Vol. 2, 2008-21
June 24, 2008
Special Issue "Using Econometrics for Assessing Economic Models"
Editor: Katarina Juselius
On the Explosive Nature of Hyper-Inflation Data
Bent Nielsen
Department of Economics, University of Oxford
Abstract:
Empirical analyses of Cagan’s money demand schedule for hyper-inflation have largely ignored the
explosive nature of hyper-inflationary data. It is argued that this contributes to an (i) inability to model
the data to the end of the hyper-inflation, and to (ii) discrepancies between “estimated” and “actual”
inflation tax. Using data from the extreme Yugoslavian hyper-inflation it is shown that a linear
analysis of levels of prices and money fails in addressing these issues even when the explosiveness is
taken into account. The explanation is that log real money has random walk behaviour while the
growth of log prices is explosive. A simple solution to these issues is found by replacing the
conventional measure of inflation by the cost of holding money.
JEL: C32, E41
Keywords: Cost of holding money, co-explosiveness, co-integration, explosive processes, hyper-
inflation.
Correspondence:
Bent Nielsen, Nuffield College, Oxford OX1 1NF, UK; e-mail: bent.nielsen@nuffield.ox.ac.uk
The data used in this paper were collected and previously analysed by Zorica Mladenovi´c and her co-
authors. I have benefitted from many discussions with her and with David Hendry, as well as from
discussions with Frédérique Bec, Aleš Bulíř, Tom Engsted, Neil Ericsson, Katarina Juselius,
Takamitsu Kurita, and John Muellbauer, and comments from two referees and one invited reviewer.
Computations were done using PcGive (Doornik and Hendry, 2001) and Ox (Doornik, 1999).
Financial support from ESRC grant RES-000-27-0179 is gratefully acknowledged.
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1Introduction
The money demand equation for hyper-inflation of Cagan (1956) is a continuous time
linear relationship between real money and the expected rate of change in prices.
Cagan’s own empirical work consists essentially of single equation regressions of log
real money, mt−pt,regressed on the changes in log prices, ∆1pt=pt−pt−1,measured
at a monthly frequency. If, as assumed in most of the literature, nominal money, mt,
and prices, pt,were integrated of order two, I(2), the money demand relation could be
found as a cointegrating relation. Here it is argued that in hyper-inflations nominal
money and prices are typically not I(2) ,but explosive, as found by Juselius and
Mladenovi´
c (2002). A different empirical analysis is called for. The problem arises
since ∆1ptas a measurement of the cost of holding money implicitly is motivated by a
Taylor expansion of the logarithmic function, which has poor mathematical properties
for large inflation rates. Using a different measure of the cost of holding money the
difficulties can be overcome.
Most empirical studies have struggled with modelling hyper-inflationary episodes
to the end. Cagan set the example of modelling for instance the German hyper-
inflation until July of 1923 although the episode continued until november. Likewise,
large discrepancies have been found between the “optimal” and the “actual” inflation
tax, and, hence only little support for Cagan’s theory for seigniorage. In the present
analysis it is shown that the explosive behaviour of the data is the main source of the
empirical problems.
The argument is based on an empirical analysis of the extreme Yugoslavian hyper-
inflation of the early 1990s. This is one of the longest and most extreme episodes
ever observed with monthly inflation rates above 50% for 24 months. Unlike the
German government in 1920s the Yugoslavian government was unable to halt the
inflation even temporarily in this period. These unfortunate features actually make
it easier to analyse the Yugoslavian case than for instance the German case which was
studied by Cagan. For a discussion of the resolution of the hyper-inflation puzzles
it is therefore convenient to focus on the Yugoslavian case. With the analysis from
thispaperitshouldbepossibletoreturntothemorecomplicatedGermanhyper-
inflation in a later study. In the present analysis two econometrics models are used.
The first model serves to show that a traditional linear econometric model linking the
logarithm of real money, mt−pt=log(Mt/Pt),and inflationmeasuredasthegrowth
of log prices, ∆1pt=pt−pt−1,is indeed unbalanced. The second model shows
that the puzzles are resolved by measuring inflation as the cost of holding money,
ct=∆1Pt/Pt=1−exp(−∆1pt).
In the first model, the conventional variables, nominal money, mt,nominal prices,
pt,and spot exchange rates, st,are analysed using a vector autoregression. Due to
the accelerating nature of the data the vector autoregression is found to be explosive.
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Using econometric methods developed in Nielsen (2005b) it is found that real money
has random walk features while changes in log prices are explosive. This contrasts
with the analyses of Sargent (1977) and Taylor (1991). A regression of real money on
changes in log prices is therefore unbalanced which explains the puzzles.
In the second model real money is instead linked to the cost of holding money,
ct.A well-specified vector autoregressive model can now be made. The cointegration
analysis leads to a linear relation between real money and the cost of holding money
as expected from the Cagan model. This model does, however, fit throughout the full
sample and the estimated “optimal” and “actual” inflation tax rates are now in line.
The outline of the paper is that §2 discusses Cagan’s empirical puzzles, in the
context of Cagan’s own analysis and later empirical studies, as well as in the context
of the Yugoslavian episode. The two econometric models are outlined in §3 and §5
with §4 describing the measure of cost of holding money. §6 concludes.
2TheHyper-inflation Puzzles
A brief outline of the empirical literature on money demand in hyper-inflations is
given. The theoretical and empirical work of Cagan (1956) is reviewed. The empirical
puzzles identified by Cagan are then traced through the literature and are finally
illustrated using data from the Yugoslavian hyper-inflation.
2.1 Cagan’s Theory for Money Demand
Cagan’stheorydescribestwoaspectsofhyper-inflations: the money demand schedule
and the seigniorage. In his empirical work he noticed puzzles associated with both.
The money demand schedule is described in his equations 2 and 5. These are
continuous time equations linking the log real cash balances with the expected rate
of change in prices:
mt−pt=−αEt−γ, (2.1)
µ∂Et
∂t ¶t
=β(Ct−Et).(2.2)
Here mtand ptrepresent the logarithm of money and prices, Ct=∂pt/∂t is the
continuous rate of change in prices, while Etrepresents an adaptive expectation of Ct.
Other variables, like output, that are usually appearing in quantity theories for money
are assumed to have a negligible influence. By solving equation (2.2) backwards from
present time, t, to an initial value, −T, the expectations term Etcan be expressed as
an exponentially weighted average of past values of C, that is
Et=Hexp (−βt)+βZt
−T
Cxexp {β(x−t)}dx. (2.3)
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Inserting this in (2.1), Cagan could then estimate αand βfrom monthly data as
follows. Letting −Trepresent the beginning of the sample and assuming that prices
had been almost constant before time −T, then Hcan be set to zero in (2.3). Cagan
then made the crucial assumption that
Ctis constant within a month, (2.4)
in which case Ct=∆1pt=pt−pt−1and the latent expectations process Etcan be
approximated by a sum. For a given value of βthe parameter αcan then be estimated
from (2.1) by regression. By varying βa joint estimate for α, β can be found.
In the empirical analysis, Cagan considered data from seven hyper-inflations. The
infamous German hyper-inflation from August 1922 to November 1923 was analysed
using data until July 1923 only, due to difficulties in fitting the data from the last
few months. This is puzzling in suggesting that the money demand schedule for
hyper-inflations is not time invariant and may not even hold when the hyper-inflation
is most extreme. In any case, he estimated the semi-elasticity αby bα=5.76.
