Hypothesis Testing in the Classical Normal Linear Regression Model. 1. Components of Hypothesis Tests

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1 ECONOMICS 35* -- NOTE 8 M.G. Abbo ECON 35* -- NOTE 8 Hypohesis Tesing in he Classical Normal Linear Regression Model. Componens of Hypohesis Tess. A esable hypohesis, which consiss of wo pars: Par : a null hypohesis, H Par 2: an alernaive hypohesis, H 2. A feasible es saisic. Definiion: A es saisic is a random variable whose value for given sample daa deermines wheher he null hypohesis H is rejeced or no rejeced. Definiion: A es saisic is feasible if i saisfies wo condiions: () Is probabiliy disribuion, or sampling disribuion, mus be known compleely when he null hypohesis H is rue, and i mus have some oher disribuion when he null hypohesis is false. (2) Is value can be calculaed from he given sample daa. 3. A decision rule or rejecion rule. Definiion: A decision rule specifies () he rejecion region and (2) he nonrejecion region of he es saisic. () Definiion: The rejecion region is he se, or range, of values of he es saisic for which he null hypohesis H is rejeced i.e., ha have a low probabiliy of occurring when he null hypohesis is rue. (2) Definiion: The nonrejecion region is he se, or range, of values of he es saisic for which he null hypohesis H is no rejeced, or reained. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page of 49 pages

2 ECONOMICS 35* -- NOTE 8 M.G. Abbo Five Basic Seps 2. Procedure for Tesing Hypoheses The procedure for esing hypoheses consiss of five basic seps. Sep : Formulae he null hypohesis H and he alernaive hypohesis H. Sep 2: Specify he es saisic and is disribuion -- specifically is disribuion when he null hypohesis H is rue. The disribuion of he es saisic when he null hypohesis H is rue is known as he null disribuion of he es saisic. Sep 3: Calculae he sample value of he es saisic under he null hypohesis H for he given sample daa. Sep 4: Selec he significance level α, and deermine he corresponding rejecion region and non-rejecion region for he es saisic. Sep 5: Apply he decision rule of he es and sae he inference, or conclusion, implied by he sample value of he es saisic. We illusrae hese five seps for an imporan class of hypohesis ess in applied economerics -- namely ess of equaliy resricions on individual regression coefficiens. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 2 of 49 pages

3 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tess of Equaliy Resricions on Individual Regression Coefficiens These ess assess he probable empirical validiy of saemens or hypoheses of he following form: β j = b j where b j is a specified consan. (j =, ) Such saemens are conjecures abou he populaion values of he regression coefficiens β j (j =, ). Examples β = E( Y X ) X, i.e., X i is unrelaed o E ( ) i i i = β =. E( Y X ) X i i i = β =.8 E( Y X ) X. 8 i i i = β =. E( Y X ) X =. i i i Y i X i Laer we will consider more general hypoheses ha ake he form of linear equaliy resricions on wo or more regression coefficiens β j (j =, ). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 3 of 49 pages

4 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP : Formulaion of he Null and Alernaive Hypoheses Sep : Formulae he null hypohesis H and he alernaive hypohesis H. Componens of a Saisical Tes A saisical hypohesis es consiss of wo opposing saemens or proposiions or conjecures abou he model parameers:. The null hypohesis, denoed by H. H is he proposiion being esed. I specifies our conjecure abou he rue value(s) of he regression coefficien(s). 2. The alernaive hypohesis, denoed by H. H is he couner-proposiion o he null hypohesis H. I specifies he se of alernaive possibiliies which is presumed o conain he ruh if he null hypohesis is false. Purpose of a Saisical Tes A saisical es is designed and consruced so as o provide sample evidence respecing he probable empirical validiy, or ruh, of he null hypohesis H. The es addresses he quesion: Are he sample esimaes of he model parameers -- consisen or inconsisen (compaible or incompaible) wih he ruh of he null hypohesis? Consisency or compaibiliy means sufficienly close o he value(s) specified by H ha we reain (do no rejec) he null hypohesis. A saisical es does no es he empirical validiy, or ruh, of he alernaive hypohesis H. Only he null hypohesis H is being subjeced o es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 4 of 49 pages

