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

The paper asks whether Divisia (expenditure-weighted) and Rotemberg Currency Equivalent monetary aggregates outperform conventional simple-sum aggregates at predicting nominal income and revealing money-income causal links, and whether any such superiority is a general property of monetary aggregation or a US-specific artifact. It tests this across seven countries — the United States, United Kingdom, Australia, Germany, Switzerland, Canada, and Japan — explicitly to see whether findings favoring Divisia in prior US-only work (Barnett 1980; Belongia 1993) generalize internationally.

The paper runs two separate, purely reduced-form frameworks with no structural shock identification. Part 1 estimates “modified St. Louis equations” — regressions of quarterly Δln(nominal GNP or GDP) on current and four lags of Δln(government spending) and Δln(money), plus Δ(Treasury bill rate) for the US — and compares two competing money measures at a time using three non-nested test statistics (AIC, the Davidson-MacKinnon J-test, and the Fisher-McAleer JA-test) plus F-tests for exclusion of money from the equation; each equation is also re-estimated with four lags of the dependent variable included. Part 2 runs Augmented Dickey-Fuller unit-root tests, Johansen cointegration analysis on a vector {money, GDP, deflator, T-bill rate}, and heteroskedasticity-robust (Hansen 1980; White 1978) Granger-type VECM exclusion tests to assess long-run and short-run causal links between money and real activity.

For narrow aggregates (M1, M1A), simple-sum dominates: the AIC favors simple-sum M1, and both the J-test and JA-test favor M1 over Divisia M1, with narrow Divisia (DM1) an insignificant predictor in the exclusion F-test (F=1.01, p=0.418). For M2 and broader aggregates the ranking reverses: Divisia M2 has “the greatest informational content of all the aggregates tested, though it is only marginally more significant than simple-sum M2” (p. 77), and both pass exclusion F-tests at conventional levels (Divisia M2 F(5,107)=4.73, p=0.001; simple-sum M2 F(5,107)=4.43, p=0.001; simple-sum M3=3.49, p=0.006; Divisia M3=4.09, p=0.002; Divisia L=3.86, p=0.003). The Currency Equivalent measure holds its own against narrow aggregates but loses to broader ones.

Australia shows what the authors call “probably the clearest case available which illustrates the domination of Divisia over simple-sum aggregates — especially for broad money measures” (p. 80): the information criterion always favors Divisia, and neither simple-sum aggregate is significant in the F-tests while both Divisia M2 and M3 are. The UK (Divisia M4 dominates simple-sum M4 on AIC, J-test, and JA-test, with “the superiority of Divisia M4 over M4… confirmed at least so far as inflation is concerned,” p. 98) and Canada (strong Divisia dominance for M2, M3, and L, with simple-sum M2/M3/L failing significance at 5% while their Divisia counterparts pass) show similarly strong patterns. Germany and Switzerland show comparatively weak informational content for money in general — Germany’s exclusion F-tests are significant only for Divisia M3 at the 10% level (p=0.087), which the authors speculate “possibly” reflects successful monetary policy suppressing variation in nominal income growth (p. 80). Japan is the clear exception: simple-sum aggregates are always favored by AIC, yet no aggregate of either type is significant in the F-tests, leading the authors to state that Japan “does not fit in at all with the pattern of other countries” (p. 82), a pattern they partly attribute to Japanese M2/M3 including negotiable CDs, unlike other countries’ aggregates.

Most monetary aggregates across the seven countries appear integrated of order one (I(1)); some — particularly six of eight Canadian monetary aggregates — appear I(2) in certain specifications, and only the Australian Treasury bill rate and the Rotemberg Currency Equivalent (RCE) appear stationary around a deterministic trend. In the Johansen cointegration analysis (Table 15), narrow US aggregates (simple-sum and Divisia M1/M1A) show no cointegrating vector, while broader aggregates (M2, M3, L and their Divisia counterparts) each produce at least one; the same pattern of “broader aggregates cointegrate, narrow ones do not” holds loosely across countries. The paper states explicitly that cointegration results show no sharp distinction between Divisia and simple-sum: “it is the broader monetary measures that produces a cointegrating set (and not the income, interest rate or inflation rate)” (p. 92).

In the US VECM exclusion tests (Table 16), Divisia M2, Divisia L, and RCE1 are significant in the output equation at 5% (Divisia M3 and RCE1 at 7%), while none of the simple-sum money terms enters significantly at 10% — a sharper Divisia advantage than in the St. Louis equations. Elsewhere the results are more mixed: in the UK, neither M4 nor Divisia M4 significantly affects real GNP, though Divisia M4 does influence inflation, which the authors call “noteworthy” (p. 98) as a limit on the case for Divisia there; in Australia, Divisia M2 and M3 are significant in the real-output equation while simple-sum is not; in Germany, neither aggregate type affects real output; in Canada, money has little significant impact on any variable except that simple-sum M3 and Divisia M3 matter in the T-bill equation; and in Japan neither aggregate type enters the output equation significantly, though both affect inflation and the T-bill rate. Across countries, the authors summarize that “there appear to be countries in which Divisia money has greater informational content than simple-sum money and this is most clear in the U.S. and Australian cases” (p. 103), attributing cross-country differences tentatively to differing paces of financial innovation.

Restricting the US sample to 1960:1-1979:3, the paper finds the Divisia advantage “still present but less pronounced,” concluding that “pre-1980 data do not show any support for Divisia” and that Divisia’s advantages “are particularly strong after 1980, the period in which financial innovation is greatest” (pp. 106-107). In this sub-sample every monetary measure produces at least one cointegrating vector (in contrast to the full sample, where narrow M1 measures show none), money has a significant impact on the deflator for every measure except RCE, and the Treasury bill rate is significant in the output equation in only two instances (weaker than in the full sample). This sub-sample pattern is the empirical anchor for the paper’s proposed mechanism (Q8).

The authors attribute simple-sum aggregates’ underperformance to the “Barnett Critique”: treating all monetary component assets as perfect substitutes (implicit in simple summation) introduces measurement error that is most severe when the components carry different and changing yields, and this measurement error is argued to generate instability in the empirical money-income relationship. Divisia aggregates are expected to internalize substitution effects between components and thus more accurately measure the flow of monetary services, drawing on the aggregation theory of Barnett (1980) and the prior US evidence in Belongia (1993). The advantage is argued to be largest after 1980, when interest payments on transaction deposits and broader liberalization caused large shifts between money components, maximizing the measurement error simple summation would incur. Despite proposing this mechanism, the authors are notably cautious about its policy implications, concluding that “the body of research supporting Divisia is not yet sufficiently large or robust that we would wish to recommend direct targeting at this stage” (p. 107).

The authors flag that the St. Louis-equation methodology has known critics and that its results speak only to the relative, not absolute, performance of competing money measures (“we are not concerned with the absolute validity of the results but only with the relative performance of different measures,” p. 76); they note the promised companion analysis using methodology “more acceptable to the econometric purist” (p. 77) is Part 2 of the paper. The non-nested J-test/JA-test pair is frequently “inconclusive” (both tests reject each other), leaving AIC as the only decisive criterion in many country-aggregate comparisons. The Currency Equivalent/RCE measure’s apparent trend-stationarity is flagged as making it “not an ideal candidate for the Johansen methodology” (p. 92). Finally, this is a policy-review publication (Federal Reserve Bank of St. Louis Review), not a peer-reviewed academic journal, so — as with other Fed Review pieces in this library, e.g. Andersen-Jordan (1968) — the results have not undergone standard academic peer review.