A MULTIVARIATE I(2) COINTEGRATION ANALYSIS OF GERMAN HYPERINFLATION*
by
Dimitris A. Georgoutsos
Department of International and European Economic Studies
Athens University of Economics and Business
GR-10434
Athens, Greece
and
Georgios P. Kouretas (corresponding author)
Department of Economics
University of Crete
University Campus
GR-74100
Rethymno, Greece
Email: kouretas@econ.soc.uoc.gr
July 2001
Abstract
This paper re-examines the Cagan model of German hyperinflation during the 1920s under
the twin hypotheses that the system contains variables that are I(2) and that a linear trend is
required in the cointegrating relations. Using the recently developed I(2) cointegration
analysis developed by Johansen (1992, 1995, 1997) extended by Paruolo (1996) and
Rahbek et al. (1999) we find that the linear trend hypothesis is rejected for the sample.
However, we provide conclusive evidence that money supply and the price level have a
common I(2) component. Then, the validity of Cagan’s model is tested via a transformation of
the I(2) to an I(1) model between real money balances and money growth or inflation. This
transformation is not imposed on the data but it is shown to satisfy the statistical property of
polynomial cointegration. Evidence is obtained in favor of cointegration between the two sets
of variables which is however weakened by the sample dependence of the trace test that the
application of the recursive stability tests for cointegrated VAR models show.
Key Words: I(2) analysis, hyperinflation, cointegration, identification, temporal stability
JEL Classification numbers: C12, C32, F31, F33
* Part of this paper was written while the second author was Visiting Fellow at
the European University Institute. The hospitality of the Department of Economics and the
Robert Schuman Centre is gratefully acknowledged. The second author also wishes to
acknowledge generous financial support provided by EUI. This paper has benefited from
discussions on the issue of I(2) analysis made in a workshop held in Bertinoro, Italy 27-29
January 2000 and in the conference on “Bridging Economics and Econometrics: Empirical
Applications and Econometric Methods, European University Institute, Florence, Italy 6-9
June 2001. We would also like to thank without implicating Michael Artis, Richard Baillie,
Anendyia Banerjee, Francis Diebold, Tom Engsted, Neils Haldrup, Soren Johansen, Katarina
Juselius and Bent Nielsen for many helpful comments and discussions.
1. Introduction
Cagan’s (1956) model on the demand for money under hyperinflation is by far the
most widely used specification in studies of inflation dynamics and the money supply process
and on issues of inflationary finance. Following this pioneering work, Sargent and Wallace
(1973) and Sargent (1977) looked at the implications of letting Cagan’s adaptive expectations
scheme be rational and Salemi and Sargent (1979) tested the cross-equation parameter
restrictions that the rational expectation imposes on a bivariate VAR model for inflation and
money growth. Flood and Garber (1980), Hamilton and Whiteman (1985) and Casella (1989)
used the Cagan model to investigate the existence of rational speculative bubbles, i.e.
situations where the price level is partly driven by self-fulfilling expectations, independently of
market fundamentals. Finally, Frenkel (1975, 1976) examined whether the authorities had
expanded the money supply at much too high a rate to maximize the inflation tax revenue.
The major assumption in the works mentioned above is that the demand
disturbances or shocks to velocity follow a random walk. This assumption implies that the
deviations from the Cagan model have an infinite population variance, which substantially
reduces the empirical content of the model from the outset (Taylor, 1991). Salemi and
Sargent (1979) study some of the classic European hyperinflation episodes and they
conclude that we can not reject the restrictions imposed by the Cagan model under rational
expectations and a random walk error term. By contrast Goodfriend (1982) finds evidence
supporting the Cagan model under rational expectations and no velocity shocks.
A main insight of Cagan’s analysis is that under the conditions of hyperinflation,
movements in prices are of a magnitude so much greater than movements in real
macroeconomic aggregates that “relations between monetary factors can be studied,
therefore, in what almost amounts to complete isolation from the real sector of the economy”
(Cagan 1956, p.25). Taylor (1991), Engsted (1993, 1994) and Michael, Nobay and Peel
(1994) suggest how this insight can be characterized formally in terms of the time properties
of the data. Thus, they argued that the assumption that money demand disturbances follow a
random walk could be tested explicitly using cointegration techniques, since the random walk
assumption implies that real balances and inflation should not cointegrate. Therefore, when
money and prices are integrated of order two, I(2), and shocks to money demand or velocity
1
are stationary, then the Cagan (1956) monetary model of hyperinflation has the implication
that real money balances cointegrate, in the sense of Engle and Granger (1987), with the rate
of inflation. In addition, Engsted (1993, 1994) shows that given that velocity shocks are
stationary, the Cagan model under rational expectations and no bubbles implies an additional
cointegrating relationship between real money balances and money growth1.
