- Email: [email protected]
Adaptive expectations, rational expectations, and money demand in hyperinflation Germany
Journal of Monetary
Economics
5 (1979) 593 604. 0
North-Holland
Publishing Company
ADAPTIVE EXPECTATIONS, RATIONAL EXPECTATIONS, AND MONEY DEMAND IN HYPERINFLATION GERMANY*
In rhe recent likrature Sargent and Wallace (I,%, June. 1973) have estimated the demand equ,ltion for money in hyperinflarion under the restriction that the adaptive formula of Phillip Cagan yields rational inflation expectations in the sense of John Muth. The present paper finds evidence IO rqcct for the Germany case the proposition that adaptive expectations are rational. The procedure employed is basically to overfit the stochastic rcprcscntation for the inflation rate implied by the ‘ad;rpti~c-is-rlti~)nal’ hypothesis. The paper also puts forward and appl~r\ a tuostep procedure IO estimate the important money demand elasticity in hypsrm~latlcw. 7‘1~ procedure returns reasonable results \%ith large estimated standard error\.
I.
Introduction
The prototype for modern research in the demand for money in hyperinflation is the model developed by Cagan (1926) in which desired real money balances are assumed inversely related to the anticipated rate of general price inflation. Cagan assumed that agents formed inflation expectations adaptively. that is, as a weighted sum of the previous period’s guess and the currently realized error in that guess. Sargent and Wallace (1973) and Sargent (1977) have based recent studies of money demand in hyperinflation upon the additional assumption that adaptively formed expectations are rational in Muth’s (1961) sense. This paper finds evidence to reject for the German case the proposition that adaptive expectations are rational. This result is important because it bears fundamentally on whether or not the important parameter of Cagan’s model, Q - the money demand elasticity with respect to expected inflation, can be consistently estimated. This paper further undertakes estimation of money demand elasticity under the assumption that expectations are rational but not necessarily adaptive and [hat the rate of inflation is statistically exogenous in the sense of Granger and Sims. Estimation is accomplished by a two-step, constrained ordinary least squares technique. Salemi and Sargent (1979) have also undertaken the estimation of x using a constrained maximum likelihood procedure. In contrast, this paper imposes throughout the assumption that xr is exogenous and estimates longer lags than those estimated in Salemi and Sargent. *This research was supported author thanks Arthur Benavie, comments.
in part by funds from the Social Science Research Council. The Richard Froyen, Roger Waud, and Warren Weber for helpful
2. A test of the ratiouality of adaptive expectations Suppose that the demand for money schedule in hyperinflation form asscnxd by Cagan (1956), ni, - p, = 7 + %rrr-i-u,,
X-CO,
has the
(1)
where nr, is the natural log of the money supply, p, is the natural log of the price level, n, is the pubiic’s expectation formed at 1 of the rate of inflation to occur between t and E+ 1, and 14, is a random error term the assumed properties of which will be discussed in what follows. Let x, z p, -17~_ 1. Ctrgan closed his model by assuming that x, was formed according to an adaptive mechanism,
R,=n,-,+(1-i.)(%,-72,._1),
(2)
and estimated J, r. and i by a form of non-linear least squares. S,argent and Wallace (1973) and Sargent (19773 assume further that the addptive mechanism (2) yields expectations raticnal in the sense of Muth (1961). That is, they
assume
where E is the linear projection operator. If z,_ , = E(x,l& _ 1 1 one may regard n,_, as the ordinary least squares prediction of xt based on a set of data available at t - 1, 4, _ 1. Adaptive expectations are rational if and only if the rate of inflation has the stochastic representation ;IJ”Zr_l
+(I -i-L&z,,
(4)
where a, is a w’iailenoise random process and L is the lag operator defined by L”‘Z,= Z,_ m.If x, evolves according to (4) then
whicil is a crucial fact for Sargent’s deduction that under (2) and (3) cxis not identtied.’ ‘?barge~t assumes (I)-@) and a restrktio?? on u,, namely Chat b(u, -u,- , I where Ksm, -m,_ ,, He shows two things. First, Cagan’s )=Q fJ-r,%-~--1C4-1,14-2,--. estimator in general sufkrs from simultaneous equat;,ons bias [Sargent (1977. p. 6811. Second, ~nlesr an arbitrary restriction is placed on the covariance between the innovations in (the first &f#krcxe~ 4) money demand and money supply it is not possible to obtain a consisknt estitnataf for ?x(Sargent (M’7, p_ 73)J. The estimates which Sargent generates are under the lrwumpriou that these innovations are uncorrelated.
MX.
Salemi, Expectations
and money demand
595
Whether the adaptive mechanism yields rational expectations of the inflation rate is directly testable. Let x:_ 1(2) be the adaptive prediction of 2, given the history of inflation and i.. If TIN_ ,(i.) yields rational expectations then the prediction error, x,-~t:_ 1(i), will be orthogonal to all the data available at C- 1 for predicting x,. The alternati’ve hypothesis is that the hi‘s. n, and c are not all zero in eq. (4), -/T)x,-,/(l
x,=(l
-/fL)--((1
-L)l,_,/(l
-X)‘)(i,-,f)
“1 +
C hiZ,-i+d’t+c’+rl,.
(6)
i= I
The first two terms on the right-hand side of (6) art: a first-order Taylor approximation to (3) about an initial &value, 2, which permits an adjustment to 2 to be estimated jointly with the hi-s. The Z’s represent data atailablc in the system at time t - 1 such as the rate of money creation. jl,, the rate of inflation. 2,. or the rate of Deutschemark exchange depreciation, K,. Each series represents data which could have been used to predict z, but which under the null hypothesis has no predictive value given nr_ 1(i.). Eq. (6) may be regarded as a way of over-fitting the stochastic representation of x,. When Z,_ i is taken to be 2, _ i, the Cagan specification is tested against the more general alternative hypothesis that lt follolvcd an unconstrained m-lag autoregression. When on the other hand Z,_ i is taken to be 11,_i (or I:,._~) the Cagan specification is tested against the alternative that the true stochastic representation of z, is comprised of an unconstrained distribution lag on past 11 (or cc:)as well as a geometric lag on past 1. It is worth repeating that the Taylor approximation in (6) is important because it provides a way of estimating i. and the other parameters simultaneously.‘*” Table 1 reports the results of fitting eq. (6) by ordinary least squares with m=2.’ The Group A and B results were obtained when the price data reported by Cagan for Germany were employed. Two values for 2 were tried: /r=O.SZ, the estimate of i. reported by Cagan for Germany. and rT=O.67, the ‘AI lirst it may stem peculiar that the test does not simply insolve regressing the prediction error from adaptive expectations, X,-R;_ ,(i.). on the lagged %‘s. a trend. and a constant. However, Durhin (1978) has shown that unless i and the hi’s are jointly estimated the test will be brased in the direction of accepting the null hypothesis. /I, = h, = . = b, d i’ =O. ‘Taylor approximation (23) permits joint estimation the h, an adjustment i.. (i. In the which were d&d>.. which equal to lag polynomial - L)(l bythelagpolynomial -(I L)(l + i 3izL2+... ‘The choice inflation and creation as for a of omitted is clear. why the of exchange is also reasonable candidate discussed in (1977) and (1978). *When lag lengths employed it true that of the omitted information at t tAs lag was increased became increasingly to rqject null hypothesis.
Cagan data series Oct. 1920 to July 1923 r: = 0.67
G:oup B
AI=O.82
.. (‘A
I
0.24 (0.21)
iI+‘;. 0.17 (0.09)
11,- ,
0.39 (0.38)
PI - 1
;I[ :; ;_ 9.40 [0.20)
-0.16 (0.45)
XI -- I
--
0.16 (0.07)
I:, . . ,
0.39 (O._iI )
11,- I
0.004 (0.003)
- 2
-0.12. (0.09)
61 - 2
- 0.88 (0.39)
t4 - 2 -_ --~----
-0.59 (0.27)
Xl - 2
- 0.09 (0.08)
4
-U.78 (0.29)
-__-
- 0.02 (0.04)
0.003 (0.002)
0.00 (0.W
con.
trend 0.001 (0.003)
- 0.05 (0.04)
con.
trend 0.009 (0.003)
- 0.03 (0.05)
0.007 (0.004)
trend
con.
con.
-..
trend
_-._
- 0.07 (0.04)
con.
- 0.03 (0.~1
0.0 IO (0.003)
I rend klcr 2 .---____.-.._
- 0.27 (0.25)
con. Ircnd xt- 2 .__... ._ ..__-.._ ____ _.______-._.._ ---
0.29 (0.23)
I
X8.
-0.15 (0.37)
in/?>.
0.16 (0.10)
....-_B----p-_
i’Ti
0.21 (0. IO)
;n,‘;i. -------_I-
0.15 (0.13)
iltpi -_-___-
2.76”
3.36”
2.43
4.28b
5.33h
3.29”
FW9b . _--..______..___.___. .__._ .__. _,_ __._ _._-__.__. ___..___
OLS estimaks
_I_.._..__.....__._.. _.“__._I__..__.,.._
Cagan data series Oct. 1920 to July 1923
Group .4
.-_ --
3.41”
4.47”
2.80
3.27”
4.89”
1.74
1.79
1.72
I .83
1.84
1.97
I .9’
D. 1% F(2.29) .._.- _._-.mm---_------ I_.
Tests of thu mtionitlity of adikplivc expectations; dcpcndunt v;rriublc p-x, - nf ,(i); cstimatetl sttindurd errors in parentheses. ____,^ _.. __... .._.__...__._____.____ ____.. ..___---.-.. ..____ _- -----PI.-<--“q_ ..-m-_Ie.--.._.^- _... --_. ...__._“I ,_... _,lyl”ll._ ._ _. . -... .._..
Table I
M.K.
Strlrmi.
Expectutions
ur~d money demund
597
MX.
598
Salemi, Expectations and money demund
value of j. which minimized the sum of squared
Given a number of hypotheses to be enumerated below, a consistent estimator of the elasticity of demand for real money balances with respect to expected inflation is available. For convenience the model may be rewritten its
u*==u,_
1
+t
I’
(9)
‘To form the sum of squared u’s_ given the 1’s and i, a, was set equal to zero. its assumed exoected value. %e rate of inflation is formed from the series of monthly averages of the daily observation of the whokzsale price index published by the Statistisches Reichsamt (1925). While Cagan begins with this series he mixes with it some point in time observations available in 1923. The rate of money creation is formed from the series of end of month numbers for the nominal money stock ako published by the Statistisehes Reichsamt. While Cagan applied different interpolation filters to diierent parts of the series and mixed the end of month numbers with some point in time data available for 1923, this section uses the data as published except for a log linear intq&tor appkd consistently to the entire series. The data on the Exchange value of the Mark were taken from Young (1925).
M.K.
Salemi.
