2. A structural VAR model of monetary policy

We begin with a typical structural model of the form:

(1) \begin{equation}{{\boldsymbol{A}}}{{\boldsymbol{z}}}_{t}={{\boldsymbol{B}}}{{\boldsymbol{z}}}_{t-1}+{{\boldsymbol{u}}}_{t} \end{equation}

where ${\boldsymbol{z}}_{t}$ is an $n\times 1$ vector of variables and ${\boldsymbol{A}}$ is $n\times n$ matrix of contemporaneous structural coefficients. ${\boldsymbol{z}}_{t-1}$ is $k\times 1$ vector (where $k=mn+1$ ) containing a constant and $m$ lags of ${\boldsymbol{z}}_{t}$ . ${\boldsymbol{u}}_{t}$ is an $n\times 1$ vector of structural disturbances with variance covariance matrix ${\boldsymbol{D}}$ .

Given (1), the reduced form is

\begin{equation*} {{\boldsymbol{z}}}_{t}=\boldsymbol{\Phi}{{\boldsymbol{z}}}_{t-1}+{\boldsymbol {\varepsilon }}_{t} \end{equation*}

where

\begin{equation*} {\boldsymbol{\Phi}}={{\boldsymbol{A}}}^{-1}{{\boldsymbol{B}}} \end{equation*}

and

\begin{equation*}{\boldsymbol {\varepsilon }}_{t}= {{\boldsymbol{A}}}^{-1} {{\boldsymbol{u}}}_{t}. \end{equation*}

We denote the vector of the unknown elements of ${\boldsymbol{A}}, \boldsymbol{\ } {\boldsymbol{B}}, \boldsymbol{\ }$ and $\boldsymbol{\ } {\boldsymbol{D}}$ by $\boldsymbol{\theta }$ . Even though the VAR parameters ( $\boldsymbol{\Phi}$ and $\boldsymbol{\Omega}=E(\boldsymbol{\varepsilon }_{t}\boldsymbol{\varepsilon }_{t}^{{\prime }})$ ) can easily be estimated using Ordinary Least Square (OLS) regression, the structural parameters in $\boldsymbol{\theta }$ will require additional assumptions regarding the structural model. There are several ways of achieving structural identification, the most popular of which is the recursive or Cholesky identification. A more recent and promising approach, championed by Faust (Reference Faust1998), Canova and De Nicolo (Reference Canova and De Nicolo2002), and Uhlig (Reference Uhlig2005), is to specify the direction of the impact of a shock of interest—see Fry and Pagan (Reference Fry and Pagan2011) for a detailed review of identification by pure sign restrictions. Despite the appealing simplicity and mostly theoretical grounding of pure sign restrictions, this approach suffers from some criticisms, including the Preston (Reference Preston1978) “model identification problem,” as noted by Fry and Pagan (Reference Fry and Pagan2011) and Danne (Reference Danne2015). As noted by Danne (Reference Danne2015, p. 11), “Uhlig’s (Reference Uhlig2005) and Rubio-Ramirez’s et al. (Reference Rubio-Ramirez, Waggoner and Zha2010) rejection method are particularly prone to this ‘model identification problem.’” This problem refers to the fact that there are many models with identified parameters that rationalize the data. Thus, summarizing the responses using for instance the median response and conventional error bands represents the spread of the responses distribution across these models. Also, Wolf (Reference Wolf2020, Reference Wolf2022) illustrates, based on the true structure being a standard new Keynesian model, how sign restrictions alone may be unable to identify the effects of shocks to monetary policy; he shows how the shocks the econometrician identifies may actually be combinations of different types of structural disturbances.

We follow Baumeister and Hamilton (Reference Baumeister and Hamilton2015, Reference Baumeister and Hamilton2018), who based on Bayesian optimal statistical decision theory utilize both information on the signs and magnitudes of the shocks of interest to achieve both parameter and model identification, thereby avoiding the Preston (Reference Preston1978) model identification problem and the Fry and Pagan (Reference Fry and Pagan2011) and Danne (Reference Danne2015) criticisms that the median impulse response functions from a pure sign restriction structural VAR are not model-specific. Also, by using sign restrictions together with other prior information, we reduce the chances of the occurrence of scenarios as noted by Wolf (Reference Wolf2020, Reference Wolf2022). In what follows, we provide a brief exposition of the Baumeister and Hamilton (Reference Baumeister and Hamilton2018) approach. This section borrows from Baumeister and Hamilton (Reference Baumeister and Hamilton2015) and Baumeister and Hamilton (Reference Baumeister and Hamilton2018), Sections 2 and 3.

