# Building new monetary services indexes: concepts, data and methods,

This is the second of two articles that describe the monetary
services index (MSI) project of the Federal Reserve Bank of St. Louis.
The projects MSI database, which contains the monetary services index
(MSI), its dual user cost index, and other related indexes and data, is
available on the Bank's World Wide Web server.(1) To facilitate
comparison with the monetary aggregates published by the Board of
Governors of the Federal Reserve System, all of the indexes in the MSI
database are provided for the same groupings of monetary assets as the
Board's M1, M2, M3, and L aggregates.(2) Indexes are provided at
monthly, quarterly, and annual frequencies. The St. Louis MSI database
also contains all non-confidential data and computer programs used to
construct the indexes.

Unlike the Board of Governor's monetary aggregates, the monetary services indexes and their dual user cost indexes are statistical index numbers, based on economic aggregation and statistical index number theory. The previous article in this Review, "Monetary Aggregation Theory and Statistical Index Numbers," surveys the literature on monetary aggregation theory and the use of statistical index number theory in monetary economics. Here, we discuss the construction of the monetary services index and related indexes.

In the first section, we define notation and introduce some key concepts that are used throughout the article. We emphasize the distinction between real and nominal monetary asset stocks and their user costs, and we review the concepts of the real monetary services index and its nominal dual user cost index. In the second section, we define each of the indexes in the monetary services indexes database, including the following: total expenditure on monetary assets; the nominal monetary services index; the real dual user cost index; the currency equivalent index; the simple sum index; and a set of indexes based on Theil's (1967) stochastic approach to index number theory. We emphasize that it is important to distinguish between real and nominal monetary index numbers: The aggregation theory underlying the monetary services indexes and related indexes is developed in terms of the real stocks of monetary assets, but actual monetary asset stock data are collected in nominal terms. We conclude that it is appropriate to construct a nominal monetary services index and thereafter to produce an approximation to the real monetary services index by deflating the nominal index.

In the third section, we describe the monetary asset stock data. We discuss the issue of weak separability, and we define the groupings of monetary assets for which we construct indexes. These groupings correspond to the assets contained in M1, M2, M3, and L, as well as the assets contained in M1A and MZM.(3) Because the aggregates are nested - each broader aggregate contains all the components of the previous, narrower aggregate - we refer to the groupings as levels of aggregation. M1A is the narrowest level of aggregation and L the broadest.

In the fourth section, we discuss the own rate of return data used in the construction of the indexes, and we detail the sources of this data. Of special importance are the methods by which we construct own rates for particular monetary assets. Specific issues include the implicit rate of return on demand deposits, fixed and variable ceiling rates for rate-regulated monetary assets, and the market rate of return on savings bonds. When the sample period of the own-rate data is shorter than that of the associated asset-stock data, we construct proxies for the missing own-rate data. Finally, we review the own rate conversions and yield curve adjustments of particular rates that are necessary because not all own rates are reported on the same basis or for the same maturity

In the fifth section, we detail the calculation of monetary asset user costs. Some published monetary asset stock data are, in fact, aggregates, or sub-indexes, of individual monetary assets with different user costs. It is necessary to obtain a single user cost index for these sub-indexes. Our solution to this problem, derived from unilateral index number theory, is described in this section. Finally, we discuss the concept of a benchmark asset and detail how we construct its rate of return.

The sixth section contains a discussion of some methodological difficulties associated with the project. The first is the introduction of new monetary assets; we implement Diewert's (1980) recommended solution to this problem. The second difficulty is created when published data for several monetary assets are combined into data for a single aggregate, or sub-index. We argue that it is inappropriate to treat this sub-index as a new monetary asset because doing so imputes economic relevance to the change in data reporting, when in fact there is none. We propose a solution to this problem that is based on the theory of splicing index numbers. The third issue is time aggregation. We use monthly data to construct the indexes. We implement Diewert's (1980) time aggregation methodology to produce quarterly and annual indexes from monthly data. Finally, we discuss seasonal adjustment.

NOTATION AND KEY CONCEPTS

In this section, we introduce notation and define some key concepts that will be used throughout the article. Readers are cautioned that this article's notation differs somewhat from that in "Monetary Aggregation Theory," because here we distinguish between real and nominal monetary assets.

Monetary Asset Stocks and User Costs

Assume that there are n monetary assets. Let [Mathematical Expression Omitted] denote the optimal real stock of monetary asset i in period t, and let

[Mathematical Expression Omitted]

be the vector of these optimal real stocks.(4) Similarly, let [Mathematical Expression Omitted] denote the optimal nominal stock of monetary asset i in period t, and let

[Mathematical Expression Omitted]

be the vector of these optimal nominal stocks. Real and nominal holdings of monetary assets are related by the identity

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is a true cost-of-living index.(5)

The user cost of an asset is the equivalent rental price of that asset. If an asset fully depreciates during the economic agent's decision period, it is said to be non-durable and the asset's user cost equals its market price. If an asset does not fully depreciate within the decision period, it is said to be durable, and the appropriate opportunity cost of the durable asset is its user cost. Monetary assets are assumed to be durable. Expressions for the user cost of monetary assets were derived for consumers in Barnett (1978), and for firms in Barnett (1987, 1990).

To define the user costs of monetary assets, we need the concept of a benchmark asset - a risk-free asset that can be used only for intertemporal transfer of wealth and provides no monetary services. Let [r.sub.it] represent the nominal holding period yield on monetary asset i in period t, and let [R.sub.t] be the nominal holding period yield on the benchmark asset, called the benchmark rate, in period t.(6) The nominal user cost of monetary asset i in period t, [Mathematical Expression Omitted], is equal to the nominal value of interest income foregone by holding a unit of that asset for one period, [Mathematical Expression Omitted], discounted by [1/(1 + [R.sub.t])] to reflect the receipt of interest at the end of the period:

[Mathematical Expression Omitted],

This form of the user cost for a monetary asset is valid for both consumers and firms. Note, however, that consumers and firms often face different market interest rates and prices; hence their user costs will differ.

The real user cost of monetary asset i in period t, [Mathematical Expression Omitted],is defined by

[Mathematical Expression Omitted],

and the nominal and real user costs are related by the identity

[Mathematical Expression Omitted].

The real stock of a durable asset multiplied by its nominal user cost is equal to the total expenditure on that asset. Thus, expenditure on monetary asset i in period t is given by the product [Mathematical Expression Omitted], and total expenditure on monetary assets in period t is given by

[Mathematical Expression Omitted].

The Monetary Services Index and the Dual User Cost

Barnett (1980, 1987, 1990) derived the conditions under which monetary quantity and dual user cost aggregates will exist. These conditions are reviewed in our previous article in this Review, "Monetary Aggregation Theory." For consumers and firms, the monetary quantity aggregate is a measure of the flow of monetary services received by the holders of the monetary assets. Barnett (1980) first suggested the use of superlative statistical index numbers to track the flow of monetary services. Statistical index numbers, which contain no unknown parameters, are specification- and estimation-free functions of the prices and optimal quantities observed in two time periods.

Diewert (1976) showed that there exists a class of statistical index numbers, which he called superlative, that can provide second-order approximations to arbitrary economic aggregates in discrete time. Although there are many superlative index numbers, the Tornqvist-Theil index number is the only one known to retain its second-order tracking properties when some common aggregation theoretic assumptions are violated (Caves, Christensen, and Diewert, 1982; Anderson, Jones, and Nesmith, 1997).

Monetary quantity index numbers have been referred to by a variety of names in the past. We label our quantity indexes as monetary services indexes because of their close connection to the flow of monetary services in microeconomic demand models. The real Tornqvist-Theil monetary services index (chained Tornqvist-Theil quantity index formula), [Mathematical Expression Omitted], is defined by

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the expenditure shares of monetary asset i in periods t and t-1, respectively, and the average expenditure share of monetary asset i in period t is [Mathematical Expression Omitted].(7)

An index number that is dual to [Mathematical Expression Omitted] can be used to measure the price of a unit of monetary services. A price index number is said to be dual to a quantity index number if their product is equal to the total expenditure on the component assets included in the indexes, a property called factor reversal. Dual to [Mathematical Expression Omitted] is the nominal dual user cost index, [Mathematical Expression Omitted], which is defined using Fisher's (1922) weak factor reversal criterion by the formula

[Mathematical Expression Omitted].

Our real monetary services index, [Mathematical Expression Omitted], and its nominal dual user cost index, [Mathematical Expression Omitted], are constructed as chained superlative indexes. They therefore have the same statistical properties as other chained superlative quantity and price indexes - such as real gross domestic product (GDP), real personal consumption expenditures (PCE), and their price deflators - that are currently published by the U.S. Department of Commerce (Triplett, 1992).

INDEXES IN THE MSI DATABASE

In this section, we define the indexes in the MSI database. The formulas and definitions for these indexes are summarized in Table 1.

In this article, we distinguish carefully between nominal and real stocks of monetary assets. Monetary asset data collected by the Federal Reserve are necessarily in nominal terms, while monetary aggregation and statistical index number theory provide conditions for the aggregation of real stocks of monetary assets (Barnett, 1978, 1980, 1987, 1990; and our previous article in this Review, "Monetary Aggregation Theory."). Aggregation of nominal, rather than real, stocks of monetary assets requires some extension of the theory.

The identities and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] can be used to demonstrate that total expenditure on monetary assets, [Y.sub.t], may be represented in two equivalent ways. Total expenditure may be expressed as either: (1) the sum of the products of the real asset stocks and their nominal user costs, or as (2) the sum of the products of the nominal monetary asset stocks and their corresponding real user costs:

[Mathematical Expression Omitted].

This result implies that the expenditure shares do not depend on the price index, [Mathematical Expression Omitted], and hence can be calculated correctly using observed nominal asset stocks and real user costs. The expenditure shares may be interpreted as either expenditure on real assets based on nominal user costs,

[Mathematical Expression Omitted],

or as expenditure on nominal assets based on real user costs,

[Mathematical Expression Omitted].

These relationships are important because they permit us to measure the total quantity of real monetary services by first constructing a quantity index from the observable nominal monetary asset stocks and then deflating that quantity index.

Specifically, we define the nominal Tornqvist-Theil monetary services index (chained Tornqvist-Theil quantity index number formula), [Mathematical Expression Omitted], by

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted]. Because the [TABULAR DATA FOR TABLE 1 OMITTED] individual expenditure shares may be interpreted as either nominal or real shares, this formula is simply the usual Tornqvist-Theil quantity index number formula applied to nominal, rather than real, stocks of monetary assets.

Similarly, define the real dual user cost index, [Mathematical Expression Omitted], by

[Mathematical Expression Omitted].

Because the total expenditure on monetary assets can be defined in terms of nominal asset stocks and real user costs, this real dual user cost index will be dual to [Mathematical Expression Omitted].

To simplify the discussion that follows, we define the log change operator as [Delta] log([z.sub.t]) = log([z.sub.t]) - log([z.sub.t-1]), where all logarithms are base e, or natural, logs. Then the real and nominal monetary services indexes, and their real and nominal dual user cost indexes, are related by

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted],

respectively.(8) The real monetary services index may be constructed by aggregating over nominal asset stocks to produce the nominal monetary service index and then deflating this index; a similar relationship holds for the nominal and real dual user cost indexes.

The St. Louis MSI database includes the nominal monetary service index, [Mathematical Expression Omitted], and its real dual user cost index, [Mathematical Expression Omitted]. Although the nominal monetary service index may be deflated to produce its real counterpart, we leave the choice of deflator to the user because choice of the appropriate deflator depends on the model being studied by the user. In consumer demand models, the appropriate price index is a measure of the true cost of living. In firm factor demand models, the appropriate price index is an index of factor input prices. There is a large set of published price indexes, any one of which may or may not be appropriate in a specific application. These indexes include the Consumer Price Index (CPI), the Producer Price Index (PPI), the GDP deflator, and the PCE deflator. It may also be appropriate to deflate the indexes by using a measure of the real wage rate. Caveat emptor.

In the remainder of this section, we discuss the additional indexes in the MSI database. Although the currency equivalent (CE) index (Rotemberg, Driscoll, and Poterba, 1995; Rotemberg, 1991) and the simple sum monetary aggregates (as published by the Federal Reserve Board) are both inferior to the Tornqvist-Theil monetary services index as measures of the flow of monetary services, they have interpretations as stock concepts. Our previous article in this Review, "Monetary Aggregation Theory," provides a more complete discussion of these concepts.

Barnett (1991) proved that, under certain assumptions, the currency equivalent index

[Mathematical Expression Omitted]

measures the discounted present value of all current and future total expenditures on monetary assets.

Under the same assumptions, the simple sum index

[Mathematical Expression Omitted]

equals the sum of the discounted present value of the expected investment yields on current and future holdings of monetary assets, plus the CE index.

Theil (1967) noted that the Tornqvist-Theil price index number is not dual to the Tornqvist-Theil quantity index number - that is, the Tornqvist-Theil index formula is not self-dual. We can define the real Tornqvist-Theil user cost index (chained Tornqvist-Theil price index formula), [Mathematical Expression Omitted], as

[Mathematical Expression Omitted].

Theil's (1967) result, applied to monetary indexes, shows that

[Mathematical Expression Omitted],

where the Tornqvist-Theil expenditure share index, [S.sub.t], is defined by

[Mathematical Expression Omitted].

Theil (1967) also defined four indexes known as Divisia second moments: the Divisia quantity growth-rate variance, Divisia user-cost growth-rate variance, Divisia expenditure share growth-rate variance, and the covariance between the quantity and user-cost growth rates. Formulas for these indexes are shown in part B of Table 1.

Barnett and Serletis (1990) propose a dispersion dependency test, based on the Divisia second moments, for the failure of the principal assumptions of aggregation theory. The Divisia second moments may, for example, contain significant information during periods of regulatory change. The latter include the phased removal of Regulation Q ceilings on depository institutions' offering rates between 1978 and 1986 and the introduction of new types of deposits, such as All Savers certificates in 1978 and money market deposit accounts in 1982.

The dispersion dependency tests are applied to U.S. monetary data in Barnett and Serletis (1990) and Barnett, Jones, and Nesmith (1996). These studies suggest that, for at least some time periods, movements in the monetary data are not consistent with the principal assumptions of aggregation theory. In this case, Barnett and Serletis (1990) suggest that including Divisia second moments in macroeconomic models might provide a correction for this aggregation error. For additional discussion, see our previous article in this Review, "Monetary Aggregation Theory."

ASSET STOCKS AND AGGREGATES

In this section, we describe our monetary asset stock data in detail, and we discuss the levels of aggregation of the indexes in the MSI database. Discussion of the own rate data is deferred until the following section.

The monetary aggregates published by the Federal Reserve Board - M1, M2, M3, and L - are constructed by summing over sets of monetary asset stocks at four nested levels of aggregation. In addition to these levels of aggregation, some economists have advocated two other levels of aggregation: M1A and MZM. M1A consists of the non-interest-bearing monetary assets in M1, and MZM includes the monetary assets in M2 which do not have a fixed maturity.(9) These levels of aggregation are summarized in Table 2.

Some economists have recently suggested that monetary indexes should contain, in addition, highly liquid capital-uncertain assets such as bond and equity mutual funds (Collins and Edwards, 1994; Orphanides, Reid, and Small, 1994). Although the theoretical procedures used in the construction of the St. Louis MSI database are valid only under the assumption of perfect certainty, extending the theory to include risk-neutral households and firms is straightforward (Barnett, 1994). Extending the theory to include risk-averse agents, however, is more difficult and requires the subtraction of a risk premium from the monetary-asset user costs (Barnett and Liu, 1994; Barnett, Liu, and Jensen, 1997). We leave as a topic for future research the measurement of monetary service indexes that include an allowance for risk aversion.

The St. Louis MSI database contains monetary services indexes constructed over the same sets of assets (levels of aggregation) as the simple sum monetary aggregates MIA, M1, MZM, M2, M3, and L. We do not test for the weak separability of these levels of aggregation, although the correct level of aggregation of monetary assets should be determined by tests for weak separability.(10) Several previous studies have examined the weak separability of the assets included in M1, M2, M3, and L (Serletis, 1987; Swofford and Whitney, 1987, 1988). More recently, Swofford and Whitney (1994) and Spencer (1994) have noted that relaxation of the assumption of continuous complete portfolio adjustment, maintained in derivation of the monetary service index, significantly complicates separability testing. Testing the separability of the included assets is a topic for future research; researchers are encouraged to conduct their own tests using the disaggregated data provided in the database.

