofUnemployment,” this yields thechange in therateofinflationas a function
of the output gap.
Unlike thePhillips Curve effect, in which the expectations term motivates the sec
ond difference specification, no theoretical reason exists for assuming that the
change in inflation depends on the gap between p and p. Although the a priori
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specification of second differences is arbitrary, there is also no theoretical reason
to rule itout. Here, economic theory provides no guide for specifying the degree
of differencing — thatremains an issue for empirical testing.
The thirdand final stepin deriving thep* model isto add dummy variables to cap
ture the effects of petroleum price shocks and the Nixon Administration’s wagepricecontrols. Combining these dummies into a singlecolumn vector, dt produces
the final form of thep* model:
(2)
C(L)A2p, = a (pl_j - p ,- 1 )+ d,Y,
where y isthe column vector of coefficients on those dummy variables.
Testing the p* restrictions
Thus, the two key assumptions underlying the p* model are: first, identical dy
namics on (p,-i - p,-i) and (jc,_i - x,_i);and second, the second-differenced spec
ification. This section subjects those restrictions to a variety of tests to determine
to what extent they are supported by the data.
We begin with a transformed version of Equation 1,which includes a white noise
error term, e(,and the setof dummy variables introduced above, d,:
^
C(L)A2p, = a i(p ,-i -pt-i) + a2(pt-2-p,-2) +
Pi(x«-i
+ P2 CX/-2 - *<-2 )+ d'ty + £/.
This equation is therefore equivalent to the original error correction model of
Equation 1, in which A(L ) and B(L) are rational transfer functions of the form
(cti + 0i2L ) C l (L) and (Pi + $2L)C~l(L), respectively.
The expanded equation generalizes the specification inHPS by including an addi
tionallagof (p -p ) and (x -x ). Including (p,-2 - p t - 2) and (x,-2 - xt~2) as additional
regressors makes possible a more general testof the various restrictions imposed
by thep* specification. Specifically, that specification requires that:
a2 = p2 = Oandai = Pi,
which reduces Equation 3 to thep* model in Equation 2.
In contrast, the narrower HPS testof thep model entails simply including in the
regression an additional lagged inflation term:
C(L)A2p, = a (p ,-i - p t-x )+ p(x,_i -x ,-x ) + d'ty + £Ap,_i + er.
Performing a testof the joint hypothesis that £ = 0 and a = p, the failure to reject
isinterpreted as support for thep* specification.
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The important thing to note isthatthe HPS testisperformed under the maintained
assumption thatthe lag structures on the ( p -p ) and ( x -x ) terms are identical, and
therefore tests only the degree of differencing, k. In the context of Equation 3 ,
this corresponds to imposing 0C2 = P2 = 0. The results which follow demonstrate
the misleading consequences of this arbitrary assumption.
Table 1displays theOLS estimatesofEquation 3 (and variations on it)from 1955:1
through 1988:1. Because this firstsetof regressions is designed to facilitate com
parison with the original HPS results, itfollows theirprocedure by including four
lags ofA 2p t as regressors, and by using the same measure ofpotentialGNP .5 Also
following HPS, these regressions use the implicit GNP deflator price series, and
the oil shock and price control dummy variables, AOSi and (PQ - PC 2 ).
The ^-statisticsassociatedwith thecoefficientson (p t- i - Pt-2 )and (xt- 2 - *t-i) lend
weak support to theirexclusion from the equation, with the additional lags signif
icant at the 12% and 7% levels, respectively. More informative are the relative
sizes of the point estimates of the coefficients on (p t-\ - p t- 1 ) and ( p t - i - Pt-i)\
equal to 0.101 and -0.091 respectively, they are nearly equal in size and opposite
in sign.
This observation suggests the specification in Column (b) of Table 1, This re
gression isidentical to the regression in Column (a),but reparameterized in terms
of A(pt-\ ~P t- 1 ), (Pm ~P t- 1 ),A(jcm - x t-\) and (xt~\
There, although the
^-statistic on A(pf_i - p t-i) remains a marginal 1.6, the ^-statistic associated with
(pt- 1 - p t- 1 ) falls to only 0.8. Once A(pr
_i - p t-\) is included in the regression,
therefore, the effect of (pt-1 - p t-\) in levels becomes insignificant.
For comparison purposes, the third column in Table 1 imposes thep* restrictions
discussed above, and embodied in Equation 2. As this regression uses the same
data, sample, and specification, the results are identical to those reported in the
HPS study.
The fourth column of Table 1 presents the estimates of an alternative model,
(4)
C(L)A2p t = a A (p t-i - p ^ 1 )+ pi(x,-i - xt-i) + p2(*:f-2 ~ xt-2)+ d'ty+ et,
inspired by the observation that the data favored a model in which {p t~\ - p t-\)
entered only in firstdifferences. Of course, because this model embodies distinct
lag structures on (p t- 1 - pM )and (xt-\ - xt-\),itisnotpossible tocombine the two
into the singlep* as itwas under the HPS restrictions.
