The Decay of cay: Dauber & Lawrenz (2026)

What this is. The paper’s core results, the model that motivates cay (the Campbell-Mankiw intertemporal budget constraint approximation developed by Lettau and Ludvigson (2001)), and the estimation and forecasting procedure with the defining equations: enough to understand what was found and why, without reading all 20 pages. To replicate or extend, read the full source at the original.

TL;DR

The paper revisits the ability of the consumption-wealth ratio (cay) to forecast US stock market excess returns and documents that its predictive power has declined substantially over the last two decades. Using comprehensive in-sample, out-of-sample, and economic significance tests on quarterly US data from 1952:1 to 2019:4, the authors show that aggregate cay has lost even its in-sample predictive ability from the perspective of the most recent data. They trace this decay to a structural shift in the underlying cointegration relationship between consumption, aggregate wealth, and labor income, as asset wealth has become increasingly detached from aggregate consumption since around 2000 due to rising wealth inequality. As a partial remedy, they propose a version of cay constructed from the top 10% richest households (cay^PCE10), which remains the most stable and significantly predictive alternative among those examined, though even this measure’s predictive advantage over a naive historical mean strategy has largely disappeared.

Core results

Magnitudes and significance are as reported; \* = 10%, \*\* = 5%, \*\*\* = 1%. Locators point into the source PDF.

#	Result	Locator	Magnitude
R1	Aggregate cay (NDS) has lost in-sample predictive power for 1Q-ahead excess returns in the full sample	Table 3, p. 8	beta = 0.241, t = 1.118, adj-R2 = 0.001; compare to Lettau and Ludvigson (2001) sample: beta = 2.165, t > 3, adj-R2 = 9%
R2	cay^PCE10 (top-10% PCE version) retains marginal in-sample significance at the 1Q horizon	Table 3, p. 8	beta = 0.629, t = 2.417**, adj-R2 = 0.020; aggregate NDS coefficient is insignificant (t = 1.118)
R3	Cointegration between consumption, wealth, and income cannot be confirmed for the full sample	Table 4 Panel B, p. 13	Phillips and Ouliaris (1990) test p-values: NDS = 0.868, PCE = 0.577 (full sample 1952:1-2019:4); p-values below critical values for the Lettau and Ludvigson (2001) sample only
R4	Asset wealth coefficient in the cointegrating vector has declined steadily since the late 1990s	Fig. 5, p. 13	beta_a (NDS) falls from approximately 0.3 in the early 1990s toward 0.0 by 2019; becomes insignificant from the perspective of the full sample (Table 4 Panel A)
R5	Out-of-sample economic performance decays across all specifications except cay^PCE10	Fig. 4, pp. 11-12	Risk-adjusted abnormal return theta starts at approximately 200bp for all specs in mid-1990s; all aggregate cay specs exhibit double-digit negative theta by 2019; cay^PCE10 ends near 10bp
R6	cay^PCE10 is the most stable and significant alternative among four proposed improvements in the full sample	Table 6, p. 17	Full sample 1Q t-statistics: cay^PCE10 = 2.417**, cday = 1.693, cay^g = 1.732, cay^unfi = 1.299

Overall (paper’s conclusion). The predictive ability of cay has fundamentally weakened over the last roughly two decades. The decay is traceable to a structural shift in the cointegrating relationship between consumption, aggregate wealth, and labor income: as asset wealth has become increasingly detached from aggregate consumption (particularly since the global financial crisis), the cointegrating parameters have drifted, undermining both the theoretical rationale and the empirical performance of cay. Focusing on the richest 10% of households via cay^PCE10 mitigates but does not eliminate the decay.

Theory / model

The model builds on the representative agent’s intertemporal budget constraint (p. 4, Eq. 1):

$$W_{t+1} = (1 + R_{w,t+1})(W_t - C_t), \tag{1}$$

where $W_t$ is aggregate wealth, $C_t$ is aggregate consumption, and $R_{w,t+1}$ is the return on total wealth between periods $t$ and $t+1$. Following Campbell and Mankiw (1989) and Lettau and Ludvigson (2001), log-linearizing around the steady state yields the log consumption-wealth approximation (p. 4, Eq. 2):

$$c_t - w_t \approx \mathbb{E}_t \sum_{i=1}^{\infty} \rho^i (r_{w,t+i} - \Delta c_{t+i}), \tag{2}$$

so cay proxies for expected future returns and expected consumption growth. Because total wealth $W_t$ includes unobservable human capital $H_t$, Lettau and Ludvigson (2001) approximate it using log labor income $y_t$ (p. 4, Eq. 3-4):

$$w_t \approx \alpha a_t + (1-\alpha) h_t, \quad r_{w,t} \approx \alpha r_{a,t} + (1-\alpha) r_{h,t}, \tag{3-4}$$

