# Money Neutrality, Monetary Aggregates and Machine Learning
## 1 Introduction  
The main monetary policy approach since the 1980s focuses on short-term interest rates; the role of monetary aggregates is reduced in the implementation of monetary policy. For decades, monetary aggregates were important in establishing central banking policies [1], but many financial innovations have significantly increased the degree of complexity of the capital and financial markets, making money demand unstable. New and diverse assets and financial instruments were introduced, rendering traditional simple sum monetary aggregates inadequate, and, thus, less important in the conduct of monetary policy. Reference [2] has linked the decline in the policy significance of monetary aggregates to the inherent problems of the naive simple sum monetary aggregates: All monetary components are considered perfect substitutes at any given time and intertemporally. This type of aggregation has been criticized in the relevant literature since [3]. Reference [4] argues that internal inconsistency between this type of aggregation used to produce monetary aggregates and the economic theory used to produce the models within which the aggregates are used are responsible for the appearance of unstable demand and supply for money. This discussion became known as the \"Barnett Critique\" [5; 6]. Despite this critique, currently, most central banks around the world still produce and use the simple sum monetary aggregates in which all financial assets are assigned a constant and equal weight. This index is \(Mt\) in,  
\[M_{t}=\sum_{j=1}^{n}x_{j,} \tag{1}\]where \(\chi_{it}\) is one of the \(n\) components of the monetary aggregate \(M_{t}\), implying that all financial assets contribute equally to the money total and all assets are perfect substitutes. References [4; 7] developed and proposed the use of the Divisia monetary aggregates that are theoretically correct. In constructing them, based on index number theory, proper weights are applied to different monetary components, such as currency, demand deposits, savings, time deposits, etc., taking into account the microeconomic aggregation theory. The Divisia index (in discrete time) is defined as,  
\[logM_{t}^{D}-logM_{t-1}^{D}=\sum_{j=1}^{n}w_{jt}^{*}\big{(}logx_{ jt}-logx_{jt-1}\big{)}. \tag{2}\]  
According to Equation (2) the growth rate of the aggregate is the weighted average of the growth rates of the component quantities, with the Divisia weight is defined as the expenditure shares averaged over the two periods of the change, \(w_{jt}^{*}=(1/2)(w_{jt}+w_{jt-1})\) for \(j=1,\ldots,n\), where  
\[w_{jt}=\pi_{ji}x_{ji}/\sum_{k=1}^{n}\pi_{kt}x_{kt}. \tag{3}\]  
Equation (3) is the expenditure share of assets \(j\) during period \(t\), and \(\pi_{ji}\) is the user cost of asset \(j\), derived in [7],  
\[\pi_{ji}=\frac{\Big{(}R_{t}-r_{jt}\Big{)}}{(1+R_{t}}, \tag{4}\]  
which is just the opportunity cost of holding a dollar's worth of the \(j\)th asset. In Equation (4), \(r_{ji}\) is the market yield on the \(j\)th asset and \(R_{t}\) is the yield available on a 'benchmark' asset that is held only to carry wealth between multiperiods [8] (For more details regarding the Divisia approach to monetary aggregation see [9].). Many studies tried to empirically compare the simple sum and the Divisia monetary aggregates to check which have a higher contribution in monetary policy. [10] use advanced panel cointegration tests to check the existence of a long-run link between the Divisia, income and interest rates in a simple Keynesian money demand function. They use data for the United States, United Kingdom, Euro Area and Japan spanning from 1980Q1 to 1993Q3. Their empirical results show a long-run link between Divisia, income and interest rates. [11] apply wavelet analysis to compare the relationship between simple sum and Divisia monetary aggregates for the US, with real GDP and CPI inflation. The results indicate stronger comovements between CPI inflation and growth in Divisia monetary aggregates than between inflation and growth rates in simple sum aggregates.