Table 2 contains results of formal F tests of the p* restrictions in the context of
Equation 3. The F tests of the first two exclusion restrictions are equivalent to t
tests based on the information in Table 1. The third test is for the joint exclusion
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Table 1
Estimates of a Generalized Error-Correction Mechanism
Unrestricted
Regressor
(a)
(b)
0.0104
(0 .8 )
1
(pt-i - p ^ )
0.1011
(2 .0 )
2
(P t-l-P t-2)
-0.0907
(1 .6 )
3
A(p,-1 -p,-i)
4
Or-i - P m
5
(JCi-1
0.1453
(2.4)
6
(Xt-2-Xt-2)
-0.1042
(1 .8 )
7
A(x, - 1 -in)
8
A 2p,_i
9
Restricted
p* Model
0.0907
(1 .6 )
Alternative
0.1171
(2 .6 )
0.0312
(4.2)
)
0.0411
(3.5)
0.1733
(3.7)
-0.1305
(2.9)
0.1042
(1 .8 )
-0.6308
(6.5)
-0.7002
(8.4)
-0.6052
(6 .6 )
A2p,-2
-0.4489
(4.2)
-0.5083
(5.5)
-0.4264
(4.1)
10
A 2p ,-3
A2p,-4
12
AOS i
13
CPC1 - P C 2 )
-0.0038
(3.5)
-0.3563
(3.9)
-0.1380
(1 .8 )
0.0059
(2 .2 )
-0.0032
(3.1)
-0.2987
(3.0)
11
-0.3140
(3.1)
-0.1131
(1.4)
0.0060
(2 .2 )
-0.1046
(1.3)
0.0061
(2.3)
-0.0039
(3.6)
R2
0.3825
0.3783
0.3846
Sampleperiodis1955:1- 1988:1,quarterlydata.
Absolutevaluesof/-statisticsappearinparentheses.
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Table 2
F-Tests of the p* Restrictions
Restriction
DF Statistic P-Value
Excluding A{pw - p t-\)
Excluding A(x,_i - x,_i)
1
2
1
3
Restrictions (1) and (2)jointly
2
2.50
3.35
1.70
4
Equal coefficients on (p,-\ - p t-\) and (x,_i
1
2.44
0 .1 2 1
5
Restrictions (1), (2) and (4)jointly
3
1.28
0.283
1
0.117
0.070
0.188
of (p t- 2 ~ Pt-i) and (xt- 2 - xt-i). The fourth imposes an equality between the coef
ficient on (pt-i - p t-i) and (xt-\ - x t-\\ while the fifth combines these two sets of
restrictions, yielding thep* specification.
Using the GN P deflator series, the imprecision of the estimates of the lagged
(p t - p t) coefficients makes itimpossible to reject any of the joint hypotheses at
traditional significance levels. (Results reported later in in the paper using other
price indices do yield strong rejections of the p restrictions, however.) On the
other hand, using the GNP deflator, it is also impossible to reject the restriction
implied by the alternative model, with (p t-\ - p t-\) in differences. Nonetheless, a
model selection process guided purely by the data and not by a setofpriors favor
ing a particularmodel probably would have favored the alternativemodel over the
p * version.
This resultcreates a predicament common toa lotofempirical work ineconomics,
in which the data lack sufficientpower to formally rejecton statisticalgrounds ei
ther of two (or more) competing models. One way out of this bind is to seek
corroboration for one model or the other using other data — the CPI or PPI, for
example. As reported later, doing so produces results unfavorable to thep* speci
fication.
Ifwe insiston modelling the behavior of the GN P deflator, the dilemma persists.
In some cases, this ambiguity may simply indicate that both models adequately
describe the data, and differ little in their underlying structure and fundamental
implications. The following section shows that this is not true in the case of the
p* and the alternative models — the two models’implications for inflationary dy
namics differ considerably.
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An analysis of error-correction dynamics
This section compares the properties of thep* model with those of the alternative
model, illustrating how seemingly small and innocuous changes in the specifica
tion can produce enormous differences in the medium- and long-run behavior of
the models.
In analyzing the models, we consider two basic attributes. The firstis the behav
ior of the steady-state error — the strength of the tendency forp t to converge to
its equilibrium level. The second property concerns the cyclicality, ifany, of the
adjustment to the new equilibrium — whether convergence is monotonic, or os
cillatory.
Answers toboth oftheseissuesliein thecharacteristicsofthe transferfunctions re
lating the endogenous variablep t to the exogenous forcing variables of the model.
The next two subsections discuss how these properties can be divined from the
model specification.
Steady-state properties
One very important featureofan errorcorrection model isthe“strength” — appro
priately defined — of the mechanism bringing the endogenous variable back into
equilibrium, and how thatconvergence depends on the characteristics of the equi
librium path. More concretely, the question is whether an error correction model
implies a zero, constant, or diverging error in the face of a given pattern of growth
of itsequilibrium, or “target” level.
This issue was firstintroduced to the economics literatureby Salmon (1982), who
analyzed the steady-state properties of a class of error correction models in re
sponse to step function, lineargrowth, and quadratic growth patterns inthe forcing
variable. The discussion inthis section parallels thatin Salmon’spaper, highlight
ing the results which are relevant for our analysis of the inflationary process.
The goal is to extract from the representation of the error correction mechanism
the relevant measure of the strength of that mechanism. To that end, suppress the
£ noise term, and rewrite the generic error correction model in Equation 1 as:
Pt = A~kLA(L)(pt - pt) + A~kLB(L)(Xt - 3c,)
=
r(L)(p,-pt) + 0(L)(xt - Xt),
defining T(L) = A kLA(L), and 0(L) = A kLB(L), and solve forp,:
r(L) .
