"A key difference between the TS and CS strategies is the threshold for taking long or short positions in an asset. The TS strategies use a fixed threshold of zero excess returns during the ranking period to assign stocks to the long and short side. In contrast the CS strategy uses average ranking period returns of the sample assets as the threshold. This difference in threshold would have a big impact on the composition of the long and short portfolios, particularly when the average ranking period return is large in magnitude. For example, if the average ranking period return were 30%, a number of stocks with positive excess returns would be on the short side in a CS strategy but on the long side in the TS strategy. We can think of such differences as being due to stock selection, or more generally due to asset selection criteria.

We show that there are also two additional differences between the TS and CS strategies, which are due to (i) risk premium and (ii) market timing. The time-series strategy takes the same dollar position in each risky asset, long if the past excess return is positive and short otherwise. If more than half the assets have positive past excess returns in any given month, the portfolio for that month would have a net long position in risky assets and net short position otherwise.2 If the average premium earned by risky assets is positive over the sample period then the TS strategy takes a net long active position on average. In contrast, the CS strategy takes a zero net active position because it invests equal amounts in long and short positions each period. Therefore, the net non-zero active investment taken by the TS strategy earns the corresponding risk premium relative to the CS strategy. We show that this component is equal the return earned by investing the average net long position in the equal weighted index of all assets over the entire sample period.

A market timing component is also inherent in the TS strategy. Generally, the difference between the number of assets with positive and negative past excess returns is larger after an up market than after a down market. So the TS strategy takes a larger net long position in the risky assets following up markets than down markets, and therefore inherently times the market. We show that market timing component equals the covariance between the net long position for a particular period and the equal weighted index return during that period. The sign of this component is in general the same as that of the correlation between equal-weighted index return during the formation period and holding period market returns, and it equals zero if equal weighted return is unpredictable. We show that this component can be earned by investing the net long active position in each period in the equal-weighted index rather than in any of the constituent stocks.

The risk premium component is earned by the TS strategy for the average active positions taken by this strategy. This component is not related to any time-series pattern of individual assets and it is not related to any behavioral model about their prices. The market timing component is earned by the time-series strategy by taking the net long or short position of the time-series strategy in the equal weighted index of all assets rather than choosing particular positions in individual assets. So this component could be relevant when we are interested in examining predictability of index returns but it is not relevant for understanding individual asset price predictability. The stock selection component, however, is driven by the behavior of individual asset prices. Therefore, if this component were a significant source of difference, then the TS and CS tests will have different implications for behavioral models related to individual stocks."

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