1. Introduction
Inflation jumped higher during 2021 and 2022, reaching levels not seen in four decades. Fig. 1 shows that this jump was preceded, in the Euro Area (EA), United Kingdom (UK), and USA, by an equally striking acceleration in money growth. Both the co-movement in money growth and inflation and temporal lead of money growth over inflation are explained easily by the quantity theory of money, summarized by Milton Friedman’s (Reference Friedman1991), p. 16) famous dictum that “inflation is always and everywhere a monetary phenomenon.”
Figure 1. Money growth and inflation. Each panel shows year-over-year percentage changes in the indicated variable.
Fig. 1 also suggests, however, that quantity-theoretic links between money growth and inflation appear strongest during episodes like that between 2020 and the present, during which both variables exhibit large movements. The connection, by contrast, seems weaker when money growth and inflation are lower and more stable.Footnote
1
Yet this fact, too, can be explained in quantity-theoretic terms, with reference to the equation of exchange
$MV=PY$
. This identity links changes in the money stock
$M$
to changes in the aggregate price level
$P$
, but also makes clear that money-price connections in the data may be disrupted by movements in velocity
$V$
, reflecting shifts in the demand for money relative to spending, and the transactions variable
$Y$
, usually taken to be real GDP, reflecting disturbances impacting on the economy from the supply side.
These considerations point to the need for a statistical model that controls for anticipated movements in
$V$
and
$Y$
and thereby isolates the quantity-theoretical effects that changes in money growth have on inflation at any given point in time. One such model—the “P-star” model—was developed by economists at the Federal Reserve in the late 1980s under the direction of then Chair Alan Greenspan. Hallman et al. (Reference Hallman, Porter and Small1991) present the P-star model to an academic audience and test the model’s implications with quarterly data from 1955:1 through 1988:4 and with annual data running back to 1870. This paper modifies and extends their analysis so as to apply the P-star model to the most recent data from the USA and from the EA and UK as well.
After briefly surveying previous work that has used the P-star model to interpret EA, UK, or USA data from older samples and then outlining the details of the P-star model itself, this paper turns its attention to several issues that show how “measurement matters” when it comes to detecting evidence of quantity-theoretic links between money growth and inflation that the model is designed to capture. Taken together, these issues apply to all four variables from the equation of exchange:
$M$
,
$V$
,
$P$
, and
$Y$
.
The first issue concerns the appropriate choice of monetary aggregate
$M$
. Barnett (Reference Barnett1980) observes that conventional monetary aggregates computed as “simple sums” of the nominal values outstanding of their component assets are flawed measures of monetary or liquidity service flows received by their owners when those component assets are imperfect substitutes, a fact easily confirmed when those assets pay interest at different rates yet are all held in private agents’ portfolios. Barnett (Reference Barnett1980) goes on to show how economic aggregation and statistical index number theory can be combined to produce Divisia aggregates of monetary services that improve, conceptually, on their simple-sum counterparts.Footnote
2
Chrystal and MacDonald (Reference Chrystal and MacDonald1994, p. 76) coin the phrase “Barnett critique” to refer to the implication that apparently fragile statistical linkages between money and other key macroeconomic variables will become much stronger when Divisia aggregates replace simple sums in econometric work. Their paper, together with Belongia (Reference Belongia1996) and Hendrickson (Reference Hendrickson2014), establish the empirical relevance of the Barnett critique by demonstrating that previously published results casting doubt on the explanatory or predictive power of money are often reversed just by replacing a simple-sum aggregate with the corresponding Divisia measure.
Also consistent with the Barnett critique, the results here provide strong evidence of quantity-theoretic links between Divisia money growth and inflation via the P-star model, regardless of the level of monetary aggregation, that is, the range of assets included in the composite. Across all three economies studied here, however, the central bank of only one—the Bank of England—provides an “official” series for Divisia money; this series for the UK is described by Fisher et al. (Reference Fisher, Hudson and Pradhan1993), Hancock (Reference Hancock2005), and Berar and Owladi (Reference Berar and Owladi2013).Footnote 3 Thus, this paper relies on Divisia monetary aggregates constructed by Darvas (Reference Darvas2014, Reference Darvas2015) for the EA and Barnett et al. (Reference Barnett, Liu, Mattson and van den Noort2013) for the USA in estimating and testing the implications of the P-star model.