Cagan also analysed the seigniorage from printing money, arguing that the revenue
from the inflation tax is the product of the rate of tax and the base
R=µdP
dt
1
P¶M
P,(2.5)
where Mand Pare levels of money and prices, and the timing is left unspecified. He
then made the counter factual assumption that the quantity of nominal money rises
at a constant rate. This would eventually imply constancy of real money balances,
which is contradicted by Cagan’s own observation that real money balances tend to
fall in hyper-inflation. It would also imply that Etcan be replaced by Ctin equation
(2.1):
M
P=exp(−αC −γ)(2.6)
Combining (2.5) and (2.6) gives a revenue of R=Cexp(−αC −γ), which has a
unique maximum, with respect to C,when
C=1
α.
The inverse of the semi-elasticity αis therefore interpreted as the rate of inflation
that maximises the revenue from seigniorage under the above assumptions.
In the empirical analysis, Cagan’s estimate for the German hyper-inflation is
ˆα−1=0.183.This is a continuously compounded rate corresponding to a monthly
tax of exp(bα−1)−1=20%.He compared this with an average monthly rate of infla-
tion of 322%, defining inflation as ∆1Pt/Pt−1.Comparingthetwoshowsapuzzling
mismatch between an “optimal” tax rate and the “actual” inflation tax.
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2.2 The I(2) Approach
WhilethetimeseriesmethodologywasinitsinfancyatthetimeofCagan’sstudy
later work on hyper-inflation has been cast in an I(2)-framework with nominal money,
mt,and prices, pt, assumed I(2)-series.
In this way Sargent and Wallace (1973) and Sargent (1977) revisited Cagan’s
analysis in part with a view towards explaining the discrepancy of the “optimal”
and the “actual” inflation tax. The model of Sargent (1977) is a bivariate model for
nominal money and prices involving a rational expectation, πt,to future inflation,
∆1pt.Unlike Cagan’s model it is discrete time model applied at a monthly frequency
in the empirical work and therefore implicitely using the discretization assumption
(2.4). Sargent further makes the assumptions:
mt,p
t∼I(2) ,m
t−pt,∆1mt,∆1pt∼I(1) ,(2.7)
for the observables, whereas the rational expectations satisfy
πt−∆1mt,π
t−∆1pt∼I(0) .
Since Sargent’s work predates the concept of co-integration this is not the focus of
the work and mt−ptand ∆1ptare not cointegrating in his model. The causality
structure in the model is that ∆1(mt−pt)and hence ∆1mtdo not Granger-causes
∆2
1pt.
Sargent went on to fit the model to the data considered by Cagan. In the case of
Germany,theestimateofαis virtually unchanged, bα=5.97, but the uncertainty is
judged differently with a standard error of 4.6 so the estimated confidence band for the
“optimal” inflation tax covers nearly the whole positive real axis. Sargent’s empirical
analysis therefore lends support, albeit only weak support, to Cagan’s model.
Around the same time Evans (1978) analysed the time series properties of mt,p
t
using Box-Jenkins analysis. That is an analysis based on inspection of the correlo-
grams rather formal testing. This analysis lead Evans to conclude that for the German
episode mtand ptare I(2) in line with Sargent. It should be noted that with the pre-
vailing definition of correlograms explosive time series have an exponentially declining
correlogram, see Nielsen (2006a). Christiano (1987) analysed variations of Sargents
model a little further within an I(2) framework. This analysis found some, but not
overwhelming, evidence against the Sargent and Wallace model. Here it should be
noted that the reported mis-specification tests are based on the Box-Pierce statis-
tic, which could suffer from the same problems as correlograms if there is explosive
behaviour in the residuals.
A partial rational expectations formulation opens up for interesting interpreta-
tions. Then Cagan’s equation is formulated in discrete time as
mt−pt=−αEt(pt+1 −pt)+ζt.(2.8)
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Here Etis the expectation conditional on (ms,p
s,ζs)s≤tand thus a construct of a
probability model rather than actually representing the expectations it is rational
to form for the agents in the economy. The equation can be solved for ptas a
function of future values of mt,see Turnovsky (2000, p. 86f & 91f) and Diba and
Grossman (1988). To do this assume (i)that α>0and α/(1 + α)<1and (ii)that
|{α/(1 + α)}jEt(˜mt+j)|vanishes at a geometric rate. Then (2.8) is solved by
pt=1
1+α
∞
X
j=0 µα
1+α¶j
Et¡mt+j−ζt+j¢+cµ1+α
α¶t
+
t
X
s=1 µ1+α
α¶t−s
ξs,(2.9)
for any c∈Rand random variables ξtsatisfying Etξt+j=0.The first term inher-
its whatever stochastic properties mtmay have while the last two components are
explosive and are referred to as an explosive bubble by Diba and Grossman (1988).
Adifficulty with the solution (2.9) is that the model for mt,p
thas to be for-
mulated into the indefinite future and it has to capture the accelerating growth of
thevariablesaswellasthesuddenhaltingrowthwhentheinflation stops. The
literature has focused on somewhat less ambitious assumptions to mtthat gives coin-
tegration properties. This is discussed in some detail in the Appendix. Here, three
interesting examples are discussed. First,ifmtisaunitrootprocessoforderI(d)
without bubble, so c=ξt=0,then pt∼I(d),mt−pt∼I{max(d−1,0)}and
mt−pt+α∆1pt∼I{max(d−2,0)}.This idea has been pursued by Taylor (1991) as
will be discussed below. Second ly,ifmtis explosive with a root ρless than 1+α−1,
without bubble, then ptand mt−ptare also explosive. Thirdly, due to the bubble
component it is possible that ptcan be explosive without mtbeing explosive. How-
ever, all three types of predictions will be contradicted by the empirical findings of
§3. It is interesting to note though that explosive bubbles appear to be suited for an
analysis stock prices and dividends as shown by Engsted (2006) using the co-explosive
analysis that will discussed in §3.
Taylor (1991) looked at the possibilities for cointegration arising from the partial
rational expectations model without bubble. He wrote the discrete time model as
mt−pt=−α∆1pe
t+1 +ζt,(2.10)
∆1pe
t+1 =∆1pt+1 +t+1,(2.11)
where the variable ∆1pe
t+1 measures the expected inflationinperiodt+1 and ζt,
t+1
are stationary error terms. He showed that ∆1pe
t+1 can be interpreted as a rational
expectation as above, or as an adaptive expectation or an extrapolative expectation,
as long as equation (2.11) is satisfied. Inserting (2.11) into (2.10), adding α∆1pton
both sides and then reorganising leads to
∆2
1pt+1 =−α−1(mt−pt+α∆1pt)−¡t+1 +α−1ζt¢.(2.12)
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Assuming that mtand ptare both I(2) variables it can be tested whether real money
mt−ptis I(1) andinturnwhethermt−pt+α∆1ptcointegrates to I(0). In this coin-
tegrated framework the coefficient to the expected inflation variable ∆1pe
t+1 therefore
showsupasthecoefficient to ∆1ptin a cointegrating relation.
In the empirical work Taylor considered six of Cagan’s episodes using 3 different
data sources. As a justification for the I(2) framework, unit root tests were applied to
levels, first, and second differences of mt−ptand ∆1pt.For instance, for Germany it
was concluded using three different data sources that ∆1ptis I(1) ,possibly I(2) .This
was based on one-sided tests against the stationary alternative ignoring any structural
breaks. Considering also the explosive alternative the test statistics of Taylor leads to
the conclusion that ∆1ptis explosive at least for two of the German data sets. Leaving
that aside, Taylor (1991) found evidence for cointegration between mt−ptand ∆1pt.
For the German case Taylor estimated αby 5.31 in line with previous results.
Frenkel (1977) suggested linking real money balances with exchange rates and
forward rates to overcome the problem of measuring expected inflation. The rationale
is that agents hold real money in foreign currency and adjust holdings of real money
to expected exchange rate depreciations. This idea was cast in Taylor’s framework by
Engsted (1996). Abel, Dornbusch, Huizinga and Marcus (1979) went one step further
in suggesting that both inflation and depreciation in exchange rates may influence real
money as in
mt−pt=−α∆1pe
t+1 −β∆1se
t+1 +γ+t.
