Question

4. Suppose that (X1, Yİ), , (XN,Yv) denotes a random sample. Let Si = a + bX, T, e+ dy, where a, b, c and d are constants. Let X ΣΧ, and σ2-NL Σ(x,-x)2, with the analogous expressions for y S, T. Let σΧΥ-ΝΤΣ (Xi-X)(X-Y), and let P:XY ƠXY/(ƠXƠY), with the analogous expressions for S, T. (a) Show that σ bbe (b) Show that ớsı, d ớx (c) Show that psT ST (d) How do the above results relate to the corresponding (population-level) results for OXY variance covariance! (e) Let /30+/31X be the best linear projection of Y on X. Similarly let 70+γ1S be the best linear projection of T on S. How do γο and γι relate to ß0 and ß? and B1, the OLS estimators of Bo and B,? How do you interpret the coefficient or estimated value β? (f) Let 70 and ชิ be the OLS estimators of 0 and 1, How do 70 and ^1 relate to ß0 (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 γ1 and estimated value γι change? How does your interpretation of the coefficient γι and estimated value change?

Please ignore part abc

Answer 1

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