The second issue involves the estimation of
$V^{*}$
, the long-run or equilibrium level to which the velocity of money is expected to return. Hallman et al. (Reference Hallman, Porter and Small1991) take
$V^{*}$
to be a constant, based on the stability of M2 velocity over the 1955–1988 period that is their primary focus. As discussed by Orphanides and Porter (Reference Orphanides and Porter2000), however, USA M2 velocity moved abruptly higher shortly after the publication of Hallman, Porter, and Small’s original article, throwing the model’s predictions off track.Footnote
4
Moreover, as discussed by Bordo & Duca (Reference Bordo and Duca2023), the velocities of Divisia monetary aggregates for the USA exhibit shifting long-run trends even in data from before 1990 and, as shown below, velocities of Divisia aggregates for the EA and UK display long-run trends as well. These considerations motivate the more flexible approach taken here, which attempts to track movements
$V^{*}$
, in real time, using a one-sided time series filter.
The third and final issue relates to the choice of transactions variable and the corresponding price aggregate referred to by the variables
$Y$
and
$P$
on the opposite side of the equation of exchange. In developing the P-star model, Hallman et al. (Reference Hallman, Porter and Small1991) use real GNP as their measure of
$Y$
and the GNP deflator as their measure of
$P$
. These are perhaps the most natural choices, as nominal GNP represents the broadest aggregate of nominal spending with which to measure
$PY$
. On the other hand, the central banks of all three economies studied here target consumer price inflation more closely. What’s more, as shown below, measures of the deflator for aggregate output appear to have been greatly distorted, especially in EA and UK, due to supply disruptions associated with the 2020 economic closures. An additional contribution of this paper is to show that, as a consequence, the P-star model fits the data much better when aggregate consumption
$C$
replaces aggregate output
$Y$
as the transactions variable and the consumption deflator replaces the output deflator in measuring the price level. Given the magnitude of the distortions—especially, again, in the EA and UK data—this same observation will likely apply to other empirical studies attempting to explain the behavior of inflation since 2020.Footnote
5
Having discussed these issues relating to measurement, the paper goes on to test the P-star model, producing as a useful byproduct time series estimates of the P-star “price gap” that indicate, for each economy, how monetary policy has applied both upward and downward pressure on inflation in each of the three economies over sample periods extending through the present. The paper concludes, therefore, with a discussion of what the P-star model reveals about the stance of monetary policy in the EA, UK, and USA today.
2. Literature review
As discussed above, this paper uses Hallman et al.’s (Reference Hallman, Porter and Small1991) P-star model to detect statistical links between money growth and inflation in the EA, UK, and USA over sample periods that include the most recent episode of rapid money growth and inflation since 2020. Humphrey (Reference Humphrey1989) traces the P-star model’s intellectual origins back through the writings of David Hume, Henry Thornton, Thomas Attwood, Irving Fisher, Holbrook Working, Carl Snyder, Milton Friedman, Michael Darby, and William Poole, all of whom emphasized that the quantity-theoretic link between money growth and inflation appears with a lag, so that movements in money growth can be used to forecast future movements in inflation. Table 1 lists previous work that has applied the P-star model to data from the EA, UK, and USA. As indicated, most of these studies use conventional, simple-sum monetary aggregates, but a few emphasize that Divisia monetary aggregates provide stronger signals of the effects that money growth is having on inflation.Footnote 6
Table 1. Previous work with the P-star model
Note: References marked with * use consumption-based measures of inflation; others use measures based on the GDP deflator.
As also noted above, this paper uses data on the Divisia monetary aggregates constructed and made publicly available by Darvas (Reference Darvas2014, Reference Darvas2015) for the EA, the Bank of England for the UK, and Barnett et al. (Reference Barnett, Liu, Mattson and van den Noort2013) for the USA. Other work constructing Divisia monetary aggregates includes Reimers (Reference Reimers2002), Stracca (Reference Stracca2004), Barnett and Gaekwad (Reference Barnett and Gaekwad2018), and Brill et al. (Reference Brill, Nautz and Sieckmann2021) for the EA, Bissoondeeal et al. (Reference Bissoondeeal, Jones, Binner and Mullineux2010), Drake and Fleissig (Reference Drake and Fleissig2006, Reference Drake and Fleissig2008), and Ezer (Reference Ezer2019) for the UK, and Anderson et al. (Reference Anderson, Jones and Nesmith1997a, Reference Anderson, Jones and Nesmithb, Reference Anderson, Jones and Nesmithc) and Anderson and Jones (Reference Anderson and Jones2011) for the USA.
Table 1 indicates, as well, that a number of previous studies depart from Hallman et al. (Reference Hallman, Porter and Small1991) original analysis by testing the P-star model using consumption-based measures of inflation instead of measures based on GDP. While most of these studies use the consumer price index, Altimari (Reference Altimari2001) and Jansen (Reference Jansen2004) use the private consumption deflator—the same choice favored here. Replacing data on the GDP deflator with data on the CPI or the consumption deflator can be justified based on the observation that the European Central Bank, the Bank of England, and the Federal Reserve all focus on consumption-based measures of inflation. The results below demonstrate, however, that statistical considerations also play a role: the P-star model draws much stronger links between money and prices when the consumption deflator replaces the GDP deflator in measuring inflation.
