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  1. 017The true model is y = x1 + x2 + noise, where x1 and x2 are standardised and have correlation 0.5. You regress y on x1 alone, then regress the residuals on x2. What coefficient do you get on x2, and how would you recover the true value of 1 in two stages?Correlation, regression and linear algebraHardQuant researchQuant trading

    Try it first

    What coefficient does the second stage give on x2?

    Show the worked solution

    You get 0.75, not 1. Regressing y on x1 alone gives a slope of 1 + 0.5 = 1.5, because x1 soaks up the half of x2 that moves with it. The residual is x2 - 0.5x1 + noise, whose slope on x2 is 1 - 0.5 squared = 0.75. To recover 1, residualise x2 on x1 as well and regress the residual of y on the residual of x2: the Frisch-Waugh-Lovell theorem.

    Where does the missing quarter go?

    Picture two salespeople who often work the same client. If you credit all joint sales to the first before looking at the second, the second looks worse than they are, because some of their work was already booked to the first. Stage one regresses y on x1 alone, and since x2 is correlated with x1, the coefficient on x1 rises to 1.5: it takes credit for 0.5 of x2. That piece has been removed from the residual, so stage two can only find what is left of x2's effect.

    x1 has already eaten the part of x2 that points its way0.5 x1: already in x1x1x2part of x2 orthogonalto x1: length 0.87cos 0.5CoefficientsStage 1: y on x11.50True effect of x21.00Residual on raw x20.75Residual on x2 orthogonal1.0010
    With a correlation of 0.5, x2 splits into 0.5 x1 plus an orthogonal part; stage one assigns the 0.5 x1 piece to x1, so regressing the residual on raw x2 gives 0.75, while regressing it on the orthogonal part of x2 recovers the true 1.

    How do you get 0.75 exactly?

    Write the residual out. y - 1.5x1 = x2 - 0.5x1 + noise, and the slope of that on x2 is its covariance with x2 over the variance of x2: (1 - 0.5 x 0.5)/1 = 0.75. The formula generalises to 1 - rho squared times the true coefficient, so the bias gets worse as the regressors get more correlated: with rho = 0.9 you would find only 0.19. A simulation of 100,000 observations gives 1.506 for stage one and 0.752 for stage two.

    The relationship
    β^2,seq=Cov⁡(x2−ρx1, x2)Var⁡(x2)=1−ρ2=0.75β^2,FWL=Cov⁡(x2−ρx1, x2−ρx1)Var⁡(x2−ρx1)=1\hat\beta_{2,\text{seq}} = \frac{\operatorname{Cov}(x_2 - \rho x_1,\ x_2)}{\operatorname{Var}(x_2)} = 1 - \rho^2 = 0.75 \qquad \hat\beta_{2,\text{FWL}} = \frac{\operatorname{Cov}(x_2 - \rho x_1,\ x_2 - \rho x_1)}{\operatorname{Var}(x_2 - \rho x_1)} = 1
    \rhothe correlation between x1 and x2, 0.5
    x_2 - \rho x_1the part of x2 left after regressing it on x1
    What it says in wordsRegressing on raw x2 shrinks the answer by one minus rho squared; regressing on the part of x2 orthogonal to x1 gives the true coefficient.

    What does Frisch-Waugh-Lovell tell you to do?

    To get a variable's coefficient from a multiple regression in stages, partial the other regressors out of both y and that variable, then regress residual on residual. Here that means regressing x2 on x1 as well, keeping the orthogonal part x2 - 0.5x1, and regressing the stage-one residual on it. The slope comes back as exactly 1; the simulation gives 1.003. This is why factor-neutralising a signal before testing it, rather than after, matters in quant research: the order of the stages changes the answer.

    Where candidates lose it

    The common answer is 1, on the belief that regressing residuals step by step is the same as a multiple regression. It is only the same when the regressors are uncorrelated, and the question gives you a correlation of 0.5 precisely to break that.

    The second loss is saying the answer is biased without saying which way or by how much. Give 1.5 for stage one, 0.75 for stage two, the 1 - rho squared rule, and the fix.

    What the interviewer asks next

    • What would the stage-two coefficient be if the correlation were -0.5?
    • In the two-stage FWL regression, how do the standard errors compare with the full multiple regression?
    • You have a new signal correlated with a known factor. How do you test whether it adds anything?