Suppose ${\boldsymbol{Y}}_{t}={\left ({{\boldsymbol{y}}}_{1}^{{\prime }},{{\boldsymbol{y}}}_{2}^{{\prime }},\dots,{{\boldsymbol{y}}}_{T}^{{\prime }}\right )}^{\prime }$ is the vector of data. The likelihood function $p\left ({{\boldsymbol{Y}}}_{t}\left \vert{\boldsymbol{\theta }}\right. \right )$ can be written as:

\begin{align*} p\left ({{\boldsymbol{Y}}}_{t}\left \vert{\boldsymbol{\theta }}\right. \right ) &=(2\pi )^{Tn/2}\left \vert \text{det}\left ({{\boldsymbol{A}}}\left ({\boldsymbol{\theta }}\right ) \right ) \right \vert ^{T}\left \vert{{\boldsymbol{D}}}(\theta )|^{-T/2}\right. \\ &\quad\times \text{exp}\left [ -\frac{1}{2}\sum _{t=1}^{T}\left ({{\boldsymbol{A}}}\left ({\boldsymbol{\theta }}\right ){{\boldsymbol{z}}}_{t}-{{\boldsymbol{B}}}\left ({\boldsymbol{\theta }}\right ){{\boldsymbol{z}}}_{t-1}\right ) ^{\prime }{{\boldsymbol{D}}}\left ({\boldsymbol{\theta }}\right ) ^{-1}\left ({{\boldsymbol{A}}}\left ({\boldsymbol{\theta }}\right ){{\boldsymbol{z}}}_{t}-{{\boldsymbol{B}}}\left ({\boldsymbol{\theta }}\right ){{\boldsymbol{z}}}_{t-1}\right ) \right ] \end{align*}

for ${\boldsymbol{u}}_{t}\sim \mathcal{N}(\boldsymbol{0},{\boldsymbol{D}})$ , where $\left \vert \text{det}\left ({{\boldsymbol{A}}}\right ) \right \vert$ is the absolute value of the determinant of ${\boldsymbol{A}}$ . For a prior distribution $p\left ({\boldsymbol{\theta }}\right )$ ,

\begin{equation*} p{\left ( {\boldsymbol {\theta }}\left \vert {{\boldsymbol{Y}}}_{t}\right. \right )} =\frac {p{\left ({{\boldsymbol{Y}}}_{t}\left \vert {\boldsymbol {\theta }}\right. \!\right )} p{\left ( {\boldsymbol {\theta }}\right )}}{\int p{\left ( {{\boldsymbol{Y}}}_{t}\left \vert {\boldsymbol {\theta }}\right. \!\right )} p{\left ( {\boldsymbol {\theta }}\right )} d {\boldsymbol {\theta }}} \end{equation*}

is the Bayesian posterior distribution. Let $\boldsymbol{\Psi}_{s}$ be the nonorthogonalized impulse response function at horizon $s$ from the VAR. Then,

\begin{equation*} {\boldsymbol{\Psi}}_{s}=\frac {\partial {{\boldsymbol{z}}}_{t+s}}{\partial {\boldsymbol {\varepsilon }}_{t}}. \end{equation*}

We are interested in the dynamic effects of the $j\text{th}$ structural shock on the $i\text{th}$ variable denoted by $h_{ij}^{s}\left ({\boldsymbol{\theta }}\right )$ which is the ( $i,j$ ) element of the matrix ${\boldsymbol{H}}_{s}$ , where

\begin{equation*} {{\boldsymbol{H}}}_{s}= {\boldsymbol{\Psi}}_{s} {{\boldsymbol{A}}}^{-1}. \end{equation*}

Given an $s\times 1$ vector $h_{ij}\left ({\boldsymbol{\theta }}\right )$ and a quadratic loss function, the estimated solution according to Baumeister and Hamilton (Reference Baumeister and Hamilton2018) is the posterior mean. That is,

\begin{equation*} \hat {h}_{ij}=\underset {\hat {h}_{ij}}{\mathrm {argmin}}\int \left [ \hat {h}_{ij}-h_{ij}\left ( {\boldsymbol {\theta }}\right ) \right ] ^{{\prime }}{{\boldsymbol{W}}}\left [ \hat {h}_{ij}-h_{ij}\left ( {\boldsymbol {\theta }}\right ) \right ] p\left ( {\boldsymbol {\theta }}\left \vert {{\boldsymbol{Y}}}_{t}\right. \right ) d {\boldsymbol {\theta }} \end{equation*}

