This section discusses the techniques employed to measure global liquidity. We employ five different methods, namely SimpleSum, GDPweighted growth rates, PCA, CE and Divisia to measure global liquidity. Each method of aggregation is elaborated below.
The simplesum method
This procedure involves the conversion of seasonally adjusted broad money of all countries into a common currency, which can be obtained by dividing the broad money by the respective country’s exchange rate, and then summing the converted series up. It can be described as:
$$ {}_{SUM_{kt}={\sum}_{i=1}^N\frac{bm_{it}}{e_{it}}\kern0.5em ;\kern0.5em i\kern0.5em \in \kern0.5em k}\kern0.5em $$
where, SUM_{kt} is the regional aggregate of region k in time t, bm_{it} is the broad money of country i in time t and e_{it} is the exchange rate of country i in time t. Exchange rate implies the value of a US dollar in terms of domestic currency.
Furthermore, the range of N varies from region to region. Its range for East Asia and the Pacific is six (N = 1, 2, …., 6), ten for Europe and Central Asia (N = 1, 2, …., 10), two for Latin America and the Caribbean (N = 1, 2), one for the Middle East and North Africa (N = 1), two again for North America (N = 1, 2), and 21 at a global level (N = 1, 2, …., 21). Hereafter, its range will remain the same in all analyses.
GDPweighted growth rates method
In this section, we follow Belke and Keil (2016), Giese and Tuxen (2007) and Beyer et al. (2001) to construct monetary aggregates at regional and global levels. This procedure of aggregation involves the conversion of nominal GDP of all countries into a common currency using purchasing power parity (PPP) exchange rates. Then each country’s GDP share in the total group GDP is calculated. The GDP share of the country is used as a weight for that country. Hence country specific weight of country i in time t is:
$$ {w}_{it}=\frac{\raisebox{1ex}{${GDP}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}^{PPP}$}\right.}{\sum \limits_{i=1}^N\left(\raisebox{1ex}{${GDP}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}^{PPP}$}\right.\right)} $$
where, w_{it} is the weight, GDP_{it} is the nominal GDP and \( {e}_{it}^{PPP} \) is the purchasing power parity exchange rate of country i in time t.
The growth rates of monetary aggregates constructed across the countries can be obtained by allotting the weights calculated above to the growth rate of broad money (in domestic currency) of the respective country.
$$ {G}_t=\sum \limits_{i=1}^N{w}_{it}{g}_{it} $$
where, g_{it} is the growth rate of broad money of country i in time t and G_{t} is the aggregate growth rate in time t. Some studies use year specific weights at this stage of aggregation because the data on GDP is generally available in annual frequency (Belke and Keil 2016; Belke et al. 2014; Baks and Kramer 1999). We also use year specific weights at this step of aggregation.
The aggregate monetary index across the countries M^{GDP} can be constructed by using an initial level 100 and multiplying it by aggregate weights computed above.
$$ {M}^{GDP}=\prod \limits_{t=2}^T\left(1+{G}_t\right).100 $$
PCAbased aggregation
The prime objective of PCA is to explain the variance of observed data by utilizing a few linear combinations of original data (Joint Research CentreEuropean Commission, 2008). The PCAbased aggregation procedure can be described as follows: Suppose we construct a monetary aggregate across the countries utilizing the data on broad money for N countries. Here, we use broad money of each country converted into a common currency based on the respective country’s exchange rate. A small number of variables (principal components) can capture a large proportion of the variation of the original N variables. Further, the P number of principal components can retain a high amount of the variability of the original variables even when P < N. However, the maximum number of principal components can be N.
$$ {\displaystyle \begin{array}{l}{Z}_1={a}_{11}{M}_1+{a}_{22}{M}_2+\dots +{a}_{1N}{M}_N\\ {}{Z}_2={a}_{21}{M}_1+{a}_{22}{M}_2+\dots +{a}_{2N}{M}_N\\ {}.\\ {}.\\ {}.\\ {}{Z}_N={a}_{N1}{M}_1+{a}_{N2}{M}_2+\dots +{a}_{NN}{M}_{\begin{array}{c}N\\ {}.\end{array}}\end{array}} $$
where, M_{i} is the broad money of country i, Z_{i} is the i^{th} principal component and a_{ij} is a weight assigned to the broad money of country j in principal component i. a_{ij} is also termed as component or factor loading and is chosen in such a way that the principal components satisfy the following conditions.

a.
The principal components are uncorrelated (orthogonal).

b.
The first principal component explains the maximum proportion of the variance of variables. The second principal component explains the maximum of the remaining variance and so on. All the remaining variances are accounted for by the last component. Further,
\( {a}_{i1}^2+{a}_{i2}^2+\dots +{a}_{iN}^2=1 \) and i = 1, 2, … , N.
