Cay Revisited : Can Optimal Scaling Resurrect the (C)Capm?

In this paper, we evaluate specification and pricing error for the Consumption (C-) CAPM in the case where the model is optimally scaled by consumption-wealth ratio (CAY). Lettau and Ludvigson (2001b) show that the C-CAPM successfully explains a large portion (about 70%) of the cross-section of expected returns on Fama and French's size and book-to-market portfolios, when the model is scaled linearly by CAY. In contrast, we use the methodology developed in Basu and Stremme (2005) to construct the optimal factor scaling as a (possibly non-linear) function of the conditioning variable (CAY), designed to minimize the model's pricing error. We use a new measure of specification error, also developed in Basu and Stremme (2005), which allows us to analyze the performance of the model both in and out-of-sample.We find that the optimal factor loadings are indeed non-linear in the instrument, in contrast to the linear specification prevalent in the literature. While our optimally scaled C-CAPM explains about 80% of the cross-section of expected returns on the size and book-to-market portfolios (thus in fact out-performing the linearly scaled model of Lettau and Ludvigson (2001b)), it fails to explain the returns on portfolios sorted by industry. Moreover, although the optimal use of CAY does dramatically improve the performance of the model, even the scaled model fails our specification test (for either set of base assets), implying that the model still has large pricing errors. Out-of-sample, the performance of the model deteriorates further, failing even to explain any significant portion of the cross-section of expected returns. For comparison, we also test a scaled version of the classic CAPM and find that it has in fact smaller pricing errors than the scaled C-CAPM