and $\hat{h}_{ij}=h_{ij}^{\ast }$ where $h_{ij}^{\ast }=\int h_{ij}\left ({\boldsymbol{\theta }}\right ) p\left ({\boldsymbol{\theta }}\left \vert{{\boldsymbol{Y}}}_{t}\right. \right ) d \boldsymbol{\theta }$ and ${\boldsymbol{W}}$ is a positive definite $s\times s$ weighting matrix. Baumeister and Hamilton (Reference Baumeister and Hamilton2018) also show that for a weighted absolute value loss function, the optimal impulse response function is the posterior median of $h_{ij}^{s}\left ({\boldsymbol{\theta }}\right )$ . Lastly, Baumeister and Hamilton (Reference Baumeister and Hamilton2018) show that conditional on $\boldsymbol{\theta }$ , the $s$ -period ahead forecast error can be written as:

\begin{equation*} {{\boldsymbol{z}}}_{t+s}-\widehat {{{\boldsymbol{z}}}}_{t+s}={{\boldsymbol{H}}}_{0}\left ( {\boldsymbol {\theta }}\right ) {{\boldsymbol{u}}}_{t+s}+{{\boldsymbol{H}}}_{1}\left ( {\boldsymbol {\theta }}\right ) {{\boldsymbol{u}}}_{t+s-1}+\dots + {{\boldsymbol{H}}}_{s-1}\left ({\boldsymbol {\theta }}\right ) {{\boldsymbol{u}}}_{t+1} \end{equation*}

with mean square errors (MSE)

\begin{equation*} E\left [ \left ( {{\boldsymbol{z}}}_{t+s}-\widehat {{\boldsymbol{z}}}_{t+s}\right ) \left ( {{\boldsymbol{z}}}_{t+s}-\widehat {{\boldsymbol{z}}}_{t+s}\right ) ^{{\prime }}\left \vert {\boldsymbol {\theta }}\right. \right ] =\sum _{j=1}^{n}{{\boldsymbol{Q}}}_{js}\left ( {\boldsymbol {\theta }}\right ) \text { and }{{\boldsymbol{Q}}}_{js}\left ( {\boldsymbol {\theta }}\right ) =d_{ij}\left ( {\boldsymbol {\theta }}\right ) \sum _{k=0}^{s-1}h_{j}(k, {\boldsymbol {\theta }})h_{j}(k,{\boldsymbol {\theta }})^{{\prime }}\end{equation*}

where $d_{ij}\left ({\boldsymbol{\theta }}\right )$ is the ( $i,j$ ) element of ${\boldsymbol{D}}$ . The contribution of shock $j$ to the $s$ -period ahead MSE of the $i\text{th}$ element of $y_{t+s}$ is the ( $i,j$ ) element of ${\boldsymbol{Q}}_{js}\left ({\boldsymbol{\theta }}\right )$ .

3. The data

We use quarterly data for the USA, over the period from 1986:q1 to 2021:q1, to investigate how augmenting a standard Taylor rule with monetary aggregates and leverage measures affects the significance of monetary policy shocks in explaining key macroeconomic variations. The Bayesian structural VAR model consists of six variables—the output gap, $y_{t}$ , inflation rate, $\pi _{t}$ , federal funds rate, $i_{t}$ , credit spread, $cs_{t}$ , growth rate of commercial bank leverage, $l_{t}$ , and the growth rate of a monetary aggregate, $\mu _{t}$ . With regard to the choice of the monetary aggregate, we follow Dery and Serletis (Reference Dery and Serletis2021) and use the Center for Financial Stability (CFS) broad Divisia monetary aggregates, Divisia M3 and Divisia M4. However, we also provide a comparison with the Fed’s broad simple-sum monetary aggregate, the Sum M2 aggregate.

It should be noted that the Divisia M4- monetary aggregate, being the broadest monetary aggregate that excludes Treasury bills, could be argued to be the most appropriate monetary aggregate for monetary policy and business cycle analysis, since excluding Treasury bills will exclude the effects of fiscal policy. However, we use the Divisia M3 aggregate, instead of the Divisia M4- aggregate, as the comprehensive empirical analysis in Dery and Serletis (Reference Dery and Serletis2021) supports the Divisia M3 aggregate. We also use the Divisia M4 aggregate, which is the broadest Divisia monetary aggregate, in order to provide a comparison with the Sum M2 monetary aggregate, which is the broadest simple-sum monetary aggregate constructed by the Federal Reserve.