5 ECONOMICS 35* -- NOTE 8 M.G. Abbo Formulaion of H and H : Equaliy Resricions on β The Null Hypohesis H H : β = b where b is a specified consan (such as or.9 or ). The Alernaive Hypohesis H For a null hypohesis of his general form, here are hree possible alernaive hypoheses. () H : β b a wo-sided alernaive hypohesis. Rejecion of he null hypohesis H : β = b implies ha β akes some oher value, and ha his oher value is eiher greaer han or less han b. Tha is, H : β b eiher β > b or β < b. (2) H : β > b a one-sided (righ-sided) alernaive hypohesis. Rejecion of he null hypohesis H : β = b in his case implies ha β akes some oher value ha is greaer han b. This alernaive hypohesis compleely discouns he possibiliy ha β < b. I implies ha values of β less han b are considered o be logically unaccepable alernaives o he null hypohesis, an implicaion ha presumably is based on economic heory. (3) H : β < b a one-sided (lef-sided) alernaive hypohesis. Rejecion of he null hypohesis H : β = b in his case implies ha β akes some oher value ha is less han b. This alernaive hypohesis compleely excludes he possibiliy ha β > b. I implies ha he value of β could no be greaer han b if in fac he null hypohesis H is false. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 5 of 49 pages

6 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 2: Specify he Tes Saisic and is Null Disribuion Sep 2: Specify he es saisic and is null disribuion when he null hypohesis H is rue. Theoreical Prerequisie Assumpions A-A9 of he CNLRM -- especially he normaliy assumpion A9. Imporan Resuls. Under assumpions A-A9 of he CNLRM, he following -saisics are feasible es saisics for he OLS coefficien esimaors ˆβ and β $ : ˆ ˆ β β β β β (ˆ β ) = = = ~ [N 2] Vâr(ˆ β ) sê(ˆ β) σˆ 2 2 ( x ) ˆ ˆ β β β β β (ˆ β ) = = = ~ [N 2] Vâr(ˆ β ) sê(ˆ β) σˆ 2 i i 2 2 / 2 2 ( X ) N ( x ) i i i i 2. For he rue (bu unknown) values of he populaion regression coefficiens β and β, each of he es saisics ( ˆβ ) and ( β $ ) has he -disribuion wih N 2 degrees of freedom, denoed as [N 2]. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 6 of 49 pages

7 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 3: Evaluae he Tes Saisic Under H Sep 3: Calculae he sample value of he es saisic under he null hypohesis H for he given sample daa. The null hypohesis is H : β = b where b is a specified consan such as or.9 or -. From Sep 2, he feasible es saisic is he -saisic for he OLS esimaor ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) To calculae he sample value of ( ˆβ ) under he null hypohesis H: β = b, simply subsiue in he above formula for ( ˆβ ) he value b for β, since b is he value of β specified by H ; he sample value of ˆβ, he poin esimae of β for he given sample daa; he sample value of sê(ˆ β ) = Vâr(ˆ β), he esimaed sandard error of ˆβ. The sample value of ( ˆ b β b (ˆ β ) = =. Vâr(ˆ β ) sê(ˆ β) ˆβ ) evaluaed under he null hypohesis H is herefore Noe: The subscrip on ( ˆβ ) indicaes he value of ( ˆβ ) under H. The null disribuion of ( ˆβ ), he calculaed sample value of ( ˆβ ), is [N 2], he -disribuion wih N 2 degrees of freedom: ˆ b β b (ˆ β ) = = ~ [N 2] under H : β = b. Vâr(ˆ β ) sê(ˆ β) ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 7 of 49 pages

8 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 4: Deermine he Rejecion and Non-Rejecion Regions Sep 4: Selec he significance level α, and deermine he corresponding rejecion region and non-rejecion region for he calculaed es saisic. Background: Type I and Type II Errors In performing any hypohesis es i.e., in deciding o rejec or reain a null hypohesis here is always some chance of making misakes. Such misakes arise whenever he decision o reain or rejec H does no reflec he rue bu unknown sae of he world. Decision Reain (do no rejec) H Rejec H H is rue Correc Decision Pr = α Type I Error Pr = α Sae of he World H is false Type II Error Pr = β Correc Decision Pr = β. A correc decision is made if: he null hypohesis H is false and he decision is o rejec i. he null hypohesis H is rue and he decision is o reain (no o rejec) i. 2. An incorrec decision is made if: he null hypohesis H is rue and he decision is o rejec i (a Type I error). he null hypohesis H is false and he decision is o reain (no o rejec) i (a Type II error). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 8 of 49 pages

9 ECONOMICS 35* -- NOTE 8 M.G. Abbo Definiions: Type I error: Type II error: rejecing H when H is rue. no rejecing H when H is false. Probabiliies of Type I and Type II Errors α Pr(Type I Error) = Pr(H is rejeced H is rue) α = Pr(H is no rejeced H is rue) = he confidence level of he es = he probabiliy of making a correc decision when he null hypohesis H is rue. β Pr(Type II Error) = Pr(H is no rejeced H is false) β = Pr(H is rejeced H is false) = he power of he es = he probabiliy of making a correc decision when he null hypohesis H is false. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 9 of 49 pages