Expectations
and money demand
599
where pI, x1, xI, E, and u are defined as before, L‘,is white noise, and 4, is information available at time t. Eq. (7) is Cagan’s money demand equation in first difference form. Eq. (8) states that expectations are formed rationally and on tr’le basis of the information set 4,. In what follows it is assumed that #, includes ,;lnly the history of inflation rates dated t and earlier. Eq. (9) states that the error in the money demand equation takes a random walk. There iIr(. two reasons for limiting the information set to past and present inflation ra’tes. First, the time series are short and the conditioning set must be limited or degrees of freedom problems will result. Salemi and Sargent (1979) used a constrained maximum likelihood procedure to estimate Q and assume that 4, includes the history of inflation rates and rates of money creation. However, the longest lag length which they estimate is 4 months compared with 8 months in the work reported here. Second, several of the German hyperinflation studies’ have reported evidence that in the sense of Granger and Sims X, is an exogenous stochastic process. If x, is statistically exogenous then all the information available at t and useful for predicting xt+ 1 is contained in past and present values of x. The results of the previous section indicated that adaptive expectations omitted information significantly useful for preuicting the one-period-ahead inflation rate. The erogeneity of 1 implies that this omitted information may be captured by predicting x[ on the basis of an unconstrained distributed lag on past x alone.’ Operating on (7) with the projection operator yields
since by assumption E (q 1c#+_1) =O. In general, (10) provides the basis for a consistent estimation of a.9 Note, however, that if adaptive expectations are rational, then E(xt+ 1 -x,1+,- 1) =0 and (10) would not be useful for recovering estimates of a. To proceed form the linear projections implicit in (lo), ‘See Frenkel (1977), Salemi (1976), and Sargent and Wallace (1973). ‘Evans (1978) using techniques proposed by Zellner and Palm (1974) studies three models of hyperinfiation and linds evidence to reject them all. In one case he too rejects the hypothesis that 1, has the ARIMA (91.1) implied by a model in which adaptive expectations are rational. In another he considers a model (termed MC by Evans) which like the one presented here is comprised of a version of Cagan’s equation and an equation imp0sir.g rational expectations. The third equation of model MC is a money supply rule of the form (in my notation) /~,=/f,_ 1 +/(L)u,. where r, is whit%and f(L) is a finitely dimensioned lag polynomial. Evans rejects MC on the basis of strong cross correlation at past lags observed between prewhitened p, and 1,. It is important to note that the model presented here is consistent with a money supply schedule of a different form. In general, the model estimated here is consistent with a money supply rule of the form ~~,=constant +h(L)~,+u,, where h(L) is a one-sided polynomial on past and present inflation rates. ‘It is worthwhile noting that eq. (lo), which is the basis for consistent estimation of z of the approach reported here, holds even if the true information set, call it $,- ,, is larger than &_ ,. This is so because of the law of iterated projections which says that if Q is a random variable, then E(E(Q1~,-,)1~,-,)=E(QI~,-,).
MX. Salemi, Expectations and money demand
i-1 L-2
x
.
+
x1-3 .
h-m-
Wit-1
[
w2t
3,
1
where w1 and w2 are the innovations respectively in pI _ 1 - xt In view of (IO), however, the Bids are not all free bul satisfy j=l,2 ,..., m.
BljzaB2j,
(11)
_1
and xt - xt _ 1.
(12)
To impose the restrictions implied by the money demand model on the parameters of (111, subtract the first equation from the second to obtain
%t-&-l=
f j=l
(B2j-Blj~Xt-l-j+w2t-wlt-l
.m =(lerI
x B2jXt-1-j+wt j= 1
(13)
m =
C j=l
‘/iXt-l-j+wtC;)
where w, E w2t -
wlr_l and *Ji=(l-E)pzj for j=l,2,...,m. The procedure employed was to recover the p,is by estimating the second equation of (1 l), to recover the 7;s by estimating (13) and to form an estimate of 2 as
(14)
The results obtained for m=4 and m= 8 with both the Cagan data and the monthly average data are summarized in table 2 and reported in detail in tables 3 and 4. For both the Cagan and the monthly average data, the two-step procedure yields appropriately signed estimates of a. By way of comparison, Cagan estimated that r equalled - 5.46 with a 90% confidence band of - 6.13 to -5.05 and Sargent (1977) estimated 01 to be - 5.97 with an estimated
M.K.
Salemi, Expectations
and money demund
601
Table 2 Two-step estimation
nf the elasticity of money demand with respect to the expected inflation m-4 --
-
-..-_-_-____.
.--
Cagan’s data
Monthly
Nov.
Sept. 1920 to June 1923 -_ --_-__---._
1920 to Aug. 1923
8
4
4
jjI, 421
- (i.26
- 1.40
-0.56
- 0.68
t ,= I
-2.16
- 2.20
- I.71
- 1.34
- 7.22
- 0.575
- 2.06
- 0.988
0.116
I.91
0.792
ci
-
.__
average data
8
Y2j
rate.
and 8.
--.-_-..
__-
___..___
Estimated 563.2
asymptotic standard error or i
asymptotic standard error of 4.62.” Using a maximum likelihood technique, Salemi and Sargent (1977) obtained estimates of zt which were sensitive to the method used to detrend the data but in most cases consistent with the results reported in table 2.” One theme that emerges from the Sargent (1977), Salemi and Sargent (1977), and the results reported here is that in contrast to the Cagan results the estimates of r are generally small (and sometimes quite small) relative to correctly estimated asymptotic standard errors. It is interesting to learn that this imprecision persists even when the lag length is permitted to extend to eight months. Indeed, it is the ability to
“The estimate of the asymptotic study by writing a first-order Taylor
‘=
I -i$,
fi/i,
S2jE
standard error of the estimator of r was obtained approximation for z. Rewrite j, as
l-@,
6i/,=F I
in this
Lr). m+l
which serves to define the 4;s. Then i G Z* + [G/c?&] I,.(4 - @*), where the asterisks designate the true parameter values and [a&/?&] is the I x 2m Jacobian matrix. As the sample size goes to Moreover, asymptotically L, T (S--a+) is approximately the weighted sum of infinity, &c;z*. normal random variables with weights equal to the probability limits of the elements of the Jacobian. It is therefore reasonable to estimate the asymptotic variance of si as Lm
“In
fact
depending in between -3.5
the
Salemi
and
Zm
Sargent
lag length and the method and - 10.5 however.
