Question

Problem 3: Absence of Intercept Consider the regression model Y, = BX,+", where , and X, satisfy Assumptions SLR1-SLR5. Y (i) Let B denote an estimator of B that is constructed as P where Y and X as are the sample means of Y,and X,, respectively. Show that B is conditionally unbiased. Derive the least squares estimator of B. Show that the estimator is conditionally unbiased. Derive the conditional variance of the estimator. (ii) (iii) (iv) 2

Answer 1

i) Observe that

ELY; X] = BX; + Elui X] = 8X,

Therefore,

E3 | x] - Εν | x] - (ΑΣΕΙ ) - (ΣΑ) - Y =

ii) Observe that the residual sum of square in this case is

R? - ΣΥ; - βX;)?

Hence, when it is minimum,

- Σ1 - βX) =0 -Σ2(Υ – βX.) = 0 = = βX = 3-Y

iii) Observe that the least square estimator is the same as $\bar{\beta}$. Hence, it is conditionally unbiased

iv) Observe that

Var(Y; X) = Var(u; X) = 0%

Therefore,

Var (| x) - rar(Υ | ) Σ'ων (5) - τομέρει -