©(L) ,
p'= T T T O p, + TTrcE)('',' * )'
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Table 3
Steady-State Error
Number of
Type of Growth Path
Linear:
Quadratic:
Unit Poles
Constant:
(k -j)
Pi = P
p, = lLt
0
1
2
Constant
OO
Constant
0
0
0
p t = liiz
OO
OO
Constant
Now, setting (xt - xt) equal to zero, the error can be written as a function ofpt:6
et = (Pt-Pt) =
1
l+ r(L)^'
The key toanalyzing the steady-statebehavior of the errorliesin the transferfunc
tion T(L), relating the levelof dieendogenous variablept tothe error. Inparticular,
the key is the number of unit roots, or poles, introduced into the denominator of
the transfer function T(z) by the differencing operation.
If the A(L) polynomial contains no (1 - L) terms, then, as T(z) = (1 - z)~*zA(z),
the number of unit roots in the denominator of T(z) simply equals k,the degree
to which the endogenous variable is differenced on the left-hand-side of the error
correction equation. However, the number of unit roots in T may be less than k
ifA(L) also contains firstdifference terms, as in the “alternative” model advanced
earlier. IfA(L) contains j (where i < j < k) firstdifference terms, then A(L) =
(1 - LyA(L). When multiplied by A-*,j of those difference terms will cancel,
leaving only k - j unit poles in the denominator.
As demonstrated by Salmon, the number of unitpoles, or integral effects,present
in the errorcorrection mechanism determines the relationshipbetween the steadystatevalue of et and the path ofp t. Specifically, invoking the Final Value Theorem
of z-transforms,7 one can findlim,_+oo et as a function ofthe number of unitpoles
and the type of growth ofp (i.e., step function, linear, quadratic, etc.).
Table 3, taken from Salmon (1982), summarizes the asymptotic properties of the
steady-state error, according to the nature of the growth of pt and the number of
unitpoles in F. Consider, for example, the conventional partial adjustment model
Aa = Hpt-i - p t- 1 ).
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with a single unitpole inT(z). According tothe table,p willconverge top , so long
asp remains constant as t — ►oo. By contrast, ifp isgrowing ata linearrate in the
steady state, then the discrepancy between p and p will not vanish, but converge
to a constant To achieve convergence in levels with linear growth would require
an additional integral effect which would require specifying A 2p t as a function of
(Pt-1
Cyclical properties
A second important consideration in the evaluation of dynamic models, like the
error correction models discussed here, is their frequency domain properties. In
particular, are the models characterized by instabilities which generate cyclical or
pseudo-cyclical behavior? Or do the models imply a smooth, monotonic conver
gence of the endogenous variable to their steady-state values?
To address thisquestion, considera simplifiedversion oftheerrorcorrectionmodel
in Equation 3:
C (L )A kp t
C (L )A kp t + <
|)pt-\
= <KPm -P m ), or
= <|>Pm ,
where § isa positiveconstant The cyclicalpropertiesofthismodel aredetermined
by the roots of the characteristic polynomial corresponding to the autoregressive
terms inp u namely, the solutions to:
C(z)(l-z)* + 4>z = 0.
Complex roots to this polynomial generally indicate pseudo-cyclical behavior.8
For well-behaved C(L) lag structures, an error correction mechanism in firstdif
ferences (i . e k = 1)produces no oscillation. However, specifying the mechanism
in second differences introduces the possibility of pseudo-cyclical behavior. A
simple example will illustrate the effectof second differencing. Setting k = 2 and
assuming C(L) = 1, the simplified version of Equation 3 reduces to:
A2pt = <|>(Pm
~P
m
).
The roots of the characteristic equation z2 - (2 - <j>)z+ 1 = 0 are:
2-4>±yTOTr 3S
2
which are complex forall<|)implying convergence (between 0 and 4). By contrast,
ifk = 1 ,the single root isalways real, equal to (1 - <j))_1.
Evaluating the cyclical behavior of models with non-trivial lag structures C(L)
is considerably more complicated, however; as all of the models analyzed here
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11
take C(L) to be a fourth-order polynomial, the relevant characteristic equations
are sixth-order polynomials. Nonetheless, it is still possible to characterize the
cyclical behavior of the models in the frequency domain by computing the models’
frequency response functions. This will be the approach in the next two sections,
which examine the models’ cyclical behavior and steady-state properties.
Properties o f the p* model
As discussed above, the p* model,
C(L)A2p , = a(p*^
-P t-i)
+ j ty,
restricts the general model in Equation 1 by imposing identical lag structures on
the monetary and output gap terms, ruling out the possibility that the ( p t~i ~ P t - i )
term may enter in differences, while the (xt~\ - x t-\ ) term enters in levels.
The combination of this restriction with the second-difference specification yields
a model with two unit poles in the denominator of T(z). This means that the model
specification enforces the convergence of to p* in the steady state when p* follows
a linear growth path, as is plausible in reality.
This powerful convergence property might be termed “strong-form” monetarism.
