Divisia-Aggregates | Macro Paper Warehouse

A reconsideration of money growth rules

This 2022 Journal of Economic Dynamics & Control paper by Michael Belongia and Peter Ireland asks whether a policy rule that steers the growth rate of money, rather than the short-term nominal interest rate, could have delivered macroeconomic stabilization comparable to the Federal Reserve’s actual interest-rate policy — including through the 2009:1-2015:4 zero-lower-bound (ZLB) episode. They build a New Keynesian DSGE model with real Divisia M2 balances entered directly in household utility (with quadratic adjustment costs), habit formation in consumption, Rotemberg price-adjustment costs with backward-looking indexation, and five structural shocks (preference, productivity growth, money demand, cost-push, and monetary policy), and estimate it by Bayesian methods on quarterly U.S. data from 1983:1-2019:4, using the Kulish et al. (2017) piecewise-linear algorithm to handle the ZLB period (during which the funds rate is dropped from the observables and time-varying expected ZLB durations, informed by survey forecasts, are estimated as parameters). Comparing log marginal likelihoods, they find that augmenting the estimated Taylor rule with a contemporaneous money growth term raises the fit only slightly (2412.7 to 2413.3, with the money-growth response coefficient’s posterior mode a modest 0.0605), while adding lagged money growth actually lowers it — money growth adds a little information but not much on top of the interest-rate rule. Variance decompositions show monetary-policy shocks account for only 0.7% of output-growth variance and 5.6% of output-gap variance, with productivity, preference, and cost-push shocks dominating instead, and the model attributes the Great Recession’s declines in inflation and interest rates mainly to adverse preference and productivity shocks. The paper’s central counterfactual result is that a flexible money growth rule of the form mu-hat_t = mu-hat_{t-1} - 0.125 x-hat_{t-1} (found by grid search to minimize macroeconomic volatility and ZLB-equivalent duration) generates standard deviations of output growth (2.5511), inflation (1.1226), and the output gap (0.7095) that closely approximate those implied by the estimated Taylor rule (2.3762, 0.9774, and 0.5481 respectively), while producing a much shorter and milder episode of negative implied interest rates than the actual seven-year ZLB period. By contrast, a strictly constant money growth rule (all feedback coefficients set to zero) sharply amplifies volatility, raising the output-growth standard deviation to 3.6423 (more than 50% larger than under the Taylor rule) and the inflation standard deviation to 1.8871, confirming earlier findings by Ireland (2000), Collard and Dellas (2005), and Galí (2015) that fixed money growth performs poorly. The authors conclude that a flexible, output-gap-responsive money growth rule — not a rigid quantity-theoretic constant-growth rule — belongs on the list of policy alternatives capable of avoiding the ZLB while matching Taylor-rule-level stabilization performance.

Economic Monetary Aggregates: An Application of Index Number and Aggregation Theory

This 1980 Journal of Econometrics paper by William A. Barnett lays out the theoretical foundation for treating monetary aggregates such as M2 as genuine economic quantities and derives the Divisia index as the theoretically correct way to measure them, arguing that the simple-sum index conventionally used by central banks “is severely defective.” Barnett shows that a monetary aggregate is meaningful as an economic variable only if the representative consumer’s utility function is blockwise weakly separable in the relevant monetary assets – utility can be written as a function of “other goods” and a subutility function over the assets to be aggregated – and that, given this separability plus linear homogeneity of the subutility function, Green’s (1964) theorem permits two-stage (or n-stage) budgeting in which total monetary expenditure is allocated first across blocks, such as transaction balances versus a passbook-savings aggregate, and then within each block across its component assets. Extending his own 1978 user-cost formula to incorporate taxation, Barnett prices each asset’s foregone return relative to a benchmark bond yield and estimates nested CES demand systems by full-information maximum likelihood on quarterly U.S. data, 1970Q1-1978Q1: across three types of passbook-savings institutions (commercial bank, savings and loan, mutual savings bank) the estimated elasticity of substitution is high, sigma = 2.66 (beta-hat = 0.62), so high that a near-linear approximation “may be a reasonable approximation” at that level, though unequal estimated weights (alpha-hat = 0.55, 0.26, 0.20) still statistically reject simple summation; but between transaction balances and the passbook aggregate substitutability is much lower, sigma = 0.28 (beta-hat = -2.53), which Barnett calls “too low to justify a linear approximation,” so simple-sum M2 is inappropriate at that level of aggregation. Barnett then shows that the Diewert-superlative Tornquist-Theil Divisia index and the Fisher Ideal index – both exact or near-exact for a broad class of well-behaved aggregator functions – agree to three decimal places when computed for M3 over 1968Q1-1978Q1, while simple-sum M3 velocity over the same period declines secularly (from 1.0 in 1968Q1 to about 0.906 in 1978Q1) even as Divisia M3 velocity stays comparatively stable (roughly 1.00-1.07); Barnett interprets this as evidence that simple-sum aggregation misreads 1970s substitution toward higher-yielding, less-regulated monetary assets as a decline in the “quantity of money,” when in fact the Divisia-measured aggregate is roughly unchanged.