Finally, in extending its application of the P-star model through the post-2020 period, this paper joins a few others in taking a quantity-theoretic view of the increase in inflation observed in all three major economies over this recent episode. Most notably, Castañeda and Congdon (Reference Castañeda and Congdon2020) provide a prescient early warning that the acceleration in money growth would lead to higher inflation through quantity-theoretic channels. As discussed here and in Greenwood and Hanke (Reference Greenwood and Hanke2021), Ireland (Reference Ireland2022, Reference Ireland2023), Bordo and Duca (Reference Bordo and Duca2023), Borio et al. (Reference Borio, Hofmann and Zakrajšek2023), Congdon (Reference Congdon2023), Hendrickson (Reference Hendrickson2023), Reynard (Reference Reynard2023), and Castañeda and Cendejas (2024), this prediction proved remarkably accurate. This work points to the usefulness of the quantity theory in general and the P-star model in particular, in gauging today whether monetary policy in the EA, UK, and USA has adjusted sufficiently to bring inflation back down to acceptable levels and in anticipating future inflationary trends in these same major economies.
3. The P-star model
The P-star model is based on the quantity theory of money and therefore takes as its starting point the equation of exchange, now written in more detail as
\begin{equation} M_{t}V_{t}^{y} = P_{t}^{y}Y_{t}, \end{equation}
where all variables are indexed by time
$t$
so as to allow the model to be estimated and tested with quarterly data. In (1),
$M_{t}$
is a monetary aggregate and, updating Hallman et al. (Reference Hallman, Porter and Small1991), the transactions variable
$Y_{t}$
is taken to be aggregate output as measured by real GDP.Footnote
7
The
$y$
superscript attached to the remaining variables is to emphasize that, with output as the transactions variable, the GDP deflator becomes the relevant measure of the aggregate price level
$P_{t}^{y}$
and monetary velocity
$V_{t}^{y}$
is computed by dividing nominal GDP
$P_{t}^{y}Y_{t}$
by
$M_{t}$
.
By itself, of course, the equation of exchange holds as a identity; that is, monetary velocity is defined to make the equality exact. The P-star model, however, adds assumptions that give (1) predictive content and testable implications. It does so by defining the variable
\begin{equation} P_{t}^{y*} = \frac{M_{t}V_{t}^{y*}}{Y_{t}^{*}} \end{equation}
that gives the model its name. In (2),
$V_{t}^{y*}$
and
$Y_{t}^{*}$
denote the natural, or equilibrium, levels to which velocity
$V_{t}^{y}$
and real GDP
$Y_{t}$
are expected to return in the long run. Thus,
$P_{t}^{y*}$
(“P-star”) represents, likewise, the long-run equilibrium nominal price level implied by the current level of the monetary aggregate
$M_{t}$
towards which the actual price level
$P_{t}^{y}$
measured by the GDP deflator will converge as velocity and real GDP return to their own long-run levels. Consistent with the quantity-theoretic views described by the authors surveyed by Humphrey (Reference Humphrey1989), therefore, the P-star model allows an increase in money
$M_{t}$
to stimulate real spending and thereby increase
$Y_{t}$
or to be held as excess cash balances relative to nominal spending as reflected in a decrease in
$V_{t}^{y}$
in the short run. In the long run, however, the increase in money should be reflected in a proportional increase in the aggregate nominal price level as these short-run real effects wear off.
Hallman et al. (Reference Hallman, Porter and Small1991) test this implication by estimating the regression
\begin{equation} \Delta \pi _{t}^{y} = \alpha + \sum _{j=1}^{q} \beta _{j}\Delta \pi _{t-j}^{y} + \gamma (p_{t-1}^{y*}-p_{t-1}^{y}) + \varepsilon _{t}. \end{equation}
In (3),
$\pi _{t}^{y} = 400[\ln (P_{t}^{y})-\ln (P_{t-1}^{y})]$
denotes the quarterly inflation rate, expressed in annualized, percentage-point terms, and
$\Delta \pi _{t}^{y} = \pi _{t}^{y}-\pi _{t-1}^{y}$
the corresponding change in inflation. The lagged “price gap” variable
$p_{t-1}^{y*}-p_{t-1}^{y}=100[\ln (P_{t-1}^{y*})-\ln (P_{t-1}^{y})]$
is computed as the percentage-point gap between the equilibrium and actual levels
$P_{t-1}^{y*}$
and
$P_{t-1}^{y}$
. Once more, the
$y$
subscript attached to all of these variables emphasizes that both inflation and the price gap are measured using data on real GDP and the GDP deflator.
