Building proxies that capture time-variation in expected returns using a VAR approach

Abstract

I use the consumer's budget constraint to derive a relationship between stock market returns, the residuals of the trend relationship among consumption, aggregate wealth and labour income, and three major sources of risk: future changes in the housing consumption share, future labour income growth and future consumption growth. I model the joint dynamics of changes in the housing consumption share, consumption–growth, wealth growth, income growth, asset returns, consumption–wealth ratio and dividend–price ratio, and show that asset returns largely reflect expectations about long-run risk. On the other hand, unexpected shocks play a negligible role in the context of forecasting future asset returns. Combining the intertemporal budget constraint and the forecasting properties of an informative Vector Autoregression (VAR), one can, therefore, generate the predictability of many economically motivated variables developed in the literature on asset pricing, and accommodate the implications of a wide class of optimal models of consumer behaviour without imposing a functional form on preferences.

Acknowledgements

I am extremely grateful to Alexander Michaelides (LSE) and Christian Julliard (LSE) for helpful comments and discussions. I also acknowledge financial support from the Portuguese Foundation for Science and Technology under Fellowship SFRH/BD/12985/2003.

Notes

¹ These authors show that Consumption CAPM (CCAPM) performs relatively better than the CAPM at longer horizons.

² See, for example, Fama and French (1988), Poterba and Summers (1988), Lettau and Ludvigson (2001a, 2004).

³ Lin et al. (2007) also analyse the issue of predictability of asset returns in the context of emerging bond markets. In contrast, Lee (2008) finds that the correlation between unpredictable stock returns and unpredictable inflation is low.

⁴ Pakos (2003) argues that there preferences are nonhomothetic.

⁵ That is,

⁶ The assumption that human capital is included in aggregate wealth explains why labour income does not appear explicitly in this equation.

⁷ Baxter and Jermann (1997) calibrate Y/H = 4.5%, which implies ρ _h  = 0.955. In this article, I set ρ _w = ρ _h = 0.95, although results do not significantly change for different values.

⁸ It can be shown that c_t  − s_t corresponds to the definition of consumption of nondurable goods and services including housing services. Denote by , the log consumption of nondurable goods and services including housing services, c_t , the log consumption of nondurable goods and services excluding housing services, and u_t , the log consumption of housing services. We can write

⁹ Note that one could also split consumption into its nondurables and durables components as in Yogo (2006). In this case, the consumer's budget constraint (Equation 1) could be written as

where W_t represents aggregate wealth, is nondurables consumption, D_t is durables consumption, is the relative price of durables consumption, S_t is the nondurables consumption share and R_{w , t+ 1} is the return on aggregate wealth between period t and t + 1.

¹⁰ The selected optimal lag length is 1, in accordance with findings from Akaike and Schwarz tests. However, the results are not sensible to different lag lengths.

¹¹ Real returns are constructed as the difference between the CRSP-VW market return index and the inflation rate. The time series are standardized to have unit variance and smoothed to facilitate the reading.

¹² I estimate cay_t and cday_t using DOLS with four lags and leads. For brevity, I only report the estimates of the coefficients associated with (dis)aggregate wealth and labour income in the cointegrating vector.

¹³ Ludvigson and Steindel (1999) suggest that the marginal propensity to consume out of stock market wealth was larger in the late 1970s and early 1980s. As a result, the estimation of cay and cday using different sub-samples could potentially improve the precision of the estimated parameters in the cointegrating relationships. Nevertheless, Lettau and Ludvigson (2005) emphasize that the difficulty with this procedure is that it can also strongly understate the predictive power of the regressor, making it difficult for cay and cday to exhibit forecasting power when the theory is true.

¹⁴ This is because the consumption–(dis)aggregate wealth ratio is computed using the DOLS approach, while the remaining empirical proxies that capture time-variation in asset returns are built upon the VAR approach. By doing so, I keep consistency with the work of Lettau and Ludvigson (2001a), which makes results comparable. Moreover, while the consumption–wealth ratio helps explaining future returns per se, in the other factors it is the long-run variation that has informational content and their construction requires the use of the VAR framework.

¹⁵ The predictive impact of cday on future returns is economically larger than that of cay: in the one-period ahead regressions, the point estimate of the coefficient on cday is about 1.549 for real returns and only 1.164 in the case of cay. Thus, a one SD increase in cday (SD is 0.019) leads to, approximately, a 82.07 basis points rise in the expected real return on CRSP-VW index, that is, a 3.32% increase at an annual rate. On the other hand, cay itself has a SD of about 0.023, implying that a one SD increase in cay leads to, approximately, a 50 basis points rise in the expected real return on CRSP-VW index, that is, a 2.02% increase at an annual rate.

¹⁶ Dunn and Singleton (1986) and Eichenbaum and Hansen (1990) also find evidence against separability of preferences, but this does not help pricing risk.

¹⁷ When cay is replaced by cday in the several specifications, the results slightly improve in terms of prediction of asset returns, but are qualitatively similar.