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Money, prices, and causality: The Chinese hyperinflation, 1945–1949
DE-PIAO TANG Columbia
Vnioersity
IEH-VVEI HU Pennsylcaniu
State
Vnitiersity
Money, Prices, and Causality: The Chinese Hyperinflation, 7945@ This paper empirically examines causality between money and prices during the Chinese hyperinflation, 1945-49. The major issue concerning the interrelationship between money and prices is the endogeneity of money supply and the rate of inflation. Both the Sims test and the Granger test indicate that there was strong feedback causation between money creation and inflation during the Chinese hyperinflation. The mutual feedback between money creation and the rate of inflation is an important cause of the development of hyperinflation.
1. Introduction Since the publication of the Sims test (I972), a number of empirical studies have examined the causality between money and prices during hyperinflation. For instance, Sargent and Wallace (1973) examined the causality issue using Cagan’s data (1956) and concluded that in several hyperinflations-Germany, Austria, and Hungary I-there was feedback from inflation to subsequent rates of money creation while only in Greece was there feedback from money creation to the inflation rate. Frenkel (1977) used the German hyperinflation data alone to examine the feedback relations between money creation and the inflation rate. No empirical examination of the causality issue has been performed for the Chinese hyperinflation. The major issue concerning the interrelationship between money and prices is the endogeneity of the money supply and the rate of inflation. It has been assumed that the accelerated rate of money creation which characterizes periods of hyperinflation is desired by government to extract real resources at a higher inflation rate in order to finance expenditures. The postulated hypothesis is that the higher the nominal price of goods and services the government de-
*The authors wish to express their gratitude to Professors Cheng Hsiao, mond Lombra, and Yash Mehra and one anonymous referee for their valuable ments on earlier versions of this paper. Joumd Copyright
of Macroeconomics, 0 1984 by Wayne
Fall 1983, Vol. State University
5, No. 4, pp. Press.
503-510
Ftaycom-
503
De-piao
Tang and Teh-wei Hu
mands, the more money the government needs to create, which in turn leads to further increase in prices. The post-World War II period in China, September 1945 to May 1949, qualifies as a hyperinflation period. Within the fortyfive-month period, the wholesale price index in China increased by 1.05 x lo”, an increase of about 78 percent per month [Hu (1971)], and the National Government deficit was approximately double the tax revenues [Chang (1958), pp. 166, 3741. The purpose of this note is to examine the causality issue by using the Chinese hyperinflation data. The empirical estimation will use both the Sims test and the Granger test [with a modified version proposed by Hsiao (1979)].
2. Empirical
Tests
The formation of the causality test between the rate of inflation (Z’) and the rate of money creation (M) and vice versa, according to Sims, is a two-sided regression model: P, = 5
ai M,-i + U, ;
(1)
bi P,-i +
(2)
i=-n
M, = ~
0, ;
i=-n
where U, and vt are error terms, and n and m denote the length of negative lags (i.e., future values) and past lags, respectively. The null hypothesis of Equation (1) indicates that no causality running from P to M is equivalent to all the coefficients on the future values of M being equal to zero. Similarly, the null hypothesis of Equation (2) indicates no causality running from M to P.’ The data sources for estimating Equations (1) and (2) are described in Hu [(1971), Table 11. T o eliminate the possible serial correlation problem, the P and M series were prefiltered, as shown ‘Money stock can be defined as currency plus demand deposit. In this paper only the currency data is used. From January 1946 through June 1946 a data series of the currency to demand deposit ratio is available. Thus, an alternative variable (currency plus demand deposit) was created. However, the results of this created variable are not statistically significant both in the individual coefficients and the F-statistic of the entire equation. As Chou [(1963), p. 2241 pointed out, during the Chinese inflationary period both price index and note issue usually expanded much faster than bank deposits. This indicated that the volume of deposits was definitely not a major force in generating the inflationary process. 504
*Estimated is the estimated
FG&W
DW F
Constant Time R2
4
3
2
0 1
I4
(absolute serial
12.20 3.93
2.16
0.274 0.519 0.016 -0.119 -0.080 0.004 0.93
t-statistics first-order
Data,
0.858 -0.171 0.490
-0.376 -0.317
value) are in parentheses. regression coefficient for
(0.98) (0.92)
(1.16)
(2.39) (4.51) (0.16)
the
fi
(3.29) (2.77) (7.66) (0.93) (2.37)
ai (1 - L)(l - fiL) In M,-i i= -* Coefficients on Coefficients on Future Rates of Lagged Rates of Money Creation Money Creation
Creation
Results (Monthly
Regressed on Money
Sims Test Regression
Inflation
1.
