Consider the equation y = 0 + 1 x + 2 x 2 +

Consider the equation

y = β₀ + β₁x + β₂x² + u

E(u|x) = 0

where the explanatory variable x has a standard normal distribution in the population. In particular, E(x) = 0, E(x²) = Var (x) = 1, and E(x³) = 0. This last condition holds because the standard normal distribution is symmetric about zero. We want to study what we can say about the OLS estimator of β₁ we omit x² and compute the simple regression estimator of the intercept and slope.

(i) Show that we can write

y α₀ + β₁x + v

where E(v) 5 0. In particular, find v and the new intercept, α₀.

(ii) Show that E(v|x) depends on x unless β₂ = 0.

(iii) Show that Cov(x, v) = 0.

(iv) If β̂₁ is the slope coefficient from regression y_i on x_i, is β̂₁ consistent for β₁? Is it unbiased? Explain.

(v) Argue that being able to estimate β₁ has some value in the following sense: β₁ is the partial effect of x on y evaluated at x 5 0, the average value of x.

(vi) Explain why being able to consistently estimate β₁ and β₁₂ is more valuable than just estimating β₁.