From the regression in (3), an estimated value of
$\gamma$
that is positive and statistically significant will confirm that inflation accelerates when the price gap is positive, so that
$P_{t-1}^{y}$
can converge to
$P_{t-1}^{y*}$
from below, and that inflation decelerates when the price gap is negative, so that
$P_{t-1}^{y}$
can converge to
$P_{t-1}^{y*}$
from above. In this case, the P-star price gap is a useful quantity-theoretic indicator of the effects that past money growth is expected to have on future inflation. The past changes of inflation, also included on the right-hand side of (3), allow these effects of money growth on inflation to appear more smoothly and with a longer lag than implied by the single lagged price gap alone.
4. Measurement matters
Several questions and problems arise in measuring all of the variables that appear in (1)–(3). First, the Barnett critique dictates the choice of Divisia over simple-sum monetary aggregates, here as in all other empirical studies that require measures of the money stock. For the EA, Darvas (Reference Darvas2014, Reference Darvas2015) provides series on Divisia money at three levels of aggregation, corresponding to the European Central Bank’s simple-sum M1, M2, and M3 measures.Footnote 8 In particular, M1 includes currency and overnight deposits, M2 adds selected savings and time deposits, and M3 adds repurchase agreements, money market mutual fund shares, and short-term debt securities issued by the banking sector.Footnote 9 As discussed by Darvas (Reference Darvas2014) and in more detail in European Central Bank (2019), the ECB adjusts its simple-sum monetary aggregates to construct “notional outstanding stocks” that correct for one-time shifts associated with “reclassifications, revaluations, and exchange rate adjustments” of financial institutions’ balance sheet items. Darvas provides similarly adjusted series for his Divisia aggregates as well; these notional stocks are used here, for the changing-composition EA economy. These quarterly data run from 2001:1 through 2023:2.
For the UK, the Bank of England provides Divisia series for the private non-financial corporations and household sector M4 aggregate, which includes notes and coin, non-interest-bearing deposits, and interest-bearing sight and time deposits. These series are described by Fisher et al. (Reference Fisher, Hudson and Pradhan1993), Hancock (Reference Hancock2005), Berar and Owladi (Reference Berar and Owladi2013), and Fleissig and Jones (Reference Fleissig and Jones2024). They run from 1977:1 through 2023:4.
For the USA, Barnett et al. (Reference Barnett, Liu, Mattson and van den Noort2013) construct series for Divisia money at various levels of aggregation, three of which are considered here.Footnote 10 Divisia M2 includes the same assets covered by the Federal Reserve’s official, simple-sum M2 aggregate: currency, checking and savings deposits including money market deposit account balances, retail money market mutual fund shares, and small time deposits. Divisia MZM, originally referred to as “nonterm M3” by Motley (Reference Motley1988) and renamed “money, zero maturity” by Poole (Reference Poole1994, p. 104), subtracts the time deposit component from M2 but adds institutional money market mutual fund shares. Divisia M4 adds small time deposits back to MZM and then includes large time deposits, overnight and term repurchase agreements, commercial paper, and USA Treasury bills as well. Though much broader than any of the other aggregates studied here, Divisia M4 has been shown by Keating et al. (Reference Keating, Kelly, Lee Smith and Valcarcel2019) and Dery and Serletis (Reference Dery and Serletis2021) to have especially high predictive power for real economic activity and inflation in recent years, reflecting the increased importance of nonbank financial intermediaries—so-called “shadow banks”—in the US financial system. These data run from 1967:1 through 2023:4.
Second, although Hallman et al. (Reference Hallman, Porter and Small1991) took equilibrium velocity
$V_{t}^{y*}$
as a constant, based on the stability of fluctuations in simple-sum M2 velocity around its mean over their 1955—1988 sample period, M2 velocity moved sharply higher shortly after the publication of their article; as discussed by Orphanides and Porter (Reference Orphanides and Porter2000), this shift in velocity threw off track the predictions of the model as originally formulated. Moreover, as discussed by Bordo and Duca (Reference Bordo and Duca2023) and as will be seen here below, the absence of a long-run trend in the velocity of simple-sum M2 before 1990 is a feature not shared by any of the Divisia monetary aggregates used here. Instead, velocities computed with Divisia money for the USA display slowly shifting trends: upwards through the 1990s and downwards since 2000. Velocities of EA and UK Divisia money, meanwhile, generally trend downwards.
