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An extended series of Divisia monetary aggregates.

The convention in monetary economics has been to create monetary aggregates by simply adding together the dollar amounts of the various financial assets included in them. This is the simple-sum method of aggregation. This procedure has been criticized because such monetary aggregates are essentially indexes that weight each component financial asset equally, a practice the is economically meaningful only under special circumstances.

A number of alternative indexes of monetary aggregates have been developed recently. The most well known are the Divisia monetary aggregates developed by Barnett (1980). This article reviews the theoretical basis for monetary aggregation and presents series of Divisia monetary aggregates for an extended sample period. The behavior of the simple-sum aggregates and their Divisia counterparts are compared over this period.

THE THEORETICAL BASIS FOR

MONETARY AGGREGATION(1)

Simple-sum aggregation stemmed directly from the classical economists' notion that the essential function of money is to facilitate transactions, that is, to serve as a medium of exchange. Assets that served as media of exchange were considered money and those that did not, were not. By this definition only two assets, currency and demand deposits, were considered money. Both assets were non-interest bearing, and individuals were free to alter the composition of their money holdings between currency and demand deposits at a fixed one-to-one ratio. Consequently the monetary value of transactions was exactly equal to the sum of the two monies.(2) Simple-sum aggregation was a natural extension of both restricting the definition of money to non-interest-bearing medium-of-exchange assets and of the fixed unitary exchange rate between the two alternative monies.(3)

In consumer demand theory, simple-sum aggregation is tantamount to treating currency and demand deposits as if they are perfect substitutes. Currency and demand deposits, however, are not equally useful for all transactions, so this assumption was clearly inappropriate. But, simple-sum aggregation of those two monetary assets was still appropriate because the assets were non-interest bearing and exchanged at a fixed one-to-one ratio. Consequently individuals would allocate their portfolio of money between the two assets until they equalized the marginal utilities of the last dollar held of each. Under these conditions, simple-sum aggregation is appropriate if it is also assumed that each agent is holding his equilibrium portfolio.

The recognition that non-interest-bearing demand deposits may have paid an implicit interest weakened the theoretical justification for simple-sum aggregation. A more serious blow to simple-sum aggregation, however, was dealt by a shift in monetary theory to emphasizing the store-of-value function of money.(4) That an asset could not be used directly to facilitate transactions was no longer a sufficient condition for excluding it from the definition of money. Instead, the asset approach to money emphasized money's role as a temporary abode of purchasing power that bridges the gap between the sale of one item and the purchase of another. Currency and checking accounts are money because they are both media of exchange and temporary abodes of purchasing power. Non-medium of exchange assets are superior to currency and non-interest-bearing checking accounts as stores of value because they earn explicit interest. This superiority typically increases with the length of time between the sale of one item and subsequent purchase of another because the cost of getting into and out of such assets and the medium of exchange assets is thought to be small and not proportional to the size of the transaction.

This shift in emphasis in monetary theory dramatically expanded the number of assets that were considered money and the number of alternative monetary aggregates proliferated.(5) Nonetheless, the method of aggregation remained the same - simple-sum aggregation.

As more financial assets came to be considered money, it became increasingly clear that it was inappropriate to treat these assets as perfect substitutes. Some financial assets have more "moneyness" than others, and hence they should receive larger weights. In what appears to be the first attempt at constructing a theoretically preferable alternative to the simple-sum monetary aggregate, Chetty (1969) added various savings-type deposits, weighted by estimates of the degree of substitution between them and the pure medium of exchange assets, to currency and demand deposits. Larger weights were given to assets with a higher estimated degree of substitution.(6)

Divisia aggregation, which also relies on consumer demand theory and the theory of economic aggregation, treats monetary assets as consumer durables such as cars, televisions and houses. They are held for the flow of utility-generating monetary services they provide. In theory, the service flow is given by the utility level. Consequently the marginal services flow of a monetary asset is its marginal utility. In equilibrium, the marginal service flow of a monetary asset is proportional to its rental rate, so the change in the value of a monetary asset's service flow per dollar of the asset held can be approximated by its user cost. The marginal monetary services of the components of Divisia aggregates are likewise proxied by the user costs of the components assets. The user cost of each component is proportional to the interest income foregone by holding it rather than a pure store-of-wealth asset - an asset that yields a high rate of return but provides no monetary services. Currency and non-interest-bearing demand deposits have the higher user cost because they earn no explicit interest income. Consequently they get the largest weights in the Divisia measure. On the other hand, pure store-of-wealth assets get zero weights.(7)

