Suppose that the true model is Y i = 1 + 2 X 2i + u i but we add an irrelevant variable

Suppose that the “true” model is

Y_i = β₁ + β₂X_2i + u_i

but we add an “irrelevant” variable X₃ to the model (irrelevant in the sense that the true β₃ coefficient attached to the variable X₃ is zero) and estimate

Y_i = β₁ + βX_2i + β₃X_3i + v_i

a. Would the R² and the adjusted R² for model (2) be larger than that for model (1)?

b. Are the estimates of β₁ and β₂ obtained from model (2) unbiased?

c. Does the inclusion of the “irrelevant” variable X₃ affect the variances of ˆ β₁ and ˆ β₂?