Re-exploring the CCAPM : The Case of US Industry Returns with Different Price Deflators

Extending US samples, this paper re-examines the classic consumption-based capital asset pricing model (CCAPM) by the generalized method of moments (GMM). Our re-exploration using US three industry returns and different price deflators supplies the following evidence. First, 1) regarding the CCAPM using the US consumption for nondurable goods and the deflator of total personal consumption expenditures (PCEs), the discount rate and risk aversion parameters show plausible values; and according to the J-tests, our above first CCAPM is generally supported. Second, 2) as for the CCAPM with the US consumption for nondurable goods and services and the deflator of total PCEs, both discount rate and risk aversion parameters generally exhibit plausible values and our J-tests show that our above second CCAPM is highly supported. Third, 3) as for the CCAPM using the US consumption for nondurable goods and the deflator of the PCEs for nondurable goods, both parameters of the discount rate and risk aversion are highly stable and our J-tests indicate that our above third CCAPM is highly supported. Finally, 4) as regards the CCAPM using the US consumption for nondurable goods and services and the calculated implicit deflator of the PCEs for nondurable goods and services, the parameters of the discount rate generally exhibit plausible values, while the risk aversion parameters are not so stable. However, according to the J-tests, our above fourth CCAPM is also highly supported.


Introduction
For understanding asset pricing mechanisms, employing the approach of generalized method of moments (GMM) (Hansen, 1982;Hansen and Singleton, 1982) is effective because by this, we are able to focus on the pricing kernels of asset pricing models (see, e.g., Epstein and Zin, 1991;Campbell and Cochrane, 1999).On the ground that it forms an essential foundation for asset pricing models in financial economics, the basic consumption-based capital asset pricing model (CCAPM) is also crucial.
From the above two viewpoints, it is valuable to re-explore the classical CCAPM by applying the GMM estimation method.Based on this motivation, in this paper, we re-test the traditional CCAPM by expanding US samples and employing Hansen and Singleton's (1982) GMM.Our exploration using US three industry returns and different price deflators supplies the following evidence.First, 1) as regards the CCAPM using the US consumption for nondurable goods and the deflator of total personal consumption expenditures (PCEs), the discount rate and risk aversion parameters exhibit plausible values; and according to the J-tests, our above first CCAPM is generally supported.Second, 2) as to the CCAPM with the US consumption for nondurable goods and services and the deflator of total PCEs, both discount rate and risk aversion parameters generally show plausible values and our J-tests indicate that our above second CCAPM is highly supported.Third, 3) as for the CCAPM using the US consumption for nondurable goods and the deflator of the PCEs for nondurable goods, both parameters of the discount rate and risk aversion are highly stable; and our J-tests indicate that our above third CCAPM is highly supported.Finally, 4) regarding the CCAPM using the US consumption for nondurable goods and services and the calculated implicit deflator of the PCEs for nondurable goods and services, the parameters of the discount rate generally show plausible values, while the risk aversion parameters are not so stable.However, according to the J-tests, our above fourth CCAPM is also highly supported.Regarding the rest of this paper, Section 2 reviews past studies; Section 3 describes our data and variables; Section 4 explains our method; Section 5 documents our results; and Section 6 presents our conclusions.

Literature Review
This section briefly reviews existing studies.There are many past interesting studies that analyzed consumption-based asset pricing models theoretically and empirically.They are such studies as those by Epstein and Zin (1991), Campbell (1996), and Lettau and Ludvigson (2001), for example.Campbell (1996) attempted better understanding of risk and return in asset pricing and Hansen et al. (2007) also attempted clearer understanding for the intertemporal substitution and risk aversion in the asset pricing framework.
An interesting paper by Epstein and Zin (1991) suggested that separating the relative risk aversion parameter and the elasticity of intertemporal substitution parameter could be a solution of the so-called, 'risk-free rate puzzle.'In a study by Bansal and Yaron (2004), they modeled dividend growth rates and consumption while maintaining the linkages of preferences shown in Epstein and Zin (1991).They suggested their model was supported by actual data and could explain the dynamic evolution of asset markets.
Further, Campbell and Cochrane (1999) proposed a consumption-based asset pricing model, and their model incorporated the time-varying risk aversion and habit formation.Lettau and Ludvigson (2001) analyzed the variable of consumption-wealth ratio, a cointegrating residual for consumption, asset wealth, and labor income.They included this variable in the pricing kernel of their asset pricing model.
From the methodological viewpoints, although there are some papers that criticized empirical studies that tested asset pricing models (see, e.g., Lewellen and Nagel, 2006;Nagel and Singleton, 2011), the GMM approach proposed by Hansen and Singleton (1982) is indeed economically meaningful.Hence, in this paper, we conduct re-examinations of the CCAPM by extending US samples, employing their methodology, and applying different price deflators in below sections.Notes.This table displays the descriptive statistics of the variables used for the analyses in this study.In this research, we have three sub-sample periods with a full sample period.

Data and Variables
This section explains data and variables for our study.Using the data of consumption, stock returns, and price deflators in the US, we construct the variables for our tests.First, as to the consumption variables, NDT is the seasonally-adjusted real per capita US PCEs for nondurable goods, which is deflated by the seasonally-adjusted deflator of the US total PCEs.ND represents the seasonally-adjusted real per capita US PCEs for nondurable goods, which is deflated by the seasonally-adjusted deflator of the US PCEs for nondurable goods.In addition, NDS denotes the seasonally-adjusted real per capita US PCEs for nondurable goods and services, which is deflated by the corresponding implicit deflator that we computed from the seasonally-adjusted deflator as to the US PCEs for nondurable goods and the seasonally-adjusted deflator as to the US PCEs for services.This is because the exact corresponding deflator for the US PCEs for nondurable goods and services was not available.
Second, as for the stock return variables, RCHEMT denotes the real US chemical industry stock return deflated by the seasonally-adjusted deflator of the US total PCEs.RTRANST denotes the real US transportation industry stock return deflated by the seasonally-adjusted deflator of the US total PCEs.RRTAILT denotes the real US retail industry stock return deflated by the seasonally-adjusted deflator of the US total PCEs.Further, RCHEMND means the real US chemical industry stock return deflated by the seasonally-adjusted deflator of the US PCEs for nondurable goods.RTRANSND denotes the real US transportation industry stock return deflated by the seasonally-adjusted deflator of the US PCEs for nondurable goods.RRTAILND denotes the real US retail industry stock return deflated by the seasonally-adjusted deflator of the US PCEs for nondurable goods.
Moreover, RCHEMNDS means the real US chemical industry stock return deflated by the corresponding implicit deflator that we computed from the seasonally-adjusted deflator as to the US PCEs for nondurable goods and the seasonally-adjusted deflator as to the US PCEs for services.RTRANSNDS denotes the real US transportation industry stock return deflated by the corresponding implicit deflator that we computed from the seasonally-adjusted deflator as to the US PCEs for nondurable goods and the seasonally-adjusted deflator as to the US PCEs for services.Finally, RRTAILNDS denotes the real US retail industry stock return deflated by the corresponding implicit deflator that we computed as explained above.
In this study, our US samples are monthly and the full sample period spans February 1959 to December 2009.In addition, the first sub-sample period spans February 1959 to December 1978, the second sub-sample period spans January 1975 to December 1994, and the third sub-sample period spans January 1990 to December 2009.Four time-series of our four kinds of deflators of the US PCEs for the above full sample period are exhibited in Panels A to D of Figure 1.Moreover, Table 1 displays the descriptive statistics of the variables we explained above.This table shows that the skewness values for the three US stock returns are generally negative except for those values in our first sub-sample period.Second, excess kurtosis values of the three kinds of US stock returns are higher in our second sub-sample period.1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 Figure 1.Three price deflators and a computed implicit deflator of the US personal consumption expenditures for testing the CCAPM in the US

Testing Method
Using the above data and the following specification by Hansen and Singleton (1982), we re-explore the traditional CCAPM in the US by updating sample periods.
{ } In the above system (1), v 1t+1 is the vector of three industry returns and v 2t+1 means the growth of consumption.Moreover, γ is the parameter of risk aversion and δ is the parameter of the discount rate.Further, z t is the vector of instrument variables and ⊗ means the Kronecker product.
Applying the above system, we estimate the CCAPMs by using 1) RCHEMT, RTRANST, RRTAILT, and NDT and 2) RCHEMT, RTRANST, RRTAILT, and NDST.We next estimate the CCAPMs by using 3) RCHEMND, RTRANSND, RRTAILND, and ND and 4) RCHEMNDS, RTRANSNDS, RRTAILNDS, and NDS.As for the instrument variables, following Hansen and Singleton (1982), lags of consumption growth and the corresponding stock return variables in each case are used.We set the lag order of instrumental variables as 1, 2, 4, or 6 as the analyses in Hansen and Singleton (1982).

Results
We first explain the estimation results of the CCAPM for our three US industry returns by using one deflator of total PCEs for the US.First, as to the CCAPM with PCEs for nondurable goods, Table 2 shows that 1) the discount rate parameters are always estimated as the values that are slightly below one with no exception.In addition, Table 2 also suggests that 2) the risk aversion parameters in the models generally take small negative values stably except for the only one case in Panel A of Table 2.Moreover, all the above estimated CCAPMs with PCEs for nondurable goods by using the deflator of the total PCEs in the US are always supported by the J-tests except for the three cases in Panel A of Table 2. Thus, our above first CCAPM for the three industry returns is considered to be generally well estimated.
We next explain the estimation results of our second CCAPM for the three US industry returns by using one deflator of total PCEs for the US.Namely, regarding the CCAPM with PCEs for nondurable goods and services, Table 2 shows that 1) the discount rate parameters are always estimated as the values that are slightly below one except for the one case in Panel B and the one case in Panel D. In addition, Table 2 also suggests that 2) the risk aversion parameters in the models generally take small negative values stably except for the two cases in Panel B and the three cases in Panel D of Table 2.Moreover, all the above estimated CCAPMs with PCEs for nondurable goods and services by using the deflator of the US total PCEs are always supported by the J-tests except for the only one case in Panel A of Table 2. Thus, our above second CCAPM for the three industry returns is considered to be rather well estimated.
Moreover, we document the estimation results of the CCAPM for the three US industry returns by using the deflator of the PCEs for nondurable goods or the implicit deflator of the PCEs for nondurable goods and services in the US.First, as for the CCAPM with PCEs for nondurable goods, Table 3 shows that 1) the discount rate parameters are always estimated as the values that are slightly below one with no exception.In addition, Table 3 also suggests that 2) the risk aversion parameters in the models generally take small negative values stably with no exception.Moreover, all the above estimated CCAPMs with PCEs for nondurable goods by using the deflator of the PCEs for nondurable goods are always supported according to the results of the J-tests except for the only one case in Panel A of Table 3.Hence, our above third CCAPM for the three industry returns is considered to be very well estimated.
Furthermore, as for the CCAPM with the computed implicit deflator of the US PCEs for nondurable goods and services, Table 3 shows that 1) the discount rate parameters are always estimated as the values that are slightly below one except for the one case in Panel B and the one case in Panel D of Table 3. Further, Table 3 also suggests that 2) the risk aversion parameters in the models generally take small negative values stably except for the one case in Panel A, three cases in Panel B, and three cases in Panel D of Table 3.Moreover, all the above estimated CCAPMs with PCEs for nondurable goods and services by using the calculated implicit deflator of the US PCEs for nondurable goods and services are always supported by the J-tests except for the only one case in Panel A of Table 3.Thus, our above fourth CCAPM for the three industry returns is considered to be very well modeled; however, as we explained, risk aversion parameters are somewhat unstable.We consider that this might be because of the goodness of fit of the deflator.

Summary and Conclusions
By extending US samples, this paper empirically re-examined the traditional CCAPMs with GMM.Our re-exploration using US three industry returns and different price deflators supplied the following evidence.First, 1) regarding the CCAPM using the US consumption for nondurable goods and the deflator of total PCEs, the discount rate parameters presented plausible values.In addition, their risk aversion parameters in the models also well exhibited plausible values.Moreover, according to the J-tests, the estimated CCAPMs for US three industry returns, which used the consumption for nondurable goods and the deflator of total PCEs, were generally supported.Second, 2) with regard to the CCAPM with the US consumption for nondurable goods and services and the deflator of total PCEs, the parameters of both the discount rate and risk aversion generally exhibited plausible values.Moreover, according to the J-test results, the estimated CCAPMs using the consumption for nondurable goods and services and the deflator of total PCEs were highly supported.
Third, 3) as to the CCAPM using the US consumption for nondurable goods and the deflator of the PCEs for nondurable goods, both the parameters of the discount rate and the risk aversion were highly stable.In addition, according to the J-test results, the estimated CCAPMs with the US consumption for nondurable goods and the deflator of PCEs for nondurable goods in the US were highly supported.Finally, 4) with regard to the CCAPM using the US consumption for nondurable goods and services and the calculated implicit deflator of the PCEs for nondurable goods and services, the parameters of the discount rate generally exhibited plausible values, while the risk aversion parameters were not so stable.However, according to the J-test results, the estimated CCAPMs with the US consumption for nondurable goods and services and the calculated implicit deflator of PCEs for nondurable goods and services in the US were highly supported.
As above, in the US, the CCAPMs using consumption for nondurable goods were generally better than the CCAPMs using consumption for nondurable goods and services.In addition, we note that the CCAPMs using consumption for nondurable goods and the deflator of the PCEs for nondurable goods were better than the CCAPMs with consumption for nondurable goods and the deflator of total PCEs.We consider that the differences of our estimation results may be because of the goodness of fit of the deflators.This is one of the very interesting findings and implications from our present study.As this paper demonstrated, Hansen and Singleton's (1982) GMM methodology matters in asset pricing research, and many extended consumption-based models and studies have recently emerged (e.g., Dreyer et al., 2013;Ghonghadze and Lux, 2016).Further investigations with this methodology and various other viewpoints are our future works.
Results for the first sub-sample period from February 1959 to December 1978 The case of the PCEs for nondurable goods NLAG δ Notes: ** and * indicate the statistical significance of the parameter or the chi-squared statistic at the 1% and 5% levels, respectively.
Notes: ** and * indicate the statistical significance of the parameter or the chi-squared statistic at the 1% and 5% levels, respectively.

Table 1 .
Descriptive statistics for real industry returns and consumption in the US Panel A. The case using the deflator of the total PCEs Statistics for the full sample period from February 1959 to December 2009

Table 2 .
Estimation results of the CCAPMs in the US: The case using the deflator of total PCEs Panel A. Results for the full sample period from February 1959 to December 2009 The case of the PCEs for nondurable goods NLAG

Table 3 .
Estimation results of the CCAPMs in the US: The case using the deflator of the PCEs for nondurable goods or the implicit deflator of the PCEs for nondurable goods and services Panel A. Results for the full sub-sample period from February 1959 to December 2009