Belongia and Ireland (Reference Belongia and Ireland2015, Reference Belongia and Ireland2017, Reference Belongia and Ireland2022) and Ireland (Reference Ireland2023) show that the one-sided variant of the Hodrick-Prescott (HP) filter, described by Stock and Watson (Reference Stock and Watson1999, p. 301), does an adequate job of tracking movements in trend velocity for the EA and USA.Footnote
11
That approach is used by extension, here, for the UK as well. Application of the same one-sided HP filter also provides an estimate of the trend in the transactions variable
$Y_{t}$
. Importantly, the one-sided property of this filter implies that only past data are used to estimate the values of
$V_{t}^{y*}$
and
$Y_{t}^{*}$
in (2). This allows the corresponding lagged price gap measure
$p_{t-1}^{y*}-p_{t-1}^{y}$
to be computed in real time and used in practice as a signal of future inflationary pressures.
Third, real GDP, as the broadest aggregate of goods and services produced and sold, usually appears to be the natural choice for the transactions variable in the equation of exchange. This conventional choice, however, raises problems when applied, especially to the EA and UK economies, over the period since 2020. To illustrate, Fig. 2 compares the behavior of three measures of inflation, measured by annualized, quarter-to-quarter changes in three aggregate price indices, for each economy. The left-hand column displays plots of inflation based on the GDP deflator—the aggregate price index implied by the choice of real GDP as the transactions variable in (1)–(3). The center column shows plots of inflation based on the deflator for the consumption component of GDP instead. Finally, the right-hand column shows plots of the inflation rate that serves as the official target for each central bank: based on the harmonized index of consumer prices for the EA, the consumer price index (CPI) for the UK, and the price index for personal consumption expenditures for the USA.
Figure 2. Inflation. Each panel shows annualized quarter-to-quarter percentage changes in the indicated price index.
Although measures of inflation based on the consumption deflator resemble quite closely the measures targeted by each central bank, for the EA and UK especially, measures of inflation based on the GDP deflator behave quite differently around the time of the 2020 economic closures. For the EA, the GDP deflator grew at an annualized rate of nearly 5 percent between the first and second quarters of 2020, even as the HIPC declined at an annualized rate of 1.5 percent. Even more striking, for the UK, the GDP deflator rose at an annualized rate of close to 30 percent, even as the CPI declined at an annualized rate of 2.9 percent. Only for the USA do all three measures of inflation behave similarly, at least over this most recent period.
To reveal the source of these divergent price movements, Table 2 reports (unannualized) quarter-to-quarter percentage changes in the deflators for real GDP and its main components over the four quarters of 2020. Clearly, for both the EA and UK, unusual behavior in the deflator for government purchases accounts for the differences. Jessop (Reference Jessop2020) elaborates, explaining that the very large increase in the UK GDP deflator reflects estimates that the real quantities of goods and services delivered in the health and education sectors of the economy fell sharply in the second quarter of 2020 even as nominal spending of those services increased.
Table 2. Quarter-to-quarter percentage changes in deflators for GDP and its components
The highly unusual dynamics displayed by the EA and UK GDP deflators should be noted in all work studying inflation over sample periods including 2020. In particular, they dictate an alternative choice made here, to re-formulate the equation of exchange (1) as
\begin{equation} M_{t}V_{t}^{c} = P_{t}^{c}C_{t}, \end{equation}
using real consumption
$C_{t}$
as the scale variable, the deflator for consumption as the corresponding price index
$P_{t}^{c}$
, and velocity
$V_{t}^{c}$
measured by dividing nominal consumption spending
$P_{t}^{c}C_{t}$
by
$M_{t}$
. The formula for the P-star variable then becomes
\begin{equation} P_{t}^{c*} = \frac{M_{t}V_{t}^{c*}}{C_{t}^{*}}, \end{equation}
where
$V_{t}^{c*}$
and
$C_{t}^{*}$
are the trend components of consumption velocity and real consumption, computed with the one-sided HP filter. In terms of these new choices, the P-star regression (3) gets reformulated as
\begin{equation} \Delta \pi _{t}^{c} = \alpha + \sum _{j=1}^{q} \beta _{j}\Delta \pi _{t-j}^{c} + \gamma (p_{t-1}^{c*}-p_{t-1}^{c}) + \varepsilon _{t}, \end{equation}
where
$\pi _{t}^{c} = 400[\ln (P_{t}^{c})-\ln (P_{t-1}^{c})]$
,
$\Delta \pi _{t}^{c} = \pi _{t}^{c}-\pi _{t-1}^{c}$
, and
$p_{t-1}^{c*}-p_{t-1}^{c}=100[\ln (P_{t-1}^{c*})-\ln (P_{t-1}^{c})]$
. Now, the
$c$
subscript attached to all of these variables emphasizes that both inflation and the price gap are measured using data on real consumption and the consumption deflator.
