All multifactor models are re-estimated in expected return–beta form (MSC, Sec. 5.4). I employ the time-series/cross-sectional regression approach proposed by Brennan et al. (2004), Cochrane (2005, Sec. 12.2), and others, that is, factor loadings (betas) are estimated by time- series regressions of excess returns on factors for each test asset in the first step. For instance, in the case of HL, I conduct the following regressions:
(B.9)
where the slope coefficients represent the factor loadings. The second step comprises a cross-sectional regression of the test assets’ average excess returns on the estimated factor loadings:
(B.10)
where the slope coefficients are the factor (beta) risk prices. The cross-sectional regressions are first conducted using the OLS methodology and the t-statistics of the factor (beta) risk
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prices are computed with Shanken’s (1992) approach to account for the estimation error in the factor loadings. The implied RRA estimate is obtained by dividing the market (beta) risk price by the variance of the market excess return:
(B.11)
Table 46 (p. 175) shows the outcomes for the ICAPM specifications. Observe that the explanatory ratios are the same as in the benchmark (first-stage GMM) estimation (Table 17, p. 105). Moreover, note that, in contrast to the benchmark tests, all models now fulfill ICAPM criterion 1, showing an RRA estimate that is between one and 10. Furthermore, observe that
is negative within P and positive within KLVN (SM25), which is in contrast to the risk
prices obtained in the benchmark tests. The same applies to the positive sign of within P and within CV (SM25). The consequences of these sign changes are that P no longer
fulfills ICAPM criteria 3a and 3b when the model is tested over SM25. Moreover, CV only meets criterion 3b when one assumes the indeterminate sign of the slope of in the volatility-predicting regressions to be negative (SM25). Table 47 (p. 176) displays the results for the models with empirical risk factors. Observe that each model’s RRA estimate is between one and 10, so that ICAPM criterion 1 is met by each model and over both sets of testing assets. Moreover, note that, in contrast to the first-stage GMM estimation results,
is negative within PS and FF5 (SBM25), as well as within C (SM25). For this reason, C meets criterion 3a over SM25. Moreover, the model meets criterion 3b over SM25 only when the indeterminate sign of the slope on in the volatility-predicting regressions is assumed to be positive.
The cross-sectional regressions are repeated using GLS. This method weights the observations according to the inverse of the full covariance matrix of the residuals from the time-series regressions, . The explanatory power of the models is now measured by the GLS coefficient of determination, which is similar to the WLS coefficient of determination, except that the demeaned pricing errors and demeaned average excess returns are weighted by the inverse of :
(B.12)
where denotes the vector of demeaned pricing errors and is the vector of demeaned
(average) excess returns. The results for the ICAPM specifications are displayed in Table 48 (p. 177). Note that the explanatory power of HL, CV (SBM25 and SM25), and P (SM25) improve considerably according to the GLS coefficient of determination. The GLS regression assigns a zero pricing error for each test asset that is also incorporated into the model as a factor. Such a test asset and factor, respectively are the market excess return in all the models I test. Observe that, for this reason, the market (beta) risk price is numerically equal to the mean excess market return in each test, that is, 1.12% per month in my sample, and each model shows the same RRA estimate (3.89). Consequently, ICAPM criterion 1 is fulfilled by each model. Furthermore, note that, in contrast to the benchmark first-stage GMM estimations, is positive within HL (SBM25), CV (SM25), and KLVN (SBM25 and
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SM25). Moreover, is positive and is negative within P (SM25), is negative
within CV (SM25), and is positive (SBM25) and negative (SM25) within KLVN. The consequences of these sign changes are that HL now fulfills ICAPM criterion 3a (SBM25) when the indeterminate sign of the slope of in the return-predicting regressions (Panel D of Table 13, p. 97) is assumed to be positive. Moreover, P no longer meets criterion 3a or 3b over SM25 and CV no longer satisfies criterion 3b over SM25. Finally, KLVN no longer satisfies criterion 3b over SBM25. Instead, the model now meets criterion 3a over SM25. Table 49 (p. 178) displays the results for the models with empirical risk factors. Observe that the models’ explanatory power, now measured by , changes noticeably with respect to the benchmark first-stage GMM tests. While the explanatory ratio of C increases over SBM25 but decreases over SM25, the ratios associated with the other three models decrease over SBM25 but increase regarding SM25. Moreover, note that all models now satisfy ICAPM criterion 1. Furthermore, one can see that, in contrast to the benchmark first-stage GMM estimations, is negative in all tests. Moreover, is negative within C and positive
within FF5 (SM25). Furthermore, is positive within PS and is positive within FF5
(SM25). The sole consequence of these sign changes is that C only meets criterion 3b over SM25 when the indeterminate signs of the slopes on and in the volatility- predicting regressions are assumed to be positive.