Question

YI = β 1X I 1 + β 2X I 2 + UI

WhereUi ∼ NID (0, σu2 ) , YI is observable random variable and theXij ' s , j =1, 2

are observable non-random (non-stochastic) variables.

The data that follows is based on a sample of size N = 120 and gives the

sums of squares and cross-products of the indicated variables

Y X1 X2Y 39 6 2X1 6 4 0X2 2 0 4

a). Compute the best linear unbiased estimates of the coefficients. (2 points)b). Give a 95% confidence interval for β1 . (2 points)Test the hypothesis H 0 : β1 + β2 = 1 against the alternative H 0: β1 + β2 ≠ 1

at the 95%confidence level.

Answer 1

a)$variance \space v=v(\Sigma \frac{xi}{n})=\frac{1}{n^{2}}\Sigma v(xi)=\frac{n}{n^{2}}σ^{2}=\frac{σ^{2}}{n}$

therefore the likelihood estimator will be

$σ^{2}=s^{2} \frac{n}{n-1}=\frac{\Sigma (xi-x)^{2}}{n-1}$

s²=constant estimator

(b)The sample median estimator of the median Xn corresponding to p = 0.5, Xn is a then a normal distribution with parameters µ and σ2