Latent Heterogeneity in the Marginal Propensity to Consume

Lewis, Melcangi, and Pilossoph estimate the unconditional distribution of the marginal propensity to consume (MPC) using the 2008 Economic Stimulus Act (ESA) rebate payments, deploying Gaussian mixture linear regression (GMLR) — a clustering regression approach — rather than the standard practice of interacting the rebate with observable household characteristics. The key methodological departure is that households are assigned to groups not by any presupposed observable, but by how well estimated group-specific MPCs describe each household’s actual consumption response; this allows recovery of the full unconditional MPC distribution, including heterogeneity driven by latent (unobservable) factors.

Data come from the 2008 Consumer Expenditure Survey (CEX), which contains household-level expenditure data and supplemental questions on ESA payments. Identification exploits the quasi-random timing of rebate receipt, determined by the last two digits of recipients’ Social Security Numbers, following the design of Parker, Souleles, Johnson, and McClelland (2013). The specification is updated following Borusyak et al. (2024) to avoid “forbidden comparisons” in staggered treatment settings. The number of groups G is selected by BIC, which selects G = 3 for total expenditures, confirmed by K-fold cross-validation.

The main finding is substantial MPC heterogeneity. For total expenditures, the three estimated group-level MPCs are 0.04, 0.23, and 1.33, with population shares of 30%, 48%, and 23% respectively. The implied aggregate (share-weighted average) MPC is 0.42, compared to 0.24 in the homogeneous Parker et al. (2013) specification estimated on the same data. Splitting by consumption category: for nondurables, two groups have MPCs of 0.09 and 0.18, with roughly equal population shares, and the lower bound of 0.09 is statistically distinguishable from zero — evidence against strict adherence to the Permanent Income Hypothesis even among the lowest-MPC group. For durables, the MPC distribution is dichotomous: about 29% of households have a durable MPC statistically indistinguishable from zero, while 21% have an MPC of 0.67. The cross-good correlation between household-level nondurable and durable predicted MPCs is only 0.13, ruling out strong substitution but indicating weak complementarity.

Turning to observable determinants, the paper finds that many household characteristics are individually correlated with estimated MPCs — including homeownership, mortgage status, income, and the average propensity to consume (APC) — despite the fact that the same dataset and similar identification strategies previously yielded insignificant relationships. Homeowners have significantly higher MPCs than renters; households with a mortgage have even higher MPCs than outright homeowners. In salary income, households in the top tercile spend 0.17 more per rebate dollar than the baseline group; households in the top tercile of non-salary income spend 0.19 more. However, in joint regressions, only two characteristics remain robustly and positively correlated with MPCs: total income (both salary and non-salary components) and the APC. The APC relationship is particularly notable: a one-percentage-point higher prior spending rate is associated with 0.19 additional cents spent per rebate dollar in the full multivariate specification.

The paper identifies three groups in the joint income-APC space: “poor savers” (low income, low APC, lowest MPCs), an intermediate group (high income or high APC but not both), and “rich spenders” (high income and high APC, highest MPCs). The “rich spender” group has received little prior attention in consumption-savings models.

Critically, observable characteristics jointly explain at most 8% of MPC variation (adjusted R-squared from a measurement-error correction). With 92% of MPC heterogeneity unexplained by standard observables, the authors conclude that a substantial share of variation reflects latent household traits — plausibly heterogeneity in discount rates or intertemporal elasticities of substitution. This finding also limits the practical scope for government targeting of fiscal transfers: because observable characteristics predict little MPC variation, any targeting strategy can exploit only a small fraction of the overall distribution.

Scope conditions: results apply to household expenditure responses (marginal propensities to spend, not to consume in the strict sense) within one quarter of rebate receipt. The income-MPC positive correlation is confined to households within the income range eligible for the 2008 ESA (phased out above $150,000 for joint filers). The sample excludes the top and bottom 1.5% of consumption changes as outliers.

Q: What is the core methodological innovation of this paper? A: The paper applies Gaussian mixture linear regression (GMLR) to the 2008 tax rebate setting, jointly estimating group-level MPCs and household group membership probabilities without imposing any prior restriction on which observable characteristics drive heterogeneity. Because groups are determined by how well group-specific MPCs explain consumption patterns rather than by presupposed observables, the method recovers the full unconditional distribution of MPCs, including latent heterogeneity. This contrasts with sample-splitting approaches that can only recover co-variation with chosen characteristics.

