At first, Lo and MacKinlay (When are Contrarian Profits Due to Stock Market Overreaction?, 1990) didn't do it for momentum precisely. However,Kyung-In Park and Dongcheol Kim (Sources of Momentum Profits in International Stock Markets, 2011) use it to identify and analyse the difference in the momentum effect across different countries. Here is the exact part that I would like to understand:
The purpose of this paper is to examine the differences in the underlying forces determining momentum profits between countries exhibiting and non-exhibiting the momentum phenomenon and to induce which component(s) drives momentum profits. In order to determine the underlying forces of momentum profits, we use the decomposition method of momentum profits by Lo and MacKinlay (1990). They decompose momentum profits into three components; (1) the first-order serial covariance of market returns, (2) the average of first-order serial covariances of all individual assets, and (3) the cross-sectional dispersion in unconditional mean returns of individual assets. The total momentum profit equals minus (1) plus (2) plus (3) [i.e., 1 2 3. That is, the first term contributes negatively, and the second and third terms contribute positively to momentum profits. The first two components reflect the intertemporal behavior [1 2], and the third component reflects the cross- sectional behavior of asset returns .
As you can see, Kim and Park have subsumed the 3 parts into 2 parts: the inter temporal behavior and the cross-sectional behaviour of asset returns.
More precisely, in the conclusion of the article we can find:
Momentum profits can be decomposed into two parts; the part reflecting the cross-sectional difference in unconditional expected returns and the part reflecting the intertemporal behavior of asset returns.
So my question can rewritten like this: what is exactly the part reflecting the cross-sectional difference in unconditional expected returns and the part reflecting the intertemporal behavior of asset returns ?