Michael, Nobay and Peel (1994) addressed Cagan’s two puzzles by adding real
economy variables, notably real wages, to the money demand schedule, but found it
necessary to separate periods of high inflation and periods of hyper-inflation. Their
analysis of the German hyper-inflation was also done in an I(2) framework, justified
with one-sided unit root tests against the stationary analysis. Once again, the unit
root statistics of their Table 1 actually show that ∆1mtand ∆1ptare explosive, and
that even for the high-inflation period prior to June 1923. It would be interesting to
follow up the idea of including real economy variables, but for simplicity the presented
analysis will ignore this aspect.
2.3 The Yugoslavian Hyper-inflation
Yugoslavia experienced two hyper-inflations in short time. The first had a long build-
up during the 1980s and peaked in 1989 briefly reaching high, but not very extreme
inflation. The second and very extreme hyper-inflation which is studied here de-
veloped from 1991 to January 1994. For the first Yugoslavian hyper-inflation, richer
data are available such as wages. Juselius and Mladenovi´
c (2002) analysed this period
seeking a link between wages and prices. They identified explosive behaviour in the
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data and set up an empirical model taking this into account. Since then econometric
techniques have been developed for this situation, and these will be used in §3.
As an empirical example it is useful to look at the extreme Yugoslavian hyper-
inflation of the 1990s. This is one of the longest and most extreme observed. Unlike
the German episode the Yugoslavian government was unsuccessful in halting the
inflation temporarily in the course of hyper-inflation. As a result the data appear
smoother and are therefore more suited for addressing the puzzles and to show how
they can be resolved. The data are taken from Petrovi´
candMladenovi
´
c (2000) and
areavailablefromtheJournal of Money, Credit and Banking online data archive.
The data has previously been analysed by Petrovi´
candVujoševi
´
c (1996), Petrovi´
c,
Bogeti´
c and Vujoševi´
c (1999) and Engsted (1998).
The institutional background for the extreme Yugoslavian hyper-inflation is de-
scribed in Petrovi´
c and Vujoševi´
c (1996) and Petrovi´
c, Bogeti´
candVujoševi
´
c (1999).
In short, the former federal republic of Yugoslavia was falling apart in 1991, the civil
war started and United Nations embargo was introduced in May 1992. Output and
fiscal revenue then decreased, while transfers to the Serbian population in Croatia
and Bosnia-Herzegovina as well as military expenditure added to fiscal problems.
The monthly inflation rose above 50% in February 1992 and accelerated further, a
pricefreezewasattemptedinAugust1993andtheinflation finally ended on 24 Janu-
ary 1994 with a currency reform after prices had risen by a factor of 1.6×1021 over 24
months. This makes it the second longest recorded hyper-inflation and therefore, from
an econometric perspective, the most promising in terms of sample length available.
Figure 1(a, d, g)shows three time series of monthly data relating to the period
1990:12 to 1994:1. The variables are the monthly retail price index, pt,narrow money
measured as M1, mt,and a black market exchange rate for German mark, st,all
reported on a logarithmic scale. The sources for the data are documented in Petrovi´
c
and Mladenovi´
c (2000). They consider the prices for 1993:12 and 1994:1 to be unre-
liable and choose to end their analyses end at the latest 1993:11, sometimes even at
1993:6. This is in line with previous studies of hyper-inflation that mostly ignore the
last few observations.
Figure 1(b, e, h)shows first differences of the series. Both in levels and in dif-
ferences the series show an exponential growth over time and hence an accelerating
inflation. Cross-plotting the variables against their lagged values would give approx-
imately straight lines with slopes in the region 1.15-1.35, which would be another
indication of explosive behaviour. This contradicts Cagan’s assumption that nomi-
nal money rises at a constant rate.
Figure 1(c, d)shows real money series, mt−ptand mt−st,where money is
discounted by the price level and the exchange rate, respectively. Both series are
falling, matching the negative sign in equation (2.1). Since German prices only
increase a few percent over the period the variable pt−stis essentially the real
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Figure 1: The series pt,m
t,s
tand linear transformations thereof
1991 1992 1993 1994
20
40
60 (a) pt
1991 1992 1993 1994
1
2
3(b) Δ1 pt
1991 1992 1993 1994
0.0
2.5
5.0 (c) mt−pt
1991 1992 1993 1994
20
40
(d) mt
1991 1992 1993 1994
1
3(e) Δ1 mt
1991 1992 1993 1994
4
6
8
(f) mt−st
1991 1992 1993 1994
0
20
40
(g) st
1991 1992 1993 1994
1
2
3
4(h) Δ1 st
0 1 2 3
0
1
2
93:10
93:6
(i) mt−pt versus Δ1 pt
Note: the growth rates in panels (b, e, h)and the cross plot in (i)is only shown until 1993:10.
exchange rate, which is mostly falling; see Figure 2(a)below.
Figure 1(i)shows a cross-plot of real money versus price growth. This illustrates
the puzzles Cagan was faced with in modelling the money demand schedule. There
is a near linear relationship between the variables until 1993:6 but then a change in
functional form. This is observed by Petrovi´
candMladenovi
´
c (2000) who makes a
linear analysis until this point and a non-linear analysis for the full sample. Michael,
Nobey and Peel (1994) make a similar split the data for their analysis of the German
hyper-inflation.
2.4 A Preliminary Analysis of the Yugoslavian Data
In the light of the structural model of Sargent (1977) it is interesting to construct
simple descriptive time series models for the real money and inflation variables.
For real money discounted by the exchange rate, mt−st,a very simple model
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fareswell. Theestimatedmodelforthefullsampleis
∆1(mt−st)=−0.15
(0.05) +0.27but,(2.13)
with standard error reported in parenthesis and butdenotes the standardised residu-
als. The residuals, but,pass mis-specification tests for normality, autocorrelation, and
autoregressive conditional heteroskedasticity. This empirical model is consistent with
the I(2) assumptions in (2.7). A similar result would be obtained for mt−ptover
the reduced period to 1993:10.The analysis presented in §5 will, however, reduce the
residual standard error by a third by more careful modeling.
Turningtothelogpricegrowth∆1pta second-order autoregression fares well for
the sample until 1993:10,
∆2
1pt=0.15
(0.09)
∆1pt−1−0.67
(0.19)
∆2
1pt−1+0.04
(0.08) +0.32but.(2.14)
Here, mis-specification tests for serial dependence pass, whereas normality cannot
be accepted. The hypothesis of a unit root can be tested from the coefficient to
∆1pt−1.The t-statistic is the augmented Dickey-Fuller test statistic, taking a value of
about 1.6, which is very large compared with the 95% quantile (against the explosive
alternative) of −0.07.This suggests that ∆1ptis an explosive process rather than a
unit root process in contrasts to the I(2) assumptions in (2.7). This issue will be
addressed more systematically through system analyses of the data.
3 A Linear Model for the Variables in Levels
In the following a linear vector autoregressive model is made for the levels of prices,
pt,money, mt,and exchange rates, st.The focus of this model is to consider the
standard I(2) assumptions within a multivariate model. Finding that these variables
are actually explosive the analysis suggested by Nielsen (2005b) is needed. It can
then be shown formally that mt,p
t,s
tco-explode showing that the real variables like
mt−ptare I(1),butleavingthegrowthrate∆1ptexplosive. The I(2) assumption
is therefore found to be unhelpful when analysing hyper-inflations. Based on these
findings an alternative way forward is found in §4 and §5. An simplified analysis along
these lines was given for the bivariate system of mt,p
tas an empirical illustration in
Nielsen (2005b).
