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Sign and size the bias from omitting market beta

Asked at DE Shaw

The true return model is r=β1x+β2m+εr = \beta_1 x + \beta_2 m + \varepsilon, where xx is your signal and mm is the market return (variables centered, ε\varepsilon exogenous). You regress rr on xx alone. Suppose the market loading is β2=0.5\beta_2 = 0.5, and the auxiliary regression of mm on xx has slope δ=Cov(x,m)/Var(x)=0.8\delta = \operatorname{Cov}(x, m)/\operatorname{Var}(x) = 0.8.

What does your estimated slope on xx converge to, and is it biased up or down?

Show a hint

Substitute the true model into the short-regression slope Cov(x,r)/Var(x)\operatorname{Cov}(x, r)/\operatorname{Var}(x) and use Cov(x,ε)=0\operatorname{Cov}(x, \varepsilon) = 0.

Your answer

This one is open-ended. Work it through, then check your reasoning against the full solution.

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