Consider two simple linear models Y1j = 1 + 1x1j + 1j , j = 1, 2, , n1 and Y2j =

Consider two simple linear models Y1j = α1 + β1x1j + ε1j , j = 1, 2, · · · , n1 and Y2j = α2 + β2x2j + ε2j , j = 1, 2, · · · , n2 Assume that β1 6= β2 the above two simple linear models intersect. Let x0 be the point on the x-axis at which the two linear models intersect. Also assume that εij are independent normal variable with a variance σ 2 . Show that

(a). x0 = α1 − α2 β1 − β2 (b). Find the maximum likelihood estimates (MLE) of x0 using the least squares estimators ˆα1, ˆα2, βˆ1, and βˆ2. (c). Show that the distribution of Z, where Z = (ˆα1 − αˆ2) + x0(βˆ1 − βˆ2), is the normal distribution with mean 0 and variance A2σ2, where A2 = x2 1j − 2x0 x1j + x2 0n1 n1 (x1j − x¯1)2 + x2 2j − 2x0 x2j + x2 0n2 n2 (x2j − x¯2)2 . (d). Show that U = Nσˆ2/σ2 is distributed as χ2(N), where N = n1 + n2 −