CAGAN’S PARADOX AND MONEY DEMAND IN HYPERINFLATION:
REVISITED AT DAILY FREQUENCY
Zorica Mladenović and Pavle Petrović
University of Belgrade
Economics Faculty,
Kamenička 6, 11 000 Belgrade
Serbia
Abstract
We have argued and found support for the claim that in advanced hyperinflation the public
adjusts at daily rather than monthly frequency. An implication is that previous monthly studies of
money demand in hyperinflation could be misleading. Using daily data Cagan (1956) money
demand is accepted for the most severe portion of Serbia’s 1992-1993 hyperinflation, i.e. its last
six months. The obtained estimates significantly differ from the prior monthly ones, thus
undermining the latter. Specifically, daily estimates reject the long-standing Cagan’s (1956)
paradox, originated from monthly studies by showing that economy has been on the correct,
increasing side of the Laffer curve almost through the end of hyperinflation. Some evidence is
offered that these results may hold for other hyperinflations, thus providing general relevance to
the findings in this paper.
JEL Classification: E 31, 41 and 51.
Key words: Hyperinflation, Money demand, Cagan’s paradox, Daily frequency,
1
1. Introduction
There has been a longstanding interest, initiated by Cagan (1956) in exploring money
demand in hyperinflation, and all relevant studies have been done at monthly frequency.
However, hyperinflations are extreme events with daily or weekly inflation rates
comparable to quarterly or annual rates in moderate inflation economies. Thus a
conjecture is that in hyperinflation the public adjusts at daily frequency, or weekly at
most, and hence that monthly observations could cover up its behavior (cf. A. Taylor,
2001). In addition, hyperinflations are short-lived episodes characteristically lasting
around 20 months. Obviously this is very small sample for sound estimation, and the
problem has been typically moderated by extending it to include the lower inflation
period preceding hyperinflation. The latter leads us to another problem with monthly
estimates which is that they encompass non-homogenous period. The issue is aggravated
by hints that even hyperinflation itself is also a non-homogenous event, with its distinct
severe portion (e.g. Michael et al. 1994). Again the latter cannot be demonstrated and
taken care of with monthly observations. All of the above suggest that previous monthly
studies of money demand could be misleading, and that it takes daily data to reveal
agents behavior in hyperinflation.
The paper examines money demand schedule at daily frequency in an advanced stage of
hyperinflation, and confronts it with monthly studies exploring the difference between the
two. In addition we ask whether money demand estimates at daily frequency can resolve
the longstanding Cagan’s paradox emanating from monthly studies.
The Cagan’s (1956) paradox states that in hyperinflation authorities tend to expand
money supply at a rate well beyond the one that would maximize their inflation tax
revenue. As early as the beginning of the 1970s Barro (1972) reported widely different
estimates of revenue-maximizing rates: Friedman (1971) came out with the maximizing
inflation rate below 20% per year, Cagan (1956) with around 20% per month, and Barro
(1972) with 140% per month.
Mainstream research has followed the Cagan model, coming out with estimates that
uniformly reinforce the paradox of non-optimal seigniorage from excessive money
creation (cf. Michael et al. 1994). Specifically, highly efficient estimates of the Cagan
money demand that hold for a wide set of expectation formation processes supported
Cagan’s results (e.g. M. Taylor 1991, and Engsted 1994). The estimated semi-elasticity
varies in the similar range as in Cagan (1956), i.e. from 4 to 6, and their inverse value
provide the revenue-maximizing inflation rates in the range of 17 to 25% per month.
Statistical tests confirm that average inflation rates across hyperinflations significantly
exceed the seigniorage maximizing ones. This suggests that for a substantial portion of
each hyperinflation economies were placed on the wrong, decreasing side of the inflation
tax Laffer curve.
2
Some alternative research however hinted that in the extreme portion of hyperinflation
semi-elasticity might decrease, placing an economy on the correct side of the Laffer
curve almost through to the end of the hyperinflation. Thus Michael et al. (1994) focused
on the most extreme period of the German hyperinflation, including the final months that
have been previously considered as outliers, and obtained a seven times smaller semielasticity than reported above, which goes long way to resolving the Cagan’s paradox.
The result also suggests that the considered extreme period represent a distinct portion of
the German hyperinflation. However, serious shortcoming of the Michael et al. (1994)
result is its reliance on a brutally small monthly sample of only 14 to 16 observations,
which severely limits its robustness.
Another strand of research abandons Cagan’s framework and opts for money demand
schedules that allow for money substitutes, where semi-elasticity decreases (elasticity
increases) as inflation accelerates. Advancing that line, Barro (1970 and 1972) obtained
estimates for the five classical hyperinflations showing that in three cases governments
were on the increasing segment of the Laffer curve through to the end of hyperinflation.
However, in the most extreme episode of Germany and Hungary II (1945-1946) the
economies were on the wrong side of the curve for the last three and four months of
hyperinflation, respectively.
Yet another variable semi-elasticity money demand schedule, which also hinges on
currency substitution (cf. Easterly et al. 1995), points to decreasing semi-elasticity in the
Serbian hyperinflation (cf. Petrović and Mladenović, 2000). The estimated schedule
captured the extreme, last portion of the Serbian episode that could not have been
explained by the Cagan model at monthly frequency. The obtained results point in the
right direction, i.e. semi-elasticity decreases as inflation accelerates, however not enough
to resolve the Cagan’s paradox.
The monthly studies of money demand, reported above, hint that hyperinflation is not a
homogenous event and hence may not be encompassed in a single schedule. Specifically
problems arise in an advanced stage of hyperinflation. This could be due to an increase in
persistency of the series at monthly frequency as hyperinflation peaks. If so, it might
explain why Cagan money demand breaks in extreme portion of hyperinflation such as in
the German and the Serbian episode. On the other hand, increased persistence, if present,
is dampened in changing semi-elasticity nonlinear schedules, and hence their better
performance in an advanced stage of hyperinflation. However, increased persistency of
monthly series may well be due to the temporal aggregation of processes that are in fact
daily (cf. A. Taylor, 2001), pointing out again that monthly observation could mask
agents behavior in hyperinflation.
This paper explores the most severe portion of the Serbian hyperinflation of 1992-93, at
daily frequency. As opposed to previous monthly studies, including Michael et al. (1994),
we are able to rely on a large sample of daily data covering the last six to seven months
of extreme hyperinflation. Importantly, the daily estimates of money demand can be then
3
confronted with various monthly estimates for the Serbian hyperinflation as the two sets,
coming from the same hyperinflation episode, are comparable.
The Serbian hyperinflation is the second (to Hungary II) highest and the second (to the
Russian in the 1920s) longest in the 20th century (cf. Petrović et al., 1999). The portion
that we shall examine is characterized by the average monthly currency depreciation rate
of 10,900% which is 34 times higher than average inflation rate (322%) in the German
hyperinflation. Apart from being distinct, the motivation for choosing this period is the
belief that the public should adjust daily in these extreme conditions.
