3.1. Data and choice of monetary aggregates

This study uses a similar set of six variables as Sims (Reference Sims1992) to model the US economy and assess money’s role in the US monetary policy analysis. The variables are listed in Appendix B, Table B.1.

Following Sims (Reference Sims1992), we include the world commodity price index (COM) to account for inflation expectations.Footnote ² The COM is expected to capture the non-policy-induced changes in inflationary pressure to which the Fed may react when setting monetary policy. As in Dery and Serletis (Reference Dery and Serletis2021), the US industrial production index (IP) and consumer price inflation (INF) are taken as the target variables of monetary policy, known as non-policy variables. Inflation is calculated by taking the annual change in the log of the consumer price index, while the log-detrended IP series is used as a proxy for the output gap. The Federal funds rate (FED) represents a measure of the short-term nominal interest rate, where the statistical innovation to the FED is associated with monetary policy disturbances (see e.g. Belongia and Ireland, Reference Belongia and Ireland2016; Dery and Serletis, Reference Dery and Serletis2021) The exchange rate is an arbitrage variable that describes the financial market equilibrium, and we employ the real narrow effective exchange rate index (RER). As highlighted by Sims (Reference Sims1992), the inclusion of COM and RER are necessary for the business cycle modeling framework as these variables influence monetary policy.

As for the choice of monetary aggregates, we follow Dery and Serletis (Reference Dery and Serletis2023) and include the Fed’s broad, simple sum monetary aggregate M2 (SumM2) and the Center for Financial Stability (CFS) broad Divisia monetary aggregates, namely Divisia M3 (DivM3) and Divisia M4 (DivM4). Currently, the broadest money supply measure reported by the Fed is only SumM2.Footnote ³ In contrast, the CFS reports broader measures where DivM3 and DivM4 include more components that serve as money in varying degrees (see for details Barnett et al. Reference Barnett, Liu, Mattson and van den Noort2013; Hanke, Reference Hanke2019; Belongia and Ireland, Reference Belongia and Ireland2019; Dery and Serletis, Reference Dery and Serletis2021).Footnote ⁴ Moreover, to derive these Divisia indices, the simple sum assumption of perfect substitutability of the monetary components is relaxed, and a weighting scheme based on monetary component user costs is used. According to Barnett et al. (Reference Barnett, Liu, Mattson and van den Noort2013), the US Divisia narrow monetary aggregates such as Divisia M2 closely follow their simple sum (SumM2) counterparts, as these series only include fewer monetary components. Therefore, in this study, to examine the role of Divisia money in the US business cycle, we focus on the broadly defined DivM3 and DivM4 with SumM2.

The vector of $\mathbf{X}_{t}$ variables is represented as

(1) \begin{equation} \mathbf{X}_{t}=[\text{COM}_{t}, \text{IP}_{t}, \text{INF}_{t}, \text{MA}_{t}, \text{FED}_{t}, \text{RER}_{t}]' \end{equation}

The MA in (1) represents either DivM3, DivM4 or SumM2. The Divisia monetary aggregates are from the Centre for Financial Statistics, while the commodity price index is the CRB/BLS spot index from Thompson Reuters. The remaining variables are from the Fed Economic Database. All data are in monthly frequency and cover the period from January 1974 to June 2022. In this paper, all variables are seasonally adjusted, in log level and detrended, using a linear trend method, except for the INF and FED, which are expressed in percentages. Appendix B provides a description of the data and sources in Table B.1; with unit root test results in Table B.2 and plots of the data series in Figure B.1.Footnote ⁵

3.2. Identification of the VARMA model

In this section, we illustrate the application of the VARMA methodology on the selected six variables of the US business cycle model listed in (1).

A $K$ dimensional VARMA $(p,q)$ process can be written as

(2) \begin{equation} \mathbf{X}_{t}= \mathbf{c}+\boldsymbol{\Phi }_{1} \mathbf{X}_{t-1}+\ldots +\boldsymbol{\Phi }_{p} \mathbf{X}_{t-p}+\boldsymbol{\upsilon }_{t} -\boldsymbol{\Theta }_{1} \boldsymbol{\upsilon }_{t-1}-\ldots -\boldsymbol{\Theta }_{q}\boldsymbol{\upsilon }_{t-q}, \end{equation}

where $\boldsymbol{\Phi }_{j}$ represent the autoregressive coefficients, $\boldsymbol{\Theta }_{i}$ represent the moving average (MA) coefficients while $\mathbf{X}_{t}$ is a ( $K\times 1$ ) vector of endogenous variables at time $\mathbf{t}$ . To identify and estimate (2), we use the Athanasopoulos and Vahid (Reference Athanasopoulos and Vahid2008b) extension of the Tiao and Tsay (Reference Tiao and Tsay1989) scalar component model (SCM) methodology. The aim of identifying scalar components is to examine whether any simplifying embedded structures underlying this process can provide a parsimonious VARMA structure.Footnote ⁶

The identification of the VARMA $(p,q)$ process is carried out in three stages. In stage 1, we identify the overall tentative order of the VARMA $(p,q)$ . Conditional on the overall tentative order $(p,q)$ , we then repeat the search process for each individual component. Embedded scalar component models are identified through a series of canonical correlation tests. The test results from identifying the overall tentative order and the individual SCMs are tabulated in what is referred to as Criterion and Root tables.

