2. Divisia monetary aggregates

In this section, we review the development of the Divisia monetary aggregates, developed by Barnett (Reference Barnett1978, Reference Barnett1980) and Barnett et al. (Reference Barnett, Chauvet, Leiva-Leon and Su2023).

2.1. Conventional Divisia aggregates

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

(1) \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, measuring the maximum expected rate of return available in the economy, and $r_{it}^{a}$ is the own rate of return on monetary asset $i$ during period $t$ . Barnett (Reference Barnett1980) argued that the simple sum monetary aggregates provided by the Federal Reserve are inconsistent with economic aggregation theory because they assume that the monetary assets are perfect substitutes with the same user cost. He developed the Divisia monetary aggregates which do not assume perfect substitution between component assets (and hence permit different user costs of the component assets).

The Divisia monetary aggregate (in discrete time) computes the growth rate of the aggregate as the share-weighted average of its monetary asset component growth rates as follows

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

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

\begin{equation*} s_{it}=\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 $i$ during period $t$ .

Over the years, most of the modern formal investigations of the impact of money on economic activity are carried out using the Divisia monetary aggregates. See, for example, Belongia (Reference Belongia1996), Serletis and Gogas (Reference Serletis and Gogas2014), Hendrickson (Reference Hendrickson2014), and Keating et al. (Reference Keating, Kelly, Smith and Valcarcel2019), among others.

2.2. Credit card-augmented Divisia aggregates

The conventional Divisia monetary aggregates exclude credit card transaction services. Barnett et al. (Reference Barnett, Chauvet, Leiva-Leon and Su2023), using economic aggregation and index number theory, 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 Su2023) derive the user cost of credit card transaction services, $\pi _{lt}^{c}$ , as

(3) \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

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

where

\begin{equation*} s_{it}=\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}=\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 $l$ , $l=1,\ldots,L$ .

2.3. Credit card-augmented Divisia inside aggregates

The conventional Divisia aggregates and the credit card-augmented Divisia monetary aggregates emphasize the demand side of liquidity services. However, conditions from the supply side of liquidity services are at least equally important. In this regard, the monetary services produced by financial firms are known as inside money and are highly relevant to the transmission mechanism of monetary policy. In fact, quantitative easing during the global financial crisis and the coronavirus pandemic, with the goal of affecting the supply of liquid assets, impacted inside money directly.

To account for the monetary services produced by deposit-based financial firms, Barnett (Reference Barnett1987) introduces Divisia supply monetary aggregates which are based upon supply-side aggregation theory, in the context of a conventional neoclassical model of financial intermediary monetary assets supply. These aggregates highlight the existence of noninterest-bearing required reserves for banks. This is important, because a regulatory wedge is created for the user of the monetary services produced by banks. Thus, the user cost of a monetary asset needs to subtract the implicit tax as follows

(5) \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 $i$ .

Barnett and Su (Reference Barnett and Su2018) construct the credit card-augmented Divisia inside monetary aggregates as in equation (4) with quantities demanded, $m_{it}^{a}$ , replaced by quantities supplied and with paid user costs, $\pi _{it}^{a}$ , replaced by received user costs calculated as in equation (5).

In what follows, we posit our empirical work on the most recently developed credit card-augmented Divisia monetary aggregates and credit card-augmented Divisia inside monetary aggregates.

3. The structural VAR

We consider a five-variable structural VAR model, consistent with the Belongia and Ireland (Reference Belongia and Ireland2021) expanded New Keynesian model, in the inflation rate, $\pi _{t}$ , the output gap, $\tilde{y}_{t}$ , the short-term nominal interest rate, $i_{t}$ , the nominal money growth rate, $\mu _{t}$ , and the user cost of money, ${uc}_{t}$ , as follows

(6) \begin{equation} \boldsymbol{A}{\boldsymbol{z}}_{t}=\boldsymbol{C}+\sum _{i=1}^{k}\boldsymbol{\Gamma}_{i}z_{t-i}+\boldsymbol{\epsilon }_{t} \end{equation}