3.1 The Unrestricted Vector Autoregressive Model
A model with a constant, a linear trend and three lags is used for Xt=(pt,m
t,s
t):
Xt=
3
X
j=1
AjXt−j+μc+μlt+εt,
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Table 1: Misspecification tests for the vector autoregressive model for p, m, s
Test pms Te st (p, m, s)
χ2
normality (2) 1.3 [0.53] 6.0 [0.05] 4.5 [0.11] χ2
normality (6) 3.1 [0.79]
FAR(1)(1,20) 1.8 [0.19] 1.0 [0.32] 0.1 [0.82] FAR(1)(9,39) 1.5 [0.20]
FAR(3)(3,18) 0.6 [0.62] 0.8 [0.53] 0.3 [0.81] FAR(3) (27,29) 1.1 [0.44]
FARCH (3)(3,15) 0.1 [0.94] 0.2 [0.92] 0.1 [0.93]
p-values are given in brackets
Table 2: Characteristic roots of unrestricted model
Re(z)1.21 -0.42 -0.42 0.02 0.02 0.75 0.75 -0.31 0.09
Im(z)00.84 -0.84 0.90 -0.90 0.33 -0.33 00
|z|1.21 0.94 0.94 0.90 0.90 0.81 0.81 0.31 0.09
where the innovations εtare assumed independent normal N3(0,Ω)distributed. The
lag length is chosen so as to ensure that the mis-specification tests pass. When
it comes to the co-explosive analysis there is then one lag to each of the random
walk component, the explosive component, and the stationary components. Adding
the linear trend appears to help in capturing the variation in the data and matches
Cagan’s potentially counter factual assumption that Mtrises a constant rate. Due to
the measurement problems of prices towards the end of the sample only the subsample
1990:12 to 1993:10 is analysed giving a sample size of T=35−3=32.On the one
hand, this gives a model that has admittedly few degrees of freedom in that each
equation has 11 mean parameters. This issue is alleviated in the subsequent general-
to-specific reduction. On the other hand, these explosively growing time series should
be rather informative.
Formal mis-specification tests are reported in Table 1. Interpreting these in
the usual way indicates that the model is well specified. Graphical tests for mis-
specification, which are not reported here, include Q-Q-plots for normality and are
likewise supportive of the model. Note that the usual asymptotic theory is valid for
general autoregressions with stationary, unit, as well as an explosive root. This has
been proved for the test for autocorrelation in the residuals, see Nielsen (2006a,b),
and for Q-Q plots for normality by Engler and Nielsen (2007). Some of the test sta-
tistics are reported in an F-form as advocated by Doornik and Hendry (2001) in an
attempt to deal with finite sample issues for these tests even though it has not yet
been argued whether this represents an improvement in the explosive case.
Table 2 reports the characteristic roots of the unrestricted vector regression. It
appears as if there is one explosive root and two unit roots as marked with bold face.
The explosive root of 1.21 is within the region of 1.15-1.35 discussed above. There
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Table 3: Cointegration rank tests
Cointegration rank, r0123
Test 79.1 [0.00] 23.1 [0.11] 9.8 [0.14]
Likelihood 15.30 43.27 49.94 54.84
p-values are given in brackets
Table 4: Characteristic roots of restricted model with rank one, r=1
Re(z)1.1911-0.37 -0.37 0.07 0.07 -0.54 0.07
Im(z)000 0.88 -0.88 0.83 -0.83 0 0
|z|1.1911 0.95 0.95 0.83 0.83 0.54 0.07
is a further set of four complex roots near the unit circle. An interpretation of a
seasonal pattern repeating itself every five months seems unlikely. In this analysis
these four roots will be ignored, but it is a matter for further research to understand
the nature of such roots.
3.2 Analysis of Cointegrating Properties
The next step of the analysis is a cointegration analysis using the approach suggested
by Johansen (1996). For this purposethemodelisre-parametrisedas
∆1Xt=(Π,Πl)X∗
t−1+
2
X
j=1
Γj∆1Xt−j+μc+εt,(3.1)
where ∆1Xt=Xt−Xt−1is the usual first difference and X∗
t−1=(X0
t−1,t
0)0.This
likelihood can be maximised analytically under the reduced rank hypothesis
rank(Π,Πl)≤r≤dim Xso (Π,Π1)=αβ∗0,
for matrices α∈Rp×r,β∗∈R(p+1)×rwith full column rank. Although the symbols
α, β were used above to describe Cagan’s model, they are used here in a different
meaning to be consistent with Johansen’s notation. The interpretation of the coin-
tegrating vectors βis now that β0Xthas no random walk component but it could
have an explosive component. This statement will be made more precise in connec-
tion with the Granger-Johansen representation in (3.2) below. The usual asymptotic
critical values are valid in the presence of explosive roots as argued by Nielsen (2001)
for the univariate case and Nielsen (2005b) for the multivariate case.
The cointegration rank ris determined using the likelihood ratio tests reported
in Table 3. It is relatively clear to conclude that ˆr=1. The characteristic roots
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Economics: The Open-Access, Open-Assessment E-Journal 13
Tabl e 5: I(2) cointegration rank tests
I(2) roots
r=rank(Π,Πl)3210
0 140 [0.00] 108 [0.00] 89.7 [0.00] 79.1 [0.00]
1 63.2 [0.00] 35.8 [0.00] 23.1 [0.11]
2 25.5 [0.00] 9.8 [0.14]
p-values are given in brackets
Table 6: Estimated cointegrating vector
pm s t
H11
(6.2) −0.35
(−6.5) −1
(−6.2) 0.065
(6.6)
H1,H
ρ1
(6.3) −0.35
(−6.8) −1
(−6.3) −0.011
(−6.7)
Cointegrating vector, ˆ
β
∗=ˆ
β
∗
1, estimated under H1and under the joint hypothesis H1,Hρ.Signed
likelihood ratio statistics, √LR, for insignificance in brackets
are only little changed by imposing this restriction as seen from comparing Table 4
with Table 2. The explosive root remains, so the cointegration rank test suggests
that is not a realisation of a unit root with multiplicity one. There is the possibility
that the explosive root could be a realisation of a unit root with multiplicity two.
Thus, Table 5 applies the system I(2) analysis proposed by Rahbek, Kongsted and
Jørgensen (1999) and shows that a non-explosive I(2) description of the data is firmly
excluded. It should be noted though that the validity of the I(2) analysis has not
been proved so far. If the I(2) restriction is successful in making the explosive root go
away then the asymptotic theory of Rahbek, Kongsted and Jørgensen (1999) applies
directly, but for the case the explosive root remains present a new asymptotic analysis
is required following the steps of Nielsen (2005b, Theorem).
Once the rank is determined we can impose restrictions on the cointegrating vector
β∗.A homogeneity restriction, H1say, between prices and exchange rates reduces the
likelihood value slightly to 43.0 and such a restriction is therefore easily accepted when
comparing the likelihood ratio statistics of 0.5 to a χ2(1) distribution. The resulting
cointegrating vector is reported in the first line of Table 6. As the cointegrating
relation β0Xtrepresents linear combinations that are explosively growing, but without
a random walk component, it can be interpreted as the relation of nominal money,
mt,and real price, pt−st, that generates the explosive trend.
