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Lecture+12+Multiple+Regression+Analysis+-+Inference

Lecture+12+Multiple+Regression+Analysis+-+Inference -...

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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-1 Lecture 12 Multiple Regression Analysis: Inference In Lecture 2 we discussed the univariate normal p.d.f. and the bivariate normal p.d.f. (B.34)
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-2 Recall: Example: Suppose X and Y have a bivariate normal distribution. Then their marginal distributions are Y - N( : Y , F Y 2 ), X - N( : X , F X 2 ), and their conditional distributions are
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-3 Y * X = x - N( : Y * X , F Y 2 * X ), X * Y = y - N( : X * Y , F X 2 * Y ), where : Y * X = : Y + D XY ( F Y / F X ) (x - : X ), F Y 2 * X = F Y 2 (1 - ), : X * Y = : X + D XY ( F X / F Y ) (y - : Y ), F X 2 * Y = F 2 X (1 - ).
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-4 In Lectures 9 and 10 we discussed the multivariate normal distribution:
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-5 Theorem: Consider the N×1 random vector Z partitioned as Z = [Z 1 N , Z 2 N ] N , where Z 1 is m×1 and Z 2 is (N - m)×1. Suppose Z - N N ( : , E ), where have been partitioned to conform with the partitioning of Z. Then: (a) The marginal distributions are: Z 1 - N m ( : 1 , E 11 ), Z 2 - N N-m ( : 2 , E 22 ).
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-6 (b) The conditional distribution of Z 1 , given Z 2 = z 2 , is Z 1 * Z 2 = z 2 - N( : 1 * 2 , E 1 * 2 ), where : 1 * 2 = E[Z 1 * Z 2 = z 2 ] = : 1 + E 12 E 22 -1 [z 2 - : 2 ], E 1 * 2 / Var[Z 1 * Z 2 = z 2 ] = E 11 - E 12 E 22 -1 E 12 N .
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-7 (c) The conditional distribution of Z 2 , given Z 1 = z 1 , is Z 2 * Z 1 = z 1 - N( : 2 * 1 , E 2 * 1 ), where : 2 * 1 = E[Z 2 * Z 1 = z 1 ] = : 2 + E 12 N [z 1 - : 1 ], E 2 * 1 = Var[Z 2 * Z 1 = z 1 ] = E 22 - E 12 N E 12 .
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-8 Properties of the Multivariate Normal Distribution: Suppose y - N( : , E ). (1) All marginal distributions are normally distributed . (2) Any two elements y i and y j are independent iff F ij = 0. (3) Ay + b - N(A : + b, A E A N ), where A and b are nonrandom. (4) For nonrandom A and B, Ay and By are independent iff A E B N = 0. (5) If : = 0, E = F 2 I, A (k×n) and idempotent B (n×n), then Ay and y N By are independent iff AB = 0. (6) If : = 0, E = F 2 I, A (n×n) and B (n×n) both idempotent, then y N Ay and y N By are independent iff AB = 0.
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Lecture 12, ECON 123A, Fall 2011 Dale J. Poirier 12-9
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Dale J. Poirier 12-10 Assumption MLR.6: The population error u is independent of the explanatory variables x 1 , x 2 , . .., x k and is normally distributed with zero mean and variance F 2 : u - N(0, F 2 ). C Assumptions E.1 - E.5 and MLR.1 - MLR.6 are essentially equivalent. C
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