(1 - L)(l - fiL) In P, = i
TABLE
residuals
of
F, do. 01)
DW F
Constant Time R”
2 3 4
0 1
I4
Equation
on Znflation
(2),
by least
-0.126 0.004 0.91 1.28 21.50 3.93
0.182 -0.251 0.087
-0.237
squares;
(2.53) (2.56) (3.76) (1.47) (1.89) (0.98)
0 =
-0.06.
0.100
0.424
0.569
0.266
0.093
(0.92) (2.60) (4.95) (3.30) (1.02)
bi (1 - L)(l - DL) In P,-i i=-* Coefficients on Coefficients on Future Rates Lagged Rates of Inflation of lnf lation
Regressed
(1 - L)(l - fiL) In M, = i
Creation
1945 to May 1949)*
Money
September
De-piao Tang and Teh-wei Hu TABLE 2.
Lags of the Manipulated
The Optimum
Controlled” Variable
‘The number in the controlled
3 4
M P
f’(6) in the parentheses variable.
indicates
and
The Optimum Lag of Manipulated Variable
Manipulated Variable
M(9)
Variables
the
order
of autoregressive
operator
in the top of Table 1, before estimating Equations (1) and (2). Since the variables are expressed in terms of the rate of changes in inflation and money creation during the hyperinflation period, the data were transformed by a second order filter so that data series could become more stationary.2 Each equation includes a time trend variable. Table 1 provides the results of the two-sided regression analysis. In each regression analysis the relevant F-statistic is provided for testing the null hypothesis that the coefficients on future values of the variable on the right-hand side of the equation are zero. The F-statistic in the equation of inflation regressed on money creation is 12.20, well above the critical value of 3.93 at the onepercent level of significance. This means that at the one-percent level of significance we must reject the hypothesis that there is no unidirectional causality running from current inflation rate to future rates of money creation. The F-statistic in the equation of money creation regressed on inflation is 21.50, meaning that the hypothesis that there is no unidirectional causality running from current money creation to future inflation is also rejected at the one-percent level of significance. These results imply that there is feedback (two-way causality) between money creation and inflation: an acceleration in the rate of money creation strongly influences the subsequent inflation rate, which in turn influences subsequent changes in money creation.3 *In an earlier version of this paper, the data series was transformed by a first order filter; however, the results from the second order filter were superior. % should be noted, however, that the DW-statistic in the equation of money creation regressed on inflation indicates a serial correlation exists in the error term. As discussed in Hsiao (1979), there is no uniform approach toward the problem for serial correlation adjustment in the Sims test. A serial correlation adjustment may reduce the power of the Sims test. On the other hand, with a very high value for the F-statistic (21.5) and low value for J (-0.96) in the equation, the adjustment of serial correlation should not alter the two-way causality conclusion. 596
Chinese Hyperinflation, the Corresponding
FPEs FPE(M)
FPE(P)
1945-49
FPE(P, M)
10.1 X 1o-s
FPE(M,P)
4.70 x 1o-s 8.62 x lo-*
1.41 X lo-*
An alternative method for testing causality follows directly from the Granger definition. Let P and M be the two stationary time series. Consider the simple causal structure: P, = i j=l
Mt = i j=l
Cj M,-j + i
dj P,-j + IA, ;
(3)
ej P,-j + C f;. M,, + 0, .
