Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

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1 Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The CLR model specfes that the relato betwee the depedet varable Y ad the depedet varable s a exact lear relato,.e. Y α β u,, K, ad that, u satsfy the assumptos

2 Assumpto : Assumpto : costats. u,, K, are radom varables wth ( ) 0,, K, are determstc,.e. o-radom, E u Assumpto 3 (Homoskedastcty) All u ' s have the same varace,.e. for, K, E( u ) Var( u ) σ Assumpto 4 (No seral correlato) The radom errors u ad u are ot correlated for all j j, K, Cov ( u, u ) E( u u ) j j 0

3 If these assumptos hold, the OLS estmators Y Y ) ( ) )( ( ˆβ Y β α ˆ ˆ of the regresso coeffcets β α, have the followg propertes 3

4 . The OLS estmators ˆ, β are ubased,.e. for the mea of the samplg dstrbuto t holds that α ˆ E ( ˆ) α α, E( ˆ β ) β For ths oly assumptos ad are eeded.. The varace of the samplg dstrbuto s ( Var ( ˆ) α σ ) 4

5 Var( ˆ) β σ ( ) Cov ( ˆ, α ˆ) β σ ( ˆ ) 3. The OLS estmators ˆ α, β are cosstet,.e. large samples the samplg dstrbuto s cocetrated α, β. The samplg varace forms us how precse the estmates are. It ca be show that uder assumptos -4, the OLS estmators are the best estmators,.e. have the smallest samplg varace, amog the ubased estmators that ca be expressed as a, lear expresso Y,.e. as a weghted average of the Y. I jargo: The OLS estmators are Best Lear Ubased Estmators (BLUE) 5

6 For cofdece tervals ad hypothess tests, the mea ad varace of the samplg dstrbuto are ot eough: we eed that the samplg dstrbuto belogs to a class of dstrbutos that we ca work wth, e.g. the ormal dstrbuto. Ths wll be the case f the followg assumpto holds Assumpto 5. The radom error terms u,, K, are radom varables wth a ormal dstrbuto. 6

7 Note that Ths ormal dstrbuto has mea 0 (assumpto ) ad varace σ (assumpto 3) for all, K,. Hece the u s are radom varables wth detcal (ormal) dstrbutos The error terms u ad u are ucorrelated (assumpto 4) ad j because they are ormal, also stochastcally depedet Why s the ormal dstrbuto a obvous choce as a dstrbuto for the error term u? 7

8 What s the samplg dstrbuto of the OLS estmators f assumpto 5 holds? Cosder a CLR model wth α 0. We have see that the OLS estmator ˆ β ca be expressed as ˆ β β W u The rght-had sde s the sum of a costat (β ) ad a weghted average of ormal radom varables W u It ca be show that such a lear combato s also a radom varable wth a ormal dstrbuto. 8

9 Cocluso: If assumpto 5 holds, the samplg dstrbuto of s ormal wth mea β ad varace σ. If assumpto 5 does ot hold the we caot coclude that the samplg dstrbuto of ˆ β s ormal. Why wll t geeral be close to ormal? βˆ 9

10 If the CLR model has a tercept we have the same result: If assumpto 5 holds, the samplg dstrbuto of the OLS estmators ˆ α, ˆ β s also ormal wth mea α, β ad varace ( Var ( ˆ) α σ Var( ˆ) β respectvely. σ ) ( ) 0

11 Now we ca fd a cofdece terval for β. As the co tossg example we stadardze the estmator Z ( ) ˆ β β σ Now Z has a ormal dstrbuto wth mea 0 ad varace.

12 We fd a 95% cofdece terval for β by cosderg the probablty. < < < <.96 ) ( ˆ.96 Pr.96).96 Pr( 95 Z σ β β Hece wth probablty.95 we have

13 ( ).96 ˆ.96 < < σ β β or equvaletly wrtte as a terval for β ( ) ( ) < <.96 ˆ.96 ˆ σ β β σ β Wth probablty.95, the ukow β s ths terval. Ths refers to repeated samples: 95% of these samples β s ths terval. 3

14 The terval s a 95% cofdece terval for β. What chages f we wat a 90% cofdece terval? Use Table of ormal dstrbuto! What problem do we have wth ths terval? Ca t be computed from the data? 4

15 Remember that σ s the varace of the radom error term u. We do ot have the s, because u u Y α β ad we do ot kow OLS resduals α, β. After we estmate α, β, we ca compute the e Y ˆ α ˆ β These are ot the same because we use estmators for Remember that the sample mea of the e,, K, s 0. α, β. 5

16 Hece, the sample varace s ˆ σ e Ths s a estmator of σ. The followg estmator s preferred s e Note that we have estmated parameters α, β to compute the e s, ad ths s the reaso we use. It ca be show that ths s a ubased ad cosstet estmator of σ. 6

17 Now we ca derve a computable cofdece terval. Start from T ( ) ˆ β β s.e. we substtute a (ubased, cosstet) estmator for σ. Because the deomator s estmated (ad hece vares samples) T does ot have a stadard ormal dstrbuto, but a Studet t dstrbuto. Ths dstrbuto has a larger varace. See Table D. for a comparso wth the stadard ormal dstrbuto (fal row). As the stadard ormal, the t dstrbuto s symmetrc aroud 0 ad has mea 0. 7

