Question

Parts e-h

Suppose that (Xi,A), , (XN,Yv) denotes a random sample. Let Si = a+bX, T, = c+ dY,, where a, b, c and with the analogous expressions for Y, ST. Let σΧΥ ρΧΥ-Oxy/(ơxdY), with the analogous expressions for S, T Σ Xi, and σ. NLī Σί (Xi-X)2, -, Σ (Xi-X)(X-Y), and let d are constants. Let X = (a) Show that σ (b) Show that 37, b d ƠXY. (c) Show that ps- pxy. (d) How do the above results relate to the corresponding (population-level) results for variance/covariance? best linear projection of T on S. How do γο and γ relate to 30 and β? and A, the OLS estimators of β0 and β1? How do you interpret the coefficient or estimated value β? (e) Let G AX be the best linear projection of Y on X. Similarly let γ0+γ1S be the (f) Let "Yo and วั be the OLS estimators of "Yo and 1. How do 70 and ^1 relate to A (g) For concreteness, say that Y represents food expenses and X income in US dollars (h) Now we convert Y in Euros to obtain T and and X in Euros to obtain S (say 1 US$ .9 Euros). How does the value of the coefficient and estimated value γι change? How does your interpretation of the coefficient γ and estimated value γ1 change?

Answer 1

Parts e and f are answered in the image below (After that part g is explained below the image)

Page No. 2 뎌 arung twth 난1) ave an exn B and Since and 1, thun anol we have eshmals i (an ear 義一c ), a

g) Y represents food expenses and X income in US dollars.

Here, Beta1 represents the amount of increase in the food expenses with an increase of one unit in the Income.