1. Art and Diseño

Journal of International Money and Finance
22 (2003) 441–451
www.elsevier.com/locate/econbase
Misspecification versus bubbles in
hyperinflation data: comment
Tom Engsted ∗
Department of Finance, The Aarhus School of Business, Fuglesangs alle´ 4, DK-8210 Aarhus V.,
Denmark
Abstract
In this article, I critically review some of the claims and analyses made by Hooker (J. Int.
Money Financ. 19 (2000) 583), in his study of the Cagan hyperinflation model. I argue that:
(i) contrary to what Hooker claims, cointegration tests can be used to discriminate between
bubbles and no bubbles; (ii) contrary to Hooker’s claim, his empirical results for the interwar
European hyperinflations do not in general imply that the Cagan model is misspecified; (iii)
although Hooker’s analyses build directly on the Durlauf and Hall (Bounds on the variances of
specification errors in models with expectations. NBER Working Paper 2936, Massachusetts,
Cambridge, 1989) methodology, he neglects an important part of that methodology, namely
the measurement of the magnitude of noise. I present such measures, and together with reported
cointegration tests, the noise measures help reinterpreting Hooker’s empirical results.
 2003 Elsevier Science Ltd. All rights reserved.
JEL classification: E31; E41; C52
Keywords: Cagan model; Cointegration; Measurement of noise
1. Introduction
Since Cagan’s (1956) seminal paper, the literature investigating money and pricelevel dynamics during hyperinflation has developed quite extensively, not least
because hyperinflation continues to dominate economies round the world, e.g. several
of the new East European countries in the 1990s. Particular focus has been on econo-
∗
Fax: +45-86151943.
E-mail address: [email protected] (T. Engsted).
0261-5606/03/$ - see front matter  2003 Elsevier Science Ltd. All rights reserved.
doi:10.1016/S0261-5606(03)00021-4
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T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
metric estimation and testing of the Cagan money demand model according to which
the demand for real balances basically depends on expected future inflation.
In a recent paper, Hooker (2000) reexamines some of the classic interwar European
hyperinflations using the Cagan model and the testing methodology from Durlauf
and Hall (1989a) and Durlauf and Hooker (1994) as a basic framework. In particular,
he investigates the small sample properties of the Durlauf–Hooker specification tests
using Monte Carlo simulations. Since the data samples in hyperinflation studies are
typically quite short, one should be careful in using asymptotic inference when testing the Cagan model and, indeed, Hooker finds that in general, the finite sample
properties of the tests are quite different from the asymptotic properties. He also
finds that the Durlauf–Hooker methodology is quite successful in detecting rational
inflationary bubbles.
The Monte Carlo study in Hooker (2000) is quite interesting and useful, and provides new evidence on the properties of some tests of the Cagan model. The purpose
of the present article is to take issue with other parts of Hooker’s analyses. First, I
will argue that contrary to what Hooker claims, cointegration tests can in principle
be used to discriminate between bubbles and no bubbles in the Cagan model. Second,
also in contrast to Hooker’s claim, his empirical results for the interwar European
hyperinflations do not in general imply that the Cagan model is misspecified, and
are in general not consistent with the results in Taylor (1991). Finally, I will point
out that although Hooker’s analysis is directly based on the methodology developed
by Durlauf and Hall (1989a), he does not take the full consequence of that methodology. An important part of Durlauf and Hall’s methodology is to measure the relative magnitude of misspecification (“noise”) in rational expectations models, and not
just to test for its statistical significance, but Hooker makes no attempt to measure
the magnitude of noise in the Cagan model.
I will refer to previous studies that have used cointegration techniques to test for
bubbles, and I present explicit estimates of the magnitude of misspecification in the
Cagan model during the hyperinflation episodes analyzed by Hooker. This additional
information proves helpful in reinterpreting Hooker’s empirical results.