10 ECONOMICS 35* -- NOTE 8 M.G. Abbo Analogy Beween Saisical Hypohesis Tess and Criminal Cour Trials Presumpion of Innocence: The accused is presumed innocen unil proven o be guily beyond a reasonable doub. H : he accused is no guily H : he accused is guily as charged The cour mus decide wheher o reain or rejec H on he basis of admissible evidence. Cour s Decision Accused is innocen Sae of he World Accused is guily Acqui (find no guily) Correc Decision Type II Error Convic (find guily) Type I Error Correc Decision. The cour makes a correc decision if: he accused is innocen and he cour s decision is o acqui. he accused is guily and he cour s decision is o convic. 2. The cour makes an incorrec decision if: he accused is innocen and he cour s decision is o convic (he cour has made a Type I error). he accused is guily and he cour s decision is o acqui (he cour has made a Type II error). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page of 49 pages

11 ECONOMICS 35* -- NOTE 8 M.G. Abbo The Significance Level of he Tes Definiion: The significance level of he es is chosen o equal α, he probabiliy of making a Type I error. Significance level of he es = α Pr(Type I Error) = Pr(H is rejeced H is rue). Confidence level of he es = α = Pr(H is no rejeced H is rue). Power of he Tes Definiion: The power of he es is defined o equal β, he probabiliy of making a correc decision when he null hypohesis H is false. Power of he es β = Pr(H is rejeced H is false) = Pr(Type II Error) Relaionship Beween Type I and Type II Errors For any given sample size N, α = Pr(Type I Error) is inversely relaed o Pr(Type II Error) α = Pr(Type I Error) is direcly relaed o he power of he es. Resul: For any given sample size N, here exiss a rade-off beween () α = Pr(Type I Error) = he significance level of he es and (2) β = Pr(Type II Error) ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page of 49 pages

12 ECONOMICS 35* -- NOTE 8 M.G. Abbo Implicaions of Trade-Off Beween α and β By choosing a lower significance level α -- and hereby reducing he Pr(Type I Error) -- we necessarily () increase β = he Pr(Type II Error), and (2) decrease β = he power of he es. Conversely, by choosing a higher significance level α -- and hereby increasing he Pr(Type I Error) -- we necessarily () decrease β = he Pr(Type II Error), and (2) increase β = he power of he es. Commens on Choosing α, he Significance Level of he Tes The value of he significance level α is usually chosen o be small. In pracice, he values mos frequenly chosen are: α =. α =.5 α =. (a % significance level); (a 5% significance level); (a % significance level). Bu he choice of value for α is more or less arbirary. I presumably reflecs he invesigaor s relaive willingness o accep, or relaive aversion o, Type I and Type II errors. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 2 of 49 pages

13 ECONOMICS 35* -- NOTE 8 M.G. Abbo Rejecion and Non-Rejecion Regions for a Tes Saisic Definiion of he Rejecion and Non-Rejecion Regions The rejecion region is he se, or range, of values of he es saisic for which he null hypohesis H is rejeced. Values of he es saisic in he rejecion region have a low probabiliy of occurring when he null hypohesis H is rue. The nonrejecion region is he se, or range, of values of he es saisic for which he null hypohesis H is no rejeced, or reained. Disinguishing Beween he Rejecion and Nonrejecion Regions Quesion: Answer: How are he rejecion and nonrejecion regions delineaed or separaed? By he criical values of he es saisic or more correcly, by he criical values of he null disribuion of he es saisic. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 3 of 49 pages

14 ECONOMICS 35* -- NOTE 8 M.G. Abbo Criical Values of a Tes Saisic Definiion: The criical values of a es saisic are defined as hose values ha separae he rejecion region from he non-rejecion region, ha pariion he sample values of a es saisic ino a rejecion region and a non-rejecion region. Deerminans The criical values of a es saisic are deermined by he following facors:. he null disribuion of he es saisic he probabiliy, or sampling, disribuion of he es saisic when he null hypohesis H is rue; 2. he chosen significance level for he es, α; 3. he naure of he hypohesis es, specifically wheher he es is () a wo-ail, or wo-sided, es or (2) a one-ail, or one-sided, es, of which here are wo ypes, (2.) a lef-ail es (2.2) a righ-ail es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 4 of 49 pages