where

\begin{eqnarray*} z_{t} &=&\left [ \begin{array}{c} \pi _{t} \\ \tilde{y}_{t} \\ i_{t} \\ \mu _{t} \\{uc}_{t}\end{array} \right ] ;\quad \boldsymbol{A}=\left [ \begin{array}{ccccc} 1 & a_{12} & 0 & 0 & 0 \\ a_{21}-a_{23} & 1 & a_{23} & -a_{21} & 0 \\ a_{31} & a_{32} & 1 & a_{34} & 0 \\ -1 & a_{42} & 0 & 1 & a_{45} \\ a_{51} & 0 & a_{53} & -a_{51} & 1\end{array}\right ] ;\quad \boldsymbol{\epsilon }_{t}=\left [ \begin{array}{c} \epsilon _{\pi,{t}} \\ \epsilon _{\tilde{y},_{t}} \\ \epsilon _{i,{t}} \\ \epsilon _{\mu,t} \\ \epsilon _{uc,t}\end{array}\right ] \text{;} \\ && \\ && \\ \boldsymbol{C} &=&\left [ \begin{array}{c} c_{1} \\ c_{2} \\ c_{3} \\ c_{4} \\ c_{5}\end{array}\right ] ;\quad \boldsymbol{\Gamma}_{i}=\left [ \begin{array}{ccccc} \gamma _{i,11} & \gamma _{i,12} & \gamma _{i,13} & \gamma _{i,14} & \gamma _{i,15} \\ \gamma _{i,21} & \gamma _{i,22} & \gamma _{i,23} & \gamma _{i,24} & \gamma _{i,25} \\ \gamma _{i,31} & \gamma _{i,32} & \gamma _{i,33} & \gamma _{i,34} & \gamma _{i,35} \\ \gamma _{i,41} & \gamma _{i,42} & \gamma _{i,43} & \gamma _{i,44} & \gamma _{i,45} \\ \gamma _{i,51} & \gamma _{i,52} & \gamma _{i,53} & \gamma _{i,54} & \gamma _{i,55}\end{array}\right ]. \end{eqnarray*}

The structural errors, $\boldsymbol{\epsilon }_{t}$ , are assumed to follow the normal distribution with zero mean and the identity matrix for their variance. The $\boldsymbol{A}$ matrix identifies the model with seven zero restrictions and three equality restrictions. Based on Belongia and Ireland (Reference Belongia and Ireland2021), the parameterized $\boldsymbol{A}$ matrix is consistent with New Keynesian theory, which is augmented by incorporating money and the cost of holding monetary balances.

We have the following structural interpretation for each equation in (6). The inflation rate equation

(7) \begin{equation} \pi _{t}=-a_{12}\tilde{y}_{t}+c_{1}+\sum _{i=1}^{k}\sum _{j=1}^{5}\gamma _{i,1j} z_{t-i,j}+\epsilon _{\pi,t} \end{equation}

mimics the New Keynesian Phillips curve. It shows a contemporaneous relationship between the inflation rate, $\pi _{t}$ , and the output gap, $\tilde{y}_{t}$ . The structural error term, $\epsilon _{\pi,t}$ , is referred to as the aggregate supply shock.

The second equation in the VAR system is

(8) \begin{equation} \tilde{y}_{t}=-a_{23}(i_{t}-\pi _{t})+a_{21}(\mu _{t}-\pi _{t})+c_{2}+\sum _{i=1}^{k}\sum _{j=1}^{5}\gamma _{i,2j} z_{t-i,j}+\epsilon _{\tilde{y},_{t}} \end{equation}

is basically the New Keynesian IS curve. It shows how the real (short-term) interest rate, $i_{t}-\pi _{t}$ , affects the output gap, $\tilde{y}_{t}$ . In general, an increase in the real interest rate will reduce the output gap. In particular, Belongia and Ireland (Reference Belongia and Ireland2021) augment the IS curve by allowing the growth rate of real money balances, $\mu _{t}-\pi _{t}$ , to impact real economic activity. This mechanism represents the classical view about the role of money in the economy. $\epsilon _{\tilde{y},_{t}}$ in equation (8) is referred to as the output shock.

The third equation gives the monetary policy rule

(9) \begin{equation} i_{t}=-a_{31}\pi _{t}-a_{32}\tilde{y}_{t}-a_{34}\mu _{t}+c_{3}+\sum _{i=1}^{k}\sum _{j=1}^{5}\gamma _{i,3j} z_{t-i,j}+\epsilon _{i,t}\text{.} \end{equation}

The monetary policy rule assumes that the central bank adjusts the interest rate by tracking inflation and output gap changes. $\epsilon _{i,t}$ is the monetary policy shock. It is to be noted that this policy rule also includes the growth rate of nominal money balances, $\mu _{t}$ . It indicates that a monetary policy shock can be considered a combined result of changes in the interest rate and the money stock. For example, an expansionary monetary shock may reflect a combination of a lower interest rate and higher nominal money growth.

