2.1. Broad Divisia measures of money versus narrow Divisia and simple-sum measures of money

Conventional (simple-sum) monetary aggregates implicitly assume that different monetary components are perfect substitutes by adding them up when aggregating. This is inconsistent with empirical estimates of elasticities of substitution from demand models for the underlying components (e.g., Jones et al. Reference Jones, Dutkowsky, Elger and Fleissig2008a, b; Fleissig and Jones, Reference Fleissig and Jones2015; Jadidzadeh and Serletis, Reference Jadidzadeh and Serletis2019; Xu and Serletis, Reference Xu and Serletis2022). Divisia monetary measures do not make implicit assumptions about substitution elasticities between the monetary assets that they include. However, the oft stated ability of Divisia aggregates to internalize pure substitution effects between its components does require weak separability without which demand for the aggregate can become unstable (see Barnett, Reference Barnett1982). Jones et al. (Reference Jones, Dutkowsky and Elger2005) found that separability tests favored broad over narrow aggregates when the monetary assets were adjusted for retail sweep accounts but not for commercial sweep accounts. Jadidzadeh and Serletis (Reference Jadidzadeh and Serletis2019) tested various separability structures within the Divisia M4 monetary aggregate published by the Center for Financial Stability (CFS), but rejected all of them, leading them to favor the broadest available monetary aggregate. Hjertstrand et al. (Reference Hjertstrand, Swofford and Whitney2016) found that M1 and a broad aggregate similar to M4 were weakly separable from consumption and leisure, but the evidence for M2 and M3 was weaker. We focus on Divisia M3, noting that movements in it are very similar to CFS’s Divisia M4- and Divisia M4.Footnote ⁵

The demand for Divisia money can be affected by changes in the liquidity of assets outside of it. This limitation suggests that broader Divisia indexes (e.g., M3) may be preferable to Divisia M2, because they are better able to internalize shifts between liquidity provided by insured deposits in M2 vis-à-vis uninsured liabilities issued by banks (e.g., large time deposits), institutional money funds, and repurchase agreements (RPs) outside of M2.Footnote ⁶ While the velocity of broad Divisia money may be affected by shifts in the liquidity of assets not spanned by them, the factors driving those changes are less numerous than for Divisia M2 and are feasible to track for Divisia M3 (Bordo and Duca, Reference Bordo and Duca2025). Indeed, consumption velocity (nominal PCE divided by money) is more mean reverting for Divisia M3 than for simple sum or Divisia M2 over the past four decades (Figure 3). As shown in Figure 3, Divisia M3 velocity does not exhibit any clear discernable trend.

Figure 3. Since the mid-1980s, the consumption velocity of broader Divisia money (M3) is more stable than that of simple-sum and Divisia M2 in the U.S.

Sources: CFS, Federal Reserve, and authors’ calculations. Shaded areas are NBER recessions.

2.2. Pre-pandemic long-run demand for Divisia monetary services

The long-run demand for money depends on overall transactions, as well as factors altering the relative liquidity of money versus that of other assets. We now provide an overview of the determinants of money demand that we include in our error-correction model of velocity. Our focus on stock loads and CFMA is very similar to Bordo and Duca (Reference Bordo and Duca2025), although we use consumption as the scale variable rather than GDP and consider several opportunity cost variables.

2.2.3. Opportunity cost of money

Monetary aggregation theory builds upon the user cost of monetary assets; see Barnett (Reference Barnett1978) and Donovan (Reference Donovan1978). From the user cost formula, the opportunity cost of the monetary services derived from holding a particular asset corresponds to the foregone interest from holding that asset relative to a non-monetary “benchmark” asset. The growth rate of a Divisia index is the average expenditure share weighted sum of the growth rates of its component monetary assets, where the expenditure on each monetary asset depends on its user cost. Thus, monetary asset user costs are a key feature of how Divisia aggregates are computed. In addition, money demand models of Divisia aggregates often feature an opportunity cost variable. Some authors, such as Belongia and Ireland (Reference Belongia and Ireland2016, Reference Belongia and Ireland2018, Reference Belongia and Ireland2019), emphasize the use of aggregation-theoretic dual user cost measures, such as those produced by CFS, to model Divisia money demand.Footnote ⁸

Others, however, use a standard interest rate as an opportunity cost variable. Barnett et al. (Reference Barnett, Ghosh and Adil2022), for example, estimate cointegration relationships between money, output, interest rates, and real effective exchange rates for several countries including the U.S. They note that short-term interest rates are typically used to model narrow money measures, but long-term interest rates are typically used for broad money, such as Divisia M3 explaining that “[t]he conventional rationale for this choice is that many components of broad money, such as time deposits, are associated with long-term interest rates.” Following this reasoning, we assess both the Divisia M3 dual user cost index (Usercost) and the ten-year Treasury yield (Bond) as additional variables in our models.Footnote ⁹ As shown in Figure 4, the Usercost series is more variable and mean reverting than the slower moving stock load series, which experiences substantial declines in the 1980s and 1990s.

Figure 4. Stock fund loads and the CFS measure of Divisia M3 user cost.

Sources: CFS and Bordo and Duca (Reference Bordo and Duca2025).