In Appendix A, we demonstrate the reading of the Criterion and Root tables. Panel A of Table A.1, is known as the “Criterion Table," allows us to identify the overall order of the model. From Panel A, the identified overall order of the system is VARMA $(2,1)$ . Conditional on the overall order of $\left ( 2,1\right )$ , canonical correlation tests described in Appendix A are performed to identify the individual orders of embedded SCMs. Panel B of Table A.1, referred to as the “Root Table," allows us to identify the individual orders of the scalar components. From Table B, the identified VARMA $(2,1)$ consists of three SCM $\left ( 1,1\right )$ and three SCM $\left (2,1\right )$ .

Implementing Stage II of the Athanasopoulos and Vahid (Reference Athanasopoulos and Vahid2008b) identification process described in Appendix A leads to additional zero restrictions on the matrix containing the contemporaneous relationships between the variables and the canonical SCM representation of the identified VARMA models.Footnote ⁷ Among the variables, as defined in (1), COM, INF and RER are found to be loading as SCM $(1,1)$ , while IP, MA and FED are loading as SCM $(2,1)$ . Similar identification outcomes are observed for models with DivM3, DivM4 and SumM2, respectively.

The identified VARMA $(2,1)$ model is

\begin{eqnarray*} \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{array} \right ] \mathbf{X}_{t}=\mathbf{c}+\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \phi _{11}^{\left ( 1 \right ) } & \phi _{12}^{\left ( 1 \right ) } & \phi _{13}^{\left ( 1 \right ) } & \phi _{14}^{\left ( 1 \right ) } & \phi _{15}^{\left ( 1 \right ) } & \phi _{16}^{\left ( 1 \right ) } \\[2pt] \phi _{21}^{\left ( 1 \right ) } & \phi _{22}^{\left ( 1 \right ) } & \phi _{23}^{\left ( 1 \right ) } & \phi _{24}^{\left ( 1 \right ) } & \phi _{25}^{\left ( 1 \right ) } & \phi _{26}^{\left ( 1 \right ) } \\[2pt] \phi _{31}^{\left ( 1 \right ) } & \phi _{32}^{\left ( 1 \right ) } & \phi _{33}^{\left ( 1 \right ) } & \phi _{34}^{\left ( 1 \right ) } & \phi _{35}^{\left ( 1 \right ) } & \phi _{36}^{\left ( 1 \right ) } \\[2pt] \phi _{41}^{\left ( 1 \right ) } & \phi _{42}^{\left ( 1 \right ) } & \phi _{43}^{\left ( 1 \right ) } & \phi _{44}^{\left ( 1 \right ) } & \phi _{45}^{\left ( 1 \right ) } & \phi _{46}^{\left ( 1 \right ) } \\[2pt] \phi _{51}^{\left ( 1 \right ) } & \phi _{52}^{\left ( 1 \right ) } & \phi _{53}^{\left ( 1 \right ) } & \phi _{54}^{\left ( 1 \right ) } & \phi _{55}^{\left ( 1 \right ) } & \phi _{56}^{\left ( 1 \right ) } \\[2pt] \phi _{61}^{\left ( 1 \right ) } & \phi _{62}^{\left ( 1 \right ) } & \phi _{63}^{\left ( 1 \right ) } & \phi _{64}^{\left ( 1 \right ) } & \phi _{65}^{\left ( 1 \right ) } & \phi _{66}^{\left ( 1 \right ) } \end{array} \right ] \mathbf{X}_{t-1} \\[2pt] + \left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 0 & 0 & 0 & 0 & 0 & 0 \\ \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } & \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } & \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } \\ \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } & \phi _{31}^{\left ( 2 \right ) } & \phi _{32}^{\left ( 2 \right ) } & \phi _{33}^{\left ( 2 \right ) } \\ 0 & 0 & 0 & 0 & 0 & 0 \end{array} \right ] \mathbf{X}_{t-2}\\ + \boldsymbol{\upsilon }_{t}-\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \theta _{11}^{\left ( 1 \right ) } & \theta _{12}^{\left ( 1 \right ) } & \theta _{13}^{\left ( 1 \right ) } & \theta _{14}^{\left ( 1 \right ) } & \theta _{15}^{\left ( 1 \right ) } & \theta _{16}^{\left ( 1 \right ) } \\[2pt] \theta _{21}^{\left ( 1 \right ) } & \theta _{22}^{\left ( 1 \right ) } & \theta _{23}^{\left ( 1 \right ) } & \theta _{24}^{\left ( 1 \right ) } & \theta _{25}^{\left ( 1 \right ) } & \theta _{26}^{\left ( 1 \right ) } \\[2pt] \theta _{31}^{\left ( 1 \right ) } & \theta _{32}^{\left ( 1 \right ) } & \theta _{33}^{\left ( 1 \right ) } & \theta _{34}^{\left ( 1 \right ) } & \theta _{35}^{\left ( 1 \right ) } & \theta _{36}^{\left ( 1 \right ) } \\[2pt] \theta _{41}^{\left ( 1 \right ) } & \theta _{42}^{\left ( 1 \right ) } & \theta _{43}^{\left ( 1 \right ) } & \theta _{44}^{\left ( 1 \right ) } & \theta _{45}^{\left ( 1 \right ) } & \theta _{46}^{\left ( 1 \right ) } \\[2pt] \theta _{51}^{\left ( 1 \right ) } & \theta _{52}^{\left ( 1 \right ) } & \theta _{53}^{\left ( 1 \right ) } & \theta _{54}^{\left ( 1 \right ) } & \theta _{55}^{\left ( 1 \right ) } & \theta _{56}^{\left ( 1 \right ) } \\[2pt] \theta _{61}^{\left ( 1 \right ) } & \theta _{62}^{\left ( 1 \right ) } & \theta _{63}^{\left ( 1 \right ) } & \theta _{64}^{\left ( 1 \right ) } & \theta _{65}^{\left ( 1 \right ) } & \theta _{66}^{\left ( 1 \right ) } \end{array} \right ] \boldsymbol{\upsilon }_{t-1}. \end{eqnarray*}