In the present paper we provide a re-examination of the empirical evidence on
Cagan’s model under rational expectations for the case of the German hyperinflationary
period of the early 1920s, by applying recent contributions to the econometrics of nonstationarities. Several novel features are included in the paper. The first feature concerns the
order of integration of the variables. We employ the recently developed testing methodology
suggested by Johansen (1992, 1995, 1997) and extended by Paruolo (1996) and Rahbek et.
al. (1999) which allows us to reveal the existence of I(2) and I(1) components in a multivariate
context. We depart from previous studies since we show that the cointegrating relationship
between real money balances and inflation can be obtained through a testing procedure
applied on a bivariate vector autoregressive, VAR, model of the monetary aggregate and the
price level. So it could be claimed that our analysis is “data driven” instead of allowing
theoretical results to determine a priori the empirical investigation. The property of
cointegrated VAR models with I(2) variables where a stationary relationship is derived from a
linear combination of the levels of the variables and their first differences is known in the
literature as polynomial cointegration or multicointegration, e.g. Granger and Lee (1990).
Second, given that at least one statistically significant cointegrating vector has been found we
examine the stability of the long-run relationships through time. Hansen and Johansen (1993,
1999) propose three tests for parameter stability in cointegrated-VAR systems that allow us to
provide evidence for the sample independence of the cointegration rank as well as of
parameter stability. Finally, following Engsted (1993, 1994) we test the Cagan specification
under rational expectations and no velocity shocks by exploiting the interesting crossequation parameter restrictions that those properties imply.
The main findings of the paper are: First the monetary aggregate and the price level
are I(2) variables and that they have a common I(2) component. Furthermore we show that a
stationary multicointegrating relationship is obtained by a linear combination of the real
2
money balances and inflation (or money growth). Second, the stability tests indicate that this
relationship is established only when the last observations, of the “true” hyperinflation period,
are included in the sample. Third, the exact rational expectations Cagan model is rejected,
which provides further doubts that the velocity shocks were negligible.
The organization of this paper is as follows. Section 2 presents the Cagan model of
hyperinflation. Section 3 discusses the cointegration methodology applied in this analysis.
Section 4 reports our empirical results. Finally, section 5 presents our concluding remarks.
2. The Cagan model of hyperinflation
In this section we present the Cagan model following Engsted (1993, 1994)2. Thus
the Cagan model under money market clearing and rational expectations is given as
mt − pt = α − β [E t pt +1 − pt ] + ut
(1)
where mt and pt are natural logarithms of the money stock and the price level, respectively,
and
α
and
β
are parameters to be estimated. E t is the expectations operator conditional
on information at time t which includes at least lagged values of pt and mt and ut is a
stochastic variable representing velocity and/or demand shocks.
If the transversality condition
lim i →∞ [ β (1 + β ) −1 Et pt +i ] = 0 holds, then by
employing the forward solution of equation (1) it can be shown that
∞
∞
mt − pt = α − ∑ b E t ∆mt + i + (1 − b) ∑ bi E t ut + i
i
i =1
where b =
(2)
i =0
β (1 + β ) −1 . The transversality condition rules out rational explosive bubbles, i.e.
situations where the price level is driven by self-fulfilling expectations, independent of the
evolution of the money supply (see e.g. Diba and Grossman, 1988) 3. If mt − pt rises, it is an
indication of expectations of future decline in money growth, which leads to lower future
inflation and therefore higher demand for real money balances today. According to eq. (2),
the level of real balances is a predictor of future money growth and/or velocity shocks. In
case of no velocity shocks, the level of balances is the optimal predictor of the money growth
discounted by the elasticity parameter b. This is a general implication of the present value
models. In the permanent income model of consumption, for example, savings predict future
3
labor income changes (Campbell, 1987), and in the expectations theory of the term structure
the spread between long and short interest rates predicts future short interest rate changes
(Campbell and Shiller, 1987).
Given that real balances and money growth need first differencing to become
stationary, it will be useful to reparameterize (2) into
∞
∞
i =1
i=0
( mt − pt ) + β∆mt − α = −(1 − b) −1 ∑ bi E t ∆2 mt + i + (1 − b) ∑ bi E t ut + i
(3)
Expression (3) shows that, if mt and pt are both integrated of order 2, I(2), and the velocity
shock, ut , is stationary, then the Cagan model under rational expectations and no bubbles
has the testable implication that real money balances, mt − pt , cointegrate in the sense of
Engle and Granger (1987), with the growth rate of money, ∆mt . By contrast, equation (1)
implies that stationarity of the velocity shocks is needed for real money balances to
cointegrate with inflation, ∆pt +1 .
Campbell and Shiller (1987) have constructed an appealing method to test the
implications of present value models when the underlying time series are I(1) processes. This
can be done by first estimating
β
in a cointegrating regression between ( mt − pt ) and ∆mt
or ∆pt +1 , and then setting up a bivariate VAR model for the two stationary variables ∆ mt
2
and S t = ( mt − pt ) +
β∆mt ( pt +1 ) .
The overidentifying restrictions implied by rational
expectations and no bubbles, which the Cagan model imposes on the parameter of this VAR
system, can then be tested formally or informally. The VAR under consideration is as follows:
∆2 mt α ( L ) b(L ) ∆2 mt −1 e1t
=
+
S t c(L ) d(L ) S t −1 e2t
(4)
where a( L ), b( L ), c(L ) and d(L ) are lag polynomials of order p . Within the context provided
by Cagan’s model, the idea is to generate the unrestricted VAR forecast of the present value
*
of future changes in the money growth rate or the expected inflation, which will be called S t .