(1977)
estimates
of detrending
do
range
the data.
between
-335
and
0.651
The estimates most often range
I
0.02 (0.29)
0.97 (0.28)
-
- 0.43 (0.27)
-0.10 (0.25)
t-Cc,-1
XI-XI-1
-0.11 (0.31)
0.09 (0.30)
-
x1-x1-1
x1- 3
-0.58 (0.29)
2
- 0.26 (0.28 )
XI -
Independent variable ‘-
X,--k-l
Dependent variable -x1-5
- 1.07 (0.31) -0.87 (0.33) - 1.13 (0.28) -0.85 (0.30)
x1-4
- 0.20 (0.31) -0.14 (0.33) - 0.54 (0.27 ) - 0.50 (0.29)
XI-
- 0.70 (0.52) -0.38 (0.55 1
X,-h
0.39 (0.41) 0.69 (0.44)
7
R: = 0.32 F(9,24)=2.70 R: = 0.47 F(5,28)=6.95 R:=0.25 F (5,28) = 3.23
- 0.05 (0.07) -0.15 (0.05) -0.11 (0.05)
0.01 (0.01) 0.02 (0.004) 0.01 (0.005 )
0.53 (0.44)
0.12 (0.55)
R: = 0.49 F(9,24)=4.55 0.02 (0.01)
0.51 (0.41)
- 0.25 (0.52)
con. -0.15 (0.06)
trend
8
XI .- 9
x1-
Regressions for the limited information estimation of r; Cagan data series; Nov. 1920 to Aug. 1923; upper part of the table for N=8. lower part for N =4; standard errors in parentheses. -
Table 3
----.--
%,-A
-0.71 (0.22)
(0.21 )
- 0.40
0.41 (0.23)
(0.2 1 I
-0.21
____
0.59 (0.28)
- ii.76 (0.25 )
-
0.04 (0.26 I
%I-- 3
0.59 (0.23)
%r z
variable -_-
information
Independent
._ .~_~_~~~~~~~~~
I
%r- 14, I
_.--
i[r - zr I
1
Dependent variable
-.--
Regressions for the limited
- 0.49 (0.37)
0.46 (0.35 1
- 0.78 (0.32)
-0.64 (0.28)
-0.18
0.2 I (0.38)
-0.10 (0.10)
(0.008)
-
Ri=0.18
F(5,28)=2.71 D.W!= 2.49
Rf,=0.21
R:: =0.27 F(5,28)=3.40 D.W=1.61
D. W = 2.69
l
1.81
R: =0.30 F(9,24)=2.61 D.W= 1.68
(0.10) F (9,24)=
-0.12
0.01
)
-0.24 (0.09)
0.26 (0.30)
0.01 (0.009
0.03 (0.008)
-0.19 (0.09)
---
(0.31)
0.40
0.15 (0.28 )
0.03 (0.008 )
(0.35)
-0.18 (0.32)
0.0 I (0.35 1
x1-7
-0.54 (0.28)
---___
- 0.66 (0.34)
-0.21 (0.32 )
- 0.27 (0.29)
-0.19 (0.27)
icf-
a!,-- 5
x,-4
b
estimation of Q; monthly average data; Sept. 1920 to June 1923; upper part of the table for N=8, lower part for N =4; standard errors in parentheses.
Table 4
2 4
9 3
% 3 P z
3 w i; z. 0 z
2 z
604
MX. Salemi, Expectations
and money demand
estimate a with these longer lags [under the constraints implied by the model of money demand (7~(9) but not by the additional assumption that adaptive expectations turn out to be rational] which particularly recommends attention to these estimates in the light of published work.
Cagan,Phillip. 1956,Themonetarydynamicsofhyperinflation,in:
Milton Friedman,ed., Studies in the quantity theory of money (University of Chicago Press, Chicago, IL). D&in, J., 1970, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables, Econometrica 38. Einzig. Paul, 1937, The theory of forwa: ? exchange (MacMillan, London). Evans. Paul, 1977, Time-series and struc;rlral analysis of the German hyperinflation, International Economic Review, Feb. Granger. C.WJ., 1969, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, July. Muth.J.ohn F.. 1961. Rational expectations and the theory of price movements, Econometrica 29, July. Salemi, Michael K., 1976, Hyperinflation, exchange depreciation, and the demand for money in post World War I Germany. Ph.D. Dissertation (University of Minnesota, Minneapolis, MN). Salemi. Michael K., 1978, Exchange depreciation and money demand in hyperinflation, Feb., mimeo. (University of North Carolina. Chapel Hill, NC). Salemi, Michael K. and Thomas J. Sargent, 1979, The demand for money during hyperinflation under rational expectations: II, International Economic Review 20, Oct. Sargent.Thomas J., 1977,Thedemand for money during hyperinflation under rational expectations: I, International Economic Review 18, Feb. Sargent, Thomas 1. and Neil Wallace, 1973, Rational expectations and the dynamics of hyperinflation, lntemational Economic Review 14. June. Sims, Christopher, 1972. Money, income, and causality, American Economic Review 62, Sept. Statistisches Reichsamt. 1923, Germany’s economic and linancial situation (Zentralverlag, Berlin). Statist&h= Reichsamt, 1924. Germany’s economy, currency and finance (Zentralverlag. Berlin 1. Stafiistisches Reichsamt, Statistisches Jahrbuch fiir das Deutsche Reich (1919 his 1924’1925) Nerlag fiir Politik und Wirtschaft, Berlin). Statistisches Reichsamt, 1925, Zahlen zur Geidentwertung in Deutschland 1914 bis 1923 (Reunar H ebbing Berlin )_ Young, John Parke, 1925 (for the Commission of Gold and Silver Inquiry, the United States Senate). Europeancurrencyandfinance,Vols.IandII(U.S.Government
PrintingOfice,
Washington,DC).
Zellner. Arnold and Franz Palm, 14;14. Time series analysis and simultaneous equation econometric models, Journal of Econometrics 2, 17-54.