The $\boldsymbol{\upsilon }_{t}$ is the vector of residuals representing unexpected movements in each variable. It is given as

(3) \begin{equation} \boldsymbol{\upsilon }_{t}=[\upsilon _{t}^{com}, \upsilon _{t}^{ip}, \upsilon _{t}^{inf}, \upsilon _{t}^{ma}, \upsilon _{t}^{fed}, \upsilon _{t}^{ex}]' \end{equation}

In Stage III, the identified VARMA model is estimated using FIML. As in Hannan and Rissanen (Reference Hannan and Rissanen1982), a long VAR is used to obtain initial values of the parameters.

3.3. Identification of the orthogonal shocks

To directly attribute the responses of variables to economically interpretable shocks, we need to transform the residuals ( $\boldsymbol{\upsilon }_{t}$ ) in equation (3) to a new set of orthogonal shocks.Footnote ⁸ It is common in the literature to use economic theory and stylized facts to identify the contemporaneous relationships between variables. The relationship between the reduced form VARMA disturbances ( $\boldsymbol{\upsilon }_{t}$ ) and the orthogonal shocks $\mathbf{u}_{t}$ is

(4) \begin{equation} \mathbf{A}_{0}\boldsymbol{\upsilon }_{t}=\mathbf{u}_{t}, \end{equation}

where $\mathbf{A}_{0}$ is an invertible square matrix, $E(\mathbf{u}_{t})=0$ , $E(\mathbf{u}_{t}\mathbf{u}^{\prime}_{t})=\boldsymbol{\Sigma }_{\mathbf{u}}$ and $\boldsymbol{\Sigma }_{\mathbf{u}}$ is a diagonal matrix. $\mathbf{A}_{0}$ is normalized across the main diagonal so that each equation in the system has a designated dependent variable.

(5) \begin{equation} \mathbf{A}_{0}=\left [ \begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 0 & 0 & 0 & 0 & 0 \\ a^{0}_{21} & 1 & 0 & 0 & 0 & 0 \\ a^{0}_{31} & a^{0}_{32} & 1 & 0 & 0 & 0 \\ 0 & a^{0}_{42} & a^{0}_{43} & 1 & 0 & 0 \\ 0 & a^{0}_{52} & a^{0}_{53} & a^{0}_{54} & 1 & 0 \\ a^{0}_{61} & a^{0}_{62} & a^{0}_{63} & a^{0}_{64} & a^{0}_{65} & 1 \\ \end{array} \right ] \end{equation}

A recursive identification structure on the contemporaneous matrix $\mathbf{A}_{0}$ is imposed with the variables ordered as in (1).Footnote ⁹