(4)
j=l
j=l
The definition of causality implies that P causes M if M, is better predicted by including P,-j on the right-hand side of the equation. The role of M and P may be reversed so that a similar definition applies to the statement that M causes P. Feedback exists if both of these events occur. The hypothesis to be tested is that cj, ej are equal to zero, and this can be done using a standard F-test. Akaike’s final prediction error (FPE) criterion is adopted by Hsiao (1979) and is used to determine the optimum order of lags. The FPE of a predictor is the mean squared prediction error. In the case of Equation (3), the FPE is defined as T+m+n+lSSR FPE = W’, - pJ2 = T _ m _ n _ 1 y-
>
where P, is the predicted value of P,, given the estimated coefficients Zj, aj; T is the total number of observations; and SSR is the sum of squared residuals of the ordinary least squares regression. Following Hsiao’s procedure, with the maximum order of lags assumed to be 10, the smallest FPEs for P and M are 6 and 9, respectively. Holding the order of the autoregressive process for the controlled variable to the one specified above, the optimum lags of M and P are 3 and 4, respectively, as shown in Table 2. Since FPE (P, M) = 4.70 X 10-s is less than FPE (P) = 10.1 X 507
CA E2
.i
(1 - L)2
1 2 3 4 5 6 7 8 9 Constant
*Estimated
f.34wl~
R2 DW
Granger
Test (with
Hsiao’s Modi$cations)
9
ej(l - L)2 In P,,
(absolute
value)
are
(0.12)
0.006
0.85 2.32 8.94 3.93
(2.56) (5.01) (2.87) (1.78)
0.406 1.188 0.888 0.507
Coefficients on Lagged Rates of Znf lation
j=l
-0.874 -0.104 0.057 -0.540 -0.476 -1.076 0.101 0.405 0.738
(4.43) (0.38) (0.17) (1.55) (1.26) (2.86) (0.29) (0.85) (1.77)
Coefficients on Lagged Rates of Money Creation
in parentheses.
= +Cf;(l-L)ZhlM+j+D+v,
i=l
M, = i
t-statistics
In
Money Creation Regressed on Znflation and Lagged Money Creation
TABLE 3.
j
Constant
L FA,(O. 1)
R2 DW F
Money
Results*
=
2.08 10.08 3.08
0.77
0.041
0.419 1.066 -0.872
(0.66)
(1.89) (2.57) (1.94)
Coefficients on Lagged Rates of Money Creation
j=l
U,
0.238 0.829 0.372
0.298
0.042 -0.197
(0.21) (0.79) (0.92) (0.59) (2.64) (1.48)
Coefficients on Lagged Rates of Znf lation
6 + C dj(I - Ly In Ptej + C +
j=l
cj(l - L)2 In M,,
Znf lation Regressed on Creation and Lagged Inflation
(1 - L)2 In P, = i
Regression .-
Chinese Hyperinf
lation,
1945-49
lo-‘, money creation is said to cause inflation; the reverse hypothesis that there is no unidirectional causality running from inflation to money creation is rejected because FPE (M) is greater than FPE (M P). A similar conclusion from the Granger test about the presence of feedback between money creation and inflation is reported in Table 3. In the left portion of Table 3, the F-statistic is a test of null hypothesis that the lagged values of money creation do not improve the forecast of the inflation rate by more than basing it on the lagged values of the inflation rate alone, and we have to reject the null hypothesis at the one-percent level of significance. In the right portion of Table 3, the lagged values of inflation are the same as the lagged values of money creation in the left portion of the Table which are significantly different from zero.
3. Concluding
Remarks
The empirical results estimated in this note indicate that there was a strong feedback causation between money creation and inflation during the Chinese hyperinflation. Both the Sims test and the Granger test yield consistent results that suggest that both the rate of inflation and the money supply are endogenous variables. The example of the Chinese hyperinflation supports the notion that during a time of hyperinflation, the government often resorts to money creation in order to finance its expenditures. The mutual feedback between money creation and the rate of inflation is an important cause of the development of hyperinflation. Receioed: January 1982 Final version received: March
1983
References Cagan, P. “The Monetary Dynamics of Hyperinflation.” In Studies in the Quantity Theory of Money. M. Friedman, ed. Chicago: University of Chicago Press, 1956. Chang, K. The Inflationary Spiral: The Experience in China 19291950. Cambridge: MIT Press, 1958. Chou, S. The Chinese Inflation, 1937-49. New York: Columbia University Press, 1963. Frenkel, J. “The Forward Exchange Rate, Expectations, and the 509
De-piao Tang and Teh-wei Hu Demand for Money: The German Hyperinflation. ” American Economic Review 67 (1977): 653-70. Granger, C. “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods. ” Econometrica 37 (1969): 424-38. Hsiao, C. “Causality Tests in Econometrics.” Journal of Economic Dynamics and Control 1 (1979): 321-46. Hu, T. “Hyperinflation and the Dynamics of the Demand for Money in China, 1945-1959. ” Journal of Political Economy 79 (1971): 186-95. Sargent, T.J. and N. Wallace. “Rational Expectations and the Dynamics of Hyperinflation. ” lnternational Economic Review 14 (1973): 328-50. Sims, C. “Money, Income, and Causality.” American Economic Review 62 (1972): 540-52.
