I am very certain that the temperature in New York's Central Park plays a super-significant role in stock returns, so I take its daily averages and I want to test it in factor model.
Cochrane (section 12.2, p. 235) says I can use cross-sectional regressions to test this whether my "factor" is a return or not. However, to use time-series regressions I must make my new factor a return. How do I make my factor a return?
Next, suppose I know each stock's headquarter's building average daily temperature. I sort the stocks every day based on their temperature. I construct the Hot-minus-Cold factor by subtracting the (equally-weighted) averaged return of the 1000 coldest stocks from the 1000 hottest stock. This Hot-minus-Cold is a return, right? Is this type of process the only way to construct factors that are returns? In the last case, when is it necessary to use excess returns?