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Economics: The Open-Access, Open-Assessment E-Journal 14
3.3 Analysis of Co-explosive Properties
To investigate the influence of the explosive trend re-parametrise the model as
∆1∆ρXt=α1β∗0
1∆ρX∗
t−1+αρβ0
ρ∆1Xt−1+ψ∆1∆ρXt−1+μc+εt,
where β∗
1=β1is the cointegrating vector from before and ∆ρXt=Xt−ρXt−1with ρ
being an unknown scale parameter representing the explosive root. The matrix αρβ0
ρ
has rank dim X−1=2due to the single explosive root. Nielsen (2005b) shows that
in this model the process Xthas Granger-Johansen representation
Xt≈C1
t
X
s=1
εs+Cρ
t
X
s=1
ρt−sεs+yt+τc+τlt+τρρt,(3.2)
where ytcan be given a stationary initial distribution. The impact matrices C1,C
ρare
functions of the parameters and satisfy β0
1C1=0and β0
ρCρ=0whereas τlsatisfies
β0
1τl+δ0
1=0and the coefficients τc,τρare functions of parameters and initial values so
β0
ρτρ=0.The explosive common trend Wt=Pt
s=1 ρ−sεsconverges almost surely to
arandomvariableWas tincreases according to the Marcinkiewicz-Zygmund result,
see for instance Lai and Wei (1983).
Simple hypotheses on the co-explosive vectors βρcan be tested using χ2-inference.
The underlying asymptotic result, due to Lai and Wei (1985) and Nielsen (2005a) is
that the stationary component, the random walk and the explosive trend are asymp-
totically uncorrelated. Nielsen (2005b) then uses this to show that simple hypotheses
on the co-explosive vectors βρcan be tested using χ2-inference under the normality
assumption to the innovations, which was checked above.
The hypothesis, that βρis known and given by
Hρ:β0
ρ=µ10−1
01−1¶,
implies that each of mt−pt,m
t−stand st−ptare co-explosive relations and are
thus non-exploding random walks. Since βρis completely specified, the model can
be estimated by reduced rank regression for each value of ρ. Thisinturnresultsina
profile likelihood in ρwhich can then be maximised by a grid search. This is done by
constructing the variables ∆ρXtand β0
ρ∆1Xt−1for a given ρ. Standard software can
then be used to perform a cointegration analysis on the variable ∆ρXtwith a linear
trend restricted to the cointegrating space and a constant and β0
ρ∆1Xt−1entered as
unrestricted regressors. Searching in the region ρ>1there appears to be a unique
maximum to the likelihood function of 41.3 with ˆρ=1.174 and a slightly changed
cointegrating vector β1as given in Table 6. The test statistic for Hρagainst H1is
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3.0 which is small compared to the χ2(2) distribution. The sign to the linear trend
component in β1appears to change but remains significantly different from zero.
In summary, this analysis indicates, that mt,p
t,s
thave common unit root com-
ponents and explosive components
mt,p
t,s
t∼I(1,x).
whereas the possibility that the variables are I(2) was rejected. Thus, differencing
removes the unit root component giving pure explosive variables
∆1mt,∆1pt,∆1st∼I(x),
while the real variables are co-explosive
mt−pt,m
t−st,s
t−pt∼I(1) .
Since I(2)-ness is rejected the Yugoslavian episode gives evidence against the I(2)
assumptions of Sargent (2.7) as well as the rational expectations solution (2.9) without
bubble component, so c=ξs=0,and where mtis assumed to be I(2) as proposed
by Taylor (1991). Since mtand ptco-explode the rational expectations solution (2.9)
without bubble component and where mtis assumed explosive is also ruled out.
The bubble solution (2.9) in which c6=0and ξs6=0was discussed by Diba and
Grossman (1988) and successfully implemented in a co-explosive analysis of stock
prices and dividends by Engsted (2006). In the context of hyperinflation the bubble
solution predicts that ptis explosive, whereas mtis not. The fact that mt,p
t,s
t
have an explosive common trend and their differences co-explode is evidence against
this hypothesis. The hypothesis could also be formulated directly as the co-exploive
relation βρ=(0,1) in an analysis of the bivariate system of X(2)
t=(mt,p
t)0.It turns
out to be awkward to estimate the model under that hypothesis, essentially because
thehypothesisdoesnotfit with the data. Since the restricted model cannot be
maximised analytically and the hypothesis does not appear to be valid the estimated
restricted model is somewhat complicated. Mimicking the analysis above for the
two variables X(2)
tgives more or less the same results: cointegration rank of one,
rejection of I(2)-ness, and mt−ptis a co-exploding relation. In particular, imposing
a cointegration rank of one gives a likelihood of 10.8andanexplosiverootof1.205.
The model satisfying βρ=(0,1) can be estimated by a reduced rank regression
of ∆1∆ρX(2)
ton ∆ρX(2)
tand a time trend correcting for intercept, ∆1∆ρX(2)
t−1and
β0
ρ∆1X(2)
t−1=∆1mt−1.For ρ=1.205 the likelihood is 1.3so twice the likelihood
distance is 19.0.While this is not the likelihood ratio statistic the statistic would
quite possibly be χ2
1if the hypothesis were valid, giving another indication that the
hypothesis is not valid. Searching over ρthe maximum is found at ρ=1with
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Economics: The Open-Access, Open-Assessment E-Journal 16
likelihood 3.7.A detailed analysis of the estimated model shows that it actually has
an I(2) root, corresponding to ρ=1,as well as an explosive root in the short term
dynamics to pick up the explosiveness of mt. The likelihood ratio statistic of 14.1
should probably be judged against an I(2)-distribution, which is not tabulated in the
literature, but then the I(2) was already rejected.
In summary, this empirical analysis follows in its principles of the I(2) analyses in
the literature in that the variables are analysed in levels with a view towards estab-
lishing cointegration and if possible (polynomial) cointegration between real money,
mt−pt,and∆1pt.As in previous studies the hyper-inflation episode is not modelled
to the end due to difficulties in capturing the properties of the data. It is found
that the three variables pt,m
t,s
thave a common explosive trend and two common
random walk trends. The series co-explode so mt−pt,m
t−stand pt−stare all
non-exploding random walks. Thereby the rational expectations solution (2.9) is con-
tradicted. The conclusion that mt−pt,m
t−stand pt−stare non-exploding random
walks is, however, in line with the assumptions of Sargent (1977) and Taylor (1991).
The differenced series ∆1pt,∆1mt,∆1stare, however, explosive with no random walk
component. This indicates that linking for instance mt−ptwith ∆1ptwill not give
a balanced regression in this situation and explains why linear modelling of the vari-
ables in levels is not giving an adequate empirical model. In the following a solution
is found by abandoning the discretization assumption (2.4).
4MeasuringInflation
The assumption (2.4) of piece wise constant rate of change in prices, Ct,appears more
and more unrealistic as the inflation progresses. This is apparent from Figure 1(b)
where the line pieces connecting the points of the time series become steeper and
steeper. By discretization of the continuous rate of change in a different way this
problem can be overcome and the puzzles resolved.
As an alternative measure of the cost of holding money consider
ct=∆1Pt
Pt
=1−Pt−1
Pt
=1−exp (−∆1pt),
showing the relative loss in purchasing power over one period and the relative gain
if ctis negative. This measure can be motivated by an argument inspired by Hendry
and von Ungern-Sternberg (1981). The nominal money stock grows according to
Mt=Mt−1+δt,
where δtrepresents net money issues. Dividing through by Ptgives
Mt
Pt
=Mt−1
Pt−1µPt−1
Pt¶+δt
Pt
,
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where the coefficient ct=1−Pt−1/Ptis the proportion of the real money stock that
is lost from period to period.
The variable ctis bounded by 1 indicating that in each period one can at most loose
all money. This fits nicely with interpreting inflation as seigniorage, giving a maximal
tax rate of 100%. When the quantity ∆1pt=pt−pt−1=log(Pt/Pt−1)is small, a
Taylor expansion shows ct≈∆1pt.Once the inflation rise above about 20% per period
there will be a substantial difference between ctand ∆1pt.Note, that ct=∆1Pt/Pt
is different from the percentage change ∆1Pt/Pt−1.The measure ctis closely related
to the inflation measure ∆1pt/(1 + ∆1pt), which, however, has an asymptote for
∆1pt=−1.Such a fall was for instance experienced in the dollars/German mark
exchange rate in the second quarter of 1920.
