2.1 Simple-sum aggregates

The monetary aggregates currently in use by most central banks around the world are based on the simple-sum index. This simple-sum index is $M_{t}$ in

(1) \begin{equation} M_{t}=\sum _{i=1}^{I}x_{it} \end{equation}

where $x_{it}$ is one of the $I$ components of the aggregate $M_{t}$ . As can be seen in equation (1), simple-sum aggregation assigns a constant and equal (unitary) weight to each monetary asset and implies that all assets contribute equally to the money total. It completely ignores the complex products and structures of modern economies, and as Barnett (Reference Barnett1980) argued it is inconsistent with economic aggregation theory, because it assumes that the monetary assets are dollar for dollar perfect substitutes with the same user cost.

2.2 Original Divisia aggregates

Barnett (Reference Barnett1978) derived the formula for the real user cost of a monetary asset as

\begin{equation*} \pi _{it}^{a}=\frac {R_{t}-r_{it}^{a}}{1+{R_{t}}} \end{equation*}

where $R_{t}$ is the rate of return on the benchmark asset and $r_{it}^{a}$ is the own rate of return on monetary asset $i$ (during period $t$ ). Barnett (Reference Barnett1980) then developed the Divisia monetary aggregates which permit different user costs of the component assets and hence do not assume perfect substitution between assets. The Divisia monetary aggregate (in discrete time) computes the growth rate of the aggregate, $d\ln M_{t}$ , as the share-weighted average of its component growth rates as follows

(2) \begin{equation} d\ln M_{t}=\sum _{i=1}^{I}s_{it}d\ln m_{it}^{a} \end{equation}

where $m_{it}^{a}$ denotes the real balances of monetary asset $i$ and

\begin{equation*} s_{it}=\frac {\pi _{it}^{a}m_{it}^{a}}{\sum _{i=1}^{I}\pi _{it}^{a}m_{it}^{a}} \end{equation*}

is the expenditure share on monetary asset $i$ .

2.3 Credit card-augmented Divisia aggregates

The original Divisia monetary aggregates, based on equation (2), do not include the services provided by credit cards. It was only recently that Barnett et al. (Reference Barnett, Chauvet, Leiva-Leon and Su2024) derived the credit card-augmented Divisia monetary aggregates which also include the transaction services of credit cards. Under the assumption of risk neutrality, Barnett et al. (Reference Barnett, Chauvet, Leiva-Leon and Su2024) derive the user cost of credit card transaction services, $\pi _{lt}^{c}$ , as

\begin{equation*} \pi _{lt}^{c}=\frac {e_{lt}-R_{t}}{1+R_{t}} \end{equation*}

where $e_{lt}$ is the expected interest on the credit card transaction $l$ and $R_{t}$ is as before the rate of return on the benchmark asset. The credit card-augmented Divisia monetary aggregate is then given by

(3) \begin{equation} d\ln M_{t}=\sum _{i=1}^{I}s_{it}d\ln m_{it}^{a}+\sum _{l=1}^{L}s_{lt}d\ln m_{lt}^{c} \end{equation}

where

\begin{equation*} s_{it}=\frac {\pi _{it}^{a}m_{it}^{a}}{\sum _{i=1}^{I}\pi _{it}^{a}m_{it}^{a}+\sum _{l=1}^{L}\pi _{lt}^{c}m_{lt}^{c}} \end{equation*}

is the user-cost-evaluated expenditure share of monetary asset $i$ , $ i=1,\ldots ,I$ , and

\begin{equation*} s_{lt}=\frac {\pi _{lt}^{c}m_{lt}^{c}}{\sum _{i=1}^{I}\pi _{it}^{a}m_{it}^{a}+\sum _{l=1}^{L}\pi _{lt}^{c}m_{lt}^{c}} \end{equation*}

is the user-cost-evaluated expenditure share of credit card transaction $l$ , $l=1,\ldots ,L$ .