The paper proceeds as follows. Section 2 gives a background of the Serbian
hyperinflation while demonstrating its severe nature in the period to be explored and
describes the evolution of the series, particularly inflation tax and real money balances. It
also explains the data set that is used. Section 3, examines whether Cagan money demand
holds for the last six months of extreme hyperinflation. In section 4 the Cagan’s paradox
is revisited at daily frequency using the obtained money demand semi-elasticity
estimates. Section 5 confronts money demand estimates at daily and monthly frequency,
especially for Serbia where the estimates are directly comparable. Barro’s (1970 and
1972) schedule is also estimated for Serbia’s episode at monthly frequency, and
compared with daily estimates of Cagan’s model as well as with Barro’s (1970 and 1972)
monthly estimates for other hyperinflations. The conclusions are offered in section 6.
2. The Extreme Portion of Serbian Hyperinflation: Background
2.1 Money Supply and Exchange Rate Dynamics
The Serbian hyperinflation is as explained above one of the most extreme episodes in the
twentieth century. It started in February 1992, when monthly inflation exceeded the
conventional 50% rate, then accelerated significantly during 1993 and was eventually
halted by end-January 1994. (Cf. Petrović et al., 1999).
We shall explore the most severe sub-period of the hyperinflation that runs from July
1993 through to its end, employing daily data on money supply (currency in circulation)
and exchange rate. The daily evolution of exchange rate depreciation (Δe) and money
growth (Δm) are shown in Figures 1 and 2, and Table 1. The exchange rate depreciation
(Δe) and the money growth (Δm) doubled in July as compared to June thus attaining a
new plateau through October. Subsequently, they accelerated significantly again in
November, December and January. The same pattern is followed by variability of these
series as shown in Figures 1 and 2, and by corresponding standard deviations in. Table 1.
Econometric tests confirm the described pattern. Specifically, the daily series of Δm and
Δe exhibited structural break by the end of June both increasing the level and persistence
i.e. switching from stationary to non-stationary series1. The latter even formally points to
July 1993 as the start of distinctive portion of hyperinflation, and motivates our choice of
the period to be examined. Additionally, variability of the series sharply increase in
4
December and January, and the corresponding tests detect the break in their variability
sometimes at end-November or early December2.
Table 1
Exchange rate Depreciation and Money Growth
Per day (%)
Δe
Mean
St. dev.
Δm
Mean
St. dev.
June
1993
July
August
Septem.
October
Novem.
Decem.
January
1994
4.1
4.7
9.2
9.5
7.7
7.9
7.2
8.7
11.6
13.5
21.5
28.3
36.4
33.1
40.8
44.8
4.1
8.5
8.1
8.7
7.6
6.7
6.7
6.4
10.1
14.5
18.0
14.9
33.4
29.1
39.8
55.0
Note: The rates above are continuous ones defined as log difference, e.g. Δe = ln(E/E -1). In December and
January some observations for money supply are missing, and we have interpolated them (cf. 2.4 The Data
Set). Data for January runs through Friday, January 21, since stabilization was introduced and
hyperinflation halted on Monday, January 24.
Figure 1
Exchange Rate Depreciation (Δe)
Figure 2
Money Growth (Δm)
To put this extreme period of the Serbian hyperinflation in comparative perspective one
should look at monthly frequency. Thus, depreciation rates (Δe) per month varied from
223% to 1,128% in December and 1,265% in January. The corresponding discrete rates
that are usually reported vary from 830% to 7,922,026% and 31,176,245%, respectively
per month. For comparison, the corresponding average monthly discrete rates were 322%
in the famous German hyperinflation versus 10,895% in the extreme portion (July
through December) of the Serbia’s hyperinflation3. In addition, the maximum monthly
rate of monetary expansion (Δm) was 660% in Germany (November 1923), 3000% in
Hungary (July 1946)4 and 1035% in Serbia (December 1993). Therefore, we shall be
exploring both distinct and indeed extreme portion of the Serbian hyperinflation.
The severity of the period to be examined strongly suggests that public adjusts their
decisions daily rather than monthly. Both the level and the variability of currency
depreciation and money growth at daily frequency (cf. Figures 1 and 2, and Table 1) are
5
large enough to invoke daily adjustment in this extreme stage of hyperinflation. Thus,
daily rates of currency depreciation and money growth are high even for period from July
through October, let alone for November and December. That is to say the scale of these
rates would be still considerable if they are to appear as quarterly or annual rates.
Additionally, the standard deviations reported in Table 1 shows that there is enough
variability within the month at daily frequency, therefore strongly supporting daily
adjustments conjecture.
Some other hyperinflations, as reported in Table 2, have extreme portions that are
comparable to the Serbian one. The average inflation and money growth rate for the last
six months in the three reported hyperinflations strongly suggest that the public adjusts
daily in their respective extreme periods. Therefore, the findings for Serbia at daily
frequency obtained in this paper may well extend to at least the German, Greek and
Hungarian (1945-46) hyperinflations.
Table 2
Inflation and Money Growth in Extreme Portion of Hyperinflation
Average for the last six months, expressed per day (%)
Inflation (Δp)
Money growth (Δm)
Germany
9.9 (14.8)
9.7 (16.4)
Greece
8.3
9.7
Hungary II
4.9
3.9
Note: Average monthly inflation and money growth for the last six months are expressed per day by
dividing monthly averages by 30. In the brackets are average rates for the last three months of the German
hyperinflation. Source: Cagan (1956) for Germany and Greece, and Barro (1972) for Hungary II.
Monthly studies of money demand, apart from an attempt by Michael et al. 1994, could
not explain the last three months of the German hyperinflation due to their severity, and
hence routinely treat them as outliers. Nevertheless, this portion of the German
hyperinflation is less severe than the corresponding three months (October - December)
period in the Serbian episode, as the respective daily rates indicate: 14.8% and 16.4% in
Germany vs. 23.2% and 20.5% in Serbia. Accordingly, explaining the extreme portion of
Serbian hyperinflation at daily frequency would provide hope for the same in the case of
Germany.
2.2 Seigniorage and Inflation Tax
Seigniorage is calculated as daily changes in cash money over exchange rate, and
inflation tax as real cash holdings multiplied by depreciation rate. These two measures
are equal only in a steady state. The results are depicted in Table 3.
6
Table 3
Seigniorage and Inflation Tax
Average per day, million German marks
June
July
August
Septem. Octob.
Novem. December
January
1-12 and 20-27
11 – 21
1993
Seigniorage
2.4
3.4
3.1
2.6
2.9
3.5
2.9
1.5
Inflation tax 2.3
3.9
3.2
2.8
3.3
3.9
3.0
1.4
Note: In December and January some observations for money supply are missing, hence we report subperiods for which data is available.
The data reported in Table 3 suggests the existence of the Laffer curve property. Namely
seigniorage and inflation tax are both relatively stable from August onwards, reaching the
maximum in November and then declining in December and January. This development
follows those of the exchange rate depreciation (Δe) and money growth (Δm) through
November (cf. Table 1). July is an outlier with seigniorage and inflation tax almost as
high as in November, but with money growth and exchange rate depreciation half of that
in November. This result might be due to the sharp rise, i.e. doubling of money growth
and exchange rate depreciation in July which perhaps had not been anticipated. Namely,
these series (Δe and Δm) exhibit structural break both in level and persistence by the end
of June. Hence the observation for July could be off the Laffer curve.