The object of a Divisia measure is to construct an index of the flow of monetary services from a group of monetary assets, where the monetary service flow per dollar of the asset held can vary from asset to assert.(8) Applying an appropriate index number to a group of assets is not sufficient, however, to get a correct measure of the flow of monetary services. The index must also be constructed from a assets that can be aggregated under conditions set by consumer demand theory. The objective of economic aggregation is to identify a group of goods that behave as if they were a single commodity. A necessary condition for this is block-wise weak separability. Block-wise weak separability requires that consumers' decisions about goods that are outside the group do not influence their preferences over the goods in the group whatsoever.(9) If this condition is satisfied, consumers behave just as though they were allocating their incomes over a single aggregate measure of monetary services and all other commodities to maximize their utility. Their total expenditure on monetary services is subsequently allocated over the various financial assets that provide such services.

The Divisia index generates such a monetary aggregate. Moreover, in continuous time it has been shown to be consistent with any unknown utility function implied by the data. In discrete time the Divisia index is in the class of superlative index numbers. Simple-sum indexes, on the other hand, do not have this desirable property. Thus they have no basis in their consumer demand theory or aggregation theory.(10)

In principle, all financial assets other than pure store-of-wealth assets provide some monetary services. Which assets can be combined into a meaningful monetary aggregate is in empirical issue because economic theory does not tell us which group of assets satisfies the condition of block-wise weak separability. Unfortunately, the most widely used test for weak separability is not powerful.(11) Consequently, it has been common simply to create Divisia indexes under the maintained hypothesis that the assets that compose the aggregate satisfy this condition. Thus the issues of the appropriate method of aggregation and the appropriate aggregate have been treated seperately.(12)

SIMPLE-SUM AND DIVISIA

MONETARY INDEXES

A simple-sum monetary aggregate is a measure of the stock of financial assets that compose it, whereas a Divisia monetary aggregate is a measure of the flow of monetary services from the stocks of financial assets that compose it.(13) For this reason alone, the methods of measurement are quite different. Simple-sum aggregates are obtained by simply adding the dollar amounts of the component assets. On the other hand, Divisia monetary aggregates are obtained by multiplying each component asset's growth rate by its share weights and adding the products. A component's share weight depends on the user costs and the quantities of all component assets.(14) Specifically, the share weight given to the [j.sup.th] component asset at time t is its share of total expenditures on monetary services; that is,

[Mathematical Expression Omitted]

where [q.sub.j] denotes the nominal quantity of the [j.sup.th] component assets, [u.sub.j] denotes the [j.sup.th] component's user cost and n donotes the number of component financial assets. The user cost is equal to [(R-r.sub.j)]p/ (1 + R), where R is the benchmark rate (that is, the rate on the pure store-of-wealth asset), [r.sub.j], is the own rate on the [j.sup.th] component, and p is the true cost-of-living price index that cancels out of the numerator and denominator of the shares. The growth rate of the [i.sup.th] Divisia monetary aggregate, [GDM.sub.i], is given by

[Mathematical Expression Omitted]

where [g.sub.jt] is the growth rate of [q.sub.jt].(15)

A Comparison of Simple-Sum and

Divisia Monetary Aggregates

Because the Divisia aggregates are an alternative to the conventional simple-sum aggregates, it is instructive to compare them. When constructing data in this section, the authors used an extension of the Farr and Johnson (1985) method. The Appendix present details of the construction of the Divisia monetary aggregates used here.

A Divisia monetary index is an approximation to a nonlinear utility function. Because it is an index, the level of utility is an arbitrary unit of measure; the level of the index has no particular meaning.(16) Nevertheless, because they are alternative measures of money, the Divisia and simple-sum aggregates are frequently compared to see how any analysis of the effects of monetary policy or other issues might be affected by the method of aggregation. The comparison of the levels of the simple-sum and Divisia measures is made by normalizing both measures so that they equal 100 at some point in the series, usually the first observation.(17) Comparisons of the levels and growth rates of the Divisia and simple-sum measures are presented in figures 1-5 for four monetary aggregates, M1A, M1, M2 and M3, and for total liquid assets, L.(18) The figures have two scales. The left-hand scale indicates the growth rate, and the right-hand scale indicates the level of the series. Both indexes equal 100 in January 1960.

M1A

M1A comprises currency and non-interest-bearing demand deposits held by households and businesses. Although neither household nor business demand deposits earn explicit interest, business demand deposits are assummed to earn an implicit own rate of return proportional to the rate paid on one-month commercial paper.(19) Consequently, additional units of business demand deposits are assumed to yield a smaller flow of monetary services than are additional units of household demand deposits. On the other hand, the simple-sum measure implicitly assumes that each unit of each component provides the same flow of monetary services. Hence the Divisia aggregate gives more weight to the growth rates of currency and household demand deposits than does the simple-sum aggregate.(20)

The average differences in the growth rates of the simple-sum and Divisia measures of M1A for the entire sample period, January 1960 to December 1992, and for selected sub-periods are presented in table 1. Because currency generally grew more rapidly than demand deposits over the sample period, the growth rate of Divisia M1A averaged about half a percentage point higher than the growth rate of simple-sum M1A over the entire period.(21) Much of this differences occurs during the latter part of the 1980s, when the growth rate of demand deposits generally slowed relative to the growth rate of currency.(22) This more rapid growth of the Divisia measure is reflected in a generally widening gap between the levels of the indexes.

M1

The behavior of simple-sum and Divisia M1 is similar to that of M1A. Indeed, the growth rates of simple-sum and Divisia M1 were similar until the late 1970's, when the growth of interest-bearing NOW accounts began to accelerate. The sharp rise in NOW accounts after their nationwide introduction on January 1, 1981, tended to increase the growth rate of the simple-sum measure relative to the Divisia measure because the growth rate of NOW accounts gets a smaller weight in the Divisia measure. As a result, the Divisia measure grew more slowly on average than the simple-sum measure from the late 1970s until the mid-1980s, after growing more rapidly previously. However, in neither period is the average difference in the growth rate of the alternative measures large.(23)

After the late 1980s the Divisia measure grew more rapidly than the simple-sum measure, reflecting the rise in the growth rate of currency relative to the growth rate of checkable deposits. Of course, the smaller average difference in the growth rates of the alternative M1 aggregates compared with 1 M1A is reflected in a smaller difference in the levels of the two indexes as well.

M2, M3 and L

Not surprisingly, larger differences arise when the monetary measures are broadened to include savings-type deposits because their explicit own rates of return are higher than those of transactions deposits. The higher own rate reduces the share weights of these components assets relative to the weights they receive in the simple-sum measures. During the sample period the growth rates of the broader simple-sum aggregates tend to be substantially larger than those of the corresponding Divisia measures. For the broader measures, the average growth rates of the simple-sum measures are about 2 percentage points greater than the corresponding Divisia measures over the entire sample period.

Much of this differences arises from the late 1970s to the mid-1980s and is likely due to financial innovation and deregulation in the period. The late 1970s witnessed a marked acceleration in the growth of money market mutual funds. These accounts paid relatively high interest rates and had limited transactions capabilities. A number of new deposit instruments that paid higher market interest rates were introduced in the early 1980s and Regulation Q interest rate ceiling were being phased out.(24) Moreover, short-term interest rates reached very high levels in the early 1980s. With share weights sensitive to the spread between an asset's own rate of return and the return on the benchmark assets, it is not surprising that the Divisia measures grew markedly slower than the corresponding simple-sum measures during this period. Nevertheless, the significantly slower growth of the broader Divisia measures during this period is more consistent with the disinflation of the period than is the growth of the simple-sum aggregates, whose growth remained fairly rapid. Although the growth rates of the broader Divisia and simple-sum aggregates have been essentially the same, on average, since a about the mid-1980s, the pattern of growth of these alte alternative measures is somewhat different.

A Comparison of Broader Divisia

Aggregates

That Divisia aggregation gives relatively small weight to less liquid assets that yield high rates of return suggests that differences in the growth rates of successively broader Divisia monetary aggregates will tend to get smaller.(25) The levels of Divisia M2, M3 and L presented in figure 6 and simple correlations of the compounded

Gannual growth rates of these Divisia aggregates presented in table 2 confirm this. The growth rates of Divisia M3 and L differ little from the growth rate in Divisia M2. This implies that adding successively less liquid assets to those in M2 adds little to the flow monetary services.(26) That the average difference in the growth rates of Divisia M2 and L is nearly zero over the entire sample period is reflected by the levels of the two Divisia aggregates, which are essentially equal by the end of the sample. Divisia M3, however, has, grown more rapidly than the other measures, so the spread between its level and the levels of Divisia M2 and L has widened over the sample period.

CONCLUDING REMARKS

Despite their theoretical advantage, Divisia and other weighted monetary aggregates have garnered relatively little attention outside of academe, and the official U.S. monetary aggregates remain simple-sum aggregates. The official reliance on simple-sum aggregates will probably continue unless the Divisia aggregates or other alternative weighted aggregates are shown to be superior in economic and policy analysis. Although nothing definitive can be said about this issue from the simple analysis of the data presented here, a few observations are offered.

First, that the growth rates of the narrow simple-sum sum and Divisia monetary aggregates are quite similar suggests that the method of aggregation may not be important at low levels of aggregation.(27) For example, it does not appear that conclusions about the long-run effects of money growth on inflation would be much different using either simple-sum or Divisia M1 or M1A. The average difference in the growth rates of narrow simple-sum and Divisia monetary aggregates is small. This observation is consistent with the empirical work of Barnett, Offenbacher and Spindt (1984) who, using a broad array of criteria, found that the difference in the performance of simple-sum and Divisia monetary aggregates was small at low levels of aggregation.

Second, the method of aggregation is likely to be more important for broader monetary aggregates. Beyond some point, however, a further broadening of the monetary aggregate makes little difference. For the United States, the differences in the average growth of Divisia M2, M3 and L are small. Consequently, long-run analysis using the growth rates of any of these Divisia aggregates is likely to produce similar results. Monthly growth rates of these Divisia aggregates are also highly correlated. Hence it would not be too surprising to find the broader Divisia aggregates perform similarly to one another in many short-run analyses as well.

These observations point to the critical need for more work to determine which financial assets should be included in the appropriate monetary aggregate. In consumer demand theory, these assets must satisfy the condition of weak separability. If analysis suggests a relatively narrow monetary aggregate such as M1, policymakers may be reluctant to adopt the theoretically superior index measure because, as a practical matter, the method of aggregation may not be empirically important.

If such test point to an aggregate that includes a much broader array of financial assets, the practical case for the weighted aggregates will be enhanced. Even casual analysis of simple-sum and Divisia monetary aggregate data show differences in both the levels and grown rates of these aggregates that are large, suggesting that the method of aggregation is important. Consequently, the method of aggregation should also be a concern for those who favor broader monetary aggregates on other grounds. The objective of the present article in publishing Divisia monetary statistics is to stimulate further empirical research both on the importance of monetary aggregation and on the role money in the economy.

(1) The discussion in this section is based on consumer demand theory. This may not be a serious limitation. For example, Feenstra (1986) has shown that money in the utility function is equivalent to other approaches. These approaches assume, however, that all of the cost and benefits of money are internalized, and it is commonly believed that there are externalities to the use of money in exchange (see Laidler [1990]). (2) This need to be true for the economy as a whole when measured over a sufficiently long time interval. In this case the amount of each from of money multiplied by its turnover velocity will equal total expenditures. This is the basis for the velocity of the demand for money. Fisher (1911) explicitly recognized that turnover velocities of currency and checkable deposits would likely be different. He circumvented this problem by assuming that there was an optimal currency-to-deposit ratio that would be a function of economic variables. Given these variables, the demand for the two alternative monetary assets was taken to be strictly proportional. Moreover, because individuals were free to adjust their money holdings between currency checkable deposits quickly and at low cost, Fisher argued that the actual ratio would deviate from the desired ratio for only short periods. For some recent evedince that the actual currency-to-deposit ratio might be determined by the policy actions of the Federal Reserve, see Garfinkel and Thornton (1991). The possibility that currency and checkable deposits have different turnover velocities is the basis for Spindt's (1985) weighted monetary aggregate, MQ.

(3) There is an issue of whether the fixed ratio was endogenous from either the perspective of supply or demand, or the result of arbitrary legal restrictions. From the demand, side this would require that these assets be perfect substitutes for all transactions. From the supply side, Pesek and Saving (1967) argued that the one-to-one exchange rate was a natural outcome of competitive pressures in the banking industry. Whether the fixed one-to-one is the endogenous outcome of a free market economy or is simply due to legal restrictions remains controversial.

Of course today some checkable deposits earn explicit interest. Consequently such deposits are a better store of wealth than currency. They are also a preferable medium of exchange for some, but not all, transactions.

(4) There has been a difference of opinion about the degree of emphasis that should be placed on the asset and transactions motives for holding money. Indeed, Laidler (1990, pp. 105-6) has noted that " . . .the most extraordinary development in monetary theory over the past fifty year is they way in which money's means-of-exchange and unit-of-account roles have vanished from what is widely regarded as the mainstream of monetary theory."

Broaching the medium-of-exchange line of demarcation between money and non-money assets also gave rise to an extensive literature on the empirical definition of money. For a critique of this literature and the idea of distinguishing between monetary and non-monetary assets based on the concept of the temporary abode of purchasing power, see Mason (1976).

(5) At one point the Federal Reserve published data on five alternative monetary aggregates. (6) Chetty's work was motivated by the Gurley/Shaw hypothesis and the general lack of agreement in the empiricial findings of Feige (1964) and others about the degree of substitutability between money and near-money assets. Gurley and Shaw (1960) suggested that the effectiveness of monetary policy was limited because of the high degree of substitutability between money (currency and demand deposits) and near-money (various bank and nonbank savings-type accounts) assets. Subsequent research has tended to support Feige's finding of a relatively low degree of substitutability between transactions media and liquid, non-medium-of-exchange assets. See Fisher (1989) for a survey of much of this literature.

(7) There does not appear to be agreement about what constitutes the best proxy measure for the theoretical pure store-of-wealth asset. Barnett, Fisher and Serletis (1992, p. 2,093) state the following, "The benchmark asset is specifically assumed to provide no liquidity or other monetary services and is held solely to transfer wealth intertemporally. In theory R (the benchmark rate) is the maximum expected holding period yield in the economy. It is usually defined in practice in such a way that the user costs for the monetary assets are (always) positive." Parentheses added. The Baa bond rate, or the highest rate paid on any of the component assets when the yield curve becomes inverted, has frequently been used to construct Divisia aggregates.

(8) See Barnett, Fisher and Serletis (1992) and Yue (1991a and b) for more detailed analyses of issues in monetary aggregation.

(9) Technically the marginal rates of substitution between any two goods inside the group must be independent of the quantities of the goods consumed that are outside of the group.

(10) Fisher (1922) was especially critical of the simple-sum index in his extensive analysis of index numbers. In particular, Fisher argued that simple-sum aggregates cannot internalize pure substitution effects associated with relative price changes. Thus changes in utility, which should occur only as a result of the income effect associated with relative price changes, occur in simple-sum aggregates because of both income and substitution effects.

(11) The most widely used test, developed by Varian (1982, 1983), is not statistical. The null hypothesis of weak separability is rejected if a single violation of the so-called regularity conditions is found. Because tests for weak separability lack power, Barnett, Fisher and Serletis (1992, p. 2,095) argue that "existing methods of conducting such tests are not . . . very effective tools of analysis," See Barnett and Choi (1989) for evidence indicating that available tests of block-wise weak separability are not very dependable. For results of tests for weak separability, see Belongia and Chalfant (1989) and Swofford and Whitney (1986), 1987).

(12) A common practice both in the United States and abroad is to construct Divisia monetary aggregates for collections of assets that are reported by the country's central bank. For example, see Yue and Fluri (1991), Belongia and Chrystal (1991) Ishida (1984).

(13) It should be noted that the accounting stock that is, the sum of the dollar amounts of all assets that are considered money, is not necessrily equal to the capital stock of money. The accounting stock is the present value of both service flow of money and the interest income (the service as a store of value). The economic capital stock of money comprises only the present value of the flow of monetary services. See Barnett (1991) for the formula for the economic capital stock of money.

(14)For the Divisia monetary aggregates, the share weight of each component's growth rate is its expenditure share of total expenditures on monetary services. Theoritically the share weights for the Divisia monetary aggregates are not a function of prices or user costs, but of quantities. The observable user costs are substituted for the unobservable marginal utilities under the implicit assumption of market-clearing equilibrium, where each consumer holds an optimal porfolio of monetary and nonmonetary assets. For the simple-sum monetary aggregates, the share weights are the components' share of the aggregate.

(15) GDM[sub.j] - In Dl[sub.t-1,] where Dl denotes the Divisia index. The index is initialized at 100, that is, Dl[sub.0]=100. See Farr and Johnson (1985) for more details.

(16) Rotemberg (1991) derives a weighted monetary aggregate stock under the conditions of risk neutrality and stationarity expectations; however, Barnett (1991) shows that this measure is the discounted value of future Divisia monetary service flows.

(17) An alternative justification for comparing the Divisia and simple-sum aggregates might come from noting that the appropriate Divisia monetary aggregate would be the simple-sum aggregate if all of the component assets had identical own rates. Such a comparison is tenuous, however, because the actual level of the simple-sum aggregate might have been different from the observed level had the user costs actually equal.

It is common to compare the levels and growth rates of simple-sum and Divisia monetary aggregates. For example, see Barnett, Fisher and Serletis (1992). Because Divisia indexes involve logarithms, the growth rate of a component asset is plus or minus infinity, respectively, when a component is introduced or eliminated. To circumvent this problem the Divisia index is replaced by Fisher's ideal index at these times and the user cost is measured by its reservation price during the period that precedes the introduction or follows the elimination of the asset. See Farr and Johnson (1985) for a discussion of this procedure.

(18) Note that the simple-sum aggregates presented here are not identical to the official published series. The official series are obtained by adding the non-seasonally adjusted components and seasonally adjusting the aggregate as a whole or by adding large subgroups of component asset that have been seasonally adjusted as a whole. The simple-sum aggregate presented here are obtained by adding the components after each component (that has a distinctive seasonal) has been seasonally adjusted. See the Appendix for details. A comparison of the series used here and the official series shows that the differences are small.

(19) Alternatively, estimates of the own rate on household demand deposits could also be used. However, such a series was not available for the entire sample period. Moreover, the desire was to follow the procedure used by Farr and Johnson (1985) as closely as possible.

(20) In both cases, the sum of the weights must equal unity.

(21) Currency grew at an annual rate of 7 percent during the entire period, whereas household and business demand deposits both grew at a 3.2 percent annual rate.

(22) This is a period of very reserve growth. Because reserves and checkable deposits are tied closely together under the present system of reserve requirements it is not surprising that is also a period of slow growth in checkable deposits, including household and business demand deposits. See Garfinkel and Thornton (1991) for a discussion of the relationship between reserves and chekable deposits under the present system of reserve requirements.

(23) We have refrained from using the phrase "statistically significant" because these observations are clearly distributed identically and independently, so the "t-statistics" reported in table 1 are biased and neither the direction nor extent of the bias is known. These statistics are presented to give the reader a rough approximation of the magnitude of the differences in the growth rates. Correlegrams of the differences in the growth rates of simple-sum and Divisia M1A and M1 show some lower level persistence through the sample period and some large spikes at seasonal frequencies after 1969. Correlegrams for the difference in the growth rates of the broader monetary aggregates reveal some higher level persistence. In any event, differences that are small in absolute value tend to be small relative to the estimated standard errors, and differences that are large in absolute value tend to be large in relative terms.

Another measure of the distance between the growth rates is the square root of the sum of the squared differences in the growth rates. These measures for the entire sample period are 58.5, 52.1, 69, 81.4 and 77.6 for M1A, M1, M2, M3 and L, respectively. These data are broadly comparable with those presented in table. 1.

(24) For a discussion of the financial innvations of this period see Gilbert (1986) and Stone and Thornton (1991).

(25) Of course this tendency also exists for the simple-sum aggregates. For the simple-sum aggregate, the growth rate of each component is weighted by the component's share of the total asset. Hence the growth rates of successively broader monetary aggregates could diverge if the marginal components were successively larger. For example, this is what happens from M1 to M2. The growth rates tend to converge, however, because the marginal components are smaller. This tendency is exacerbated in the Divisia measures because of smaller weights associated with higher own rates of return on successively less liquid assets.

(26) The average differences in the growth rates of Divisia M2, M3 and L over the sample period are small (less than 0.12 percentage points in absolute value). The absolute values of the average differences in the growth rates of simplesum M2, M3 and L are larger than those of the corresponding Divisia measures; the standard errors are also much larger.

(27) There may be some differences in the levels, however, because the levels of the simple-sum and Divisia measures do not appear to be cointegrated at any level of aggregation.
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Author:Thornton, Daniel L.; Yue, Piyu
Publication:Federal Reserve Bank of St. Louis Review
Date:Nov 1, 1992
Words:5233
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