The zero lower bound on short-term interest rates binds in much of the latter part of our sample (2008–2015 and 2020–2021), which by compressing user costs to a low positive range, could complicate the interpretation of dual user costs over those periods. Mattson and Valcarcel (Reference Mattson and Valcarcel2016) show that the fall of the federal funds rate “likely passed through to the user costs of many of the assets” in Divisia M4 resulting in “unprecedented compression in user costs” and to convergence between Divisia and simple-sum M4 starting in 2009. Nevertheless, the dual user cost does not hit the zero lower bound for any of the CFS Divisia measures (Chen and Valcarcel, Reference Chen and Valcarcel2024).Footnote ¹⁰

2.3. Modelling the consumption velocity of Divisia M3

Our analysis focuses on the consumption velocity of CFS’s Divisia M3 aggregate, which measures the monetary services from the components of the Federal Reserve’s simple-sum M2 aggregate as well as from large time deposits, institutional money market mutual funds, and repurchase agreements (RPs). The Federal Reserve’s simple sum M3 aggregate, which was published through early 2006, added institutional money funds, total large time deposits, total repurchase agreements and total Eurodollar deposits to M2; making its components roughly comparable, but not identical, to CFS’s Divisia M3 measure. As Bordo and Duca (Reference Bordo and Duca2025) stress, Divisia M3 helps internalize substitution between the liquidity provided by M2 and non-M2 liabilities, where the latter help track shifts in the role of liquidity provided by instruments funding the shadow banking sector. Of the three broad Divisia measures (M3, M4-, and M4) constructed by CFS, Bordo and Duca (Reference Bordo and Duca2025) were slightly better able to track the GDP velocity of Divisia M3.

The preceding discussion implies that the long-run demand for the log of real Divisia M3 ( $\textit{LDM}$ ), takes the form:

(1) \begin{equation} \textit{LDM}^{*}=\gamma _{0}+\gamma _{1}\textit{CFMA}+\gamma _{2}\textit{LSLoad}+\gamma _{3}\textit{LPCE}+\gamma _{4}\textit{LOC}\end{equation}

where $\textit{CFMA}$ is a level shift dummy equal to one following passage of CFMA and zero for earlier periods. The remaining variables are the logs of the stock load variable ( $\textit{LSLoad}$ ), real consumption ( $\textit{LPCE}$ ), and the opportunity cost ( $\textit{LOC}$ ), which is either the long-term bond yield or the dual user cost index. For consistency, Divisia M3 should be converted to real terms using the PCE deflator. As discussed above, CFMA was associated with increased money holdings implying that $\gamma _{1}\gt 0$ . Higher transfer costs should increase money demand implying that $\gamma _{2}\gt 0$ by reducing the liquidity of assets outside of Divisia M3 and higher opportunity costs are expected to reduce money demand implying $\gamma _{4}\lt 0$ . Finally, the elasticity of money demand with respect to consumption, $\gamma _{3}$ , should be positive. For later reference, omitting the opportunity cost variable results in the simplified expression:

(1′) \begin{equation}\textit{LDM}^{*}=\gamma _{0}+\gamma _{1}\textit{CFMA}+\gamma _{2}\textit{LSLoad}+\gamma _{3}\textit{LPCE} \end{equation}

If the consumption elasticity of long-run Divisia money demand is unity, $\gamma _{3}=1$ , (as Bordo and Duca (Reference Bordo and Duca2025) impose for GDP velocity), then eqs. (1) and (1’) can be rearranged into corresponding models of long-run consumption velocity (in logs):

(2) \begin{equation}\textit{LV}^{*}=\theta _{0}+\theta _{1}\textit{CFMA}+\theta _{2}\textit{LSLoad}+\theta _{3}\textit{LOC}\end{equation}

and

(2’) \begin{equation}\textit{LV}^{*}=\theta _{0}+\theta _{1}\textit{CFMA}+\theta _{2}\textit{LSLoad}\end{equation}

respectively, since $\textit{LV}=\textit{LPCE}-\textit{LDM}$ . Assuming that CFMA bolstered Divisia M3, as argued earlier, then it would have correspondingly lowered its velocity implying that $\theta _{1}\lt 0$ . Previous arguments likewise imply that $\theta _{2}\lt 0$ (higher transfer costs raise money demand and hence lower velocity) and $\theta _{3}\gt 0$ (higher opportunity costs lower money demand and hence increase velocity).

We posit that velocity (real money) adjusts gradually to the deviation of it from its long-run equilibrium, $\textit{LV}_{t}-\textit{LV}_{t}^{*}$ ( $\textit{LDM}_{t}-\textit{LDM}_{t}^{*}$ ), which gives rise to an error-correction model. However, including the CFMA level shift dummy in the long-run relationship must be properly incorporated into the estimation, and there can be transitional short-run dynamics immediately following passage of the Act. To handle these issues, we estimate broken constant models from Johansen et al. (Reference Johansen, Rosconi and Nielson2000). These models generalize the standard cointegration models (see, e.g., Johansen, Reference Johansen1996) to allow for structural breaks in the deterministic components where the breakpoints are known. Since we are concerned with breaks associated with the adoption of the CFMA in our analysis, this is an attractive framework for our empirical design.

2.4. Technical details

The standard VECM framework consists of a model in the form:

(3) \begin{equation}{\Delta} X_{t}=\Pi X_{t-1}+\sum\nolimits _{i=1}^{k-1}{\Gamma} _{i}{\Delta} X_{t-i}+\Pi _{1}\mathrm{t}+\mu +\varepsilon _{t}\end{equation}

where $k$ is the number of lags of the underlying VAR model (in levels) and the vector X includes LV, LSload, and LOC. Let $p$ be the number of variables in the VAR. Table 1 reports unit root tests for each of the variables over our sample period (1985–2022). These tests indicate that LV and LSLoad are I(1), but that both LBond and LUsercost may be I(0). While the VECM model can include stationary variables, doing so would increase the rank of the system. The table also includes tests for real Divisia M3 and real PCE (both in logs), which are used in Appendix 1.

Table 1. Unit root tests

Notes: ⁺, ^* and ^** denote 90%, 95% and 99% significance. Lag length or bandwidth statistics are in parentheses. Tests for SupPress begin in 1998 when the series starts. Lag lengths for ADF tests selected using the SIC. Phillips–Perron tests use the quadratic spectral kernel with an Andrews Bandwidth. All tests include an intercept and a time trend.