The asset stock data used to produce the indexes in the St. Louis MSI database are shown in Table 2. They include both seasonally adjusted and unadjusted data, except for the non-M3 components of L and Super NOW accounts at commercial banks and at thrift institutions, which are included only on an unadjusted basis. Most data were originally published by the Federal Reserve Board and have been later revised by Board staff. For discussion of Federal Reserve monetary aggregates and their components, see Anderson and Kavajecz (1994).

The data in Table 2 are reported at the most disaggregate level feasible.(11) Super NOW accounts have been separated from other checkable deposits from 1983.01-1985.12, the period in which separate data are available. Similarly, savings deposits and money market deposit accounts are separated during 1960.01-1991.08. In addition, the following asset categories are separated into thrift institution and commercial bank [TABULAR DATA FOR TABLE 2 OMITTED] categories: other checkable deposits, Super NOW accounts, small-denomination time deposits, savings deposits, and money market deposit accounts. The MSI database contains two exceptions to our practice of reporting data at its finest level of disaggregation. The first is that the sum of overnight and term Eurodollar deposits is included as total Eurodollar deposits, and the sum of overnight and term repurchase agreements is included as total repurchase agreements; the second is that large-denomination time deposits are not separated with respect to commercial banks and thrift institutions.(12)

OWN RATES OF MONETARY ASSETS

User costs of monetary assets are constructed from the assets' own rates of return. For most periods beginning October 1983, we base the user costs of deposits at commercial banks and thrift institutions on actual rates paid by depositories. These data have recently been revised and documented by Board staff. Although the Board of Governors has published deposit rate data for periods prior to October 1983, we choose to use the Regulation Q statutory rate ceilings due to (our) uncertainty whether the survey data are representative of rates generally paid.

In this section, we provide a detailed discussion of several aspects of the data, including our procedures for measuring the implicit rate of return on demand deposits and for estimating proxies for missing values, the market interest rates available on U.S. savings bonds, a set of own-rate conversions required prior to calculating user costs, and our yield curve adjustment of the own-rate data.(13) Table 3 lists the own-rate data used to calculate the indexes in the St. Louis MSI database.

The Implicit Rate of Return on Demand Deposits

To construct a user cost for demand deposits, we need to specify its own rate of return. Appropriate measures of this rate have been widely debated among economists because the Banking Acts of 1933 and 1935 forbade banks from paying explicit interest on demand deposits. Regardless, economists recognize that most financial institutions, during at least some recent time periods, have paid implicit interest on demand deposits in the form of free or reduced-cost bank services, or perhaps easier access to credit. Some economists have suggested that such non-price competition has allowed depositories to evade the prohibition of explicit interest on demand deposits. Startz (1979) discusses three competing hypotheses: the "traditional" hypothesis, which maintains that the prohibition on interest paid to demand deposits has been fully effective; the "competitive" hypothesis, which maintains that the prohibition of interest on demand deposits has been completely ineffective; and the modified competitive hypothesis, which maintains that the prohibition was partially effective.

Klein (1974) derived an expression for the fully competitive implicit rate of return on demand deposits. Assuming that banks earn no profit on demand deposits and that banks face perfectly competitive markets, the implicit rate of return for demand deposits is defined by

[r.sub.A] - [r.sub.D] = M[C.sub.D],

where [r.sub.D] is the implicit interest rate on demand deposits, [r.sub.A] is the interest rate on an alternative assets, and M[C.sub.D] is the marginal cost of producing demand deposits. Under additional assumptions, Klein shows that this is equivalent to

[r.sub.D] = (1- c) [r.sub.A],

where c is the ratio of reserves to deposits.

Startz (1979) advocates a modified competitive hypothesis. He argues, using functional cost analysis data, that the implicit demand deposit rate has been positive, well below the fully competitive Klein rate, and responsive to market interest rates.(14) Empirical evidence on the various hypotheses has been mixed (see Rush, 1980; Carlson and Frew, 1980; Allen, 1983; and Rossiter and Lee, 1987).

[TABULAR DATA FOR TABLE 3 OMITTED]

In previous constructions of monetary index numbers, it has been assumed that the prohibition of interest on demand deposits is completely ineffective for business demand deposits and is fully effective for household demand deposits. In the calculation of the implicit rate of return on business demand deposits, the alternative asset was assumed to be commercial paper with one month remaining to maturity. In Farr and Johnson (1985) and Thornton and Yue (1992), the distinction between household and business demand deposits was based on the Federal Reserve's Demand Deposit Ownership Survey (Board of Governors, 1971-1991). Because that survey has been discontinued, we cannot base our indexes on the methods used in previous studies.

We apply the modified competitive hypothesis to all demand deposits. Startz (1979) has argued that the implicit rate of return on demand deposits is between 0.34 and 0.58 times the fully competitive Klein rate, using five-year Treasury notes as the alternative asset. Thus, the implicit rate of return on demand deposits is proxied as

[r.sub.D] = (1 - [Tau])([r.sub.A])([Alpha]),

where [r.sub.A] is the rate on 5-year Treasury notes, [Tau] is (an estimate of) the maximum reserve requirement on demand deposits, and [Alpha] is between 0.34 and 0.58. In this article, we set [Alpha] equal to its maximum value of 0.58. This is equivalent to assuming that all demand deposits were issued by large banks, and that the deposit holders regarded deposits at different-size banks as perfect substitutes. Our estimates of [Tau], the maximum reserve-requirement ratio on demand deposits, are shown in Table 4.

Regression-Based Proxies for Own-Rate Data

For some monetary assets, the asset stock data shown in Table 2 are available for dates before the earliest corresponding own-rate data shown in Table 3. Rather than discard these quantity data, we created proxies for the missing (unrecorded) own-rate data. For each such stock, we regressed the asset's available own-rate data for the later periods on one or more closely related rates and used the predicted values from the regression for earlier periods as proxies for the missing own-rate data. Our proxies, summarized in Table 5, are robust to reasonable alternative regression specifications.

Regulation Q Ceilings as Own-Rate Proxies

The regression method cannot be used for some commercial bank and thrift institution deposits prior to 1986. In these cases, we proxy the missing deposit own rates with the maximum rate that depositories were legally permitted to offer. These fixed and variable ceiling rates are summarized in parts A and B, respectively, of Table 6.

Negotiable order-of-withdrawal (NOW) accounts are checkable deposits currently included in the Federal Reserve's M1 monetary aggregate. Introduced in 1972 by a Massachusetts savings bank, NOW accounts spread rapidly: to all commercial banks and thrift institutions in Massachusetts and New Hampshire in January 1974, to the rest of the New England states in February 1976, to New York in November 1978, to New Jersey in December 1979, and nationwide in December 1980. Rates paid on NOW [TABULAR DATA FOR TABLE 5 OMITTED] accounts became subject to legal ceilings beginning in January 1974. Ceiling rates also were in effect for savings deposits during most of our sample through April 1986.

Small-denomination time deposits were subject to either fixed or variable ceiling rates during various periods of our sample; the latter were tied to Treasury market interest rates. We have constructed fixed ceiling-rate series for one-year maturity small-denomination time deposits at both commercial banks and thrift institutions. Because these rates, shown in Table 6, actually applied to deposits with a wide range of maturities, we caution the reader against overly precise interpretations. For large-denomination time deposits, we have assumed that depository institutions' offering rates, for dates since June 1964, have been approximately equal to secondary-market yields on negotiable certificates of deposit. The [TABULAR DATA FOR TABLE 6 OMITTED] latter are not available in the Federal Reserve's databases for dates prior to June 1964, however, and we have used as a proxy the ceiling rate permitted by the Federal Reserve's Regulation Q on time deposits payable from six months to one year. The Regulation Q ceiling was generally binding during this period (Federal Reserve Bulletin, 1963a,b, 1964a,b).

With the introduction of money market certificates in June 1978, some small-denomination time deposits were subject to variable ceiling rates that were tied to market interest rates (part B, Table 6).(15) We constructed these variable ceiling rates for small-denomination time deposits from information contained in various issues of the Federal Reserve Board's Annual Statistical Digest covering 1970-82.12 (16)

Market Interest Rate on Savings Bands

Investment yields to maturity for series-E savings bonds are available for January 1960-October 1982. Starting in November 1982, the Treasury Department issued bonds that paid a variable, market-based interest rate. This market rate is constructed according to the following procedure: The monthly five-year Treasury securities yield is averaged over six months, with six-month blocks beginning either on May 1 or November 1. The market-based savings bond rate for the next six months is equal to 85 percent of the average.(17)

Own-Rate Conversion

The application of aggregation theory and index-number methods to monetary data requires that all the own rates of return for the component assets be measured on the same basis. This is generally not true in published data because different sources have different reporting conventions, and because own rates are reported for a variety of different maturities. For monetary aggregation, the choice of a common measurement basis is arbitrary; that is, the information content of the index numbers is unaffected by the choice. We have chosen to convert all available rate data to an annualized monthly yield, calculated on a bond (or, coupon) equivalent basis, primarily for consistency with past monetary aggregation research. In this subsection, we describe general procedures for adjusting various own rates to this common basis. Our adjustments of own-rate data are summarized in Table 7 by type of adjustment. In each case, r is the unadjusted own rate of the asset, [r.sup.adj] is the adjusted own rate, and n is the maturity in months.

The simplest adjustment is to convert annualized one-month yields, quoted on a 360-day bank interest basis, to annualized one-month yields quoted on a 365-day bond coupon-equivalent basis. In this case, we simply multiply the unadjusted own rate by 365/360.

The second type of adjustment is to convert an annual effective yield, quoted in percentage points on a bond interest basis, to an annualized one-month holding-period yield on a bond interest basis. In this procedure, we convert the annual effective yield to a daily rate, compound that daily rate to a monthly rate, and then, assuming that all months contain 30 days, annualize the rate.

The third type of adjustment is to convert an annual effective yield on a bank interest basis to an annualized one-month holding-period yield on a bond interest basis, a procedure similar to the second one. We convert the bank-interest-basis annual effective yield to a daily rate, compound that daily rate to a monthly rate, and then, assuming a 30-day month, annualize to a bond interest basis.

In the fourth type of adjustment, we convert a rate quoted on a bank discount basis, for a monetary asset with a maturity of n months, to an annualized one-month holding-period yield. This conversion, which is discussed in detail by Farr and Johnson (1985), is valid only for rates with maturity of less than six months, and it assumes that each month has 30 days.

Yield Curve Adjustment

Own rates for monetary assets that have different maturities may have different term premiums, and hence are not directly comparable. Therefore, in addition to making the above adjustments, we need to remove a liquidity, or term, premium from each own rate. We yield curve adjust monetary assets' own rates by using the yield curve for U.S. Treasury securities. These adjustments of the own-rate data are summarized in Table 8.

We adjust the own rates by subtracting, from each own rate, an estimate of the liquidity premium obtained from the yield curve for Treasury securities. (Because these securities have no default risk, the slope of the Treasury yield curve provides a relatively "pure" estimate of the term premium.) The following discussion of yield curve adjustment assumes that all own rates (including Treasury bill rates) have been converted to an annualized one-month holding-period yield, on a bond interest basis.

Let [r.sub.n] be an own rate for a monetary asset with a maturity of n months, let [Mathematical Expression Omitted] be the own rate on Treasury securities that mature in n months, and let [Mathematical Expression Omitted] be the one-month secondary-market Treasury bill rate. The own rate, [r.sub.n], is yield curve adjusted by subtracting the estimated liquidity premium [Mathematical Expression Omitted] from the own rate, such that the yield curve adjusted own rate, [Mathematical Expression Omitted]. For a Treasury security that matures in n months [Mathematical Expression Omitted] is the n-month secondary-market Treasury bill rate, adjusted from a bank discount basis to an annualized one-month holding-period yield on a bond interest basis. If maturity is in n years [Mathematical Expression Omitted] is the corresponding constant-maturity Treasury security. Other values of [Mathematical Expression Omitted] may be interpolated from the Treasury's constant-maturity yield curve.

If a single monetary asset stock contains components with a range of maturities, we yield-curve adjust the own rate using the yield on a Treasury security with a maturity that falls within that range.

USER COSTS OF MONETARY ASSETS

In this section, we discuss in detail how we construct the user costs for monetary assets from the previous section's adjusted own-rate data. We address the problem that reported monetary asset stock data, such as the Federal Reserve Board's data on small-denomination time deposits, do not distinguish between monetary assets with different terms to maturity, and we construct the user costs of such assets as unilateral user cost sub-indexes.

Monetary Assets With Different Maturities

The definition of the real user cost of a monetary asset assumes that, in each period, each asset has a single applicable own rate and, hence, a unique user cost. Published Federal Reserve Board deposit data for commercial banks and thrift institutions do not distinguish adequately among monetary assets with different maturity. Only total dollar amounts, summed across all maturities, are reported for the following categories: small-denomination time deposits at commercial banks and thrift institutions; large-denomination time deposits; total Eurodollar deposits; total repurchase agreements; bankers acceptances; short-term Treasury securities; and commercial paper.(18) This bundling of assets with different maturities into monetary asset sub-indexes causes difficulty. Prior to measuring the sub-index's user cost, one should remove a liquidity (or term) premium from each component monetary asset's user cost. Because the own rates of the unobserved subcomponents may differ even after the yield curve adjustment, several user costs may apply to the sub-index.(19) A method must be found to combine the various user costs into a single user cost sub-index that corresponds to the reported asset stock.

A similar problem, in which a single price index is constructed from multiple individual prices without the use of quantity data, has been examined by Diewert (1995).(20) Price indexes constructed solely from price data, without quantity data, are called unilateral price indexes. Diewert (1995) advocates the use of a particular unilateral price index formula, called the Jevons unilateral price index. We construct such unilateral user cost indexes, based on the Jevons formula, for the following monetary asset categories: small-denomination time deposits at commercial banks and thrift institutions; large-denomination time deposits; total Eurodollar deposits; bankers' acceptances; and commercial paper.

User Costs by Component

In Table 9, we summarize the own rates used in the construction of each user cost. The own rates are (1) adjusted to a common basis, (2) yield curve adjusted, and (3) proxied, where appropriate (see Tables 5, 8, and 9). The own-rate series refer either to the own rate data shown in Table 4, or to the data discussed in the previous section of this article.

The construction of real user costs also requires the rate of return for a benchmark asset. The benchmark asset is a theoretical construct: It provides no monetary services, has no default risk, and is used by economic agents only to transfer wealth between periods. A theoretical lower bound for the benchmark asset can be identified; because monetary services are valued by households and firms, the user costs of monetary assets must be positive. Thus, the benchmark asset's rate of return must exceed the own rates on all assets that furnish monetary services.

A theoretical way of constructing the benchmark rate is to set it equal to the maximum rate of return over a large class of assets, both financial and non-financial. This method is inappropriate, however, because (unadjusted) rates of return on [TABULAR DATA FOR TABLE 9 OMITTED] debt and equity contain risk premia. In empirical work, the traditional approach has been to identify the benchmark rate during each time period, t, as the "envelope" of the own rates of return on monetary assets and the rate on Moody's seasoned BAA bonds, [r.sub.BAA,t]:

[R.sub.t] = max{[r.sub.it] (i = 1, 2, ..., n), [r.sub.BAA,t]}

(Barnett and Spindt, 1982; Farr and Johnson, 1985; and, Thornton and Yue, 1992). We adopt this practice, with a minor modification, and define the benchmark rate as

[Mathematical Expression Omitted],

where c is a small constant. Although we typically set the value of the constant at one basis point or less, its inclusion guarantees that the benchmark rate is strictly greater than the rate on any monetary asset, and it allows us, in a previous section, to define Divisia second moments of our indexes. The indexes are robust experimentally to a large range of values for the constant.

Unilateral Index Number Theory

In this section, we provide the reader with a discussion of unilateral index number theory in the context of monetary aggregation, and we define and discuss our use of the Jevons unilateral price index formula.

Bilateral index numbers, such as the Tornqvist-Theil index number, are functions, in each period, of both observed prices and quantities. Unilateral index numbers are functions, in each period, of either the observed prices or the observed quantities, but not both. Unilateral indexes may be useful, therefore, when some of the price or quantity data required for a bilateral index have not been recorded. As previously noted, the Federal Reserve Board's monetary asset stock data often do not separate monetary assets with different maturities. The reported asset data - total dollar amounts summed across all maturities - are unilateral quantity indexes. We refer to these aggregates as monetary quantity sub-indexes, and the unreported individual assets with differing maturities as sub-components.

In an ideal world, we would treat each sub-component of each monetary quantity sub-index as a separate asset with its own user cost. In practice, the data collection process forces us to treat each monetary quantity sub-index as if it were a single asset. If the user costs of the sub-components are observed, we can construct a unilateral user cost sub-index, which can be viewed as the "user cost" of the monetary quantity sub-index.

Let [Mathematical Expression Omitted] be the value of a monetary quantity sub-index in period s, let [Mathematical Expression Omitted] be a vector of M own rates that apply to the sub-components of the monetary quantity sub-index. Then, [Mathematical Expression Omitted] where j = 1, ..., M, are the real user costs that apply to the sub-components of the monetary quantity sub-index, [Mathematical Expression Omitted]. (We remind the reader that the own rates in the vector [Mathematical Expression Omitted] must all be converted to a common basis, and yield curve adjusted.) Diewert (1995) defines the Jevons and the Dutot unilateral price indexes.(21) For monetary aggregation, the Jevons user cost sub-index is defined by

[Mathematical Expression Omitted],

and the Dutot user cost sub-index is defined by

[Mathematical Expression Omitted].

Diewert (1995) defined a set of axioms that a reasonable unilateral price index should satisfy, and he showed that both the Jevons and the Dutot price indexes satisfy these axioms.(22) It can also be shown that a third unilateral price index, which we call the Leontief price index, satisfies weak versions of these same axioms.(23) In the present context, the Leontief user cost sub-index, [Mathematical Expression Omitted] is defined by

[Mathematical Expression Omitted].

Because the Jevons, Dutot, and Leontief user cost sub-indexes all satisfy Diewert's axioms, we can compare the economic justifications of the indexes.

Diewert (1976) showed that the members of a class of bilateral statistical index numbers, called superlative, have strong economic justification as approximations of aggregator functions. Unfortunately, unilateral price indexes have only weak economic justifications; Diewert (1995) gives the stringent conditions under which the Jevons and Dutot user cost sub-indexes will be correct (exact).(24) The Jevons user cost sub-index will be correct only if two conditions hold: (1) the elasticities of substitution between the sub-components are unity, once the liquidity premium has been extracted; and (2) the expenditure shares on each sub-component are constant. The Dutot user cost sub-index will be correct if two conditions hold: (1) the elasticities substitution between the sub-components are zero, once the liquidity premium has been extracted; and (2) in each period the quantities of the sub-components are equal.

The Leontief user cost sub-index will be correct if the sub-components are, in fact, perfect substitutes after the liquidity premium has been removed. This assumption underlies the user cost sub-indexes that have been constructed, for certain subsets of assets, by Barnett and Spindt (1982), Farr and Johnson (1986), and Thornton and Yue (1992). In these articles, Barnett's (1978) user cost formula was applied, in each time period, to the maximum of the yield curve-adjusted own rates for the assets included in each subset, a procedure that is equivalent to the Leontief user cost sub-index. These sub-indexes were subsequently aggregated with other assets and user costs, using superlative index number formulas such as the Tornqvist-Theil or Fisher Ideal index formula.

Diewert (1995) argued that, for calculating price indexes, the Jevons index formula is superior to the Dutot index formula because the Jevons index's conditions - unit elasticity and constant (or proportional) expenditures - are more plausible than the Dutot index's conditions: zero elasticity and constant (or proportional) quantities. The Advisory Commission to Study the Consumer Price Index (1996) also advocated the use of the Jevons index formula to calculate lower-level price indexes for sub-components of the CPI. The Jevons index has been widely used as the benchmark for studying bias in lower-level price indexes in a number of countries; see Diewert (1995) for a review of these studies. The current consensus is that the Jevons index number formula should be used to calculate unilateral price indexes.

For measuring the user costs of our monetary service index, we concur with Diewert's preference for the Jevons index. We further argue that the Jevons index's unit elasticity condition is more reasonable than the Leontief index's assumption that monetary assets are perfect substitutes. The perfect substitutes condition would imply that, unless all user costs applying to the sub-index are equal, economic agents will hold only the least-expensive sub-component of each monetary sub-index - an implication that is usually rejected when it can be tested.(25) In our MSI database, we use the Jevons formula to create user cost sub-indexes for small-denomination time deposits at commercial banks and thrift institutions, large-denomination time deposits, total Eurodollar deposits, bankers acceptances, and commercial paper. The growth rate of the Jevons user cost sub-index, in two adjacent periods, is the ratio of the geometric means of the applicable user costs. We cardinalize the Jevons user cost sub-index by setting the initial value of the sub-index equal to the geometric mean of the user costs during the initial period.

After selection of the Jevons index, one more important difficulty remains: The number of own rates that are observed for the sub-components of a monetary quantity sub-index may change, from period to period, due to a number of factors, including changes in regulations and data-collection practices. In these cases, we calculate the growth rate of the Jevons user cost sub-index from the subset of user costs that are observed in the adjacent periods. This procedure is based on Diewert's (1980) new goods procedure, which is discussed in the section of this paper titled, "Introduction of New Monetary Assets." In a few cases, the set of observed sub-component user costs in adjacent periods changes completely. In such cases, we calculate, for both periods, the geometric means of the observed user costs and then calculate the Jevons index as the ratio of the current period's geometric mean divided by the geometric mean in the previous period.(26)

ADDITIONAL PROBLEMS

Several additional problems that arise in the construction of monetary services indexes are discussed in the following sub-sections: (1) the introduction of new monetary assets, (2) changes in the definitions of underlying monetary asset stock data, (3) the calculation of monetary service indexs and related indexes at different frequencies, and (4) seasonal adjustment of the indexes.

Introduction of New Monetary Assets

There have been many financial innovations during the time span of our monetary services indexes. New monetary assets have been created at various dates, and the indexes must be modified to include them.

The nominal Tornqvist-Theil monetary services index, [Mathematical Expression Omitted], and its real dual price index, [Mathematical Expression Omitted], are not well defined when new assets enter the indexes. The real Fisher Ideal user cost index,

[Mathematical Expression Omitted],

is well defined, and a corresponding quantity index may be obtained by Fisher's factoral reversal formula. We therefore switch to the Fisher Ideal index in periods when new monetary assets are introduced.(27)

To implement this approach, we need to develop an estimator for the new asset's user cost during the period prior to its introduction. Theoretically, the correct solution is to define a user cost, called the reservation user cost, that is sufficient to ensure that a zero quantity of the new asset would have been demanded at that user cost during the prior period if the asset had, in fact, existed. In practice, doing this correctly requires econometric estimation of the aggregator function (Diewert, 1980), whereas our primary motive for the use of statistical index numbers is to avoid such estimation.

Rather than estimate the reservation user cost, we use the following method, introduced by Diewert (1980) and used in Diewert and Smith (1994).(28) In the period when a new monetary asset is introduced, we calculate the Fisher Ideal real user cost index over all monetary assets except the new one, which we will call [Mathematical Expression Omitted]. If monetary asset i is introduced in period t, [Mathematical Expression Omitted] will be defined by

[Mathematical Expression Omitted].

Diewert (1980) shows that this procedure will, in general, have lower bias than the other available alternatives, in the absence of strong information about the reservation user cost. The procedure is exactly correct in a special case: If the actual user cost of the new asset i in period t divided by the reservation user cost is equal to

[Mathematical Expression Omitted],

then [Mathematical Expression Omitted] will be exactly correct.

We form our real user cost indexes by switching to a Fisher Ideal index, calculated according to Diewert's recommended approach, during periods in which new monetary assets enter the indexes. The dual monetary services index is then defined implicitly by Fisher's weak factor reversal formula.

In Table 10, we list the periods in which new monetary assets are introduced.

Changes in the Definitions of Asset Stock Data

In the preceding section, we discussed the introduction of new monetary assets. A related problem is that, at times, the Federal Reserve has changed the definitions and the manner of reporting the components of its monetary aggregates. This happens twice in our series: (1) after 1985.12, Super NOW accounts are included in other checkable deposits (OCD), and (2) after 1991.08, money market deposit accounts (MMDA) and savings deposits are reported only on a combined basis, for thrift institutions and for commercial banks. In both of these cases, monetary assets that had been reported separately were combined into sub-indexes, and the sub-component data were no longer available.

These changes represent a redefinition of the asset stocks (and consequently the monetary services indexes), but they do not represent a meaningful change in the structure of the economy; in other words, these data-reporting changes are not economically relevant. The Federal Reserve Board's monetary aggregates are invariant to such changes because their aggregates are themselves sums of all the component data. Tornqvist-Theil monetary services indexes are not invariant to these changes because the change in reporting, from a group of assets to a single sub-index, represents a loss of information. In this section, we describe our approach to this problem.(29)

From 1983.01 through 1985.12, Super NOW accounts and savings deposits are included in our Tornqvist-Theil monetary services indexes as separate assets. Beginning in 1986.01, however, Super NOW accounts and savings deposits were reported only as a combined total. In response to this change, we define a second Tornqvist-Theil monetary services index that begins in 1985.12 and contains the total of Super NOW accounts and other checkable deposits as a single asset. The value of the second index in its initial period, December 1985, is arbitrary, which permits us to scale the second index so that it equals the first index in 1985.12. This splices the two indexes so as to produce a single Tornqvist-Theil monetary services index over the entire period. We perform an analogous splice in 1991.08 when money market deposit accounts and savings deposits begin to be reported only on a combined basis.(30)

Data reporting changes that are not based on economic reasons, such as financial innovation or regulatory changes, represent a loss of information. We preserve as much information as possible during the periods when disaggregate data are available and avoid imputing economic relevancy to the data reporting change when it occurs. Our method draws on the literature of index number splicing (Hill and Fox, 1995).

Indexes at Different Frequencies

The disaggregated data in the MSI database are reported monthly In some applications, monetary aggregates must be available at quarterly or annual frequency In this section, we discuss a method developed by Diewert (1980) for constructing, from monthly data, indexes at quarterly and annual frequencies.

In the problem of constructing annual indexes from monthly indexes, the solution is to treat each asset, in each month, as a separate asset and then to aggregate over these assets. For example, demand deposits held in January and in February will be treated as different assets. Formally, let [Mathematical Expression Omitted] be the nominal stock of monetary asset i in month r of year t. Similarly let

[Mathematical Expression Omitted]

be the real user cost associated with, [Mathematical Expression Omitted], where [Mathematical Expression Omitted] is the rate of return on the benchmark asset in month r of year t, and [Mathematical Expression Omitted] is the rate of return on the nominal stock of monetary asset i in the month r of year t. Then the log change of the annual Tornqvist-Theil nominal monetary services index, [Mathematical Expression Omitted] is defined by

[Mathematical Expression Omitted],

where

[Mathematical Expression Omitted].

An analogous method can be used to define the quarterly indexes. In the MSI database, this method is used to produce both annual and quarterly indexes. Dual user cost indexes are obtained by Fisher's weak factor reversal criterion.(31)

Seasonal Adjustment

The issue of seasonal adjustment is a difficult one. Index number theoretic methods for dealing with seasonality, which are related to the issues discussed in the section of this paper dealing with indexes at different frequencies, can be found in Diewert (1980, 1983, 1996). Our approach is more traditional. We produce the indexes in the database by using both seasonally adjusted and unadjusted asset stock data, except for the non-M3 components of L, which are not seasonally adjusted in either set of indexes. Our seasonally adjusted data are produced with the Bureau of the Census X11 program, using default values for all options. We urge users of our unadjusted data to experiment with alternative seasonal adjustment methods.

CONCLUSION

The St. Louis MSI database is an important resource for economists and policymakers studying the role of money in the economy. Monetary services are an important aspect of the economic behavior of households and firms, and the monetary service indexes provide new up-to-date measures of the flows of monetary services. The database also contains dual measures of the opportunity cost of monetary services and related stock and total expenditure variables.

The indexes in the MSI database are consistent with microeconomic aggregation theory and have the same statistical properties as commonly used macroeconomic indexes such as GDP and its deflator. In general, the monetary service index and its dual user cost index can be modeled in the same way as other macroeconomic quantity and price indexes and, in particular, models of money demand can be estimated by using the monetary service index.

In addition to our aggregate indexes, the MSI database contains disaggregate asset-stock and user-cost data that will allow researchers to study the demand for the disaggregated monetary assets in a way that is consistent with microeconomic models of decision making. The database is also comprehensive enough to allow researchers to experiment with alternative levels of aggregation, different measures of assets' own rates, and various seasonal adjustment techniques.

These data provide numerous opportunities for applied monetary research. Although monetary services indexes have been produced before by Barnett and Spindt (1982), Farr and Johnson (1985), and Thornton and Yue (1992), none of these studies furnished a broad enough set of indexes, the underlying data, or the computer programs necessary to build the indexes.

We offer special thanks to Professor William A. Barnett, Washington University in St. Louis, for his generous and invaluable guidance during this project. We also thank Professors W. Erwin Diewert, University of British Columbia, and Adrian R. Fleissig, St. Louis University, for advice and comments. For assistance with data, we thank the Savings Bond Operations Office of the U.S. Department of the Treasury and the Division of Monetary Affairs of the Board of Governors of the Federal Reserve System.

1 The address is www.stls.frb.org/research.

2 The Board of Governors' monetary aggregates are published weekly in the statistical release, Money Stock, Liquid Assets, and Debt Measures, and monthly in the Federal Reserve Bulletin, Table 1.21.

3 Simple sum M1A (non-interest bearing M1) was produced as an official monetary aggregate from 1960 through April 1971 (Kavajecz, 1994). MZM was suggested by William Poole.

4 We assume that these stocks were chosen by an optimizing economic agent.

5 The correct price index to be used depends on the context. We state the appropriate index for the consumer case. Barnett (1987) uses the same index for firms and financial intermediaries, although he notes that this is not strictly correct.

6 All holding period yields are assumed to be reported on a common basis. This issue is discussed in the section of this article titled "Own Rate Data."

7 Because of its connection with Divisia's (1925) continuous time index number, Barnett (1980) referred to this index as the Divisia Index.

8 In discrete time, these equalities are true up to a third-order error when the true cost of living index is measured by a superlative index number. If a non-superlative price index is used, the equality will be true only up to the tracking ability of the index.

9 William Poole's MZM included institutional money market mutual funds. We exclude these funds because they do not follow the same accounting rules as retail money market funds, and are marketed only to larger investors.

10 A weakly separable block could contain both monetary assets and consumption goods, but an aggregate formed over such a block would not usually be interpreted as a monetary service flow.

11 The criterion for feasibility is that disaggregated data for the desired asset stock must be available and of good quality. In addition, reliable own-rate data for the category must exist.

12 We combine overnight and term repurchase agreements and Eurodollar deposits because we have no separate, reasonable own rates for their components. We combine large-denomination time deposits at commercial banks and thrift institutions for the same reason.

13 Additional discussion can be found in Barnett and Spindt (1982), Farr and Johnson (1985), Thornton and Yue (1992), and Belongia (1995).

14 Other implicit rates of return are discussed by Becker (1975) and Barro and Santomero (1972).

15 Ceiling on small-denomination time deposits with original maturity of three and one-half years or longer were removed in May 1982, and on all other small-denomination time deposits on October 1, 1983.

16 The linkages between variable ceiling rates and auction-average Treasury rates are discussed in Mahoney (1987) and Annual Statistical Digest, editions for 1970-1979 and 1980, 1981, and 1982.

17 This methodology was supplied to us by the Savings Bond Operations Office of the U.S. Department of the Treasury.

18 Data are not available on the outstanding quantities of monetary assets by remaining time to maturity. Data published by the Board of Governors regarding the outstanding stocks of small-denomination time deposits by original-issue maturity are not appropriate for calculating index numbers (see Table 1.22, Federal Reserve Bulletin, February 1997).

19 Total repurchase agreements have a unique user cost in each period because we use the only available rate, that on overnight agreements, for oil maturities. Small-denomination time deposits also have several applicable own rates due to the existence of both variable and fixed ceiling-rate time deposits from 1978.06 to 1983.9.

20 The Advisory Commission to Study The Consumer Price Index (1996) discusses a similar problem in the construction of lower-level price indexes, in which multiple prices are combined into a single price index.

21 These indexes also are defined in Diewert (1992).

22 The growth rate of the Dutot price index is the ratio of the averages of prices in adjacent periods. An index based on the average of the ratios of prices in adjacent periods, the Carli (Diewert, 1992), does not satisfy the time reversal test, hence is not a reasonable unilateral price index.

23 Specifically, the Leontief index satisfies the axiomatic tests if strict monotonicity is weakened to monotonicity

24 These unilateral index numbers are based on particular constant elasticity of substitution (CES) aggregator functions, which are not flexible functional forms. Hence the unilateral indexes are not superlative.

25 Prior to 1991:08, for example, economic agents held non-zero quantities of both MMDA and saving deposits assets, even though the user costs differed.

26 Our calculations assume that each unit of a specific monetary asset stock has the same user cost, after necessary conversions and yield curve adjustment. Aggregation error will occur if deposit own rates vary with the size of the deposit. Some evidence on this practice is analyzed by Collins (1991).

27 Farr and Johnson (1985) advocate the Fisher Ideal index because it is well-defined even when new assets are introduced. For all periods when data are available, the Tornqvist-Theil index is superior to the Fisher Ideal index because it is superlative in a stronger sense (Caves, Christensen, and Diewert, 1982).

28 Diewert suggests this procedure in the general case; we state it here in the case of monetary aggregation.

29 The CE and simple sum indexes are well-defined and do not require any modifications when asset stocks are redefined. The real user cost index that is dual to the Tornqvist-Theil monetary services index is calculated by factor reversal. The Tornqvist-Theil real user cost index and the Tornqvist-Theil expenditure share index are calculated according to the procedure described in this section.

30 In 1985.12, the combined OCD asset stock is the sum of the asset stocks of its subcomponents, and the user cost for OCD is constructed from the weighted average of its sub-components' own rates. A similar procedure is followed in August 1991 for savings deposits and MMDA.

31 At quarterly and annual frequencies, the splicing procedure described in the preceding sub-section needs to be modified in a straightforward way.

REFERENCES

Advisory Commission to Study the Consumer Price Index. Toward a More Accurate Measure of the Cost of Living, Final Report to the Senate Finance Committee (December 1996).

Allen, Stuart D. "A Note on the Implicit Interest Rate on Demand Deposits," Journal of Macroeconomics (Spring 1983), pp. 233-39.

Anderson, Richard G., and Kenneth A. Kavajecz. "A Historical Perspective on the Federal Reserve's Monetary Aggregates: Definition, Construction and Targeting," this Review (March/April 1994), pp. 233-39.

----- and Robert H. Rosche. "Measuring the Adjusted Monetary Base in an Era of Financial Change," this Review (November/December 1996), pp. 3-37.

-----, Barry E. Jones, and Travis D. Nesmith. "An Introduction to Monetary Aggregation Theory and Statistical Index Numbers," this Review, this issue.

Barnett, William A. "Reply [to Julio J. Rotemberg]," Monetary Policy on the 75th Anniversary of the Federal Reserve System: Proceedings of the Fourteenth Annual Economic Policy Conference of the Federal Reserve Book of St. Louis, Michael T. Belongia, ed., Kluwer Academic Publishers, 1991, pp. 232-43.

-----. "Developments in Monetary Aggregation Theory," Journal of Policy Marketing (Summer 1990), pp. 205-57.

-----. "The Microeconomic Theory of Monetary Aggregation," New Approaches to Monetary Economics: Proceedings of the Second International Symposium in Economic Theory and Econometrics, William A. Barnett and Kenneth J. Singleton, eds., Cambridge University Press, 1987, pp. 115-68.

-----. "Economic Monetary Aggregates: An Application of index Number and Aggregation Theory," Journal of Econometrics (Summer 1980), pp. 11-48.

-----. "The User Cost of Money," Economic Letters (vol.1, 1978), pp. 145-49.

----- and Yi Liu. "Beyond the Risk Neutral Utility Function," paper presented at the University of Mississippi conference, "Divisia Monetary Aggregates: Theory and Practice," October 1994; conference volume, Macmillan, forthcoming.

----- and Apostolos Serletis. "A Dispersion-Dependency Diagnostic Test for Aggregation Error: With Applications to Monetary Economics and Income Distribution," Journal of Econometrics (January/February 1990), pp. 5-34.

----- and Paul A. Spindt. "Divisia Monetary Aggregates: Compilation, Data, and Historical Behavior," Staff Study 116, Board of Governors of the Federal Reserve System, May 1982.

-----, Barry E. Jones, and Travis D. Nesmith. "Divisia Second Moments: An Application of Stochastic Index Number Theory," International Review of Comparative Public Policy (vol. 8, 1996), pp. 115-38.

-----, Yi Liu, and Mark Jensen. "The CAPM Risk Adjustment far Exact Aggregation over Financial Assets," Macroeconomic Dynamics (May 1997), forthcoming.

Barro, Robert J., and Anthony J. Santomero. "Household Money Holdings and the Demand Deposit Rate," Journal of Money, Credit, and Banking (May 1972), pp. 397-413.

Becker, William E., Jr. "Determinants of the United States Currency-Demand Deposit Ratio," The Journal of Finance (March 1975), pp. 57-74.

Belongia, Michael T. "Weighted Monetary Aggregates: A Historical Survey," Journal of International and Comparative Economics (vol. 4, 1995), pp. 87-114.

Board of Governors of the Federal Reserve System. Annual Statistical Digest, editions for 1970-79, 1980-89, and annual editions for 1980 through 1995.

-----. Money Stock, Liquid Assets, and Debt Measures, H.6 statistical release.

-----. Selected Interest and Exchange Rates, N.13 statistical release (various issues).

-----. Selected Interest Rates, H.15 statistical release (various issues).

-----. Federal Reserve Bulletin (various issues).

-----. "Demand Deposit Ownership Survey," Federal Reserve Bulletin, July issues, 1971-91.

-----. Banking and Monetary Statistics: 1941-1970, September 1976.

-----. "Survey of Time and Savings Deposits at Commercial Banks," Federal Reserve Bulletin, vols. 52 through 68.

-----. "Flows Through Financial Intermediaries," Federal Reserve Bulletin (May 1964a), pp. 549-57.

-----. "Bank Credit and Money in 1963," Federal Reserve Bulletin (February 1964b), pp. 141-47.

-----. "Interest Rates on Time Deposits, Mid-February 1963," Federal Reserve Bulletin (June 1963b), pp. 766-72.

-----. "Negotiable Time Certificates of Deposit," Federal Reserve Bulletin (April 1963b), pp. 458-68.

Brennan, Michael J., and Eduardo S. Schwartz. "Savings Bonds: Theory and Empirical Evidence," Monograph Series in Finance and Economics, Monograph 1979-4. Saloman Brothers Center for the Study of Financial Institutions, 1979.

Caves, Douglas W., Laurits R. Christensen, and W. Erwin Diewert. "The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity," Econometrica (November 1982), pp. 1393-414.

Carlson, John A., and James R. Frew. "Money Demand Responsiveness to the Rate of Return on Money: A Methodological Critique," Journal of Political Economy (June 1980), pp. 598-607.

Collins, Sean. "Do Banks Price Discriminate Among Their Retail Customers? A Test Based on Tiered Deposit Rates," working paper, Division of Monetary Affairs, Board of Governors of the Federal Reserve System, September 1991.

----- and Cheryl L. Edwards. "An Alternative Monetary Aggregate: M2 Plus Household Holdings of Bond and Equity Mutual Funds," this Review (November/December 1994), pp. 7-29.

Diewert, W. E. "Seasonal Commodities, High Inflation and Index Number Theory," Working Paper DP 96-06, University of British Columbia, January 1996.

----- "Axiomatic and Economic Approaches to Elementary Price Indexes," Working Paper No. 5104, National Bureau of Economic Research, May 1995.

----- "The Treatment of Seasonality in a Cost-Of-Living-Index," Price Level Measurement, W.E. Diewert and C. Montmarquette, eds., Ottawa: Statistics Canada, 1983, pp. 1019-45.

----- "Index Numbers," The New Palgrave Dictionary of Money & Finance, Vol. 2, Peter Newman, Murray Milgate, and John Eatwell, eds., Stockton Press, 1992, pp. 364-79.

----- "Aggregation Problems in the Measurement of Capital," The Measurement of Capital, Dan Usher, ed., University of Chicago Press, 1980, pp. 433-538.

----- "Exact and Superlative Index Numbers," Journal of Econometrics (May 1976), pp. 115-45.

----- and Ann Marie Smith. "Productivity Measurement for a Distribution Firm," Journal of Productivity Analysis (vol. 5,1994), pp. 335-47.

Divisia, Francois. "L'lndice Monetaire et la Theorie de la Monnaie," Revue d'Economie Politique (1925), pp. 980-1008.

Farr, Helen T., and Deborah Johnson. "Revisions in the Monetary Services (Divisia) Indexes of the Monetary Aggregates," Staff Study 147, Board of Governors of the Federal Reserve System, December 1985.

Fisher, Irving. The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability, Houghton Mifflin Company, 1922.

Hill, Robert J., and Kevin J. Fox. "Splicing Index Numbers," working paper, School of Economics, The University of New South Wales, 1995; Journal of Business and Economic Statistics, forthcoming.

Kavajecz, Kenneth A. "The Evolution of the Federal Reserve's Monetary Aggregates: A Timeline," this Review (March/April 1994), pp. 32-66.

Klein, Benjamin. "Competitive Interest Payments on Bank Deposits and the Long-Run Demand for Money," American Economic Review (December 1974), pp. 931-49.

Mahoney, Patrick I., et.al., "Responses to Deregulation: Retail Deposit Pricing from 1983 through 1985," Staff Study 151, Board of Governors of the Federal Reserve System, January 1987.

OECD Financial Statistics. Interest Rates 1960-1974, 1976.

Orphanides, Athanasios, Brian Reid, and David H. Small. "The Empirical Properties of a Monetary Aggregate That Adds Bond and Stock Funds to M2," this Review (November/December 1994), pp. 31-51.

Rossiter, Rosemary, and Tong Hun Lee. "Implicit Returns on Conventional Demand Deposits: An Empirical Comparison," Journal of Macroeconomics (Fall 1987), pp. 613-24.

Rotemberg, Julio J. "Commentary: Monetary Aggregates and Their Uses," Monetary Policy on the 75th Anniversary of the Federal Reserve System: Proceedings of the Second International Symposium in Economic Theory and Econometrics, Michael T. Belongia, ed., Kluwer Academic Publishers, 1991, pp. 223-31.

-----, John C. Driscoll, and James M. Poterba. "Money, Output, and Prices: Evidence from a New Monetary Aggregate," Journal of Business and Economic Statistics (January 1995), pp. 67-83.

Rush, Mark. "Comment and Further Evidence on 'Implicit Interest on Demand Deposits'," Journal of Monetary Economics (July 1980), pp. 437-51.

Serletis, Apostolos. "Monetary Asset Separability Tests," New Approaches to Monetary Economics: Proceedings of the Second International Symposium in Economic Theory and Econometrics, William A. Barnett and Kenneth J. Singleton, eds., Cambridge University Press, 1987, pp. 169-82.

Spencer, Peter. "Portfolio Disequilibrium: Implications for the Divisia Approach to Monetary Aggregation," The Manchester School of Economic and Social Studies (June 1994), pp. 125-50.

Startz, Richard. "Implicit Interest on Demand Deposits," Journal of Monetary Economics (October 1979), pp. 515-34.

Swofford, James L., and Gerald A. Whitney. "A Revealed Preference Test for Weakly Separable Utility Maximization with Incomplete Adjustment," Journal of Econometrics (January/February 1994), pp. 235-49.

----- and -----. "A Comparison of Nonparametric Tests of Weak Separability for Annual and Quarterly Data on Consumption, Leisure, and Money," Journal of Business and Economic Statistics (April 1988), pp. 241-46.

----- and ----- "Nonparametric Tests of Utility Maximization and Weak Separability for Consumption, Leisure, and Money," Review of Economics and Statistics (August 1987), pp. 458-64.

Theil, Henri. Economics and Information Theory. Amsterdam: North Holland, 1967.

Thornton, Daniel L., and Piyu Yue. "An Extended Series of Divisia Monetary Aggregates," this Review (November/December 1992), pp. 35-52.

Triplett, Jack E. "Economic Theory and BEA's Alternative Quantity and Price Indexes," Survey of Current Business (April 1992), pp. 32-48.

Whitesell, William C., and Seen Collins. "A Minor Redefinition of M2," Finance and Economics Discussion Series, no. 96-7, Board of Governors of the Federal Reserve System, February 1996.

Table 7

Own Rate Adjustments

A. To convert an annualized 1-month holding period yield on a bank interest (360-day) basis to an annualized 1-month holding period yield on a bond interest (365-day) basis.

Adjustment Formula

[r.sup.adj] = r x (365 / 360)

Own Rates Adjusted

Eurodollar deposits: 1-month, 3-month, 6-month Certificates of deposits: 1-month, 3-month, 6-month

B. To convert an annual effective yield on a bond interest basis to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [[(1 + (r/100) / 365).sup.30] - 1] x (365 / 30) x 100

Own Rates Adjusted

NOW accounts: thrift institutions, commercial banks Super NOW accounts: thrift institutions, commercial banks

Small-denomination time deposits at commercial banks and thrift institutions:

7-day to 91-day 92-day to 182-day 183-day to 1-year 1-year to 2.5-year 2.5-year and longer

MMDAs at commercial banks and thrift institutions Savings deposits at commercial banks and thrift institutions

C. To convert an annual effective yield on a bank interest basis to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [[(1 + (r/100) / 360).sup.30] - 1] x (365 / 30) x 100

Own Rates Adjusted

Overnight repurchase agreements Overnight Eurodollars Overnight federal funds

D. To convert an n-month bank discount basis rate to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [365(r/100) / 360 - 30n(r/100)] x 100

Own Rates Adjusted

Secondary market Treasury bill rate: 1-month, 3-month, 6-month Commercial paper rate: 1-month, 3-month, 6-month Bankers acceptance rate: 3-month, 6-month

Table 8

Yield Curve Adjustments for Own Rates on Five Groups of Monetary Assets (by Treasury security used for adjustment)

A. Three-Month Secondary-Market Treasury Bill Rate

Eurodollar deposits, 3-month maturity

Commercial paper, 3-month maturity

Bankers acceptances, 3-month maturity

Negotiable certificates of deposit, 3-month maturity

Small-denomination deposits at commercial banks and thrift institutions, 7-day to 91-day maturity

Small-denomination time deposits at commercial and thrift institutions, variable ceiling rates on 91-day maturity

B. Six-Month Secondary-Market Treasury Bill Rate

Eurodollar deposits, 6-month maturity

Commercial paper, 6-month maturity

Bankers acceptances, 6-month maturity

Certificates of deposit, secondary-market, 6-month maturity

Small-denomination time deposits at commercial banks and thrift institution, 92- to 182-day maturity

Money market time deposits at commercial banks and thrift institutions, variable ceiling rates on 6-month maturity

C. One-Year Constant-Maturity Treasury Security Rate

Time deposits at commercial banks, 1-year maturity

Small-denomination time deposits at commercial banks and thrift institutions, 183-day to 1-year maturity

Small-denomination time deposits at banks and that institution, fixed ceiling rate on 1-year maturity

All Savers certificate, variable ceiling rate on 12-month maturity

D. Two-Year Constant-Maturity Treasury Security Rate

Small-denomination time deposits at commercial banks and thrift institutions, 1- to 2.5-year maturity

E. Three-Year Constant-Maturity Treasury Security Rate

Small-denomination time deposit at commercial banks and thrift institution, 2.5-year and longer maturity

NOTE: All rates are adjusted to an annualized one-month yield on a bond interest (365-day, coupon equivalent) basis.

Richard G. Anderson is an assistant vice president at the Federal Reserve Bank of St. Louis. Barry E. Jones and Travis D. Nesmith are Ph.D. candidates at Washington University in St. Louis and visiting scholars at the Federal Reserve Bank of St. Louis. Mary C. Lohmann, Kelly M. Morris, and Cindy A. Gleit provided research assistance.

Unlike the Board of Governor's monetary aggregates, the monetary services indexes and their dual user cost indexes are statistical index numbers, based on economic aggregation and statistical index number theory. The previous article in this Review, "Monetary Aggregation Theory and Statistical Index Numbers," surveys the literature on monetary aggregation theory and the use of statistical index number theory in monetary economics. Here, we discuss the construction of the monetary services index and related indexes.

In the first section, we define notation and introduce some key concepts that are used throughout the article. We emphasize the distinction between real and nominal monetary asset stocks and their user costs, and we review the concepts of the real monetary services index and its nominal dual user cost index. In the second section, we define each of the indexes in the monetary services indexes database, including the following: total expenditure on monetary assets; the nominal monetary services index; the real dual user cost index; the currency equivalent index; the simple sum index; and a set of indexes based on Theil's (1967) stochastic approach to index number theory. We emphasize that it is important to distinguish between real and nominal monetary index numbers: The aggregation theory underlying the monetary services indexes and related indexes is developed in terms of the real stocks of monetary assets, but actual monetary asset stock data are collected in nominal terms. We conclude that it is appropriate to construct a nominal monetary services index and thereafter to produce an approximation to the real monetary services index by deflating the nominal index.

In the third section, we describe the monetary asset stock data. We discuss the issue of weak separability, and we define the groupings of monetary assets for which we construct indexes. These groupings correspond to the assets contained in M1, M2, M3, and L, as well as the assets contained in M1A and MZM.(3) Because the aggregates are nested - each broader aggregate contains all the components of the previous, narrower aggregate - we refer to the groupings as levels of aggregation. M1A is the narrowest level of aggregation and L the broadest.

In the fourth section, we discuss the own rate of return data used in the construction of the indexes, and we detail the sources of this data. Of special importance are the methods by which we construct own rates for particular monetary assets. Specific issues include the implicit rate of return on demand deposits, fixed and variable ceiling rates for rate-regulated monetary assets, and the market rate of return on savings bonds. When the sample period of the own-rate data is shorter than that of the associated asset-stock data, we construct proxies for the missing own-rate data. Finally, we review the own rate conversions and yield curve adjustments of particular rates that are necessary because not all own rates are reported on the same basis or for the same maturity

In the fifth section, we detail the calculation of monetary asset user costs. Some published monetary asset stock data are, in fact, aggregates, or sub-indexes, of individual monetary assets with different user costs. It is necessary to obtain a single user cost index for these sub-indexes. Our solution to this problem, derived from unilateral index number theory, is described in this section. Finally, we discuss the concept of a benchmark asset and detail how we construct its rate of return.

The sixth section contains a discussion of some methodological difficulties associated with the project. The first is the introduction of new monetary assets; we implement Diewert's (1980) recommended solution to this problem. The second difficulty is created when published data for several monetary assets are combined into data for a single aggregate, or sub-index. We argue that it is inappropriate to treat this sub-index as a new monetary asset because doing so imputes economic relevance to the change in data reporting, when in fact there is none. We propose a solution to this problem that is based on the theory of splicing index numbers. The third issue is time aggregation. We use monthly data to construct the indexes. We implement Diewert's (1980) time aggregation methodology to produce quarterly and annual indexes from monthly data. Finally, we discuss seasonal adjustment.

NOTATION AND KEY CONCEPTS

In this section, we introduce notation and define some key concepts that will be used throughout the article. Readers are cautioned that this article's notation differs somewhat from that in "Monetary Aggregation Theory," because here we distinguish between real and nominal monetary assets.

Monetary Asset Stocks and User Costs

Assume that there are n monetary assets. Let [Mathematical Expression Omitted] denote the optimal real stock of monetary asset i in period t, and let

[Mathematical Expression Omitted]

be the vector of these optimal real stocks.(4) Similarly, let [Mathematical Expression Omitted] denote the optimal nominal stock of monetary asset i in period t, and let

[Mathematical Expression Omitted]

be the vector of these optimal nominal stocks. Real and nominal holdings of monetary assets are related by the identity

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] is a true cost-of-living index.(5)

The user cost of an asset is the equivalent rental price of that asset. If an asset fully depreciates during the economic agent's decision period, it is said to be non-durable and the asset's user cost equals its market price. If an asset does not fully depreciate within the decision period, it is said to be durable, and the appropriate opportunity cost of the durable asset is its user cost. Monetary assets are assumed to be durable. Expressions for the user cost of monetary assets were derived for consumers in Barnett (1978), and for firms in Barnett (1987, 1990).

To define the user costs of monetary assets, we need the concept of a benchmark asset - a risk-free asset that can be used only for intertemporal transfer of wealth and provides no monetary services. Let [r.sub.it] represent the nominal holding period yield on monetary asset i in period t, and let [R.sub.t] be the nominal holding period yield on the benchmark asset, called the benchmark rate, in period t.(6) The nominal user cost of monetary asset i in period t, [Mathematical Expression Omitted], is equal to the nominal value of interest income foregone by holding a unit of that asset for one period, [Mathematical Expression Omitted], discounted by [1/(1 + [R.sub.t])] to reflect the receipt of interest at the end of the period:

[Mathematical Expression Omitted],

This form of the user cost for a monetary asset is valid for both consumers and firms. Note, however, that consumers and firms often face different market interest rates and prices; hence their user costs will differ.

The real user cost of monetary asset i in period t, [Mathematical Expression Omitted],is defined by

[Mathematical Expression Omitted],

and the nominal and real user costs are related by the identity

[Mathematical Expression Omitted].

The real stock of a durable asset multiplied by its nominal user cost is equal to the total expenditure on that asset. Thus, expenditure on monetary asset i in period t is given by the product [Mathematical Expression Omitted], and total expenditure on monetary assets in period t is given by

[Mathematical Expression Omitted].

The Monetary Services Index and the Dual User Cost

Barnett (1980, 1987, 1990) derived the conditions under which monetary quantity and dual user cost aggregates will exist. These conditions are reviewed in our previous article in this Review, "Monetary Aggregation Theory." For consumers and firms, the monetary quantity aggregate is a measure of the flow of monetary services received by the holders of the monetary assets. Barnett (1980) first suggested the use of superlative statistical index numbers to track the flow of monetary services. Statistical index numbers, which contain no unknown parameters, are specification- and estimation-free functions of the prices and optimal quantities observed in two time periods.

Diewert (1976) showed that there exists a class of statistical index numbers, which he called superlative, that can provide second-order approximations to arbitrary economic aggregates in discrete time. Although there are many superlative index numbers, the Tornqvist-Theil index number is the only one known to retain its second-order tracking properties when some common aggregation theoretic assumptions are violated (Caves, Christensen, and Diewert, 1982; Anderson, Jones, and Nesmith, 1997).

Monetary quantity index numbers have been referred to by a variety of names in the past. We label our quantity indexes as monetary services indexes because of their close connection to the flow of monetary services in microeconomic demand models. The real Tornqvist-Theil monetary services index (chained Tornqvist-Theil quantity index formula), [Mathematical Expression Omitted], is defined by

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted] and [Mathematical Expression Omitted] are the expenditure shares of monetary asset i in periods t and t-1, respectively, and the average expenditure share of monetary asset i in period t is [Mathematical Expression Omitted].(7)

An index number that is dual to [Mathematical Expression Omitted] can be used to measure the price of a unit of monetary services. A price index number is said to be dual to a quantity index number if their product is equal to the total expenditure on the component assets included in the indexes, a property called factor reversal. Dual to [Mathematical Expression Omitted] is the nominal dual user cost index, [Mathematical Expression Omitted], which is defined using Fisher's (1922) weak factor reversal criterion by the formula

[Mathematical Expression Omitted].

Our real monetary services index, [Mathematical Expression Omitted], and its nominal dual user cost index, [Mathematical Expression Omitted], are constructed as chained superlative indexes. They therefore have the same statistical properties as other chained superlative quantity and price indexes - such as real gross domestic product (GDP), real personal consumption expenditures (PCE), and their price deflators - that are currently published by the U.S. Department of Commerce (Triplett, 1992).

INDEXES IN THE MSI DATABASE

In this section, we define the indexes in the MSI database. The formulas and definitions for these indexes are summarized in Table 1.

In this article, we distinguish carefully between nominal and real stocks of monetary assets. Monetary asset data collected by the Federal Reserve are necessarily in nominal terms, while monetary aggregation and statistical index number theory provide conditions for the aggregation of real stocks of monetary assets (Barnett, 1978, 1980, 1987, 1990; and our previous article in this Review, "Monetary Aggregation Theory."). Aggregation of nominal, rather than real, stocks of monetary assets requires some extension of the theory.

The identities and [Mathematical Expression Omitted] and [Mathematical Expression Omitted] can be used to demonstrate that total expenditure on monetary assets, [Y.sub.t], may be represented in two equivalent ways. Total expenditure may be expressed as either: (1) the sum of the products of the real asset stocks and their nominal user costs, or as (2) the sum of the products of the nominal monetary asset stocks and their corresponding real user costs:

[Mathematical Expression Omitted].

This result implies that the expenditure shares do not depend on the price index, [Mathematical Expression Omitted], and hence can be calculated correctly using observed nominal asset stocks and real user costs. The expenditure shares may be interpreted as either expenditure on real assets based on nominal user costs,

[Mathematical Expression Omitted],

or as expenditure on nominal assets based on real user costs,

[Mathematical Expression Omitted].

These relationships are important because they permit us to measure the total quantity of real monetary services by first constructing a quantity index from the observable nominal monetary asset stocks and then deflating that quantity index.

Specifically, we define the nominal Tornqvist-Theil monetary services index (chained Tornqvist-Theil quantity index number formula), [Mathematical Expression Omitted], by

[Mathematical Expression Omitted],

where [Mathematical Expression Omitted]. Because the [TABULAR DATA FOR TABLE 1 OMITTED] individual expenditure shares may be interpreted as either nominal or real shares, this formula is simply the usual Tornqvist-Theil quantity index number formula applied to nominal, rather than real, stocks of monetary assets.

Similarly, define the real dual user cost index, [Mathematical Expression Omitted], by

[Mathematical Expression Omitted].

Because the total expenditure on monetary assets can be defined in terms of nominal asset stocks and real user costs, this real dual user cost index will be dual to [Mathematical Expression Omitted].

To simplify the discussion that follows, we define the log change operator as [Delta] log([z.sub.t]) = log([z.sub.t]) - log([z.sub.t-1]), where all logarithms are base e, or natural, logs. Then the real and nominal monetary services indexes, and their real and nominal dual user cost indexes, are related by

[Mathematical Expression Omitted]

and

[Mathematical Expression Omitted],

respectively.(8) The real monetary services index may be constructed by aggregating over nominal asset stocks to produce the nominal monetary service index and then deflating this index; a similar relationship holds for the nominal and real dual user cost indexes.

The St. Louis MSI database includes the nominal monetary service index, [Mathematical Expression Omitted], and its real dual user cost index, [Mathematical Expression Omitted]. Although the nominal monetary service index may be deflated to produce its real counterpart, we leave the choice of deflator to the user because choice of the appropriate deflator depends on the model being studied by the user. In consumer demand models, the appropriate price index is a measure of the true cost of living. In firm factor demand models, the appropriate price index is an index of factor input prices. There is a large set of published price indexes, any one of which may or may not be appropriate in a specific application. These indexes include the Consumer Price Index (CPI), the Producer Price Index (PPI), the GDP deflator, and the PCE deflator. It may also be appropriate to deflate the indexes by using a measure of the real wage rate. Caveat emptor.

In the remainder of this section, we discuss the additional indexes in the MSI database. Although the currency equivalent (CE) index (Rotemberg, Driscoll, and Poterba, 1995; Rotemberg, 1991) and the simple sum monetary aggregates (as published by the Federal Reserve Board) are both inferior to the Tornqvist-Theil monetary services index as measures of the flow of monetary services, they have interpretations as stock concepts. Our previous article in this Review, "Monetary Aggregation Theory," provides a more complete discussion of these concepts.

Barnett (1991) proved that, under certain assumptions, the currency equivalent index

[Mathematical Expression Omitted]

measures the discounted present value of all current and future total expenditures on monetary assets.

Under the same assumptions, the simple sum index

[Mathematical Expression Omitted]

equals the sum of the discounted present value of the expected investment yields on current and future holdings of monetary assets, plus the CE index.

Theil (1967) noted that the Tornqvist-Theil price index number is not dual to the Tornqvist-Theil quantity index number - that is, the Tornqvist-Theil index formula is not self-dual. We can define the real Tornqvist-Theil user cost index (chained Tornqvist-Theil price index formula), [Mathematical Expression Omitted], as

[Mathematical Expression Omitted].

Theil's (1967) result, applied to monetary indexes, shows that

[Mathematical Expression Omitted],

where the Tornqvist-Theil expenditure share index, [S.sub.t], is defined by

[Mathematical Expression Omitted].

Theil (1967) also defined four indexes known as Divisia second moments: the Divisia quantity growth-rate variance, Divisia user-cost growth-rate variance, Divisia expenditure share growth-rate variance, and the covariance between the quantity and user-cost growth rates. Formulas for these indexes are shown in part B of Table 1.

Barnett and Serletis (1990) propose a dispersion dependency test, based on the Divisia second moments, for the failure of the principal assumptions of aggregation theory. The Divisia second moments may, for example, contain significant information during periods of regulatory change. The latter include the phased removal of Regulation Q ceilings on depository institutions' offering rates between 1978 and 1986 and the introduction of new types of deposits, such as All Savers certificates in 1978 and money market deposit accounts in 1982.

The dispersion dependency tests are applied to U.S. monetary data in Barnett and Serletis (1990) and Barnett, Jones, and Nesmith (1996). These studies suggest that, for at least some time periods, movements in the monetary data are not consistent with the principal assumptions of aggregation theory. In this case, Barnett and Serletis (1990) suggest that including Divisia second moments in macroeconomic models might provide a correction for this aggregation error. For additional discussion, see our previous article in this Review, "Monetary Aggregation Theory."

ASSET STOCKS AND AGGREGATES

In this section, we describe our monetary asset stock data in detail, and we discuss the levels of aggregation of the indexes in the MSI database. Discussion of the own rate data is deferred until the following section.

The monetary aggregates published by the Federal Reserve Board - M1, M2, M3, and L - are constructed by summing over sets of monetary asset stocks at four nested levels of aggregation. In addition to these levels of aggregation, some economists have advocated two other levels of aggregation: M1A and MZM. M1A consists of the non-interest-bearing monetary assets in M1, and MZM includes the monetary assets in M2 which do not have a fixed maturity.(9) These levels of aggregation are summarized in Table 2.

Some economists have recently suggested that monetary indexes should contain, in addition, highly liquid capital-uncertain assets such as bond and equity mutual funds (Collins and Edwards, 1994; Orphanides, Reid, and Small, 1994). Although the theoretical procedures used in the construction of the St. Louis MSI database are valid only under the assumption of perfect certainty, extending the theory to include risk-neutral households and firms is straightforward (Barnett, 1994). Extending the theory to include risk-averse agents, however, is more difficult and requires the subtraction of a risk premium from the monetary-asset user costs (Barnett and Liu, 1994; Barnett, Liu, and Jensen, 1997). We leave as a topic for future research the measurement of monetary service indexes that include an allowance for risk aversion.

The St. Louis MSI database contains monetary services indexes constructed over the same sets of assets (levels of aggregation) as the simple sum monetary aggregates MIA, M1, MZM, M2, M3, and L. We do not test for the weak separability of these levels of aggregation, although the correct level of aggregation of monetary assets should be determined by tests for weak separability.(10) Several previous studies have examined the weak separability of the assets included in M1, M2, M3, and L (Serletis, 1987; Swofford and Whitney, 1987, 1988). More recently, Swofford and Whitney (1994) and Spencer (1994) have noted that relaxation of the assumption of continuous complete portfolio adjustment, maintained in derivation of the monetary service index, significantly complicates separability testing. Testing the separability of the included assets is a topic for future research; researchers are encouraged to conduct their own tests using the disaggregated data provided in the database.

The asset stock data used to produce the indexes in the St. Louis MSI database are shown in Table 2. They include both seasonally adjusted and unadjusted data, except for the non-M3 components of L and Super NOW accounts at commercial banks and at thrift institutions, which are included only on an unadjusted basis. Most data were originally published by the Federal Reserve Board and have been later revised by Board staff. For discussion of Federal Reserve monetary aggregates and their components, see Anderson and Kavajecz (1994).

The data in Table 2 are reported at the most disaggregate level feasible.(11) Super NOW accounts have been separated from other checkable deposits from 1983.01-1985.12, the period in which separate data are available. Similarly, savings deposits and money market deposit accounts are separated during 1960.01-1991.08. In addition, the following asset categories are separated into thrift institution and commercial bank [TABULAR DATA FOR TABLE 2 OMITTED] categories: other checkable deposits, Super NOW accounts, small-denomination time deposits, savings deposits, and money market deposit accounts. The MSI database contains two exceptions to our practice of reporting data at its finest level of disaggregation. The first is that the sum of overnight and term Eurodollar deposits is included as total Eurodollar deposits, and the sum of overnight and term repurchase agreements is included as total repurchase agreements; the second is that large-denomination time deposits are not separated with respect to commercial banks and thrift institutions.(12)

OWN RATES OF MONETARY ASSETS

User costs of monetary assets are constructed from the assets' own rates of return. For most periods beginning October 1983, we base the user costs of deposits at commercial banks and thrift institutions on actual rates paid by depositories. These data have recently been revised and documented by Board staff. Although the Board of Governors has published deposit rate data for periods prior to October 1983, we choose to use the Regulation Q statutory rate ceilings due to (our) uncertainty whether the survey data are representative of rates generally paid.

In this section, we provide a detailed discussion of several aspects of the data, including our procedures for measuring the implicit rate of return on demand deposits and for estimating proxies for missing values, the market interest rates available on U.S. savings bonds, a set of own-rate conversions required prior to calculating user costs, and our yield curve adjustment of the own-rate data.(13) Table 3 lists the own-rate data used to calculate the indexes in the St. Louis MSI database.

The Implicit Rate of Return on Demand Deposits

To construct a user cost for demand deposits, we need to specify its own rate of return. Appropriate measures of this rate have been widely debated among economists because the Banking Acts of 1933 and 1935 forbade banks from paying explicit interest on demand deposits. Regardless, economists recognize that most financial institutions, during at least some recent time periods, have paid implicit interest on demand deposits in the form of free or reduced-cost bank services, or perhaps easier access to credit. Some economists have suggested that such non-price competition has allowed depositories to evade the prohibition of explicit interest on demand deposits. Startz (1979) discusses three competing hypotheses: the "traditional" hypothesis, which maintains that the prohibition on interest paid to demand deposits has been fully effective; the "competitive" hypothesis, which maintains that the prohibition of interest on demand deposits has been completely ineffective; and the modified competitive hypothesis, which maintains that the prohibition was partially effective.

Klein (1974) derived an expression for the fully competitive implicit rate of return on demand deposits. Assuming that banks earn no profit on demand deposits and that banks face perfectly competitive markets, the implicit rate of return for demand deposits is defined by

[r.sub.A] - [r.sub.D] = M[C.sub.D],

where [r.sub.D] is the implicit interest rate on demand deposits, [r.sub.A] is the interest rate on an alternative assets, and M[C.sub.D] is the marginal cost of producing demand deposits. Under additional assumptions, Klein shows that this is equivalent to

[r.sub.D] = (1- c) [r.sub.A],

where c is the ratio of reserves to deposits.

Startz (1979) advocates a modified competitive hypothesis. He argues, using functional cost analysis data, that the implicit demand deposit rate has been positive, well below the fully competitive Klein rate, and responsive to market interest rates.(14) Empirical evidence on the various hypotheses has been mixed (see Rush, 1980; Carlson and Frew, 1980; Allen, 1983; and Rossiter and Lee, 1987).

[TABULAR DATA FOR TABLE 3 OMITTED]

In previous constructions of monetary index numbers, it has been assumed that the prohibition of interest on demand deposits is completely ineffective for business demand deposits and is fully effective for household demand deposits. In the calculation of the implicit rate of return on business demand deposits, the alternative asset was assumed to be commercial paper with one month remaining to maturity. In Farr and Johnson (1985) and Thornton and Yue (1992), the distinction between household and business demand deposits was based on the Federal Reserve's Demand Deposit Ownership Survey (Board of Governors, 1971-1991). Because that survey has been discontinued, we cannot base our indexes on the methods used in previous studies.

We apply the modified competitive hypothesis to all demand deposits. Startz (1979) has argued that the implicit rate of return on demand deposits is between 0.34 and 0.58 times the fully competitive Klein rate, using five-year Treasury notes as the alternative asset. Thus, the implicit rate of return on demand deposits is proxied as

[r.sub.D] = (1 - [Tau])([r.sub.A])([Alpha]),

where [r.sub.A] is the rate on 5-year Treasury notes, [Tau] is (an estimate of) the maximum reserve requirement on demand deposits, and [Alpha] is between 0.34 and 0.58. In this article, we set [Alpha] equal to its maximum value of 0.58. This is equivalent to assuming that all demand deposits were issued by large banks, and that the deposit holders regarded deposits at different-size banks as perfect substitutes. Our estimates of [Tau], the maximum reserve-requirement ratio on demand deposits, are shown in Table 4.

Table 4 Statutory Maximum Reserve-Requirement Ratios for Transactions Deposits (percentage) Reserve-Requirement Ratio Applicable Dates 16.50(a) 1960.01-1967.12 17.00(a) 1968.01-1969.03 17.50(b) 1969.04-1973.06 18.00(b) 1973.07-1974.11 17.50(b) 1974.12-1975.01 16.50(c) 1975.02-1976.12 16.25(c) 1977.01-1980.10 12.00(d) 1980.11-1992.03 10.00 1992.04-present a On net demand deposits at reserve city banks, from Table 10.4, Banking and Monetary Statistics: 1941-1970 (1976). b On net demand deposits over $5 million at reserve city banks, from Annual Statistical Digest. c On net demand deposits over $400 million, from Annual Statistical Digest. d On net transaction deposits after implementation of the Monetary Control Act, from Annual Statistical Digest. Because the act's reserve requirements were phased in, member banks faced a marginal ratio above 12 percent through January 1984, and nonmember institutions faced a ratio below 12 percent through 1987 (1991 in Hawaii). We use 12 percent because no data exist on average effective marginal reserve-requirement ratios during the phase-in period. For discussion, see Anderson and Rasche (1996).

Regression-Based Proxies for Own-Rate Data

For some monetary assets, the asset stock data shown in Table 2 are available for dates before the earliest corresponding own-rate data shown in Table 3. Rather than discard these quantity data, we created proxies for the missing (unrecorded) own-rate data. For each such stock, we regressed the asset's available own-rate data for the later periods on one or more closely related rates and used the predicted values from the regression for earlier periods as proxies for the missing own-rate data. Our proxies, summarized in Table 5, are robust to reasonable alternative regression specifications.

Regulation Q Ceilings as Own-Rate Proxies

The regression method cannot be used for some commercial bank and thrift institution deposits prior to 1986. In these cases, we proxy the missing deposit own rates with the maximum rate that depositories were legally permitted to offer. These fixed and variable ceiling rates are summarized in parts A and B, respectively, of Table 6.

Negotiable order-of-withdrawal (NOW) accounts are checkable deposits currently included in the Federal Reserve's M1 monetary aggregate. Introduced in 1972 by a Massachusetts savings bank, NOW accounts spread rapidly: to all commercial banks and thrift institutions in Massachusetts and New Hampshire in January 1974, to the rest of the New England states in February 1976, to New York in November 1978, to New Jersey in December 1979, and nationwide in December 1980. Rates paid on NOW [TABULAR DATA FOR TABLE 5 OMITTED] accounts became subject to legal ceilings beginning in January 1974. Ceiling rates also were in effect for savings deposits during most of our sample through April 1986.

Small-denomination time deposits were subject to either fixed or variable ceiling rates during various periods of our sample; the latter were tied to Treasury market interest rates. We have constructed fixed ceiling-rate series for one-year maturity small-denomination time deposits at both commercial banks and thrift institutions. Because these rates, shown in Table 6, actually applied to deposits with a wide range of maturities, we caution the reader against overly precise interpretations. For large-denomination time deposits, we have assumed that depository institutions' offering rates, for dates since June 1964, have been approximately equal to secondary-market yields on negotiable certificates of deposit. The [TABULAR DATA FOR TABLE 6 OMITTED] latter are not available in the Federal Reserve's databases for dates prior to June 1964, however, and we have used as a proxy the ceiling rate permitted by the Federal Reserve's Regulation Q on time deposits payable from six months to one year. The Regulation Q ceiling was generally binding during this period (Federal Reserve Bulletin, 1963a,b, 1964a,b).

With the introduction of money market certificates in June 1978, some small-denomination time deposits were subject to variable ceiling rates that were tied to market interest rates (part B, Table 6).(15) We constructed these variable ceiling rates for small-denomination time deposits from information contained in various issues of the Federal Reserve Board's Annual Statistical Digest covering 1970-82.12 (16)

Market Interest Rate on Savings Bands

Investment yields to maturity for series-E savings bonds are available for January 1960-October 1982. Starting in November 1982, the Treasury Department issued bonds that paid a variable, market-based interest rate. This market rate is constructed according to the following procedure: The monthly five-year Treasury securities yield is averaged over six months, with six-month blocks beginning either on May 1 or November 1. The market-based savings bond rate for the next six months is equal to 85 percent of the average.(17)

Own-Rate Conversion

The application of aggregation theory and index-number methods to monetary data requires that all the own rates of return for the component assets be measured on the same basis. This is generally not true in published data because different sources have different reporting conventions, and because own rates are reported for a variety of different maturities. For monetary aggregation, the choice of a common measurement basis is arbitrary; that is, the information content of the index numbers is unaffected by the choice. We have chosen to convert all available rate data to an annualized monthly yield, calculated on a bond (or, coupon) equivalent basis, primarily for consistency with past monetary aggregation research. In this subsection, we describe general procedures for adjusting various own rates to this common basis. Our adjustments of own-rate data are summarized in Table 7 by type of adjustment. In each case, r is the unadjusted own rate of the asset, [r.sup.adj] is the adjusted own rate, and n is the maturity in months.

The simplest adjustment is to convert annualized one-month yields, quoted on a 360-day bank interest basis, to annualized one-month yields quoted on a 365-day bond coupon-equivalent basis. In this case, we simply multiply the unadjusted own rate by 365/360.

The second type of adjustment is to convert an annual effective yield, quoted in percentage points on a bond interest basis, to an annualized one-month holding-period yield on a bond interest basis. In this procedure, we convert the annual effective yield to a daily rate, compound that daily rate to a monthly rate, and then, assuming that all months contain 30 days, annualize the rate.

The third type of adjustment is to convert an annual effective yield on a bank interest basis to an annualized one-month holding-period yield on a bond interest basis, a procedure similar to the second one. We convert the bank-interest-basis annual effective yield to a daily rate, compound that daily rate to a monthly rate, and then, assuming a 30-day month, annualize to a bond interest basis.

In the fourth type of adjustment, we convert a rate quoted on a bank discount basis, for a monetary asset with a maturity of n months, to an annualized one-month holding-period yield. This conversion, which is discussed in detail by Farr and Johnson (1985), is valid only for rates with maturity of less than six months, and it assumes that each month has 30 days.

Yield Curve Adjustment

Own rates for monetary assets that have different maturities may have different term premiums, and hence are not directly comparable. Therefore, in addition to making the above adjustments, we need to remove a liquidity, or term, premium from each own rate. We yield curve adjust monetary assets' own rates by using the yield curve for U.S. Treasury securities. These adjustments of the own-rate data are summarized in Table 8.

We adjust the own rates by subtracting, from each own rate, an estimate of the liquidity premium obtained from the yield curve for Treasury securities. (Because these securities have no default risk, the slope of the Treasury yield curve provides a relatively "pure" estimate of the term premium.) The following discussion of yield curve adjustment assumes that all own rates (including Treasury bill rates) have been converted to an annualized one-month holding-period yield, on a bond interest basis.

Let [r.sub.n] be an own rate for a monetary asset with a maturity of n months, let [Mathematical Expression Omitted] be the own rate on Treasury securities that mature in n months, and let [Mathematical Expression Omitted] be the one-month secondary-market Treasury bill rate. The own rate, [r.sub.n], is yield curve adjusted by subtracting the estimated liquidity premium [Mathematical Expression Omitted] from the own rate, such that the yield curve adjusted own rate, [Mathematical Expression Omitted]. For a Treasury security that matures in n months [Mathematical Expression Omitted] is the n-month secondary-market Treasury bill rate, adjusted from a bank discount basis to an annualized one-month holding-period yield on a bond interest basis. If maturity is in n years [Mathematical Expression Omitted] is the corresponding constant-maturity Treasury security. Other values of [Mathematical Expression Omitted] may be interpolated from the Treasury's constant-maturity yield curve.

If a single monetary asset stock contains components with a range of maturities, we yield-curve adjust the own rate using the yield on a Treasury security with a maturity that falls within that range.

USER COSTS OF MONETARY ASSETS

In this section, we discuss in detail how we construct the user costs for monetary assets from the previous section's adjusted own-rate data. We address the problem that reported monetary asset stock data, such as the Federal Reserve Board's data on small-denomination time deposits, do not distinguish between monetary assets with different terms to maturity, and we construct the user costs of such assets as unilateral user cost sub-indexes.

Monetary Assets With Different Maturities

The definition of the real user cost of a monetary asset assumes that, in each period, each asset has a single applicable own rate and, hence, a unique user cost. Published Federal Reserve Board deposit data for commercial banks and thrift institutions do not distinguish adequately among monetary assets with different maturity. Only total dollar amounts, summed across all maturities, are reported for the following categories: small-denomination time deposits at commercial banks and thrift institutions; large-denomination time deposits; total Eurodollar deposits; total repurchase agreements; bankers acceptances; short-term Treasury securities; and commercial paper.(18) This bundling of assets with different maturities into monetary asset sub-indexes causes difficulty. Prior to measuring the sub-index's user cost, one should remove a liquidity (or term) premium from each component monetary asset's user cost. Because the own rates of the unobserved subcomponents may differ even after the yield curve adjustment, several user costs may apply to the sub-index.(19) A method must be found to combine the various user costs into a single user cost sub-index that corresponds to the reported asset stock.

A similar problem, in which a single price index is constructed from multiple individual prices without the use of quantity data, has been examined by Diewert (1995).(20) Price indexes constructed solely from price data, without quantity data, are called unilateral price indexes. Diewert (1995) advocates the use of a particular unilateral price index formula, called the Jevons unilateral price index. We construct such unilateral user cost indexes, based on the Jevons formula, for the following monetary asset categories: small-denomination time deposits at commercial banks and thrift institutions; large-denomination time deposits; total Eurodollar deposits; bankers' acceptances; and commercial paper.

User Costs by Component

In Table 9, we summarize the own rates used in the construction of each user cost. The own rates are (1) adjusted to a common basis, (2) yield curve adjusted, and (3) proxied, where appropriate (see Tables 5, 8, and 9). The own-rate series refer either to the own rate data shown in Table 4, or to the data discussed in the previous section of this article.

The construction of real user costs also requires the rate of return for a benchmark asset. The benchmark asset is a theoretical construct: It provides no monetary services, has no default risk, and is used by economic agents only to transfer wealth between periods. A theoretical lower bound for the benchmark asset can be identified; because monetary services are valued by households and firms, the user costs of monetary assets must be positive. Thus, the benchmark asset's rate of return must exceed the own rates on all assets that furnish monetary services.

A theoretical way of constructing the benchmark rate is to set it equal to the maximum rate of return over a large class of assets, both financial and non-financial. This method is inappropriate, however, because (unadjusted) rates of return on [TABULAR DATA FOR TABLE 9 OMITTED] debt and equity contain risk premia. In empirical work, the traditional approach has been to identify the benchmark rate during each time period, t, as the "envelope" of the own rates of return on monetary assets and the rate on Moody's seasoned BAA bonds, [r.sub.BAA,t]:

[R.sub.t] = max{[r.sub.it] (i = 1, 2, ..., n), [r.sub.BAA,t]}

(Barnett and Spindt, 1982; Farr and Johnson, 1985; and, Thornton and Yue, 1992). We adopt this practice, with a minor modification, and define the benchmark rate as

[Mathematical Expression Omitted],

where c is a small constant. Although we typically set the value of the constant at one basis point or less, its inclusion guarantees that the benchmark rate is strictly greater than the rate on any monetary asset, and it allows us, in a previous section, to define Divisia second moments of our indexes. The indexes are robust experimentally to a large range of values for the constant.

Unilateral Index Number Theory

In this section, we provide the reader with a discussion of unilateral index number theory in the context of monetary aggregation, and we define and discuss our use of the Jevons unilateral price index formula.

Bilateral index numbers, such as the Tornqvist-Theil index number, are functions, in each period, of both observed prices and quantities. Unilateral index numbers are functions, in each period, of either the observed prices or the observed quantities, but not both. Unilateral indexes may be useful, therefore, when some of the price or quantity data required for a bilateral index have not been recorded. As previously noted, the Federal Reserve Board's monetary asset stock data often do not separate monetary assets with different maturities. The reported asset data - total dollar amounts summed across all maturities - are unilateral quantity indexes. We refer to these aggregates as monetary quantity sub-indexes, and the unreported individual assets with differing maturities as sub-components.

In an ideal world, we would treat each sub-component of each monetary quantity sub-index as a separate asset with its own user cost. In practice, the data collection process forces us to treat each monetary quantity sub-index as if it were a single asset. If the user costs of the sub-components are observed, we can construct a unilateral user cost sub-index, which can be viewed as the "user cost" of the monetary quantity sub-index.

Let [Mathematical Expression Omitted] be the value of a monetary quantity sub-index in period s, let [Mathematical Expression Omitted] be a vector of M own rates that apply to the sub-components of the monetary quantity sub-index. Then, [Mathematical Expression Omitted] where j = 1, ..., M, are the real user costs that apply to the sub-components of the monetary quantity sub-index, [Mathematical Expression Omitted]. (We remind the reader that the own rates in the vector [Mathematical Expression Omitted] must all be converted to a common basis, and yield curve adjusted.) Diewert (1995) defines the Jevons and the Dutot unilateral price indexes.(21) For monetary aggregation, the Jevons user cost sub-index is defined by

[Mathematical Expression Omitted],

and the Dutot user cost sub-index is defined by

[Mathematical Expression Omitted].

Diewert (1995) defined a set of axioms that a reasonable unilateral price index should satisfy, and he showed that both the Jevons and the Dutot price indexes satisfy these axioms.(22) It can also be shown that a third unilateral price index, which we call the Leontief price index, satisfies weak versions of these same axioms.(23) In the present context, the Leontief user cost sub-index, [Mathematical Expression Omitted] is defined by

[Mathematical Expression Omitted].

Because the Jevons, Dutot, and Leontief user cost sub-indexes all satisfy Diewert's axioms, we can compare the economic justifications of the indexes.

Diewert (1976) showed that the members of a class of bilateral statistical index numbers, called superlative, have strong economic justification as approximations of aggregator functions. Unfortunately, unilateral price indexes have only weak economic justifications; Diewert (1995) gives the stringent conditions under which the Jevons and Dutot user cost sub-indexes will be correct (exact).(24) The Jevons user cost sub-index will be correct only if two conditions hold: (1) the elasticities of substitution between the sub-components are unity, once the liquidity premium has been extracted; and (2) the expenditure shares on each sub-component are constant. The Dutot user cost sub-index will be correct if two conditions hold: (1) the elasticities substitution between the sub-components are zero, once the liquidity premium has been extracted; and (2) in each period the quantities of the sub-components are equal.

The Leontief user cost sub-index will be correct if the sub-components are, in fact, perfect substitutes after the liquidity premium has been removed. This assumption underlies the user cost sub-indexes that have been constructed, for certain subsets of assets, by Barnett and Spindt (1982), Farr and Johnson (1986), and Thornton and Yue (1992). In these articles, Barnett's (1978) user cost formula was applied, in each time period, to the maximum of the yield curve-adjusted own rates for the assets included in each subset, a procedure that is equivalent to the Leontief user cost sub-index. These sub-indexes were subsequently aggregated with other assets and user costs, using superlative index number formulas such as the Tornqvist-Theil or Fisher Ideal index formula.

Diewert (1995) argued that, for calculating price indexes, the Jevons index formula is superior to the Dutot index formula because the Jevons index's conditions - unit elasticity and constant (or proportional) expenditures - are more plausible than the Dutot index's conditions: zero elasticity and constant (or proportional) quantities. The Advisory Commission to Study the Consumer Price Index (1996) also advocated the use of the Jevons index formula to calculate lower-level price indexes for sub-components of the CPI. The Jevons index has been widely used as the benchmark for studying bias in lower-level price indexes in a number of countries; see Diewert (1995) for a review of these studies. The current consensus is that the Jevons index number formula should be used to calculate unilateral price indexes.

For measuring the user costs of our monetary service index, we concur with Diewert's preference for the Jevons index. We further argue that the Jevons index's unit elasticity condition is more reasonable than the Leontief index's assumption that monetary assets are perfect substitutes. The perfect substitutes condition would imply that, unless all user costs applying to the sub-index are equal, economic agents will hold only the least-expensive sub-component of each monetary sub-index - an implication that is usually rejected when it can be tested.(25) In our MSI database, we use the Jevons formula to create user cost sub-indexes for small-denomination time deposits at commercial banks and thrift institutions, large-denomination time deposits, total Eurodollar deposits, bankers acceptances, and commercial paper. The growth rate of the Jevons user cost sub-index, in two adjacent periods, is the ratio of the geometric means of the applicable user costs. We cardinalize the Jevons user cost sub-index by setting the initial value of the sub-index equal to the geometric mean of the user costs during the initial period.

After selection of the Jevons index, one more important difficulty remains: The number of own rates that are observed for the sub-components of a monetary quantity sub-index may change, from period to period, due to a number of factors, including changes in regulations and data-collection practices. In these cases, we calculate the growth rate of the Jevons user cost sub-index from the subset of user costs that are observed in the adjacent periods. This procedure is based on Diewert's (1980) new goods procedure, which is discussed in the section of this paper titled, "Introduction of New Monetary Assets." In a few cases, the set of observed sub-component user costs in adjacent periods changes completely. In such cases, we calculate, for both periods, the geometric means of the observed user costs and then calculate the Jevons index as the ratio of the current period's geometric mean divided by the geometric mean in the previous period.(26)

ADDITIONAL PROBLEMS

Several additional problems that arise in the construction of monetary services indexes are discussed in the following sub-sections: (1) the introduction of new monetary assets, (2) changes in the definitions of underlying monetary asset stock data, (3) the calculation of monetary service indexs and related indexes at different frequencies, and (4) seasonal adjustment of the indexes.

Introduction of New Monetary Assets

There have been many financial innovations during the time span of our monetary services indexes. New monetary assets have been created at various dates, and the indexes must be modified to include them.

The nominal Tornqvist-Theil monetary services index, [Mathematical Expression Omitted], and its real dual price index, [Mathematical Expression Omitted], are not well defined when new assets enter the indexes. The real Fisher Ideal user cost index,

[Mathematical Expression Omitted],

is well defined, and a corresponding quantity index may be obtained by Fisher's factoral reversal formula. We therefore switch to the Fisher Ideal index in periods when new monetary assets are introduced.(27)

To implement this approach, we need to develop an estimator for the new asset's user cost during the period prior to its introduction. Theoretically, the correct solution is to define a user cost, called the reservation user cost, that is sufficient to ensure that a zero quantity of the new asset would have been demanded at that user cost during the prior period if the asset had, in fact, existed. In practice, doing this correctly requires econometric estimation of the aggregator function (Diewert, 1980), whereas our primary motive for the use of statistical index numbers is to avoid such estimation.

Rather than estimate the reservation user cost, we use the following method, introduced by Diewert (1980) and used in Diewert and Smith (1994).(28) In the period when a new monetary asset is introduced, we calculate the Fisher Ideal real user cost index over all monetary assets except the new one, which we will call [Mathematical Expression Omitted]. If monetary asset i is introduced in period t, [Mathematical Expression Omitted] will be defined by

[Mathematical Expression Omitted].

Diewert (1980) shows that this procedure will, in general, have lower bias than the other available alternatives, in the absence of strong information about the reservation user cost. The procedure is exactly correct in a special case: If the actual user cost of the new asset i in period t divided by the reservation user cost is equal to

[Mathematical Expression Omitted],

then [Mathematical Expression Omitted] will be exactly correct.

We form our real user cost indexes by switching to a Fisher Ideal index, calculated according to Diewert's recommended approach, during periods in which new monetary assets enter the indexes. The dual monetary services index is then defined implicitly by Fisher's weak factor reversal formula.

In Table 10, we list the periods in which new monetary assets are introduced.

Changes in the Definitions of Asset Stock Data

In the preceding section, we discussed the introduction of new monetary assets. A related problem is that, at times, the Federal Reserve has changed the definitions and the manner of reporting the components of its monetary aggregates. This happens twice in our series: (1) after 1985.12, Super NOW accounts are included in other checkable deposits (OCD), and (2) after 1991.08, money market deposit accounts (MMDA) and savings deposits are reported only on a combined basis, for thrift institutions and for commercial banks. In both of these cases, monetary assets that had been reported separately were combined into sub-indexes, and the sub-component data were no longer available.

These changes represent a redefinition of the asset stocks (and consequently the monetary services indexes), but they do not represent a meaningful change in the structure of the economy; in other words, these data-reporting changes are not economically relevant. The Federal Reserve Board's monetary aggregates are invariant to such changes because their aggregates are themselves sums of all the component data. Tornqvist-Theil monetary services indexes are not invariant to these changes because the change in reporting, from a group of assets to a single sub-index, represents a loss of information. In this section, we describe our approach to this problem.(29)

From 1983.01 through 1985.12, Super NOW accounts and savings deposits are included in our Tornqvist-Theil monetary services indexes as separate assets. Beginning in 1986.01, however, Super NOW accounts and savings deposits were reported only as a combined total. In response to this change, we define a second Tornqvist-Theil monetary services index that begins in 1985.12 and contains the total of Super NOW accounts and other checkable deposits as a single asset. The value of the second index in its initial period, December 1985, is arbitrary, which permits us to scale the second index so that it equals the first index in 1985.12. This splices the two indexes so as to produce a single Tornqvist-Theil monetary services index over the entire period. We perform an analogous splice in 1991.08 when money market deposit accounts and savings deposits begin to be reported only on a combined basis.(30)

Data reporting changes that are not based on economic reasons, such as financial innovation or regulatory changes, represent a loss of information. We preserve as much information as possible during the periods when disaggregate data are available and avoid imputing economic relevancy to the data reporting change when it occurs. Our method draws on the literature of index number splicing (Hill and Fox, 1995).

Indexes at Different Frequencies

The disaggregated data in the MSI database are reported monthly In some applications, monetary aggregates must be available at quarterly or annual frequency In this section, we discuss a method developed by Diewert (1980) for constructing, from monthly data, indexes at quarterly and annual frequencies.

In the problem of constructing annual indexes from monthly indexes, the solution is to treat each asset, in each month, as a separate asset and then to aggregate over these assets. For example, demand deposits held in January and in February will be treated as different assets. Formally, let [Mathematical Expression Omitted] be the nominal stock of monetary asset i in month r of year t. Similarly let

[Mathematical Expression Omitted]

be the real user cost associated with, [Mathematical Expression Omitted], where [Mathematical Expression Omitted] is the rate of return on the benchmark asset in month r of year t, and [Mathematical Expression Omitted] is the rate of return on the nominal stock of monetary asset i in the month r of year t. Then the log change of the annual Tornqvist-Theil nominal monetary services index, [Mathematical Expression Omitted] is defined by

[Mathematical Expression Omitted],

where

[Mathematical Expression Omitted].

Table 10 Introduction of New Assets New Asset Introduction Date Total repurchase 1969.10 agreements Retail money funds 1973.02 Other checkable deposits 1974.01 at commercial banks Institutional money funds 1974.01 Money Market deposit 1982.12 accounts at commercial banks Money Market deposit accounts 1982.12 at thrift institutions Super NOW accounts 1982.12 at commercial banks Super NOW accounts 1982.12 at thrift institutions

An analogous method can be used to define the quarterly indexes. In the MSI database, this method is used to produce both annual and quarterly indexes. Dual user cost indexes are obtained by Fisher's weak factor reversal criterion.(31)

Seasonal Adjustment

The issue of seasonal adjustment is a difficult one. Index number theoretic methods for dealing with seasonality, which are related to the issues discussed in the section of this paper dealing with indexes at different frequencies, can be found in Diewert (1980, 1983, 1996). Our approach is more traditional. We produce the indexes in the database by using both seasonally adjusted and unadjusted asset stock data, except for the non-M3 components of L, which are not seasonally adjusted in either set of indexes. Our seasonally adjusted data are produced with the Bureau of the Census X11 program, using default values for all options. We urge users of our unadjusted data to experiment with alternative seasonal adjustment methods.

CONCLUSION

The St. Louis MSI database is an important resource for economists and policymakers studying the role of money in the economy. Monetary services are an important aspect of the economic behavior of households and firms, and the monetary service indexes provide new up-to-date measures of the flows of monetary services. The database also contains dual measures of the opportunity cost of monetary services and related stock and total expenditure variables.

The indexes in the MSI database are consistent with microeconomic aggregation theory and have the same statistical properties as commonly used macroeconomic indexes such as GDP and its deflator. In general, the monetary service index and its dual user cost index can be modeled in the same way as other macroeconomic quantity and price indexes and, in particular, models of money demand can be estimated by using the monetary service index.

In addition to our aggregate indexes, the MSI database contains disaggregate asset-stock and user-cost data that will allow researchers to study the demand for the disaggregated monetary assets in a way that is consistent with microeconomic models of decision making. The database is also comprehensive enough to allow researchers to experiment with alternative levels of aggregation, different measures of assets' own rates, and various seasonal adjustment techniques.

These data provide numerous opportunities for applied monetary research. Although monetary services indexes have been produced before by Barnett and Spindt (1982), Farr and Johnson (1985), and Thornton and Yue (1992), none of these studies furnished a broad enough set of indexes, the underlying data, or the computer programs necessary to build the indexes.

We offer special thanks to Professor William A. Barnett, Washington University in St. Louis, for his generous and invaluable guidance during this project. We also thank Professors W. Erwin Diewert, University of British Columbia, and Adrian R. Fleissig, St. Louis University, for advice and comments. For assistance with data, we thank the Savings Bond Operations Office of the U.S. Department of the Treasury and the Division of Monetary Affairs of the Board of Governors of the Federal Reserve System.

1 The address is www.stls.frb.org/research.

2 The Board of Governors' monetary aggregates are published weekly in the statistical release, Money Stock, Liquid Assets, and Debt Measures, and monthly in the Federal Reserve Bulletin, Table 1.21.

3 Simple sum M1A (non-interest bearing M1) was produced as an official monetary aggregate from 1960 through April 1971 (Kavajecz, 1994). MZM was suggested by William Poole.

4 We assume that these stocks were chosen by an optimizing economic agent.

5 The correct price index to be used depends on the context. We state the appropriate index for the consumer case. Barnett (1987) uses the same index for firms and financial intermediaries, although he notes that this is not strictly correct.

6 All holding period yields are assumed to be reported on a common basis. This issue is discussed in the section of this article titled "Own Rate Data."

7 Because of its connection with Divisia's (1925) continuous time index number, Barnett (1980) referred to this index as the Divisia Index.

8 In discrete time, these equalities are true up to a third-order error when the true cost of living index is measured by a superlative index number. If a non-superlative price index is used, the equality will be true only up to the tracking ability of the index.

9 William Poole's MZM included institutional money market mutual funds. We exclude these funds because they do not follow the same accounting rules as retail money market funds, and are marketed only to larger investors.

10 A weakly separable block could contain both monetary assets and consumption goods, but an aggregate formed over such a block would not usually be interpreted as a monetary service flow.

11 The criterion for feasibility is that disaggregated data for the desired asset stock must be available and of good quality. In addition, reliable own-rate data for the category must exist.

12 We combine overnight and term repurchase agreements and Eurodollar deposits because we have no separate, reasonable own rates for their components. We combine large-denomination time deposits at commercial banks and thrift institutions for the same reason.

13 Additional discussion can be found in Barnett and Spindt (1982), Farr and Johnson (1985), Thornton and Yue (1992), and Belongia (1995).

14 Other implicit rates of return are discussed by Becker (1975) and Barro and Santomero (1972).

15 Ceiling on small-denomination time deposits with original maturity of three and one-half years or longer were removed in May 1982, and on all other small-denomination time deposits on October 1, 1983.

16 The linkages between variable ceiling rates and auction-average Treasury rates are discussed in Mahoney (1987) and Annual Statistical Digest, editions for 1970-1979 and 1980, 1981, and 1982.

17 This methodology was supplied to us by the Savings Bond Operations Office of the U.S. Department of the Treasury.

18 Data are not available on the outstanding quantities of monetary assets by remaining time to maturity. Data published by the Board of Governors regarding the outstanding stocks of small-denomination time deposits by original-issue maturity are not appropriate for calculating index numbers (see Table 1.22, Federal Reserve Bulletin, February 1997).

19 Total repurchase agreements have a unique user cost in each period because we use the only available rate, that on overnight agreements, for oil maturities. Small-denomination time deposits also have several applicable own rates due to the existence of both variable and fixed ceiling-rate time deposits from 1978.06 to 1983.9.

20 The Advisory Commission to Study The Consumer Price Index (1996) discusses a similar problem in the construction of lower-level price indexes, in which multiple prices are combined into a single price index.

21 These indexes also are defined in Diewert (1992).

22 The growth rate of the Dutot price index is the ratio of the averages of prices in adjacent periods. An index based on the average of the ratios of prices in adjacent periods, the Carli (Diewert, 1992), does not satisfy the time reversal test, hence is not a reasonable unilateral price index.

23 Specifically, the Leontief index satisfies the axiomatic tests if strict monotonicity is weakened to monotonicity

24 These unilateral index numbers are based on particular constant elasticity of substitution (CES) aggregator functions, which are not flexible functional forms. Hence the unilateral indexes are not superlative.

25 Prior to 1991:08, for example, economic agents held non-zero quantities of both MMDA and saving deposits assets, even though the user costs differed.

26 Our calculations assume that each unit of a specific monetary asset stock has the same user cost, after necessary conversions and yield curve adjustment. Aggregation error will occur if deposit own rates vary with the size of the deposit. Some evidence on this practice is analyzed by Collins (1991).

27 Farr and Johnson (1985) advocate the Fisher Ideal index because it is well-defined even when new assets are introduced. For all periods when data are available, the Tornqvist-Theil index is superior to the Fisher Ideal index because it is superlative in a stronger sense (Caves, Christensen, and Diewert, 1982).

28 Diewert suggests this procedure in the general case; we state it here in the case of monetary aggregation.

29 The CE and simple sum indexes are well-defined and do not require any modifications when asset stocks are redefined. The real user cost index that is dual to the Tornqvist-Theil monetary services index is calculated by factor reversal. The Tornqvist-Theil real user cost index and the Tornqvist-Theil expenditure share index are calculated according to the procedure described in this section.

30 In 1985.12, the combined OCD asset stock is the sum of the asset stocks of its subcomponents, and the user cost for OCD is constructed from the weighted average of its sub-components' own rates. A similar procedure is followed in August 1991 for savings deposits and MMDA.

31 At quarterly and annual frequencies, the splicing procedure described in the preceding sub-section needs to be modified in a straightforward way.

REFERENCES

Advisory Commission to Study the Consumer Price Index. Toward a More Accurate Measure of the Cost of Living, Final Report to the Senate Finance Committee (December 1996).

Allen, Stuart D. "A Note on the Implicit Interest Rate on Demand Deposits," Journal of Macroeconomics (Spring 1983), pp. 233-39.

Anderson, Richard G., and Kenneth A. Kavajecz. "A Historical Perspective on the Federal Reserve's Monetary Aggregates: Definition, Construction and Targeting," this Review (March/April 1994), pp. 233-39.

----- and Robert H. Rosche. "Measuring the Adjusted Monetary Base in an Era of Financial Change," this Review (November/December 1996), pp. 3-37.

-----, Barry E. Jones, and Travis D. Nesmith. "An Introduction to Monetary Aggregation Theory and Statistical Index Numbers," this Review, this issue.

Barnett, William A. "Reply [to Julio J. Rotemberg]," Monetary Policy on the 75th Anniversary of the Federal Reserve System: Proceedings of the Fourteenth Annual Economic Policy Conference of the Federal Reserve Book of St. Louis, Michael T. Belongia, ed., Kluwer Academic Publishers, 1991, pp. 232-43.

-----. "Developments in Monetary Aggregation Theory," Journal of Policy Marketing (Summer 1990), pp. 205-57.

-----. "The Microeconomic Theory of Monetary Aggregation," New Approaches to Monetary Economics: Proceedings of the Second International Symposium in Economic Theory and Econometrics, William A. Barnett and Kenneth J. Singleton, eds., Cambridge University Press, 1987, pp. 115-68.

-----. "Economic Monetary Aggregates: An Application of index Number and Aggregation Theory," Journal of Econometrics (Summer 1980), pp. 11-48.

-----. "The User Cost of Money," Economic Letters (vol.1, 1978), pp. 145-49.

----- and Yi Liu. "Beyond the Risk Neutral Utility Function," paper presented at the University of Mississippi conference, "Divisia Monetary Aggregates: Theory and Practice," October 1994; conference volume, Macmillan, forthcoming.

----- and Apostolos Serletis. "A Dispersion-Dependency Diagnostic Test for Aggregation Error: With Applications to Monetary Economics and Income Distribution," Journal of Econometrics (January/February 1990), pp. 5-34.

----- and Paul A. Spindt. "Divisia Monetary Aggregates: Compilation, Data, and Historical Behavior," Staff Study 116, Board of Governors of the Federal Reserve System, May 1982.

-----, Barry E. Jones, and Travis D. Nesmith. "Divisia Second Moments: An Application of Stochastic Index Number Theory," International Review of Comparative Public Policy (vol. 8, 1996), pp. 115-38.

-----, Yi Liu, and Mark Jensen. "The CAPM Risk Adjustment far Exact Aggregation over Financial Assets," Macroeconomic Dynamics (May 1997), forthcoming.

Barro, Robert J., and Anthony J. Santomero. "Household Money Holdings and the Demand Deposit Rate," Journal of Money, Credit, and Banking (May 1972), pp. 397-413.

Becker, William E., Jr. "Determinants of the United States Currency-Demand Deposit Ratio," The Journal of Finance (March 1975), pp. 57-74.

Belongia, Michael T. "Weighted Monetary Aggregates: A Historical Survey," Journal of International and Comparative Economics (vol. 4, 1995), pp. 87-114.

Board of Governors of the Federal Reserve System. Annual Statistical Digest, editions for 1970-79, 1980-89, and annual editions for 1980 through 1995.

-----. Money Stock, Liquid Assets, and Debt Measures, H.6 statistical release.

-----. Selected Interest and Exchange Rates, N.13 statistical release (various issues).

-----. Selected Interest Rates, H.15 statistical release (various issues).

-----. Federal Reserve Bulletin (various issues).

-----. "Demand Deposit Ownership Survey," Federal Reserve Bulletin, July issues, 1971-91.

-----. Banking and Monetary Statistics: 1941-1970, September 1976.

-----. "Survey of Time and Savings Deposits at Commercial Banks," Federal Reserve Bulletin, vols. 52 through 68.

-----. "Flows Through Financial Intermediaries," Federal Reserve Bulletin (May 1964a), pp. 549-57.

-----. "Bank Credit and Money in 1963," Federal Reserve Bulletin (February 1964b), pp. 141-47.

-----. "Interest Rates on Time Deposits, Mid-February 1963," Federal Reserve Bulletin (June 1963b), pp. 766-72.

-----. "Negotiable Time Certificates of Deposit," Federal Reserve Bulletin (April 1963b), pp. 458-68.

Brennan, Michael J., and Eduardo S. Schwartz. "Savings Bonds: Theory and Empirical Evidence," Monograph Series in Finance and Economics, Monograph 1979-4. Saloman Brothers Center for the Study of Financial Institutions, 1979.

Caves, Douglas W., Laurits R. Christensen, and W. Erwin Diewert. "The Economic Theory of Index Numbers and the Measurement of Input, Output, and Productivity," Econometrica (November 1982), pp. 1393-414.

Carlson, John A., and James R. Frew. "Money Demand Responsiveness to the Rate of Return on Money: A Methodological Critique," Journal of Political Economy (June 1980), pp. 598-607.

Collins, Sean. "Do Banks Price Discriminate Among Their Retail Customers? A Test Based on Tiered Deposit Rates," working paper, Division of Monetary Affairs, Board of Governors of the Federal Reserve System, September 1991.

----- and Cheryl L. Edwards. "An Alternative Monetary Aggregate: M2 Plus Household Holdings of Bond and Equity Mutual Funds," this Review (November/December 1994), pp. 7-29.

Diewert, W. E. "Seasonal Commodities, High Inflation and Index Number Theory," Working Paper DP 96-06, University of British Columbia, January 1996.

----- "Axiomatic and Economic Approaches to Elementary Price Indexes," Working Paper No. 5104, National Bureau of Economic Research, May 1995.

----- "The Treatment of Seasonality in a Cost-Of-Living-Index," Price Level Measurement, W.E. Diewert and C. Montmarquette, eds., Ottawa: Statistics Canada, 1983, pp. 1019-45.

----- "Index Numbers," The New Palgrave Dictionary of Money & Finance, Vol. 2, Peter Newman, Murray Milgate, and John Eatwell, eds., Stockton Press, 1992, pp. 364-79.

----- "Aggregation Problems in the Measurement of Capital," The Measurement of Capital, Dan Usher, ed., University of Chicago Press, 1980, pp. 433-538.

----- "Exact and Superlative Index Numbers," Journal of Econometrics (May 1976), pp. 115-45.

----- and Ann Marie Smith. "Productivity Measurement for a Distribution Firm," Journal of Productivity Analysis (vol. 5,1994), pp. 335-47.

Divisia, Francois. "L'lndice Monetaire et la Theorie de la Monnaie," Revue d'Economie Politique (1925), pp. 980-1008.

Farr, Helen T., and Deborah Johnson. "Revisions in the Monetary Services (Divisia) Indexes of the Monetary Aggregates," Staff Study 147, Board of Governors of the Federal Reserve System, December 1985.

Fisher, Irving. The Making of Index Numbers: A Study of Their Varieties, Tests, and Reliability, Houghton Mifflin Company, 1922.

Hill, Robert J., and Kevin J. Fox. "Splicing Index Numbers," working paper, School of Economics, The University of New South Wales, 1995; Journal of Business and Economic Statistics, forthcoming.

Kavajecz, Kenneth A. "The Evolution of the Federal Reserve's Monetary Aggregates: A Timeline," this Review (March/April 1994), pp. 32-66.

Klein, Benjamin. "Competitive Interest Payments on Bank Deposits and the Long-Run Demand for Money," American Economic Review (December 1974), pp. 931-49.

Mahoney, Patrick I., et.al., "Responses to Deregulation: Retail Deposit Pricing from 1983 through 1985," Staff Study 151, Board of Governors of the Federal Reserve System, January 1987.

OECD Financial Statistics. Interest Rates 1960-1974, 1976.

Orphanides, Athanasios, Brian Reid, and David H. Small. "The Empirical Properties of a Monetary Aggregate That Adds Bond and Stock Funds to M2," this Review (November/December 1994), pp. 31-51.

Rossiter, Rosemary, and Tong Hun Lee. "Implicit Returns on Conventional Demand Deposits: An Empirical Comparison," Journal of Macroeconomics (Fall 1987), pp. 613-24.

Rotemberg, Julio J. "Commentary: Monetary Aggregates and Their Uses," Monetary Policy on the 75th Anniversary of the Federal Reserve System: Proceedings of the Second International Symposium in Economic Theory and Econometrics, Michael T. Belongia, ed., Kluwer Academic Publishers, 1991, pp. 223-31.

-----, John C. Driscoll, and James M. Poterba. "Money, Output, and Prices: Evidence from a New Monetary Aggregate," Journal of Business and Economic Statistics (January 1995), pp. 67-83.

Rush, Mark. "Comment and Further Evidence on 'Implicit Interest on Demand Deposits'," Journal of Monetary Economics (July 1980), pp. 437-51.

Serletis, Apostolos. "Monetary Asset Separability Tests," New Approaches to Monetary Economics: Proceedings of the Second International Symposium in Economic Theory and Econometrics, William A. Barnett and Kenneth J. Singleton, eds., Cambridge University Press, 1987, pp. 169-82.

Spencer, Peter. "Portfolio Disequilibrium: Implications for the Divisia Approach to Monetary Aggregation," The Manchester School of Economic and Social Studies (June 1994), pp. 125-50.

Startz, Richard. "Implicit Interest on Demand Deposits," Journal of Monetary Economics (October 1979), pp. 515-34.

Swofford, James L., and Gerald A. Whitney. "A Revealed Preference Test for Weakly Separable Utility Maximization with Incomplete Adjustment," Journal of Econometrics (January/February 1994), pp. 235-49.

----- and -----. "A Comparison of Nonparametric Tests of Weak Separability for Annual and Quarterly Data on Consumption, Leisure, and Money," Journal of Business and Economic Statistics (April 1988), pp. 241-46.

----- and ----- "Nonparametric Tests of Utility Maximization and Weak Separability for Consumption, Leisure, and Money," Review of Economics and Statistics (August 1987), pp. 458-64.

Theil, Henri. Economics and Information Theory. Amsterdam: North Holland, 1967.

Thornton, Daniel L., and Piyu Yue. "An Extended Series of Divisia Monetary Aggregates," this Review (November/December 1992), pp. 35-52.

Triplett, Jack E. "Economic Theory and BEA's Alternative Quantity and Price Indexes," Survey of Current Business (April 1992), pp. 32-48.

Whitesell, William C., and Seen Collins. "A Minor Redefinition of M2," Finance and Economics Discussion Series, no. 96-7, Board of Governors of the Federal Reserve System, February 1996.

Table 7

Own Rate Adjustments

A. To convert an annualized 1-month holding period yield on a bank interest (360-day) basis to an annualized 1-month holding period yield on a bond interest (365-day) basis.

Adjustment Formula

[r.sup.adj] = r x (365 / 360)

Own Rates Adjusted

Eurodollar deposits: 1-month, 3-month, 6-month Certificates of deposits: 1-month, 3-month, 6-month

B. To convert an annual effective yield on a bond interest basis to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [[(1 + (r/100) / 365).sup.30] - 1] x (365 / 30) x 100

Own Rates Adjusted

NOW accounts: thrift institutions, commercial banks Super NOW accounts: thrift institutions, commercial banks

Small-denomination time deposits at commercial banks and thrift institutions:

7-day to 91-day 92-day to 182-day 183-day to 1-year 1-year to 2.5-year 2.5-year and longer

MMDAs at commercial banks and thrift institutions Savings deposits at commercial banks and thrift institutions

C. To convert an annual effective yield on a bank interest basis to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [[(1 + (r/100) / 360).sup.30] - 1] x (365 / 30) x 100

Own Rates Adjusted

Overnight repurchase agreements Overnight Eurodollars Overnight federal funds

D. To convert an n-month bank discount basis rate to an annualized 1-month holding period yield on a bond interest basis:

Adjustment Formula

[r.sup.adj] = [365(r/100) / 360 - 30n(r/100)] x 100

Own Rates Adjusted

Secondary market Treasury bill rate: 1-month, 3-month, 6-month Commercial paper rate: 1-month, 3-month, 6-month Bankers acceptance rate: 3-month, 6-month

Table 8

Yield Curve Adjustments for Own Rates on Five Groups of Monetary Assets (by Treasury security used for adjustment)

A. Three-Month Secondary-Market Treasury Bill Rate

Eurodollar deposits, 3-month maturity

Commercial paper, 3-month maturity

Bankers acceptances, 3-month maturity

Negotiable certificates of deposit, 3-month maturity

Small-denomination deposits at commercial banks and thrift institutions, 7-day to 91-day maturity

Small-denomination time deposits at commercial and thrift institutions, variable ceiling rates on 91-day maturity

B. Six-Month Secondary-Market Treasury Bill Rate

Eurodollar deposits, 6-month maturity

Commercial paper, 6-month maturity

Bankers acceptances, 6-month maturity

Certificates of deposit, secondary-market, 6-month maturity

Small-denomination time deposits at commercial banks and thrift institution, 92- to 182-day maturity

Money market time deposits at commercial banks and thrift institutions, variable ceiling rates on 6-month maturity

C. One-Year Constant-Maturity Treasury Security Rate

Time deposits at commercial banks, 1-year maturity

Small-denomination time deposits at commercial banks and thrift institutions, 183-day to 1-year maturity

Small-denomination time deposits at banks and that institution, fixed ceiling rate on 1-year maturity

All Savers certificate, variable ceiling rate on 12-month maturity

D. Two-Year Constant-Maturity Treasury Security Rate

Small-denomination time deposits at commercial banks and thrift institutions, 1- to 2.5-year maturity

E. Three-Year Constant-Maturity Treasury Security Rate

Small-denomination time deposit at commercial banks and thrift institution, 2.5-year and longer maturity

NOTE: All rates are adjusted to an annualized one-month yield on a bond interest (365-day, coupon equivalent) basis.

Richard G. Anderson is an assistant vice president at the Federal Reserve Bank of St. Louis. Barry E. Jones and Travis D. Nesmith are Ph.D. candidates at Washington University in St. Louis and visiting scholars at the Federal Reserve Bank of St. Louis. Mary C. Lohmann, Kelly M. Morris, and Cindy A. Gleit provided research assistance.

Printer friendly Cite/link Email Feedback | |

Author: | Anderson, Richard G.; Jones, Barry E.; Nesmith, Travis D. |
---|---|

Publication: | Federal Reserve Bank of St. Louis Review |

Date: | Jan 1, 1997 |

Words: | 12295 |

Previous Article: | Monetary aggregation theory and statistical index numbers. |

Next Article: | The business cycle and chain-weighted GDP: has our perspective changed? |

Topics: |