Question

4. Suppose that(x,y), ,(XN,Yv) denotes a random sample. Let Si-a+bX, T, e+ dy, where a, b, c and d are constants. Let X = Σ x, and with the analogous expressions for Y, S, T. Let ớXY = N- ρχ Y-σχ Y/(σχσΥ), with the analogous expressions for S, T. = NT Σ(X,-X)2, . Σ(X,-X)(X-Y), and let (a) Show that σ = b20% (b) Show that ớST-b d ƠXY. (c) Show that psT sT- d) How do the above results relate to the corresponding (population-level) results for variance/covariance? e Let Bo+BiX be the best linear projection of Y on X. Similarly let 7o +mS be the best linear projection of T on S. How do γ0 and γ1 relate to ß0 and ß? Let ^o and ชิ be the OLS estimators of "Yo and 11-How do 70 and ชิ relate to A and A, the OLS estimators of β0 and β?

I have solved the questions (a) to (c). Could you please help me with questions (d),(e),(f)? Thank you!

Answer 1

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