Under cointegration, $\Pi$ is of reduced rank and is usually written as $\Pi =\alpha \beta '$ , with $\alpha ,\beta$ both being $p$ by $r$ . In that case, $\beta 'X_{t-1}$ represents the ( $r$ ) cointegration relations and $\alpha$ represents the corresponding impact coefficients of those relations on the first differences of each of the system variables in the VAR. $\Pi _{1}$ is typically imposed to be of the form $\Pi _{1}=\alpha\gamma '$ , so that the model does not have a quadratic trend in levels (see Johansen et al. Reference Johansen, Rosconi and Nielson2000). Consequently, with the reduced rank condition imposed, the cointegration relations would be trend stationary, since $\Pi X_{t-1}+\Pi _{1}t=\alpha (\beta 'X_{t-1}+\gamma 't)$ . In applications, $\gamma =0$ can be further imposed, which eliminates the trend from the cointegration space, but not from the levels. Assuming $\gamma =0$ , if it is further imposed that $\mu =\alpha \rho '$ then the constant is restricted to the cointegrating relationship, and the model will not have a linear trend in any direction (Johansen et al. Reference Johansen, Rosconi and Nielson2000, p. 218).

Johansen et al. (Reference Johansen, Rosconi and Nielson2000) generalize this framework by introducing indicator variables for different sample periods:

(4) \begin{equation}E_{j,t}=\begin{cases} 1 & \text{for }T_{j-1}+1+k\leq t\leq T_{j}\\ 0 & \text{o.w.}\, \end{cases}\end{equation}

where $j=1,\ldots ,q$ . Setting $T_{0}=0$ for convenience, the effective sample is $t=1+k,\ldots ,T_{q}=T$ with $X_{1},\ldots ,X_{k}$ being used as initial observations and there are $q$ subsamples with effective sample periods running from $1+k$ to $T_{1}$ , $T_{1}+1+k$ to $T_{2}$ , and so on. Following Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 219), we define the vector of sub-sample indicator variables as $E_{t}=(E_{1,t},E_{2,t},\ldots ,E_{q,t})'$ .

Using this notation, Johansen et al. (Reference Johansen, Rosconi and Nielson2000) proposes several models that allow for breaks in the deterministic components. We impose $\gamma =0$ , which eliminates the possibility of trends entering the long-run relationship and we focus on the broken constant specification from Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 225). That specification is as follows:

(5) \begin{equation}{\Delta} X_{t}=\left(\Pi ,{\unicode{x03BC}} \right)\left(\begin{array}{c} X_{t-1}\\ E_{t} \end{array}\right)+\sum\nolimits _{i=1}^{k-1}{\Gamma} _{i}{\Delta} X_{t-i}+\sum\nolimits _{i=1}^{k}\sum\nolimits _{j=2}^{q}\kappa _{j,i}D_{j,t-i}+\varepsilon _{t}\end{equation}

where $D_{j,t-i}=1$ if $t=T_{j-1}+i$ (and zero otherwise) and $\mu =(\mu _{1},\ldots ,\mu _{q})$ is a $p$ by $q$ matrix. The term, $\mu E_{t}$ , allows for the constant in the VAR to differ across subsamples. As Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 220) explain, the model dummies out the $k$ initial observations following each breakpoint, $X_{{T_{1}}+1},\ldots ,X_{{T_{1}}+k}$ , $X_{{T_{2}}+1},\ldots ,X_{{T_{2}}+k},\ldots$ rendering “the corresponding residuals zero thereby essentially eliminating the corresponding factors from the likelihood function, and hence producing the conditional likelihood function given the initial values in each period.” The relevant reduced rank cointegration hypothesis (Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 225) is

(6) \begin{equation}H_{c}(r)\,\colon \, \textit{rank}\, \left(\Pi ,\mu \right)\leq r\end{equation}

so that $(\Pi ,\mu )=\alpha (\beta ',\rho ')$ , where $\alpha ,\beta$ are both $p$ by $r$ matrices and $\rho$ is $q$ by $r$ . With the reduced rank condition imposed, $\Pi X_{t-1}+{\unicode{x03BC}} E_{t}=\alpha (\beta 'X_{t-1}+\rho 'E_{t}),$ corresponding to a set of $r$ cointegration relations with different constants in each of the $q$ subsamples. Within the same specification, Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 219) also consider an alternative reduced rank hypothesis:

(7) \begin{equation}H_{\ell c}(r)\,\colon \, rank\, \Pi \leq r\end{equation}

Johansen et al. (Reference Johansen, Rosconi and Nielson2000, pp. 218–220) consider the following specification as well:

(8) \begin{equation}{\Delta} X_{t}=\left(\Pi ,\Pi _{1},\ldots ,\Pi _{q}\right)\left(\begin{array}{c} X_{t-1}\\ tE_{t} \end{array}\right)+\sum\nolimits _{i=1}^{k-1}{\Gamma} _{i}{\Delta} X_{t-i}+\sum\nolimits _{i=1}^{k}\sum\nolimits _{j=2}^{q}\kappa _{j,i}D_{j,t-i}+\mu E_{t}+\varepsilon _{t}\end{equation}

For this specification, the reduced rank cointegration hypothesis is

(9) \begin{equation}H_{\ell}(r)\,\colon \, rank\, \left(\Pi ,\Pi _{1},\ldots ,\Pi _{q}\right)\leq r\end{equation}

so that

(10) \begin{equation}\left(\Pi ,\Pi _{1},\ldots ,\Pi _{q}\right)=\alpha \left(\begin{array}{c} \beta \\ \gamma _{1}\\ \begin{array}{c} \vdots \\ \gamma _{q} \end{array} \end{array}\right)'\end{equation}

where $\gamma _{j}$ is 1 by $r$ for all $j$ . Imposing the reduced rank condition, the cointegration relations have broken trends denoted by $\gamma _{j}'t$ in each subsample $j$ . We do not consider this specification further in this paper.

