ECON 203 Practice Exercises Week 10 (Solutions) PDF
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University of Illinois at Urbana-Champaign
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This document contains practice exercises for ECON 203, a course on econometrics offered at the University of Illinois at Urbana-Champaign. The exercises cover various concepts related to linear regression models, and include detailed solutions.
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Practice Exercises – Tenth Week ECON 203 University of Illinois at Urbana-Champaign 1. Consider a random sample (Yi , Xi ), i = 1,... , n, from a population such that m(x) := E(Yi |Xi = x) = β0 + β1 x. Write the li...
Practice Exercises – Tenth Week ECON 203 University of Illinois at Urbana-Champaign 1. Consider a random sample (Yi , Xi ), i = 1,... , n, from a population such that m(x) := E(Yi |Xi = x) = β0 + β1 x. Write the linear regression model as Yi = E(Yi |Xi = x) + Ui = β0 + β1 x + Ui , where Ui is the random error. Assume that E(Ui2 ) = σ 2. Show that n " # 1 X (X − X̄) m(x) b = β0 + β1 x + 1 + (x − X̄) i 2 Ui , n i=1 sX Pn where s2X = 1 n i=1 (Xi − X̄)2. Solution. Start by writing m(x) b = βb0 + βb1 x = Ȳ − βb1 X̄ + βb1 x = Ȳ + βb1 (x − X̄), Remember that Pn i=1 (Xi − X̄)(Yi − Ȳ ) βb1 = Pn 2 i=1 (Xi − X̄) Pn i=1 (Xi − X̄)(Yi − Ȳ ) = ns2X Pn i=1 (Xi − X̄)Yi = ns2 Pn X (X − X̄)Ui = β1 + i=1 i 2 nsX n X (Xi − X̄)Ui = β1 + i=1 ns2X 1 Now, writing m(x) b = Ȳ − βb1 x = Ȳ + (x − X̄)βb1 n X (Xi − X̄)Ui = Ȳ + (x − X̄) β1 + i=1 ns2X n n 1 X X (Xi − X̄)Ui = (β0 + β1 Xi ) + (x − X̄) β1 + n i=1 i=1 ns2X n n X (Xi − X̄)Ui 1X = β0 + β1 X̄ + Ui + (x − X̄)β1 + (x − X̄) n i=1 i=1 ns2X n " # 1X (Xi − X̄) = β0 + β1 x + 1 + (x − X̄) Ui. n i=1 s2X 2