The world commodity price index (COM) is assumed to be contemporaneously exogenous to any variable in the domestic economy, while no restrictions are imposed on its lagged structural parameters. In contrast, in the Divisia monetary literature, COM is treated as an information market variable, implying that it is contemporaneously affected by all the variables in the model. As in (Kim and Roubini, Reference Kim and Roubini2000; Dungey and Osborn, Reference Dungey and Osborn2014), we treat the exchange rate (RER) as an information market variable while COM as a proxy for negative and inflationary supply shocks.Footnote ¹⁰ The RER is seen as responding quickly to all relevant economic disturbances and, therefore, is assumed to be contemporaneously affected by all the variables in the model. Since commodities are a crucial input for most economic sectors and account for inflation expectations, the COM is assumed to affect the domestic real sector (IP), inflation (INF) and RER contemporaneously.

The interest rate (FED), money (MA), and exchange rate are assumed not to affect the level of real activity and inflation contemporaneously but affect them with a one-period lag. As in Kim and Roubini (Reference Kim and Roubini2000) we assume businesses do not change their output and prices unexpectedly in response to unanticipated changes in monetary policy or financial signals within a month due to inertia, adjustment costs and planning delays. IP and INF are assumed to respond immediately to COM following their mark-up rule but not with the RER, as the exchange pass-through effect is not instantaneous. In line with Dungey and Osborn (Reference Dungey and Osborn2014), we assume output immediately affects inflation, but inflation does not enter the output equation contemporaneously.

According to Kim and Roubini (Reference Kim and Roubini2000) and Sims and Zha (Reference Sims and Zha2006), real demand money balances depend on real income and nominal interest rates, which represent the opportunity costs of holding money. Similarly, we also assume the money demand equation depends on economic activities and thus includes IP and INF contemporaneously while allowing the FED to influence it in the lag structure. As in Belongia and Ireland (Reference Belongia and Ireland2015) and Dery and Serletis (Reference Dery and Serletis2021), the US monetary policy equation is assumed to be the reaction function of the Fed, which sets the interest rate after observing the current IP, INF and MA. We include these variables in the monetary policy reaction function to control for current systematic responses of monetary policy to the state of the economy. We tried removing the monetary aggregate in the interest rate equation to represent the generalized Taylor restrictions studied within a New Keynesian framework. We found the results did not vary that much.

The contemporaneous structure identified in (5) is used to estimate the orthogonal shocks where the sizes of the shocks represent the one standard deviation of the corresponding orthogonal errors obtained from the SVARMA models. The vector of $\mathbf{u}_{t}$ orthogonal shocks in (4) is represented as

(6) \begin{equation} \mathbf{u}_{t}=[u_{t}^{com}, u_{t}^{ip}, u_{t}^{inf}, u_{t}^{ma}, u_{t}^{fed}, u_{t}^{ex}]' \end{equation}

where $u_{t}^{com}, u_{t}^{ip}, u_{t}^{inf}, u_{t}^{ma}, u_{t}^{fed}, u_{t}^{ex}$ are structural disturbances represented as commodity market shocks, output shocks, inflation shocks, money demand shocks, monetary policy shocks and exchange rate shocks respectively. The sizes of the shocks from the SVARMA models with different monetary aggregates are reported in Table 1.

Table 1. Magnitude of one standard deviation shocks from the SVARMA models with different monetary aggregates

With the exception of the money demand shock, the sizes of the other orthogonal shocks across the three models appear to be similar.

4.1. Impulse responses of business cycle variables to various shocks

The dynamic effects of the identified monetary and economic shocks in (6) are analyzed from impulse response functions, which are derived from pure vector moving average representations (VMA) of the model. The VMA representation of (2) is given by

(7) \begin{equation} \mathbf{X}_{t}=\mathbf{A}_{0}^{-1}\mathbf{u}_{t}+\sum _{i=1}^{\infty }\boldsymbol{\Psi }_{i}\mathbf{A}_{0}^{-1}\mathbf{u}_{t-i}. \end{equation}

where

\begin{equation*} \mathbf {\Psi }_{i}=\mathbf {\Theta }_{i}+\sum _{j=1}^{i}\mathbf {\Phi }_{j}\mathbf {\Psi }_{i-j} \end{equation*}

and $\boldsymbol{\Psi }_{0}=\mathbf{I}_{k}$ , $\boldsymbol{\Phi }_{j}=0$ for $j\gt p$ and $\boldsymbol{\Theta }_{i}=0$ for $i\gt q$ . To attribute the responses of business cycle variables to an economically interpretable shock, we replace the reduced form errors ( $\boldsymbol{\upsilon }_{t}$ ) with a $(K\times 1)$ orthogonal shocks ( $\mathbf{u}_{t}$ ) as in (4).