Economics
5 (1979) 593 604. 0
North-Holland
Publishing Company
ADAPTIVE EXPECTATIONS, RATIONAL EXPECTATIONS, AND MONEY DEMAND IN HYPERINFLATION GERMANY*
In rhe recent likrature Sargent and Wallace (I,%, June. 1973) have estimated the demand equ,ltion for money in hyperinflarion under the restriction that the adaptive formula of Phillip Cagan yields rational inflation expectations in the sense of John Muth. The present paper finds evidence IO rqcct for the Germany case the proposition that adaptive expectations are rational. The procedure employed is basically to overfit the stochastic rcprcscntation for the inflation rate implied by the ‘ad;rpti~c-is-rlti~)nal’ hypothesis. The paper also puts forward and appl~r\ a tuostep procedure IO estimate the important money demand elasticity in hypsrm~latlcw. 7‘1~ procedure returns reasonable results \%ith large estimated standard error\.
I.
Introduction
The prototype for modern research in the demand for money in hyperinflation is the model developed by Cagan (1926) in which desired real money balances are assumed inversely related to the anticipated rate of general price inflation. Cagan assumed that agents formed inflation expectations adaptively. that is, as a weighted sum of the previous period’s guess and the currently realized error in that guess. Sargent and Wallace (1973) and Sargent (1977) have based recent studies of money demand in hyperinflation upon the additional assumption that adaptively formed expectations are rational in Muth’s (1961) sense. This paper finds evidence to reject for the German case the proposition that adaptive expectations are rational. This result is important because it bears fundamentally on whether or not the important parameter of Cagan’s model, Q - the money demand elasticity with respect to expected inflation, can be consistently estimated. This paper further undertakes estimation of money demand elasticity under the assumption that expectations are rational but not necessarily adaptive and [hat the rate of inflation is statistically exogenous in the sense of Granger and Sims. Estimation is accomplished by a two-step, constrained ordinary least squares technique. Salemi and Sargent (1979) have also undertaken the estimation of x using a constrained maximum likelihood procedure. In contrast, this paper imposes throughout the assumption that xr is exogenous and estimates longer lags than those estimated in Salemi and Sargent. *This research was supported author thanks Arthur Benavie, comments.
in part by funds from the Social Science Research Council. The Richard Froyen, Roger Waud, and Warren Weber for helpful
2. A test of the ratiouality of adaptive expectations Suppose that the demand for money schedule in hyperinflation form asscnxd by Cagan (1956), ni, - p, = 7 + %rrr-i-u,,
X-CO,
has the
(1)
where nr, is the natural log of the money supply, p, is the natural log of the price level, n, is the pubiic’s expectation formed at 1 of the rate of inflation to occur between t and E+ 1, and 14, is a random error term the assumed properties of which will be discussed in what follows. Let x, z p, -17~_ 1. Ctrgan closed his model by assuming that x, was formed according to an adaptive mechanism,
R,=n,-,+(1-i.)(%,-72,._1),
(2)
and estimated J, r. and i by a form of non-linear least squares. S,argent and Wallace (1973) and Sargent (19773 assume further that the addptive mechanism (2) yields expectations raticnal in the sense of Muth (1961). That is, they
assume
where E is the linear projection operator. If z,_ , = E(x,l& _ 1 1 one may regard n,_, as the ordinary least squares prediction of xt based on a set of data available at t - 1, 4, _ 1. Adaptive expectations are rational if and only if the rate of inflation has the stochastic representation ;IJ”Zr_l
+(I -i-L&z,,
(4)
where a, is a w’iailenoise random process and L is the lag operator defined by L”‘Z,= Z,_ m.If x, evolves according to (4) then
whicil is a crucial fact for Sargent’s deduction that under (2) and (3) cxis not identtied.’ ‘?barge~t assumes (I)-@) and a restrktio?? on u,, namely Chat b(u, -u,- , I where Ksm, -m,_ ,, He shows two things. First, Cagan’s )=Q fJ-r,%-~--1C4-1,14-2,--. estimator in general sufkrs from simultaneous equat;,ons bias [Sargent (1977. p. 6811. Second, ~nlesr an arbitrary restriction is placed on the covariance between the innovations in (the first &f#krcxe~ 4) money demand and money supply it is not possible to obtain a consisknt estitnataf for ?x(Sargent (M’7, p_ 73)J. The estimates which Sargent generates are under the lrwumpriou that these innovations are uncorrelated.
MX.
Salemi, Expectations
and money demand
595
Whether the adaptive mechanism yields rational expectations of the inflation rate is directly testable. Let x:_ 1(2) be the adaptive prediction of 2, given the history of inflation and i.. If TIN_ ,(i.) yields rational expectations then the prediction error, x,-~t:_ 1(i), will be orthogonal to all the data available at C- 1 for predicting x,. The alternati’ve hypothesis is that the hi‘s. n, and c are not all zero in eq. (4), -/T)x,-,/(l
x,=(l
-/fL)--((1
-L)l,_,/(l
-X)‘)(i,-,f)
“1 +
C hiZ,-i+d’t+c’+rl,.