where $a_t$ is log asset wealth, $\alpha$ is the average share of assets in total wealth, and $r_{a,t}$ ($r_{h,t}$) is the return on assets (human capital). Together with a stationarity argument, this gives the defining identity (p. 4, Eq. 6):

$$\text{cay}_t = c_t - \alpha a_t - (1-\alpha) y_t. \tag{6}$$

If $c_t$, $a_t$, and $y_t$ are cointegrated with the vector $(1, -\alpha, -(1-\alpha))$, then $\text{cay}_t$ is stationary and forecasts future asset returns. The paper’s central finding is that this cointegrating relationship has become unstable: the coefficient $\alpha$ on asset wealth has drifted steadily toward zero since the late 1990s, which it attributes to rising wealth inequality making asset wealth increasingly detached from aggregate consumption.

Top-10% version. To better capture the representative investor, the paper constructs a version of cay using consumption, wealth, and income of the top 10% richest households. Wealth share $AS^{\text{top10}}_t$ and income share $YS^{\text{top10}}_t$ of this group are regressed on the capital share $KS_t$ (p. 5, Eqs. 7-8):

$$AS^{\text{top10}}_t = \alpha_A + \beta_A \, KS_t + \varepsilon_t, \quad YS^{\text{top10}}_t = \alpha_Y + \beta_Y \, KS_t + \varepsilon_t. \tag{7-8}$$

The fitted values $\widehat{AS}^{\text{top10}}_t$ and $\widehat{YS}^{\text{top10}}_t$ are then used to construct top-10% series for aggregate consumption, income, and wealth. The top-10% version of cay is then (p. 5, Eq. 9):

$$\text{cay}^{\text{top10}}_t = c^{\text{top10}}_t - \alpha^{\text{top10}} a^{\text{top10}}_t - (1-\alpha^{\text{top10}}) y^{\text{top10}}_t. \tag{9}$$

The rationale, following Lettau et al. (2019), is that rich households own a disproportionate share of stock market wealth and better approximate the marginal investor whose expectations drive equity premia.

Method

Cointegration estimation (DLS). The paper follows Lettau and Ludvigson (2001) in using the dynamic-least-squares (DLS) technique of Stock and Watson (1993), which builds on time-series-forecasting and the proposed dynamic-least-squares technique. To estimate the cointegrating parameters $\beta_a$ and $\beta_y$, the following regression is run (p. 6, Eq. 10):

$$c_t = \alpha + \beta_a a_t + \beta_y y_t + \sum_{i=-8}^{8} b_{a,i} \Delta a_{t-i} + \sum_{i=-8}^{8} b_{y,i} \Delta y_{t-i} + \varepsilon_t, \tag{10}$$

where $\Delta$ denotes the first-difference operator. The 8-lead and 8-lag augmentation removes regressor endogeneity that would otherwise cause OLS to be inconsistent. The estimated cay is then the residual from the cointegrating equation (p. 6, Eq. 11):

$$\widehat{\text{cay}}_t = c_t - \hat{\beta}_a a_t - \hat{\beta}_y y_t. \tag{11}$$

Stability analysis. The paper also re-estimates the cointegration relationship in an expanding window beginning in 1990, tracking the time-varying behavior of $\hat{\beta}_a$ and $\hat{\beta}_y$. A Vector Error Correction Model (VECM) is estimated to assess the short-term dynamics of the cointegration relationship for the two sample periods.

OOS economic significance. To measure economic performance, the paper follows the approach of Della Corte et al. (2010), adapted for a short-selling constrained mean-variance investor who allocates between the risk-free asset and the market portfolio. The optimal weight on the market portfolio at time $t$ is (p. 11, Eq. 13):

$$w_t = \frac{1}{\lambda} \frac{\mathbb{E}_t[r_{t+1} - r_{f,t+1}]}{\text{Var}_t[r_{t+1} - r_{f,t+1}]}, \tag{13}$$

with $\lambda = 3$ as the coefficient of relative risk aversion. The ex-post performance of the cay-timing strategy relative to a rolling historical mean strategy is evaluated by the risk-adjusted abnormal return $\theta$ (Goetzmann et al. (2007)), measured in basis points.

Empirical specifications

All results use quarterly data, 1952:1-2019:4. The main baseline sample excludes the Covid-19 episode (post-2019:4); a robustness check through 2022:4 is in Appendix H and does not affect conclusions.

