Fit a linear regression model and plot your estimates

Question

Simulate N = 10⁴ outcomes from the model

Y = -2 + X -X² + ε

where X ~ U[0, 1] and ε ~ N(0, 0.1²) independently. From your simulated outcomes of (X, Y ).

(a) Fit a linear regression model as in Q5, and plot your estimates for β₀ and β₁ as N increases, together with a line indicating their true values. Supply your code.

(b) Fit a quadratic regression model y = β₀ +β₁x + β₂x₂ , and plot your estimates for β₀, β₁, and β₂ as N increases, together with a line indicating their true values. Supply your code.