For illustration, we'll use the output variable y that takes on the value 1 if a stock return is positive over a given time horizon d, and 0 otherwise:

We could easily extend y to three categories, where 0 and 2 reflect negative and positive price moves beyond a certain threshold, and 1 otherwise. Rather than modeling the output variable y, however, logistic regression models the probability that y belongs to either of the categories given a vector of alpha factors or features . In other words, the logistic regression models the probability that the stock price goes up, conditional on the values of the variables included in the model: