Section 3 - Section3-Econl40 GSI: Edson Severnini MODEL i...

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THE TWO-VARIABLE REGRESSIO}{ MODEL i THE MODEL Section3-Econl40 GSI: Edson Severnini Yi= d l BXi* ei Assumptions I through 5 constitute the classical linear regression model. l. The relationship between Y and X is linear, as given by Eq' (3.1). 2. The X's are nonstochastic variables whose values are fixed. 3. The error hus ,.ro expected value:lE(e)' : O. : * 4. The error term has constant variance for,all observations, i-e., : n@\ : 02-(homoscedasticttY\.i : 5. The random variables el are stdtistically independent. Thus, '6.TheerrotterrnisnormaIlydistributed.,.,l. . or 3'. The random variable Y has expected value o * E(Yi) = E(a *'FXi + e;) : d + FXi + E(e;) 4'. The random variabie Yhas constant variance' 5'. The random variables Yi ar€ independent- Heteroscedasticity Y FXi E(Xie):XiE(e):0 Bx :q. I Serial correlation, (a) Negative Serial Correlation (b) Positive Serial Correlation
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BEST LINEAR UNBIASED ESTIMATION Yi:d+BXi+ei ? = a+ BX'E Y1 - V = B(x; - t) + (si - 6) or yi= fu+ ei-E li : Bi + 6i assumption that e = 0 ordinory least-squ:afts (OLS) estimators ^ 4v, r s,2 ^ E(B\ = p E(a) = a ^o' Var (F) = -r var (d, : o-;./L -v\z z.^i r\-\ i-^l Gauss-Markov Theorem Given assumpdons I it[ough 5, the estimatorsd and p are rhe b€st (most effiaient')'1i!.ea.r unbiased.esttnators of o and prin the sense that they hav€ thJ ininiYhum vadance of aU $near unbiased estimators. ' : With assumPtion 6 p-,(p,*) ^-r(",*f*r) -xo' uov (4, p) = .s,-2 or ., $2 where s2 = &2 = -?'6? - ->(\ --a -: En' s;= S7 N-z N-z 4i . . ./ >X? \ (t, SER, is call€d the standard error of the regression.l s; = r\tv>"?/ 6i" taPr = -5 z'^i Even if the 1.,'s are not normally distributed, the distribution of 1i can be shown to be asymptotically normal (under reasonable conditions) by appeal to the central limit theorem oI sratistics.
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This note was uploaded on 08/06/2009 for the course ECON 140 taught by Professor Duncan during the Summer '08 term at University of California, Berkeley.

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Section 3 - Section3-Econl40 GSI: Edson Severnini MODEL i...

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