The fourth equation is the money demand equation, which shows that the growth rate of real money balances is determined by the output gap (the scale variable) and the user cost of money (an opportunity cost variable), as follows

(10) \begin{equation} \mu _{t}-\pi _{t}=-a_{42}\tilde{y}_{t}-a_{45}{uc}_{t}+c_{4}+\sum _{i=1}^{k}\sum _{j=1}^{5}\gamma _{i,4j} z_{t-i,j}+\epsilon _{\left ( \mu -\pi \right ),t} \end{equation}

where $\epsilon _{\left ( \mu -\pi \right ),t}$ is the money demand shock.

The fifth equation characterizes the opportunity cost of holding money balances which is the user cost, ${uc}_{t}$ , as follows

(11) \begin{equation}{uc}_{t}=-a_{53}i_{t}+a_{51}(\mu _{t}-\pi _{t})+c_{5}+\sum _{i=1}^{k}\sum _{j=1}^{5}\gamma _{i,5j} z_{t-i,j}+\epsilon _{uc,t}. \end{equation}

According to Belongia and Ireland (Reference Belongia and Ireland2014, Reference Belongia and Ireland2021), an increase in the interest rate raises the cost of holding reserves from the viewpoint of commercial banks. The banks then pass the cost to consumers by lowering the return on deposits and other monetary assets, which directly increases the user cost. Moreover, the increase in the interest rate also signals a higher return on non-monetary assets in the financial market. Therefore, the opportunity cost of holding monetary assets should increase, also leading to a higher user cost. The second term in equation (11) tells that the additional creation of real money balances may impact its user cost. Belongia and Ireland (Reference Belongia and Ireland2021) refer to $\epsilon _{uc,t}$ as the monetary system shock.

The reduced form of model (6) is

(12) \begin{equation} z_{t}=\widetilde{\boldsymbol{C}}+\sum _{i=1}^{k}\widetilde{\boldsymbol{\Gamma}}_{t-i}z_{t-i}+\widetilde{\boldsymbol{\epsilon }}_{t} \end{equation}

where

\begin{eqnarray*} \widetilde{\boldsymbol{C}} &=& \boldsymbol{A}^{-1}\boldsymbol{C}\\ \widetilde{\mathbf{\Gamma }}_{t-i} &=& \boldsymbol{A}^{-1}\mathbf{\Gamma }_{t-i} \\ \widetilde{\boldsymbol{\epsilon }}_{t} &=&\boldsymbol{A}^{-1}\boldsymbol{\epsilon }_{t} \end{eqnarray*}

with the reduced from errors, $\widetilde{\boldsymbol{\epsilon }}_{t}$ , following the normal distribution, $\widetilde{\boldsymbol{\epsilon }}_{t}\sim N(\mathbf{0},\boldsymbol{\Omega })$ , where $\boldsymbol{\Omega }=\boldsymbol{A}^{-1}\mathbf{I}\left (\boldsymbol{A}^{-1}\right ) ^{\prime }$ .

4. The data

We use quarterly data for the United States over the period from 2006:q3 to 2023:q2.Footnote ¹ For the real output series, $y_{t}$ , we use the real GDP series GDPC1 from the Federal Reserve Economic Database (FRED) maintained by the Federal Reserve Bank of St. Louis. We use the Congressional Budget Office’s estimate for potential real GDP, $\bar{y}_{t}$ , which is also retrieved from FRED. We then calculate the output gap, $\tilde{y}_{t}$ , as the difference between observed real GDP and potential real GDP expressed as a percentage of potential real GDP, $\tilde{y}_{t}=(y_{t}-\bar{y}_{t})/\bar{y}_{t}$ .

Figure 1.

Impulse response functions to a money demand shock based on M1A and M1AI.

The inflation rate, $\pi _{t}$ , is the year-over-year percentage change in the personal consumption expenditures price index, retrieved from FRED (series CPIAUCSL). The nominal interest rate, $i_{t}$ , is the effective federal funds rate from FRED. However, we use the Wu and Xia (Reference Wu and Xia2016) shadow federal funds rate for the time period from 2009:q1 to 2015:q4, since the federal funds rate hit the zero lower bound during this period.