It goes beyond asserting that inflation is always a monetary phenomenon, insisting
that the price level itself is also determined solely by the money stock. Alterna
tively, it states that for non-accelerating price growth, there is no path dependence,
or hysteresis, in the price level; the steady-state error will converge to zero, re
gardless of the path followed by potential GNP and the money stock. This prop
erty sqares with classical models of price level determination, in which the money
stock uniquely determines the level of prices.
The cyclical properties of the p* model are also rather distinctive. Figure 1 displays
the simulated time paths of inflation generated by thep* inflation model in response
to a reduction in the rate of growth of M2 to a rate consistent with zero inflation
(i.e., equal to the assumed growth rate of potential real GNP of 2.5% per annum).
The p* model produces wide oscillations, with inflation fluctuating between -4%
and 3%. Only slightly damped, these oscillations continue to display a sizeable
magnitude for over a century.9
The reason for this cyclical behavior can be traced directly to the second-difference
specification employed by the p* model. Assuming that the change in the rate of
inflation depends on (p*_l - p t-\) builds in an “accelerationist” dynamic— inflation
does not abate when a balance is reached between p and p * . Instead, it continues
even after p reaches p * , causing p to overshoot its target Only once p exceeds
p* does inflation start to slow down. Although the specification that produced
this behavior is not rejected (using GNP deflator data), it does conflict with most
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Figure 1
Simulated Inflation Path — p* Model
economists’ understanding of the inflation process — one simply doesn’t observe
the kinds of large swings in the inflation rate implied by the model.
This oscillatory pattern can be examined more precisely in Figure 2, which plots
(in the frequency domain) the gain of the p* model as a function of the period (in
quarters) of an arbitrary input. The estimated values of a and the elements of the
C(L) lag polynomial are used for the computation.
The gain function shows a large peak corresponding to a period of approximately
60 quarters, implying that any input at that frequency is amplified tremendously,
yielding an inflation rate with five-year cycles. No evidence of this kind of oscilla
tion is found in spectrum of the GNP deflator inflation rate, shown in Figure 3 .10
Properties o f the alternative model
In contrast, the alternative model,
C (L )A 2p t = a A (p t-i - p t - i) + Pi (xt- i - x t-i) + p2(*,_2 - x t- 2) + d'ty,
displays no such pseudo-cyclical properties. Figure 4 displays its simulated re
sponse to the same experiment depicted in Figure 1, in which M2 growth is set
to a rate consistent with zero inflation in the steady state. 11 Again, the actual pa
rameter estimates are used. In contrast to the p* model, here the rate of inflation
converges smoothly to its steady-state level of zero.
FRB CHICAGO Working Paper
August 1989, W P -1989-15
13
Figure 2
Amplitude Response of p* Model
Figure 3
Log Spectrum of GNP Deflator Inflation
Again, examining the frequency domain properties of this model provides an alter
native way of discerning die cyclical properties. As indicated by die plot of its gain
function, shown in Figure 5, die alternative model displays no cyclical tendencies
at all.
The steady-state properties of the alternative model also differ substantially from
those of the p model. Because the ( p t-\ - pt~i) term enters the alternative model in
first differences, one finds only a single unit pole in the denominator of the transfer
function F, compared with the two unit poles in the p* model. This difference is
FRB CHICAGO Working Paper
August 1989, WP-1989-15
14
Figure 4
Q
t
N
t
N
O
INFLATION
B
O
N
Simulated Inflation Path — Alternative Model
w
-1 9 0 0
-1 9 0 0
2000
2 0 -1 0
YEAR
2020
2030
2 0 4 -0
Figure 5
Amplitude Response of Alternative Model
0 .6 -
what produces the contrasting steady-state behavior of p t. Specifically, it implies
that in the presence of a linearly growing p, the error (p - p ) will not converge
to zero in the steady state. Instead, the error will converge to a constant which
depends on the growth rate of p (which depends, in turn, on the growth rate of
M2) and the speed of adjustment, a.
FRB CHICAGO Working Paper
August 1989, W P-1989-15
15
Another interpretation of the alternative model is as an error correction model be
tween inflation and the growth rate of M2. A linear growth path of p implies a
constant rate of inflation, so the alternative model does have the inflation rate con
verging in the steady-state to the growth rate of money.
This does not, however, imply that the price level p converges to the unique price
level implicit in a given money stock and mean velocity. Because it takes some
time for inflation to catch up to a change in the rate of money growth, a non-zero
discrepancy between p and p may persist over time. In the quantity equation, this
(constant) divergence would appear as a shift in the velocity of circulation. There
fore, it may be only coincidental that the M2 velocity has returned to its historical
average over the past 33 years, owing to a fortuitous balancing of increases and
decreases in the rate of M2 growth.
A model that displays this kind of price-level hysteresis in response to changes
in the rate of money growth might be labelled “weak-form” monetarist Here,
inflation is indeed a monetary phenomenon in the long run, as it converges to the
rate of money growth. But in the short run, there is considerable slack between
the monetary aggregate and the price level. In fact the precise value of the price
level in the steady state will exhibit path dependence, and vary with the growth
path followed by the money stock on the way to the steady state.
While it is impossible to formally dismiss either model on the basis of its steadystate properties or its cyclical dynamics, a consideration of these properties illumi
nates the radical differences implied by the subtle changes in their specification.
