A regression on a 0/1 dummy is just a difference of means

Question

You regress a trade's slippage y on a constant and a single dummy variable D, where D = 1 for algorithmic orders and D = 0 for manual orders: y = b0 + b1 D + \varepsilon. Show that the fitted coefficients are just group means, and connect the slope's test to a two-sample comparison.

Accepted Answer

Fit the two possible predictions. With only a 0/1 regressor, the model can output exactly two fitted values: - For manual orders (D = 0): \hat y = b0. - For algorithmic orders (D = 1): \hat y = b0 + b1. Least squares minimizes squared error, and the best single number for each group is that group's mean. Therefore b0 = \bar y{	ext{manual}}, \qquad b0 + b1 = \bar y{	ext{algo}} \implies b1 = \bar y{	ext{algo}} - \bar y{	ext{manual}}. The intercept is the baseline group's mean; the slope is the difference between the two group means. That is the entire content of the regression. Connection t…

A regression on a 0/1 dummy is just a difference of means

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