510
Vnioersity
IEH-VVEI HU Pennsylcaniu
State
Vnitiersity
Money, Prices, and Causality: The Chinese Hyperinflation, 7945@ This paper empirically examines causality between money and prices during the Chinese hyperinflation, 1945-49. The major issue concerning the interrelationship between money and prices is the endogeneity of money supply and the rate of inflation. Both the Sims test and the Granger test indicate that there was strong feedback causation between money creation and inflation during the Chinese hyperinflation. The mutual feedback between money creation and the rate of inflation is an important cause of the development of hyperinflation.
1. Introduction Since the publication of the Sims test (I972), a number of empirical studies have examined the causality between money and prices during hyperinflation. For instance, Sargent and Wallace (1973) examined the causality issue using Cagan’s data (1956) and concluded that in several hyperinflations-Germany, Austria, and Hungary I-there was feedback from inflation to subsequent rates of money creation while only in Greece was there feedback from money creation to the inflation rate. Frenkel (1977) used the German hyperinflation data alone to examine the feedback relations between money creation and the inflation rate. No empirical examination of the causality issue has been performed for the Chinese hyperinflation. The major issue concerning the interrelationship between money and prices is the endogeneity of the money supply and the rate of inflation. It has been assumed that the accelerated rate of money creation which characterizes periods of hyperinflation is desired by government to extract real resources at a higher inflation rate in order to finance expenditures. The postulated hypothesis is that the higher the nominal price of goods and services the government de-
*The authors wish to express their gratitude to Professors Cheng Hsiao, mond Lombra, and Yash Mehra and one anonymous referee for their valuable ments on earlier versions of this paper. Joumd Copyright
of Macroeconomics, 0 1984 by Wayne
Fall 1983, Vol. State University
5, No. 4, pp. Press.
503-510
Ftaycom-
503
De-piao
Tang and Teh-wei Hu
mands, the more money the government needs to create, which in turn leads to further increase in prices. The post-World War II period in China, September 1945 to May 1949, qualifies as a hyperinflation period. Within the fortyfive-month period, the wholesale price index in China increased by 1.05 x lo”, an increase of about 78 percent per month [Hu (1971)], and the National Government deficit was approximately double the tax revenues [Chang (1958), pp. 166, 3741. The purpose of this note is to examine the causality issue by using the Chinese hyperinflation data. The empirical estimation will use both the Sims test and the Granger test [with a modified version proposed by Hsiao (1979)].
2. Empirical
Tests
The formation of the causality test between the rate of inflation (Z’) and the rate of money creation (M) and vice versa, according to Sims, is a two-sided regression model: P, = 5
ai M,-i + U, ;
(1)
bi P,-i +
(2)
i=-n
M, = ~
0, ;
i=-n
where U, and vt are error terms, and n and m denote the length of negative lags (i.e., future values) and past lags, respectively. The null hypothesis of Equation (1) indicates that no causality running from P to M is equivalent to all the coefficients on the future values of M being equal to zero. Similarly, the null hypothesis of Equation (2) indicates no causality running from M to P.’ The data sources for estimating Equations (1) and (2) are described in Hu [(1971), Table 11. T o eliminate the possible serial correlation problem, the P and M series were prefiltered, as shown ‘Money stock can be defined as currency plus demand deposit. In this paper only the currency data is used. From January 1946 through June 1946 a data series of the currency to demand deposit ratio is available. Thus, an alternative variable (currency plus demand deposit) was created. However, the results of this created variable are not statistically significant both in the individual coefficients and the F-statistic of the entire equation. As Chou [(1963), p. 2241 pointed out, during the Chinese inflationary period both price index and note issue usually expanded much faster than bank deposits. This indicated that the volume of deposits was definitely not a major force in generating the inflationary process. 504
*Estimated is the estimated
FG&W
DW F
Constant Time R2
4
3
2
0 1
I4
(absolute serial
12.20 3.93
2.16
0.274 0.519 0.016 -0.119 -0.080 0.004 0.93
t-statistics first-order
Data,
0.858 -0.171 0.490
-0.376 -0.317
value) are in parentheses. regression coefficient for
(0.98) (0.92)
(1.16)
(2.39) (4.51) (0.16)
the
fi
(3.29) (2.77) (7.66) (0.93) (2.37)
ai (1 - L)(l - fiL) In M,-i i= -* Coefficients on Coefficients on Future Rates of Lagged Rates of Money Creation Money Creation
Creation
Results (Monthly
Regressed on Money
Sims Test Regression
Inflation
1.