Evaluating these properties in the light of theory, or independent pieces of evi
dence, can, therefore help to choose between statistically indistinguishable mod
els.
In particular, one might be reluctant to insist on thep* specification when it implies
such large, underdamped oscillations in the inflation rate. On the other hand, the
conflicting implications for the convergence of the price level to its equilibrium
target should be borne in mind, and judged according to their congruence to formal
theories of price level determination.
O i l s h o c k s , p r ic e c o n tr o ls , a n d a lte r n a tiv e p r ic e in d ic e s
This section analyzes the p* and the alternative models using two additional price
indices: the Consumer Price Index (CPI), and the final goods component of the
Producer Price Index (PPI). Because the GNP deflator is a by-product of the con
struction of the real GNP figures, it reflects not only price changes, but also changes
in the composition of GNP. The Bureau of Economic Analysis, recognizing its
flaws, states that “its use as a measure of price change should be avoided.” 12 This
undesirable aspect of the GNP deflator suggests using the alternative price mea
sures to check on the robustness of the original results.
FRB CHICAGO Working Paper
August 1989, WP-1989-15
16
Before beginning this comparison, one additional feature of the original p* specifi
cation deserves closer scrutiny: the use of dummy variables to capture the effects
of oil shocks and price controls. The following two sections are a short detour
through this issue.
Dummy variables and oil price shocks
Using dummy variables to capture a shock present in a single time period intro
duces a danger of spurious correlation, or overfitting. Often, such dummy vari
ables end up fitting a specific historical episode in a way that has no plausible
economic interpretation.
A strong case can be made that the HPS oil shock dummies, OSi and OS 2 , are
guilty of this charge. The OSi variable is defined to equal unity in 1973:4, and zero
elsewhere, and is supposed to capture the impact of the first oil shock. Likewise,
the OS 2 variable, representing the second oil shock, is set equal to unity in 1979:4,
and zero elsewhere. By including them in first differences, they would imply a
once-and-for-all jump in inflation in the relevant quarter, and a once-and-for-all
decline in the following quarter.
Stated in this way, it apparent that such a specification might fail to capture accu
rately the effects of the oil price shocks. First, most of the shocks’ impact on the
various price indices did not really appear until the following quarters. Second,
such a one-time-only effect might not realistically reflect the dynamics through
which oil price increases feed through to the prices of other goods. These reserva
tions suggest that the oil shock dummies’ significance may be spurious. One way
to investigate this possibility is to estimate the oil shock effects using alternative
price indices, such as the CPI and PPI.
Table 4 displays the results of estimating the p* and the alternative models on the
three alternative price indices: the GNP deflator (as in the original results), the
CPI, and the PPI. The models include both AOSi and AOS 2 , even though AOS 2
was deemed insignificant using the GNP deflator. For the sake of brevity, the table
omits the estimates of the coefficients on lagged A2p t.
The results show that there is indeed good reason to suspect the validity of the oil
shock dummies. While AOSi is statistically significant with the correct (positive)
sign using the GNP deflator, it is statistically significant and negative when using
the PPI, paradoxically indicating a deflationary effect. Similarly, AOS 2 is large
and negative, and nearly statistically significant at the 10 % level when using the
CPI in the alternative model.13
Based on these results, we eliminate the oil shock dummies altogether in the re
gressions which follow. In place of the dummies, we include a direct measure of
the price of crude petroleum.
FRB CHICAGO Working Paper
August 1989, WP-1989-15
17
Table 4
The Robustness of Oil Shock Dummies
p* Model
Regressor
Alternative Model
GNP
CPI
PPI
0.0312
(4.2)
0.0231
(4.1)
0.0191
(3.0)
GNP
CPI
PPI
0.2133
(5.3)
0.3237
(5.0)
1
(Pm “ P m )
2
(x,_i - i n )
0.1758
(3.7)
3
(Xf-2 —Xt-f)
-0.1329 -0.1703 -0.2603
(4.1)
(2.9)
(4.3)
4
M P t-i - P t - i )
0.1190
(2 .6 )
9
(PC 1 - P C 2)
-0.0032 -0.0025 -0.0048
(2.3)
(2 .6 )
(3.1)
-0.0039 -0.0036 -0.0072
(3.6)
(3.4)
(3.9)
10
AOSi
0.0059
(2 .2 )
11
0.1429
(3.7)
0.1990
(3.7)
0.0003
(0 . 1 )
-0.0209
(4.3)
0.0061
(2.3)
AOS 2
-0.0005 -0.0029
(0 .2 )
( 1 .1 )
0.0006
(0 . 1 )
-0.0013 -0.0040 -0.0014
(0.5)
(0.3)
( 1 -6 )
R2
0.3735
0.3308
0.3807
0.2605
0 .0 0 0 2
(0 . 1 )
0.3407
-0 .0 2 0 2
(4.4)
0.4175
A b s o lu te v a lu e s o f th e /- s ta tis tic s a p p e a r in p a re n th e s e s .
What did price controls really accomplish?
The HPS price control dummies, PCi and PC 2, are subject to criticism on sim
ilar grounds. The PCi variable represents Phase I of the Nixon Adminstration’s
price control program, and equals unity from 1971:3 through 1972:4. Phase II is
represented by the variable PC 2, and equals unity from 1973:1 through 1974:4.