The parameters on current and lagged $\mathbf{u}_{t}$ represent one standard deviation orthogonalized impulse response functions. To aid comparison, the impulse responses of the US variables across the three models are normalized by dividing them by the standard deviations estimated using the model with DivM3. The impulse responses of the US variables are observed over a period of 36 months.Footnote ¹¹

In Figures 1 and 2, we present the impulse responses of the business cycle variables to monetary policy shocks when we allow the DivM3, the DivM4 and the SumM2 monetary aggregates to enter the monetary policy rule (as shown in the contemporaneous structure identified in 5). The responses of MA, FED and RER are shown in Figure 1. A positive monetary policy shock, defined as an unanticipated contractionary monetary policy, leads to a rise in the US interest rate by 20 basis points within the first three months. This increase decays over three years and is consistent across three models.

Figure 1. Impulse responses of MA, FED and RER to a monetary policy shock.

Note: Impulse responses of monetary models with DivM3, DivM4 and SumM2 respectively. The 68% confidence bands are shown as dashed lines.

The rise in interest rates creates excess demand for US currency in the foreign exchange market, resulting in the dollar’s appreciation. The appreciation is more pronounced when the narrowly defined SumM2 aggregate is included in the monetary policy rule. As expected, monetary aggregates DivM3 and DivM4 decline with the rise in interest rates, aligning with liquidity market expectations. A one standard deviation monetary policy shock causes DivM3 to fall by 4% in six months, nearly four times the decline observed for SumM2. These dynamic effects are distinguishable for up to 48 months, with the model including SumM2 indicating a positive response after six months, leading to a liquidity puzzle.

Figure 2 shows the responses of COM, IP, and INF to a contractionary monetary policy shock. The effects on COM are short-lived for models with DivM3 and DivM4 compared to the model with SumM2. Statistically negative effects on IP are observed across all three models after 3 months. Though the deflationary episode associated with a contractionary monetary policy shock appears to be consistent with theoretical expectations across the three models, the policy rule with DivM3 appears to have a more significant and pronounced effect. As the Fed adopted a more hawkish stance to control inflation, the sharp increases in US interest rates and the fall in monetary aggregates have been effectively captured by the Fed’s reaction function with the monetary rule that includes DivM3.

Figure 2. Impulse responses of COM, IP and INF to a monetary policy shock.

Note: Impulse responses of monetary models with DivM3, DivM4 and SumM2 respectively. The 68% confidence bands are shown as dashed lines.

Figure 3 displays the responses of the IP, INF and FED to the identified money demand shock for models with DivM3, DivM4 and SumM2, respectively. A positive monetary shock is expected to trigger an increase in demand for money and a rise in demand for goods and services, leading to a temporary rise in output and more persistent inflationary pressure, causing a rise in the Fed rate. As shown in Figure 3, these outcomes are clearly observed in the model with DivM3, where IP increases and reaches a peak of 6% above the trend 18 months after the shock before declining. Inflation peaks at 4% around 36 months and remains at that level. On the other hand, in the SumM2 model, inflation does not respond significantly, and the output increase is more permanent. A one standard deviation money demand shock causes a notable rise in the Fed rate for DivM3, almost double that of SumM2, highlighting the transitory nature of the IP response in the DivM3 model.

Figure 3. Impulse responses of IP, INF, and FED to a money demand shock.

Note: Impulse responses of monetary models with DivM3, DivM4 and SumM2 respectively. The 68% confidence bands are shown as dashed lines.

Figure 4 illustrates the Fed’s rate responses to various domestic shocks. A positive commodity price shock induces inflationary pressure on the economy, prompting the Fed to increase the policy rate, as seen across all three models. In response to a positive output shock, the Fed raises rates, consistent with its typical tightening policy in an expanding economy. This output expansion also generates inflationary pressure. When faced with a positive inflation shock, which represents unexpected inflationary pressure, the Fed’s response is gradual and positive. The models with DivM3 and DivM4 show a less pronounced response, while the model with SumM2 displays a sharp increase followed by a gradual decline. Regarding a positive exchange rate shock, policymakers typically counteract currency appreciation. Figure 4 shows that the Fed rate falls for the DivM3 model, lasting about two years before returning to a steady state. In contrast, the response for the SumM2 model is positive but insignificant.

Figure 4. Impulse responses of FED to commodity price, output, inflation and exchange rate shocks.

Note: Impulse responses of monetary models with DivM3, DivM4 and SumM2 respectively. The 68% confidence bands are shown as dashed lines.

