1996issue C101-2
Constructing linear-regression alpha from a yield gap
A two-coefficient market regression treats the intercept as an alpha term built from a defined yield-gap input. The published pair is rebuilt and checked with the least-squares identities rather than copied as a constant.
- A two-variable linear regression sets the dependent observation equal to a slope times the independent observation plus an intercept, and that intercept is the fitted value when the independent term is zero.
- This construction names that intercept the alpha coefficient and uses a yield-gap, the gap between a bond yield and its six-month moving average, as the independent series.
- A published pair with intercept 0.859 percent and slope -5.947 maps a one-percentage-point positive yield gap to -5.088 percent.
- Least-squares slope and intercept minimize the least-squares criterion, and alpha can be recovered from the two series averages; reversing independent and dependent series is a documented construction error.
A two-coefficient line with an intercept
A two-variable linear regression is constructed so the dependent observation equals a slope times the independent observation plus an intercept. That intercept is the fitted value when the independent term is zero.
Linear regression, in this archive construction, is a fitted line that maps an ordered independent series onto a dependent series through one slope and one intercept. Slope is the change in the dependent series associated with a one-unit change in the independent series.
The intercept named as the alpha coefficient
In this construction the intercept is named the alpha coefficient. The alpha coefficient is the constructed value of the dependent series when the independent term equals zero.
A published index and yield-gap pair
One published construction used monthly percent change of a stock index as the dependent series and the gap between a bond yield and its six-month moving average as the independent series, with intercept 0.859 percent and slope -5.947.
Substituting a defined yield-gap
Substituting a one-percentage-point positive yield gap into that constructed equation produces 0.859 percent minus 5.947 percentage points, which equals -5.088 percent.
Least-squares identities for the same pair
Least-squares is the fitting rule that chooses slope and intercept to minimize the sum of squared residuals. Least-squares slope and intercept are the pair that minimize the least-squares criterion, a result attributed to differential calculus.
The alpha intercept can be recovered as the average of the dependent series minus the slope times the average of the independent series.
Editorial reading of the construction
Editorial interpretation, not an archive claim: rebuild the published intercept as an alpha term from the defined yield-gap input, then check that pair with the least-squares identities, rather than copy the intercept as a black-box constant. The archive presents a historical workflow for constructing the two coefficients.