Estimation using the GNP deflator produces a negative, statistically significant co
efficient on (PCi - PC 2), contrary to the widespread perception of the inefficacy
of the controls. Combined with the second-difference specification, this result im
plies afalling (not just lower) rate of inflation under Phase I, and a rising (not just
higher) rate of inflation under Phase II; inflation then falls back to the “normal”
level (i.e., driven entirely by lagged inflation changes and (p \Lj
at the end
of Phase II. While attempting to model the impact of the Nixon Administration’s
FRB CHICAGO Working Paper
August 1989, W P-1989-15
18
Table 5
The Robustness of Price Control Dummies
Variables Significant at the 10% Level
p* Model
Alternative
Model
GNP
CPI
PPI
(PC1 - P C 2)
APC2
apc2
PCi
PC 2
PC 2
apc2
PC 2
apc2
Unrestricted
Model
(P Q -P Q O
PC 2
PC 2
APC 2
wage-price control policies is certainly a worthwhile endeavor, the very odd es
timated response to the controls suggests that the price control dummies may be
capturing some other phenomenon.
As with the oil shock dummies, one possible check on the robustness of the results
is to compare the results for alternative price indices. (If anything, due to the
nature of the controls, the dummies should have an even stronger effect on the CPI
inflation.) Table 5 displays a check on the robustness of the price control dummy
variables. Versions of the p* model, the alternative model, and the unrestricted
model of Equation 3 were estimated on each of the three price indices. The table
indicates which price control variables were individually statistically significant at
the 10% level. The table shows no consistent pattern at all with regard to which
price control dummies ought to be included in the regression. It is only with the
GNP deflator that the P Q variable ever enters significantly. Yet, if P Q has an
insignificant effect on inflation, it is not clear what role PC 2 has in the regression;
one would imagine the Phase II relaxation to be inflationary only in relation to the
Phase I controls.
Results with alternative price indices
Finally, after the foregoing examination of the performance of the oil shock and
price control dummy variables, we are in a position to test the p model against
the alternative using the CPI and PPI price series. These results differ substantially
from those which used the GNP deflator; specifically, the p * specification fails in
the statistical tests which use the CPI and PPI.
FRB CHICAGO Working Paper
August 1989, W P-1989-15
19
In constructing a revised framework for estimating and testing the competing mod
els, we draw on the results of the previous sections, which cast doubt on the oil
shock and price control dummy variables as originally used. For lack of a satisfac
tory alternative to the price control dummies, we drop them from the regression. In
lieu of the oil' shock dummies, we model the inflationary impact of the oil shocks
directly by including the change in oil price inflation, A 2pl_s, as an additional ex
ogenous variable:
C (L )A 2p t =
(5)
~ P t -i ) + VfiA(Pt-\ - p t- 1 ) +
s
+ $2(xi-2 - *t-2)+
SfApl.s+
e(.
S
The oil price used is the crude petroleum component of the Producer Price Index.
One question about this specification is whether this second-differenced oil price
ought to be considered contemporaneously exogenous, i.e., uncorrelated with the
noise term e, in the error correction model. If so, then current values of A2p ° can
be included on the right-hand-side of Equation 5, and the index s runs from 0. If
not, the appropriate procedure is to include only lagged (predetermined) oil prices,
and s starts at 1 .
To remain agnostic on this exogeniety issue, we estimate three versions of Equa
tion 5: one with both current and lagged oil prices, the second with only lagged
oil prices, and the third with no oil prices at all. The results of the F tests based on
these regressions appear in Table 6 . Starting from Equation 5, the p* model and
our alternative model can be represented by simple linear coefficient restrictions:
\|/2 = p2 = 0 and\|/i = Pi. By contrast, the alternative model involves only a single
restriction: \|/i = 0 .
As shown on line 4 of the table, the joint p* restrictions always fail at the 2%
level (and usually at the 1% level) with the CPI and the PPI. Lines 1 through 3
show the results of the tests of the exclusion restrictions, t|/2 = 0 and p2 = 0 , both
individually and jointly. The form in which the oil price appears seems largely
irrelevant to the test results. Only the GNP deflator results fail to reject the p*
restrictions; the GNP deflator data are unable to reject either hypothesis at the
usual significance levels. In contrast, the statistics on line 5 are all too small to
reject the alternative model’s restriction, \j/i = 0, at the usual levels. The smallest
P-value is 0.22.
Thus, the statistical tests of the p* restrictions are highly sensitive to the choice
of price index, passing with the GNP deflator, and failing with the CPI and PPI.
Given the well-known peculiarities of the GNP deflator, a specification based on
the GNP deflator results seems untenable.