(1 - L)(l - fiL) In P, = i
TABLE
residuals
of
F, do. 01)
DW F
Constant Time R”
2 3 4
0 1
I4
Equation
on Znflation
(2),
by least
-0.126 0.004 0.91 1.28 21.50 3.93
0.182 -0.251 0.087
-0.237
squares;
(2.53) (2.56) (3.76) (1.47) (1.89) (0.98)
0 =
-0.06.
0.100
0.424
0.569
0.266
0.093
(0.92) (2.60) (4.95) (3.30) (1.02)
bi (1 - L)(l - DL) In P,-i i=-* Coefficients on Coefficients on Future Rates Lagged Rates of Inflation of lnf lation
Regressed
(1 - L)(l - fiL) In M, = i
Creation
1945 to May 1949)*
Money
September
De-piao Tang and Teh-wei Hu TABLE 2.
Lags of the Manipulated
The Optimum
Controlled” Variable
‘The number in the controlled
3 4
M P
f’(6) in the parentheses variable.
indicates
and
The Optimum Lag of Manipulated Variable
Manipulated Variable
M(9)
Variables
the
order
of autoregressive
operator
in the top of Table 1, before estimating Equations (1) and (2). Since the variables are expressed in terms of the rate of changes in inflation and money creation during the hyperinflation period, the data were transformed by a second order filter so that data series could become more stationary.2 Each equation includes a time trend variable. Table 1 provides the results of the two-sided regression analysis. In each regression analysis the relevant F-statistic is provided for testing the null hypothesis that the coefficients on future values of the variable on the right-hand side of the equation are zero. The F-statistic in the equation of inflation regressed on money creation is 12.20, well above the critical value of 3.93 at the onepercent level of significance. This means that at the one-percent level of significance we must reject the hypothesis that there is no unidirectional causality running from current inflation rate to future rates of money creation. The F-statistic in the equation of money creation regressed on inflation is 21.50, meaning that the hypothesis that there is no unidirectional causality running from current money creation to future inflation is also rejected at the one-percent level of significance. These results imply that there is feedback (two-way causality) between money creation and inflation: an acceleration in the rate of money creation strongly influences the subsequent inflation rate, which in turn influences subsequent changes in money creation.3 *In an earlier version of this paper, the data series was transformed by a first order filter; however, the results from the second order filter were superior. % should be noted, however, that the DW-statistic in the equation of money creation regressed on inflation indicates a serial correlation exists in the error term. As discussed in Hsiao (1979), there is no uniform approach toward the problem for serial correlation adjustment in the Sims test. A serial correlation adjustment may reduce the power of the Sims test. On the other hand, with a very high value for the F-statistic (21.5) and low value for J (-0.96) in the equation, the adjustment of serial correlation should not alter the two-way causality conclusion. 596
Chinese Hyperinflation, the Corresponding
FPEs FPE(M)
FPE(P)
1945-49
FPE(P, M)
10.1 X 1o-s
FPE(M,P)
4.70 x 1o-s 8.62 x lo-*
1.41 X lo-*
An alternative method for testing causality follows directly from the Granger definition. Let P and M be the two stationary time series. Consider the simple causal structure: P, = i j=l
Mt = i j=l
Cj M,-j + i
dj P,-j + IA, ;
(3)
ej P,-j + C f;. M,, + 0, .
(4)
j=l
j=l
The definition of causality implies that P causes M if M, is better predicted by including P,-j on the right-hand side of the equation. The role of M and P may be reversed so that a similar definition applies to the statement that M causes P. Feedback exists if both of these events occur. The hypothesis to be tested is that cj, ej are equal to zero, and this can be done using a standard F-test. Akaike’s final prediction error (FPE) criterion is adopted by Hsiao (1979) and is used to determine the optimum order of lags. The FPE of a predictor is the mean squared prediction error. In the case of Equation (3), the FPE is defined as T+m+n+lSSR FPE = W’, - pJ2 = T _ m _ n _ 1 y-
>
where P, is the predicted value of P,, given the estimated coefficients Zj, aj; T is the total number of observations; and SSR is the sum of squared residuals of the ordinary least squares regression. Following Hsiao’s procedure, with the maximum order of lags assumed to be 10, the smallest FPEs for P and M are 6 and 9, respectively. Holding the order of the autoregressive process for the controlled variable to the one specified above, the optimum lags of M and P are 3 and 4, respectively, as shown in Table 2. Since FPE (P, M) = 4.70 X 10-s is less than FPE (P) = 10.1 X 507
CA E2
.i
(1 - L)2
1 2 3 4 5 6 7 8 9 Constant
*Estimated
f.34wl~
R2 DW
Granger
Test (with
Hsiao’s Modi$cations)
9
ej(l - L)2 In P,,
(absolute
value)
are
(0.12)
0.006
0.85 2.32 8.94 3.93
(2.56) (5.01) (2.87) (1.78)
0.406 1.188 0.888 0.507
Coefficients on Lagged Rates of Znf lation
j=l
-0.874 -0.104 0.057 -0.540 -0.476 -1.076 0.101 0.405 0.738
(4.43) (0.38) (0.17) (1.55) (1.26) (2.86) (0.29) (0.85) (1.77)
Coefficients on Lagged Rates of Money Creation
in parentheses.
= +Cf;(l-L)ZhlM+j+D+v,
i=l
M, = i
t-statistics
In
Money Creation Regressed on Znflation and Lagged Money Creation
TABLE 3.
j
Constant
L FA,(O. 1)
R2 DW F
Money
Results*
=
2.08 10.08 3.08
0.77
0.041
0.419 1.066 -0.872
(0.66)
(1.89) (2.57) (1.94)
Coefficients on Lagged Rates of Money Creation
j=l
U,
0.238 0.829 0.372
0.298
0.042 -0.197
(0.21) (0.79) (0.92) (0.59) (2.64) (1.48)
Coefficients on Lagged Rates of Znf lation
6 + C dj(I - Ly In Ptej + C +
j=l
cj(l - L)2 In M,,
Znf lation Regressed on Creation and Lagged Inflation
(1 - L)2 In P, = i
Regression .-
Chinese Hyperinf
lation,
1945-49
lo-‘, money creation is said to cause inflation; the reverse hypothesis that there is no unidirectional causality running from inflation to money creation is rejected because FPE (M) is greater than FPE (M P). A similar conclusion from the Granger test about the presence of feedback between money creation and inflation is reported in Table 3. In the left portion of Table 3, the F-statistic is a test of null hypothesis that the lagged values of money creation do not improve the forecast of the inflation rate by more than basing it on the lagged values of the inflation rate alone, and we have to reject the null hypothesis at the one-percent level of significance. In the right portion of Table 3, the lagged values of inflation are the same as the lagged values of money creation in the left portion of the Table which are significantly different from zero.
3. Concluding
Remarks
The empirical results estimated in this note indicate that there was a strong feedback causation between money creation and inflation during the Chinese hyperinflation. Both the Sims test and the Granger test yield consistent results that suggest that both the rate of inflation and the money supply are endogenous variables. The example of the Chinese hyperinflation supports the notion that during a time of hyperinflation, the government often resorts to money creation in order to finance its expenditures. The mutual feedback between money creation and the rate of inflation is an important cause of the development of hyperinflation. Receioed: January 1982 Final version received: March
1983
References Cagan, P. “The Monetary Dynamics of Hyperinflation.” In Studies in the Quantity Theory of Money. M. Friedman, ed. Chicago: University of Chicago Press, 1956. Chang, K. The Inflationary Spiral: The Experience in China 19291950. Cambridge: MIT Press, 1958. Chou, S. The Chinese Inflation, 1937-49. New York: Columbia University Press, 1963. Frenkel, J. “The Forward Exchange Rate, Expectations, and the 509
De-piao Tang and Teh-wei Hu Demand for Money: The German Hyperinflation. ” American Economic Review 67 (1977): 653-70. Granger, C. “Investigating Causal Relations by Econometric Models and Cross-Spectral Methods. ” Econometrica 37 (1969): 424-38. Hsiao, C. “Causality Tests in Econometrics.” Journal of Economic Dynamics and Control 1 (1979): 321-46. Hu, T. “Hyperinflation and the Dynamics of the Demand for Money in China, 1945-1959. ” Journal of Political Economy 79 (1971): 186-95. Sargent, T.J. and N. Wallace. “Rational Expectations and the Dynamics of Hyperinflation. ” lnternational Economic Review 14 (1973): 328-50. Sims, C. “Money, Income, and Causality.” American Economic Review 62 (1972): 540-52.
510