Overall, the impulse responses provided evidence in support of broad Divisia M3 aggregates in monetary policy and business cycle analysis, consistent with Dery and Serletis (Reference Dery and Serletis2021) and Dery and Serletis (Reference Dery and Serletis2023). This is particularly so when the Taylor rule is augmented with the Divisia M3 monetary aggregate, suggesting that money measurement issues are important in the identification and transmission of shocks.

4.2. Robustness checks

In this section, we performed robustness checks to evaluate the crucial role of monetary aggregates in monetary policy analysis, the validity of our contemporaneous identification scheme and the significance of SVARMA models as opposed to the commonly used structural vector autoregressive (SVAR) models.

First, to demonstrate the importance of Divisia M3 and money broadly in identifying the monetary policy shock, in Figure 5, we present the impulse responses for the model with and without monetary aggregates entering the monetary policy rule. Compared to the model without money, the dynamic responses of COM, IP, INF and RER for the model with Divisia M3 appear to be more responsive and consistent with theoretical expectations.

Figure 5. Responses to monetary policy shocks identified for SVARMA models with and without money.

Note: Impulse responses of monetary models with and without DivM3 are shown as unbroken black lines and broken blue lines, respectively.

In our model, COM is ordered first and assumed to be contemporaneously exogenous to any variable in the domestic economy, while a commodity price shock is assumed to contemporaneously affect IP, INF, and RER. In our alternative model, consistent with the Divisia monetary literature, we treat COM as an information market variable, implying it is contemporaneously affected by all structural shocks in the model. Consequently, we assume that COM does not contemporaneously affect IP, INF, and RER. As shown in Figure 6, the results remain largely unchanged, indicating the robustness of our findings despite the different ordering and identification schemes.

Figure 6. Responses to monetary policy and money demand shocks identified for SVARMA models with different contemporaneous restrictions.

Note: Impulse responses of the original and alternative models with DivM3 are shown as unbroken black and blue lines respectively with confidence bands shown as dashed lines.

We compare the performance of our identified SVARMA(2,1) with the commonly used SVAR(12) for models with DivM3. Figure 7 shows that the SVARMA model has notable advantages over the SVAR model. While both models produce qualitatively similar impulse response functions, only the SVARMA model exhibits an immediate fall in the response of inflation to a contractionary monetary policy shock. In contrast, the SVAR model displays a price puzzle.Footnote ¹² Additionally, the confidence bands around the SVARMA responses are narrower than those around the SVAR responses, indicating that the parsimonious SVARMA model yields more precise impulse response functions.

Figure 7. Responses to monetary policy and money demand shocks identified for SVARMA vs SVAR models.

Note: SVARMA and SVAR impulse responses are shown as unbroken black and blue lines respectively with confidence bands shown as dashed lines.

Overall, it is important to note that with the SVARMA method of identification, coupled with appropriate contemporaneous restrictions, by allowing money to enter the Taylor-type policy rule, we are able to produce theoretically consistent results marked by the absence of the usual price, liquidity and exchange rate puzzles that have plagued previous empirical work in the literature.

Figure 8. Variance decomposition of MA and FED.

Note: VDC of monetary models with DivM3, DivM4 and SumM2 respectively.

4.3. Forecast error variance decomposition

The forecast error variance decomposition (FEVD) determines how much of each variable can be explained by exogenous shocks to the other variables in the system. In Figures 8 and 9, the FEVD of the domestic variables for the models with DivM3, DivM4 and SumM2 are presented. Results are reported for forecast horizons 0 to 60 months ahead. The results appear to have a varied path between the three models.

Focusing first on the decomposition of the monetary aggregates, as shown in Figure 8, that over a twelve-month horizon, more than 15% of the variances are attributable to the output shocks. There is evidence that in the longer horizon of five years, both output and exchange rate shocks are important sources of fluctuations, while commodity price and inflationary shocks have minimal effects for all three monetary aggregates. Monetary policy shocks play a larger role when the Taylor rule is augmented with the DivM3. Its contribution is felt even in the shorter horizon, that is 12% over 12 months and the effects continue across the longer horizon. On the other hand, monetary policy shock is hardly a source of fluctuations for DivM4. Its effects on SumM2 are felt only after three years, and the size of the fluctuations is half of that attributed to DivM3.

Overall, the decomposition of the Fed rate across all three models, as presented in the right panel of Figure 8 show that the commodity price shock is an important source of fluctuation while the Fed rate is hardly affected by the exchange rate. However, different results are projected for the output, inflation and monetary shocks when the Fed rate is augmented with different monetary aggregate. Over a one-year horizon, the fluctuations in Fed rate due to DivM3 shocks are more than two times that due to SumM2. The monetary rule that includes DivM3 also appears to be more sensitive to output fluctuations compared to the models with SumM2 and DivM4. Over the five-year horizon, monetary rule with SumM2 saw Fed rate fluctuating to inflation shock around 4% compared to only 2% for the model with DivM3 and DivM4.