The Divisia monetary aggregates, as constructed by Barnett (Reference Barnett1980), are based on Diewert’s (Reference Diewert1976) class of superlative quantity index numbers. In deriving the 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 instead—see Barnett et al. (Reference Barnett, Liu, Mattson and van den Noort2013) for more details. The commercial bank leverage series is calculated using data from the Board of Governors of the Federal Reserve System. All the other variables, except for the Divisia monetary aggregates, are obtained from the Federal Reserve Economic Database (FRED). The Divisia monetary aggregates are from the Centre for Financial Stability.

The output gap, $y_{t}$ , is defined as the log difference between actual output and potential output scaled by 100; we use real GDP to measure output. The inflation rate, $\pi _{t}$ , is the year-on-year log difference of the consumer price index scaled by 100, while $i_{t}$ and $cs_{t}$ are the yearly percentage changes of the Wu and Xia (Reference Wu and Xia2016) effective shadow federal funds rate and the credit spread, respectively. We define the credit spread as the difference between Moody’s seasoned Baa corporate bond yield and the yield on 10-year Treasury constant maturity. $\mu _{t}$ is the yearly growth rate of the relevant monetary aggregate. $l_{t}$ is measured as the yearly growth rate of commercial bank leverage. We follow Adrian et al. (Reference Adrian, Etula and Muir2014) and define leverage as the ratio of total financial assets to total net financial assets. In calculating net financial assets, we follow Istiak and Serletis (Reference Istiak and Serletis2017) and exclude total miscellaneous liabilities from total liabilities of commercial banks. As Istiak and Serletis (Reference Istiak and Serletis2017) argue, this avoids extreme leverage values as well as negative leverage in some quarters.

In Figures 1 and 2, we present the levels (relative to the 1986:q1 value) and growth rates, respectively, of the Divisia M3, Divisia M4, and Sum M2 monetary aggregates over the sample period from 1986:q1 to 2021:q1. As can be seen, the aggregates are clearly distinguishable in both log levels and growth rates. In what follows, we investigate whether the differences in levels and growth rates of these aggregates, as shown in Figures 1 and 2, matter for the identification of monetary policy shocks. In a similar fashion, in Figures 3 and 4, we present the levels (again relative to the 1986:q1 value) and growth rates, respectively, of the commercial bank leverage, as well as broker-dealer leverage, shadow bank leverage, and household leverage.

Figure 1. Broad monetary aggregates. This figure plots the level of the Divisia M3, Divisia M4, and Sum M2 monetary aggregates. Vertical shaded bars are National Bureau of Economic Research (NBER) recession dates.

Figure 2. Annualized growth rates of broad monetary aggregates. Vertical shaded bars are NBER recession dates.

Figure 3. Leverage measures. This figure plots the level of four alternative leverage measures relative to their 1986:q1 value. Vertical shaded bars are NBER recession dates.

Figure 4. Annualized growth rates of leverage measures. Shaded bars are NBER recession dates.

4. Estimation

We estimate the structural monetary VAR model with four (quarterly) lags. In using the Baumeister and Hamilton (Reference Baumeister and Hamilton2015, Reference Baumeister and Hamilton2018) approach, we are able to combine sign restrictions and other prior information, as well as zero restrictions to identify aggregate supply shocks, aggregate demand shocks, and monetary policy shocks. Specifically,

\begin{equation*} {{\boldsymbol{z}}}_{t}=\left [ \begin {array} [c]{c}y_{t}\\ \pi _{t}\\ i_{t}\\ cs_{t}\\ l_{t}\\ \mu _{t}\end {array} \right ]\ \text { and }\ {{\boldsymbol{A}}} = \left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & -a_{12} & 0 & 0 & 0 & 0\\ 1 & -a_{22} & -a_{23} & 0 & 0 & 0\\ -a_{31} & -a_{32} & 1 & 0 & -a_{35} & -a_{36}\\ -a_{41} & -a_{42} & -a_{43} & 1 & -a_{45} & -a_{46}\\ -a_{51} & -a_{52} & -a_{53} & -a_{54} & 1 & -a_{56}\\ -a_{65} & -a_{62} & -a_{63} & -a_{64} & -a_{65} & 1 \end{array}\right] \end{equation*}

so that the system of equations of interest consists of a Phillips curve, an aggregate demand curve, and a monetary policy rule given by:

(2) \begin{align} y_{t} & =c^{s}+a_{12}\pi _{t}+[b^{s}]^{{\prime }}{{\boldsymbol{z}}}_{t-1}+u_{t}^{s} \end{align}