In-sample predictive regression (R1, R2, R6). The standard regression forecasting excess returns at horizon $H$ is (p. 8, Eq. 12):

$$\bar{r}_{t,H} = \alpha^k + \beta^k_{\text{cay}} \widehat{\text{cay}}^k_t + \varepsilon^k_{t,H}, \quad k \in \{\text{NDS, PCE, NDS10, PCE10}\}, \tag{12}$$

where $\bar{r}_{t,H} = r_{t+1} - r_{f,t+1} + \cdots + r_{t+H} - r_{f,t+H}$ is the H-period cumulative log excess return. Newey and West (1987) corrected t-statistics are reported. Table 3 (p. 8) covers horizons $H \in \{1, 2, 4, 8, 12, 16, 20\}$ quarters. A time-varying version re-estimates the regression recursively in an expanding window from 1990 to track how the coefficient $\hat{\beta}_{\text{cay}}$ has evolved (Fig. 3, pp. 9-10).

Cointegration stability tests (R3, R4). The paper re-estimates the cointegrating regression in an expanding window from 1990, reporting the resulting $\hat{\beta}_a$ and $\hat{\beta}_y$ paths (Fig. 5, p. 13). Cointegration tests use the Phillips and Ouliaris (1990) test; results for Engle and Granger (1987) and Johansen (1988, 1991) tests appear in Appendix C.2. Table 4 (p. 13) compares cointegrating parameters and test statistics across sample periods.

Structural shift with time trend (R4 extended). Section 4.2 augments Eq. (10) with a deterministic time trend $\hat{\pi} t$ to assess whether the shift in $\hat{\beta}_a$ is driven by an omitted trend. The cointegrating parameters are re-estimated with $c_t - \hat{\pi} t$ as the dependent variable; the resulting time paths of $\hat{\beta}_a$, $\hat{\beta}_y$ and $\hat{\pi}$ are tracked in an expanding window (Fig. 7, p. 16).

OOS economic performance (R5). The cay-timing strategy is initialized with a 43-year training period (through 1994:4) and evaluated from 1995:1 onward. The risk-adjusted abnormal return $\theta_t$ is plotted over time for all four cay specifications (Fig. 4, pp. 11-12).

Comparison with alternatives (R6). Table 6 (p. 17) runs Eq. (12) at the 1Q horizon for four competing cay specifications (cay^PCE10, cday from Sousa (2010), cay^g from Guo (2006), and cay^{unfi} from Kroencke (2017)) using PCE consumption, reporting coefficient estimates, Newey-West t-statistics, and adjusted R2 statistics for both the Lettau and Ludvigson (2001) sample period (through 1998:3) and the full sample (through 2019:4).

Datasets used

Dataset	Role in paper	Wiki page
BEA NIPA tables: nondurables and services (NDS) and PCE consumption; labor income	Consumption and income series for cay estimation	FRED
Federal Reserve Financial Accounts (Flow of Funds)	Aggregate asset wealth series for cay estimation	FRED
World Inequality Database (WID) / US Distributional National Accounts	Wealth and income shares of top 10% households (AS^top10, YS^top10)	no page yet
CRSP NYSE/NYSE MKT/NASDAQ/Arca Value-Weighted Market Index	Stock market excess return (dependent variable)	WRDS / CRSP (licensed)
BLS nonfarm business labor share	Labor share for KS construction; interest rate proxy	FRED
Federal Reserve H.15: 3-Month Treasury Bill rate	Risk-free rate proxy	FRED

Sample: quarterly, 1952:1-2019:4 (272 quarters). All nominal series deflated using the PCE deflator from the BEA. Inequality shares from the World Inequality Database (Saez and Zucman (2016) series, continued in the World Inequality Database (WID)).

When to read the full paper

Read the original if you are: examining the structural stability of cay over time (Sections 4-4.2 and Fig. 5-7 give the fullest treatment); comparing Welch and Goyal (2008) out-of-sample failure results against IS evidence in a single paper; assessing whether adjusting cay for wealth inequality (Lettau et al. (2019) argument) restores predictability; or using the Brennan and Xia (2005) look-ahead bias critique and want the authors’ detailed response. The Appendices (C-H) cover additional cointegration tests, the direct multivariate regression, consumption predictability, OOS tests, alternative specifications, and Covid-19 robustness.

Attribution and rights

Source: peer-reviewed, Journal of Empirical Finance 85 (2026) 101668. This distillation was extracted by an LLM on 2026-06-25 and is not human-verified or independently reproduced. The CC BY 4.0 licence permits mirroring; the verbatim PDF is not hosted in this batch.

Attribution (CC BY 4.0). Dauber, Moritz, and Jochen Lawrenz. “The Decay of cay.” Journal of Empirical Finance 85 (2026): 101668. DOI: 10.1016/j.jempfin.2025.101668. © 2025 The Author(s). Licensed under CC BY 4.0. This page is an adaptation by the Institute for Automated Research: core results extracted and re-expressed; changes were made.