Q: What are the three group-level MPCs for total expenditures, and what shares of the population do they represent? A: The three estimated MPCs are 0.04 (30% of households), 0.23 (48%), and 1.33 (23%), all with precisely estimated group shares (standard errors of 0.01). The largest MPC of 1.33 is statistically significant at the 1% level. The lowest MPC of 0.04 is not statistically different from zero even under the more favorable conditional standard errors that treat group assignment as known.

Q: How does the average MPC implied by the GMLR distribution compare to the homogeneous specification? A: The share-weighted average MPC from the three-group GMLR is 0.42, compared to 0.24 from the homogeneous (G=1) specification on the same data and identification strategy. This gap arises partly because the homogeneous estimate averages across households with very heterogeneous responses, and partly because the distribution has a right-skewed tail with a meaningful mass at MPC above 1.

Q: What are the MPC distributions for nondurable and durable goods separately? A: For nondurables, BIC selects two groups with MPCs of 0.09 and 0.18 and roughly equal population shares (48% and 52%); crucially, the lower bound of 0.09 is statistically distinguishable from zero at the 5% level, providing evidence that no household strictly follows the Permanent Income Hypothesis for nondurables. For durables, BIC selects three groups: MPCs of 0.03 (not distinguishable from zero, 29% of households), 0.15 (50%), and 0.67 (21%), reflecting the discrete, lumpy nature of durable goods purchases.

Q: How correlated are nondurable and durable MPCs at the household level? A: The correlation between household-level posterior predicted MPCs for nondurables and durables is 0.13, statistically significant at the 1% level. This rules out substitution between goods categories, but the positive complementarity is quantitatively small. The authors interpret this as possibly reflecting a small share of “spender” types who adjust multiple consumption categories in response to transitory income shocks.

Q: Which observable characteristics are individually correlated with MPCs? A: Homeowners have significantly higher MPCs than renters; households with a mortgage display even greater MPCs than outright homeowners. Both salary and non-salary income are positively correlated: households in the top tercile of salary income have MPCs about 0.13 higher than the omitted group, and top-tercile non-salary income households have MPCs about 0.015 higher (though the latter is individually less precisely estimated). The average propensity to consume (APC) is significantly positively correlated with the MPC, with a coefficient of 0.075 in univariate regression and 0.166 in the full joint specification.

Q: Which observable characteristics remain significant in the joint (multivariate) regression? A: When all household characteristics are included jointly, only income (both salary and non-salary components) and the APC remain robustly and positively correlated with MPCs. Top-tercile salary income is associated with 0.112 higher MPCs and top-tercile non-salary income with 0.049 higher MPCs, while the APC coefficient rises to 0.166 (from 0.075 univariate). Homeownership, age, education, and most demographic controls become statistically insignificant in the joint specification.

Q: What fraction of MPC variation is explained by observable characteristics? A: The adjusted R-squared from the full multivariate regression of predicted MPCs on all observable characteristics is approximately 6%. After a measurement-error correction proposed in Supplement A.6 to account for noise in estimated posterior MPCs, the corrected R-squared rises to 8%. Either way, the vast majority — over 90% — of MPC heterogeneity is unexplained by standard observables, implicating latent household traits such as heterogeneous discount rates or intertemporal elasticities of substitution.

Q: How does the extent of MPC heterogeneity recovered by GMLR compare to sample-splitting on observables? A: Table 4 shows that splitting by age terciles yields MPC estimates ranging from 0.13 to 0.34; splitting by total income yields a range of 0.18 to 0.45; splitting by the APC yields 0.06 to 0.21. All of these ranges are far narrower than the GMLR-recovered range of 0.04 to 1.33. The authors argue that sample-splitting on individual observables, which are noisy and correlated with only a portion of MPC heterogeneity, systematically understates the true extent of heterogeneity.

Q: What is the “rich spender” finding and why is it theoretically notable? A: Households with both high total income and a high prior average propensity to consume have the largest MPCs. This “rich spender” group is poorly accommodated by standard consumption-savings models: the canonical one-asset incomplete markets model typically predicts a negative MPC-APC correlation conditional on income, and the two-asset Kaplan-Violante (2014) model can generate wealthy hand-to-mouth households with high income and high MPCs, but not necessarily high APCs. Preference heterogeneity — e.g., heterogeneous intertemporal elasticities of substitution as in Aguiar, Boar, and Bils (2019) — can rationalize the positive income-APC-MPC nexus.