2.4 Credit card-augmented Divisia inside aggregates

The credit card-augmented Divisia monetary aggregates are demand-side measures of the services from monetary assets and credit cards. They treat monetary assets and credit card services as goods demanded by households and firms, and aggregate them using Divisia weights based on user costs actually paid by holders. Barnett and Su (Reference Barnett and Su2018) also constructed credit card-augmented Divisia inside monetary aggregates, which measure the services of inside money as produced and supplied by financial firms. These aggregates are constructed as in equation (3), with quantities demanded, $m_{it}^{a}$ , replaced by quantities supplied and with paid user costs, $\pi _{it}^{a}$ , replaced by user costs received by financial intermediaries, calculated as follows

\begin{equation*} \pi _{it}^{a}=\frac {(1-k_{i})R_{t}-r_{it}}{1+R_{t}} \end{equation*}

where $k_{i}$ is the required reserve ratio on monetary asset $i$ .

It is to be noted that the inside aggregates exclude instruments that are not privately produced inside money (such as, for example, currency and Treasury bills) and include the credit card services supplied by private financial firms, so they focus on private inside money plus credit cards. Also, in constructing the credit-augmented Divisia inside aggregates, the required reserve ratios, $k_{i}$ , for all component assets $i$ are set equal to zero starting in March 2020, when the Federal Reserve eliminated reserve requirements for all classes of bank deposits.

In what follows, we posit our empirical work on the two simple-sum monetary aggregates published by the Fed, the original Divisia monetary aggregates, and the most recently developed credit card-augmented Divisia monetary aggregates and credit card-augmented Divisia inside monetary aggregates.

4.1 Granger causality tests

Hall and Noble (Reference Hall and Noble1987), Brocato and Smith (Reference Brocato and Smith1989), Mehra (Reference Mehra1989), Fisher and Serletis (Reference Fisher and Serletis1989), and Thornton (Reference Thornton1995) all use the Granger-causality method to test Friedman’s hypothesis that money growth volatility is causal factor in changes in velocity. In doing so, they assume that the relevant information is contained in the present and past values of these variables and use the following bivariate autoregressive representation

(4) \begin{equation} v_{t}=\alpha _{0}+\sum _{j=1}^{r}\mathbf{\alpha }_{j}v_{t-j}+\sum _{j=1}^{s} \mathbf{\beta }_{j}\text{VOL}_{t-j}+\varepsilon _{t} \end{equation}

where $\Delta$ is the difference operator, $v_{t}=\Delta \log V_{t}$ , $ V_{t}$ is the level of money velocity, VOL $_{t}$ is the level of money growth volatility, and $\varepsilon _{t}$ is a white noise disturbance. Here, VOL $_{t}$ is a 12-month moving standard deviation of money growth. Given the monthly frequency of the data, the 12-month window captures annual cyclical variation while smoothing short-term fluctuations, a choice commonly adopted in the literature to balance responsiveness and stability in volatility measurement.

In this framework, the Granger procedure requires that we first estimate equation (4) by ordinary least squares to obtain the unrestricted sum of squared residuals, $SSR_{u}$ . Then, by running another regression under the restriction that all $\beta _{j}$ ’s are zero, the restricted sum of squared residuals, $SSR_{r}$ , is obtained. If $\varepsilon _{t}$ is white noise, then the statistic

\begin{equation*} \frac {\left ( SSR_{r}-SSR_{u}\right ) /s}{SSR_{u}/\left ( T-r-s-1\right ) } \end{equation*}

has an asymptotic $F$ -distribution with the numerator having degrees of freedom of $s$ and the denominator of $T-r-s-1$ where $T$ is the number of observations and 1 is subtracted out to account for the constant term in equation (4). If the null hypothesis of $\widehat {\beta } _{j}=0$ for $j=1,\cdot \cdot \cdot ,s$ cannot be rejected, then the conclusion is that the data do not show causality. If the null hypothesis is rejected, then the conclusion is that the data do show causality.

For monetary variability, the literature has produced mixed results. In particular, Hall and Noble (Reference Hall and Noble1987) using quarterly data (from 1963:1 to 1984:2), define monetary variability as an eight-quarter standard deviation of M1 growth, and find a causal relation from money growth variability to velocity. Fisher and Serletis (Reference Fisher and Serletis1989) test the same hypothesis that Hall and Noble (Reference Hall and Noble1987) did, using monthly U.S. data (covering the period 1970 through mid-1985), and search for the relationship over nine simple-sum and Divisia measures of the money stock, in an attempt to deal with anomalies that arise because of different definitions of money. Although they find some slight differences across the different monetary aggregates of equal breadth, they claim that the influence of money growth variability shows up on velocity in a (Granger-) causal sense.

However, Brocato and Smith (Reference Brocato and Smith1989), using monthly U.S. data from 1962:2 through 1985:9, reestimate the Hall and Noble (Reference Hall and Noble1987) equation over the full period, as well as over pre- and post-October 1979 periods. They find evidence to support the Friedman hypothesis in the pre-October 1979 period but find no evidence that the variability of money growth influenced velocity thereafter. They conclude that the change in Federal Reserve operating procedures in October 1979 contributed to a break down in the money growth/velocity growth relationship and that this finding runs counter to the Friedman hypothesis which would suggest stronger causality results after 1979, given the increase in money growth variability that took place. The Brocato and Smith (Reference Brocato and Smith1989) finding has also been confirmed by Mehra (Reference Mehra1989) who shows that the Granger-causality result reported by Hall and Noble (Reference Hall and Noble1987) is not robust to some changes of specification and the sample period. Moreover, Thornton (Reference Thornton1995) extends this work by testing Friedman’s hypothesis using quarterly data (over the period from 1961 to 1990) for nine countries—Australia, Austria, Canada, France, Germany, Italy, Japan, Switzerland, and the United Kingdom. He finds no evidence of the hypothesis that money supply volatility causes income velocity to change and concludes that “the Friedman hypothesis would appear to have little general applicability.”

We report our results, using monthly data over the 2006:m7 to 2025:m6 period, in the second and third columns of Table 4. For each monetary aggregate, we select the model using the SIC to obtain a parsimonious lag structure. As shown in the second column of Table 4, the selected lag structure for Sum M1 and Sum M2 is (1,1) and (1,2), respectively. For the original Divisia aggregates, the lag structure is (1,1) for Divisia M1 and Divisia M4 and (2,1) for Divisia M2 and Divisia M3. For all the credit card-augmented Divisia aggregates and credit card-augmented Divisia inside monetary aggregates the chosen lag structure is (2,1), except for Divisia M4A for which the SIC suggests a lag structure of (1,2). After estimating each model with its selected lag structure, we conduct the Granger-causality tests based on equation (4). The $p$ -values indicate that monetary variability affects the consumption velocity of money only with the Divisia M4 and Divisia M4A aggregates with $p$ -values of $0.009$ and $0.001$ , respectively. Thus, only two of the fourteen monetary aggregates show a statistically significant link between the variability of money growth and the consumption velocity.

Table 4.

Granger causality tests from the variability of money growth to the velocity of money

Table 4 Long description

A table titled ’Granger causality tests from the variability of money growth to the velocity of money’ presents data on the relationship between monetary aggregates and their consumption and income velocities. The table has 14 rows and 6 columns, including headers. The columns are labeled ’Monetary aggregate’, ’Consumption velocity Lag’, ’Consumption velocity F-statistic’, ’Income velocity Lag’, and ’Income velocity F-statistic’. Each row corresponds to a different monetary aggregate, such as Sum M1, Sum M2, Divisia M1, etc. The table reports lag structures and F-statistic values with corresponding p-values in parentheses. For example, Sum M1 has a lag structure of (1,1) for both consumption and income velocity, with F-statistic values of 0.607 and 0.356, respectively. Divisia M4 shows significant results with an F-statistic of 6.918 for consumption velocity and 4.056 for income velocity. The table notes that the sample period is from 2006:m7 to 2025:m6, and lag structures are selected based on the minimum SIC value by evaluating 144 regressions.

