申論題

Problem 23.(10%) For a simple linear regression model
Yᵢ = α + βXᵢ + εᵢ    i = 1,2,....., n
Where Yᵢ is the dependent variable in the iᵗʰ case, Xᵢ is the value of the independent variable in the iᵗʰ case and assumed to be a known constant, α and β are parameters, and εᵢ is the error term for the iᵗʰ case.

(A) Please demonstrate that the regression line will pass the centroid of all observation cases, i.e., , where α̂ and β̂ are the estimated values of α and β respectively by using least squares method., and (X̄, Ȳ) is the centroid of all observation cases. (7%)