It seems more conceivable that ce
t−ctis stationary than ∆e
1pt−∆1ptis stationary.
Likewise, agents in the economy can handle and perhaps even forecast a variable like
ctrather than ∆1pt. This is illustrated by a numerical example in which prices could
go up 10- or 20-fold. This translates into a ctof 0.9or 0.95 and a ∆1ptof log 10 = 2.3
or log 20 = 3.0. In the latter case the uncertainty is exploding with ∆1ptwhereas the
bounded nature of ctensures that the increasing uncertainty about the economy has
a bounded impact.
The proposal is therefore to use ctas a discrete time proxy for the continuous
time cost of holding money, Ct,appearing in Cagan’s model. Inspired by the setup of
Taylor, see §2.2, the testable assumptions are that mt−ptand ctare I(1) .The usual
rational expectations machinary does no longer apply since the difference variable
∆1ptis now replaced by ctwhich is not linear in ptand pt−1.It is, however, possible
that the agent’s of the economy may form somewhat accurate forecasts, ce
t+1,of the
cost of holding money that cointegrate with the actual cost, ct+1 ,as long as the
inflation runs. This leads to the following discrete time version of Cagan’s model
mt−pt=−αce
t+1 +ζt,
ce
t+1 =ct+1 +t+1,
where αcorresponds to Cagan’s continuous time semi-elasticity. Following the ma-
nipulations of Taylor this implies the equilibrium correction model
∆1ct+1 =−α−1(mt−pt+αct)+¡t+1 +α−1ζt¢.
While ∆1ptis the standard inflation measure when analysing economies without
severe inflation the choice of measure becomes increasingly important as the inflation
accelerates. As the price series ptaccelerates, ctapproaches 1 indicating a nearly
complete loss in value of money. This type of transformation is related to the non-
linear models suggested by Frenkel (1977) linking real money, mt−pt,with either
log(∆1pt)or (∆1pt)γ.These measures do, however, not approximate ∆1pteven for
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small values of inflation, so they do not fit easily with the Cagan setup. A measure
like ctappears to give a more direct measure of the cost of holding money and can
more easily be used in a linear model. Finally, the cost of holding money has the
added benefit of reducing the impact of measurement error as prices accelerate. In
the Yugoslavian case the measurement issues for the last few observations of ptcan
therefore be ignored when using ctrather than ∆1ptas inflation measure.
Figure 2: The series mt−pt,mt−st,c
t,dtand linear transformations thereof
1991 1992 1993 1994
5
6
7(a) pt−st
1991 1992 1993 1994
0.5
1.0 (b) ct
1991 1992 1993 1994
−0.3
0.0
0.3 (c) ct − dt
1991 1992 1993 1994
4
6
8
(d) mt−st
1991 1992 1993 1994
0.5
1.0 (e) dt
1991 1992 1993 1994
−2.5
0.0
2.5
(f) ecmt
1991 1992 1993 1994
0.5
1.0 (g)
c
s−m
1991 1992 1993 1994
0.0
0.5
1.0 (h)
c
p−m
0.0 0.5 1.0
4
6
8
(i) mt−st versus ct
Note: the series in (a, h)are shown only until 93:10.
The transformed variable ctas well as a depreciation rate dt=1−exp (−∆1st)are
plottedinFigure2(b, e). Real money will be measured as mt−strather than mt−pt.
This is partly due to measurement problems in prices as shown in Figure 1(c),and
partly due to a considerable currency substitution. Moreover, the exchange rate is in
effect a price index for a single ‘good’, whereas the price index ptis an average over
goods which will have very different inflation rates if there are price controls on some
of the goods. The cross-plot in Figure 2(i)shows a near linear relationship between
mt−stand ctin contrast to Figure 1(i). Concentrating on the variables mt−st,c
t,d
t
at first it is possible to set up a model for the entire period up to 1994:1. This will
be done in the following.
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5 A Linear Model for Transformed Variables
A vector autoregressive model is set up for the transformed variables mt−st,c
t,
dt.This model can be analysed using standard I(1) cointegration techniques. Here
money is chosen to be deflated by the exchange rate instead of the price index, partly
because the exchange rate measures the price of just one ‘good’, namely German
mark, and partly to avoid measurement problems for ptin the end of the sample.
5.1 Model and Rank Determination
A third order vector autoregression with a restricted constant is fitted to the data
1991:1 to 1994:1 giving a sample size of T=37−3=34.The lag length is chosen
so as to ensure that the mis-specification tests pass. As the explosive component
will now be eliminated two lags would perhaps have been preferable on grounds of
parsimony. Modelling the data right until the end of the hyper-inflationin1994:1
resolves the first puzzle set out in §2.1. While this only represents a modest gain in
degrees of freedom, the importance lies in the ability to analyse the hyper-inflation
to the end. This is where Cagan’s theory is meant to work best. Mis-specification
tests supporting the model are reported Table 7. Graphical tests, not reported here,
include recursive tests and they are likewise supportive of the model. This shows that
a well-specified joint model with time-invariant parameters can be established
Table 8 shows tests for stationarity of individual variables. No variable can be
considered stationary on its own, not even the linear combination ct−dt.
Table 7: Misspecification tests for model for transformed data
Test mt−stctdtTes t (mt−st,c
t,d
t)
χ2
normality (2) 0.1 [0.95] 1.2 [0.54] 1.9 [0.38] χ2
normality (6) 2.8 [0.83]
FAR(1)(1,23) 0.1 [0.71] 0.1 [0.70] 1.4 [0.25] FAR(1)(9,46) 0.5 [0.87]
FAR(3)(3,21) 0.8 [0.49] 1.3 [0.31] 2.1 [0.13] FAR(3) (27,38) 0.9 [0.59]
FARCH (3)(3,18) 1.4 [0.28] 0.2 [0.91] 0.2 [0.88]
p-values are given in brackets
Table 8: Test for stationarity of individual variables
mt−stctdtct−dt
32.4 32.4 30.6 21.9
The stationarity tests are fore the restriction that β=(e3,1)0where e3is a unit vector of
dimension three. All tests are asymptotically χ2
2,and therefore strongly rejected
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There is now one characteristic root at 1.035 while the remaining roots are well
inside the unit circle, see Table 9. The cointegration rank tests reported in Table 10
point to a rank of 1. Under that hypothesis the slightly explosive root is restricted to
1 and all characteristic roots, but two unit roots, are well inside the unit circle, see
Table 11. In other words the apparent explosive root in the unrestricted model is not
significantly different from one. The issue of explosiveness then disappears and the
standard cointegration analysis of Johansen (1996) is applicable with the conventional
interpretation.
Table 9: Characteristic roots of unrestricted model for transformed data
Re(z)1.035 0.61 0.61 -0.01 -0.01 -0.04 -0.33 -0.33 -0.44
Im(z)0 0.21 -0.21 0.57 -0.57 0 0.64 -0.64 0
|z|1.035 0.65 0.65 0.57 0.57 -0.04 0.72 0.72 0.44
Table 10: Cointegration rank tests for transformed data
Hypothesis H(0) H(1) H(2) H(3)
Test 60.1 [0.00] 15.5 [0.20] 4.2 [0.40]
Likelihood 80.03 102.31 107.97 110.06
p-values are given in brackets
Table 11: Characteristic roots when the rank is restricted to one
Re(z)110.810.09-0.14-0.14-0.17-0.40-0.40
Im(z)0 0 0 0 0.68 -0.68 0 0.54 -0.54
|z|1 1 0.81 0.09 0.70 0.70 0.17 0.67 0.67
5.2 The Cointegrating Vector
The cointegrating relation estimated from the Johansen approach is given by
ecmt
(√LR)
=1
(2.8) (mt−st)+13.5
(5.1) ct−10.3
(5.7) dt−8.48
(−2.7) (5.1)
=1
(2.8) (mt−st)+3.26
(2.0) ct−10.3
(5.7) (dt−ct)−8.48
(−2.7) (5.2)
=1
(2.8) (mt−st)+3.26
(2.0) dt−13.5
(5.1) (dt−ct)−8.48
(−2.7) (5.3)
The signed log-likelihood ratio test statistics for individual exclusion restrictions are
reported in brackets and are asymptotically standard normal distributed, so one-
sided tests 5% level tests would have a critical value of about plus or minus 1.65.