2. Hooker’s methodology
First, it will be instructive to briefly recapitulate Hooker’s methodology. The
Cagan model is given as
mt⫺pt ⫽ b ⫹ aEt(pt+1⫺pt) ⫹ et, a ⬍ 0
(1)
where mt and pt are logs of nominal money and prices, respectively, Et is the conditional expectations operator, εt is a stochastic “money demand shock”, a is the
semi-elasticity of the demand for real balances w.r.t. expected inflation, and b is an
inessential constant which will be abstracted from in the following. The fundamental
price solution, pft, to this expectational difference equation is obtained by solving Eq.
(1) recursively forward for pt and imposing the non-bubble transversality condition
limi→⬁[a / (a⫺1)]iEtpt + i = 0:
T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
pft ⬅
冉 冊冘冉 冊
1
1⫺a
⬁
j⫽0
a j
E (m ⫺e ).
a⫺1 t t+j t+j
443
(2)
The perfect foresight fundamental price, p∗t , is defined as
pt∗ ⬅
冉 冊冘冉 冊
1
1⫺a
⬁
j⫽0
a j
(mt+j⫺et+j).
a⫺1
(3)
If there is a rational bubble, bt, it satisfies
bt ⫽
冉 冊
a
Eb .
a⫺1 t t+1
(4)
Hooker also adds “a model noise” term, st, to denote specification error. Thus, the
observed price-level, pt, comprises the three components pt = pft + bt + st, and by
defining the rational forecast error vt ⬅ pft⫺p∗t , it follows that
pt⫺p∗t ⫽ vt ⫹ bt ⫹ st.
(5)
Finally, Hooker defines the forward quasi-difference operator ⌽ ⬅ ⫺[1⫺(a / (a⫺
1))L⫺1], which, when applied to Eq. (5), results in the following expression for
rt + 1 ⬅ ⌽(pt⫺p∗t ):1
rt+1 ⫽
a
1
1
pt+1⫺pt ⫹
mt ⫺
e.
a⫺1
1⫺a
1⫺a t
(6)
The testing methodology is now based on the orthogonality properties characterizing pt⫺pt∗ and rt+1 under the Cagan model. The linear projections of pt⫺pt∗ and rt+1
onto time-t information are denoted stock tests and flow tests, respectively. If both
pt⫺pt∗ and rt+1 are orthogonal to variables dated t and earlier, the price-level obeys
the fundamentals solution (Eq. (2)). If the pt⫺pt∗ projection is non-zero while the
rt+1 projection is zero, that is evidence for the presence of a bubble. If both projections are non-zero, the noise term is non-negligible so that the Cagan model is misspecified, and in that case we cannot infer whether or not there are bubbles in the
price-level.
Since the money demand disturbance, εt, is unobservable, one has to impose parametric restrictions on this term in order to apply the above methodology. Hooker
considers three different specifications for εt: (i) et = 0 ∀t (the exact case); (ii) εt is
a stationary AR(1), et = ret⫺1 + ut; and (iii) εt is a random walk (r = 1). Each of
these specifications implies different transformations of pt⫺pt∗ and rt+1—and different
datings of the information sets—to be used in the linear projections.
The methodology also requires a value for the parameter a (and for r in the
AR(1) case). In the Monte Carlo simulations, Hooker estimates a in a cointegrating
regression of real balances onto inflation under the exact and AR(1) specifications
for εt, while for the random walk specification, a is estimated using an instrumental
1
Note that rt+1 is just the rational forecast error in predicting pt+l, c.f. Eq. (1).
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T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
variables technique. In the empirical application, however, Hooker estimates a using
GMM in the exact and random walk cases, and cointegration in the AR(1) case. r
is estimated from simple autoregressions on the cointegrating residuals.2
Finally, the procedure requires truncation of the infinite sum in Eq. (3). In his
empirical analysis, Hooker truncates using values of pt from after the hyperinflations
have terminated, thereby avoiding the potential problem that pt∗ contains a bubble.
3. Discussion of Hooker’s methodology and empirical results
In the following sub-sections, I take issue with some of the claims and analyses
made by Hooker, and I present additional empirical results for the interwar European hyperinflations.
3.1. Cointegration tests for bubbles
The presence of a bubble makes the price-level explosive so that no finite number
of differences will make it stationary. This leads Hooker to conclude that cointegration tests cannot be used to test for bubbles: “…in the latter (case, i.e. bubbles),
no transformation of prices… will render them stationary, so cointegration tests cannot be implemented” (p. 589). However, as Campbell and Shiller (1987) and Diba
and Grossman (1988) have noted earlier, testing for cointegration can in principle
be used to discriminate between bubbles and no bubbles. They examined the relationship between stock prices and dividends, and they explicitly argued that the finding
of cointegration between stock prices and dividends excludes rational bubbles
(Campbell and Shiller, 1987, p. 1077; Diba and Grossman, 1988, pp. 520–521).
In Engsted (1993), I used arguments similar to those in Campbell and Shiller and
Diba and Grossman to argue that cointegration can be used to test for bubbles in
the Cagan model. First, observe that when both mt and pt are second-order integrated,
I(2), which is a natural identifying assumption under hyperinflation and no bubbles
(and one almost unanimously made in the empirical hyperinflation literature, including Hooker), then a convenient way to express the Cagan model is in the following
way (which is a direct reparameterization of Eq. (2), see Engsted (1993) p. 352)
冘冉 冊
⬁
(mt⫺pt)⫺a⌬mt ⫽ (a⫺1)
j⫽1
冉 冊冘冉 冊
a j 2
1
Et⌬ mt+j ⫹
a⫺1
1⫺a
⬁
j⫽0
a j
Ee ,
a⫺1 t t+j
(7)
Eq. (7) holds only when there are no rational bubbles. Thus, this expression shows
that given that εt is stationary and under the hypothesis of no bubbles, real balances
mt⫺pt cointegrate with money growth ⌬mt.
2
Under the null that st = 0, the error term in the cointegrating regression will be an ARMA(1,1)
process (a sum of the AR(1) εt process and the white noise forecast error in predicting pt+1). Thus, a
more efficient estimate of r can be obtained by taking into account the MA(1) component. Incidentally,
Hooker does not report the estimated values of r.
T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
445
Secondly, Eq. (1) shows that when εt is stationary, the linear combination between
mt⫺pt and ⌬pt + 1 ⬅ pt + 1⫺pt will be stationary whether or not there are bubbles in
prices: a rational bubble will make both mt⫺pt and ⌬pt + 1 explosive so that, although
they do not satisfy the formal definition of cointegration, they still possess the key
property that a unique linear combination between them is stationary and has finite
variance.3 Thus, even if there is a bubble in pt, regressing real balances onto inflation
will produce stationary residuals if the Cagan model holds and εt is stationary.
This leads to the following cointegration based testing strategy for bubbles in the
Cagan model: in two regressions involving mt⫺pt and ⌬pt + 1, and mt⫺pt and ⌬mt,
respectively, test for the stationarity of the residuals. If both sets of residuals are
stationary, this is evidence of no bubbles. If the residuals are stationary in the first
regression and non-stationary in the second regression, it points to the presence of
bubbles. If residuals are non-stationary in both regressions, it reflects either a fundamental failure of the Cagan model or that εt is non-stationary.4 Of course, as documented by e.g. Evans (1991), in finite samples, cointegration tests may work poorly
in detecting certain types of bubbles. But the point here is that, in contrast to what
Hooker claims, cointegration tests can in principle be used to discriminate between
bubbles and no bubbles.
Note that the above testing methodology is conceptually quite similar to Hooker’s
methodology described in Section 2. Both methodologies exploit the fact that a nonbubble transversality condition is imposed in Eq. (2) but not in Eq. (1). This is
similar to the specification tests for bubbles developed by West (1987) and Casella
(1989). However, in contrast to these tests, Hooker’s methodology and the cointegration methodology do not require a detailed specification of the process governing
the forcing variable, mt. The difference between the latter two methodologies is then
that Hooker’s methodology exploits the high-frequency properties of the data,
whereas the cointegration methodology exploits the low-frequency properties.
The description of the cointegration methodology above implicitly assumes that
the explosiveness of pt induced by a bubble does not carry over to mt. If money is
endogenous, however, mt will share the explosive component in pt, thus making real
balances stationary, which is an easily testable restriction.5
Hooker carries out his empirical analysis on data from the interwar hyperinflations
in Austria, Germany, Hungary, and Poland. Despite his claim that cointegration cannot be used to test for bubbles, he allegedly tests for cointegration between real
balances and inflation (see the end of Section 3 in his paper), but he does not report
3
Campbell (1987, footnote 11 and pp. 1258–1259)) makes the same point when discussing the Permanent Income Hypothesis in the case where consumption is explosive.
4
In many studies from the 1970s and 1980s using the Cagan model, εt is assumed to be a random
walk. However, as argued by Taylor (1991), such an assumption substantially reduces the empirical
content of the model from the outset.
5
The properties of the West and Casella bubble tests when fundamentals are endogenous, and under
the explosive alternative, are intriguing and not straightforward, see West (1985) (i.e. the working paper
version of West (1987)), Casella (1989, footnotes 6 and 12), and Durlauf and Hall (1989c, pp. 18–22).
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T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
the results of the cointegration tests.6 However, Taylor (1991) and Engsted (1993,
1994) have previously reported cointegration tests on these hyperinflation data, and
although the results are not completely unambiguous for Austria, Germany, and Hungary, they in general imply that real balances cointegrate with both inflation and
money growth. This does not indicate the presence of rational bubbles during these
three hyperinflations, in accordance with Hooker’s conclusion based on his Table 4
(which I return to in the next section).7
For Poland, the results in Taylor (1991, Table 4) clearly indicate that real balances
cointegrate with inflation, while the results in Engsted (1994, Table 2) imply that
real balances do not cointegrate with money growth. This is consistent with the
Cagan model with a stationary or negligible εt, and rational bubbles! Hooker concludes from his results for Poland (Table 4) that the Cagan model holds without
bubbles and εt being a random walk. This conclusion is inconsistent with the cointegration results in Taylor (1991). An alternative interpretation of Hooker’s results for
Poland, which is consistent with the cointegration results, is that the exact Cagan
model holds with rational bubbles (in the next section, I take a closer look at Hooker’s interpretation of the results in his Table 4).
3.2. Reinterpreting Hooker’s results, and comparison with Taylor (1991)
Based on Table 4 in his paper, Hooker claims that the Cagan model is misspecified
for Austria, Germany, and Hungary, while the model without a rational bubble, but
with money demand shocks being a random walk, fits the data for Poland. He also
states (pp. 597–598) that the results are broadly consistent with the results in Taylor
(1991, Table 5). For Austria and Germany, these claims are unjustified. In fact,
Hooker’s results are fundamentally different from Taylor’s results in that Hooker’s
flow tests across all countries and all information sets do not reject the exact Cagan
model (thereby lending support to the results in Goodfriend (1982) for Germany,
Hungary, and Poland), while Taylor strongly rejects the exact version of the model
for Austria and Germany.8 Instead, Hooker rejects the non-exact Cagan model. This
leaves a puzzle since the non-exact version is more general than the exact version.
6
Also he does not explain how the standard errors reported in his Table 4 for the cointegration estimates of a in the AR(1) case are computed. It is well-known that the usual standard errors from OLS
cointegrating regressions are invalid in general.
7
The null hypothesis in the cointegration tests is that of no cointegration. With a rational bubble the
relevant alternative is not stationarity but explosiveness. This is why Diba and Grossman (1988) use the
Bhargava (1986) Durbin–Watson statistic which allows investigation of the explosive alternative. With
an (Augmented) Dickey–Fuller test, in case the null is rejected, one should explicitly be aware of ADFvalues in the right tail of the distribution (see Haldrup, 1998, p. 608 for elaboration on this point in the
context of testing for unit roots). From Table 3 in Engsted (1994), it is seen that in regressions involving
real balances and money growth, ADF-values for the residuals are in every case less than ⫺3.0. Thus,
it is clear that the null of no cointegration is rejected in favor of stationarity, not explosiveness, thereby
lending support to the no-bubble alternative.
8
Engsted (1994), using a different methodology, also strongly rejects the exact Cagan model for
Austria and Germany.
T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
447
An obvious explanation is that the non-exact version is rejected because the specific
parameterizations for εt (stationary AR(1) and random walk) are invalid. For Hungary, Hooker’s non-rejection of the exact model is consistent with the results in
Taylor (1991) and Engsted (1994). Regarding Poland, in the exact case, Hooker’s
flow tests are insignificant while the stock tests are significant. This constitutes evidence of a bubble according to Hooker’s own methodology!
Thus, for Austria, Germany, and Hungary, the only consistent and unambiguous
conclusion that can be made from the results reported by Hooker in his Table 4, is
that the exact Cagan model with no bubbles holds in all three countries, and that
the results for the AR(1) and random walk specifications simply reflect the invalidity
of these specifications. This interpretation of the results is directly opposite to that
given by Hooker. For Poland, the results are consistent with the no-bubble random
walk specification, but they are equally consistent with the exact bubble specification,
and the latter interpretation is more in line with the cointegration tests reported by
Taylor (1991) and Engsted (1994).
In the next section, I will obtain an explicit estimate of the magnitude of specification error in the four hyperinflation episodes. That will prove useful in further
assessing the Cagan-model’s ability to explain hyperinflation data.
3.3. Measuring noise in the Cagan model
One reason for the difficulty in interpreting Hooker’s results is that his methodology operates with two unobservable variables, “money demand shocks”, εt, and
“specification error”, st. As he himself notes (footnote 7), this may lead to identification problems which make interpretation difficult. Hooker ends up concluding that
there is significant misspecification in the Cagan model for Austria, Germany, and
Hungary, despite the fact that his results clearly do not reject the model with εt =
st = 0.
Hooker’s methodology is directly built on the methodology in Durlauf and Hall
(1989a, 1989b, 1989c). However, the Durlauf–Hall approach explicitly sets εt equal
to zero such that only the exact rational expectations model is considered. st then
measures any stochastic deviation from the exact relationship, and the purpose is
then not just to test whether st is statistically significant, but equally important, to
measure the relative magnitude of st. The reason for this focus is that although there
may be statistically significant deviations from the model, these deviations may not
be economically significant: statistical rejection of a model at a 5% level does not
necessarily imply that the model may not be able to describe important aspects of
the data (this view is articulated in particular in Durlauf and Hall (1989c, pp. 2–3)).
Similarly, non-rejection of the model may simply be due to low power of the test,
and does not necessarily imply that st is small. Therefore, Durlauf and Hall argue
for the importance of supplementing statistical tests for the significance of st with
explicit measures of the magnitude of st, and they propose various such measures.
In particular, Durlauf and Hall (1989b) construct a “noise ratio” that measures the
fraction of the movements of the actual price explained by noise (see also Durlauf
and Maccini (1995) for an alternative noise ratio).
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T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
In my opinion, since in the Cagan model εt is an unobservable variable which can
be labeled anything (e.g. “money demand shocks”), there is really no point in introducing it along with another unobservable variable, st. Alternatively, st can be skipped
and εt can be labeled “noise”. Such an approach would be fully consistent with the
Durlauf–Hall methodology, whereby estimates of the magnitude of st are readily
obtainable. Surprisingly, although Hooker’s methodology is directly based on the
Durlauf–Hall methodology, he does not report estimates of the magnitude of st. This
is unfortunate since it would be of great help in interpreting his results, especially
for the exact specification. As noted above, Hooker does not reject statistically the
exact Cagan model for Austria and Germany, which is in sharp contrast to the results
in Taylor (1991) and Engsted (1993, 1994). One possible explanation is that the nonrejection in Hooker is based on small sample critical values, whereas the rejections
in Taylor and Engsted are based on asymptotic critical values that may be highly
misleading in samples as small as during hyperinflation. However, if this explanation
is true, the estimated magnitude of st should be relatively small.
In order to investigate this possibility, I now construct noise measures in the spirit
of Durlauf and Hall, using data from the Austrian, German, Hungarian, and Polish
hyperinflations. I estimate a “noise ratio” using the VAR-based methodology set
forth in Engsted (1998) on the same data as those used in Engsted (1994).9 Engsted
(1998) and Engsted and Haldrup (1999) show that the VAR-based methodology
gives a noise measure that is conceptually identical to the noise measure suggested
by Durlauf and Hall. The idea is to estimate a VAR-model for the two variables
Xt ⬅ (mt⫺pt)⫺a⌬mt and ⌬2mt, where a comes from cointegrating regressions involving mt⫺pt and ⌬mt. From the estimated VAR-parameters, a VAR-forecast of the
right-hand side of Eq. (7) is constructed, setting et + j = 0 ∀t + j. Denote this VARforecast X⬘t; then Xt⫺X⬘t is a measure of model noise, and the noise ratio NR =
Var(Xt⫺X⬘t) / Var(Xt) is an estimate of the relative magnitude of st.10 NR = 0 means
that there is no noise and that the Cagan model with st = 0 is exactly true. NR =
1 means that noise, st, explains all (100%) of the variability of Xt, i.e. the Cagan
model has no empirical content whatsoever (see Engsted (1998, section 6), Engsted
and Haldrup (1999, section 3), and Engsted (2002, section 4) for more details on the
construction of this noise ratio and its relation to the original Durlauf–Hall measure).
Table 1 reports estimates of NR. These are based on the same data and VARmodels as those underlying Table 4 in Engsted (1994), but here only for the four
countries considered by Hooker: Austria, Germany, Hungary, and Poland. As seen,
noise accounts for 43% in Austria, 25% in Germany,11 14% in Hungary, and 70%
9
Engsted (1998) applied the methodology using data from the Chinese 1946–1949, Hungarian 1945–
1946, and Yugoslav/Serbian 1991–1993 hyperinflations.
10
Let zt = Azt⫺1 + ut be the VAR model, where zt and A contain the two variables Xt and ⌬2mt, and
the VAR-parameters, respectively. Then X⬘t is generated as X⬘t = agA(I⫺(a(a⫺1)⫺1)A)⫺1zt, where g is
a row vector that picks out ⌬2mt from the VAR.
11
Based on the results in Table 4 in Engsted (1993) for Germany, one can infer a noise ratio of 28%.
This slight difference is due to differences in the data used for Germany in the two papers (Engsted,
1993, 1994).
T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
449
Table 1
Estimated noise ratio’s for the Cagan model
Austria
a
NR
⫺4.074
0.433
Germany
Hungary
Poland
⫺5.437
0.247
⫺4.616
0.144
⫺2.411
0.701
Notes: The values of a are the cointegration estimates from Engsted (1994). NR denotes “noise ratio”
and these values are computed from the same three-lag VAR models underlying Table 4 in Engsted (1994).
in Poland. For Austria and Germany, these estimates do not imply that st is small
and, thus, do not support the conclusion from Hooker’s Table 4 that the exact nobubble Cagan model holds. The noise measure NR is based on Eq. (7) which assumes
the absence of bubbles. Thus, if there are bubbles, NR may be large even if the
exact model is true. However, the cointegration results preclude this explanation for
Austria and Germany. For Poland, though, the large value of NR (70%) and the
apparent lack of cointegration between real balances and money growth, is consistent
with Hooker’s stock and flow tests that indicate bubbles in the exact Cagan model
for this country.12 For Hungary, the relatively low size of the noise component (14%)
may be regarded consistent with the statistical non-rejection of the exact no-bubble model.
4. Concluding remarks
In this article, I have argued that testing for bubbles using cointegration techniques,
and estimating the magnitude of specification error provide useful information on
the Cagan-model’s ability to explain money demand during hyperinflation. I have
also pointed out that some of the claims made by Hooker (2000) in his study of the
Cagan model are unjustified, and I have reinterpreted his empirical results for the
interwar European hyperinflations.
Hooker claims that his results for the Austrian, German, and Hungarian hyperinflations imply that the Cagan model is fundamentally misspecified for these countries.
I have shown that this claim is unjustified. The only consistent conclusion one can
draw from Hooker’s results is that the exact rational expectations Cagan model with
no bubbles actually fit the data during these hyperinflations. This conclusion is consistent with the cointegration findings in earlier studies, which precludes rational
bubbles. However, in the case of Austria and Germany, the non-rejection of the exact
model is inconsistent with the finding of a large specification error (noise component)
12
Further evidence in support of this interpretation comes from the R2-value in a regression of rt+1,
defined as in Eq. (6), onto current and lagged values of Xt and ⌬2mt, i.e. essentially the R2-value from
Hooker’s flow test. Durlauf and Maccini (1995) show that this R2-value is an alternative noise ratio (see
also Engsted, 1998, p. 546). This value will be low if the exact Cagan model holds, even if there are
bubbles in pt. On the Polish data, an R2-value of 0.198 is obtained, which is substantially less than 0.701.
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T. Engsted / Journal of International Money and Finance 22 (2003) 441–451
in the model. This indicates that Hooker’s statistical tests of the exact model may
lack power towards finding deviations from the model.
Hooker’s results for Hungary are consistent with the finding of a relatively small
noise component in the exact Cagan model. Similarly, his results for Poland, which
indicate the presence of a bubble, are consistent with previous cointegration studies
and with the finding of a large noise component in the exact no-bubble Cagan model.
However, his interpretation of the Polish results as implying a non-stationary money
demand shock, is inconsistent with previous cointegration studies. A more natural
interpretation is that the exact Cagan model with rational bubbles holds.
My interpretation of the many empirical hyperinflation studies that have accumulated over the years, is that the Cagan model (Eq. (1)) contains an important element
of truth in that real balances and inflation appear to be cointegrated such that εt (or
st) is stationary. In most cases, real balances also appear to cointegrate with money
growth, which precludes rational bubbles. However, in most cases deviations from
the exact rational expectations Cagan model are non-negligible, so either expectations
are formed non-rationally, money and prices are measured with error, and/or other
factors than just expected inflation determine money demand. An important topic
for future research will be to investigate in more detail these possibilities.13
5. Acknowledgments
Comments and suggestions from two anonymous referees are gratefully acknowledged.
References
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Cagan, P., 1956. The monetary dynamics of hyperinflation. In: Friedman, M. (Ed.), Studies in the Quantity
Theory of Money. The University of Chicago Press, Chicago, pp. 25–117.
Campbell, J.Y., 1987. Does saving predict future labor income? An alternative test of the Permanent
Income Hypothesis. Econornetrica 55, 1249–1273.
Campbell, J.Y., Shiller, R.J., 1987. Cointegration and test of present value models. Journal of Political
Economy 95, 1062–1088.
Casella, A., 1989. Testing for rational bubbles with exogenous and endogenous fundamentals: the German
hyperinflation once more. Journal of Monetary Economics 24, 109–122.
Diba, B.T., Grossman, H.I., 1988. Explosive rational bubbles in stock prices? American Economic Review
78, 520–530.
13
Engsted (1996) finds for the German hyperinflation that replacing prices with exchange rates, and
invoking the instantaneous purchasing power parity condition, substantially reduce deviations from the
exact Cagan model. Similar results are reported by Petrovic and Mladenovic (2000) for the
Yugoslav/Serbian 1990–1993 hyperinflation. This points to substantial measurement errors in the reported
price series as a possible explanation for the finding of a large noise component in the standard Cagan
model, measurement errors that do not contaminate reported exchange rate series.
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