We can re-write the VAR model in first-order companion form as
4
Z t = A Z t −1 + et ,
where Z t = [ ∆mt ,....., ∆ mt − p +1 , S t ,.....S t − p +1 ] and A is the companion matrix of VAR
2
parameters. It can then be shown that this forecast equals to:
S = −(1 − b)
*
t
−1
∞
∑b E ∆ m
i
2
t
i =1
t +i
∞
= −(1 − b) −1 g ∑ b i A i Z t
(5)
i =1
= − βgA(1 − bA) −1 Z t
where g is a (1x2 p) vector that picks out ∆ mt from the VAR model and is assumed that
2
u t = 0 , for all t.
Setting (5) equal to (3) the cross-equation parameter restrictions implied by the rational
expectations Cagan model with no velocity shocks, the case investigated by Goodfriend
(1982) who argued that velocity shocks were negligible during the German hyperinflation, are
ci = (1 + β )ai , i = 1,...... p
d1 = (1 + β )( β
−1
+ b1 )
(6)
di = (1 + β )bi , i = 2,....... p
These restrictions can be tested using a Wald or Likelihood ratio test (Johansen and
Swensen, 1999). Equivalently the hypothesis can be tested by imposing the restrictions (5)
on
the
VAR
model
(4).
A
simple
transformation
gives
the
variable
X t = S t − (1 + β ) β −1S t −1 − (1 + β ) ∆2 mt which is shown to be uncorrelated with
information at time t-1 when velocity shocks are negligible. Next, by regressing X t on lagged
∆2 mt and S t we can test for the statistical significance of those variables. As Engsted (1993,
1994) argues, a problem with formal testing of these restrictions is that a statistical rejection
of the restrictions is difficult to interpret economically. However the difference between S t and
S t* is easily shown to measure the noise in the model so that by plotting S t together with S t*
in a diagram one can get an informative picture of the Cagan model ability’s to explain the
data. Campell and Shiller (1987) argue that when deviations from the exact linear rational
*
expectations model are transitory, S t and S t
5
will be highly positively correlated.
Simultaneous statistical rejection of the parameter restrictions (6) and a close comovement of
S t and S t* therefore implies the presence of non-negligible but stationary velocity shocks. On
*
the other hand, if S t and S t do not exhibit a high correlation this is indication that the velocity
shock follows a random walk.
In much of the previous rational expectations literature on hyperinflation (e.g.
Burmeister and Wall, 1982, Flood et al. 1984), money is assumed to be exogenous, in the
sense that no feedback from prices to money is allowed. An important feature of the
cointegrated VAR model described above is that such feedback is allowed ( S t Granger
causes ∆ mt ). The intuitive explanation for this is that, if agents use information besides
2
current and lagged money growth to forecast future money growth, then, according to the
exact Cagan model under rational expectations, S t summarizes this additional information.
To summarize, if velocity shocks are either negligible or stationary, (1) shows that the real
balances cointegrate with the inflation rate. This holds regardless of the presence of bubbles
in pt . In contrast, real balances only cointegrate with money growth if the no bubble
transversality condition holds. Therefore, if it is found that real balances cointegrate with both
inflation and money growth, it in effect precludes bubbles. On the other hand, if real balances
cointegrate with inflation, but not with money growth, it could be due to the presence of
bubbles.
3. Econometric methodology
Our cointegration analysis is based on the Full Information Maximum Likelihood
(FIML) multivariate cointegration technique developed by Johansen (1988, 1991), extended
by Johansen (1992, 1992, 1997) and Paruolo (1996) and Rahbek et al. (1999) to incorporate
the analysis of I(2) variables.
Consider a p-dimensional vector autoregressive model which in error correction form
is given by
k −1
∆z t = ∑ Γi ∆z t − i + Πz t −1 +γDt + µ + εt ,
i =1
6
t=1,...,T
(7)
[
]
where z = m, p , z k +1 ,...., z 0 are fixed and
t
ε t ~Niidp (0,Σ). The adjustment of the variables
to the values implied by the steady state relationship is not immediate due to a number of
reasons like imperfect information or costly arbitrage. Therefore, the correct specification of
the dynamic structure of the model, as expressed by the parameters (Γ1,.........,Γk−1,γ ) , is
important in order that the equilibrium relationship be revealed. The matrix Π = αβ ' defines
β
the cointegrating relationships,
, and the rate of adjustment,α , of the endogenous
variables to their steady state values. D t is a vector of non-stochastic variables, such as
centered seasonal dummies which sum to zero over a full year by construction and are
necessary to account for short-run effects which could otherwise violate the Gaussian
assumption, and/or intervention dummies; µ = µ0 + µ1 t account for the constant and the
deterministic trend (Rahbek et al. 1999) and T is the sample size.
If we allow the parameters of the model
θ = (Γ1,......,Γk−1,Π,γ , µ ,Σ)
to vary
unrestrictedly then model (7) corresponds to the I(0) model. The I(1) and I(2) models are
obtained if certain restrictions are satisfied. Thus, the higher-order models are nested within
the more general I(0) model.
It has been shown (Johansen, 1991) that if z t ~ I (1) , then that matrix Π has
reduced rank r < p, and there exist pxr matrices
Furthermore, Ψ = α ⊥ (Γ) β ⊥
'
α
β
and
k −1
has full rank, where Γ = Ι − ∑ Γi
i =1
such that Π = αβ ' .
and a⊥ and
β⊥
are
px( p − r) matrices orthogonal to α and β , respectively.
Following this parameterization, there are r linearly-independent stationary relations
given by the cointegrating vectors
β
and p − r linearly-independent non-stationary relations.
These last relations define the common stochastic trends of the system and the MA
representation shows how they contribute to the various variables. By contrast the AR
representation of model (7) is useful for the analysis of the long-run relations of the data.
The I(2) model is defined by the first reduced rank condition of the I(1) model and
that Ψ = α ⊥ Γβ ⊥ = ϕη is of reduced rank s1 , where
'
'
s1 < ( p − r) .
7
ϕ
and
η
are ( p − r) xs1 matrices and
Under these conditions we may re-write (7) as
∆2 z = Πz
t
t −1
− Γ∆z
k −2
+ ∑ Ψi ∆2 z t −1 +γDt + µ + εt
t −1
(8)
i =1
k −1
where Ψ = − ∑ Γi ,
i
j = i +1
i =1,...,k − 2
Johansen (1997) shows that the space spanned by the vector z t can be decomposed
into r stationary directions,
the
β
(β ⊥1 , β ⊥2 ),
directions
β ⊥2 = β ⊥ ( β ⊥' β ⊥ ) −1η ⊥
, and p − r nonstationary directions,
where
β ⊥1 = β ⊥η
is
of
β ⊥,
and the latter into
dimension
pxs1
and
is of dimension pxs2 and s1 + s2 = p − r . The properties of the
process are described by:
I (2):{ β ⊥2' z t },
I (1):{ β ' z t },{ β ⊥1' z t },
I (0):{ β ⊥1' ∆zt },{ β ⊥2' ∆2zt },{ β 'zt + ω '∆zt }
where
ω
is a pxr matrix of weights, designed to pick out the I(2) components of z t
(Johansen, 1992, 1995). Thus, we have that the cointegrating vectors
β ' zt
are actually I(1)
and require a linear combination of the differenced process ∆z t to achieve stationarity.
Johansen (1991) shows how the model can be written in moving average form, while
Johansen (1997) derives the FIML solution to the estimation problem for the I(2) model.
Furthermore, Johansen (1995) provides an asymptotically equivalent two-step procedure
which computationally is simpler (Paruolo, 2000). It applies the standard eigenvalue
procedure derived for the I(1) model twice, first to estimate the reduced rank of the Π matrix,
and then for given estimates of
α
and
β
^'
, to estimate the reduced rank of
^
α ⊥Γβ
⊥,
(Juselius, 1994, 1995, 1998). In both steps a likelihood ratio test for the associated rank of
either Π or
^'
^
α ⊥Γβ
⊥
are provided. The sum of the two likelihood ratio tests for all possible
values of r forms the basis of the testing procedure. Paruolo (1996) and Rahbek et. al. (1999)
8
have extended
model (7) to allow for linear deterministic trends. Both of them apply
restrictions so that quadratic trends are excluded in the solution for zt, the difference however
between them is that in the specification followed by Paruolo (1996) deterministic trends in
the cointegrating vector are not allowed while in the Rahbek et al. (1999) study this is
feasible. Another interesting result of those two studies is that the joint test for I(1) and I(2)
cointegrating ranks is asymptotically similar with respect to the drift terms which implies that it
is not necessary to determine the rank together with the trend specification.
An equally important issue, along with the existence of at least one cointegration
vector, is the issue of the stability of such a relationship through time as well as the stability of
the estimated coefficients of such a relationship. Thus, Septhon and Larsen (1991) have
shown that Johansen’s test may be characterized by sample dependency. Hansen and
Johansen (1993,1999) have suggested methods for the evaluation of parameter constancy in
cointegrated VAR models, formally using estimates obtained from the Johansen FIML
technique. Three tests have been constructed under the two VAR representations. In the “Zrepresentation” all the parameters of model (7) are re-estimated during the recursions while
under the “R-representation” the short-run parameters Γi = 1,..., k -1 are fixed to their full
sample values and only the long-run parameters α and
β
are re-estimated.
The first test is called the Rank test and we examine the null hypothesis of sample
independency of the cointegration rank of the system. This is accomplished by first estimating
the model over the full sample, and the residuals corresponding to each recursive subsample
are used to form the standard sample moments associated with Johansen’s reduced rank.
The eigenvalue problem is then solved directly from these subsample moment matrices. The
obtained sequence of trace statistics is scaled by the corresponding critical values, and we
accept the null hypothesis that the chosen rank is maintained regardless of the subperiod for
which it has been estimated if it takes values greater than one.
A second test deals with the null hypothesis of constancy of the cointegration space
for a given cointegration rank. Hansen and Johansen (1993, 1999) propose a likelihood ratio
test that is constructed by comparing the likelihood function from each recursive subsample
to the likelihood function computed under the restriction that the cointegrating vector
estimated from the full sample falls within the space spanned by the estimated vectors of
9
each individual sample. The test statistic is a
χ2
distributed with ( p − r) r degrees of
freedom.