Empirical Evidence on the Recent Behavior and Usefulness of Simple-Sum and Weighted Measures of the Money Stock

This 1994 Federal Reserve Bank of St. Louis Review paper by K. Alec Chrystal and Ronald MacDonald asks whether replacing simple-sum monetary aggregates with expenditure-weighted Divisia (or Rotemberg Currency Equivalent) aggregates actually improves money’s empirical usefulness for predicting nominal income and for detecting causal links with real activity, and it tests this question reduced-form, with no structural identification, across seven countries: the United States, United Kingdom, Australia, Germany, Switzerland, Canada, and Japan. The comparison runs on two tracks: modified St. Louis equations (regressing quarterly log-differenced GNP/GDP on current and four lags of log-differenced government spending and money, plus the Treasury bill rate for the US), scored with three non-nested test statistics (the Akaike Information Criterion, the Davidson-MacKinnon J-test, and the Fisher-McAleer JA-test) and exclusion F-tests; and a time-series causality track using Augmented Dickey-Fuller unit-root tests, Johansen cointegration analysis, and heteroskedasticity-robust Granger-type VECM exclusion tests, with country-specific quarterly samples spanning roughly the late 1960s/1970s through 1987-1992. The results are country-specific rather than uniform: for the US, Divisia dominates simple-sum for M2 and broader aggregates (Divisia M2’s F-test for exclusion is F(5,107)=4.73, p=0.001, versus simple-sum M2’s F(5,107)=4.43, p=0.001) but simple-sum still beats Divisia for the narrow M1/M1A aggregates; the UK and Australia show clearer, broader Divisia superiority (Australia is called “probably the clearest case” of Divisia dominance, especially for broad money); Germany and Switzerland show weak informational content for money generally; Canada shows strong Divisia dominance for M2, M3, and L; and Japan is an explicit outlier where simple-sum is favored by the information criterion yet no aggregate, weighted or unweighted, is significant in the F-tests. A US pre-1980 sub-sample (1960:1-1979:3) shows the Divisia advantage present but markedly weaker, consistent with the authors’ interpretation that Divisia’s edge stems from measurement error in simple-sum aggregates that is largest during the post-1980 period of rapid financial innovation. The authors are explicit that their St. Louis-equation results speak only to relative, not absolute, performance of competing money measures, and they conclude that the evidence for Divisia, while real in several countries, is not yet robust enough to recommend it as a direct policy-targeting variable.