Notes: The sample period is 2006:m7 to 2025:m6.

Lag structures are selected based on the minimum SIC value by evaluating 144 regressions.

Numbers in parentheses are p-values.

Next, we compare the above findings with the results reported in the third and fourth columns of Table 4 which use the income velocity of money. In most cases, the selected lag structures differ from those based on the consumption velocity. For the simple-sum aggregates, the chosen lag structure is (1,1) for Sum M1 and (1,3) for Sum M2. For the original Divisia aggregates, the optimal lag structure is (1,3) for Divisia M1, Divisia M2, and Divisia M4, while the model for Divisia M3 is (1,1). In the case of the credit card-augmented Divisia aggregates, the optimal lag structure is (1,1) for Divisia M1A, Divisia M2A, and Divisia M4A, which differ from their corresponding lag structures based on the consumption velocity. The lag structure for Divisia M3A is (2,1), which coincides with the specification based on the consumption velocity. For the credit card-augmented Divisia inside aggregates, the selected lag structure is (1,1) for M1AI, (3,1) for M2AI, (2,1) for M3AI, and (2,2) for M4AI, again differing from most specifications based on the consumption velocity, except for the Divisia M3AI aggregate. The Granger-causality test $p$ -values in fifth column of Table 4 show that monetary variability based on Sum M2, all four original Divisia aggregates, and Divisia M4A Granger causes the income velocity at the 5 percent level of significance. Hence, six of the fourteen monetary aggregates exhibit a statistically significant relationship between money growth variability and the income velocity.

Overall, we conclude that the Granger-causality results provide weak support for Friedman’s hypothesis across most monetary aggregates. This pattern suggests that the velocity of money remains largely stable in the presence of money growth volatility.

4.2 A structural GARCH-in-mean VAR model

The Granger causality approach relies on ad hoc non-parametric moving standard deviation to measure monetary variability. In this section we use a structural model, based on Elder and Serletis (Reference Elder and Serletis2010). In the model, a vector autoregression (VAR) captures the joint dynamics of the variables by allowing each variable to depend on its own past values as well as the past values of the other variables in the system. This approach addresses the dynamic interactions between money growth and the velocity of money. To account for time-varying uncertainty, we augment the VAR with a generalized autoregressive conditional heteroskedasticity (GARCH) structure, which models monetary uncertainty as the conditional standard deviation of the one-step-ahead forecast error given the contemporaneous information set.

This combined framework allows us to jointly analyze the evolution of monetary dynamics and the role of uncertainty. Here, the model is a bivariate structural VAR in the monetary growth rate and the growth rate of the velocity of money, as follows

(5) \begin{align} \boldsymbol{B}\boldsymbol{z}_{t}=\boldsymbol{C}+\sum _{j=1}^{p}\mathbf{\Gamma }_{j} \boldsymbol{z}_{t-j}+\boldsymbol{\Lambda } \boldsymbol{H}_{t}^{1/2}+\boldsymbol{\epsilon }_{t} \\[-35pt] \nonumber \end{align}

\begin{align*} \boldsymbol{\epsilon }_{t}\left \vert \Omega _{t-1}\right . \sim \text{iid }N\left ( \boldsymbol{0}, \boldsymbol{H}_{t}\right ) ,\quad \boldsymbol{H}_{t}= \begin{bmatrix} h_{v_{t},t} &\quad 0 \\ 0 &\quad h_{\mu _{t},t} \end{bmatrix} \end{align*}

where the vector $\boldsymbol{z}$ $_{t}$ includes the growth rate of the velocity of money, $v_{t}=\Delta \log V_{t}$ , and the money growth rate, $\mu _{t}=\Delta \ln M_{t}$ , $\Omega _{t-1}$ is the information set at $t-1$ , and $\boldsymbol{\Lambda }$ captures the effect of the conditional standard deviations on the mean. The lag length $p$ is chosen based on SIC from a homoskedastic VAR. The matrices $\boldsymbol{B}$ and $\boldsymbol{\Gamma }_{\boldsymbol{j}}$ are $2\times 2$ and $\boldsymbol{B}$ is lower triangular normalized on the diagonal

\begin{equation*} \boldsymbol{B}= \begin{bmatrix} 1 &\quad 0 \\ b_{21} &\quad 1 \end{bmatrix} \text{,}\quad \boldsymbol{\Lambda }= \begin{bmatrix} 0 &\quad 0 \\ \lambda &\quad 0 \end{bmatrix} \end{equation*}

