Investigating the Impact of Consumption Distribution on CRRA Estimation: Quantile-CCAPM-Based Approach

Sofia B. Ramos; Abderrahim Taamouti; Helena Veiga

doi:10.1515/snde-2023-0005

Article

Investigating the Impact of Consumption Distribution on CRRA Estimation: Quantile-CCAPM-Based Approach

Sofia B. Ramos , Abderrahim Taamouti and Helena Veiga

Published/Copyright: January 8, 2024

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From the journal Studies in Nonlinear Dynamics & Econometrics Volume 29 Issue 1

Abstract

Using quantile maximization decision theory, this paper considers a quantile-based Euler equation that states that the asset price is a function of the quantiles of the payoff, consumption growth, the stochastic discount factor, risk aversion, and the distribution of the consumption growth rate. We use a more general distribution assumption (log-elliptical distributions) than the log-normality of the consumption growth rate assumed in the literature. The simulation results show that: (1) the higher the downside risk aversion, the lower the constant relative risk aversion; (2) the heavier the tails of the Student-t distribution, the higher the risk aversion for each level of downside risk aversion; and (3) the curve of the relationship between risk aversion and downside risk aversion shifts upward when the normality assumption is dropped, and the magnitude of this shift is high even for high degrees of freedom of the Student-t distribution. Our results suggest that using normally distributed errors to model stock returns and consumption growth rates could lead to an underestimation of the risk aversion coefficient.

Keywords: CCAPM; consumption volatility; downside risk aversion; quantile-based Euler equation; risk aversion; stochastic volatility

Corresponding author: Abderrahim Taamouti, Department of Economics, University of Liverpool Management School, University of Liverpool, Chatham St., Liverpool L69 7ZH, UK, E-mail: Abderrahim.Taamouti@liverpool.ac.uk

Acknowledgment

We thank seminar participants at the twelfth Annual SoFiE Conference for helpful comments.

Research funding: The third author acknowledges financial support from the Agencia Estatal de Investigación PID2021-122919NB-I00 and PID2022-139614NB-C22, and from Fundação para a Ciência e a Tecnologia, grant UIDB/00315/2020.

Appendix

6.1 Proof of Propositions

Proof of Proposition 1. Since the consumption growth rate is log-elliptically distributed, we have

(10) Q t α g t + 1 = Q t α ln C t + 1 C t = μ + σ c , t D − 1 α .

Thus, using Equation (2) and the result in (10), we get

Q t α r t + 1 = − ln δ + γ Q t α ln C t + 1 C t = − ln δ + γ μ + σ c , t D − 1 α = − ln δ + γ μ + γ σ c , t D − 1 α = μ r + β r σ c , t ,

where μ r = − ln δ + γ μ and β r = γ D − 1 α .

6.2 Data Description

U.S. Consumption data is taken from Table 7 in Appendix.. Selected Per Capita Product and Income Series in Current and Chained Dollars. We sum Nondurable goods (A796RC0) + Services (A797RC0). More details can be seen in Table 4.
U.S. market indexes are obtained from Datastream. Stock market series are presented in Table 5. Series are nominal and were then deflated using one of the deflators and then compute the rate of change in logs.
We have used as deflators PCE: Personal Consumption Expenditures index (the base year is 2009). More details can be seen in Table 7.

Considering the consumption growth rate from U.S., we observe that the unconditional means of consumption growth rate and stock market indices are positive; see Table 6. Furthermore, the unconditional distributions of both consumption growth rate and stock market indices exhibit excess kurtosis and negative skewness. The sample kurtosis for all variables is greater than three, the kurtosis of a Gaussian random variable. Finally, Figure 2 shows the autocorrelation of the squares of consumption growth rate. We observe that the autocorrelation function decays very slowly towards zero, which suggest that the volatility of the consumption growth rate is quite persistent,^[9] which we obtain from the U.S. national accounts. Following prior work (e.g. Feunou et al. 2014; Hansen and Singleton 1983; Yogo 2006), aggregate consumption is measured as the seasonally adjusted real per capita consumption of non-durables plus services. The quarterly real per capita consumption data are taken from the NIPA tables available from the Bureau of Economic Analysis.^[10] Regarding stock market data, we use the return of a stock market index that represents the whole total stock market index. All these data are withdrawn from Datastream. The starting periods of these indexes are presented on Table 5. Furthermore, all variables are measured at constant prices, nominal values are adjusted using the associated Personal Consumption Expenditures (PCE) from the NIPA tables. All variables are also multiplied by 100 and expressed in logarithmic form (Table 7).

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Received: 2023-01-13

Accepted: 2023-11-29

Published Online: 2024-01-08

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Articles in the same Issue

https://doi.org/10.1515/snde-2023-0005

Keywords for this article

CCAPM; consumption volatility; downside risk aversion; quantile-based Euler equation; risk aversion; stochastic volatility