15 ECONOMICS 35* -- NOTE 8 M.G. Abbo Two-Tail and One-Tail Tess: Which is i? Imporan Poin: Wheher a wo-ail or one-ail es is appropriae depends on he naure of he alernaive hypohesis H. Definiions A wo-ail es is one for which he alernaive hypohesis H is a wo-sided hypohesis ha incorporaes he no equal condiion. Example: H : β = b where b is a specified consan H : β b is a wo-sided alernaive hypohesis a wo-ail es a wo-ail rejecion region. A one-ail es is one for which he alernaive hypohesis H is a one-sided hypohesis ha incorporaes eiher he less han condiion < or he greaer han condiion >. Case : A lef-ail es is one in which H incorporaes he less han condiion <. Example: H : β = b where b is a specified consan H : β < b is a lef-sided alernaive hypohesis a one-ail es, specifically a lef-ail es. a lef-ail rejecion region. Case 2: A righ-ail es is one in which H incorporaes he greaer han condiion >. Example: H : β = b where b is a specified consan H : β > b is a righ-sided alernaive hypohesis a one-ail es, specifically a righ-ail es. a righ-ail rejecion region. Noe: The form of he alernaive hypohesis H specifically he direcion of he inequaliy in he alernaive hypohesis H deermines wheher a oneail es is a righ-ail es or a lef-ail es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 5 of 49 pages

16 ECONOMICS 35* -- NOTE 8 M.G. Abbo Deermining Criical Values for Two-Tail Tess Problem: Deermine he criical values for he following wo-ail -es: H : β = b or β b = where b is a specified consan H : β b or β b a wo-sided alernaive hypohesis.. The appropriae es saisic is he -saisic for ˆβ, he OLS esimae of he slope coefficien β: ˆ β β β (ˆ β ) = =. Vâr(ˆ β ) sê(ˆ β) 2. The sample value of he es saisic ( ˆβ ) is calculaed by evaluaing ( ˆβ ) under he null hypohesis H. Tha is, in he expression for ( ˆβ ), se β equal o b, which is he value of β specified by H. The resuling sample value of he es saisic ( ˆβ ) under H is b (ˆ β ) =. sê(ˆ β ) 3. The null disribuion of ( ˆβ ) is [N 2], he -disribuion wih N 2 degrees of freedom. In oher words, if he null hypohesis H: β = b is rue, hen b (ˆ β ) = ~ [N 2] under H : β = b. sê(ˆ β ) ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 6 of 49 pages

17 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion : Wha values of ˆβ and (ˆ β ) would lead us o rejec H agains H? Answer: Examine he numeraor of he calculaed -saisic: Remember: (ˆ β ) = b sê(ˆ β ) ˆβ = he esimaed value of β b = he hypohesized value of β ê(ˆ β ) > ( ê(ˆ β ) is always a posiive number) s s We would rejec H agains H if ˆβ, he esimaed value of β, is very differen from b, he hypohesized value of β. More specifically, we would rejec H agains H in eiher of he following wo cases:. ˆβ >> b ˆβ b >> b (ˆ β ) = >> sê(ˆ β ) Values of ˆβ much greaer han b imply large posiive values of ( ˆβ ); or 2. ˆβ << b ˆβ b << b (ˆ β ) = << sê(ˆ β ) Values of ˆβ much less han b imply large negaive values of ( ˆβ ). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 7 of 49 pages

18 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion 2: How much larger or smaller han zero does he value of o be for us o rejec H in favour of H? β (ˆ ) have Answer: I depends on he significance level we choose for he es and he null disribuion of our es saisic. Le α = he chosen significance level for he es (e.g.,.,.5, or.) = he probabiliy of making a Type I error. Allocae α equally beween large posiive values of and large negaive values of. We herefore have boh an upper criical value and a lower criical value of he null disribuion of our es saisic, which is he [N 2]-disribuion. α 2[N 2] = he upper α/2 criical value of he [N 2]-disribuion; α [N 2] = he lower α/2 criical value of he [N 2]-disribuion. 2 Implicaions: If H : β = b is rue, hen he following wo probabiliy saemens hold. () ( [N 2] β ) [N 2] ) = α Pr α / 2 (ˆ α / 2 () (2) ( (ˆ β ) < [N 2] or (ˆ β ) > [N 2] ) where Pr α / 2 α / 2 = Pr (ˆ β) > α / 2[N 2] = (2) ( ) α (ˆ β ) = he calculaed sample value of he -saisic under H; (ˆ β ) = he absolue value of ( ˆβ ); α 2[N 2] = he upper α/2 criical value of he [N 2]-disribuion; α [N 2] = he lower α/2 criical value of he [N 2]-disribuion; 2 α = he significance level for he es; α = he confidence level for he es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 8 of 49 pages