The primary coefficient of interest is the GARCH-in-Mean coefficient, $ \lambda$ . This coefficient captures the effect of uncertainty in money growth on velocity. Thus, we test the null hypothesis $\widehat {\lambda }=0$ to examine the significance of the coefficient on the conditional standard deviation of the money growth rate in the second mean equation. Here, the GARCH-in-Mean specification provides a direct link between monetary uncertainty and the behavior of velocity. A statistically significant $ \lambda$ would indicate the idea that when monetary growth becomes more uncertain, economic agents may adjust their money holdings and portfolio allocations, thereby affecting the velocity of money. This mechanism is closely related to Friedman’s hypothesis, which suggests that increased monetary instability can influence the demand for money and, consequently, the dynamics of velocity.

The conditional variance matrix $\boldsymbol{H}$ $_{t}$ is parametrized using a diagonal multivariate GARCH(1,1) specification

(6) \begin{equation} \text{diag}\left ( \boldsymbol{H}_{t}\right ) = \boldsymbol{C}+\boldsymbol{F} \mathrm{diag}\left ( \boldsymbol{\epsilon }_{t-1}\boldsymbol{\epsilon }_{t-1}^{\prime }\right ) + \boldsymbol{G}\mathrm{diag} \left ( \boldsymbol{H}_{t-1}\right ) \end{equation}

where diag is the operator that extracts the diagonal from a square matrix. We impose the additional restriction that the conditional variance of $ z_{i,t}$ depends only on its own past squared errors and its own past conditional variances, so that the parameter matrices $\boldsymbol{F}$ and $\boldsymbol{G}$ are also diagonal. This assumption is not restrictive, given the focus of the paper.

The parameters of the model consisting of equations (5) and (6) are estimated by Full Information Maximum Likelihood (FIML), maximizing the log-likelihood over $T$ observations

\begin{equation*} \mathcal{L}=-\left ( \frac {N}{2}\right ) \ln (2\pi )+\sum _{t=1}^{T}\frac {1}{2} \ln \left \vert \boldsymbol{B}\right \vert ^{2}-\frac {1}{2}\ln \left \vert \boldsymbol{H} \right \vert -\frac {1}{2} \boldsymbol{\epsilon }_{t}^{^{\prime }} \boldsymbol{H}_{t}^{-1} \boldsymbol{\epsilon }_{t} \end{equation*}

The presample values of the conditional variance matrix $\boldsymbol{H}$ $_{0}$ are set to their unconditional expectation and condition on the presample values $\boldsymbol{z}$ $_{0}$ , $\boldsymbol{z}$ $_{-1},\dots ,$ $\boldsymbol{z}$ $_{-p+1}$ . By imposing element-wise positivity on $\boldsymbol{C}$ , element-wise non-negativity on $\boldsymbol{F}$ and $\boldsymbol{G}$ , and requiring that, under the diagonal specification, the sum of the corresponding diagonal elements of $\boldsymbol{F}$ and $\boldsymbol{G}$ be less than one to ensure covariance stationarity. This stationarity condition is imposed during estimation to guarantee a well-defined conditional variance process, ensuring that $\boldsymbol{H}$ $_{t}$ is positive definite and that $\boldsymbol{\epsilon }$ $_{t}$ is covariance stationary. Under these conditions, and with the asymptotic variance given by the inverse of the Fisher information matrix, the FIML estimates are asymptotically normal and efficient, provided that the standard regularity conditions hold. We use the statistical package RATS 9.0 to estimate the structural GARCH-in-Mean VAR model in this paper.

Table 5.

Model specification tests for the GARCH-in-mean VAR

Table 5 Long description

A table comparing model specification tests for GARCH-in-mean VAR across different monetary aggregates. The table has 16 rows and 6 columns. The columns are labeled as follows: Monetary aggregate, Lag, SIC for VAR, SIC for GARCH-in-Mean VAR, Lag, SIC for VAR, SIC for GARCH-in-Mean VAR. The rows are labeled with different monetary aggregates: Sum M1, Sum M2, Divisia M1, Divisia M2, Divisia M3, Divisia M4, Divisia M1A, Divisia M2A, Divisia M3A, Divisia M4A, Divisia M1AI, Divisia M2AI, Divisia M3AI, Divisia M4AI. Each row provides values for the lag, SIC for VAR, and SIC for GARCH-in-Mean VAR for both consumption velocity and income velocity. Notable trends include varying lag selections and SIC values across different monetary aggregates. The table also includes notes on the sample period and lag selection criteria.

Notes: The sample period is 2006:m7 to 2025:m6.

Lags are selected based on minimum SIC value for the conventional VAR.

$^\#$ Lag is selected among twelve lags based on the minimum SIC for conventional VAR subject to stable GARCH-in-Mean VAR.

Tables 5 and 6 summarize the estimates from the GARCH-in-Mean VAR using both the consumption and income velocity of money. The tables document model specification tests based on the SIC for each monetary aggregate. We select the lag length based on the minimum SIC from the conventional VAR associated with each stable (i.e. model with convergence) GARCH-in-Mean VAR model. Thus, the optimal lag is chosen from the convergent GARCH-in-Mean VAR specifications. Using the consumption velocity, we select a lag length of 2 with the Sum M1 and Sum M2 aggregates, as reported in the second column of Table 5. Similarly, we select a lag length of 2 for all of the (original, credit card-augmented, and credit card-augmented inside) Divisia M2, Divisia M3, and Divisia M4 aggregates. A lag length of 3 is selected for the narrow (at the M1 level of aggregation) Divisia aggregates.

Table 6.

GARCH-in-mean VAR $\lambda$ coefficient estimates

Table 6 Long description

A table comparing GARCH-in-mean VAR lambda coefficient estimates for different monetary aggregates. The table has 16 rows and 3 columns. The columns are labeled ’Monetary aggregate’, ’Consumption velocity’, and ’Income velocity’. Each row lists a different monetary aggregate and its corresponding lambda coefficient estimates with p-values in parentheses. The monetary aggregates include Sum M1, Sum M2, Divisia M1, Divisia M2, Divisia M3, Divisia M4, Divisia M1A, Divisia M2A, Divisia M3A, Divisia M4A, Divisia M1AI, Divisia M2AI, Divisia M3AI, and Divisia M4AI. The table provides estimates for both consumption velocity and income velocity. Notable trends include varying lambda coefficient estimates and p-values across different monetary aggregates.

Notes: The sample period is 2006:m7 to 2025:m6.

Numbers in parentheses are p-values.

The third and fourth columns of Table 5 report the SIC values for the conventional VAR and the GARCH-in-Mean VAR, respectively. For each aggregate and selected lag, the GARCH-in-Mean VAR yields a lower SIC than the conventional VAR when using the consumption velocity, underscoring the superiority of the structural GARCH-in-Mean VAR model over the conventional VAR. After fixing the lag length, we estimate the GARCH-in-Mean VAR to identify the coefficient capturing the effect of money growth volatility on the consumption velocity, denoted by $\lambda$ . The second column of Table 6 reports the estimates of $\lambda$ and $p$ -values for the null hypothesis $H_{0} \, : \, \hat {\lambda }=0$ . The coefficients are statistically significant at the 5% level only with the Sum M2 aggregate ( $\hat {\lambda }=-1.070$ with a $ p$ -value of $0.022$ ) and the Divisia M1A aggregate ( $\hat {\lambda }=-0.029$ with a $p$ -value of $0.000$ ), with the negative signs indicating the negative effect of monetary volatility on the velocity of money.

