A linear regression line is usually determined quantitatively by a best-fit procedure such as least squares (i.e. the distance between the regression line and every observation is minimized). In linear regression, one variable is plotted on the X axis and the other on the Y. The X variable is said to be the independent variable, and the Y is said to be the dependent variable. When analyzing two random variables, you must choose which variable is independent and which is dependent.

The choice of independent and dependent follows from the hypothesis - for many examples, this distinction should be intuitive. The most popular use of regression analysis is on investment returns, where the market index is independent while the individual security or mutual fund is dependent on the market. In essence, regression analysis formulates a hypothesis that the movement in one variable (Y) depends on the movement in the other (X).

Regression Equation

The regression equation describes the relationship between two variables and is given by the general format.

Where: Y = dependent variable; X = independent variable,
a = intercept of regression line; b = slope of regression line,
ε = error term

In this format, given that Y is dependent on X, the slope b indicates the unit changes in Y for every unit change in X. If b = 0.66, it means that every time X increases (or decreases) by a certain amount, Y increases (or decreases) by 0.66*that amount. The intercept a indicates the value of Y at the point where X = 0. Thus if X indicated market returns, the intercept would show how the dependent variable performs when the market has a flat quarter where returns are 0. In investment parlance, a manager has a positive alpha because a linear regression between the manager's performance and the performance of the market has an intercept number a greater than 0.