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The variable you left out · Part 3 of 3

The true model is y=x1+2x2+εy = x_1 + 2x_2 + \varepsilon, with Var⁡(x1)=Var⁡(x2)=1\operatorname{Var}(x_1) = \operatorname{Var}(x_2) = 1 and ε\varepsilon independent of both regressors. The sample is large enough that estimates sit at their probability limits.

Back to Cov⁡(x1,x2)=0.5\operatorname{Cov}(x_1, x_2) = 0.5. You now include x2x_2, but you only observe a noisy proxy x~2=x2+u\tilde x_2 = x_2 + u with Var⁡(u)=1\operatorname{Var}(u) = 1, independent of everything else. What coefficient on x1x_1 do you estimate, to three decimal places?

Answer with a number. Fractions, powers and expressions like 23/6 or C(52,5) are read correctly in practice.

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The variable you left out, part 3 of 3 · QuantMax