Johansen et al. (Reference Johansen, Rosconi and Nielson2000) derive likelihood ratio tests for $H_{c}(r)$ and $H_{\ell c}(r)$ against $H_{\ell c}(p)=H_{c}(p)$ and for $H_{\ell }(r)$ against $H_{\ell }(p)$ , which are in the usual trace test form: i.e. $\textit{LR}\{H_{c}(r)|H_{c}(p)=-T\sum\nolimits_{i=r+1}^{p}\ln (1-\hat{\lambda }_{i})$ . Corresponding tables of critical values for $H_{c}(r)$ and $H_{\ell }(r)$ can be found in Giles and Godwin (Reference Giles and Godwin2012). The critical values depend on the relative breakpoints, $v_{j}=T_{j}/T$ , where $T_{j}$ is the last observation of the jth subsample and $T$ is the full sample size. Johansen et al. (Reference Johansen, Rosconi and Nielson2000, p. 219) find the $H_{\ell c}(r)$ hypothesis to be “less attractive” than the other two hypotheses, however, in part due to their finding that “the asymptotic analysis is heavily burdened with nuisance parameters” in that case. Consequently, we focus on testing the $H_{c}(r)$ hypothesis.

In our application, the only structural break we consider is associated with the adoption of CFMA. To align with our notation, we set the number of sub-samples, $q$ , equal to 2 and set $T_{1}$ to be 2000Q3 corresponding to the end of the pre-CFMA period, since CFMA passed in 2000Q4. As in eq. (4), $E_{2,t}$ is an indicator variable that is one from $T_{1}+1+k$ through the end of the sample, where $k$ is the number of lags in the VAR, and zero for earlier periods. The inclusion of the $D_{j,t-i}$ dummies in eq. (5) implies that we do not need to model the transitional impacts of CFMA and that $E_{2,t}$ is, therefore, effectively an indicator for the post-CFMA period. We follow Johansen et al. (Reference Johansen, Rosconi and Nielson2000) as laid out above with one exception. We set $E_{1,t}=1$ over the entire sample rather than over just the sub-sample covering the pre-CFMA period. This allows us to present the results in terms of a constant for the full sample and a shift in the constant for the post-CFMA period as in eq. (2). This minor change does not affect the estimation of any other parameters nor does it affect the test procedures employed. We do this to simplify exposition regarding the effect of CFMA.

Estimating eq. (5) and setting $r=1$ yields coefficients on the X vector components for a single long-run relationship, $\beta 'X_{t-1}$ , with constant term, $\rho 'E_{t}$ . The corresponding impact coefficients, $\alpha$ , determine how the endogenous variables each adjust to the long-run relationship. Since we focus on velocity, the impact of the long-run relationship on the stock load, the long-term Treasury rate, or the user cost are not of particular interest and are not reported. Similarly, we only report first-order short-run dynamics for the change in velocity, as captured by the corresponding row of $\Gamma _{1}$ (full results are available upon request). By normalizing the $\beta$ vector on the element corresponding to log velocity, $\beta 'X_{t-1}$ can be represented in the following form:

(11) \begin{equation}\textit{LV}_{t-1}+\beta _{2}\textit{LSLoad}_{t-1}+\beta _{3}\textit{LOC}_{t-1}\end{equation}

Ignoring the constants in the cointegration, we can define long-run equilibrium velocity as

\begin{equation*} \textit{LV}_{t-1}^{*}\equiv -\beta _{2}\textit{LSLoad}_{t-1}-\beta _{3}\textit{LOC}_{t-1}=\theta _{2}\textit{LSLoad}_{t-1}+\theta _{3}\textit{LOC}_{t-1} \end{equation*}

so that the corresponding error-correction term is as follows:

(12) \begin{equation}\textit{EC}_{t-1}\equiv \textit{LV}_{t-1}-\textit{LV}_{t-1}^{*}=\textit{LV}_{t-1}+\beta _{2}\textit{LSLoad}_{t-1}+\beta _{3}\textit{LOC}_{t-1}\end{equation}

which are the deviations from long-run equilibrium. Incorporating the constants in the cointegration relations results in

(13) \begin{equation}\textit{LV}^{*}=\theta _{0}+\theta _{1}\textit{CFMA}+\theta _{2}\textit{LSLoad}+\theta _{3}\textit{LOC}\end{equation}

as in eq. (2).Footnote ¹¹ As discussed previously, theoretical priors imply $\theta _{1},\, \theta _{2}\lt 0$ and $\theta _{3}\gt 0$ .

Our model of consumption velocity builds on Bordo and Duca (Reference Bordo and Duca2025) who model the GDP velocity of broad Divisia but differs in three important respects: First, we use PCE to track the aggregate transactions instead of GDP as discussed above. Second, we adopt the technically more correct approach of estimating the broken constant specification rather than treating CFMA as a regular variable within the VECM as in Bordo and Duca (Reference Bordo and Duca2025). The trace test for the broken constant model can be compared to tables of critical values and are interpreted in the usual way as described above. Third, Bordo and Duca (Reference Bordo and Duca2025) model the impact of COVID-19 with a continuous variable that they construct from an index of government mobility restrictions and vaccination rates. For the sample that includes post-2019 observations, we instead just include simple impact dummies for each of the quarters of 2020, which are sufficient to produce stable long-run relationships over our full sample. Modelling velocity in terms of PCE is more attractive than GDP, because the Fed’s price stability goal is in terms of PCE inflation. We later incorporate estimates from our model of consumption velocity into a P-Star model of PCE inflation.

3.1. Sample periods

We estimate our consumption velocity model over four sample periods that all start in 1985Q3 but have different end-of-sample periods. Earlier observations are used as initial starting values in the models. We generally find clean residuals for our estimated models with lag lengths of 7 quarters or less as described below. For 7 lags, the 1985Q3 start-of-sample implies that eq. (5) is estimated using only data (in differences) going back to 1984Q1. The starting point is chosen to avoid using earlier data that could be affected by deposit deregulation and financial innovations that led to the introduction of new types of monetary assets.

During the 1970s and early 1980s, there were shifts away from savings and time deposits with binding deposit interest rate ceilings (whose measured user costs were high, because Regulation Q capped deposit interest rates below market levels), to money market mutual funds (MMMFs), and later to money market deposit accounts (MMDAs). Super NOW accounts and MMDAs were both introduced beginning in late 1982. Super NOWs, MMDAs, and MMMFs all paid higher interest rates than regulated deposits and hence had user costs that were relatively low and MMDAs and MMMFs also had checking features unlike traditional savings and small-time deposits. In light of this, the shifts out of regulated deposits would be weighted by a Divisia index using expenditure shares based on relatively high user costs. In contrast, the movement of funds into the assets paying higher rates would receive expenditure share weights based on relatively low user costs. As a result, these shifts would be associated with a distorted decline in measured Divisia (but not simple-sum) money (see Bordo and Duca, Reference Bordo and Duca2025, for a discussion). Indeed, over this period, there was a corresponding upward level shift in Divisia velocity in the late-1970s and early 1980s. Chen and Valcarcel (Reference Chen and Valcarcel2024) find structural breaks in Divisia M2 and Divisia M3 around 1980 and subsequent breaks in 2012 and 1988 respectively. They also find structural breaks in the corresponding user cost measures in 2008, and that “[o]verall, these statistical tests show a preponderance of evidence for a structural break in 1980.”Footnote ¹²

The earliest end of sample period we consider is 2005Q4 (Model 1), which is just before the bursting of the subprime mortgage bubble and the GFC. Model 2 ends in 2012Q4 to allow ample time for the adjustment of money to the staggered implementation of new financial regulations under the Dodd-Frank Act, through which the U.S. implemented the Basel III reforms. Model 3 ends in 2019Q4, to span the pre-COVID period, while Model 4 uses a full sample ending in 2022Q4. The full sample is based on the availability of the underlying data used to construct the stock load variable. The full sample also differs in that we include dummies for each quarter in 2020 to control for the unprecedented impact of the pandemic during its acute initial phase.

3.2. Estimation results

Our specification of long-run equilibrium consumption velocity is spelled out in eq. (13) with the corresponding error-correction term in eq. (12) which enters into the short-run dynamics of the model. We estimate three versions of the model that differ in terms of opportunity costs. We first assess models that include stock fund loads and the long-term bond yield. We then assess models including stock fund loads and the Divisia M3 user cost measure from the CFS. Finally, we present models that just include stock mutual fund loads. In our estimations, we select lag lengths using two criteria: First, we estimate a simple unrestricted VAR in levels and employ LM tests for serial correlation (up to four quarters) for lag lengths from 2 to 8. From models having clean residuals, we select the lag length that minimizes the Schwartz Information Criterion (SIC).

Table 2. Quarterly models of U.S. consumption velocity including long treasury yields.

Notes: Significance is determined using critical values for the trace test from Giles and Godwin (Reference Giles and Godwin2012) for various values of $v_{1}$ as described in footnote 13. +, *, and ** denote 90%, 95%, and 99% significance respectively. $\dagger$ indicates significance at the 90% level for the lower bracketing value only. Long-term coefficient estimates are for $r=1$ . Absolute t-statistics are in parentheses. For the short-run model, all coefficients are for the velocity equation. ${\Gamma} _{1}(\mathrm{i},\mathrm{j})$ is the $i,j$ element of ${\Gamma} _{1}$ in eq. (5). Short run coefficients corresponding to higher order lags ( ${\Gamma} _{2},\, \ldots {\Gamma} _{k-1}$ ) are omitted to save space.

3.2.3. Estimation results excluding the opportunity cost variables

Building on our previous results, we now estimate more parsimonious models that include just the stock load variable as an explanatory factor while accounting for CFMA as in Bordo and Duca (Reference Bordo and Duca2025), although we estimate our models differently as described above. In these models, $p=2$ . As reported in Table 3, for Models 1 and 2, which use the pre-GFC (1985Q3 to 2005Q4) and the 1985Q3 to 2012Q4 subsamples respectively, we find a significant cointegrating vector ( $r=0$ is rejected at the 95% significance level or higher and $r=1$ cannot be rejected at the 90% significance level). For Model 3, which uses the pre-pandemic subsample (1985Q3 to 2019Q4), $r=0$ is rejected at the 99% significance level, but $r=1$ is rejected at the 90% significance level, so there is some ambiguity about whether the system is cointegrated or just stationary for that sample. Setting $r=1$ and normalizing on velocity yields similar coefficient estimates across all three sub-samples in Models 1–3, with coefficients on the load variable near −0.3 and those on CFMA near −0.1. The estimates for these sample periods are all based on VARs with 6 lags. We also estimated VARs with 4 and 8 lags and found similar coefficient estimates.Footnote ¹⁸

Table 3. Quarterly models of U.S. consumption velocity excluding long treasury yields.

Notes: Significance is determined using critical values for the trace test from Giles and Godwin (Reference Giles and Godwin2012) for various values of $v_{1}$ as described in footnote 13. +, *, and ** denote 90%, 95%, and 99% significance respectively. Long-term coefficient estimates are for $r=1$ . Absolute t-statistics are in parentheses. For the short-run model, all coefficients are for the velocity equation. ${\Gamma} _{1}(\mathrm{i},\mathrm{j})$ is the $i,j$ element of ${\Gamma} _{1}$ in eq. (5). Short run coefficients corresponding to higher order lags ( ${\Gamma} _{2},\, \ldots {\Gamma} _{k-1}$ ) are omitted to save space.

The estimated impact coefficient on the error-correction term for log velocity is −0.071 with a t-stat of 2.56 for the pre-GFC subsample indicating that velocity significantly error-corrects. The corresponding coefficients are around −0.12 with t-stats above 4 for the other two subsamples indicating faster adjustment of velocity. The model also performs well out-of-sample insofar as the error-correction terms based on the subsample ending in 2012Q4 are very similar to the ones estimated over the pre-pandemic sample period from 2013 on. This is an improvement vis-à-vis the models that included the long-term rate. When we included the long-term bond rate, we found that the error-correction terms based on the estimates from the subsample ending in 2012Q4 were higher than the ones estimated over the pre-pandemic sample period from 2013 on. Turning to the full sample (Model 4), we identify a significant cointegrating vector (i.e., $r=1)$ with sensible coefficients on LSload (−0.320) and CFMA (−0.091) and the impact coefficient on the error-correction term for log velocity (−0.09) is reasonable as well.

To provide more perspective on our results, Figure 5 plots the estimated long-run equilibrium consumption velocities from samples ending in 2012Q4 (Model 2) and 2022Q4 (Model 4). These line up well with each other and with long-run movements in log velocity. The slight lead of the long-run equilibrium relative to actual log velocity reflects the importance of accounting for partial adjustment toward long-run equilibrium for tracking velocity in the short-run and buttresses the credibility of the error-correction models. As explained above, velocity fell initially during the onset of the pandemic when the money supply surged.

Figure 5. Estimated equilibrium consumption velocity trends with long-run velocity.

Sources: CFS, BEA, and author’s calculations.

Overall, including stock loads without either the bond yield or the dual user cost measure works well for modeling consumption velocity. Our model also performs well if we estimate our specification allowing for an unrestricted broken constant (corresponding to $H_{\ell c}(1)\,$ rather than $H_{c}(1)$ ). Furthermore, we found that estimating real Divisia M3 money demand versions of our model yielded similar results when imposing a single cointegrating vector. These additional results are presented in Appendix 1, parts A and B respectively. Accordingly, we adopt our parsimonious model of long-run equilibrium velocity for our P-Star models of inflation.Footnote ¹⁹

3.2.4. Broader levels of aggregation

Finally, we briefly consider the simple velocity model using the broader Divisia M4- and M4 aggregates, both of which include commercial paper. We estimated Models 1–3 excluding opportunity cost variables using both of these aggregates for lag lengths of 4, 6, and 8 similar to Divisia M3 as discussed above. For M4, we found generally similar coefficient estimates for the pre-GFC sample period (Model 1) and the sample period ending in 2012Q4 (Model 2) across the various lag lengths. For these two sample periods, the estimated coefficients on the load variable ranged from −0.25 to −0.21 and the estimated coefficients on CFMA ranged from −0.09 to −0.075 (with t-stats above 6.0 in all cases) consistent with our theoretical priors. The estimated coefficients differed for the pre-pandemic sample period (Model 3). At 4 lags, we found that the coefficient on the load variable was −0.27 (t-stat. of 5.93) and the coefficient on CFMA was −0.054 (with a t-stat of 2.52). At 6 and 8 lags, the coefficients on the load variable were around −0.3 with t-stats above 5.75 and the coefficients on CFMA were −0.038 and −0.021 (with t-stats of 1.49 and 0.97) respectively.Footnote ²⁰ Thus, there is evidence of parameter instability especially regarding the impact of CFMA between the sample period ending in 2012Q4 and the pre-pandemic sample period.

For M4-, there is additional evidence of parameter instability. For the pre-GFC sample period (Model 1), the coefficient estimates at 6 and 8 lags are around −0.28 for the load variable and around −0.08 for CFMA. At 4 lags, the corresponding estimates are −0.153 for the load variable and −0.103 for CFMA. For Model 2, however, the coefficient estimates ranged from −0.322 to −0.275 for the load variable and from −0.056 to −0.03 for CFMA. At 6 lags, the coefficient estimate on CFMA (−0.03) had a t-stat of just 0.92. For the pre-pandemic sample period (Model 3), the estimated coefficients on the load variable were around −0.37, but the coefficients on CFMA were wrongly signed (estimates ranged from 0.002 to 0.024), although the t-stats were below 0.7 for all three lag lengths.Footnote ²¹

Thus, for Divisia M4, and even more so for Divisia M4-, there is some evidence of parameter instability particularly with respect to the impact of CFMA on long-run velocity vis-à-vis Models 2 and 3. Throughout the remainder of the paper, our focus will be on prediction via the P-Star model and the relative stability of the parameter estimates for Divisia M3 is an attractive feature in that respect. Consequently, we will focus on Divisia M3 for subsequent analysis.

A plausible explanation for the difficulty in modeling the demand for Divisia M4- and Divisia M4 is that they include the monetary services from assets that have not been transformed by intermediaries into broadly accepted means of payment. As described earlier, Divisia M4- augments Divisia M3 by including commercial paper. When scaled by trend potential nominal GDP, commercial paper holdings shifted up with the rise of asset-backed commercial paper from the mid-1990s to the mid-2000s (Figure 6).Footnote ²² This occurred when shadow banks made greater use of leverage in funding their assets by issuing commercial paper much, but not all, of which was held by money funds. This rise unwound sharply starting with the onset of the Global Financial Crisis and amplified after Lehman Brothers failed. Even though the opportunity cost of commercial paper is small, the large swings in asset holdings could have loosened the overall link between Divisia M4- and nominal spending. In addition to including commercial paper, M4 also includes Treasury bills held outside of banks and money funds, which may further affect the ability to model the velocity of M4.

Figure 6. Commercial paper not held by money funds scaled by potential nominal GDP.

Sources: Financial Accounts of the U.S., CBO, and authors’ calculations.