The FEVD of the output is presented in Figure (9). Over the five-year horizon, inflation and money demand shocks have larger effects on output across the three models, while the monetary policy and commodity price shocks have minimal effects. This demonstrates that monetary aggregates play an important role compared to the Fed rate in influencing output variation. According to the model with DivM3, the exchange rate shock plays an important role in affecting output, especially between the medium to the long-run horizon.

Different results are projected for inflation variation, which is largely influenced by commodity price shocks across the three models. From these results, we can deduce that commodity price shocks affect output indirectly through the inflation channel. For the model with SumM2, it appears that inflation is not largely affected by other variables. However, if we switch that with DivM3, money, output, and exchange rate contribute to inflation variation over a longer horizon. Overall, monetary policy shock has minimal effect on inflation. If the monetary rule is augmented with DivM3, the Fed rate could be affecting inflation via other channels, mainly through output and monetary aggregate.

Figure 9. Variance decomposition of IP and INF.

Note: VDC of monetary models with DivM3, DivM4 and SumM2 respectively.

4.4. Historical decomposition

A useful reorganization of an estimated VARMA is the historical decomposition, which exposes the contribution of shocks to the observed outcome at each point in time. The historical decomposition of a variable utilizes a representation of any variable in terms of the product of its impulse responses with estimates of the structural shocks. The structural VMA representation of (7) is given by

(8) \begin{equation} \mathbf{X}_{t}=\sum _{i=0}^{\infty }\boldsymbol{\Xi }_{i}\mathbf{u}_{t-i} \end{equation}

where $\boldsymbol{\Xi }_{i}=\boldsymbol{\Psi }_{i}\mathbf{A}_{0}^{-1}$ and the historical decompositions can be derived by simply recognizing that the VARMA form allows for any variable to be written as a weighted sum of previous shocks plus the effects of an initial condition, that is

(9) \begin{equation} {}_{t}=\text{initial conditions} + \sum _{i=0}^{t}\boldsymbol{\Xi }_{i}\mathbf{u}_{t-i} \end{equation}

and the contribution of the $k$ th structural shock to the $j$ th variable can be represented as

(10) \begin{equation} x_{jt}^{(k)}=\text{initial conditions} + \sum _{i=0}^{t}{\xi }_{jk,i}{u}_{k,t-i} \end{equation}

Ideally, plotting the $x_{jt}^{(k)}$ for $k=1,2,\ldots, K$ , throughout the sample period, we could interpret and analyze the relative contributions of the different structural shocks to the $j$ th variable.

First, we delve into a comprehensive historical decomposition analysis, unraveling the complex interactions between selected macroeconomic variables (COM, IP, INF, FED, and RER) and the dynamic of DivM3, DivM4, and SumM2 from January 1974 to June 2022. Figure 10 unveils the historical decomposition of DivM3, DivM4, and SumM2 series, where the solid black line traces the demeaned monetary series. It is obvious from Figure 10 that the time paths of the five shocks significantly shape the movements in DivM3, overshadowing the impact on DivM4 and SumM2. Notably, SumM2 appears detached from the five shocks, exposing a disconnect between the simple sum monetary aggregates and the macroeconomic variables. Therefore, attention is directed towards examining the movements within the Divisia monetary aggregates, focusing specifically on DivM3.

We navigate through key periods, starting from mid-1974 to mid-1978, where the positive surge in the demeaned DivM3 series is predominantly attributed to the Fed rate. Transitioning to the period from mid-1978 to 1985, a decline in the demeaned DivM3 series confirms the influence of the Fed rate, a phenomenon unsurprising given the era of Volcker’s transformative policies. Moving into the era spanning the Great Moderation period, the early 1980s to the late 1990s, the trajectory of DivM3 aligns with contributions from commodity prices, output, and exchange rates. In contrast, inflation and interest rates serve as dampeners. The narrative shifts to the new millennium, from 2000 to 2010, witnessing a positive upswing linked to output and exchange rates, while the post-global financial crisis (GFC) era till the pre-COVID period is marked by negative movements attributed to inflation and the Fed rate. As we navigate the post-COVID environment, a resurgence in positive movements in DivM3 is observed, intricately tied to the dynamics of output, inflation and exchange rates. The unfolding narrative paints dynamic economic interactions, with DivM3 serving as an indicator, capturing the rebounds of policy decisions, global shocks, and the ever-evolving economic landscape.

Figure 10. Historical decomposition of MA for various domestic shocks.