PCA involves tracing the eigenvalues which requires covariance matrix. So, the sample covariance matrix CM can be expressed as:
$$ CM=\left[\begin{array}{ccc}{cm}_{11}& \cdots & {cm}_{1N}\\ {}\vdots & \ddots & \vdots \\ {}{cm}_{N1}& \cdots & {cm}_{NN}\end{array}\right] $$
where, cm_{ii} is the variance of the monetary aggregate (broad money) of country i and cm_{ij} is the covariance of monetary aggregates of country i and j when i ≠ j. The eigenvalues of the matrix CM show variances of the principal components and can be obtained by solving the characteristic equation. The characteristic equation can be obtained from:
$$ \left CM\lambda I\right=0 $$
where, I is the identity matrix of the same order as that of CM and λ is the vector of eigenvalues.^{Footnote 12}
Currency equivalent (CE) method
The roots of this method of monetary aggregation can be traced back to the rigorous work of Hutt and Keynes (1963), and Rotemberg et al. (1995). In this method, we first construct the currency equivalent of the broad money of each country and then derive their weighted sum after converting them into a common currency. The aggregation within countries can be sketched as:
$$ {cebm}_{it}=\sum \limits_{j=1}^J\left(\frac{R_{it}{r}_{ijt}}{R_{it}}\right){m}_{ijt} $$
For aggregation across the countries, Chung et al. (2014) use the simple sum method. But, at this stage of aggregation we follow Barnett (2007) where he suggests the aggregation procedure of Divisia index across the countries. We follow him so that the aggregation procedure may be based on theory even at the stage of aggregation across countries. Through the heterogenouscountries approach this procedure can be described as:
$$ {CE}_{kt}=\sum \limits_{i=1}^N{W}_{it}\left(\frac{cebm_{it}}{e_{it}}\right)\kern0.5em ;i\in k $$
$$ \mathrm{where},{W}_{it}=\frac{\raisebox{1ex}{${ce}_{it}^{\ast }{\Pi}_{it}{p}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.}{\sum \limits_{i=1}^N\left[\raisebox{1ex}{${ce}_{it}^{\ast }{\Pi}_{it}{p}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right]}=\frac{\raisebox{1ex}{${ce}_{it}{\Pi}_{it}{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.}{\sum \limits_{i=1}^N\left[\raisebox{1ex}{${ce}_{it}{\Pi}_{it}{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right]} $$
and, cebm_{it} is the currency equivalent measure of the broad money of country i in time t, R_{it} is the benchmark rate of return of country i in time t, r_{ijt} is the rate of return of component j in country i at time t, m_{ijt} is the value of component j of the broad money of country i in time t, CE_{kt} is the regional currency equivalent aggregate of region k in time t, e_{it} is currency exchange rate of country i in time t, H_{it} is the population of country i in time t and Π_{it} is the user cost price aggregate of country i in time t.
The Fisher’s factor reversal test claims the existence of a user cost aggregate price dual to the quantity aggregate in a way that their product is equal to the total expenditures on all components. Hence;
$$ {\Pi}_{it}{M}_{it}=\sum \limits_{j=1}^J{\pi}_{ijt}{m}_{ijt} $$
$$ \mathrm{So},{\Pi}_{it}=\frac{\sum \limits_{j=1}^J{\pi}_{ijt}{m}_{ijt}}{M_{it}} $$
where, Π_{it} is the user cost aggregate price for country i at time t and M_{it} is the quantity aggregate (broad money) of country i at time t. We use this method to calculate country specific user cost aggregate price.
We use three to four components of broad money for each country as stated in section 3. So, J ranges from one to four (J = 1, 2, … . . , 4) and the currency in circulation or M1 bears no interest. Deposit rate is imputed to transferable deposits included in broad money. The treasury bill rate (hereafter TBR) or money market rate (if TBR is not available) is assigned to other deposits or the constituents of M2 but not of M1. Shortterm or medium term (if shortterm is not available) government bond yield is allocated to deposits other than securities included in broad money or the assets included in M3 but not in M1. However, for the US, we assign TBR to the components incorporated in M1 other than currency in circulation, and shortterm bond yield to the components included in M2 but not in M1.
Benchmark rate of return
The benchmark rate of return (R_{t}), as defined by Barnett (1987), is the return obtained on an investment asset, purely held for accumulating wealth, and which does not perform any other service such as liquidity. Researchers use different proxies for it. Usually, the investigators follow the envelope approach: the highest rate of interest is used as a proxy (Alkhareif and Barnett 2012; Serletis and Molik 2000). Some other researchers construct its proxy by adding liquidity premia to the selected rate of interest (Stracca 2004).^{Footnote 13} This study also follows the envelope approach in most of the cases but only in a very few cases we have constructed a benchmark rate of return by adding liquidity premia.^{Footnote 14} In most of the sampled countries, the highest rate of interest is the lending rate, so, we use that as the benchmark rate of interest.
Divisia index
The Divisia index meets the standards of the class of superlative index numbers as defined by Diewert (1976). Its footing primarily rests on the seminal works of Diewert (1976, 1978) and Barnett (1978, 1980). However, Barnett (1980) succeeded in constructing Divisia monetary aggregates consistent with the microeconomic theory. Therefore, the Divisia monetary index is attributed to Barnett (1980). It entertains three functions of money (medium of exchange, store of value and unit of account) and dismisses investment motives, hence measuring other monetary services related to liquidity (Hancock 2005). This method involves the construction of Divisia indices within countries and then aggregation of the indices across the countries.
Aggregation within countries
Let m_{ijt} be the per capita value of asset type j (component of broad money) in country i at time t and J be the total number of asset types in country i. Also, let the rate of return of asset j in country i at time t be r_{ijt}, and the true cost of living in country i at time t be \( {p}_{it}^{\ast } \). Then, the discretetime approximation to continuoustime Divisia monetary index can then be expressed as:
$$ {M}_{it}=\prod \limits_{j=1}^J{\left(\frac{m_{ij t}}{m_{ij,t1}}\right)}^{w_{ij t}}.{M}_{i,t1} $$
$$ \mathrm{where},{w}_{ij t}=\frac{1}{2}\left({s}_{ij t}+{s}_{ij,t1}\right) $$
$$ \mathrm{and},{s}_{ijt}=\frac{\left({R}_{it}{r}_{ijt}\right){m}_{ijt}}{\sum \limits_{j=1}^J\left({R}_{it}{r}_{ijt}\right){m}_{ijt}}=\frac{\pi_{ijt}{m}_{ijt}}{\sum \limits_{j=1}^J{\pi}_{ijt}{m}_{ijt}} $$
where, M_{it} is the Divisia monetary index of the broad money of country i in time t, w_{ijt} is the weight allotted to component j of the broad money of country i in time t, s_{ijt} is the expenditure share of component j of the broad money of country i in time t, R_{it} is the benchmark rate of return of country i in time t and π_{ijt} is the user cost of component j of broad money of country i in time t. The user cost is the return given up due to holding a monetary asset instead of holding an asset with higher return (Barnett 1978). In other words, user cost is the opportunity cost of asset and a representation of its price. It can be calculated as:
$$ {\pi}_{ijt}=\frac{\left({R}_{it}{r}_{ijt}\right)}{1+{R}_{it}} $$
The logarithmic transformation of the Divisia index can be expressed as:
$$ {logM}_{it}\mathit{\log}{M}_{i,t1}=\sum \limits_{j=1}^J{w}_{ij t}\left(\mathit{\log}{m}_{ij t}\log {m}_{ij,t1}\right) $$
Aggregation across the countries
We follow the heterogenous countries approach of Barnett (2007) to construct Divisia index across the sampled countries. Let N be the total number of countries in the group. The population of country i at time t is H_{it}. The discretetime approximation to continuoustime Divisia monetary index, aggregated across the countries, can be expressed as:
\( {DIV}_{kt}=\prod \limits_{i=1}^N{\left(\raisebox{1ex}{${h}_{it}{M}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right)}^{W_{it}^{\ast }}.{DIV}_{k,t1} \) and i ∈ k
$$ \mathrm{where},{h}_{it}=\frac{H_{it}}{\sum \limits_{i=1}^N{H}_{it}} $$
$$ \mathrm{and},{W}_{it}=\frac{\raisebox{1ex}{${M}_{it}^{\ast }{\Pi}_{it}^{\ast }{p}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.}{\sum \limits_{i=1}^N\left[\raisebox{1ex}{${M}_{it}^{\ast }{\Pi}_{it}^{\ast }{p}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right]}=\frac{\raisebox{1ex}{${M}_{it}{\Pi}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.}{\sum \limits_{i=1}^N\left[\raisebox{1ex}{${M}_{it}{\Pi}_{it}^{\ast }{H}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right]} $$
$$ {W}_{it}^{\ast }=\frac{1}{2}\left({W}_{it}+{W}_{i,t1}\right) $$
From the above expression it is apparent that 0 ≤ W_{i} ≤ 1 for all i, and \( \sum \limits_{i=1}^N{W}_i=1 \). Thereby, we may consider {W_{1}, … … …, W_{N}} as a probability distribution in constructing Divisia means across the countries. DIV_{kt} is the Divisia monetary index for region k at time t. The remaining notations are the same as those in the previous section of CE measure.
The logarithmic transformation of Divisia indices can be expressed as:
$$ {logDIV}_{kt}\mathit{\log}{DIV}_{k,t1}=\sum \limits_{i=1}^N{W}_{it}\left[\mathit{\log}\left(\frac{h_{it}{M}_{it}}{e_{it}}\right)\mathit{\log}\left(\frac{h_{i,t1}{M}_{i,t1}}{e_{i,t1}}\right)\right] $$
Through Fisher’s factor reversal property of user cost aggregates and monetary quantity the user cost aggregate across the countries can be obtained as:
$$ {\Pi}_t=\frac{\sum \limits_{i=1}^N\left(\raisebox{1ex}{${M}_{it}{s}_{it}{\Pi}_{it}$}\!\left/ \!\raisebox{1ex}{${e}_{it}$}\right.\right)}{M_t} $$