(6)
i= I
The first two terms on the right-hand side of (6) art: a first-order Taylor approximation to (3) about an initial &value, 2, which permits an adjustment to 2 to be estimated jointly with the hi-s. The Z’s represent data atailablc in the system at time t - 1 such as the rate of money creation. jl,, the rate of inflation. 2,. or the rate of Deutschemark exchange depreciation, K,. Each series represents data which could have been used to predict z, but which under the null hypothesis has no predictive value given nr_ 1(i.). Eq. (6) may be regarded as a way of over-fitting the stochastic representation of x,. When Z,_ i is taken to be 2, _ i, the Cagan specification is tested against the more general alternative hypothesis that lt follolvcd an unconstrained m-lag autoregression. When on the other hand Z,_ i is taken to be 11,_i (or I:,._~) the Cagan specification is tested against the alternative that the true stochastic representation of z, is comprised of an unconstrained distribution lag on past 11 (or cc:)as well as a geometric lag on past 1. It is worth repeating that the Taylor approximation in (6) is important because it provides a way of estimating i. and the other parameters simultaneously.‘*” Table 1 reports the results of fitting eq. (6) by ordinary least squares with m=2.’ The Group A and B results were obtained when the price data reported by Cagan for Germany were employed. Two values for 2 were tried: /r=O.SZ, the estimate of i. reported by Cagan for Germany. and rT=O.67, the ‘AI lirst it may stem peculiar that the test does not simply insolve regressing the prediction error from adaptive expectations, X,-R;_ ,(i.). on the lagged %‘s. a trend. and a constant. However, Durhin (1978) has shown that unless i and the hi’s are jointly estimated the test will be brased in the direction of accepting the null hypothesis. /I, = h, = . = b, d i’ =O. ‘Taylor approximation (23) permits joint estimation the h, an adjustment i.. (i. In the which were d&d>.. which equal to lag polynomial - L)(l bythelagpolynomial -(I L)(l + i 3izL2+... ‘The choice inflation and creation as for a of omitted is clear. why the of exchange is also reasonable candidate discussed in (1977) and (1978). *When lag lengths employed it true that of the omitted information at t tAs lag was increased became increasingly to rqject null hypothesis.
Cagan data series Oct. 1920 to July 1923 r: = 0.67
G:oup B
AI=O.82
.. (‘A
I
0.24 (0.21)
iI+‘;. 0.17 (0.09)
11,- ,
0.39 (0.38)
PI - 1
;I[ :; ;_ 9.40 [0.20)
-0.16 (0.45)
XI -- I
--
0.16 (0.07)
I:, . . ,
0.39 (O._iI )
11,- I
0.004 (0.003)
- 2
-0.12. (0.09)
61 - 2
- 0.88 (0.39)
t4 - 2 -_ --~----
-0.59 (0.27)
Xl - 2
- 0.09 (0.08)
4
-U.78 (0.29)
-__-
- 0.02 (0.04)
0.003 (0.002)
0.00 (0.W
con.
trend 0.001 (0.003)
- 0.05 (0.04)
con.
trend 0.009 (0.003)
- 0.03 (0.05)
0.007 (0.004)
trend
con.
con.
-..
trend
_-._
- 0.07 (0.04)
con.
- 0.03 (0.~1
0.0 IO (0.003)
I rend klcr 2 .---____.-.._
- 0.27 (0.25)
con. Ircnd xt- 2 .__... ._ ..__-.._ ____ _.______-._.._ ---
0.29 (0.23)
I
X8.
-0.15 (0.37)
in/?>.
0.16 (0.10)
....-_B----p-_
i’Ti
0.21 (0. IO)
;n,‘;i. -------_I-
0.15 (0.13)
iltpi -_-___-
2.76”
3.36”
2.43
4.28b
5.33h
3.29”
FW9b . _--..______..___.___. .__._ .__. _,_ __._ _._-__.__. ___..___
OLS estimaks
_I_.._..__.....__._.. _.“__._I__..__.,.._
Cagan data series Oct. 1920 to July 1923
Group .4
.-_ --
3.41”
4.47”
2.80
3.27”
4.89”
1.74
1.79
1.72
I .83
1.84
1.97
I .9’
D. 1% F(2.29) .._.- _._-.mm---_------ I_.
Tests of thu mtionitlity of adikplivc expectations; dcpcndunt v;rriublc p-x, - nf ,(i); cstimatetl sttindurd errors in parentheses. ____,^ _.. __... .._.__...__._____.____ ____.. ..___---.-.. ..____ _- -----PI.-<--“q_ ..-m-_Ie.--.._.^- _... --_. ...__._“I ,_... _,lyl”ll._ ._ _. . -... .._..
Table I
M.K.
Strlrmi.
Expectutions
ur~d money demund
597
MX.
598
Salemi, Expectations and money demund
value of j. which minimized the sum of squared
Given a number of hypotheses to be enumerated below, a consistent estimator of the elasticity of demand for real money balances with respect to expected inflation is available. For convenience the model may be rewritten its
u*==u,_
1
+t
I’
(9)
‘To form the sum of squared u’s_ given the 1’s and i, a, was set equal to zero. its assumed exoected value. %e rate of inflation is formed from the series of monthly averages of the daily observation of the whokzsale price index published by the Statistisches Reichsamt (1925). While Cagan begins with this series he mixes with it some point in time observations available in 1923. The rate of money creation is formed from the series of end of month numbers for the nominal money stock ako published by the Statistisehes Reichsamt. While Cagan applied different interpolation filters to diierent parts of the series and mixed the end of month numbers with some point in time data available for 1923, this section uses the data as published except for a log linear intq&tor appkd consistently to the entire series. The data on the Exchange value of the Mark were taken from Young (1925).
M.K.
Salemi.