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This cointegrating vector shows that real money, deflated by exchange rates, moves
both with ctand dt. It is formulated in three equivalent ways involving two of the
three variables ct,dtand dt−ct.By construction the coefficients to ctand dtin (5.2)
and (5.3) are identical and are interpreted as the semi-elasticity for the expected
future cost of holding money as discussed in §4. Exclusion of the differential dt−ct
is strongly rejected whereas the decision to keep ctor dtis marginal. In order not
to distort the subsequent analysis by making a marginal decision no restrictions are
made on the cointegrating relation.
The cointegrating equation is approximately of the same form as Cagan’s with real
money stock measured in foreign currency falling with depreciation rate dt.Indeed,
ignoring the significant component dt−ctin (5.3) and replacing dtby ∆1stthe relation
(5.3) appears close to the relation mt−st=−3.4∆1st+8.4found by Petrovi´
cand
Mladenovi´
c (2000, Table 2) in an analysis until 1993:6. A similar analysis using dt
gives mt−st=−6.0∆1st+8.9with the difference stemming from the discrepancy
between dtand ∆1stculminating at ∆1s93:6 =1.22 and d93:6 =0.70 in this sub-sample.
The term dt−ctcan be interpreted as the real appreciation rate of the German
mark. It enters positively so that if the German mark appreciates faster than prices
rise goods become relative cheaper and the real money circulation rises. This is
a variation of the combination of transactions and portfolio demand discussed by
Ando and Shell (1975), Goldfeld and Sichel (1990), Baba, Hendry, and Starr (1992).
Comparing the Figures 2(c, d)shows how the sign of ct−dtvaries over time so mt−st
tends to increase when ct−dtis negative. The cointegrating relation itself, normalised
on real money is plotted in Figure 2(f).
5.3 The Inflation Tax
Ignoring the differential of the cost of holding money and the depreciation, Cagan’s
semi-elasticity αcan be estimated by bα=3.26.ThisvalueisinlinewithbothCagan’s
and Sargent’s estimates for the German hyper-inflation. According to Cagan the
maximal revenue from seigniorage, assuming money rises at a constant rate, is then
estimated by exp(bα−1)−1=36%. It seems natural to compare this with the average
cost of holding money for a month, ct=∆1Pt/Pt,rather than the average of inflation
measure through ∆1Pt/Pt−1since the former is precisely a measure for how much
value is lost over a month. For the full sample this average is 42.6%. The likelihood
ratio test statistic for the hypothesis that the coefficient to ctis {log(1 + 0.426)}−1
is 0.43 [p=0.51]. Likewise the average of dtis 44.9%. The test statistic for the
coefficient to dtbeing {log(1 + 0.449)}−1is 0.58 [p=0.45]. While the assumption
underlying Cagan’s theory of money rising at a constant rate is violated and the
idea of taking average over time of a trending variable is somewhat contrived, the
predictions of his theory are not rejected this way.
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Table12: Theadjustmentvectorˆαfor the transformed model
mt−stctdtct−dt
0.31
(4.6) −0.092
(−5.3) −0.065
(−1.7) −0.028
(−0.7)
Signed likelihood ratio statistic, √LR, for insignificance is given in brackets
5.4 Weak Exogeneity Properties
Having the cointegrating relation in place, the short term dynamics of the system
can be analysed in order to understand how the variables adjust. The notion of weak
exogeneity introduced by Engle, Hendry and Richard (1983) is helpful and can be
implemented in the cointegration analysis by restricting the adjustment vector α,see
Johansen (1996, §8). After exploration of weak exogeneity properties the approach of
Hendry (1995, §16.8) is followed in obtaining parsimonious vector autoregressions by
simultaneous equation methods using the estimated cointegrating relation as regres-
sor. This will go a step towards uncovering the causality structure.
An advantage of Johansen’s method for cointegration analysis is its invariance to
linear transformations of the variables, hence it is equivalent to consider the variable
vectors (mt−st,c
t,d
t)and (mt−st,c
t,c
t−dt).Table 12 reports the four different
adjustment coefficients related to this model. While it is rejected that real money,
mt−st,or the cost of holding money, ct,could be weakly exogeneous, there is a
marginal indication that the depreciation rate, dt,could be weakly exogenous, and
stronger evidence that the real depreciation rate, ct−dtcould be weakly exogenous.
In the following weak exogeneity is imposed for ct−dt. This has the interpretation
that the fluctuations in the foreign exchange rate, ct−dt,are exogenous to the demand
for money.
In the sub-sequent analysis weak exogeneity of ct−dtis imposed in the context of
amodelfor(mt−st,d
t,c
t−dt).In this way the endogenous variables mt−stand dt
are balanced in that dtis the cost, in terms of the depreciation rate, of holding money
deflated by the exchange rate. When weak exogeneity is imposed the cointegrating
vector (5.3) changes slightly to
ecmd
t=mt−st+3.22dt−13.5(dt−ct)−8.50.
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Including this as a regressor the conditional system can be reduced to
∆1dt=−0.087
(0.009)ecmt−1+0.12
(0.04)
∆1(m−s)t−1+0.17
(0.03)
∆1(m−s)t−2
−0.89
(0.05)
∆2
1(c−d)t−0.27
(0.03)
∆2
1(c−d)t−1+0.044ˆεt,(5.4)
∆1(m−s)t=+0.29
(0.04)ecmt−1−0.64
(0.16)
∆1(m−s)t−1+0.74
(0.22)
∆1(c−d)t
−1.22
(0.28)
∆1(c−d)t−1−0.90
(0.21)
∆1(c−d)t−2+0.177ˆεt,(5.5)
where the over-all likelihood ratio test statistic is 6.8 [p=0.34]comparedtoaχ2(6)-
distribution. The marginal model for (ct−dt)can likewise be reduced to
∆1(c−d)t=−0.47
(0.14)
∆1(c−d)t−1−0.42
(0.14)
∆1(c−d)t−2−0.25
(0.08)
∆1(m−s)t−1+0.139ˆεt,
(5.6)
where the likelihood ratio for the reduction is 1.3 [p=0.74]comparedtoaχ2(3)-
distribution. The weak exogeneity of ct−dtfits with the combined transactions and
portfolio demand interpretation of the cointegrating vector discussed in §5.2 with the
real depreciation rate dt−ctbeing a driving force for inflation.
The empirical model indicates that the (weakly) endogenous variables, real money
and the cost of holding money, are determined simultaneously. This suggests a more
complicated relationship than in single cause models like Sargent’s model where infla-
tion causes money and models where money causes inflation. Moreover, the equation
for the exogenous variable ct−dtshows an ongoing feedback from the changes in real
money into the foreign exchange market, which is not unreasonable. The residual
standard error in the equation for mt−stis 0.18 compared to 0.27 in the simple
time series model in §2.4 that form the basis for Sargent’s model. It is interesting to
note that due to the new measures ctand dtof the cost of holding money and the
depreciation rate the emphasis in this model is on real money, whereas in Sargent’s
model the role of nominal and real money is more interchangeable.
