1082 The Journal of finance The risk premium can be written in general as E-Eo=∑R where l indexes individual agents, W is the proportion of total wealth held by gent l, Ri is his coefficient of relative risk aversion is the partial elasticity of his consumption with respect to changes in the jth factor, ando, is the variance of the jth factor. Not very much is known about the term in parentheses and, all other things being equal, about all we can conclude is that risk premia should be larger, the larger the own variance of the factor. We would not expect this result to be specialized to the diffusion model and, in general, we would expect, with beta weights appropriately normalized, that factors with larger own variances would have larger associated risk premia Let us return now to the general APt model and aggregate it to a testable arket relationship. The key point in aggregation is to make strong enough assumptions on the homogeneity of individual anticipations to produce a testable theory. To do so with the aPt we need to assume that individuals agree on both he factor coefficients, by, and the expected returns, Er. It now follows that the pricing relationship(2)which holds for each individual holds at the market level as well. Notice that individual, and aggregate risk premia must coincide when here are homogenous beliefs on the expected returns and the factor coefficients As with the CAPM, the purpose of assuming homogenous anticipations is not to facilitate the algebra of aggregation. Rather, it is to take the final step to a testable theory. We can now make the rational anticipations assumption that(1) not only describes the ex ante individual perceptions of the returns process but also that ex post returns are described by the same equation. This fundamental intertemporal rationality assumption permits the ex ante theory to be tested by examining ex post data. In the next section we will discuss the possibilities for empirical testing which derive from this assumption. B. Testing the apT Our empirical tests of the APT will follow a two step procedure. In the first tep, the expected returns and the factor coefficients are estimated from time eries data on individual asset returns. The second step uses these estimates to test the basic cross-sectional pricing conclusion, (2), of the APT. This procedure is analogous to familiar CAPM empirical work in which time series analysis is used to obtain market betas, and cross-sectional regressions are the expected returns, estimated for various time periods, on the estimated betas While flawed in some respects, the two step procedure is free of some major conceptual difficulties in CAPM tests. In particular, the aPt applies to subsets course, developed a complete rational anticipations model in diffusio from this outline that the aPt is compatible with the more specific results of Merton [35], Lucas [31], Cox, Ingersoll, and Ross[], and Ross[48]