Fig. 3 plots series on money growth and consumption price inflation over the longest sample available for each of the three economies: extending back to 2001 for the EA, 1977 for the UK, and 1967 for the USA.Footnote 12 Figs. 4 and 5 show that for each economy and each monetary aggregate, both consumption and income velocities display considerable long-run drift. In each case however, these movements are tracked quite closely, in real time, by the estimate of trend provided by the one-sided HP filter. Finally, Fig. 6 shows that the consumption and output gaps, defined as the percentage-point deviation of each variable from its estimated trend component, behave similarly in all three economies. Taken together, therefore, these figures imply that the primary reason for replacing output with consumption in moving from the model described by (1)–(3) to that described by (4)–(6) is to minimize distortions associated with the measurement of prices and quantities of government services during the 2020 economic closures, which make GDP price inflation a misleading measure of inflation compared to those most relevant to consumers and central bankers.
Figure 3. Money growth and inflation. Each panel shows year-over-year percentage changes in the indicated variable.
Figure 4. Monetary velocity based on consumption. Each panel shows the consumption velocity of the indicated monetary aggregate (solid blue line) together with its long-run trend (dashed red line), computed using the one-sided Hodrick-Prescott filter.
Figure 5. Monetary velocity based on output. Each panel shows the income velocity of the indicated monetary aggregate (solid blue line) together with its long-run trend (dashed red line), computed using the one-sided Hodrick-Prescott filter.
Figure 6. Consumption and output gaps. Each panel shows the percentage-point deviation of the indicated variable from its long-run trend, computed using the one-sided Hodrick-Prescott filter.
Sources for the non-monetary data used here are standard. For the EA, price and quantity data on GDP and its components are from the Eurostat database and apply, as do Darvas (Reference Darvas2014, Reference Darvas2015) Divisia monetary aggregates, to the changing-composition Euro Area. Data on the HIPC are from the European Central Bank’s data portal. For the UK, all non-monetary series come from the Office of National Statistics. Finally, for the USA, all non-monetary data are from the Federal Reserve Bank of St. Louis’ FRED database. All of the series available from these sources are seasonally adjusted, except for the UK CPI. Thus, the measures of inflation for the UK CPI shown in Figs. 1 and 2 are de-seasonalized via a regression using quarterly dummy variables.
Finally, the P-star regressions (3) and (6), whether cast in terms of GDP or consumption, take the form of the more general autoregressive distributed lag models described and discussed by Pesaran (Reference Pesaran2015, pp. 121-3). Hence, following Pesaran’s suggestion, the Schwarz (Reference Schwarz1978) information criterion is used to select the optional value for
$q$
, the number of lagged changes in inflation included in each specification. Values ranging from
$q=1$
through
$q=8$
are considered. Across all of the results reported below, the Schwarz criterion typically selects models with just one or two lags and never favors a model with more than
$q=4$
, the value originally used by Hallman et al. (Reference Hallman, Porter and Small1991) when estimating (3) with quarterly data from 1955:1 through 1988:4.
5. Results
Tables 3 (Tables 4, 5) through 6 present the main results, which are easily described. Tables 3 and 4 report estimates of (6), using consumption as the transactions variable. The results in Table 3 come from the common sample of data, running from 2001:1 through 2023:2, available for all three economies. Those in Table 4 use data from the longest sample of data available for each economy: 2001:1–2023:2 for the EA, 1977:1–2023:4 for the UK, and 1967:1–2023:4 for the USA. In both tables, the lagged price gap always enters the regression with a positive coefficient that is highly significant. In every single case, the
$p$
value rejecting the null hypothesis that
$\gamma =0$
is 0.01 or smaller. The range of estimates for
$\gamma$
, from 0.09 to 0.24, cleanly encompasses the value of 0.148 first reported by Hallman et al. (Reference Hallman, Porter and Small1991, p. 847).
Table 3. Estimated P-star models: common sample period, consumption as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 4. Estimated P-star models: longest sample periods, consumption as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 5. Estimated P-star models: common sample period, output as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 6. Estimated P-star models: longest sample periods, output as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 5 shows how, for the common sample period 2001:2–2023:2, these positive and statistically significant estimates of
$\gamma$
disappear for the EA and UK when output replaces consumption as the transactions variable. Only for the USA do the estimates of
$\gamma$
remain positive and significant. The contrasting results in Tables 3 and 5 underscore the importance of measurement in assessing the quantity-theoretic links between money and prices. Together with the close look at the data themselves in Table 2, they serve more generally to warn future researchers that distortions in the measurement of price indices for government purchases in the EA and UK around the time of the 2020 economic closures are large enough to affect statistical estimates of and hypothesis tests about the magnitudes of key model coefficients.
In Table 6, also reporting results with output as the scale variable but from the longest sample period available for each economy, the coefficient on the lagged price gap regains its size and significance when, in particular, the UK sample used to estimate it is extended back to 1977. This raises the question of how important the 2020 data are in shaping the results: does the choice of scale variable for the P-star model matter for any other periods as well?