(3) \begin{align} y_{t} & =c^{d}+a_{22}\pi _{t}+a_{23}i_{t}+[b^{d}]^{{\prime }}{{\boldsymbol{z}}}_{t-1}+u_{t}^{d} \end{align}

(4) \begin{align} i_{t} & =c^{mp}+a_{31}y_{t}+a_{32}\pi _{t}+a_{35}l_{t}+a_{36}\mu _{t} +[b^{mp}]^{{\prime }}{{\boldsymbol{z}}}_{t-1}+u_{t}^{mp} \end{align}

where $u_{t}^{s}$ is the supply shock, $u_{t}^{d}$ is the demand shock, and $u_{t}^{mp}$ is the monetary policy shock. In equation (2), supply depends on the inflation rate, $\pi _{t}$ , while in equation (3) we distinguish demand by making it depend on the inflation rate, $\pi _{t}$ , as well as the interest rate, $i_{t}$ . In equation (4), the monetary policy shock is interpreted as one which causes the Fed to adjust its policy rate in response to changes in the output gap, $y_{t}$ , the inflation rate, $\pi _{t}$ , the leverage growth rate, $l_{t}$ , and the money growth rate, $\mu _{t}$ . As already noted, we use the Fed’s Sum M2 aggregate and the CFS broad Divisia M3 and Divisia M4 aggregates as our measures of money and commercial bank leverage as our measure of leverage.

The version of equation (4) that is estimated is

(5) \begin{equation} i_{t}-\bar{\imath }=(1-\rho )\psi ^{y}y_{t}+(1-\rho )\psi ^{\pi }(\pi _{t}-\pi ^{\ast })+\rho (i_{t-1}-\bar{\imath })+(1-\rho )\psi ^{l}l_{t}+(1-\rho )\psi ^{\mu }\mu _{t}+u_{t}^{mp} \end{equation}

where $\psi ^{y}$ , $\ \psi ^{\pi }$ , $\ \psi ^{l}$ , and $\psi ^{\mu }$ are the Fed’s long run responses to output, inflation, leverage growth, and money growth, respectively. $\pi ^{\ast }$ and $\bar{\imath }$ are the inflation rate and federal funds rate targets, respectively. $\rho$ is the smoothing parameter reflecting the Fed’s desire to implement changes gradually. Equation (5) is the same as in Baumeister and Hamilton (Reference Baumeister and Hamilton2018) except that it is augmented with leverage and a monetary aggregate.

In Table 1, we summarize the sign restrictions and prior information of the model. We follow Baumeister and Hamilton (Reference Baumeister and Hamilton2018), and references therein, to use the prior information as shown in Table 1 except for the prior regarding the Fed’s response to leverage growth and money growth. The priors for the Fed’s response to leverage growth and money growth are obtained from OLS regressions of the effective shadow federal funds rate on the output gap, inflation, leverage, and money. These OLS regressions show that the Fed’s response to output is roughly 0.5, the response to inflation is 1.5, the response to leverage growth is 0.02, and the Fed’s response to money growth is 0.2.

Table 1. Summary of sign restrictions and prior information of the model. Distribution is Student’s $t$ test with 3 degrees of freedom

In addition to the priors on the parameter mode, we also use prior information on signs as shown in the last column of Table 1. Specifically, we restrict the effect of inflation on supply to be nonnegative, while the effect of the federal funds rate on demand is negative. Also, the responses of the Fed to output, inflation, and money growth are expected to be nonnegative, while the response of the Fed to leverage growth is unconstrained. For the interest rate smoothing parameter, we follow Baumeister and Hamilton (Reference Baumeister and Hamilton2018), Lubik and Schorfheide (Reference Lubik and Schorfheide2004), and Del Negro and Schorfheide (Reference Del Negro and Schorfheide2004) and use a beta distribution with mean 0.5 and standard deviation 0.2. For all the other parameters not shown in Table 1, we follow Baumeister and Hamilton (Reference Baumeister and Hamilton2018) and use Student’s $t$ test prior centered at $0$ and an uninformative scale of $1$ .

All estimated results are based on 2 million draws from the posterior distribution of the structural parameters with the first 1 million as burn-in. We conducted the Geweke (Reference Geweke and Geweke1992) convergence diagnostics of the relevant parameters of the model as indicated in Table 1 and confirm that we are drawing from the same stationary distribution. The Geweke diagnostics essentially take two nonoverlapping parts of the Markov chain and compare the means of both parts. Basically, we tested for equality of means of the first $10\%$ of the Markov chain and the last $50\%$ of the chain. The test statistic is a standard $Z$ -score with the standard errors adjusted for autocorrelation. Thus, being unable to reject the null of equality of means (say at the $5\%$ significance level) implies that we are drawing from the same stationary distribution.

