Problem 291 of 333MediumLinear AlgebraP291

Orthogonalizing a factor before the regression

Over four days you observe returns (in %) for the market $m$ , a momentum factor $f$ , and a stock $y$ . Treat all series as mean zero and fit without an intercept.
$m = (1, 1, -1, -1), \quad f = (2, 0, 0, -2), \quad y = (5.5, -0.5, -1.5, -3.5)$
- Regress $y$ on $f$ alone. What is the slope?
- Remove from $f$ its projection onto $m$ , call the result $f_\perp$ , and regress $y$ on $f_\perp$ . What is the slope?
- Fit $y$ on $m$ and $f$ together. What are the two coefficients and the residual vector?
Which of the first two slopes equals the joint momentum coefficient, and why?
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