We repeat the analysis using the income velocity of money. The corresponding results appear in columns five to seven of Table 5 and the third column of Table 6. The fifth column of Table 5 shows that the selected lag length for the Sum M1 and Sum M2 aggregates remains 2, matching the specifications based on the consumption velocity. As in the corresponding consumption velocity-based specifications, the same lag length of 2 also applies to Divisia M2, Divisia M3, Divisia M2A, Divisia M4A, and all the augmented Divisia inside aggregates; the optimal lag length for Divisia M1, Divisia M1A, Divisia M3A, and Divisia M1AI is 3, and that for Divisia M4 is 1, differing from the consumption-based specifications. Nevertheless, the SIC values in the sixth and seventh columns of Table 5 indicate that the GARCH-in-Mean VAR outperforms the conventional VAR for all aggregates.

After fixing the lag length, we estimate the model using the income velocity of money. The estimated $\lambda$ coefficients in the third column of Table 6 show that at the 5% level of significance money growth volatility has no statistically significant effect on the income velocity for any of the monetary aggregates. This result broadly aligns with the findings based on the consumption velocity of money, and we conclude that the structural model provides strong evidence against Friedman’s hypothesis that monetary variability affects the velocity of money.

4.3 Robustness

In the analysis so far we used monthly data over the period from 2006:m7 to 2025:m6 for which all monetary aggregates are available. Since the simple-sum aggregates and the original Divisia aggregates are available since 1967:m1, in this section we use a longer sample. However, because the monthly NGDP index from S&P Global Market Intelligence that we used in the previous section to construct the income velocity of money is not available over this longer period, we use quarterly data and a real GDP series from FRED to construct the income velocity of money. Thus, in doing so, we investigate the robustness of our results to the use of a longer sample using the consumption velocity of money (over the period from 1967:m1 to 2025:m6) and also to the use of quarterly data using the income velocity of money based on real GDP (over the period from 1967:q1 to 2025:q2). It is to be noted that this longer sample, and the comparison between the simple-sum aggregates and the original Divisia aggregates, attracted most of the attention so far in the literature. Figures 15–20 show the logged monetary aggregates (Sum M1, Sum M2, Divisia M1, Divisia M2, Divisia M3, and Divisia M4, respectively) together with their corresponding consumption and income velocities, in the same fashion as in Figures 1–14.

Figure 15.

The logged Sum Ml aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.

Figure 16.

The logged Sum M2 aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.

Figure 17.

The logged Divisia Ml aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.

Figure 17. Long description

The image contains two line graphs. The main graph shows the Divisia M1 aggregate and consumption velocity over time from 1967 to 2026. The horizontal axis represents the years, while the left vertical axis represents the Divisia M1 values and the right vertical axis represents the Divisia M1 consumption velocity. The red solid line represents the Divisia M1 aggregate, and the blue dashed line represents the consumption velocity. Key events such as the Global financial crisis and the COVID pandemic are highlighted with vertical blue bands. The inset graph shows the Divisia M1 aggregate and income velocity over the same period. The horizontal axis represents the years, while the left vertical axis represents the Divisia M1 values and the right vertical axis represents the Divisia M1 income velocity. The red solid line represents the Divisia M1 aggregate, and the black dashed line represents the income velocity. The graphs illustrate the trends and relationships between these economic indicators over time.

Figure 18.

The logged Divisia M2 aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.

Figure 19.

The logged Divisia M3 aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.

