ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages QUEE'S UIVERSITY AT KIGSTO Department of Economcs ECOOMICS 35* - Secton A Introductory Econometrcs Fall Term 000 MID-TERM EAM ASWERS MG Abbott DATE: Monday October 3, 000 TIME: ISTRUCTIOS: MARKIG: 80 mnutes; :30 pm - 3:50 pm The exam conssts of FIVE (5) questons Students are requred to answer ALL FIVE (5) questons Answer all questons n the exam booklets provded Be sure your name and student number are prnted clearly on the front of all exam booklets used Do not wrte answers to questons on the front page of the frst exam booklet Please label clearly each of your answers n the exam booklets wth the approprate number and letter Please wrte legbly The marks for each queston are ndcated n parentheses mmedately above each queston Total marks for the exam equal 00 QUESTIOS: Answer ALL FIVE questons All questons pertan to the smple (two-varable) lnear regresson model for whch the populaton regresson equaton can be wrtten n conventonal notaton as: Y β + β + u () where Y and are observable varables, β and β are unknown (constant) regresson coeffcents, and u s an unobservable random error term The Ordnary Least Squares (OLS) sample regresson equaton correspondng to regresson equaton () s Y β! +! β + u! (,, ) () where! β s the OLS estmator of the ntercept coeffcent β,! β s the OLS estmator of the slope coeffcent β,!u s the OLS resdual for the -th sample observaton, and s sample sze (the number of observatons n the sample) ECO 35*-A (Fall Term 000): Md-Term Exam Page of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages (5 marks) State the Ordnary Least Squares (OLS) estmaton crteron State the OLS normal equatons Derve the OLS normal equatons from the OLS estmaton crteron ASWER: (3 marks) State the Ordnary Least Squares (OLS) estmaton crteron (3 marks) The OLS coeffcent estmators are those formulas or expressons for β and β that mnmze the sum of squared resduals RSS for any gven sample of sze The OLS estmaton crteron s therefore: Mnmze RSS { β j } ( β, β ) û ( Y β β ) State the OLS normal equatons The frst OLS normal equaton can be wrtten n any one of the followng forms: Y β β β β β + β 0 Y Y () The second OLS normal equaton can be wrtten n any one of the followng forms: Y β β β β β + β 0 Y Y () ECO 35*-A (Fall Term 000): Md-Term Exam Page of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 3 of 3 pages Queston (contnued) (8 marks) Show how the OLS normal equatons are derved from the OLS estmaton crteron Step : Partally dfferentate the RSS (, β ) β functon wth respect to β and β, usng û û Y β β β and û β RSS β û û β û ( ) û ( Y β β ) () RSS β û û β û ( Y β β ) ( ) ( Y β β ) snce û û Y β β () Step : Obtan the frst-order condtons (FOCs) for a mnmum of the RSS functon by settng the partal dervatves () and () equal to zero and then dvdng each equaton by and re-arrangng: RSS 0 β û 0 ( Y β β ) 0 ( Y β β ) 0 Y β β 0 Y β + β () RSS 0 β û 0 ( Y β β ) 0 ( Y β β ) 0 Y β β ( ) 0 Y β β 0 Y β + β () ECO 35*-A (Fall Term 000): Md-Term Exam Page 3 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 4 of 3 pages (5 marks) Answer parts (a), (b) and (c) below (3 marks) (a) Explan the meanng of the followng statement: The estmator β! s an unbased estmator of the slope coeffcent β (3 marks) The mean of the samplng dstrbuton of! β s equal to β : E(! β ) β (An approprate dagram would also be suffcent) (b) Show that the OLS slope coeffcent estmator β! s a lnear functon of the Y sample values β x y x Σ k x x Y Y x (Y Y) x because where x Y x k x x Σ x Y 0 x x (8 marks) (c) Statng explctly all requred assumptons, prove that the OLS slope coeffcent estmator β! s an unbased estmator of the slope coeffcent β () Substtute for Y the expresson Y β + β + u from the populaton regresson equaton (or PRE) β k Y k ( β + β ( β k β β k + β + k u, + u ) + β k + k u ) k + k u sn ce Y snce β + β + u by assumpton (A) k 0 and k () ow take expectatons of the above expresson for β : E( β ) E( β β β β ) + E[ k 0 k u + k E(u ) + ] snce β s a constant and the k snce E(u ) 0 by assumpton (A) are nonstochastc ECO 35*-A (Fall Term 000): Md-Term Exam Page 4 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 5 of 3 pages (0 marks) 3 Explan what s meant by each of the followng statements about the estmator θ of the populaton parameter θ, and explan the dfference between the two statements (a) θ s a mnmum varance estmator of θ (b) θ s an effcent estmator of θ ASWER: (a) θ s a mnmum varance estmator of θ The varance of the estmator θ s smaller than the varance of any other estmator of the parameter θ If θ ~ s any other estmator of θ, then θ s a mnmum varance estmator of θ f Var( θ ) Var( θ ~ ) (b) θ s an effcent estmator of θ The estmator θ s an effcent estmator f t s unbased and has smaller varance than any other unbased estmator of the parameter θ If θ ~ s any other unbased estmator of θ, then θ s an effcent estmator of θ f Var( θ ) Var( θ ~ ) where E(θ ) θ and E( θ ~ ) θ ( marks) The mportant dfference between statements (a) and (b) s that an effcent estmator must be unbased whereas a mnmum varance estmator may be based or unbased An effcent estmator s the mnmum varance estmator n the class of all unbased estmators of the parameter θ ECO 35*-A (Fall Term 000): Md-Term Exam Page 5 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 6 of 3 pages (0 marks) 4 State the error normalty assumpton State and explan the mplcatons of the error normalty assumpton for () the dstrbuton of the Y sample values and () the samplng dstrbuton of the OLS slope coeffcent estmator β! ASWER: (3 marks) Statement of Error ormalty Assumpton (A9): The random error terms u are ndependently and dentcally dstrbuted (d) as the normal dstrbuton wth zero mean and constant varance σ OR The random error terms u are normally and ndependently dstrbuted (ID) wth zero mean and constant varance σ the u are d as ( 0, σ ) for all OR the u are ID( 0 ), σ (7 marks) Two Implcatons of (A9): Follow from the lnearty property of the normal dstrbuton Lnearty property of the normal dstrbuton: any lnear functon of a normally dstrbuted random varable s tself normally dstrbuted ( mark) Two mplcatons of error normalty assumpton (A9): (3 marks each) (3 marks) () The Y values are normally dstrbuted: Y are ID ( β + β σ ), ( marks) Why? Because the PRE states that the Y values are lnear functons of the u : Y β + β + u ( mark) (3 marks) () The OLS slope coeffcent estmator β s normally dstrbuted: β ~ (,Var( ) ) β β σ where Var( β ) ( marks) x Why? Because β can be wrtten as a lnear functon of the Y values: β ky ( mark) ECO 35*-A (Fall Term 000): Md-Term Exam Page 6 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 7 of 3 pages (50 marks) 5 A researcher s usng data for a sample of 56 pad workers to nvestgate the relatonshp between hourly wage rates Y (measured n dollars per hour) and years of formal educaton (measured n years) Prelmnary analyss of the sample data produces the followng sample nformaton: 56 Y 30 35 y 870400 76044 66080 Y 54469 44065 x y 79 04 x Y 40543 û 598068 where x and y Y Y for,, Use the above sample nformaton to answer all the followng questons Show explctly all formulas and calculatons (0 marks) (a) Use the above nformaton to compute OLS estmates of the ntercept coeffcent β and the slope coeffcent β x y 7904 0543593 05436 (5 marks) β x 40543 β Y β Y Y 3035 56 589603 and 66080 56 5674 Therefore β Y β 589603 (05436)(5674) 589603 6800965 090486 (5 marks) (5 marks) (b) Interpret the slope coeffcent estmate you calculated n part (a) -- e, explan what the numerc value you calculated for β! means ote:! β 05436 Y s measured n dollars per hour, and s measured n years The estmate 05436 of β means that a one-year ncrease (decrease) n years of formal educaton s assocated on average wth an ncrease (decrease) n hourly wage rate of 05436 dollars per hour, or 5436 cents per hour ECO 35*-A (Fall Term 000): Md-Term Exam Page 7 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 8 of 3 pages (5 marks) (c) Calculate an estmate of σ, the error varance σ RSS û 598068 598068 4355 435 56 54 (6 marks) (d) Compute the value of R, the coeffcent of determnaton for the