Measurement Matters: Recent Results from Monetary Economics Reexamined

This 1996 Journal of Political Economy paper by Michael T. Belongia asks a narrow but consequential question: do the qualitative conclusions of recent monetary economics depend on whether money is measured as a conventional simple-sum aggregate (just adding up the dollar face values of currency, checking deposits, savings deposits, and so on) or as a Divisia aggregate (which weights each component by its expenditure share in the flow of monetary services, using the Barnett 1981 user-cost formula, so that a dollar of a low-yield, highly liquid asset counts differently from a dollar of a higher-yield, less liquid one)? Rather than proposing new theory, the paper is a comparative replication study: it takes five previously published, prominent empirical results in monetary economics and re-runs each one twice, once with the original simple-sum aggregate and once substituting a Divisia (or Divisia M1-plus) aggregate in its place, using U.S. quarterly data mostly spanning 1960:2-1992:4 (monthly for the fifth case). In four of the five cases the qualitative conclusion flips or is substantially altered by the choice of aggregate: extracted money innovations from the De Long-Summers (1988)/Rotemberg (1993) equation fit similarly well by R-squared (0.36 for simple-sum M1 versus 0.23 for Divisia M1) but diverge sharply at business-cycle turning points (1989:3 and 1990:3); Cover’s (1992) test of symmetric effects of positive versus negative money shocks rejects the symmetry null in 5 of 6 specifications for simple-sum M1 (Wald chi-square roughly 25.3-43.5) but cannot reject it in any specification for Divisia M1 or Divisia M1-plus (chi-square roughly 1.01-1.03); the Kydland-Prescott (1990) cyclicality exercise finds that simple-sum M1 does not behave procyclically (correlations with the cycle are small but uniformly positive, peaking around 0.35) while Divisia M2 and Divisia M1-plus are strongly procyclical (contemporaneous correlations of about 0.48 and 0.47 with real output, peaking near 0.70 and 0.65 when leading the cycle by two quarters), a divergence that Belongia shows is stable across the 1960-79 and 1970-89 subsamples for the procyclical measures; and in Stock-Watson/Friedman-Kuttner predictive-power VARs using the T-bill rate, Divisia M1-plus is the only aggregate with statistically significant F-statistics for predicting output (industrial production) at multiple sample end-dates (F = 4.23 for the sample through 1985:12 and F = 3.76 for the longer sample through 1990:12, both with six lags of money), with simple-sum aggregates performing poorly, though none of the money measures — including Divisia M1-plus — shows a significant effect when the commercial paper rate is used instead of the T-bill rate. Belongia argues the mechanism is that simple-sum aggregation implicitly assumes perfect substitutability among monetary assets, a condition that Hicks (1946) aggregation theory requires but that is violated once components pay different yields, and that this measurement error became especially acute after the 1980 Depository Institutions Deregulation and Monetary Control Act introduced interest-bearing NOW accounts. The paper is explicit that this is comparative replication of five results the author judged prominent, not a representative or exhaustive sample, that it identifies no structural monetary policy shock, and that its sample ends in 1992:4, well before the near-zero-rate period when simple-sum and Divisia measures would be expected to diverge most.

Monetary transmission in money markets: The not-so-elusive missing piece of the puzzle

This 2021 Journal of Economic Dynamics and Control paper by Zhengyang Chen and Victor Valcarcel shows that the Wu-Xia (2016) shadow federal funds rate — a standard modern-sample proxy for the stance of monetary policy that extends below the zero lower bound — produces a persistent, statistically significant price puzzle in a time-varying-parameter VAR estimated on 1988-2020 U.S. data, and that this puzzle survives the standard fixes (adding commodity prices, federal funds futures, or forward rates) that resolved the price puzzle in earlier, pre-1988 samples. In its place, the authors propose Divisia monetary aggregates — weighted monetary aggregates that account for the different liquidity services of component assets, rather than simply summing dollar balances — as an alternative policy indicator: using Divisia M4 (and, with more muted magnitudes, the narrower Divisia M2) in place of the shadow rate produces no puzzling price response in the first three months and the theoretically correct price-level increase at 18-, 30-, and 60-month horizons following an expansionary shock, a correction the authors show is not an artifact of their time-varying estimation approach since it also holds in a constant-parameter VAR. Extending the analysis to the transmission of monetary shocks into 14 disaggregated money-market components (spanning currency, deposits, retail and institutional money-market funds, time deposits, repurchase agreements, commercial paper, and Treasury bills), the paper documents that transmission strengthened substantially after the 2007 financial crisis, with patterns in the responses of savings deposits and less-liquid institutional instruments that the authors interpret as evidence of a “flight-to-safety” effect among both households and firms during and after the crisis. The authors attribute the shadow rate’s breakdown as a policy indicator to the Federal Reserve’s increased forward-looking transparency (making it harder to generate a true interest-rate “surprise”) and to the post-crisis shift from reserve scarcity to reserve abundance, concluding that reintroducing monetary aggregates — measured correctly via the Divisia index rather than simple summation — may be “the missing piece of the puzzle” in a low-rate environment where a key short-term policy rate is highly persistent.

Targeting Constant Money Growth at the Zero Lower Bound

This 2018 International Journal of Central Banking paper by Michael T. Belongia and Peter N. Ireland asks two linked questions: could the Federal Reserve have maintained a constant rate of broad (Divisia) money growth once the federal funds rate hit its zero lower bound in 2008-09, and would doing so have produced a stronger, more rapid recovery than the unconventional policies (quantitative easing, forward guidance) actually pursued? The authors build a six-variable structural VAR over 2000:Q1-2016:Q2 (66 quarterly observations, 2 lags) in the GDP deflator, real GDP, an interest rate (either the Wu-Xia (2016) shadow federal funds rate or the two-year Treasury yield), Divisia M1 or M2 (Center for Financial Stability measures), the Divisia user-cost index, and the Gilchrist-Zakrajsek excess bond premium, estimating the system four times (two interest-rate measures times two money measures). Rather than the standard lower-triangular (Cholesky) identification – whose money-demand equation produces an implausible negative income elasticity, a supply-side rather than demand-side pattern – they identify the model with a non-recursive scheme (extending Belongia and Ireland 2015, 2016b) that lets the policy rule respond to money alongside prices and output, ties money demand to real balances relative to income with the Divisia user cost (not the interest rate) as opportunity cost, and lets a monetary-system equation link user cost to both interest rates and real balances; this non-recursive model imposes three testable over-identifying restrictions that a likelihood-ratio test fails to reject in all four specifications (p-values 0.28-0.71). Using the non-recursive SVAR to simulate constant-money-growth counterfactuals from 2008:Q1 forward while holding all other historical shocks at their realized values, the authors find that, although the faster-money-growth counterfactual would have required interest rates to fall more quickly at the onset of the recession than they actually did, the counterfactual rate paths lie above the historical path from 2011 onward, and (in the milder constant-historical-rate-growth scenario) the shadow rate never falls as far below zero as the actual historical path eventually does. A faster-money-growth counterfactual (12% for M1, 8.5% for M2, versus average historical rates of about 9% and 6.75%) raises average real GDP growth from its historical 1.19% to 1.67% per year over 2008:Q1-2016:Q2 (from 2.04% to 2.59% during the 2010:Q1-2016:Q2 recovery period specifically, in the benchmark shadow-FFR/M1 specification) while leaving inflation essentially unchanged (GDP deflator growth of 1.63% under both the historical and counterfactual paths in the recovery period); a slower-money-growth counterfactual (6% M1 / 5% M2), by contrast, would have cut average real GDP growth to roughly 0.57% and would have required even lower nominal rates than actually observed, which the authors interpret – following Friedman and Schwartz (1963) on the Great Depression – as reflecting excessively tight monetary policy despite low rates. These conclusions are robust across all four VAR specifications, though a price puzzle (a short-run rise in the price level after a contractionary shock) persists throughout, and monetary policy shocks account for only about 3-4% of GDP-deflator forecast-error variance versus roughly 20% of real GDP variance at four-to-five-year horizons, leaving the model’s inflation channel comparatively weak.