FRB CHICAGO Working Paper
August 1989, W P-1989-15
20
Table 6
F-Tests of Alternative Inflation Models
L a g g e d
N o O i l P r ic e s
R e s tr ic tio n
1
D F
E x c lu d in g
1
M p r - i
2
E x c lu d in g
fat— 2
3
1
2)
R e s tr ic tio n s
2
1 and 2
4
5
p
M o d e l
A lt e r n a t iv e
M o d e l
P - v a lu e s
a p p e a r in
3
1
G N P
C P I
0 .4 8
4 .8 0
( 0 .4 9 )
( 0 .0 3 )
0 .7 2
8 .0 1
(0 .4 0 )
( 0 .0 1 )
0 .3 6
4 .1 0
( 0 .7 0 )
(0 .0 2 )
0 .4 2
3 .5 0
( 0 .7 4 )
(0 .0 2 )
1 .5 4
0 .8 3
(0 .2 2 )
( 0 .3 6 )
P P I
2 .4 0
( 0 .1 2 )
7 .0 2
( 0 .0 1 )
3 .5 3
( 0 .0 3 )
4 .0 5
( 0 .0 1 )
0 .9 0
(0 .3 4 )
G N P
C u rre n t &
O il
C P I
1 .0 6
4 .4 5
(0 .3 1 )
( 0 .0 4 )
1 .1 2
7 .2 9
(0 .2 9 )
( 0 .0 1 )
0 .6 2
3 .7 8
( 0 .5 4 )
( 0 .0 3 )
0 .5 2
3 .6 5
(0 .6 7 )
( 0 .0 1 )
1 .1 2
0 .5 8
( 0 .2 9 )
(0 .4 5 )
P P I
2 .3 4
(0 .1 3 )
6 .2 0
( 0 .0 1 )
3 .1 9
( 0 .0 4 )
4 .0 8
( 0 .0 1 )
0 .8 1
(0 .3 7 )
G N P
L a g g e d O il
C P I
P P I
0 .6 9
3 .0 6
0 .8 3
( 0 .4 1 )
( 0 .0 8 )
(0 .3 6 )
0 .8 7
6 .5 0
5 .6 8
(0 .3 5 )
(0 .0 1 )
(0 .0 2 )
0 .4 5
3 .2 6
2 .8 8
( 0 .6 4 )
( 0 .0 4 )
(0 .0 6 )
0 .4 6
3 .8 0
4 .2 3
( 0 .7 1 )
( 0 .0 1 )
( 0 .0 1 )
1 .1 9
0 .3 5
0 .6 2
(0 .2 8 )
(0 .5 5 )
( 0 .4 3 )
p a r e n th e s e s .
Table 7 presents the parameter estimates of what is, in light of the tests in Table 6,
the preferred specification: the alternative model earlier, with {p t-\ - Pt-\) in first
differences. Taking a conservative approach to the exogeniety of the petroleum
price, these regressions include only lagged oil prices.
The choice of price index seems to make very little difference in terms of the
point estimates of interest; each index yields an estimate of the coefficient on
A( p t-i —pt—\) very close to 0.1, with a slightly stronger effect found using the CPI.
One striking aspect of these estimates is that although the alternative model in
some sense embodies a weaker link between money and prices than the p* model,
the large coefficient on A (p t-\ - p t-0 indicates a higher speed of adjustment to
changes in the M2 growth rate. This more rapid adjustment is also reflected in the
simulated inflation paths in Figures 1 and 4.
As with the earlier GNP deflator results, additional lags of (xt - xt) enter signifi
cantly in the regression. Comparing the magnitudes of the coefficients on (xt - x t\
the strength of the Phillips Curve effect appears considerably stronger using the
final goods PPI than with the other indices.
While the alternative model represents an improvement over the p* specification,
it is only a first step towards a satisfactory inflation model. One important check
on it will be an analysis of its out-of-sample forecasting performance. In addition,
FRB CHICAGO Working Paper
August 1989, W P-1989-15
21
Table 7
Estimates of the Preferred Specification
Price Index
Regressor
GNP
CPI
PPI
0.0953
(2.1)
0.1374
(2.9)
0.1034
(2.7)
0.1699
(4.3)
-0.0969
(2.1)
-0.1287
(3.2)
4
A2/»(-i
-0.5200
(5.3)
-0.2838
(3.0)
0.1087
(1.9)
0.2711
(3.9)
-0.2077
(3.1)
-0.5264
(5.0)
5
& 2P t - 2
-0.3171
(2.9)
6
A2P(_3
7
aV m
-0.2816
(2.7)
-0.0624
(0.8)
-0.3416
(3.3)
-0.1542
(1.7)
-0.0231
(0.3)
-0.3138
(2.8)
-0.0544
(0.5)
-0.0632
(0.7)
8
A2/>ti
-0.0101
(2.1)
0.0081
(1.8)
0.0184
(2.0)
9
^ P ll
0.0073
(1.5)
0.0151
(3.2)
0.0073
(0.8)
R2
0.3357
0.3336
0.2744
1
M P t-i - P t -
2
(Xf-J JCf-1)
3
S a m p le p e r io d is
1 9 5 5 :1 -
1)
1 9 8 8 :1 , q u a r te r ly d a ta .
A b s o lu te v a lu e s o f th e /- s ta tis tic s a p p e a r in p a r e n th e s e s .
it retains certain unsatisfactory ad hoc aspects which deserve more thoughtful con
sideration.
One of these is the naive modelling of inflationary expectations in the Phillips
Curve portion of the model. Another is the crude way of introducing the effects
of exogenous supply shocks. Through an aggregate cost or price function, it may
be possible to motivate another error correction term incorporating the oil supply
price as one of the inputs. Furthermore, by re-introducing real GNP as a deter
minant of inflation, the alternative model is not closed, but requires an additional
FRB CHICAGO Working Paper
August 1989, WP-1989-15
22
equation relating real output to a set of exogenous variables, including money.
Clearly, more work remains.
C o n c lu s io n s
There is little doubt that some monetary aggregate — appropriately measured —
is an important long-run determinant of inflation and the price level. Inspired by
the apparent stability of the M2 velocity for the past 33 years, Hallman, Porter
and Small propose and estimate an error correction model whose sole exogenous
variable is p *9a function of only M2 and potential GNP.
The first section of this paper showed how the p* restrictions are equivalent to re
quiring that the output gaps and money supply changes affect prices in exactly the
same way. The main finding of this paper is that thesep* restrictions are suspect on
two grounds. As shown earlier, they produce implausible, underdamped oscilla
tions in the price level and inflation. Second, they can be rejected outright in direct
statistical tests of a generalized specification, when the Consumer Price Index or
the Producer Price Index is used as the price variable. These results appeared in
the second main section of the paper.
The acceptance of the p* specification, therefore, depends both on the use of the
GNP deflator, and on a specific set of maintained assumptions within which the
HPS model selection process proceeded. A careful analysis of the error correction
mechanism between the price level and M2 shows that the inflationary effects of
an increase in the money stock differ qualitatively from those resulting from an
output gap. It therefore seems inappropriate to combine the two in a way that
makes p* (and implicitly M2) the only determinant of inflation.
Despite the sharp differences between the p* model and the alternative, their pol
icy implications are not fundamentally dissimilar. Assuming that the observed
empirical relationship between M2 and prices continues to endure, both establish
a link between long-run inflation and the growth rate of the money stock. What
distinguishes the alternative model is its separate inclusion of the output gap as
an additional, distinct source of inflation. Because of this additional factor, it sug
gests that controlling M2 is a necessary, but insufficient, condition for controlling
inflation.
FRB CHICAGO Working Paper
August 1989, W P-1989-15
23
F o o tn o te s
1. See Cooley and LeRoy (1985).
2. “M2 per Unit of Potential GNP as an Anchor for the Price Level,” Board of
Governors of the Federal Reserve System Staff Study #157.
3. More specifically, if money and income are integrated random variables while
velocity is stationary, an error correction mechanism can be shown to exist.
See Engle and Granger (1987).
4. Friedman and Kuttner (1989) examine the behavior of the aggregates’ veloc
ities, finding that the recent deterioration of the M2 velocity has been mild
compared with that of the other aggregates.
5. The relevant measure is the series constructed by staff members of the Board
of Governors, from the work of Peter K. Clark, in “Okun’s Law and Potential
GNP,” Board of Governors of the Federal Reserve System, October 1982.
6. In the context of error correction models, the term “error” refers to the gap
between the endogenous variable and its equilibrium level, and should not be
confused with the stochastic noise term appended to the estimating equation.
7. See Gabel and Roberts (1980).
8. Sargent (1979) and Harvey (1981) are useful introductions to the properties of
second-order stochastic difference equations.
9. Similar simulations appear in the HPS study (pp. 21-23).
10. See also Sargent (1979, p. 217).
11. Because x, real output, does not fall out of this specification, it is also necessary
to assume something about its time path. For the purposes of this simulation,
x is simply set to x.
12. Bureau of Economic Analysis, “GNP Report and Tables.”
13. Similar results can be found in the HPS study itself. In their Table 1, the
estimate of the coefficient on AOSi using the fixed-weight GNP deflator is
also negative and statistically significant
FRB CHICAGO Working Paper
August 1989, W P-1989-15
24
R e fe r e n c e s
Bureau of Economic Analysis, Commerce Department (1989), “GNP Report and
Tables,” June 22.
Clark, RK. (1982), “Okun’s Law and Potential GNP,” Board of Governors of the
Federal Reserve System.
Cooley, T.F. and S.F. LeRoy (1985), “Atheoretical Macroeconomics: A Critique,”
Journal of Monetary Economics 16, November, 283-308.
Davidson, J.E.H., D.F. Hendry, F. Srba, and S. Yeo (1978), “Econometric Mod
elling of the Aggregate Time-Series Relationship Between Consumers’ Ex
penditure and Income in the United Kingdom,” Economic Journal 88, De
cember, 661-692.
Engle, R.F. and C.W J. Granger (1987), “Co-Integration and Error Correction:
Representation, Estimation, and Testing,” Econometrica 55, March, 251276.
Friedman, B.M. and K.N. Kuttner (1989), “Money, Income and Prices After the
1980s,” NBER Working Paper # 2852, Cambridge: National Bureau of Eco
nomic Research.
Gabel, R.A. and R.A. Roberts (1980), Signals and Linear Systems. New York:
John Wiley and Sons.
Hallman, J.J., R.D. Porter, and D.H. Small (1989), “M2 per Unit of GNP as an
Anchor for the Price Level,” Board of Governors of the Federal Reserve
System: Staff Study #157.
Harvey, A.C. (1981), Time Series Models. New York: John Wiley and Sons.
Salmon, M. (1982), “Error Correction Mechanisms,” Economic Journal 92, Sep
tember, 615-629.
Sargent, TJ. (1979), Macroeconomic Theory. New York: Academic Press.
FRB CHICAGO Working Paper
August 1989, W P-1989-15
25
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