With weekly observations, one finds that the maximum seigniorage is achieved in the last
week of November (22–28), while the maximum for inflation tax is attained a week later,
i.e. beginning of December. Again there is one outlier week in July.
Summarizing the trends explored above, we see that money growth and exchange rate
depreciation abruptly rose in July 1993, remained relatively stable through October and then
started to increase again. Accordingly, seigniorage and inflation tax were also initially
stable, then increased and reached the maximum by the end of November 1993, and
subsequently largely declined in December and January. This suggests that the economy
was on the efficient, increasing side of the Laffer curve all way through November, and on
the wrong side in December and January. The latter has been reflected in the movement of
exchange rate depreciation (Δe) and money growth (Δm). Namely, the daily data indicates
that, after surpassing the maximum of the Laffer curve by the end of November, the
exchange rate depreciation (Δe) and money growth (Δm) surged and became considerably
more volatile.
Thus the data suggests the existence of the inflation tax Laffer curve, and that the
economy was on the efficient side of the curve throughout the severe hyperinflation, with
the exception of the last two months (seven weeks). This challenges Cagan’s paradox and
supports the hypothesis that the hyperinflation could have been driven by seigniorage
collection almost to its end.
7
2.3 Evolution of the Real Money Balances
Real money balances, defined as currency in circulation over exchange rate, or in logs,
(m-e), exhibit movement consistent with the Cagan money demand schedule.
Figure 3
Real Money Balances (m-e)
As shown in Figure 3, real money balances are relatively stable in July through
September 1993 period, when money growth (Δm) and exchange rate depreciation (Δe)
are also stable (cf. Table 1). The sharp decrease in real money holdings during November
and December coincides with the surge in hyperinflation. Thus developments of real
money balances from July through December seem consistent with Cagan money
demand schedule. However, demonetization was halted in January 1994 despite extreme
rates of money growth and currency depreciation. The latter may be due to the
announcement of stabilization, first for the beginning of January and then postponed until
the January 24th when it was enacted. Additionally half of the money data points in the
January are missing, and the interpolated observations used instead might be prone to
measurement error. All this motivates to skip January from the sample while estimating
money demand.
2.4 The Data Set
The data to be used are black market exchange rates and the currency in circulation
(cash) as money supply, both with a daily frequency. This data is relatively sound
compared to other data in hyperinflation. Exchange rates were generated by the black
market and recorded daily, as opposed to price indices that were calculated magnitudes
based on monthly surveys and hence distorted in hyperinflation. The source for the daily
cash series is the central bank of Serbia5. Again this series is relatively sound. Namely,
the central bank had direct control and evidence of cash expansion, by printing and
shipping it, and thus was able to record its magnitude quite precisely6.
An additional reason to opt for cash money is that the average share of cash in the base
money was around 80%, representing the main source for seigniorage in the period
considered. Finally, most of the classical hyperinflations are studied also employing notes
in circulations (e.g. Barro, 1970 and 1972).
The daily data for exchange rates is available for the whole period, i.e. through January
1994. However some observations on the cash money supply are missing in December
and January, in particular December 13-18, and December 28 – January 10. We have
interpolated them (see Fig. 2), and used this money series when analyzing the whole
8
period as in Figure 3. However, partly due to missing data the sample used for estimation
runs through December. Also, it covers five working days per week, since the money
supply did not change over weekends.
3. Estimating Cagan Money Demand
The Cagan money demand model we shall be looking at is:
mt - et = -αEt (et+1 - et) + ut
The mt and et are the natural logarithms of money and exchange rate, respectively, Et is
the conditional expectation operator, coefficient α is the semi-elasticity of money
demand, and ut is stationary velocity shock.
The model differs from the standard one in replacing the price level with the exchange
rate. It can be thought of as the reduced form obtained by substituting out prices in the
standard model using the purchasing power parity hypothesis. Alternatively, one may
argue that exchange rate is determined directly in the money market and not through
relative price levels. There is empirical evidence on the Serbian (Petrović and
Mladenović, 2000) and the German (Engsted, 1996) hyperinflations that support the latter
hypothesis. Interestingly enough, some support is found even in a panel of 19 lowinflation developed economies (Mark and Sul, 2001)7.
In monthly studies of high and hyperinflation (e.g. Taylor 1991, Phylaktis and Taylor
1993, Engsted 1994, Petrović and Mladenović 2000) it has been well documented that
real money balances cointegrate with inflation rate. This gives a super-consistent estimate
of money demand semi-elasticity that holds for a wide array of expectations formation
schemes. We shall now explore whether the same pattern emerges at daily frequencies in
the severe portion of hyperinflation.
Accordingly, we shall test first if money (m) and exchange rate (e) are I(2) processes, and
whether they cointegrate such that real money balances (m-e) are I(1) process. If so, one
can carry on and estimate Cagan money demand above by testing for cointegration
between real balances (m-e) and exchange rate depreciation (Δe), and obtaining
cointegrating vector. Alternatively, semi-elasticity of money demand (α) estimate can be
obtained from cointegrating vector between real balances (m-e) and money growth (Δm).
The latter cointegration precludes the presence of bubbles and indicates forward looking
behavior (cf. Engsted 1994).
As explained in section 2 the sample runs from July 1 to December 31, 1993. Since the
surge in the currency depreciation and monetary expansion started from November 26,
we have also explored the sub-period running through November 25. The data set covers
five working days per week. Table 5 summarizes the results on unit root testing.
9
Table 5
Unit Root Testing
Augmented Dickey-Fuller
Period: July 1– November 25, 1993
et
mt
mt-et
Ho:I(3)
-15.10
-14.33
H1:I(2)
Ho:I(2)
-2.15
-1.22
-9.35
H1:I(1)
Ho:I(1)
-2.31
H1:I(0)
Kwiatkowski-Phillips-Schmidt-Shin
Period: July 1–November 25, 1993
mt
mt-et
et
Ho:I(2)
0.006
0.006
H1:I(3)
Ho:I(1)
0.18
0.20
0.02
H1:I(2)
6.59
Ho:I(0)
H1:I(1)
Period: July 1– December 31, 1993
et
mt
mt-et
-18.55
-15.89
-1.23
-1.03
-8.35
-3.02
Period: July 1–December 31, 1993
et
mt
mt-et
0.005
0.009
0.29
0.29
0.018
1.78
Note: The number of lags in ADF and KPSS tests is chosen as a minimum number of lags that eliminates
autocorrelation; the values of information criteria, SC and the modified SC (Ng and Perron, 2001), are also
taken into account. For the sample July 1 – November 25, 1993, the number of corrections is equal to 11 for
the exchange rate and money, and 0 for real money. For the July 1 – December 31, 1993 sample the number of
corrections is equal to 9 for exchange rate and money, and 0 for real money. The unit root tests for exchange
rate and money are based on the model with constant and trend with the 5% critical value for ADF test –
3.45 (MacKinnon, 1991). The unit-root presence in real money for the July 1 – November 25, 1993 sample
is tested within the model without a trend, such that the 5% critical values is –2.89 for ADF test, and
within the model with constant and trend for the July 1 – December 31, 1993 sample. The 5% critical values
for the KPSS test is 0.15 for the model with constant and trend, and 0.46 for model with constant
(Kwiatkowski, Phillips, Schmidt and Shin, 1992). The 5% critical value for the right tail of the ADF
distribution is –0.90 in the model with constant and trend, and –0.05 in model without a trend (Fuller,
1976).