Figure 5. Impulse responses of the Baumeister and Hamilton (Reference Baumeister and Hamilton2018) type model without money and leverage. For each response, the shaded area shows the 68% credibility region, while the dashed lines show the 95% confidence bands of the median response (solid blue line).

5. Empirical evidence

We begin by showing results of a Baumeister and Hamilton (Reference Baumeister and Hamilton2018) type model without money and leverage. This is a representative model according to which the behavior of the monetary authority is captured by the standard Taylor rule. Specifically, the Fed adjusts its policy rate in response to changes in output and inflation.

Figure 5 presents the impulse responses of a Baumeister and Hamilton (Reference Baumeister and Hamilton2018) type model without money and leverage. The model is estimated with the output gap, the inflation rate, the shadow federal funds rate, the credit spread, the CRB commodity spot price index, and average hourly earnings of production and nonsupervisory employees, over the period from 1986:q1 to 2021:q1. For each response, the shaded area shows the $68\%$ credibility region, while the dashed lines show the $95\%$ confidence bands of the median response. The response of the output gap to a contractionary monetary policy shock is not statistically significant at the $5\%$ level. This is also the case for the response of the inflation rate to a monetary policy shock. The variance decomposition of the model attributes 2% of the variation in output and 1% of the variation in inflation to the monetary policy shock after 1 year (see the top panel of Table 3). Using this standard Taylor rule to capture the behavior of the monetary authority, it appears that monetary policy shocks are trivial in accounting for macroeconomic variations.

Our aim in this paper is to show that augmenting the policy function of the monetary authority with measures of money and leverage improves the role of monetary policy shocks in accounting for macroeconomic variations. By construction, sign restrictions and in particular the use of prior information eliminate the price puzzle that most of the monetary VAR literature has been plagued with. For this reason we do not use the commodity price index, since this variable is mostly included in VAR analyses as a means of dealing with the price puzzle—see Sims (Reference Sims1992) and Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2016). Instead, we use commercial bank leverage as a means of capturing the importance of financial intermediaries in monetary policy and business cycles. In our monetary model, we also replace the average hourly earnings of production and nonsupervisory employees with a monetary aggregate reflecting our desire to give a more prominent role to monetary aggregates in explaining business cycles. Thus, the only difference between our preferred monetary model and the Baumeister and Hamilton (Reference Baumeister and Hamilton2018) type model is the presence of money and leverage in the standard Taylor rule.

5.1. Impulse response functions

We now turn to an analysis of the effects of money and leverage in the identification of monetary policy shocks and assess the relative importance of supply, demand, and monetary policy shocks in affecting key macroeconomic variations. In doing so, while we estimated the augmented model with the Divisia M3, Divisia M4-, Divisia M4, and Sum M2 monetary aggregates, for brevity we only report results with the Divisia M3 and Sum M2 aggregates, as the results with the Divisia M4- and Divisia M4 monetary aggregates are similar to those with the Divisia M3 aggregate. In Figures 6 and 7, we present the impulse responses of the output gap, the inflation rate, and the federal funds rate to demand, supply, and monetary policy shocks when we allow the Divisia M3 (in Figure 6) and the Sum M2 (in Figure 7) monetary aggregates and commercial bank leverage to enter the monetary policy rule. As in Figure 5, for each response, the shaded area shows the $68\%$ credibility region, while the dashed lines show the $95\%$ confidence band of the median response.

Figure 6. Impulse responses of the monetary model with Divisia M3. For each response, the solid blue line is the median response, the shaded area shows the 68% credibility region, while the dashed lines show the 95% confidence bands of the median response.

Figure 7. Impulse responses of the monetary model with Sum M2. For each response, the shaded area shows the 68% credibility region, while the dashed lines show the 95% confidence bands of the median response (solid blue line).

