Professor Chris Murray. Midterm Exam
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1 Econ 7 Econometrcs Sprng 4 Professor Chrs Murray McElhnney D cjmurray@uh.edu Mdterm Exam Wrte your answers on one sde of the blank whte paper that I have gven you.. Do not wrte your answers on ths exam. You must explan your answers. If you are confused about a queston or you thnk t s unclear, please ask for clarfcaton before answerng. When testng a hypothess, make sure to wrte down the null and alternatve, the crtcal value(s), the test statstc, and your decson (reject or fal to reject). Use tests wth 5% sze. Unless otherwse specfed, use -sded alternatve hypotheses.. Statstcal Propertes of the OLS Estmator Consder the classcal lnear regresson model and u ~ (, σ ). I n y = Xβ + u, wth determnstc regressors a. Derve the mean and covarance matrx of ˆOLS β. (5 ponts) b. Is βˆ OLS effcent? Be explct. (5 ponts) Now assume that the error terms are Gaussan; u ~ N(, σ ). c. Does your answer to part b change? ( ponts) ( ) [ ] ( ) ˆ Rβ r ' R( X ' X ) R' R ˆ β r d. Consder the statstc, σ dstrbuton n fnte samples, but s not feasble snce I n, whch we know has a χ j σ s unknown. Dscuss how to deal wth ths nusance parameter problem to get the feasble F-statstc. ( ponts) e. Wrte down (.e. do not derve) the F-statstc n terms of restrcted and unrestrcted sums of squared resduals and R-squares. (5 ponts)
2 . Numercal Propertes of the OLS Estmator Consder the OLS decomposton of y nto ts explaned and unexplaned components: y = yˆ + uˆ. a. Wrte ŷ and û n terms of projecton matrces. (5 ponts) b. Interpret these projecton matrces. (5 ponts) c. In what sense s the OLS decomposton an orthogonal decomposton? Be specfc. (5 ponts). The Frsch-Waugh Theorem Consder the regresson model Y = α + βx + u, () compared wth the model y = β x + u, () where y Y Y and x X X. a. Argue that the dependent and ndependent varables n () can be thought of as resduals from (). What regressons generate these resduals? What s the resdual maker whch generates these resduals? Interpret ths resdual maker. (5 ponts) b. Wth these n mnd, what s the formula for the least squares estmator of the slope coeffcent n ()? (5 ponts)
3 4. Consder the followng regresson model wth Normal errors: Y + u = β + βx + β X + β X where u ~ dn(, σ ). a. Construct a 95% confdence nterval for β. Use ths confdence nterval to test the followng hypothess: (5 ponts) : β = 4 : β 4 b. Test the followng hypothess usng an F-statstc: (5 ponts) : β + β = 4 : β + β 4 c. Test the followng hypotheses: (5 ponts) : β =, β = : β, β d. Usng a p-value, test the null that β = aganst the -sded alternatve. (5 ponts)
4 Regresson Output for Problem 4 Dependent Varable: Y Date: //8 Tme: 5: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. C X X X R-squared.9958 Mean dependent var Adjusted R-squared S.D. dependent var 8.85 S.E. of regresson.474 Akake nfo crt.649 Sum squared resd 9.47 Schwarz crteron.85 Log lkelhood F-statstc Durbn-Watson stat.4894 Prob(F-statstc). Coeffcent Covarance Matrx C X X X C X X X
5 Dependent Varable: Y Date: //8 Tme: 6: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. C R-squared. Mean dependent var Adjusted R-squared. S.D. dependent var 8.85 S.E. of regresson 8.85 Akake nfo crt Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat.6955 Dependent Varable: Y Date: //8 Tme: 6: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. C X X R-squared Mean dependent var Adjusted R-squared S.D. dependent var 8.85 S.E. of regresson Akake nfo crt Sum squared resd Schwarz crteron Log lkelhood F-statstc 8.8 Durbn-Watson stat.578 Prob(F-statstc).9 5
6 Dependent Varable: Y Date: //8 Tme: 6: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. X X R-squared Mean dependent var Adjusted R-squared.856 S.D. dependent var 8.85 S.E. of regresson Akake nfo crt Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat Dependent Varable: Y Date: //8 Tme: 6: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. X X R-squared.994 Mean dependent var Adjusted R-squared.9946 S.D. dependent var 8.85 S.E. of regresson.7697 Akake nfo crt 4. Sum squared resd Schwarz crteron Log lkelhood Durbn-Watson stat.79 6
7 Dependent Varable: Y Date: //8 Tme: 6: Sample: 5 Included observatons: 5 Varable Coeffcent Std. Error t-statstc Prob. C X X R-squared.9978 Mean dependent var Adjusted R-squared.996 S.D. dependent var 8.85 S.E. of regresson Akake nfo crt Sum squared resd Schwarz crteron Log lkelhood F-statstc 7.44 Durbn-Watson stat.696 Prob(F-statstc). 7
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