To answer this question, Tables 7 and 8 compare estimates based on consumption to those based on GDP over the longest pre-2020 samples available for each economy. Table 7 shows that, using consumption as the transactions variable, the lagged price gap enters positively and significantly in all of the estimated equations, except when Divisia M3 is used as the measure of money for the EA. Table 8 reveals, by contrast, that the lagged price gap in most cases loses its significance when output is used as the transactions variable instead. Notably, this time, the results weaken considerably for the USA in addition to the EA. Once more, these results highlight the importance of measurement—particularly in the measurement of inflation—when looking for quantity-theoretic links between money and prices.
Table 7. Estimated P-star models: Pre-2020 sample periods, consumption as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 8. Estimated P-star models: Pre-2020 sample periods, output as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
It is important to ask, as well, whether the stronger relationships between money and consumption-based measures of prices drawn by the P-star model may have changed over time. The short sample of available data makes it difficult to do this for the EA, but the longer samples available for the UK and USA can be split and used to re-estimate the models for two distinct subsamples. For the UK, 1993 presents itself as the natural date for a sample split, since the Bank of England began targeting inflation at that time. And while the Federal Reserve did not official announce an explicit inflation target until much later, a look back at future 3 shows that, by 1993, inflation in the USA had also returned to the low and stable levels that characterize other economies’ experiences under inflation targeting.
Accordingly, Tables 9 and 10 compare results when the consumption-based P-star regressions are estimated over pre- and post-1993 data for both the UK and USA. Although the estimated coefficient on the lagged price gap remains sizable when estimated with pre-1993 data from the UK, a large standard error prevents this estimate from achieving statistical significance. The UK coefficient increases further in magnitude and regains its significance for the post-1993 period. For the USA, estimated values of
$\gamma$
are positive and significant across both subsamples, but as for the UK, these values also become larger and more significant when estimated with post-1993 data. Thus, the results in Tables 9 and 10 consistently indicate that the predictive power of money for inflation has actually strengthened in the post-1993 period.
Table 9. Estimated P-star models: Pre-1993 sample periods, consumption as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 10. Estimated P-star models: Post-1993 sample periods, consumption as transactions variable
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Tables 11 and 12 revisit the Barnett critique, by re-running the benchmark regressions from Tables 3 and 4 using simple sums in place of Divisia monetary aggregates. For the changing-composition EA, the simple-sum series are for notional outstanding stocks of M1, M2, and M3, also used by Darvas (Reference Darvas2014) and supplied in the same dataset, available through Bruegel, cited above. The Bank of England provides the series for simple-sum M4 for the non-financial and household sector for the UK, and the FRED database provides a long sample of data on simple-sum M2 for the USA.Footnote 13 Somewhat surprisingly, the same strong statistical relationships between the P-star price gap and future changes in inflation evident in Tables 3 and 4 with Divisia money also appear in Tables 11 and 12 with the corresponding simple sums.
Table 11. Estimated P-star models: common sample period, simple-sum money
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 12. Estimated P-star models; longest sample periods, simple-sum money
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Fig. 7 sheds some light on the source of these results. The left-hand panels confirm that, in all three economies, Divisia and simple-sum monetary aggregates have sometimes behaved quite differently. And for many purposes, these differences do matter. For the USA, in particular, simple-sum M2 growth greatly and persistently exceeded the rate of Divisia M2 growth from 1979 through 1984. Barnett (Reference Barnett2012, pp. 102-11) describes this episode, during which prominent monetarists, including Milton Friedman, were misled by high rates of simple-sum money growth and incorrectly predicted the return of higher rates of inflation. The right-hand panels of Fig. 7 show, however, that P-star price gaps behave quite similarly, whether based on Divisia or simple-sum monetary aggregates. Thus, the results from Table 11 and 12 can be interpreted as showing that, after accounting for slow-moving trends in velocity using the one-sided Hodrick-Prescott filter, the P-star model can glean the “right” monetary signal even from the “wrong” monetary aggregate.
Figure 7. Money growth and price gaps. Left-hand panels compare the year-over-year percentage changes in a Divisia monetary aggregate to those in the corresponding simple-sum aggregate. Right-hand panels compare the P-star price gaps computed using the same Divisia and simple-sum aggregates, using consumption as the transactions variable. Positive price gaps indicate periods when monetary policy puts upward pressure on inflation; negative values indicate periods when monetary policy puts downward pressure on inflation.