Q: What explains the positive income-MPC correlation, and how does the paper relate it to the prior literature? A: The paper notes that this positive correlation is consistent with Kueng (2018), who finds higher spending propensities among high-income recipients of Alaska Permanent Fund payments, and rationalizes it via near-rationality or mental accounting: when a rebate is small relative to income, the perceived cost of deviating from consumption smoothing is low. The authors also note that low-income households still exhibit large absolute MPCs, suggesting sizable deviations from consumption smoothing at the bottom of the income distribution, even if relatively lower than for high-income households.

Q: What are the policy implications for targeting fiscal transfers? A: The paper finds that the 2008 ESA increased spending for all households in partial equilibrium (minimum group MPC of 0.04, nondurable lower bound 0.09, all statistically positive or near-positive). Among observable characteristics, targeting relatively higher-income households (including retirees and entrepreneurs via non-salary income) would maximize aggregate consumption effects. However, since observables explain only 8% of MPC variation, any targeting strategy can exploit only a small fraction of the overall heterogeneity; the government faces fundamental limits on feasible targeting. This also implies a tension between stimulus and distributional/insurance motives for transfer programs.

Q: How does the paper confirm that recovered heterogeneity is not spurious? A: The authors generate 250 Monte Carlo samples from the estimated homogeneous model, impose G=3, and re-run the GMLR and observable regressions; they find significant relationships with observable characteristics in virtually none of these samples. Additionally, applying the BIC to homogeneous Monte Carlo samples, the BIC selects G=1 in all 250 samples, confirming that the selected G=3 in actual data reflects genuine heterogeneity rather than overfitting.

Q: How does GMLR compare to quantile regression for recovering the MPC distribution? A: Quantile regression (as used by Misra and Surico (2014) on the same data) recovers relationships at percentiles of the overall conditional distribution of consumption changes, so the ranking of households is driven by all sources of variation in consumption, not just the rebate response. If factors unrelated to the rebate dominate the conditional distribution, MPC heterogeneity will be underestimated in the presence of noise. The authors illustrate this formally in Supplement B and note that Misra and Surico (2014) find a substantial share of MPCs at or below zero for nondurables, in contrast to the GMLR lower bound of 0.09 that is statistically positive.

Q: What do the longer-run (lagged) MPC estimates show? A: The specification includes up to two lags of rebate indicators, allowing measurement of spending responses in subsequent quarters after rebate receipt. The paper reports these results (Section 4.4) but the text provided does not fully detail them; the heterogeneous structure is maintained across horizons.

Gaussian Mixture Linear Regression (GMLR): A probabilistic clustering regression approach that jointly estimates group-specific regression coefficients (here, MPCs) and population group shares by maximizing an expected log-likelihood via the EM algorithm. Households receive continuous posterior weights (gamma_{jg}) reflecting uncertainty about their group membership rather than binary hard assignment, with identification from a Gaussianity assumption on within-group errors.

Unconditional MPC Distribution: The full marginal distribution of MPCs across all households in the population, capturing heterogeneity from both observable and latent (unobservable) sources. Contrasted in the paper with the conditional distributions recovered by sample-splitting on observables, which by construction can only reflect co-variation with the chosen splitting variable.

Posterior Predicted MPC: For each household, the expectation of the group-specific MPC weighted by the household’s posterior group membership probabilities (lambda-tilde_{0,j} = sum_g gamma_{jg} lambda_{0g}). This object is the optimal (MSE-minimizing) individual-level MPC prediction and is the relevant input for targeted fiscal policy design.

Latent Heterogeneity: MPC variation that cannot be attributed to any observable household characteristic and is instead driven by unobserved traits — plausibly heterogeneous discount rates, intertemporal elasticities of substitution, or other preference parameters. Operationalized as the share of MPC variance unexplained by observable regressors (approximately 92% in this paper).

Rich Spenders: A group identified jointly in the APC-income space: households with both high total income and a high average propensity to consume, displaying the largest marginal propensities to consume out of the rebate. This group is not well-accommodated by standard one-asset or two-asset incomplete markets models under homogeneous preferences.

Average Propensity to Consume (APC): Defined empirically as average lagged consumption expenditures divided by total income, intended to capture persistent preference heterogeneity — a “spender type” — by measuring how much of income a household habitually spends before receiving the rebate. A one-percentage-point higher APC is associated with 0.19 additional cents spent per rebate dollar in the full multivariate specification.

Forbidden Comparisons: A bias identified by Borusyak et al. (2024) in event-study designs with staggered treatment, arising when newly treated units are compared to previously treated units rather than true controls. The paper addresses this by regressing consumption changes on rebate receipt indicators (iota_{jl}) directly rather than on rebate amounts, and including lagged rebate indicators to account for persistent effects.