Reported results do confirm that both money (m) and exchange rate (e) are I(2) processes,
and hence respective first differences Δm and Δe are I(1) processes. Additionally, results
in Table 5 show that money (m) and exchange rate (e) cointegrate making real money
balances (m-e) an I(1) process. Furthermore, these processes are not explosive. Namely,
the right tail critical value of the ADF test can be used to test whether the processes (me), Δm and Δe are I(1) or explosive. Since all calculated values of the ADF test are less
than the corresponding 5% right tail critical value, we may conclude that the time series
considered do not contain an explosive root. Specifically, ADF test for Δm and Δe, being 1.03 and -1.23 respectively are lower than matching 5% right tail (-0.90); in case of real
10
money balances ADF test (-3.02) is also lower than corresponding 5% critical value (0.05).
The results reported in Table 5 clear the way for estimating Cagan money demand.
Namely, one can proceed and explore cointergration of real money balances (m-e) with
exchange rate depreciation (Δe) and money growth (Δm) respectively. The results are
presented in Table 6 and Table 78.
Table 6
Cointegration Test and Estimation of Cointegrating Vectors
Period: July 1–November 25, 1993
Cointegration between (m-e) and Δe
Rank
Eigenvalue Trace test
Cointegrating vector
1
Δet
mt-et
r=0
r≤1
0.160
0.046
21.23
4.50
1
4.26
-4.07
The largest roots of the process:
1.00, 0.91, 0.91, 0.90, 0.90, 0.86, 0.86, 0.84, 0.83, 0.83
Cointegration between (m-e) and Δm
Rank
Eigenvalue Trace test
mt -et
Cointegrating vector
1
Δmt
5.88
26.03
1
0.05
The largest roots of the process:
1.00, 0.80, 0.79, 0.79, 0.79, 0.79, 0.77, 0.77, 0.76, 0.76
r=0
r≤1
0.233
0.001
-4.22
Note: A constant term is restricted to entering a cointegration vector only. There are eight lags in the VAR
model of (m-e) and Δm, and ten lags in the VAR model of (m-e) and Δe. Dummy variables are included in the
VAR to model outliers that are identified as extreme values of standardized VAR residuals. The first VAR
contains the following dummy variables: D1, D2, D3, D4, D5 and D6.These dummy variables are defined as
follows: D1 = 1 for 1993:07:26 and 0 otherwise, D2 = -1 for 1993:10:1, 1993:10:4 and 0 otherwise, D3 = 1
for 1993:10:11 and 0 otherwise, D4 = 1 for 1993:11:1 and 0 otherwise, D5 = 1 for 1993:11:15 and 0 otherwise
and D6 = 1 for 1993:09:13 and 0 otherwise. The second VAR contains the following dummy variables: D4,
D6, D7, D8, and D9. New dummy variables are defined as follows: D7 = 1 for 1993:10:4 and 0 otherwise,
D8 = 1 for 1993:11:8, 1993:11:9 and 0 otherwise, and D9 = 1 for 1993:11:17, 1993:11:18 and 0 otherwise.
The 5% critical values for the trace test are simulated with 10000 replications using CATS in RATS 2.0
(Dennis, 2006). In the VAR model of (m-e) and Δm, the 5% critical values are: 20.55 for r=0 and 9.20 for r≤1.
In the VAR model of (m-e) and Δe, the 5% critical values are: 20.58 for r=0 and 9.41 for r≤1.
Table 7
Cointegration Test and Estimation of Cointegrating Vectors
11
Period: July 1–December 31, 1993
Cointegration between (m-e) and Δe
Rank
Eigenvalue Trace test
Cointegrating vector
1
Δet
mt-et
r=0
r≤1
0.145
0.028
22.17
3.37
1
5.00
-4.13
The largest roots of the process :
1.00, 0.94, 0.94, 0.94, 0.93, 0.93, 0.91, 0.91, 0.89, 0.89
Cointegration between (m-e) and Δm
Rank
Eigenvalue Trace test
Cointegrating vector
1
Δmt
mt -et
r=0
r≤1
0.168
0.010
24.09
1.28
1
5.37
-4.13
The largest roots of the process:
1.00, 0.89, 0.89, 0.85, 0.85, 0.78, 0.78, 0.77, 0.76, 0.76
Note: A constant term is restricted to entering a cointegration vector only. There are eight lags in the VAR
model of (m-e) and Δm, and twelve lags in the VAR model of (m-e) and Δe. Dummy variables are included in
the VAR to model outliers that are identified as extreme values of standardized VAR residuals. The first
VAR contains the following dummy variables: D1, D2, D3, D4, D6, D10, D11, D12 and D13. For the largest
sample considered dummy variables D10, D11, D12 and D13 are included. They are defined as follows: D10
= 1 for 1993:11:29 and 0 otherwise, D11 = 1 for 1993:12:6 and 0 otherwise, D12 = 1 for 1993:12:16 and 0
otherwise and D13=1 for 1993:12:24, -1 for 1993:12:27 and 0 otherwise. The second VAR contains the
following dummy variables: D4, D6, D7, D8 and D9. The 5% critical values for the trace test are simulated
with 10000 replications using CATS in RATS 2.0 (Dennis, 2006). In the VAR model of (m-e) and Δm, the
5% critical values are: 19.93 for r=0 and 9.19 for r≤1. In the VAR model of (m-e) and Δe, the 5% critical
values are: 20.33 for r=0 and 9.31 for r≤1.
As reported in Appendix, misspecification tests support VAR models used for
cointegration testing.
In each case, for both periods, tests do confirm the presence of cointegration. Neither
process is explosive, i.e. there is no root larger than one, which confirms results reported
above. Semi-elasticity estimates are very close to each other. The estimated Cagan money
demand is thus stable even through December 31, thus encompassing the period of the
great surge in money growth and exchange rate depreciation as well as rise in their
volatility.
In summary, the Cagan money demand is accepted at daily frequency for the most severe
period of the Serbian hyperinflation since real money balances (m-e) cointegrate with
currency depreciation (Δe) and money growth (Δm) respectively. The acceptance of the
12
model is independent of expectations formation, and may encompass both adaptive and
rational expectations.
13
4. The Maximum of the Laffer Curve and When It Was Attained: Cagan’s Paradox
Revisited
We shall now look at how the Cagan model, specifically the estimated money demand
semi-elasticity, concurs with the observed pattern of inflation tax, i.e. its bell-shape with
non-decreasing inflation tax through November 1993 and subsequent drop in December
and January (cf. Table 3).
Following Drazen (1985), the standard measures of the revenues arising from inflation in
our case would be, respectively, monetary growth Δm and depreciation rate Δe multiplied
by real money holdings (m-e).
As to the former (Δm), the corresponding estimates of money demand semi-elasticity are
5.88 and 5.37, respectively through November 25 and December 31. They imply that the
money growth rates of 17.0% and 18.6% per day maximize revenue from inflation.
Actual rate of monetary expansion (cf. Table 1) was below the maximizing one through
October, reached it in November (18%), and surpassed the maximizing rate in December
and January. The latter suggests that through November the economy was on the
increasing segment of the Laffer curve and subsequently, in December and January, on
the decreasing segment. The same result is obtained for the alternative measure that
employs exchange rate depreciation (Δe). Specifically, corresponding semi-elasticity
estimates for the two periods are 4.26 and 5.00 respectively, leading to Laffer’s curve
maximizing rates of 23.5% and 20.0%. The comparison with the actual average currency
depreciation rates (cf. Table 1) in November (21.5%) and December (36.4%) shows
again that the economy reached the maximum of the Laffer curve in November, and then
switched to the wrong, decreasing segment of the curve.
Looking more closely i.e. at the weekly data, one finds that monetary expansion exceeded
maximizing rate (18.6%) in the week of November 15 – 21 (23% per day), then dropped
below it in the subsequent week (10.5%), and finally surpassed it in the week of
November 29 – December 5 (35.5%). The same pattern, only lagging one week, is
observed for the alternative measure, i.e. inflation tax. Thus the actual rate (27.1%)
exceeded conclusively the maximizing rate (20%) in the week of December 6 – 12.
Accordingly the estimated Cagan model suggests that the maximum of the Laffer curve is
attained sometimes at end-November or early December 1993. Thus the economy was on
the increasing side of the Laffer curve through the end of November and on the wrong
side only for the subsequent, last two months (seven weeks) of hyperinflation. This
pattern concurs with the actual evolution of seigniorage and inflation tax as shown in
section 2.
Therefore, the government’s demand for seigniorage could have driven monetary
expansion and hence hyperinflation in Serbia.
14
5. Comparisons with Monthly Studies: Can They Explain Extreme Portion of
Hyperinflation
We shall now confront our daily data estimates of money demand with monthly studies,
both for Serbia and other hyperinflations. Specifically we shall look at Cagan’s paradox
and related issue of whether single money demand schedule can encompass whole
hyperinflation including its most extreme stage. Comparing daily and monthly estimates
is particular saying for Serbia as they come from the same episode. Hence for the sake of
comparison we have also estimated Barro (1970 and 1972) schedule for the Serbian
episode at monthly frequency.
5.1 Shifts in the Cagan Money Demand
Comparable monthly studies of Cagan money demand, reported in Table 8, imply that
Cagan’s paradox exists, but also that the demand schedule is not stable.
Table 8
Semi-elasticity of Money Demand in High and Hyperinflation
High inflation
Per month
Semi-elasticity
Chile
Argentina
Peru
Brazil
Bolivia
16.9
12.7
11.8
11.2
7.4
Average monthly
inflation rate
5.4%
10.3
6.3
4.7
6.6
Seigniorage-maximizing
inflation rate
5.9%
7.9
8.5
8.9
13.5
Source: Phylaktis and Taylor (1993)
Hyperinflation
Austria
Germany
Hungary I
Poland
Taiwan
Greece
Russia
Serbia (Δe)
Germany (Δe)
Per month
Semielasticity
3.8
5.3
8.3
3.4
4.7
3.0
3.1
3.4
6.1
Average monthly inflation rate
Discrete 1)
continuous 1)
47%
38.5%
322
144
46
37.8
81
59.3
22
19.9
365
154
57
45.1
58.3
45.9
28.9
25.4
Seigniorage-maximizing
inflation rate
26.3%
18.9
12.0
29.4
21.3
33.3
32.3
29.4
16.4
15
Note: Average monthly inflation rates for the six hyperinflations in the 1920s are taken from Cagan (1956,
Table 1) and they respectively cover the hyperinflation periods determined by Cagan, and not the samples
used for estimation. Semi-elasticity estimate for Austria, Germany, Hungary I and Poland are from Taylor
(1991); for Greece and Russia from Engsted (1994). The last two studies are exchange rate models: for
Serbia, the sample includes the year 1991 preceding hyperinflation and runs through June 1993, i.e. short of
the last seven months of extreme hyperinflation, Petrović and Mladenović (2000); for Germany also pre1)
hyperinflation period is included in the sample, Engsted (1996). Cf. endnote 3.
Estimates in Table 8 do support Cagan’s paradox. Namely, the reported monthly studies
of hyperinflation imply, as a rule, that economies are placed on the decreasing segment of
the Laffer curve for significant period of hyperinflation. As can be seen above, the
average inflation rate is higher than the one that maximizes the Laffer curve in all but one
case, and statistical tests confirm this (cf. Taylor, 1991). It would follow that in
hyperinflation governments carry on expanding the money supply at an increasing rate
even when the resulting seigniorage is declining, hence creating the Cagan’s paradox.
There is, however, another important point stemming from Table 8, i.e. as inflation
progresses from high to hyperinflation, semi-elasticity of money demand tends to decline.
This suggests that semi-elasticity might further decrease as the economy moves to the
severe portion of hyperinflation.
Actually, Michael et al. (1994) documented that money demand semi-elasticity decreases
substantially in the most extreme period of the German hyperinflation. They argued that
high inflation and hyperinflation episodes should be carefully distinguished, suggesting
that the German hyperinflation lasted only from July 1922 to November 1923. The latter
differs from the universally used estimation sample: January 1921 – July 1923, when the
remaining months are treated as outliers. Nevertheless, Michael et al. (1994) treatment
concurs with the approach taken in this paper where we explore the most severe period of
the Serbian hyperinflation. Estimating augmented Cagan model, with wages to take care
of economic activity, they (Michael et al. (1994)) obtained cointegration even through
October 1923, with semi-elasticity being only 0.61. Furthermore, as they moved the
ending point of the sample from July to November 1923, the estimated semi-elasticity
decreased from 0.81 to 0.58, indicating that something is happening in the severe period
of hyperinflation.
Michael et al. (1994) attribute the obtained decrease in semi-elasticity to the inclusion of
an economic activity variable that has been arguably missing in the previous studies.
However we obtained the same result for the German hyperinflation with a standard
version of the Cagan model used in this paper i.e. by regressing (m-p) on Δm. The results
are reported in Table 9.
16
Table 9
Cointegration between (m-p) and Δm
July1922-Aug. 1923
July1922– Sep.1923
July 1922– Oct.1923
Semi-elasticity
1.44
0.68
0.60
DFR
-3.35
-3.33
-3.13
PPR(1)
-3.55
-3.54
-3.29
Note: Estimated with OLS. DFR and PPR denote Dickey-Fuller and Phillips-Perron cointegration tests,
respectively. The 5% critical value of DFR and PPR test is –3.29 (Engle and Yoo, 1987; used by Michael,
Nobay and Peel, 1994).
The series are cointegrated through September or October, and the semi-elasticity estimates
are very close to the estimates obtained by Michael et al. (1994), and far lower than those
reported in Table 8. Thus, even without an economic activity variable one obtains that
semi-elasticity decreases in severe portion of hyperinflation.
Summarizing the results above, three points are due. First, both Michael et al. (1994) and
our monthly estimates of the German hyperinflation make a big step towards resolving
the long-standing paradox of non-optimal seigniorage for this well known episode.
