# Divisia monetary aggregates index

In econometrics and official statistics, and particularly in banking, the Divisia monetary aggregates index is an index of money supply. It is a particular application of a Divisia index to monetary aggregates.

## Background

The monetary aggregates currently in use by the Federal Reserve (and most other central banks around the world) are simple-sum indexes, in which all monetary components are assigned a unitary weight, as follows

$M_{t}=\sum_{j=1}^{n}x_{jt},$

where xjt is one of the n monetary components of the monetary aggregate Mt. This summation index implies that all monetary components contribute equally to the money total, and it views all components as dollar for dollar perfect substitutes. It has been argued, however, that such an index cannot, in general, represent a valid structural economic variable for the services of the quantity of money.

Over the years, there have been many attempts at properly weighting monetary components within a simple-sum aggregate. Without theory, however, any weighting scheme was questionable. Since 1980, attention has been focused on the gains that can be achieved by a rigorous use of microeconomic- and aggregation-theoretic foundations in the construction of monetary aggregates. This new approach to monetary aggregation was derived and advocated by William A. Barnett (1980) and has led to the construction of monetary aggregates based on Diewert's (1976) class of superlative quantity index numbers. The new aggregates are called the Divisia aggregates or Monetary Services Indexes. Early research with those aggregates using American data was done by Salam Fayyad in his 1986 PhD dissertation.

The Divisia index (in discrete time) is defined as

$\log M_{t}^{D}-\log M_{t-1}^{D}=\sum_{j=1}^{n}s_{jt}^{*}(\log x_{jt}-\log x_{j,t-1}),$

according to which the growth rate of the aggregate is the weighted average of the growth rates of the component quantities. The original continuous time Divisia index formula for consumer goods was derived by Francois Divisia in his classic paper published in French in 1925 in the Revue d'Economie Politique. The discrete time Divisia weights are defined as the expenditure shares averaged over the two periods of the change

$s_{jt}^{*}=\frac{1}{2}(s_{jt}+s_{j,t-1}),$

for j = 1,...,n, where

$s_{jt}=\frac{\pi _{jt}x_{jt}}{\sum_{k=1}^{n}\pi _{kt}x_{kt}},$

is the expenditure share of asset j during period t, and πjt is the user cost of asset j, derived by Barnett (1978),

$\pi _{jt}=\frac{R_{t}-r_{jt}}{1+R_{t}},$

which is just the opportunity cost of holding a dollar's worth of the jth asset. In the last equation, rjt is the market yield on the jth asset, and Rt is the yield available on a 'benchmark' asset that is held only to carry wealth between different time periods.

In the literature on aggregation and index number theory, the Divisia approach to monetary aggregation, Bank of Japan, the Bank of Israel, and the International Monetary Fund.

## References

• Barnett, William A. and Apostolos Serletis. The Theory of Monetary Aggregation. Contributions to Economic Analysis 245. Amsterdam: North-Holland (2000).
• Barnett, William A., Douglas Fisher, and Apostolos Serletis. "Consumer Theory and the Demand for Money". Journal of Economic Literature 30 (1992), 2086-2119.
• Diewert, W. Erwin. "Exact and Superlative Index Numbers". Journal of Econometrics 4 (1976), 115-146.
• Divisia, Francois. "L'Indice Monétaire et la Théorie de la Monnaie," Revue D'Économie Politique 39 (1925), 842-864.
• Fayyad, Salam. "Monetary Asset Component Grouping and Aggregation: An Inquiry into the Definition of Money". PhD Dissertation. University of Texas at Austin (1986).
• International Monetary Fund. "Monetary and Financial Statistics Compilation Guide." (2008), 183-184.
• Journal of Econometrics, special issue on "Measurement with Theory," Elsevier journal, Amsterdam, vol. 161, no. 1, March (2011).
• Macroeconomic Dynamics, special issue on "Measurement with Theory," Cambridge University Press journal, Cambridge, UK, vol 13, supplement 2 (2009).

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