Note: HDC of DivM3, DivM4 and SumM2 respectively. The solid line represents the demeaned series of the respective monetary aggregates.

Figure 11. Historical decomposition of IP and INF for monetary policy and monetary aggregate shocks.

Note: HDC of DivM3, DivM4 and SumM2 monetary models, with the blue lines representing monetary policy shocks and the red bars representing monetary aggregates shocks.

Next, we carry out historical decompositions to examine the relative contributions of the three identified monetary aggregates (DivM3, DivM4, and SumM2) shocks and monetary policy shocks to fluctuations in key domestic variables of output and inflation. Figure 11, illustrates noticeable differences in how these shocks influence IP and INF over time. DivM3 and DivM4 exert a more substantial influence on domestic variables, while SumM2 plays a comparatively minor role. These movements emphasize the pivotal role that Divisia monetary aggregates play in steering economic outcomes.

In the 1970s, particularly in 1974, concerns arose about a structural shift in money markets, indicated by a dramatic downward shift in the velocity function. By the early 1970s, evidence supported a stable demand for money function, which underpinned a monetary policy strategy based on intermediate targets for money supply growth rates (Belongia and Ireland, Reference Belongia and Ireland2019). Following the inflationary 1970s, Fed Chairman Paul Volcker sought to control inflation by reducing the money supply growth rate. As shown in Figure 11, this “Monetarist Experiment” successfully curbed inflation but led to an unintended recession (Barnett and Chauvet, Reference Barnett and Chauvet2011).

In the early 1980s, the stability of money demand remained questionable. Accumulating evidence indicated instability due to financial innovations affecting the velocity of simple sum M1 and M2. Consequently, as highlighted the minimal role of SumM2 in Figure 11, it appears the Fed abandoned its strategy of using money aggregate as an intermediate target. This period’s findings led to a consensus that money measures could be excluded from assessing monetary policy effects on economic activity. Later, the Taylor rule formalized this shift, outlining how the Fed could manipulate short-term interest rates to achieve economic goals.

During the Great Moderation, the growth rate measured by the Divisia index was higher than perceived, indicating that monetary policy was more contractionary before 1984 and more expansive afterwards than suggested by simple sum measures. The Divisia index exhibited a business cycle pattern more consistent with monetary policy, implying that more contractionary policies before recessions could have contributed to weak economic activity. During the Great Moderation period, the reactions of IP and INF to monetary policy shocks appeared subdued. According to Barnett and Chauvet (Reference Barnett and Chauvet2011), the Great Moderation may be unrelated to monetary policy, with factors such as increased US productivity, improved technology and communication, financial innovation, the rise of China and reduced size and volatility of shocks potentially playing significant roles.

The zero lower bound on nominal interest rates from 2008 to 2015 highlights the limitations of using the Federal funds rate alone to gauge monetary policy. This period suggests reconsidering monetary aggregates in policy analysis, especially for capturing the effects of unconventional policies like large-scale asset purchases and quantitative easing on money growth. Despite a decline in the user costs of all Divisia aggregates after the 2007–2009 financial crisis, they appear to contribute positively to economic growth, as demonstrated in Figure 11. As Belongia and Ireland (Reference Belongia and Ireland2019) notes, the Fed could have supplemented its unconventional policies with a targeting scheme for Divisia monetary aggregates, potentially stabilizing inflation even at the zero lower bound. During the post-2008 global financial crisis (GFC), DivM4 exhibited more significant effects than DivM3. This discrepancy can be attributed to the Fed’s implementation of quantitative easing, where the significant quantitative components of DivM4 effectively capture the repercussions of its monetary disturbances on domestic variables.

In response to the COVID-19 pandemic, the Fed implemented unprecedented monetary and fiscal measures, such as massive asset purchases, fiscal stimulus, and prolonged low interest rates, to stabilize the economy. These actions highlight the critical role and impact of unconventional monetary policies. The inflation resurgence in 2021–2022, with rates reaching multi-decade highs, has prompted a reevaluation of monetary policy. The Fed has begun tightening policy by raising interest rates and reducing asset purchases. The Divisia aggregates play a pivotal role in signaling the inflation value, with DivM3 and DivM4 acting in offsetting directions. This dichotomy reflects the inertia of the inflation rate, particularly evident up until late 2021. This shift highlights the dynamic nature of monetary policy and the necessity for the Fed to remain vigilant and responsive to evolving economic conditions.

In summary, the analysis provides compelling evidence that Divisia monetary shocks dominate in explaining variations in macroeconomic variables. The sources of these shocks exhibit considerable fluctuations over the sample period, at times asserting dominance in a positive direction while at other times acting as a dampening force.