Specifically, estimates in Table 9 imply that the revenue maximizing rates of monetary
expansion equal to 147% and 167%. Comparing the latter to the actual rate of monetary
expansion, the conclusion is that Germany was on the increasing segment of the Laffer
curve through August 1923, when money expanded at the rate of 150%. Hence,
according to these results, as well as Michael et al. (1994) ones, the economy was placed
on the wrong side of the Laffer curve for the last three months of hyperinflation, which is
by far shorter than implied by estimates in Table 8. In a different setup as showed below,
Barro (1972) also obtained that Germany was on the increasing segment of the Laffer
curve through August 1923, thus reinforcing the findings reported above.
Second, as demonstrated in Table 9, a drop in semi-elasticity occurs even when an
activity variable is skipped. Thus, the inclusion of this variable does not seem to be
necessary in order to explain the decrease in semi-elasticity in German hyperinflation.
The latter is in accordance with the findings for Serbia in this paper at daily frequency.
Third, the obvious limitation of the reported semi-elasticity estimates is the very small
sample used to obtain them. Some support for using it comes from the result that unit root
tests in small samples have low power and hence are inclined towards rejecting
cointegration. Therefore if in a very small sample cointegration is accepted, one may not
outright dismiss the obtained estimates (cf. Michael et al.1994). Nevertheless, the results
above are just conjectures. It takes an abundant daily data sample to test properly whether
semi-elasticity falls in extreme portion of hyperinflation, and whether this could be
explained by inclusion of an economic activity variable. It is exactly what we have done in
17
this paper, and found that semi-elasticity decreased even though the activity variable was not
included.
5.2 Variable Semi-elasticity Money Demand Schedules
One might recourse to variable semi-elasticity money demand schedules in order to capture
hyperinflation dynamics in its advance stage and possibly resolve Cagan’s paradox. As it
turns out, these monthly nonlinear schedules have led to considerable improvements in
explaining hyperinflation dynamics but still well short of daily analysis presented in this
paper.
Thus, Barro (1970 and 1972) advanced the money demand schedule, which allowed for
money substitutes and hence the flight from domestic currency as inflation progresses.
The elasticity of money demand increases as inflation accelerates, and therefore semielasticity decreases. Estimates for the five classical hyperinflations show that in cases of
Austria, Hungary I and Poland, governments were on the increasing segment of the
Laffer curve through the end of hyperinflation. However, in the more extreme episodes of
Germany and Hungary II (1945-46) actual money growth exceeded the revenue
maximizing money growth during the last three and four months of hyperinflation,
respectively. Corresponding monthly inflation revenue maximizing rates are estimated to
be 154% (k = 1.30) for Germany and 117% (k = 1.49) for Hungary II (Barro 1972).
Although this represent considerable improvement over results reported in Table 8, the
estimated money schedule still implies 3 to 4 months on the wrong side of the Laffer
curve for extreme German and Hungary II hyperinflation respectively.
As the Serbian hyperinflation is as severe as those of Germany and Hungary II, it is
relevant to examine how Barro’s (1970 and 1972) money demand schedule fares in the
Serbian episode, and even more importantly how it compares with daily estimates.
Accordingly, a version of Barro’s (1970 and 1972) money demand schedule for the
Serbian hyperinflation is estimated at monthly frequency, and results are reported in
Table 10.
Table 10
Estimates of Barro’s Money Demand Schedule for Serbia
a2
Feb.1991 Jan. 1994
Jan. 1992Jan. 1994
Feb.1991 Jan. 1994
Jan. 1992Jan. 1994
K
-0.59 1.09
R2
86%
DFR
- 4.62
(-3.98)
-0.42 1.10
89%
- 3.30
(-4.10)
-0.5
1.20
83%
- 4.47
(-3.98)
-0.5
0.88
85%
- 5.02
(-4.10)
Revenue
maximizing rate:
Δmmax= 2.6/k2
219% per month
7.3% per day
215%
7.2%
180%
6%
336%
11.2%
Wrong side of
Laffer curve
6 months
6 months
7 months
3 months
18
Note: In brackets are 5% critical values (MacKinnon, 1991), indicating that residuals are stationary in all
but the second case.
We followed Barro (1970 and 1972) in estimating model both with unrestricted
coefficient a2 and restricting it to be a2 = -0.5 as underlying theory suggests. However, we
set the expected rate of inflation to equal actual money growth rate Δm, as oppose to
Barro’s (1970) effective inflation rate, and the estimated model reads as follows:
(m - e) = a1 +a2 lnΔm + ln(1 + k√Δm) - k√Δm+ error term
A general motivation for using actual money growth rate Δm is that forward looking agents
look at money growth while predicting inflation. The latter also motivates the inclusion of
money growth in place of expected inflation in Cagan model (cf. section 3), and daily
estimates reported above do support that specification (cf. Tables 6 and 7). Hence, putting
money growth in Barro’s schedule would as well facilitate comparisons with daily Cagan
model. Additional motivation is that Barro (1972), as described above, compares money
growth rates – actual and revenue maximizing one, rather than inflation rates, while
determining the position of an economy on the Laffer curve. The version of the model used
in this paper gives directly the estimate of the revenue maximizing money growth rate rather
than the inflation one. Anyhow, the two series, effective inflation rate and money growth,
should not differ substantially, thus making the results comparable.
Two samples used for estimation – one covering hyperinflation proper (Jan. 1992- Jan.
1994) and the other extended to include the preceding year (Feb.1991- Jan. 1994),
indicate that estimates are not stable. Either a2 varies while k is stable, or when a2 is
restricted to be -0.5, the estimate of k varies across the samples. In the former case, the
estimated k implies that the Serbian economy has been on the wrong, decreasing side of
the Laffer curve for as much as six months. However when a2 is restricted, estimate of k
substantially decreases as one switches from the whole sample to the sub-sample
covering the hyperinflation proper. The two estimates imply that the economy has been
on the wrong side of Laffer curve for seven or three months respectively. Thus three out
of four estimates suggest that the economy is on decreasing side of Laffer curve for
substantial period of time, thus sharply contradicting the estimates of Cagan money
demand at daily frequency.
There are also some econometric issues involved when applying Barro’s money demand
schedule in hyperinflation, implying that all its estimates should be taken with some
caution. Namely, as explained above (section 3) variables such as real money balances,
inflation, money growth and etc., are non-stationary I(1) processes in hyperinflation.
Thus while estimating the model we looked to obtain at least stationary residuals, and in
three out of four cases they are so (cf. DFR in Table 10). However, non-linear least
squares (NLLS) commonly used to estimate non-linear schedules, generally yields
inefficient estimators and invalid tests in nonstationary regressions (cf. Chang, Park and
Phillips, 2001).
For the Serbian episode, another version of a schedule relaying on currency substitution
and allowing for both variable elasticity and semi-elasticity of money demand has been
19
estimated9. As above, the monthly sample includes the last seven months of severe
hyperinflation and the implied semi-elasticity estimates, which fall as depreciation rate
increases, are reported in Table 11.