Figure 19. Long description

The image contains two line graphs. The first graph shows the Divisia M3 aggregate and consumption velocity over time from 1967 to 2026. The x-axis represents the years, and the y-axis on the left represents the Divisia M3 values, while the y-axis on the right represents the consumption velocity. The red solid line represents the Divisia M3 aggregate, and the blue dashed line represents the consumption velocity. Key events such as the Global financial crisis and the COVID pandemic are highlighted with vertical blue bars. The second graph, inset within the first, shows the Divisia M3 aggregate and income velocity over the same period. The x-axis represents the years, and the y-axis on the left represents the Divisia M3 values, while the y-axis on the right represents the income velocity. The red solid line represents the Divisia M3 aggregate, and the black dashed line represents the income velocity. The graphs illustrate the trends and relationships between the Divisia M3 aggregate, consumption velocity, and income velocity over time, highlighting significant economic events.

We estimate the GARCH-in-Mean VAR (in the same fashion as in the previous section) and report the results in Tables 7 and 8. With the monthly data and the consumption velocity of money, the selected lag length is 4 for both Sum M1 and Sum M2, 3 for Divisia M1, and 2 for each of Divisia M2, Divisia M3, and Divisia M4. With the quarterly data and the income velocity of money, the selected lag length is 1 for Sum M1 and 2 for all of the other aggregates. Table 7 also shows that the structural specification outperforms the conventional VAR across all monetary aggregates considered for both the monthly and quarterly data. Finally, Table 8 indicates a negative and statistically significant (at the 5% level) effect of money growth variability on the consumption velocity of money only with the Sum M2 ( $\hat { \lambda }=-0.387$ with a $p$ -value of $0.006$ ) and Divisia M3 ( $\hat {\lambda } =-0.430$ with a $p$ -value of $0.004$ ) aggregates. We find no negative and statistically significant effect with the quarterly data and the income velocity of money.

Table 7.

Model specification tests for the GARCH-in-mean VAR for the 1967–2025 sample

Table 7 Long description

A table comparing model specification tests for GARCH-in-mean VAR across different monetary aggregates. The table has two main sections: Consumption velocity and Income velocity, each with three columns: Lag, SIC for VAR, and SIC for GARCH-in-Mean VAR. The rows are labeled with different monetary aggregates: Sum M1, Sum M2, Divisia M1, Divisia M2, Divisia M3, and Divisia M4. Row 1: Sum M1, Lag 4, SIC for VAR 5,365.479, SIC for GARCH-in-Mean VAR 3,701.035, Lag 1, SIC for VAR 2,158.330, SIC for GARCH-in-Mean VAR 2,138.011. Row 2: Sum M2, Lag 4, SIC for VAR 2,207.853, SIC for GARCH-in-Mean VAR 1587.707, Lag 2, SIC for VAR 1,276.406, SIC for GARCH-in-Mean VAR 1,226.219. Row 3: Divisia M1, Lag 3, SIC for VAR 3,264.771, SIC for GARCH-in-Mean VAR 2,482.470, Lag 2, SIC for VAR 1,482.791, SIC for GARCH-in-Mean VAR 1,312.411. Row 4: Divisia M2, Lag 2, SIC for VAR 2,448.932, SIC for GARCH-in-Mean VAR 1864.232, Lag 2, SIC for VAR 1,332.817, SIC for GARCH-in-Mean VAR 1,227.410. Row 5: Divisia M3, Lag 2, SIC for VAR 2,613.001, SIC for GARCH-in-Mean VAR 1985.372, Lag 2, SIC for VAR 1,329.556, SIC for GARCH-in-Mean VAR 1,242.572. Row 6: Divisia M4, Lag 2, SIC for VAR 2,784.246, SIC for GARCH-in-Mean VAR 2,072.821, Lag 2, SIC for VAR 1,415.222, SIC for GARCH-in-Mean VAR 1,378.704.

Notes: The sample period with the consumption velocity is 1967:m1 to 2025:m6 and that with the income velocity is 1967:q1 to 2025:q2.

Lags are selected based on the minimum SIC value for the conventional VAR.

$^\#$ Lag is selected among twelve lags based on the minimum SIC for conventional VAR subject to stable GARCH-in-Mean VAR.

Figure 20.

The logged Divisia M4 aggregate and its consumption and income velocities since 1967.

Note: All series are normalized to 1 in 1967 ml.