The third test examines the constancy of the individual elements of the cointegrating
vectors β and the loadings
α . However, when the cointegration rank is greater than one, the
elements of those vectors can not be identified, except under restrictions. Fortunately, one
can exploit the fact that there is a unique relationship between the eigenvalues and the
cointegrating vectors. Therefore, when the cointegrating vectors or the loadings have
undergone a structural change this will be reflected in the estimated eigenvalues. Hansen
and Johansen (1993, 1999) have derived the asymptotic distribution of the estimated
eigenvalues.
4. Empirical results for the German hyperinflation
The data used in our analysis are monthly observations on prices and money from
the German hyperinflation episode, 1920-23 and are depicted on diagrams 1 and 2. The
wholesale price index (1913-14 = 100) is a monthly average reported in International
Economic Statistics 1919-1930 (1934, pp. 82-84, International Conference of Economic
Services, London), and is the same series used to construct the inflation series reported in
Cagan (1956). The time series on nominal money is taken directly from the appendix in Flood
and Garber (1980). The sample runs from January 1920 to June 1923, giving a total of fortytwo observations. This is the same sample period used by Casella (1989) and Engsted
(1993).
4.1 Determination of the cointegration rank and the order of integration
The first step in the analysis is the determination of the order of integration of German
money and prices. Since the data employed in our study have been subjected to careful
scrutiny before we present the main findings of previous studies. Taylor (1991) and Engsted
(1993), among others, have tested real balances, money growth and inflation for being non
stationary stochastic processes and they concluded that all variables appear to be I(1)
processes. The problem however with the sequential use of the Dickey – Fuller test statistic,
10
for the identification of the number of unit roots, is that it has a low power against the
explosive alternative which have properties mimicking those of the I(2) processes (Haldrup,
1998). It has been suggested that the Hazda and Fuller (1979) I(2)-test statistic is more
appropriate in those cases since it tests jointly for double unit roots by applying a two sided
test where the alternative hypothesis is quite general as it covers situations where z t is either
explosive, I(0) or I(1). Haldrup (1998) has applied this methodology and he failed to reject the
I(2) null hypothesis for money and prices on the same period as the one used in the present
paper.
As a first check of the statistical adequacy of model (7) we report some multivariate
and univariate misspecification tests in Table 1. We note that our chosen VAR model with six
lags is well specified. The multivariate Ljung-Box test and the multivariate LM test for first and
fourth order residual autocorrelations as well as the multivariate normality test are not
significant. The univariate residual tests also show no signs of misspecification. ARCH(6)
tests for sixth order autoregressive heteroscedasticity and could not be rejected for both
equations. The R2 measures show that with the chosen specification we can explain quite a
large proportion of the variation in the money growth and the inflation rate.
The Johansen - Juselius multivariate cointegration technique, as explained in section
3, is applicable only in the presence of variables that are realizations of I(1) processes and/or
a mixture of I(1) and I(0) processes, in systems used for testing for the order of cointegration
rank. Until recently the order of integration of each series was determined via the standard
unit root tests. However, it has been made clear by now that if the data are being determined
in a multivariate framework, a univariate model is at best a bad approximation of the
multivariate counterpart, while at worst, it is completely misspecified leading to arbitrary
conclusions. Thus, in the presence of I(1) series, Johansen and Juselius (1990) developed a
multivariate stationarity test which has become the standard tool for determining the order of
integration of the series within the multivariate context.
Additionally, when the data are I(2), one also has to determine the number of I(2)
trends, s2 , among the p − r common trends. The two-step procedure discussed in section 3
is used to determine the order of integration and the rank of the two matrices. The hypothesis
that the number of I(1) trends = s1 and the rank = r is tested against the unrestricted H 0
11
model based on a likelihood ratio test procedure discussed in Johansen (1992, 1995, 1997)
and extended by Paruolo (1996) and Rahbek et al. (1999).
Table 2 reports the trace test statistics for all possible values of r and s1 = p − r − s2 ,
under the assumption that the data contain linear but no quadratic trends. We have included
in the estimation eleven centered seasonal dummies, which are necessary to account for
short-run effects which could otherwise violate the Gaussian assumption. The 95% critical
test values reported in italics below the calculated test values are taken from the asymptotic
distributions reported in Rahbek et al. (1999, Table 1). Starting from the most restricted
hypothesis {r = 0, s1 = 0, s2 = 2} and testing successively less and less restricted hypotheses
according to the Pantula (1989) principle, it is shown that the case in favor of the presence
I(2) components can not be rejected at the 5% level, since the hypothesis of {r = 1, s1 = 0, s2
= 1} can not be rejected.4,5,6
We then tested for the significance of the deterministic trend in the multicointegrating
relation. Rahbek et al. (1999) have shown that this hypothesis can be tested with a likelihood
ratio test constructed from the r largest eigenvalues of two models; in the first the
deterministic trend appears in z t while in the second it is excluded. The null hypothesis is
that the linear trend does not enter significantly in the cointegration vector and the test
statistic under the null is a likelihood ratio test is asymptotically distributed as
χ 2 (r ) .
The
test statistic in our case is equal to 1.4 with a p-value of 0.25 and thus we reject the presence
of a deterministic trend in the multicointegrating relation.
4.2. Interpreting the I(2) results
In Table 2 the estimated results are reported for the case r = 1, s1 = 0, and s 2 = 1.