As shown in column 1 and 2 of Figures 6 and 7, both a positive supply shock and an expansionary demand shock generate statistically significant responses in the output gap and inflation similar to those in Figure 5. The dynamic effects of a contractionary monetary policy shock on output and inflation are shown in column 3 of Figures 6 and 7. In comparison to the nonmonetary model in Figure 5, the effects of a monetary policy shock on output and inflation are statistically significant with persistence varying from one to three-quarters depending on the measure of money. The statistically significant output contraction and deflationary episode associated with a contractionary monetary policy shock in our monetary model are more consistent with theory and many empirical studies—see, for example, Sims (Reference Sims1992), Leeper et al. (Reference Leeper, Sims and Zha1996), Bagliano and Favero (Reference Bagliano and Favero1998), Christiano et al. (Reference Christiano, Eichenbaum and Evans1999), Kim (Reference Kim1999), Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2016, Reference Belongia and Ireland2018), and Arias et al. (Reference Arias, Caldara and Rubio-Ramirez2019), among others.

It is important to notice that the method of identification using sign restrictions, short-run restrictions, prior information, and allowing money and leverage to enter the Taylor-type policy rule is by construction devoid of the usual price puzzle that many empirical models normally generate in this literature. For instance, Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2016, Reference Belongia and Ireland2018) allow Divisia money to enter into a typical Taylor rule and produce theoretically consistent results with regard to the response of output to a contractionary monetary policy shock, but their model exhibits a price puzzle. Dery and Serletis (Reference Dery and Serletis2021) also estimate a Taylor-type policy rule with Divisia money allowing for GARCH behavior in the structural shocks. Their results regarding the effect of a monetary policy shock on prices shows a puzzle. Using the Baumeister and Hamilton (Reference Baumeister and Hamilton2015, Reference Baumeister and Hamilton2018) approach, we achieve both model and parameter identification, thereby avoiding the Preston (Reference Preston1978) “model identification problem,” which as noted by Fry and Pagan (Reference Fry and Pagan2011) and Danne (Reference Danne2015) is one of the major failings of the pure sign restrictions approach. The adopted empirical approach produces theoretically consistent results marked by the absence of the usual price puzzle. The absence of the price puzzle is by construction and a characteristic of the Baumeister and Hamilton (Reference Baumeister and Hamilton2018) approach. Even though our analysis and conclusions are based on 95% confidence intervals, if one were to focus instead on the 68% credibility region (shaded regions in Figures 5–7), we note that a contractionary monetary policy generates a statistically significant increase in output at longer forecast horizons. We also observe a price puzzle at longer horizons in the nonmonetary model (Standard Taylor rule), but the incidence of a statistically significant price puzzle at longer horizons is not recorded in the models augmented with money and leverage. We think that this supports the findings of Chen and Valcarcel (Reference Chen and Valcarcel2021) who concluded that the identification of monetary policy shocks based on innovations in the shadow fed funds rate produces a price puzzle at longer horizons, but the use of monetary aggregates like the Divisia M4 eliminates such a puzzle. However, output puzzles do exist, as the output responses to a contractionary monetary policy shock are significant and positive at longer horizons. This could be because the contractionary policy might be associated with a decline in expected inflation. If the expected inflation effect is dominant, interest rates will eventually decline and cause positive output responses at longer horizons.

In the first subplot of Figure 8, we present the cumulative responses of output and inflation to a contractionary monetary policy shock with the Divisia M3 and Sum M2 monetary aggregates and commercial bank leverage entering into the monetary policy rule. We also show the responses of output and inflation to a contractionary monetary policy shock emanating from the standard Taylor rule. As can be seen in the top panel of Figure 8, the contraction in output is more pronounced in the models with money and leverage compared to the model without money and leverage (standard Taylor rule). We present a similar comparison in the bottom panel of Figure 8 in terms of the response of inflation to a contractionary monetary policy shock. As can be seen, all the models with money and leverage generate more pronounced deflation relative to the standard Taylor rule. In unreported results, we compared each monetary model’s output and inflation responses to the standard Taylor rule responses with their corresponding 95% confidence bands and confirm that there is significant overlap of the confidence bands and that there are also differences in the confidence regions. In Table 2, we provide a $t$ test of differences in means of the relevant impulse responses. Specifically, we are testing the null hypothesis that the impulse response of each monetary model on average is statistically different from the impulse response of the standard Taylor rule without money and leverage. Hence, this test assists us in examining the assertion that the mean of an impulse response from the augmented model is not equal to the mean of an impulse response of the standard Taylor rule.

Figure 8. Comparison of the median output and inflation responses to a contractionary monetary policy shock.

Both Figure 8 and Table 2 provide evidence that empirical models measuring the macroeconomic effects of central bank policy should capture a measure of money and leverage. Our evidence is supportive of Leeper and Roush (Reference Leeper and Roush2003), and Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2016, Reference Belongia and Ireland2018) who have argued for or provided evidence supporting the inclusion of monetary aggregates in interest rate-based monetary policy rules.