4.2. Modeling overall and core PCE inflation

We use the framework in eq. (22) to model overall and core PCE inflation: ΔLPPCE and ΔLPcorePCE respectively. We use the estimated coefficients from Model 2 in Table 3 corresponding to the 1985Q3-2012Q4 sample period, to calculate $v^{*}$ . Using these coefficients, we calculate a static forecast of $v^{*}$ out-of-sample from 2013 until the end of our sample period and the corresponding velocity gap term, $v^{*}-v$ (vgap). As previously noted, we extend the out-of-sample period to 2024Q3. The in-sample period ends in 2012Q4, which reflects two factors. First, as Figure 5 shows, this provides ample time for both equilibrium and actual velocity to recover from the large negative shock associated with the subprime and global financial crisis (GFC) of 2007–11. Moreover, the regulatory response to the GFC took the form of the Dodd-Frank Act, which was passed in mid-2010, and which came out in phases and affected money creation. We then estimate the P-Star model eq. (22) from 2013Q1 to 2024Q3. All regressions include separate dummies for 2020Q2, 2020Q3, and 2020Q4 to control for the initial phase of the pandemic.

To eq. (22), we add two variables to control for other unusual shocks. All of our estimated models include DPEnergy, which is a dummy equal to 1 in any quarter in which the PCE price index for energy goods and services falls by more than 10 percent relative to a PCE price index which excludes energy goods and services. Including this dummy addresses large outliers that are reflective of global shocks to energy prices while avoiding introducing multicollinearity with smoother energy price variables, such as the continuous percent change in relative energy prices that may partly reflect general trends in overall inflation. The dummy picks up 2020Q2, but we set it to zero in that period to avoid conflating the coefficient on it with the worst quarter of the pandemic.Footnote ²⁶ Consequently, the dummy is de facto a dummy for 2015Q1, when energy prices sagged during a major downshift in the long-term growth rate of the Chinese economy. Owing to its construction, DPEnergy is expected to have a negative coefficient, which is expected to be larger in magnitude for overall than for core PCE inflation. The coefficient in core PCE inflation models may be significant and negative owing to the pass-through of energy price drops into the prices of other consumer items.

To shed light on how disruptions to supply chains affected inflation during the recovery from the COVID Recession, we test versions of eq. (22) that also include the t-1 level of the quarterly average of the FRBNY’s Index of Global Supply Chain Pressures (SupPress). This index draws information from several indicators of supply pressures and effort is made to remove the influence of demand side factors from the index. During the pandemic, spikes in SupPress reflect breakdowns in market functioning associated with supply chain disruptions that are not easily captured by our measure of the consumption gap and are plausibly inflationary in nature. Over a sample from when SupPress begins in 1998Q1 to 2024Q3, the unit root tests (Table 1) indicate that it is I(0), implying that the level of SupPress could be tested in a regression with other variables that are stationary. We separately add the t-1 change and the t-1 level of the index to different models because it is unclear a priori which should enter the P-Star model. For overall PCE inflation, we include both the t-1 and t-2 lags of ΔSupPress when testing for the effects of changes.

Table 4. Quarterly P-star models of U.S. PCE (consumer) inflation, 2013Q1-2024Q3.

Notes: ⁺,^*,^** denote 90%, 95% & 99% significance. Absolute t-statistics are in parentheses. vgap is formed using estimates of v ^* from Model 2, Table 3, estimated over 1985Q3-2012Q4. cgap is formed from real PCE and an estimate of trend real PCE from a one-sided HP filter.

Table 5. Quarterly P-Star models of U.S. core PCE (consumer) inflation, 2013Q1-2024Q3.

Notes: ⁺,^*,^** denote 90%, 95% & 99% significance. Absolute t-statistics are in parentheses. vgap is formed using estimates of v ^* from Model 2, Table 3, estimated over 1985Q3-2012Q4. cgap is formed from real PCE and an estimate of trend real PCE from a one-sided HP filter.

We estimate six P-Star models for quarterly overall PCE inflation and core PCE inflation (annualized) over a 2013Q1-2024Q3 sample, which are reported in Tables 4 and 5 respectively. By extending the sample through 2024Q3, we are able to bolster the limited number of observations and to pull in more data from the recent period when SupPress moves a lot. In each table, Model 1 is a P-Star model that follows eq. (22) allowing for differences in the coefficients on vgap and cgap as in Ireland (Reference Ireland2024). Model 2 combines the velocity (vgap) and consumption (cgap) gaps into the price gap (pgap = vgap − cgap) thus imposing equality of the coefficients (in absolute value) as in the standard formulation of the model (e.g., Belongia and Ireland, Reference Belongia and Ireland2017). Given the potential errors in measuring consumption gaps in real time during the pandemic, it may be more accurate to focus on the overall price gap rather than its subcomponents. Models 1 and 2 omit SupPress. Models 3 and 4 repeat Models 1 and 2, respectively, except that they add the lagged level of SupPress, while Models 5 and 6 add the lagged change in SupPress to the models of core inflation and both the first and second lags of the change in SupPress to the models of overall PCE inflation.

4.3. Estimation results for overall and core PCE inflation

Several notable patterns emerge in Tables 4 and 5. In the even-numbered models, the combined price gap term is statistically significant. In the odd-numbered models that decompose the price gap, the velocity gap terms are significant with the expected positive sign.Footnote ²⁷ The real consumption gaps have insignificant coefficients in all cases with unexpected positive coefficients in two cases. Moreover, the coefficient on the consumption gap is lower in absolute magnitude than the coefficient on the velocity gap across all specifications. The lagged level of the supply chain index has a positive and significant coefficient for all models in Tables 4 and 5, while the coefficient on the lagged change in the index is significant and positive as expected for models of core PCE inflation in Table 5. For overall PCE inflation, the first lag of the change is insignificant, but positive, while the second lag is significant and positive. The evidence that the velocity and price gaps have statistically significant and positive effects, while the coefficients on the level and first difference of SupPress are positive and significant, together imply roles for both aggregate demand and supply pressures in driving the rise and partial ebbing of inflation during the COVID recovery. The COVID impact dummies for 2020Q2 and 2020Q4 are highly significant in all specifications.

Figure 7 illustrates the estimated role of these factors on core PCE inflation using Model 4 from Table 5. Actual core inflation is plotted along with an adjustment for the COVID dummy effects. The figure plots the estimated effect of global supply pressures and the sum of the estimated effects of the lag of pgap and of the level of supply pressures, where the latter is the product of the lag of SupPress and its estimated coefficient. The gap between the latter two lines reflects how much the lagged deviation between equilibrium prices and actual prices (pgap) affects inflation beyond that captured by including five quarterly lags of inflation (and the energy dummy). It is unclear how much of the lagged changes in inflation reflect past aggregate demand versus supply shocks. The chart illustrates how the standard P-Star price gap and changes in supply conditions are directly linked to recent movements in core inflation rates. While the estimated contribution of supply pressures to the temporary run-up of core inflation is notable, more of it is attributable to aggregate demand pressures reflected in the gap between the two lower plotted lines.

Figure 8 repeats Figure 7 except it uses estimated coefficients from Model 6 in Table 5 and also gauges supply chain pressure effects using the change instead of the level of SupPress. The only notable difference in results is that the estimated effect of global supply chain pressures is much smaller, implying that there was an even larger relative role for aggregate demand pressures in explaining the 2021−22 run-up in core PCE inflation and its retreat in 2023 to mid-2024.

Figure 7. Effects of the price gap and supply chain (level) pressures on U.S. core inflation.

Sources: BEA, FRBNY, and authors’ calculations.

Figure 8. Effects of the price gap and supply chain (changes) pressures on U.S. core inflation.

Sources: BEA, FRBNY, and authors’ calculations.

4.4. Estimation results using the HP filter to model long-run velocity

Tables 6 and 7 report results from overall and core PCE inflation models that correspond to the models in Tables 4 and 5, respectively, except that they use a one-sided HP filter to calculate the velocity gap, as in Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2017, Reference Belongia and Ireland2021) and Ireland (Reference Ireland2024), and consistent with how we calculated cgap. We use vgapHP and pgapHP to distinguish these from the earlier estimates in Tables 4 and 5. We refer to models in Tables 4 and 5 as our “modified approach.”

Table 6. Quarterly P-star models of U.S. PCE inflation (HP filter), 2013Q1-2024Q3.

Notes: ⁺,^*,^** denote 90%, 95% & 99% significance. Absolute t-statistics are in parentheses. vgapHP is formed from velocity and an estimate of trend velocity from a one-sided HP filter. cgap is formed from real PCE and an estimate of trend real PCE from a one-sided HP filter.

Table 7. Quarterly P-star models of U.S. core PCE inflation (HP filter), 2013Q1-2024Q3.

Notes: ⁺,^*,^** denote 90%, 95% & 99% significance. Absolute t-statistics are in parentheses. vgapHP is formed from velocity and an estimate of trend velocity from a one-sided HP filter. cgap is formed from real PCE and an estimate of trend real PCE from a one-sided HP filter.

There are several patterns across the overall PCE inflation results in Tables 4 and 6. First, in the odd numbered models, the velocity gap is either less significant or not at all for the HP filtered velocity gaps (Table 6), whereas vgap is always significant at the 99% level in our modified models (Table 4). Second, and similarly, the price gap terms in the even-numbered models are more significant in the modified models in Table 4 than in the corresponding models in Table 6 except for Model 6. Third, the fit of the modified models in Table 4 is uniformly better (including for Model 6) as evidenced by the higher adjusted R²’s across the board that range from 0.092 higher for Model 4 (0.751 vs. 0.843) to 0.188 higher for Model 5 (0.683 vs. 0.871).Footnote ²⁸

The differences in significance of the price and velocity gaps and overall fit of the model arise from the differences in how the velocity gap is calculated. These are compared in Figures 9 and 10. During much of the period from 2010 to 2019, the one-sided HP filter trend is above velocity leading to small, but often positive, velocity gaps, indicating upward pressure on inflation. In contrast, the velocity gap from our model is generally slightly negative over 2010-19 (since velocity is usually above its long-run equilibrium over this period), implying that velocity was putting downward pressure on inflation in a period when inflation was generally slightly below the Fed’s 2 percent target.

Figure 9. HP trend velocity lags actual U.S. velocity in the early-2020s.

Sources: CFS, Federal Reserve, BEA, and author’s calculations.

Figure 10. Estimated velocity gap implies more plausible swings in U.S. inflationary pressures than an HP filter-based measure.

Sources: CFS, Federal Reserve, BEA, and author’s calculations.

From 2020 through 2024, the one-sided HP trend lags behind the actual u-shaped velocity trajectory as shown in Figure 9. This produces a short-lived upward spike in the velocity gap as shown in Figure 10 after which the velocity gap turns negative in mid-2021 and bottoms out in early 2023—suggesting downward pressure on inflation precisely when inflation was surging. In contrast, our velocity gap also spikes and then declines, but much more gradually, and remains positive through early 2023. Then, small negative velocity gaps and negative money growth accompany the sharp deceleration of inflation since early 2023. Thus, our velocity gap measure appears more consistent with actual inflation.

Similar qualitative differences arise when modeling core PCE inflation, Tables 5 and 7, except that they are smaller in magnitude in several ways. While the modified models in Table 5 fit better across the board, the improvement in fit relative to the corresponding models in Table 7 is not as large for core inflation. This partly reflects the fact that vgapHP and pgapHP are both statistically significant at the 99 percent level in two of core PCE inflation models in Table 7 and have notably higher t-statistics than in the corresponding models of overall inflation in Table 6. Despite some of these nuanced differences, however, the model fits are better, the velocity and price gap terms are more statistically significant, and the magnitude of the effects of supply pressures is smaller for core inflation when the velocity gap is calculated out-of-sample from our error-correction model versus when it is calculated via a one-sided HP filter.

Taken together, these results suggest that our modified model enhances the ability of the P-star model to gauge inflationary pressures over this period and that by doing so, it helps us to avoid overstating the role of supply pressures in contributing to the rise and the partial ebbing of inflation over 2021-2024.