We have decomposed the vector z t into the cointegrating ( β ) and the non-cointegrating
components ( β ⊥ ) and moreover the cointegrating components into the r0 = 0 directly
stationary relations, and the
r1 = 1 multicointegrated relation, and the remaining
nonstationary components into the I(1) and the I(2) relations as discussed in section 3.
12
Given that there are only two variables under consideration, there are no directly
stationary relations, but only one multicointegration,
β 1' zt + ω ' ∆zt ,
which is homogeneous
relation between money supply, prices and money growth or inflation. This seems to support
the choice of p − r = 1 common stochastic trend, which is second-order nonstationary, being
in line with the evidence from the roots of the characteristic polynomial.
Given that s1 = 0 , no estimates of
α 1⊥ and β ⊥1
are provided since these elements
are associated with the I(1) stochastic trend. The estimate of
β ⊥2
shows how the I(2)
stochastic trend affects nominal money stock and prices. Thus, this estimate describes the
weight with which the I(2) trend component influences the variables of the system. Hence, a
β ⊥2ij ≠ 0 for j = 1,...., s 2 .
condition that the variable z t is I(2) is, therefore,
The estimate of
α ⊥2 shows that the twice cumulated shocks to the nominal money stock are of importance for
the I(2) trend
α ⊥2 ' ∑∑ ε i ,i = 1,2. This
indicates that the second order stochastic trend in
nominal prices derives from unanticipated shocks to the nominal money supply.
4.3. A data transformation from I(2) to I(1)
Since the statistical inference of the I(2) model is not yet as developed as that of the
I(1) model, a data transformation that allows us to move to the I(1) model will simplify the
empirical analysis considerably. A natural hypothesis which follows from the I(2)-ness of the
money supply and price level is that the real money balances {mt − pt } is a first-order
nonstationary process. The implication of this hypothesis is that the money supply and prices
are
cointegrating
from
I(2)
to
I(1),
and
use
of
the
transformed
data
vector
~
z t ' = [mt − pt , ∆mt ( ∆pt +1 )] would then allow us to move to the I(1) model. The validity of this
transformation is based on the assumption that {mt − pt } ~ I (1) , {∆mt or ∆pt +1 } ~ I (1) , and
that {mt − pt } is a valid restriction on the long-run structure, but not necessarily on the shortrun structure. Price homogeneity is directly tested by imposing the linear restriction (1,-1) to
the accepted cointegrating vector. The test statistic which is asymptotically distributed as
χ 2 (1)
is equal to 1.04 and therefore we fail to reject the hypothesis of long run price
13
homogeneity and this approach contrast with some of the literature on the subject that takes
this hypothesis as given, (Cagan, 1956; Taylor, 1991).7
The second requirement needed for the transformation to the I(1) model is that
{mt − pt } ~ I (1) , {∆mt or ∆pt +1 } ~ I (1) . This will be directly checked in the remaining
analysis that will be performed in the I(1) model, containing long-run but not short-run price
homogeneity, based on the transformed vector [ m − p, ∆m( ∆pt +1 )].
To assess the statistical properties of the chosen variables in the transformed model
the test statistics reported in Table 3 are useful. The test of long-run exclusion is a check of
the adequacy of the chosen measurements and show that none of the variables can be
excluded from the cointegration space. The tests for stationarity indicate that none of the
variables can be considered stationary under any reasonable choice of
r . Finally, the test of
weak exogeneity shows that none of the variables can be considered weakly exogenous for
the long-run parameters
β
independently of the choice of
r.
All three tests are
χ2
distributed and are constructed following Johansen and Juselius (1990, 1992). Furthermore,
Table 3 presents diagnostics on the residuals from the cointegrated VAR model which
indicate that they are i.i.d. processes, since no evidence of serial correlation or non-normality
was detected. This provides further support for the hypothesis of a correctly specified model.8
Finally, Table 3 also reports the estimated cointegrating vectors of the transformed
model which are based upon eigenvectors obtained from an eigenvalue problem resulting
from Johansen’s reduced rank regression approach. The estimates of the elasticity parameter
β
in the Cagan model are 7.826 and 6.616 respectively, which are very reasonable and
within the expected theoretical range.
The final stage of the cointegration analysis involves the stability analysis of our
cointegration results. Figures 3(a)-(b), and 4(a)-(b) present the Hansen-Johansen
(1993,1999) recursive analysis on the parameter stability of the cointegrated-VAR models.
The first set of graphs shows that the rank of the cointegration space depends on the sample
size from which it has been estimated, since the null hypothesis of a constant rank is
rejected. This result is quite important for making inference about the validity of the Cagan
model under rational expectations and no speculative bubbles, provided that we require
cointegration between real money balances and money growth and between real money
14
balances and inflation, in order to preclude bubbles and to show that the velocity shocks were
negligible. Thus, from these figures we note that in fact cointegration in both cases is
established in the last two months and even in this case this has occurred marginally. The
evidence of the recursive analysis is in agreement with the cointegration results reported
above and we argue that the results show very weak evidence in favour of cointegration. The
result we present here indicates that, for the Cagan model to be established, we need the
observations of the late months of 1923 when the inflationary pressures are exacerbated due
to the suspension of reparations by Germany and the French invasion of the Ruhr. The
previous works by Taylor (1991) and Engsted (1993, 1994) do not consider a small sample
adjustment in the estimated trace test statistic. In a sample of 42 observations, as in this
case, this is a requirement. The second set of graphs indicates that we are always unable to
reject the null hypothesis for the sample independence of the cointegration space for a given
cointegration rank. Therefore, we can conclude that the estimated coefficients do not display
instabilities in recursive estimates.
Since we have found cointegration between the real money balances and money
growth and real money balances and inflation, the next step of our analysis is to estimate the
VAR model in (4), which takes into account the cointegrating properties of the data. Table 4
reports the results of estimating a sixth-order VAR model for the two I(0) variables
S t = ( mt − pt ) + 7.826∆mt and ∆2 mt . This estimation yields two main findings. First, it is
shown that S t strongly-causes
∆2 mt . Second, the exact rational expectation restrictions (6)
which are imposed on the VAR are strongly rejected providing further evidence against the
view taken by Goodfriend (1982), that velocity shocks were negligible in the German
hyperinflation episode.
Finally, to examine whether the rejection of the Cagan model is due to a model
misspecification or is caused by transitory deviations from the model, we construct the
*
variable S t given in (5) using the unrestricted VAR-parameter and a
β
-value of 7.826. Then,
*
regressing S t and S t gives a slope coefficient of 1.95 which is statistically significantly
different from unity. Thus, it is obvious that there are significant differences between the two
variables, implying that the deviations from the model are non-negligible.
15
5. Conclusions
In this paper we provided a re-examination of the Cagan model of hyperinflation for
the German case of the 1920s by applying recent contributions in the econometrics of nonstationarities and cointegration. First, we examined the order of integration and the
cointegration rank in a multivariate context using the recently developed testing methodology
suggested by Johansen (1992, 1995, 1997) and extended by Paruolo (1995) and Rahbek et
al. (1999) it was shown that both money supply and prices are I(2) processes while we were
able to identify one statistically significant cointegrating vector as well as one I(2) component
between money supply and prices. Second, given that the variables of interest are I(2) we
estimated the multi-cointegrating relationship of the transformed I(1) model between real
money balances and money growth and real money balances and inflation. This is proved to
be the necessary and sufficient condition for excluding the presence of rational bubbles.
Third, although cointegration was established on both cases, the evidence is rather weak
given that the rank of the cointegration space exhibits sample dependence, a result obtained
from the application of the recursive tests of Hansen-Johansen (1993,1999). This evidence
implies that the Cagan model is statistically established only when the “true” hyperinflation
period, which began in June 1922 and lasted until the end of 1923 when stabilization was
achieved, is included in the sample. Finally, the exact rational expectations restrictions
implied by the Cagan model with rational expectations and no transitory shocks were
rejected, providing further evidence against this model specification of hyperinflation.
16
Footnotes
1. Phylaktis and Taylor (1992, 1993), Frenkel and Taylor (1993), Engsted (1996, 1998),
Petrovic and Yujosevic (1996), and Choudhry (1998) are among the recent studies which
apply cointegration methods to test the Cagan model for several other countries that have
experienced hyperinflation in different periods of time. Recently, Lee et al. (2000) exploited
the main insight of Cagan’s model in order to examine the relationship between stock returns
and inflation during the German hyperinflation period.
2. Recently Laidler and Stadler, (1998) provide evidence that there was a small minority of
German economists who at the time of the Weimar hyperinflation favored a monetary
explanation of the phenomenon.
3. As Timmerman, (1994) has shown in present value models with feedback relations,
rational explosive bubbles can be ruled out without invoking the transversality condition. The
intuition is that in the presence of a feedback from prices to money, the bubble component in
the endogenous price process which grows asymptotically will come to dominate the forcing
variable (money) such that a growing difference between those two variables cannot exist.
This result implies that cointegration tests for the presence of explosive bubbles make no
sense once the presence of feedback from prices to money is established. Since there is no
unanimity in the literature on the issue of exogeneity or not of the money supply (Flood et. al.,
1984, Sargent and Wallace, 1973) we proceed with the assumption that bubbles are
excluded if the transversality condition holds.
4. The estimation of the eigenvectors as well as the stability tests have been performed using
the program CATS in RATS 4.20 developed by Katarina Juselius and Henrik Hansen, Estima
Inc., Illinois, 1995.
5. A small sample adjustment has been made in all the likelihood ratio statistics, equal to
k
−2 ln Q = −(T − kp) ∑ ln(1 − λ )
i = r0 +1
^
i
as suggested by Reimers (1992).
6. Gonzalo (1994) shows that the performance of the maximum likelihood estimator of the
cointegrating vectors is little affected by non-normal errors. Lee and Tse (1996) have shown
similar results when conditional heteroskedasticity is present.
17
7. Johansen (1995) shows that such an analysis is valid when mt and pt are I(2) variables.
Specifically, the tables of Osterwald-Lenum (1992) can be used to test for cointegration, and
inference concerning the cointegrating vectors can be conducted using the chi-squared
distribution.
8. The application of the Rahbek et al. (1999) test on the two systems shows that the adopted
transformation removes all signs of the I(2) component from the data since it is shown that all
I(2) hypotheses can be rejected at the 5% critical level. To save space these results are
available upon request.