A similar analysis could also be carried out with mt−ptinstead of mt−stas
measure for real money, were it not for the measurement errors of ptin the end of the
sample and an attempted prize freeze in July 1990. Even when taking these issues
into account the cointegration analysis is less clear. This point can be illustrated
graphically. In Figure 2(g, h),thenegativeofthetherealmoneyvariables,st−mt
and pt−mt, respectively, are plotted with ctwith ranges and means adjusted to
the latter. It is clear that st−mtfollows ctnicely with discrepancies matched by
dt−ctof Figure 2(c)as in the analysis above while pt−mtdoes not track ctwell.
Further research would be needed to see whether this is a feature particular to the
Yugoslavian case, or whether the relative ease of measuring exchange rates rather
than prices makes mt−sta better measure for real money in hyper-inflations. The
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issue at hand could be that mt−stinvolves the price of a single "good", whereas
mt−ptinvolves the price index, which is constructed by averaging over goods which
can have very different inflation rates in a hyper-inflation.
6 Discussion
The discretization assumption (2.4) and the use of ∆1ptas the cost of holding money
have been identified as the main sources for the puzzles in the empirical analysis
of hyper-inflations. Since mt−pthas random walk-like behaviour while ∆1pthas
explosive behaviour regressions of mt−pton ∆1ptwill be unbalanced. The proposed
solution is straightforward in replacing ∆1ptby the cost of holding money, ct.This
variable has desirable statistical properties in that it is bounded and it has random
walk-like behaviour. Its interpretation is simple and similar to that of the rate of
change in prices appearing in Cagan’s continuous time model.
The rational expectations model of the kind discussed by Taylor (1991) and
analysed in some detail by Engsted (1993) does not appear to match these data.
That model essentially expresses ptas a function of mt, so that the properties of pt
are derived from whatever properties mtis thought to have. The appendix analyses
four different assumptions for mtwith and without bubbles, and none of these exam-
ples appear to be supported by the data. Thus, the Yugoslavian hyper-inflation does
not appear to give evidence in favour of rational expectations model.
With the new inflation measure various lines of future research are opened up.
First, a comparative analysis of hyper-inflationepisodesindifferent countries using
the cost of holding money as inflation measure can provide new insights, notably for
the classic episodes studied by Cagan. Secondly, Cagan’s assertion that variables like
productivity and wages are irrelevant in hyper-inflation can be reviewed as done in
the work by Michael, Nobay and Peel (1994) and Juselius and Mladenovi´
c (2002).
Thirdly, on the structural side it would be interesting to reconsider the structural
models in literature, recognising that it does not appear valid to exploit that the cost
of holding money is linear in ptand pt−1.
Appendix: Explosive Bubbles
The rational expectations model (2.8), that is
mt−pt=−αEt(pt+1 −pt)+ζt,(A.1)
allows expressing prices, ptin terms of expectation to future money stocks, ˜mt=
mt−ζt.In the following it is discussed how properties assumed for ˜mttransmits into
properties pt.
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The solution to (A.1) was given in (2.9) as
pt=1
1+α
∞
X
j=0 µα
1+α¶j
Et(˜mt+j)+cµ1+α
α¶t
+
t
X
s=1 µ1+α
α¶t−s
ξs.(A.2)
This is seen to be a solution by deriving Etpt+1 from (A.2) and inserting in (A.1), see
also Turnovsky (2000, p. 86f & 91f) and Diba and Grossmann (1988). The solution
is well-defined when α>0and |{α/(1 + α)}jEt(˜mt+j)|vanishes at a geometric rate.
It is convenient to derive equations for pt−˜mtand ˜mt−pt+α∆1pt+1.First,
subtracting ˜mton both sides of (2.9) and using that P∞
j=0{α/(1 + α)}j=1+αit
follows that
pt−˜mt=1
1+α
∞
X
j=0 µα
1+α¶j
et,j +cµ1+α
α¶t
+
t
X
s=1 µ1+α
α¶t−s
ξs,(A.3)
where et,j =Et(˜mt+j−˜mt),whereas differencing gives
α∆1pt+1 =∞
X
j=0 µα
1+α¶j+1
ft,j +cµ1+α
α¶t
+
t
X
s=1 µ1+α
α¶t−s
ξs+αξt+1 .(A.4)
where ft,j =Et+1 (˜mt+1+j)−Et(˜mt+j).This implies the observable version of (A.1) is
˜mt−pt+α∆1pt+1 =1
1+α
∞
X
j=0 µα
1+α¶j
(αft,j −et,j )+αξt+1.(A.5)
To describe the properties of ptsome assumptions must be made to ˜mt,c, and ξs.It
is interesting to consider a few examples.
Example 1. An I(1) model without bubble. Assume that ∆1˜mt=ηtwith Etηt+j=
0,andthatc=ξs=0.The equation for ˜mtimplies
˜mt+j=
j
X
u=1
ηt+u+˜mt,
so Et(˜mt+j)= ˜mtimplying et,j =0and ft,j =∆1˜mt+1.Then (A.2) shows
pt=1
1+α
∞
X
j=0 µα
1+α¶j
˜mt=˜mt
since P∞
j=0{α/(1 + α)}j=1+αso pt∼I(1) .In the same way (A.3) and (A.5) show
that ˜mt−pt,˜mt−pt+α∆1pt+1 ∼I(0)
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Example 2. An I(2) model without bubble as in model (2.12), originally suggested
by Taylor (1991). Assume, for simplicity that ∆2
1˜mt=ηtwith Etηt+j=0,andthat
c=0and ξs=0.The equation for ˜mtimplies
˜mt+j=
j
X
v=1
v
X
u=1
ηt+u+˜mt+j∆1˜mt,
so Et(˜mt+j)= ˜mt+j∆1˜mtimplying et,j =j∆1˜mtand ft,j =∆1˜mt+1 +j∆2
1˜mt+1.Then
(A.2) shows that pt∼I(2),(A.3)showsthat ˜mt−pt∼I(1) so ∆1˜mt−∆1pt∼I(0),
whereas (A.5) shows ˜mt−pt+α∆1pt+1 ∼I(0) since P∞
j=0{α/(1 + α)}j(α−j)=0.
Combining these results shows that also ˜mt−pt+α∆1mt+1 ∼I(0) as pointed out by
Engsted (1993).
Example 3. An I(1) model with bubble. Assume that ∆1˜mt=ηtwith Etηt+j=0,
and that c6=0and ξs6=0.Then ptand ˜mt−ptand ˜mt−pt+α∆1pt+1 have the same
number of unit roots as in Example 1 and ptand ˜mt−ptand ˜mt−pt+α∆1ptalso
have explosive components, whereas ˜mt−pt+α∆1pt+1 has no explosive component,
see also Engsted (1993, p. 354).
Example 4. An I(1) model with explosive roots, that is I(1,x)say, but no bubble.
Assume ∆1∆ρ˜mt=ηtwhere 1<ρ<(1 + α)/α with Etηt+j=0,andthatc=ξs=0.
Then
˜mt+j=1
1−ρ
j
X
u=1
ηt+u+1
ρ−1ρj
j
X
u=1
ρ−uηt+u+˜mt+∆1˜mtρ1−ρj
1−ρ,
so (1 −ρ)Et(˜mt+j)=∆ρ˜mt−ρj+1∆1˜mtimplying (1 −ρ)et,j =∆1˜mtρ(1 −ρj)and
(1−ρ)ft,j =(1−ρj+1)∆1∆ρ˜mt+1 +(1−ρ)ρj+1∆1˜mt.Then pt∼I(1,x),˜mt−pt∼I(x),
and ˜mt−pt+α∆1pt+1 ∼I(x)since P∞
j=0{α/(1 + α)}j{α(1 −ρ)ρj+1 −ρ(1 −ρj)}=
(1+ α)(1 −ρ)6=0.So the variables have the same number of unit roots as in Example
1 combined with an explosive component.
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