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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?

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When he is saying that the factor analysis requires returns, he is considering how the change in one asset would imply a change in another asset. You can model this the same way by modeling a change in temperature day over day, in which the "returns" would be $temp_t - temp_{t-1}$ or $\frac{temp_t}{temp_{t-1}}-1$.

However, if you believe the stock market performs worse at lower temperatures in general, you could just model the temperature and not consider just the change. In this case, you would be predicting stock price movements by looking at the actual temperature. In this case, the explanatory variable would just be $temp$. Finally, you could model the significance of the effect on returns under adverse conditions by using a Boolean for extreme temperatures. For instance, you could use 1 as your explanatory variable if the temperature is above 100 degrees or below 32, and 0 if it's a nice comfortable (32, 100).

The hot minus cold index that you are looking at is interesting, but I believe is unnecessary, and I'm not sure what it would add.

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