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1baltagi_b_h_solutions_manual_for_econometrics.pdf. GRADED A+.

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2.1 Variance and Covariance of Linear Combinations of Random Variables. a. Let Y D a C bX, then E.Y/ D E.a C bX/ D a C bE.X/. Hence, var.Y/ D EŒY E.Y/2 D EŒa C bX a bE.X/2 D EŒb.X E.X//2 D b... 2EŒX E.X/2 D b2 var.X/. Only the multiplicative constant b matters for the variance, not the additive constant a. b. Let Z D a C bX C cY, then E.Z/ D a C bE.X/ C cE.Y/ and var.Z/ D EŒZ E.Z/2 D EŒa C bX C cY a bE.X/ cE.Y/2 D EŒb.X E.X// C c.Y E.Y//2 D b2EŒXE.X/2Cc2EŒYE.Y/2C2bc EŒXE.X/ŒYE.Y/ D b2var.X/ C c2var.Y/ C 2bc cov.X, Y/. c. Let Z D aCbXCcY, and W D dCeXCfY, then E.Z/ D aCbE.X/CcE.Y/ E.W/ D d C eE.X/ C fE.Y/ and cov.Z, W/ D EŒZ E.Z/ŒW E.W/ D EŒb.XE.X//Cc.YE.Y//Œe.XE.X//Cf.YE.Y// D be var.X/ C cf var.Y/ C .bf C ce/ cov.X, Y/. 2.2 Independence and Simple Correlation. a. Assume that X and Y are continuous random variables. The proof is similar if X and Y are discrete random variables and is left to the reader. If X and Y are independent, then f.x, y/ D f1.x/f2.y/ where f1.x/ is the marginal probability density function (p.d.f.) of X and f2.y/ is the marginal p.d.f. of Y. In this case, E.XY/ D ’ xyf.x, y/dxdy D ’ xyf1.x/f2.y/dxdy D . R xf1.x/dx/.R yf2.y/dy/ D E.X/E.Y/ B.H. Baltagi, Solutions Manual for Econometrics, Springer Texts in Business and Economics, DOI 10.1007/978-3-642-54548-1 2, © Springer-Verlag Berlin Heidelberg 2015 5 [Show More]

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