18
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21
Table 1. Residual misspecification tests of the model with k = 6
(a) Univariate Residual Misspecification Tests:
ARCH(6)
NORM
η3
η4
LB(36)
σε
R2
Notes:
∆mt
∆pt
11.82
0.95
0.34
5.27
2.33
-0.38
2.68
2.30
26.28
0.02
30.10
0.05
0.98
0.91
σ ε is the standard error of the residuals, η 3
and
η4
are the skewness and kurtosis
statistics. The LB is the test for serial correlation, ARCH is the test for the presence of
conditional heteroskedasticity, and NORM the Jarque-Bera test for normality. The ARCH and
NORM statistics are distributed as
LB statistic is distributed as
χ
2
χ 2 with 6
and 2 degrees of freedom, respectively and the
with 36 degrees of freedom.
(b) Multivariate Residual Misspecification Tests
Residual autocorrelation.
LB(9)
Residual autocorrelation.
LM(1)
Residual autocorrelation.
LM(4)
Normality
χ 2 (138) = 36.9
p-value = 0.01
χ 2 (4) = 4.23
p-value = 0.38
χ 2 (4) =10.72
p-value = 0.03
χ 2 (4)=3.75
p-value=0.48
LB is the multivariate version of the Ljung-Box test for autocorrelation based on the estimated
auto and cross – correlations of the first (T/4=9) lags distributed as a χ with 138 degrees of
freedom. LM(1) and LM(4) are the tests for the first and fourth-order autocorrelation
2
distributed as a χ with 4 degrees of freedom. The Normality test is a multivariate version of
the Shenton-Bowman test and is distributed with 4 degrees of freedom.
2
22
Table 2. Testing the Rank in the I(1) and I(2) Model
Testing the joint hypothesis H ( s1 ∩ r)
p−r
Q ( s1 ∩ r / H 0 )
r
2
0
1
1
Qr
61.2*
47.6
40.6*
34.4
9.7
19.9
1
2
s2
26.3*
25.4
2.1
12.5
0
Notes: p is the number of variables, r denotes the number of cointegrating vectors, s1 and
s 2 denote the number of I(1) and I(2) components respectively. In performing the Johansen
test, a structure of six lags was chosen according to a likelihood ratio test, corrected for the
degrees of freedom, (Sims, 1980) and the Ljung-Box Q statistic for detecting serial correlation
in the residuals of the equations of the VAR. A model with an unrestricted constant and a
linear trend in the cointegrating vector is estimated according to the Johansen (1992) testing
methodology. The numbers in italics are the 95% critical values (Rahbek et. al., 1999, Table
1). (*) denotes statistical significance at the five percent critical level.
Vector z t decomposed into the I(0), I(1), and I(2) directions
Cointegration Space
^ 0
^ 0
^
β1
ω
-
- 4.080
102.1
-
7.664
154.2
β1
β
m
-
p
-
2
The space spanned by
β⊥
The space spanned by α ⊥
β ⊥1
β ⊥2
α ⊥1
α ⊥2
m
-
3.1
-
0.05
p
-
2.2
-
-0.08
mt and pt , are, respectively, the nominal money supply and price level
23
Table 3. Estimation of the Transformed I(1) Model
(a) Tests for Long-Run Exclusion, Stationarity, and Weak Exogeneity
Test
Long-Run Exclusion
p-r
1
Stationarity
1
Weak Exogeneity
1
mt − pt
14.80*
15.62*
20.46*
15.20*
12.16*
7.06*
∆mt
20.46*
∆pt +1
15.20*
14.80*
15.62*
6.70*
4.96*
Notes: m t − pt , ∆m t , ∆pt +1, are respectively, the real money balances, the money growth
and the next period’s inflation rate. The long-run exclusion restriction test is χ distributed
2
with r degrees of freedom, the multivariate stationarity test is χ distributed with ( p − r )
2
degrees of freedom and the weak-exogeneity test is χ distributed with r degrees of
freedom. In our case all the tests are distributed with one degree of freedom and the 5%
critical value is 3.84. (*) denotes statistical significance at the five percent critical level.
2
(b) Multivariate Residuals Diagnostics
Case
L-B(9)
m t − pt , ∆m t
m t − pt , ∆pt +1
LM(1)
LM(4)
χ 2 (4)
34.28(0.05)
0.93(0.92)
5.69(0.22)
3.73(0.44)
35.19(0.05)
3.91(0.42)
4.67(0.36)
2.89(0.58)
Notes: as in Table 1.
(c) Estimated coefficients
m t − pt
∆mt
1.0
7.826
mt − p t
∆pt +1
1.0
6.616
24
Table 4. Summary Statistics from VAR model
Sample : January 1920 - June 1923
Sims likelihood ratio criterion selects five - lags VAR
− 2
∆2 mt equation: R = 0.89
F test for the null hypothesis that S t Granger-causes ∆2 mt : F = 15.26(0.00)
− 2
S t equation: R = 0.86
F test of rational expectations restriction (eq. 6) on VAR parameters : F = 35.66(0.00)
*
Correlation coefficient ( S t , S t ) = 0.99
*
Slope coefficient in regression of S t on S t : 1.95 (0.034)
25
26
27
28
29
30
31