Table 11
Variable Semi-elasticity Estimates for Serbia
Per day
Semielasticity
July
1993
13.1
August
September October
15.3
16.3
10.4
November December January
1994
6.3
4.0
3.6
These semi-elasticity estimates give support to the daily estimates of severe
hyperinflation obtained in this paper. When hyperinflation is moving towards its peak,
between November and January, these estimates approach the daily ones obtained in this
paper. However the July-December average of these semi-elasticities (10.9) is twice as
high as one(s) found in this paper. Consequently the revenue maximizing rate implied by
this model is only Δemax = 64% per month10 (2.1% per day), implying that the economy
has been on the decreasing segment of the inflation tax Laffer curve for 14 months of
advanced hyperinflation. This is in sharp contrast to both the actual pattern of the
inflation tax (Table 3) and the estimates at daily frequency reported in this paper. Thus,
the employed variable semi-elasticity model at the monthly frequency points in the right
direction. Nonetheless it is inferior to the Cagan model at daily frequency in explaining
dynamics of severe hyperinflation outlined by data (cf. Table 3).
5.3 Why Monthly Estimates Deviate from Daily: Increase in Persistency and Temporal
Aggregation
As demonstrated above, monthly estimates of money demand do differ from daily ones
and particularly perform poorly in an advanced stage of hyperinflation. The latter might
be due to an increase in persistency of monthly series that, as oppose to daily series, even
approach explosiveness in a severe portion of hyperinflation. Again data on the Serbian
hyperinflation lends opportunity to explore this.
The values of ADF tests reported in Table 12 indicate that the persistence of both Δet and
Δmt at monthly frequency have increased when the sample is extended through the end of
hyperinflation (January 1994).
Table 12
Persistence of Exchange Rate Depreciation (Δe) and Money Growth (Δm):
Augmented Dickey-Fuller test
20
Monthly Series
Period
Ho:I(1)
H1:I(x)
December 1990– June 1993
Δet
Δmt
-1.34
-1.67
December 1990– January 1994
Δet
Δmt
-1.14
-0.70
Note: The alternative hypothesis H1:I(x) states that the series is explosive, and is accepted if ADF test is greater
than the right tail critical value: -0.8 at 5% (Fuller, 1976). For the December 1990– June 1993 sample the
number of corrections is equal to 3 for the exchange rate and 1 for money. For the December 1990– January
1994 sample the number of corrections is equal to 0. The test is based on the model with a constant and
trend.
Through June 1993 both Δe and Δm are clear-cut I(1) processes (cf. Table 12). Nonetheless,
when seven months of extreme hyperinflation are added, the ADF test for Δm has increased
from -1.67 to -0.70 thus just surpassing the right tail 5% critical value: -0.80. The latter
implies that money growth is on the verge of being explosive process. ADF test for Δe
has also increased indicating the raise in persistency, but not far enough to propose
explosiveness (i.e. -1.14 < -0.80).
This increased persistence, almost explosiveness, of the monthly series could account for the
observed break of the Cagan money demand in the extreme portion of the Serbian
hyperinflation at monthly frequency (cf. Petrović and Mladenović, 2000). Furthermore, it
can also explain better performance of non-linear money demand schedules over the
Cagan’s linear one, as the former dampen persistence of Δe and Δm by taking logs and/or
some roots of these variables.
Comparisons with daily processes indicate that daily Δe and Δm series are I(0) processes
through June 199311, as oppose to monthly, which are I(1) processes (Table 12). In the
subsequent period of extreme hyperinflation daily series also increase their persistence
becoming I(1) processes, and remain so even when hyperinflation peaks in December (cf.
Table 5). Thus for the July –December 1993 period daily series are still clear-cut I(1)
processes while the monthly are on the verge of being explosive.
Accordingly there is some tentative evidence proposing that temporal aggregation of a daily
process into monthly might lead to an increase in persistence of the latter. This would
concur with A. Taylor (2001) result showing that if daily process is sampled at monthly
frequency it will, in certain cases, lead to an increase in persistency. The latter might then
prevent one from revealing the true, daily process from monthly observations, and could
account for the difference in money demand estimates at monthly and daily frequency.
21
8. Conclusions
Daily data for an extreme stage of the Serbia’s 1992-93 hyperinflation, i.e. 7 months
towards its end, strongly supports a conjecture that agents adjust their decisions daily
rather than monthly. Namely, daily money growth and currency depreciation, and their
respective variability, are of the order of quarterly or annual changes in moderate
inflation countries (cf. Table 1). The acceptance of the Cagan (1956) money demand at
daily frequency for the last six months of the Serbia’s hyperinflation strongly speaks in
favor of daily adjustments. The obtained semi-elasticity estimate at daily frequency is in
sharp contrast to the prior monthly estimates, and rejects the Cagan’s paradox implied by
the latter. All this challenges previous studies of money demand in hyperinflation carried
out at monthly frequency.
Semi-elasticity of money demand obtained at daily frequency is by far lower than the
ones attained in monthly studies. Thus, when our daily semi-elasticity estimate (5.37) is
expressed at monthly frequency one gets 0.18 (0.015 annually), which is about twenty
times lower than previous monthly estimates for hyperinflation (cf. Table 8). The inverse
of semi-elasticity gives inflation tax maximizing rate, and it is equal to 559% per month.
The corresponding monthly discrete rate, which is commonly reported, is as high as
26,582% per month.
The semi-elasticity estimates are robust, i.e. they are obtained from cointegration vectors
using a large sample of around 130 daily observations that cover the most extreme
portion of hyperinflation. This is in sharp contrast with the previous monthly
hyperinflation studies that used only 20 to 35 observations including relatively low rates
of inflation. The inclusion of these low rates in the sample may have led to
underestimation of inflation tax maximizing rate (cf. Barro, 1972).
The low value of the money demand semi-elasticity might be crucial in addressing the
Cagan’s paradox. In the case of Serbia, the estimated semi-elasticity implies that money
printing and the consequent currency depreciation did not result in the decrease of
inflation tax through the severe period of hyperinflation, short of the last two months
(seven weeks). The actual inflation tax path concurs with the pattern predicted by the
model, both concerning the bell shape of the Laffer curve and the timing of the
maximum. These results point out that the government’s demand for seigniorage forced
the Serbian economy into hyperinflation and it lasted almost as long as non-decreasing
seigniorage could be extracted. This resolves the Cagan’s paradox in the case of the
Serbian hyperinflation.
There are some hints from monthly studies supporting our low semi-elasticity result
obtained at daily frequency. Thus by comparing Cagan’s money demand estimates for
high and hyperinflation in a number of episodes a clear pattern of decrease in semielasticity emerges when one switches from high to hyperinflation (cf. Table 8). Therefore
one may conjecture that semi-elasticity has further dropped as these economies entered
severe hyperinflation.
22
The conjecture above that semi-elasticity decreases as hyperinflation progresses is
supported by estimates of the Cagan’s model for the extreme portion of the German
hyperinflation, equally in Michael et al. (1994) and in this paper. Namely, both estimates
indicate a sharp drop in semi-elasticity. However, a very short span of monthly data (14
to 16) seriously constrains the results obtained in the German case. Therefore it takes
daily data, used in this paper, to get conclusive results on the extreme portion of
hyperinflation.
Non-linear money demand schedules at monthly frequency, such as Barro (1970, 1972)
also suggest decreasing semi-elasticity as inflation progresses thus placing three out of
five classical hyperinflations on the correct, increasing side of the Laffer curve through
their respective ends. However, in the more extreme episodes of Germany and Hungary
(1945-46) the economies were on the wrong, decreasing side of the Laffer curve during
the last three and four months of hyperinflation, respectively.
Serbia’s hyperinflation enables one to confront directly money demand schedule
estimated at daily and monthly frequency respectively, as the data are available at both
frequencies. At monthly frequency, Cagan money demand holds apart from the last seven
months of extreme hyperinflation, when the model breaks down (cf. Petrović and
Mladenović, 2000). On the other hand, for this very period of extreme hyperinflation
Cagan money demand holds at daily frequency. Monthly semi-elasticity estimate (3.4) is
far above the comparable daily one (0.18), and strongly supports Cagan’s paradox.
Nevertheless, some variable semi-elasticity money demand schedules at monthly
frequency hold through the end of the Serbian hyperinflation, and give rise to decreasing
semi-elasticity. The latter points in direction of daily estimates, but not far enough to be
reconciled with them. Specifically, various estimates of Barro’s (1970 and 1972) money
demand place the Serbian economy on the wrong side of the Laffer curve for as much
three to seven months, compared with two months (seven weeks) in the case of daily
estimates.
Thus the Serbian episode shows that monthly estimates of money demand do differ from
daily ones, and that former perform poorly in an advanced stage of hyperinflation. The
disparity may be due to temporal aggregation of essentially daily processes into monthly
ones. There is some evidence from Serbia that this aggregation leads to increase in
persistence of monthly series which even approach explosiveness as hyperinflation peaks.
This concurs with certain theoretical findings on temporal aggregation (A. Taylor, 2001),
and could account for both the break of Cagan model and somewhat better performance
of non-linear schedule at monthly frequency in an advanced stage of hyperinflation.
The most severe portion of a few other hyperinflations, notably those of Germany,
Greece and Hungary (1945-46), is similar to that of Serbia hence implying daily
adjustment in those episodes. Therefore, disparity between monthly and daily estimates
of money demand and consequent rejection of Cagan’s paradox found in the Serbian
episode may well extend to these hyperinflations.
23
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25
Appendix
The VAR models that are used for cointegration testing statistically perform well. In the
following tables we report some misspecification multivariate and univariate tests of the
VAR of (m-e) and Δm for both samples considered. The multivariate tests for residual
normality and first and fourth order residual correlation do not suggest misspecification.
This is confirmed by the univariate ARCH and Jarque-Bera normality tests. These tests are
valid irrespective of whether the variables are I(0), I(1) or explosive, and hence can be
performed as a first step independently of subsequent unit root testing (cf. Nielsen, 2006,
Engler and Nielsen, 2007).
Misspecification Tests in the VAR of (m-e) and Δm
Period: July 1–November 25, 1993
Multivariate Tests
Residual autocorrelation: LM1 , CHISQ(4)
6.92(p-value=0.12)
Residual autocorrelation: LM2 , CHISQ(4)
4.47(p-value=0.35)
Normality: CHISQ(4)
2.74(p-value=0.60)
Univariate tests
Equation
ARCH(7)
Skewness
Kurtosis
Normality (2)
R2
Δ(m-e)
10.23
-0.27
3.19
1.63
0.44
Δ2m
8.14
0.03
3.14
0.76
0.81
Estimated cointegration relation between (m-e) and Δm
Estimated cointegration relation between (m-e) and
Δm corrected for short-run effects and outliers
26
Period: July 1–December 31, 1993
Multivariate Tests
Residual autocorrelation: LM1 , CHISQ(4)
1.96(p-value=0.74)
Residual autocorrelation: LM2 , CHISQ(4)
5.92(p-value=0.21)
Normality: CHISQ(4)
8.96(p-value=0.06)
Univariate Tests
Equation
ARCH(8)
Skewness
Kurtosis
Normality (2)
R2
Δ(m-e)
5.48
-0.24
4.18
8.80
0.68
Δ2m
10.37
0.32
3.10
2.20
0.82
Estimated cointegration relation between (m-e) and Δm
Estimated cointegration relation between (m-e) and
Δm corrected for short-run effects and outliers
27
Figure 1
Exchange Rate Depreciation (Δe)
2.0
1.5
1.0
0.5
0.0
-0.5
93M07 93M08 93M09 93M10 93M11 93M12 94M01
Depreciation rate
Figure 2
Money Growth (Δm)
2.4
2.0
1.6
1.2
0.8
0.4
0.0
-0.4
93M07 93M08 93M09 93M10 93M11 93M12 94M01
Money growth
Interpolated data
28
Figure 3
Real money (m – e)
4
3
2
1
0
93M07 93M08 93M09 93M10 93M11 93M12 94M01
Real money
1
Various tests are employed all suggesting a break in persistence by the end of June, first in money growth
and then in exchange rate depreciation. The null hypothesis that the series (Δ m and Δe) are I(1) throughout
the sample against the alternative that the number of unit roots changes from 0 to 1 (cf. Leybourne, et al., 2003)
is rejected in both cases. The test is based on the Elliot-Rothenberg-Stock type of DF t-ratio (cf. Elliott et al.,
1996). A two-step procedure (cf. Busetti and Taylor, 2004) estimating the break point in level and then testing
change in persistence confirmed that both occurred by the end of June. The results are available from the
authors upon request.
2
The results are available from the authors upon request.
The discrete rate is defined as x = [(E/E -1) – 1], while continues as Δe = ln(E/E -1), where E is the
exchange rate. Comparing in terms of the latter, the German average inflation rate equals Δp = 144% vs.
the extreme portion Serbian one Δe = 470%. This still indicates that the latter is three times as severe as the
former. However, as suggested by Cagan (1956), comparisons are regularly made in terms of discrete rates.
3
4
For Germany and Hungary see Table 1, Barro (1972)
5
At the time of hyperinflation it was the central bank of FR Yugoslavia (Serbia and Montenegro) and both
money supply and exchange rate correspond to FR Yugoslavia. However, since Montenegro accounts for
only 5% of joint GDP, we are referring to this hyperinflation as the Serbian. Furthermore the central bank,
which generated hyperinflation, was effectively under Serbian control.
29
6
This was not, however, the case with M1. (cf. Petrović et al., 1999).
7
Mark and Sul conclude that an explanation for their results “… may be that the long-run nominal
exchange rate is determined directly by monetary fundamentals and not by relative price levels”. Cf. Mark
and Sul, 2001, p.
8
The VAR models that are used for cointegration testing are reported in the Appendix.
9
We followed Easterly, Mauro, and Schmidt-Hebell (1995), and estimated the subsequent money demand:
R2 = 85%
(m - e) = 14.2 - 8.15Δe0.13
Although variables are I(1), NLLS estimators for this particular specification are consistent. (cf. Petrović
and Mladenović, 2000)
10
Cf. Petrović and Mladenović, 2000.
11
Cf. endnote 1.
30