19 ECONOMICS 35* -- NOTE 8 M.G. Abbo Deermine he Rejecion and Nonrejecion Regions Two-Tail Tes The non-rejecion region for ( ˆβ ) is defined by he double inequaliy in probabiliy saemen () above. I consiss of all values of ( ˆβ ) such ha 2 α / 2[N 2] (ˆ β) α / [N 2] non-rejecion region for H : β = b. The rejecion region for ( ˆβ ) is he wo-sided region or wo-ail region defined in probabiliy saemen (2) above. I consiss of all values of ( ˆβ ) such ha or (ˆ β ) < α [N 2] or (ˆ β ) > [N 2] / 2 α / 2 / 2 (ˆ β ) > α [N 2] rejecion region for H : β = b is a wo-ail rejecion region. NOTE: For a wo-ail es, he rejecion region is a wo-ail rejecion region consising of wo pars. ) The lower or lef-ail rejecion region ( ˆβ ) < α/2[n 2], which conains unexpecedly small values of ( ˆβ ) under H i.e., values ha we would only expec o obain wih small probabiliy α/2 if he null hypohesis H : β = b is rue. 2) The upper or righ-ail rejecion region ( ˆβ ) > α/2[n 2], which conains unexpecedly large values of ( ˆβ ) under H i.e., values ha we would only expec o obain wih small probabiliy α/2 if he null hypohesis H : β = b is rue. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 9 of 49 pages

20 ECONOMICS 35* -- NOTE 8 M.G. Abbo 3) The rejecion region for a wo-ail es hus consiss of boh he lower lefhand ail and he upper righ-hand ail of he [N 2]-disribuion. The area in each ail under he [N 2]-disribuion is α/2. The sum of hese wo ail area probabiliies equals he significance level α. Thus, for a wo-ail es, he significance level α is allocaed equally beween he lower α/2 rejecion region and he upper α/2 rejecion region. Rejecion and Nonrejecion Regions for a Two-Tail Tes lef-ail rejecion nonrejecion region righ-ail rejecion region region area = α/2 area = α area = α/ α/2 α/2 Pr( < α/2 ) = α/2 Pr( α/2 α/2 ) = α Pr( > α/2 ) = α/2 ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 2 of 49 pages

21 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 5: Apply he Decision Rule and Sae Inference Sep 5: Apply he decision rule of he es and sae he inference, or conclusion, implied by he sample value of he es saisic.. Formulaion of he Decision Rule for a Two-Tail Tes Formulaion : Deermine if he sample value of he es saisic lies in he rejecion or nonrejecion region a he chosen significance level α. Decision Rule for a Two-Tail Tes Formulaion. If he sample value of he es saisic lies in he rejecion region a he chosen significance level, hen rejec he null hypohesis H. For a wo-ail es: > α [N 2] rejec H a significance level α. / 2 Rejec H in favour of H a significance level α if () > α [N 2] meaning lies in he upper ail rejecion area; or / 2 (2) < α [N 2] meaning lies in he lower ail rejecion area. / 2 Inference: Rejec H in favour of H a significance level α. 2. If he sample value of he es saisic lies in he nonrejecion region a he chosen significance level, hen reain (do no rejec) he null hypohesis H. For a wo-ail es: α [N 2] reain H a significance level α. / 2 Reain H agains H a significance level α if 2 α / 2[N 2] α / [N 2] meaning lies in he nonrejecion area. Inference: Reain H agains H a significance level α. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 2 of 49 pages

22 ECONOMICS 35* -- NOTE 8 M.G. Abbo Examples of Two-Tail Hypohesis Tess The Model: DATA: auo.da (a Saa-forma daa file) MODEL: price i = β + β weigh i + u i (i =,..., N). regress price weigh Source SS df MS Number of obs = F(, 72) = Model Prob > F =. Residual R-squared = Adj R-squared =.282 Toal Roo MSE = price Coef. Sd. Err. P> [95% Conf. Inerval] weigh _cons N = 74 N 2 = 74 2 = 72 ˆβ = 2.44 ê(ˆ β ) = s α =.5 α/2 = α / 2[N 2] =. [72] =.9935 ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 22 of 49 pages

23 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tes : Tes he proposiion ha weigh i is unrelaed o price i a he 5 percen significance level (α =.5). Null and Alernaive Hypoheses H : β = H : β a wo-sided alernaive hypohesis. The feasible es saisic is he -saisic for ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) Compue he sample value of ( ˆβ 2 ) under he null hypohesis H: Se ