# 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 uses Divisia index methods.

## Background

The monetary aggregates used by most central banks (notably the US Federal Reserve) are simple-sum indexes in which all monetary components are assigned the same weight:

${\displaystyle M_{t}=\sum _{j=1}^{n}x_{jt},}$

in which ${\displaystyle x_{jt}}$ is one of the ${\displaystyle n}$ monetary components of the monetary aggregate ${\displaystyle M_{t}}$. The 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 that such an index does not weight such components in a way that properly summarizes the services of the quantities of money.

There have been many attempts at weighting monetary components within a simple-sum aggregate. An index can rigorously apply microeconomic- and aggregation-theoretic foundations in the construction of monetary aggregates. That 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. Salam Fayyad's 1986 PhD dissertation did early research with those aggregates using U.S. data.

This index is a discrete-time approximation with this definition:

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

Here, the growth rate of the aggregate is the weighted average of the growth rates of the component quantities. The discrete time Divisia weights are defined as the expenditure shares averaged over the two periods of the change

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

for ${\displaystyle j=1,...,n}$, where

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

is the expenditure share of asset ${\displaystyle j}$ during period ${\displaystyle t}$, and ${\displaystyle \pi _{jt}}$ is the user cost of asset ${\displaystyle j}$, derived by Barnett (1978),

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

Which is the opportunity cost of holding a dollar's worth of the ${\displaystyle j}$th asset. In the last equation, ${\displaystyle r_{jt}}$ is the market yield on the ${\displaystyle j}$th asset, and ${\displaystyle R_{t}}$ is the yield available on a benchmark asset, held only to carry wealth between different time periods.

In the literature on aggregation and index number theory, the Divisia approach to monetary aggregation, ${\displaystyle M_{t}^{D}}$, is widely viewed as a viable and theoretically appropriate alternative to the simple-sum approach. See, for example, International Monetary Fund (2008), Macroeconomic Dynamics (2009), and Journal of Econometrics (2011). The simple-sum approach, ${\displaystyle M_{t}}$, which is still in use by some central banks, adds up imperfect substitutes, such as currency and non-negotiable certificates of deposit, without weights reflecting differences in their contributions to the economy's liquidity. A primary source of theory, applications, and data from the aggregation-theoretic approach to monetary aggregation is the Center for Financial Stability in New York City. More details regarding the Divisia approach to monetary aggregation are provided by Barnett, Fisher, and Serletis (1992), Barnett and Serletis (2000), and Serletis (2007). Divisia Monetary Aggregates are available for the United Kingdom by the Bank of England, for the United States by the Federal Reserve Bank of St. Louis, and for Poland by the National Bank of Poland. Divisia monetary aggregates are maintained for internal use by the European Central Bank, the Bank of Japan, the Bank of Israel, and the International Monetary Fund.

Recent research by Chen and Valcarcel (2021) and Keating et al. (2019) has investigated using Divisia monetary aggregates as alternatives to the federal funds rate for analyzing monetary policy shocks. Chen and Valcarcel examined effects of policy shocks over 1988-2020, finding persistent price puzzles from shocks based on the Wu and Xia (2016) shadow federal funds rate, even when including commonly suggested fixes. However, replacing the shadow rate with Divisia monetary aggregates consistently resolved these puzzles. Their analysis suggests substantial expansionary monetary policy effects, especially in money markets post-2007-09 crisis. Additionally, Keating et al. develop a model examining monetary policy shock effects, comparing the federal funds rate to Divisia M4 from 1960-2017. They find Divisia M4 outperforms the federal funds rate, providing superior counterfactuals during crises and normal periods, while the federal funds rate frequently yielded puzzles. Their model with Divisia M4 better captures time-variation in shocks' effects. Together, these studies provide evidence for the value of Divisia monetary aggregates over the traditional federal funds rate as indicators of monetary policy.

## References

• Barnett, William A. "The User Cost of Money". Economics Letters (1978), 145-149.
• Barnett, William A. "Economic Monetary Aggregates: An Application of Aggregation and Index Number Theory," Journal of Econometrics 14 (1980), 11-48.
• 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". (Salam Fayyad's Ph.D. thesis, University of Texas, 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).
• Serletis, Apostolos. Archived 1 October 2006 at the Wayback Machine The Demand for Money: Theoretical and Empirical Approaches. Springer (2007).
• Chen, Zhengyang and Valcarcel, Victor J. "Monetary Transmission in Money Markets: The Not-So-Elusive Missing Piece of the Puzzle," Journal of Economic Dynamics and Control (2021).
• Keating, John W., Kelly, Logan J., Smith, A. Lee, and Valcarcel, Victor J. "A Model of Monetary Policy Shocks for Financial Crises and Normal Conditions," Journal of Money, Credit and Banking 51 (2019), 227-259.