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Assume I have $y$, $x_1$ and $x_2$. I regress

  • $y\sim\alpha_0 + \alpha_1 x_1$,
  • $y\sim\beta_0 + \beta_1 x_2$ and
  • $y\sim\gamma_0 + \gamma_1 x_1 + \gamma_2 x_2$

using Ordinary Least Squares.

Does $\alpha_1 \geq \beta_1$ imply $\gamma_1 \geq \gamma_2$? If not, are there "simple" conditions under which it would?

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No. I don't know under what conditions your conjecture would hold, but I found a counterexample to its most general form by brute force.

If $x_1 = (0, 2, 1)$, $x_2 = (2, 1, 2)$, and $y = (0, 1, 2)$, then the model coefficients are:

  • $α_0 = \frac{1}{2}$, $α_1 = \frac{1}{2}$
  • $β_0 = 1$, $β_1 = 0$
  • $γ_0 = -6$, $γ_1 = 2$, $γ_2 = 3$
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