5.2. Variance decomposition

Another key area of interest is the examination of the proportion of the variation in a variable of interest that can be explained by a particular identified shock. In Table 3, we present the estimated contribution of each structural shock (supply shocks (SS), demand shocks (DS), and monetary policy shocks (MPS)) to the $4$ - and $20$ -quarters ahead median squared forecast error of each variable of interest (output gap, the inflation rate, and the federal funds rate) in bold and also expressed as a percent of the total MSE in squared brackets. We also provide the 95% credibility intervals in parentheses. Our format for reporting the variance decomposition follows closely the format of Baumeister and Hamilton (Reference Baumeister and Hamilton2018), since our paper is related to theirs and we intend to report the results in a comparable manner.

Table 2. Test of significant difference between impulse responses of the various models. This is a t test testing the null hypothesis of the difference between the impulse response of any two models is on average indistinguishable

Table 3. Variance decomposition of the monetary and nonmonetary models. The estimated contribution of each structural shock to the 4- and 20-quarter-ahead median squared forecast error of each variable in bold and expressed as a percent of total MSE in brackets. 95% credibility intervals are provided in parentheses. The top panel (Standard Taylor rule) is the Baumeister and Hamilton (Reference Baumeister and Hamilton2018) type model without money and leverage. Divisia M3 and Sum M2 are the monetary models with Divisia M3 and Sum M2, respectively, entering the policy function. The measure of leverage is commercial bank leverage

As can be seen in Table 3, the model with the Divisia M3 aggregate and commercial bank leverage allows monetary policy shocks to explain thrice (6%) as much of the variation in output compared to the standard Taylor rule which would only ascribe 2% of the variation in output to monetary policy shocks after 1 year. In fact, 5 years after the shock, monetary policy shocks identified with Divisia M3 and commercial bank leverage account for 8% of the variation in output compared to 5% in the case of the standard Taylor rule. It should be noted that there are overlaps in the 95% credibility intervals among the competing models. Generally, the models with money and leverage indicate that monetary policy shocks explain a higher proportion of macroeconomic variations. Also, the model with the Divisia M3 monetary aggregate marginally improves the explanatory power of monetary policy shocks for both output and inflation variations compared to the models with Sum M2. Generally, a model augmented with any measure of money (Divisia or Sum M2) and leverage explains a slightly higher variation in output and inflation than the standard Taylor rule without money and leverage.

Monetary policy shocks when the Divisia M3 monetary aggregate and leverage are included in the policy rule are almost as important as supply shocks are in explaining the mean-squared error of the 5-year ahead forecast of output. While demand shocks and supply shocks dominate monetary policy shocks in explaining the variations in output and inflation, the explanatory power of monetary policy shocks generally improves if the monetary authority policy function is augmented with a monetary aggregate and leverage. Monetary policy shocks also explain a slightly higher percentage of the variation in the fed funds rate when the model is augmented with money and leverage. We conclude that a model with money is preferred to a model without money in terms of its ability to explain key macroeconomic variations. We provide evidence in support of Belongia and Ireland (Reference Belongia and Ireland2015, p. 268) who “call into question the conventional view that the stance of monetary policy can be described with exclusive reference to its effects on interest rates and without consideration of simultaneous movements in the monetary aggregates” and Chen and Valcarcel (Reference Chen and Valcarcel2021) who advocate for Divisia monetary aggregates as policy indicators (not necessarily as instruments) in order to reconcile empirical results with macroeconomic theoretical predictions.

We note that our sample period (1986:q1 to 2021:q1) represents a period during which the Fed de-emphasized money growth targets and has gradually and finally discontinued reporting target ranges for all other monetary aggregates since 2000. However, as Chen and Valcarcel (Reference Chen and Valcarcel2021, pp. 2) recently put it, “the 2007 Financial Crisis and the following protracted effective-lower-bound (ELB) period highlighted some shortcomings of the information content that the federal funds rate alone provides about monetary transmission.” As such, we have considered and shown how an augmented policy rule can capture the behavior of the central bank relative to the standard Taylor rule. Also with the emergence of new policy tools (unconventional tools such as quantitative easing/tightening), it is possible to discount any apparent shortcomings of interest rate targeting. However, as Belongia and Ireland (Reference Belongia and Ireland2015, p. 255) recently put it, “the new policy initiatives can be characterized simply as conventional attempts to increase money growth,” and it is thus worth exploring augmented Taylor rules for monetary and business cycle analysis.