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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,

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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.

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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 semi-elasticity 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

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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

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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 (%) June

1993 July August Septem. October Novem. Decem. January

1994 Δe Mean St. dev.

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

Δm Mean St. dev.

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

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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 (%) Germany Greece Hungary II Inflation (Δp) 9.9 (14.8) 8.3 4.9 Money growth (Δm) 9.7 (16.4) 9.7 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.

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Table 3 Seigniorage and Inflation Tax Average per day, million German marks

June 1993

July August Septem. Octob. Novem. December 1-12 and 20-27

January 11 – 21

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 sub-periods 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.

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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

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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 low-inflation 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.

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Table 5 Unit Root Testing Augmented Dickey-Fuller Period: July 1– November 25, 1993 Period: July 1– December 31, 1993 et mt mt-et et mt mt-et

Ho:I(3) H1:I(2)

-15.10 -14.33 -18.55 -15.89

Ho:I(2) H1:I(1)

-2.15 -1.22 -9.35 -1.23 -1.03 -8.35

Ho:I(1) H1:I(0)

-2.31 -3.02

Kwiatkowski-Phillips-Schmidt-Shin Period: July 1–November 25, 1993 Period: July 1–December 31, 1993 et mt mt-et et mt mt-et

Ho:I(2) H1:I(3)

0.006 0.006 0.005 0.009

Ho:I(1) H1:I(2)

0.18 0.20 0.02 0.29 0.29 0.018

Ho:I(0) H1:I(1)

6.59 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 (m-e), 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

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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

mt-et Δet 1 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 Cointegrating vector

mt -et Δmt 1 r=0 r≤1

0.233 0.001

26.03 0.05

1 5.88 -4.22

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

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

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Period: July 1–December 31, 1993

Cointegration between (m-e) and Δe Rank Eigenvalue Trace test Cointegrating vector

mt-et Δet 1 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

mt -et Δmt 1 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

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model is independent of expectations formation, and may encompass both adaptive and rational expectations.

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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.

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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 Average monthly

inflation rate Seigniorage-maximizing inflation rate

Chile 16.9 5.4% 5.9% Argentina 12.7 10.3 7.9 Peru 11.8 6.3 8.5 Brazil 11.2 4.7 8.9 Bolivia 7.4 6.6 13.5 Source: Phylaktis and Taylor (1993) Hyperinflation Per month

Average monthly inflation rate Semi-elasticity Discrete 1) continuous 1)

Seigniorage-maximizing inflation rate

Austria 3.8 47% 38.5% 26.3% Germany 5.3 322 144 18.9 Hungary I 8.3 46 37.8 12.0 Poland 3.4 81 59.3 29.4 Taiwan 4.7 22 19.9 21.3 Greece 3.0 365 154 33.3 Russia 3.1 57 45.1 32.3 Serbia (Δe) 3.4 58.3 45.9 29.4 Germany (Δe) 6.1 28.9 25.4 16.4

16

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 pre-hyperinflation period is included in the sample, Engsted (1996). 1) 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.

17

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

18

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 semi-elasticity 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 K R2 DFR Revenue

maximizing rate: Δmmax= 2.6/k2

Wrong side of Laffer curve

Feb.1991 -Jan. 1994

-0.59 1.09 86% - 4.62 (-3.98)

219% per month 7.3% per day

6 months

Jan. 1992- Jan. 1994

-0.42 1.10 89% - 3.30 (-4.10)

215% 7.2%

6 months

Feb.1991 -Jan. 1994

-0.5 1.20 83% - 4.47 (-3.98)

180% 6%

7 months

Jan. 1992- Jan. 1994

-0.5 0.88 85% - 5.02 (-4.10)

336% 11.2%

3 months

19

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

20

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 July

1993 August September October November December January

1994 Semi-elasticity

13.1 15.3 16.3 10.4 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

21

Monthly Series Period December 1990– June 1993 December 1990– January 1994 Δet Δmt

Δet Δmt

Ho:I(1) H1:I(x)

-1.34 -1.67 -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.

22

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 semi-elasticity 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.

23

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.

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References Barro, Robert J. (1970). "Inflation, Payments Period, and the Demand for Money." Journal of Political Economy 78, 1228-1263. Barro, Robert J. (1972). "Inflationary Finance and the Welfare Cost of Inflation." Journal of Political Economy 80, 978-1001. Busetti, Fabio and A. M. Robert Taylor. (2004). "Tests of Stationarity Against a Change in Persistance." Journal of Econometrics 123, 33-66. Cagan, Phillip. (1956). "The Monetary Dynamics of Hyperinflation." In Studies in the Quantity Theory of Money, edited by Milton Friedman. Chicago: Chicago University Press, 1956. Chang, Yoosoon, Joon Y. Park and Peter C.B. Phillips. (2001). "Nonlinear Econometric Models with Cointegrated and Deterministically Trending Regressors." Econometrics Journal 4, 1-36. Dennis, Jonathan, G. (2006). CATS in RATS. Evanston, IL: Estima. Drazen, Allan. (1985). "A General Measure of Inflation Tax Revenues." Economics Letters, 17, 327-330. Easterly, William, Paolo Mauro, and Klaus Schmidt-Hebell. (1995). "Money Demand and Seigniorage-Maximizing Inflation". Journal of Money, Credit, and Banking 25, 584-603. Elliott, Graham., James, H. Stock and Thomas Rothenberg. (1996). "Efficient Tests for an Autoregressive Unit Root." Econometrica 64, 813 – 836. Engler, Eric and Bent Nielsen. (2007). "The Empirical Process of Autoregressive Residuals". Discussion Paper, Nuffield College, Oxford. Engsted, Tom. (1994). "The Classic European Hyperinflations Revisited: Testing the Cagan Model Using a Cointegrated VAR Approach." Economica 61, 331-343. Engsted, Tom. (1996). "The Monetary Model of the Exchange Rate under Hyperinflation." Economics Letters 51, 37-44. Engle, Robert F. and Byung Sam Yoo. (1987). "Forecasting and Testing in Cointegrated Systems". Journal of Econometrics 35, 143 – 159. Friedman, Milton. (1971). "Government Revenue from Inflation." Journal of Political Economy 79, 846-56. Fuller, Wayne, A. (1976). Introduction to Statistical Time Series. New York: Wiley.

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Kwiatkowski, Dennis, Peter C.B. Phillips, Peter Schmidt and Yoon Shin.(1992)."Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure are we that Economic Time Series Have a Unit Root ?" Journal of Econometrics 54, 159 – 178. Leybourne, Stephen, Tae-Hwan Kim, Vanessa Smith and Paul Newbold. (2003)."Tests for a Change in Persistence Against the Null of Difference-stationarity." Econometrics Journal 6, 290-310. MacKinnon, James G. (1991). "Critical Values for Cointegration Tests." In Long-Run Economic Relationships: Readings in Cointegration, edited by Robert F. Engle and Clive W.J. Granger, pp. 267-276, Oxford: Oxford University Press Mark, Nelson C. and Donggyu Sul. (2001). "Nominal Exchange Rates and Monetary Fundamentals: Evidence from a Small Post-Bretton Woods Panel." Journal of International Economics 53, 29-52. Michael, P., A. R. Nobay and D. A. Peel. (1994). "The German Hyperinflation and the Demand for Money Revisited." International Economic Review 35, 1-22. Ng, Serena and Pierre Perron. (2001). "Lag Length Selection and the Construction of Unit Root Test with Good Size and Power." Econometrica 69, 1519 – 1554. Nielsen, Bent. (2006). "Order Determination in General Vector Autoregressions". In Time Series and Related Topics: In Memory of Ching-Zong Wei, edited by Ho, H.C., Ing, C.K., and Lai, T.L, pp. 93-112, IMS Lecture Notes and Monograph Series 58. Phylaktis, Kate and Mark P. Taylor. (1993). "Money Demand, The Cagan Model, and Inflation Tax: Some Latin American Evidence." Review of Economics and Statistics 75, 32-37. Petrović, Pavle, Zeljko Bogetić and Zorica Vujošević (Mladenović). (1999)."The Yugoslav Hyperinflation of 1992-1994: Causes, Dynamics and Money Supply Process." Journal of Comparative Economics 27, 335-353. Petrović, Pavle and Zorica Mladenović.(2000)."Money Demand and Exchange Rate Determination under Hyperinflation: Conceptual Issues and Evidence from Yugoslavia." Journal of Money, Credit and Banking 32, 785-806. Taylor,Alan M.(2001)."Potential Pitfalls for the Purchasing-power-parity Puzzle? Sampling and Specification Biases in Mean-reversion tests of the Law of One Price", Econometrica 69, 473-498. Taylor, Mark P. (1991). "The Hyperinflation Model of Money Demand Revisited", Journal of Money, Credit and Banking 23, 327-351.

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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

27

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

28

Figure 1 Exchange Rate Depreciation (Δe)

-0.5

0.0

0.5

1.0

1.5

2.0

93M07 93M08 93M09 93M10 93M11 93M12 94M01

Depreciation rate

Figure 2 Money Growth (Δm)

-0.4

0.0

0.4

0.8

1.2

1.6

2.0

2.4

93M07 93M08 93M09 93M10 93M11 93M12 94M01

Money growth Interpolated data

29

Figure 3 Real money (m – e)

0

1

2

3

4

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. 3 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. 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.

30

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: (m - e) = 14.2 - 8.15Δe0.13 R2 = 85% 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.