18 The t dstrbuto depeds o the umber of observatos. If s large t s close to the stadard ormal, but f s small t s much more dspersed. If we have observatos, the we have a t dstrbuto wth parameter. I jargo ths s umber of degrees of freedom of the t dstrbuto. Note that f > 60the error that oe makes usg the stadard ormal stead of the t dstrbuto s small. 8

19 The startg pot for fdg a 95% cofdece terval s ow. < < < < ) ( ˆ Pr ) Pr( 95 c s c c T c β β If we have 30 observatos, the 048. c. 9

20 What s f we have 6 observatos or f we wat a 90% cofdece terval? c For 30 observatos the 95% cofdece terval for β s ( ) ( ) < < s s.048 ˆ.048 ˆ β β β Ths ca be computed from the data. 0

21 The estmated stadard devato of the OLS estmator βˆ s ( ) s Ths s called the stadard error of (of course we ca also compute the stadard error of αˆ ) It s customary to report both the OLS estmate ad ts stadard error. Reaso s clear from formula for cofdece terval: boudares of cofdece terval are OLS estmate plus/mus a multple of the stadard error. Multple close to for 95% (exactly for 6). βˆ

22 I graphs the samplg dstrbuto of T s gve for the lear regresso model of lecture 6 wth α β 3,, K, are uform [0,] radom umbers u has a stadard ormal dstrbuto or a uform dstrbuto wth the same mea ad varace The graphs are base o 0000 samples.

23 If s small, the dstrbuto s more dspersed tha stadard ormal Eve f the error has a uform dstrbuto we ca use the t ad stadard ormal dstrbuto ( large) as a approxmato to the samplg dstrbuto of T 3

24 4

25 5

26 Explas why we do ot worry much about the valdty of assumpto 5. Next we cosder testg hypotheses o regresso parameters. We use the same setup as the co tossg example: Null ad alteratve hypothess Decso rule that chooses betwee these hypotheses 6

27 Null hypothess H 0 : β β 0 Alteratve hypothess or or H : β β 0 (two-sded alteratve H : β > β 0 (oe-sded alteratve) H : β < β0 (oe-sded alteratve) For stace f β s a prce elastcty we may test H : β 0 agast H : β < 0 0 H β 0 s the ull hypothess that has o effect o Y. 0 : 0 7

28 Decso rule Cosder T ( ) ˆ β β 0 s If H0 s true, the T has a t dstrbuto wth degrees of freedom. Ths dstrbuto s symmetrc aroud 0 ad most of the tme ( repeated samples) we get a value ot too far from 0. If H0 s ot true, the the dstrbuto of β > β 0 ) or to the left (f β < β0). T wll shft to the rght (f 8

29 The samplg dstrbuto of T for β 0 3 (correct value) ad for β (correct value) s plotted ( 30 ad 0000 samples)

30 β 0 ( 30)

31 We choose H (ad reject H0) f we obta a uexpectedly large postve or large egatve value for T, otherwse we accept H 0. The decso rule or test s Reject H 0 T How do we choose c? s ( ) ˆ β β 0 > c 3

32 Remember we ca make two errors the decso False rejecto of or Type I error H 0 False acceptace of or Type II error H 0 As co tossg example we caot make both small. Usual approach: Fx probablty of Type I error at small value. Ths gves c. Now f H0 s correct the T has a t dstrbuto wth df. The α Pr( Type I error) Pr( T > c) Pr( T < c) 3

33 From cofdece terval we kow that for α. 05 ad 30, we have c Hece we reject H 0 : β β0 f ad oly f T ˆ β β ( ) s 0 >.048 Note f β 0 0 we just look at the rato of the OLS estmate ad ts stadard error. If ths s greater tha about, we reject H : β

34 If the alteratve s H : β > β0, we may use a dfferet decso rule. I that case we reject oly f we get a uexpectedly large postve value for T. We reject H : β 0 f ad oly f 0 T s ( ) ˆ β β 0 > c Now c.70 for a 5% probablty of a Type I error. Ths s a oesded test. 34

35 Predcto After we have computed the OLS estmates ˆ, β usg the data Y,,, K, we ca use the estmated regresso model to predct Y for values of that have ot bee observed. α ˆ Example: predcto of sales prce of house that s ew o market. If that house has lvg area, the we predct Yˆ ˆ ˆ α β How close s ths to Y? 35

36 Predcto error Y ˆ) ( ˆ) ( ˆ ˆ ˆ u u Y β β α α β α β α Estmato error Ukow future omtted vars. Estmato error s small f we have e.g. may observatos ( large) We caot reduce u 36

37 We ca obta a predcto terval for. Ths s a cofdece terval. A 95% terval s f 30 Y 048. ˆ.048 ˆ < < s Y Y s Y (ote we use the t dstrbuto wth df) wth s s ) ( ) ( Note that f s large ths s close to s. Hece a rule of thumb for the predcto terval s s Y Y s Y ˆ ˆ < < 37

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UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS

UNIVERSITY OF OSLO DEPARTMENT OF ECONOMICS Exam: ECON430 Statstcs Date of exam: Frday, December 8, 07 Grades are gve: Jauary 4, 08 Tme for exam: 0900 am 00 oo The problem set covers 5 pages Resources allowed: