Journal of Econometrics | Macro Paper Warehouse

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.

Joint Bayesian inference about impulse responses in VAR models

This 2022 Journal of Econometrics paper by Atsushi Inoue and Lutz Kilian develops a Bayesian decision-theoretic framework for reporting inference about an entire vector of structural VAR impulse responses jointly, rather than horizon by horizon, and shows that the conventional practice of reporting the posterior median or mean impulse response at each horizon together with pointwise credible intervals is not in general the Bayes estimator and can be economically misleading. Formally, letting theta = g(lambda) denote the n_irf-dimensional vector of structural impulse responses implied by the underlying VAR parameters lambda, the Bayes estimator minimizes posterior expected loss over the feasible response space Theta under a chosen loss function – quadratic, absolute, Dirac delta (which yields the posterior mode), or an angular loss invariant to the scaling of the data and structural shocks – and the corresponding joint credible set is constructed as the region of lowest posterior risk (for quadratic or absolute loss) or highest posterior density (for Dirac delta loss), visualized as a “shot-gun” trajectory plot of complete retained impulse-response paths rather than as a separate interval at each horizon. The key theoretical result is that when the number of estimated impulse responses n_irf exceeds the number of underlying structural parameters n_p – the empirically typical case – the space of feasible response vectors Theta is a lower-dimensional manifold embedded in R^{n_irf}, so the vector of pointwise posterior medians or means need not lie in Theta at all; a stylized AR(1) illustration with posterior rho ~ N(0.7, 1/5) yields a mean-response vector across horizons 1-4 of [0.70, 0.69, 0.76, 0.95], a shape that first falls and then rises and is infeasible for any AR(1) coefficient. In two empirical illustrations – a quarterly VAR(4) identifying aggregate demand, aggregate supply, and monetary policy shocks for U.S. real GNP growth, the federal funds rate, and GNP deflator inflation over 1954:IV-2007:IV, exactly identified via short- and long-run exclusion restrictions, and a monthly VAR(24) for the global crude oil market over 1973:2-2018:6 with flow supply, flow demand, and storage demand shocks set-identified via narrative, sign, and impact-elasticity bound restrictions – the paper finds that conventional pointwise 68% error bands “grossly understate” joint estimation uncertainty about the response vector, with the range spanned by the responses in the joint credible set in some cases three times as wide as the conventional bands. Across both examples the absolute, quadratic, and Dirac delta loss functions produce Bayes estimates that are quite similar to one another, and angular loss gives similar answers too, though at greater computational cost, so the paper concludes there is “little to choose” between the loss functions in practice and recommends absolute or quadratic loss as computationally cheap defaults.

Large Bayesian vector autoregressions with stochastic volatility and non-conjugate priors

This 2019 Journal of Econometrics paper by Andrea Carriero, Todd Clark, and Massimiliano Marcellino develops a computationally efficient Markov Chain Monte Carlo (MCMC) algorithm for estimating large Bayesian vector autoregressions (VARs) – models with many included variables – that feature stochastic volatility (time-varying error variances) together with non-conjugate priors, meaning economically motivated shrinkage priors (Minnesota, Sims-Zha, cross-variable shrinkage) whose independent Normal-Wishart form breaks the Kronecker structure that the standard closed-form conjugate posterior requires. The key device is a triangular (Cholesky-type) factorization of the time-varying error covariance matrix, used purely as a computational tool rather than as a structural identification assumption – the authors argue (Section 3.1, “The role of variable ordering”) that the draw of the VAR coefficients from their conditional posterior via this triangular recursion is invariant to the ordering of variables used to build A – which lets the VAR coefficients be drawn equation-by-equation with standard GLS formulas instead of requiring the full system-wide posterior precision matrix to be formed and inverted. This lowers the computational complexity of the coefficient draw from O(N^6) to O(N^4): for a 20-variable VAR the paper reports that estimation using the traditional system-wide algorithm was about 356 times slower than the triangular algorithm (Section 4.1), with the advantage growing further as the number of variables rises. The algorithm makes estimation of very large systems tractable: the paper estimates a 125-variable monthly VAR with 13 lags (roughly 203,250 mean coefficients) on the FRED-MD dataset, drawing 5,000 posterior draws in about seven hours, and documents substantial heterogeneity in stochastic volatility across financial, real, and price variable groups; evidence consistent with the Great Moderation, with posterior median volatilities declining from the 1970s/early 1980s into the mid-1980s; a first principal component of log-volatilities that accounts for roughly 45% of total volatility variance; and recursively identified monetary-policy-shock responses (with the federal funds rate ordered last) consistent with Banbura, Giannone, and Reichlin (2010): a contractionary shock lowers output and inflation with no price puzzle. In a second, forecasting application using four macroeconomic variables over 1960:3-2014:5 with 531 recursive out-of-sample forecasting exercises, the paper finds that a large cross-section improves point forecasts (RMSE) while stochastic volatility adds little further gain there, but stochastic volatility matters for density-forecast accuracy (log scores), and that combining a large cross-section with stochastic volatility produces gains that are super-additive – larger than the sum of the gains from each ingredient taken separately – holding across variables and forecast horizons of 1, 3, 6, and 12 months.

Local Projections vs. VARs: Lessons from Thousands of DGPs

This 2024 Journal of Econometrics paper by Dake Li, Mikkel Plagborg-Møller, and Christian K. Wolf asks a purely practical question rather than proposing a new estimator or identification scheme: when researchers estimate structural impulse responses, does the local-projection (LP) estimator or the vector-autoregression (VAR) estimator perform better in realistic macroeconomic settings, and under what conditions does the ranking flip? Because no single real-world dataset can answer this — the true DGP is never known — the authors build an “encompassing model,” a non-stationary dynamic factor model with six latent factors estimated on the 207-series Stock and Watson (2016) quarterly U.S. dataset (1959Q1-2014Q4), and use it to generate 6,000 simulated economies (3,000 built around a monetary policy shock with the federal funds rate as instrument, 3,000 around a fiscal policy shock with government spending as instrument), each simulated for T=200 quarters with 5,000 Monte Carlo draws. Across this population of realistically calibrated DGPs, comparing least-squares, bias-corrected, and penalized LP against least-squares, bias-corrected, Bayesian, and model-averaged VARs (plus SVAR-IV for the instrumented case), the paper documents a clear and pervasive bias-variance trade-off: at short horizons (h ≤ the p=4 lag length) LP and VAR have similar bias, but at longer horizons VAR bias grows substantially larger than LP bias, while LP’s standard deviation rises steeply with horizon — by h=20 roughly double the VAR’s. Bias-corrected LP removes only about a third of LP’s bias while adding variance, so it is preferred over uncorrected LP only when a researcher places very high weight (ω ≥ 0.9 in the paper’s bias-variance loss function) on bias at intermediate horizons; otherwise VAR-type methods, especially a Bayesian VAR with a Minnesota-type prior, dominate essentially throughout. A further headline finding concerns SVAR-IV: because roughly 90% of the simulated DGPs exhibit a degree of shock invertibility below 49%, the external-instrument SVAR-IV estimator carries substantially higher bias than internal-instrument alternatives at every horizon, though it also has notably lower dispersion. The authors are explicit that these are simulation-based lessons conditional on the choice of encompassing model and DGP class (quarterly, five-variable systems, no sign or long-run restrictions), not universal theorems.

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.