Belongia and Ireland (Reference Belongia and Ireland2021) take advantage of the Divisia aggregates and their theoretically coherent measures of the user cost and use the Divisia M2 monetary aggregate. In this paper, we focus on the new credit card-augmented Divisia and credit card-augmented Divisia inside monetary aggregates. In particular, we consider eight credit card-augmented Divisia monetary aggregates. They are the credit card-augmented Divisia M1 monetary aggregate (M1A), the credit card-augmented Divisia M2 monetary aggregate (M2A), the credit card-augmented Divisia M3 monetary aggregate (M3A), the credit card-augmented Divisia M4 monetary aggregate (M4A), the credit card-augmented Divisia M1 inside monetary aggregate (M1AI), the credit card-augmented Divisia M2 inside monetary aggregate (M2AI), the credit card-augmented Divisia M3 inside monetary aggregate (M3AI), and the credit card-augmented Divisia M4 inside monetary aggregate (M4AIM).Footnote ²

For a graphical presentation and brief discussion of the monthly growth rates of the conventional Divisia aggregates, the credit card-augmented Divisia monetary aggregates, and the credit card-augmented Divisia inside monetary aggregates, see Figure 1 in Andreadis et al. (Reference Andreadis, Fragkou, Karakasidis and Serletis2023).

Table 1.

Bayesian priors

Table 2.

Estimates of the $a_{21}$ and $a_{34}$ parameters

Notes: Sample period, quarterly data, 2006:q3 to 2023:q2.

The posterior distribution for the parameters is reported with the median, 16th percentile, and 84th percentile.

5. Empirical evidence

We follow Belongia and Ireland (Reference Belongia and Ireland2021) and perform Bayesian estimation. In particular, we calibrate Bayesian prior distributions for the structural parameters of the model that reflect the New Keynesian view of the cycle, combine them with information contained in the data, and through the likelihood function characterize the posterior distributions of the same parameters. Next, we examine whether the posterior distributions of the structural parameters support specifications that allow classical channels of monetary transmission. That is, whether changes in the growth rate of nominal money balances signal more clearly than changes in the nominal interest rate whether monetary policy is contractionary or expansionary and whether changes in real money balances, together with changes in the real interest rate, affect real output.

The restrictions and other prior information of the model are presented in Table 1. For example, the $a_{21}$ coefficient in the augmented version of the New Keynesian aggregate demand curve (8), which captures the impact of nominal money growth, $\mu _{t}$ , on the output gap (via changes in the growth rate of real money balances, $\mu _{t}-\pi _{t}$ ), is assigned a zero prior mean to be consistent with the New Keynesian interest rate channel of monetary policy transmission that links the output gap only to policy-induced changes in the real short-term interest rate. Similarly, the coefficient $a_{34}$ in equation (9), which captures the effect of changes in nominal money growth on the short-term nominal interest rate, is assigned a zero prior mean to be consistent with the New Keynesian view that changes in nominal money play no role in formulation of monetary policy. We follow Belongia and Ireland (Reference Belongia and Ireland2021) in assigning Bayesian priors for all the other parameters not shown in Table 1—see Belongia and Ireland (Reference Belongia and Ireland2021) for more details.

Figure 2.

Impulse response functions to a money demand shock based on M2A and M2AI.

In Table 2, we summarize posterior estimates (medians together with 16th and 84th percentiles) for the two coefficients that characterize the potential role of money in the economy— $a_{21}$ in the augmented New Keynesian aggregate demand equation (8) and $a_{34}$ in the monetary policy rule (9). For comparison purposes, we also include the Belongia and Ireland (Reference Belongia and Ireland2021) estimates with the original Divisia M2 monetary aggregate over their sample period, from 1967:q1 to 2017:q4, as well as our estimates with the original Divisia M2 monetary aggregate over our sample period, from 2006:q3 to 2023:q2. Consistent with the evidence in Belongia and Ireland (Reference Belongia and Ireland2021), our estimates suggest that the data prefer a more classical version of both the augmented New Keynesian aggregate demand equation (8) and the monetary policy rule (9).

Moreover, the posterior median for the coefficient on real money balances in the augmented New Keynesian aggregate demand equation (8) is much higher than the one reported by Belongia and Ireland (Reference Belongia and Ireland2021). In particular, it is $0.62$ with the credit card-augmented Divisia M2 (Divisia M2A) aggregate and $0.61$ with the credit card-augmented Divisia M2 inside (Divisia M2AI) monetary aggregate. On the other hand, the posterior median for the $a_{34}$ coefficient in the monetary policy rule (9) is much smaller than the estimate in Belongia and Ireland (Reference Belongia and Ireland2021) although similar to our estimate when the model is estimated with Divisia M2 over the 2006:q3 to 2023:q2 sample period. It is $0.28$ with Divisia M2A and $0.27$ with Divisia M2AI compared to $1.23$ in Belongia and Ireland (Reference Belongia and Ireland2021), suggesting a weaker (but nontrivial) policy-rate response to changes in the growth rate of the credit-augmented Divisia monetary aggregates. Overall, our estimates suggest that the credit-augmented Divisia aggregates play an even more important role in shaping the monetary business cycle than the original Divisia aggregates. This result can be linked to the incorporation of credit payments. Credit card transaction services have been an important part of the general economic activity and include additional information when measuring monetary services. Our results highlight how such information enhances the understanding of money’s role in the business cycle.

Figure 3.

Impulse response functions to a money demand shock based on M3A and M3AI.

Figure 4.

Impulse response functions to a money demand shock based on M4A and M4AI.

Figure 5.

Impulse response functions to a monetary system shock based on M1A and M1AI.

Figure 6.

Impulse response functions to a monetary system shock based on M2A and M2AI.

Figure 7.

Impulse response functions to a monetary system shock based on M3A and M3AI.

Figure 8.

Impulse response functions to a monetary system shock based on M4A and M4AI.

Figure 9.

Variance decomposition for money demand shocks based on M1A and M1AI.

Figure 10.

Variance decomposition for money demand shocks based on M2A and M2AI.

Figure 11.

Variance decomposition for money demand shocks based on M3A and M3AI.

Figure 12.

Variance decomposition for money demand shocks based on M4A and M4AIM.

Figure 13.

Variance decomposition for monetary system shocks based on M1A and M1AI.

Figure 14.

Variance decomposition for monetary system shocks based on M2A and M2AI.

Figure 15.

Variance decomposition for monetary system shocks based on M3A and M3AI.

Figure 16.

Variance decomposition for monetary system shocks based on M4A and M4AIM.

To access the dynamic role of the credit-augmented Divisia aggregates in the monetary business cycle, we report selected impulse response functions in Figures 1–4.Footnote ³ They show the responses of inflation and the output gap to a one-standard deviation money demand shock. As can be seen, a positive money demand shock is likely to decrease inflation after about eight quarters with the narrow Divisia M1A and Divisia M1AI monetary aggregates and much earlier (about two quarters) with the broader monetary aggregates. The effect, however, on the output gap is generally statistically insignificant.

Figures 5–8 plot the responses of inflation and the output gap to a one-standard deviation monetary system shock. The monetary system shock captures the impacts of the creation of monetary assets on the supply side. It is interesting to see that an unexpected increase in user costs will cause a decline in inflation and the output gap. This effect is likely to appear after two quarters, and all Figures 5–8 show the same pattern. The intuition is that an increase in the user cost is associated with a higher interest rate which slows down economic activity, reducing inflation and output.

Finally, we report the variance decompositions in Figures 9–16. As can be seen (in Figures 9–12), money demand shocks explain less than 10% of inflation and output gap variations in the long run. On the other hand, monetary system shocks (see Figures 13–16) explain around 10% of inflation variation and play an even larger role in explaining output fluctuations. As in Belongia and Ireland (Reference Belongia and Ireland2021), these results point to additional classical channels of monetary transmission, captured by movements in the credit-augmented Divisia aggregates above and beyond movements in the real interest rate.

In the Appendix, we report the responses of inflation and the output gap to a positive monetary shock as well as the full set of variance decompositions based on all the credit-augmented Divisia monetary aggregates. As can be seen in Appendix Figures A1–A4, the responses of inflation and the output gap to a positive monetary shock are all consistent with economic theory. In particular, a contractionary monetary policy (an increase in the federal fund rate) reduces inflation and limits economic activities, and there are no output and price puzzles. Appendix Figures A5–A44 show the full set of variance decompositions with all the credit-augmented Divisia aggregates. There is robust evidence (see Appendix Figures A5, A10, A15, A20, A25, A30, A35, and A40) suggesting that the aggregate supply shock dominates the fluctuations in inflation in the long run. On the other hand, the aggregate demand shock is an important factor that explains the variations in all five macroeconomic variables in the long run, based on Appendix Figures A6, A11, A16, A21, A26, A31, A36, and A41.