Finally, (4) and (5) imply that the consumption-based P-star price gap can be decomposed into two components,
\begin{equation} \begin{split} p_{t}^{c*} -p_{t}^{c} & = 100[\ln (P_{t}^{c*})-\ln (P_{t}^{c})] \\ & = 100[\ln (V_{t}^{c*})-\ln (V_{t}^{c})] + 100[\ln (C_{t})-\ln (C_{t}^{*})] \\ & = (v_{t}^{c*} - v_{t}^{c}) + (c_{t}-c_{t}^{*}), \end{split} \end{equation}
the first measuring the percentage-point deviation of equilibrium velocity
$V_{t}^{c*}$
from its actual level
$V_{t}^{*}$
and the second measuring the percentage-point deviation of consumption
$C_{t}$
from its equilibrium level
$C_{t}$
.Footnote
14
This decomposition highlights the quantity-theoretic foundations of the P-star model, which allow an increase in the money stock
$M_{t}$
to be held temporarily as excess real balances, thereby reducing
$V_{t}^{c}$
relative to
$V_{t}^{c*}$
. or to yield a transitory increase in real spending, thereby increasing
$C_{t}$
relative to
$C_{t}^{*}$
. Then, in the long run, the price level
$P_{t}$
will rise to meet
$P_{t}^{*}$
as these velocity and consumption gaps are closed. Using (7), the P-star regression (6) can be expanded to include both the velocity and consumption gaps instead of the price gap alone,
\begin{equation} \Delta \pi _{t}^{c} = \alpha + \sum _{j=1}^{q} \beta _{j}\Delta \pi _{t-j}^{c} + \gamma _{1}(v_{t-1}^{c*}-v_{t-1}^{c}) + \gamma _{2}(c_{t-1}-c_{t-1}^{*}) + \varepsilon _{t}, \end{equation}
and thereby used to gauge the importance of each term separately.
Tables 13 and 14 confirm that, for all three economies, lagged velocity gaps always enter positively and significantly into the expanded regression (8). Lagged consumption gaps generally do so as well, except for the USA in cases where Divisia MZM or Divisia M4 is used to measure money. Thus, both margins of adjustment—but especially that measured by the velocity gap—appear important in accounting for the delayed response of prices that then give changes in money their predictive power for inflation.
Table 13. Estimated P-star models: common sample periods, consumption and velocity gaps
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
Table 14. Estimated P-star models: longest sample periods, consumption, and velocity gaps
Note: Maximal lag length for changes in inflation selected based on the Schwarz Information Criterion.
6. Conclusion
Taken together, the results presented here highlight several ways in which “measurement matters” in assessing the quantity-theoretic links between money and prices using the P-star model. Most importantly, distortions in the measurement of the GDP price deflator, especially for the EA and UK economies during 2020 but for all three economies and at other times as well, make consumption-based measures of transactions and inflation preferable. This finding should be noted by researchers, even those working with alternative models, attempting to explain the dramatic movements in inflation seen across many of the world’s economies since 2020.
The results show, in addition, that adjusting measures of monetary ease or restraint by accounting for slowly shifting long-run trends in velocity and real economic growth is crucial when using those measures to forecast changes in inflation. In fact, the approach taken here, using the one-side Hodrick-Prescott filter, makes these adjustments satisfactorily even when, ignoring the importance of the Barnett critique, simple-sum measures are used instead of Divisia monetary aggregates.
Returning to Fig. 7, plotting the P-star price gap variables for each economy using the consumption-based measure of prices as implied by (4)–(6), several more conclusions emerge. First, price gaps near zero in the USA and below zero in the EA and UK signal that, despite central banks’ efforts to provide additional stimulus through “forward guidance” promising to hold short-term interest rates near zero for extended periods of time and multiple rounds of “quantitative easing” or large-scale asset purchases, slow rates of money growth put little if any upward pressure on inflation during the sluggish recoveries that, in all three economies, followed the 2008-9 global financial crisis and recession.
Second, price gaps rivaling or exceeding their historical peaks in 2020-1 show that, by sharp contrast, low interest rates and large-scale asset purchases did generate rapid money growth and strong upward pressure on inflation during and after the 2020 economic closures. Taken together, these observations point to the usefulness of the quantity-theoretic perspective provided by the P-star model, as an alternative to or at least a cross-check against more conventional analyses that focus on the behavior of short and long-term interest rates alone.
Third, even as unusually large and positive price gap measures signal the episode of vigorous monetary expansion that presaged the rise in inflation in 2021 and 2022, large (in magnitude) but negative (in sign) values for the same price gap measures suggest that monetary policies in all three countries are putting strong downward pressure on inflation today. These results illustrate, once again, how a quantity-theoretic approach, combined with more careful attempts at measurement, can help guide central bankers in their quest to restore and maintain a more favorable environment of low and stable inflation.
Acknowledgement
I would like to thank two anonymous referees for extremely helpful comments on a previous draft of this paper.
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