estmated OLS sample regresson equaton Brefly explan what the calculated value of R means () ESS TSS RSS y û 76044 598068 7973 or () R R ESS TSS ŷ 7973 0647575 0648 76044 y RSS TSS û 598068 0835 0648 76044 y Interpretaton of R 0648: The value of 0648 ndcates that 648 percent of the total sample (or observed) varaton n Y (hourly wage rates) s attrbutable to, or explaned by, the regressor (years of formal educaton) ( marks) ECO 35*-A (Fall Term 000): Md-Term Exam Page 8 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 9 of 3 pages ( marks) (e) Perform a test of the null hypothess H 0 : β 0 aganst the alternatve hypothess H : β 0 at the 5% sgnfcance level (e, for sgnfcance level α 005) Show how you calculated the test statstc State the decson rule you use, and the nference you would draw from the test Brefly explan what the test outcome means H 0 : β 0 H : β 0 a two-sded alternatve hypothess a two-taled test β β Test statstc s t( β ) ~ t[ ] () sê( β ) Calculate the estmated standard error of β : σ 4355 Vâr( β ) 00083535 x 40543 s ê( β ) Vâr( β ) 00083535 005348 ( mark) Calculate the sample value of the t-statstc () under H 0 : set β 0 n () t β β 05436 00 05436 ( β ) 06676 067 (3 marks) sê( β ) 005348 005348 0 ull dstrbuton of ( β ) s t[ ] t[54] ( mark) t0 Decson Rule -- Formulaton : At sgnfcance level α, ( marks) reject H 0 f t 0 ( β ) > t α [54], e, f ether () ( β ) > t [54] or () ( β ) < t [54]; t 0 α t 0 α retan H 0 f ( β ) t [54], e, f [54] t ( β ) t [54] t 0 α t α 0 α Crtcal value of t[54]-dstrbuton: from t-table, use df two-taled 5 percent crtcal value ] t [54 α t 005 [ ] 960 ( mark) ECO 35*-A (Fall Term 000): Md-Term Exam Page 9 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 0 of 3 pages Queston 5(e) -- contnued Inference: At 5 percent sgnfcance level, e, for α 005, (3 marks) t 0( β ) 067 > 960 t 005[ ] reject H 0 vs H at 5 percent level Inference: At the 5% sgnfcance level, the null hypothess β 0 s rejected n favour of the alternatve hypothess β 0 Meanng of test outcome: Rejecton of the null hypothess β 0 n favour of the alternatve hypothess β 0 means that the sample evdence favours the exstence of a relatonshp between wage rates and years of educaton ( mark) ECO 35*-A (Fall Term 000): Md-Term Exam Page 0 of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages ( marks) (f) Compute the two-sded 95% confdence nterval for the slope coeffcent β Would the two-sded 99% confdence nterval be wder or narrower than the two-sded 95% confdence nterval for β? Why? The two-sded ( α)-level, or 00( α) percent, confdence nterval for β s computed as β t α / [ ] sê( β ) β β + t α [ ] sê( β ) ( marks) where β α ]sê( the lower 00( α)% confdence lmt for β β α ]sê( the upper 00( α)% confdence lmt for β L β t [ β) U β + t [ β) t [ ] α the α/ crtcal value of the t-dstrbuton wth degrees of freedom!β 05436 ê( β ) Vâr( β ) 0 0083535 005348 s α 095 α 005 α/ 005: α [ ] t [54] 960 t 0 05 t α [ ] sê( β ) t 005[54] sê( β ) t 005[ ] sê( β ) 960(005348) 004366 Lower 95% confdence lmt for β s: β L β tα [ ]sê( β) β t 005[ ]sê( β) 05436 960(005348) 0 5436 004366 0436994 04370 Upper 95% confdence lmt for β s: β t [ ]sê( ) U β + α β β + t 005[5]sê( β ) 0 5436 + 960(005348) 0 5436 + 004366 064576 06457 Result: The two-sded 95% confdence nterval for β s: [04370, 06457] ECO 35*-A (Fall Term 000): Md-Term Exam Page of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page of 3 pages Queston 5(f) -- contnued ( marks) The two-sded 99% confdence nterval for β would be wder than the two-sded 95% confdence nterval Reason: The crtcal value of the t[54] t[ ] dstrbuton s greater for the confdence level α 099 than for the confdence level α 095 For the two-sded 95% confdence nterval: α 095 α 005 α/ 005: α [ ] t [54] t 005 [ ] 960 For the two-sded 99% confdence nterval: t 0 05 α 099 α 00 α/ 0005: α [ ] t [54] t 0005 [ ] 576 t 0 005 ECO 35*-A (Fall Term 000): Md-Term Exam Page of 3 pages
ECOOMICS 35*-A Md-Term Exam -- Fall Term 000 Page 3 of 3 pages Percentage Ponts of the t-dstrbuton ECO 35*-A (Fall Term 000): Md-Term Exam Page 3 of 3 pages