Expectations
and money demand
599
where pI, x1, xI, E, and u are defined as before, L‘,is white noise, and 4, is information available at time t. Eq. (7) is Cagan’s money demand equation in first difference form. Eq. (8) states that expectations are formed rationally and on tr’le basis of the information set 4,. In what follows it is assumed that #, includes ,;lnly the history of inflation rates dated t and earlier. Eq. (9) states that the error in the money demand equation takes a random walk. There iIr(. two reasons for limiting the information set to past and present inflation ra’tes. First, the time series are short and the conditioning set must be limited or degrees of freedom problems will result. Salemi and Sargent (1979) used a constrained maximum likelihood procedure to estimate Q and assume that 4, includes the history of inflation rates and rates of money creation. However, the longest lag length which they estimate is 4 months compared with 8 months in the work reported here. Second, several of the German hyperinflation studies’ have reported evidence that in the sense of Granger and Sims X, is an exogenous stochastic process. If x, is statistically exogenous then all the information available at t and useful for predicting xt+ 1 is contained in past and present values of x. The results of the previous section indicated that adaptive expectations omitted information significantly useful for preuicting the one-period-ahead inflation rate. The erogeneity of 1 implies that this omitted information may be captured by predicting x[ on the basis of an unconstrained distributed lag on past x alone.’ Operating on (7) with the projection operator yields
since by assumption E (q 1c#+_1) =O. In general, (10) provides the basis for a consistent estimation of a.9 Note, however, that if adaptive expectations are rational, then E(xt+ 1 -x,1+,- 1) =0 and (10) would not be useful for recovering estimates of a. To proceed form the linear projections implicit in (lo), ‘See Frenkel (1977), Salemi (1976), and Sargent and Wallace (1973). ‘Evans (1978) using techniques proposed by Zellner and Palm (1974) studies three models of hyperinfiation and linds evidence to reject them all. In one case he too rejects the hypothesis that 1, has the ARIMA (91.1) implied by a model in which adaptive expectations are rational. In another he considers a model (termed MC by Evans) which like the one presented here is comprised of a version of Cagan’s equation and an equation imp0sir.g rational expectations. The third equation of model MC is a money supply rule of the form (in my notation) /~,=/f,_ 1 +/(L)u,. where r, is whit%and f(L) is a finitely dimensioned lag polynomial. Evans rejects MC on the basis of strong cross correlation at past lags observed between prewhitened p, and 1,. It is important to note that the model presented here is consistent with a money supply schedule of a different form. In general, the model estimated here is consistent with a money supply rule of the form ~~,=constant +h(L)~,+u,, where h(L) is a one-sided polynomial on past and present inflation rates. ‘It is worthwhile noting that eq. (lo), which is the basis for consistent estimation of z of the approach reported here, holds even if the true information set, call it $,- ,, is larger than &_ ,. This is so because of the law of iterated projections which says that if Q is a random variable, then E(E(Q1~,-,)1~,-,)=E(QI~,-,).
MX. Salemi, Expectations and money demand
i-1 L-2
x
.
+
x1-3 .
h-m-
Wit-1
[
w2t
3,
1
where w1 and w2 are the innovations respectively in pI _ 1 - xt In view of (IO), however, the Bids are not all free bul satisfy j=l,2 ,..., m.
BljzaB2j,
(11)
_1
and xt - xt _ 1.
(12)
To impose the restrictions implied by the money demand model on the parameters of (111, subtract the first equation from the second to obtain
%t-&-l=
f j=l
(B2j-Blj~Xt-l-j+w2t-wlt-l
.m =(lerI
x B2jXt-1-j+wt j= 1
(13)
m =
C j=l
‘/iXt-l-j+wtC;)
where w, E w2t -
wlr_l and *Ji=(l-E)pzj for j=l,2,...,m. The procedure employed was to recover the p,is by estimating the second equation of (1 l), to recover the 7;s by estimating (13) and to form an estimate of 2 as
(14)
The results obtained for m=4 and m= 8 with both the Cagan data and the monthly average data are summarized in table 2 and reported in detail in tables 3 and 4. For both the Cagan and the monthly average data, the two-step procedure yields appropriately signed estimates of a. By way of comparison, Cagan estimated that r equalled - 5.46 with a 90% confidence band of - 6.13 to -5.05 and Sargent (1977) estimated 01 to be - 5.97 with an estimated
M.K.
Salemi, Expectations
and money demund
601
Table 2 Two-step estimation
nf the elasticity of money demand with respect to the expected inflation m-4 --
-
-..-_-_-____.
.--
Cagan’s data
Monthly
Nov.
Sept. 1920 to June 1923 -_ --_-__---._
1920 to Aug. 1923
8
4
4
jjI, 421
- (i.26
- 1.40
-0.56
- 0.68
t ,= I
-2.16
- 2.20
- I.71
- 1.34
- 7.22
- 0.575
- 2.06
- 0.988
0.116
I.91
0.792
ci
-
.__
average data
8
Y2j
rate.
and 8.
--.-_-..
__-
___..___
Estimated 563.2
asymptotic standard error or i
asymptotic standard error of 4.62.” Using a maximum likelihood technique, Salemi and Sargent (1977) obtained estimates of zt which were sensitive to the method used to detrend the data but in most cases consistent with the results reported in table 2.” One theme that emerges from the Sargent (1977), Salemi and Sargent (1977), and the results reported here is that in contrast to the Cagan results the estimates of r are generally small (and sometimes quite small) relative to correctly estimated asymptotic standard errors. It is interesting to learn that this imprecision persists even when the lag length is permitted to extend to eight months. Indeed, it is the ability to
“The estimate of the asymptotic study by writing a first-order Taylor
‘=
I -i$,
fi/i,
S2jE
standard error of the estimator of r was obtained approximation for z. Rewrite j, as
l-@,
6i/,=F I
in this
Lr). m+l
which serves to define the 4;s. Then i G Z* + [G/c?&] I,.(4 - @*), where the asterisks designate the true parameter values and [a&/?&] is the I x 2m Jacobian matrix. As the sample size goes to Moreover, asymptotically L, T (S--a+) is approximately the weighted sum of infinity, &c;z*. normal random variables with weights equal to the probability limits of the elements of the Jacobian. It is therefore reasonable to estimate the asymptotic variance of si as Lm
“In
fact
depending in between -3.5
the
Salemi
and
Zm
Sargent
lag length and the method and - 10.5 however.
(1977)
estimates
of detrending
do
range
the data.
between
-335
and
0.651
The estimates most often range
I
0.02 (0.29)
0.97 (0.28)
-
- 0.43 (0.27)
-0.10 (0.25)
t-Cc,-1
XI-XI-1
-0.11 (0.31)
0.09 (0.30)
-
x1-x1-1
x1- 3
-0.58 (0.29)
2
- 0.26 (0.28 )
XI -
Independent variable ‘-
X,--k-l
Dependent variable -x1-5
- 1.07 (0.31) -0.87 (0.33) - 1.13 (0.28) -0.85 (0.30)
x1-4
- 0.20 (0.31) -0.14 (0.33) - 0.54 (0.27 ) - 0.50 (0.29)
XI-
- 0.70 (0.52) -0.38 (0.55 1
X,-h
0.39 (0.41) 0.69 (0.44)
7
R: = 0.32 F(9,24)=2.70 R: = 0.47 F(5,28)=6.95 R:=0.25 F (5,28) = 3.23
- 0.05 (0.07) -0.15 (0.05) -0.11 (0.05)
0.01 (0.01) 0.02 (0.004) 0.01 (0.005 )
0.53 (0.44)
0.12 (0.55)
R: = 0.49 F(9,24)=4.55 0.02 (0.01)
0.51 (0.41)
- 0.25 (0.52)
con. -0.15 (0.06)
trend
8
XI .- 9
x1-
Regressions for the limited information estimation of r; Cagan data series; Nov. 1920 to Aug. 1923; upper part of the table for N=8. lower part for N =4; standard errors in parentheses. -
Table 3
----.--
%,-A
-0.71 (0.22)
(0.21 )
- 0.40
0.41 (0.23)
(0.2 1 I
-0.21
____
0.59 (0.28)
- ii.76 (0.25 )
-
0.04 (0.26 I
%I-- 3
0.59 (0.23)
%r z
variable -_-
information
Independent
._ .~_~_~~~~~~~~~
I
%r- 14, I
_.--
i[r - zr I
1
Dependent variable
-.--
Regressions for the limited
- 0.49 (0.37)
0.46 (0.35 1
- 0.78 (0.32)
-0.64 (0.28)
-0.18
0.2 I (0.38)
-0.10 (0.10)
(0.008)
-
Ri=0.18
F(5,28)=2.71 D.W!= 2.49
Rf,=0.21
R:: =0.27 F(5,28)=3.40 D.W=1.61
D. W = 2.69
l
1.81
R: =0.30 F(9,24)=2.61 D.W= 1.68
(0.10) F (9,24)=
-0.12
0.01
)
-0.24 (0.09)
0.26 (0.30)
0.01 (0.009
0.03 (0.008)
-0.19 (0.09)
---
(0.31)
0.40
0.15 (0.28 )
0.03 (0.008 )
(0.35)
-0.18 (0.32)
0.0 I (0.35 1
x1-7
-0.54 (0.28)
---___
- 0.66 (0.34)
-0.21 (0.32 )
- 0.27 (0.29)
-0.19 (0.27)
icf-
a!,-- 5
x,-4
b
estimation of Q; monthly average data; Sept. 1920 to June 1923; upper part of the table for N=8, lower part for N =4; standard errors in parentheses.
Table 4
2 4
9 3
% 3 P z
3 w i; z. 0 z
2 z
604
MX. Salemi, Expectations
and money demand
estimate a with these longer lags [under the constraints implied by the model of money demand (7~(9) but not by the additional assumption that adaptive expectations turn out to be rational] which particularly recommends attention to these estimates in the light of published work.
Cagan,Phillip. 1956,Themonetarydynamicsofhyperinflation,in:
Milton Friedman,ed., Studies in the quantity theory of money (University of Chicago Press, Chicago, IL). D&in, J., 1970, Testing for serial correlation in least-squares regression when some of the regressors are lagged dependent variables, Econometrica 38. Einzig. Paul, 1937, The theory of forwa: ? exchange (MacMillan, London). Evans. Paul, 1977, Time-series and struc;rlral analysis of the German hyperinflation, International Economic Review, Feb. Granger. C.WJ., 1969, Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, July. Muth.J.ohn F.. 1961. Rational expectations and the theory of price movements, Econometrica 29, July. Salemi, Michael K., 1976, Hyperinflation, exchange depreciation, and the demand for money in post World War I Germany. Ph.D. Dissertation (University of Minnesota, Minneapolis, MN). Salemi. Michael K., 1978, Exchange depreciation and money demand in hyperinflation, Feb., mimeo. (University of North Carolina. Chapel Hill, NC). Salemi, Michael K. and Thomas J. Sargent, 1979, The demand for money during hyperinflation under rational expectations: II, International Economic Review 20, Oct. Sargent.Thomas J., 1977,Thedemand for money during hyperinflation under rational expectations: I, International Economic Review 18, Feb. Sargent, Thomas 1. and Neil Wallace, 1973, Rational expectations and the dynamics of hyperinflation, lntemational Economic Review 14. June. Sims, Christopher, 1972. Money, income, and causality, American Economic Review 62, Sept. Statistisches Reichsamt. 1923, Germany’s economic and linancial situation (Zentralverlag, Berlin). Statist&h= Reichsamt, 1924. Germany’s economy, currency and finance (Zentralverlag. Berlin 1. Stafiistisches Reichsamt, Statistisches Jahrbuch fiir das Deutsche Reich (1919 his 1924’1925) Nerlag fiir Politik und Wirtschaft, Berlin). Statistisches Reichsamt, 1925, Zahlen zur Geidentwertung in Deutschland 1914 bis 1923 (Reunar H ebbing Berlin )_ Young, John Parke, 1925 (for the Commission of Gold and Silver Inquiry, the United States Senate). Europeancurrencyandfinance,Vols.IandII(U.S.Government
PrintingOfice,
Washington,DC).
Zellner. Arnold and Franz Palm, 14;14. Time series analysis and simultaneous equation econometric models, Journal of Econometrics 2, 17-54.