The Barnett critique after three decades: A New Keynesian analysis

This 2014 Journal of Econometrics paper by Michael Belongia and Peter Ireland asks whether Barnett’s (1980) decades-old critique of simple-sum monetary aggregation – that adding up the nominal values of different liquid assets as if they were perfect substitutes is theoretically inconsistent with monetary aggregation theory – still applies inside a fully modern, dynamic, stochastic New Keynesian (DSGE) model. The authors build a calibrated (not estimated) NK model in which a representative household derives liquidity services from both currency and interest-bearing bank deposits through a CES aggregator (governed by a substitution elasticity omega and a steady-state currency-share parameter v), and in which a representative bank sets deposit rates through a zero-profit condition tied to a reserve ratio and a financial-sector cost shock. Using a standard quarterly Kydland-Prescott calibration (beta=0.99, a markup of 20%, benchmark omega=1.5, v matched to US M2-to-consumption and currency-to-M2 ratios over 1959-2009, and a benchmark Taylor rule with interest-smoothing 0.75 and an inflation-response coefficient of 0.30), the authors simulate impulse responses to six structural shocks – money demand, household preference, technology, bank reserve-ratio (“reserves demand”), deposit-servicing cost, and monetary policy – and compare three measures of money: the model’s true theoretical aggregate, a Divisia (Tornqvist-Theil) index built from time-varying expenditure shares, and a conventional simple-sum aggregate. The central finding is that the properly weighted Divisia quantity index tracks the true monetary aggregate’s impulse responses so closely that, plotted together, “the two lines would appear indistinguishable” (p. 11), and this holds across all six shocks and across a wide range of substitution elasticities (omega = 0.10 to 5.0) – a result the authors attribute to Diewert’s (1978) theorem that the Tornqvist-Theil Divisia index is a second-order approximation to any linear-homogeneous aggregator regardless of its true functional form or parameter values. By contrast, the simple-sum aggregate – which implicitly weights currency and deposits 1:1 as perfect substitutes – diverges substantially from the true aggregate, and can even differ in the sign of its response, especially following reserve-ratio shocks, deposit-cost shocks, and monetary policy shocks; an analogous Divisia-versus-simple-weighted-average contrast holds on the price (opportunity-cost) side of money as well. The simulations further show that under the benchmark backward-looking Taylor rule, financial-sector shocks generate persistent output declines that ordinary interest-rate policy cannot offset, because the underlying disturbance is a change in the money multiplier rather than in inflation; adding a very large output-gap response coefficient to a backward-looking Taylor rule can insulate output from all five non-technology shocks while still allowing an efficient output response to a technology shock, whereas pushing the same coefficient into a forward-looking rule specification instead produces indeterminacy. All reported results are calibration/simulation outcomes from a theoretical model – the paper contains no empirical estimation.

The User Cost of Money

This 1978 Economics Letters paper by William A. Barnett asks what the correct price of holding a monetary asset actually is – the “user cost” or equivalent rental price of the liquidity services a monetary asset provides – and derives that price rigorously from an explicit consumer optimization model rather than from informal reasoning about stocks and flows. Barnett sets up a discrete-time Fisherine intertemporal consumption-allocation model in which a representative consumer, over a planning horizon of T+1 periods, chooses paths of goods consumption, holdings of n monetary assets, and bond holdings subject to a period-by-period budget constraint; monetary assets are assumed blockwise weakly separable from the other arguments of utility, and labor supply is exogenous. Solving the period constraints backward from the terminal bond holding yields a single intertemporal wealth constraint whose left-hand side prices monetary-asset holdings each period at their user cost. For the current period this user cost reduces to p_it = p*_t (R_t - r_it)/(1+R_t), where p*_t is the aggregate price index, R_t is the yield on the benchmark bond, and r_it is the nominal own-yield on monetary asset i – the same formula Donovan (1977) had proposed on informal grounds. Barnett’s contribution is to show this formula is in fact correct and unique within an explicit optimizing model, without needing to assume any particular functional relationship between asset stocks and the service flows they yield. The formula measures the opportunity cost of holding asset i rather than the benchmark bond, deflated by the gross benchmark rate; it does not depend directly on the inflation rate, though nominal yields can be expected to embed expected inflation, and the paper notes (in a footnote) that the user cost of monetary assets relative to durables rises as expected inflation rises. As a pure theory paper, it reports no empirical estimates; its results are the derivation itself and the scope conditions – discrete time (approximating a continuous-time model over longer intervals), constant within-period portfolio stocks, end-of-period interest payment, and exogenous labor supply – under which the formula holds. This user-cost formula subsequently became the foundation for Barnett’s (1980) Divisia monetary aggregation.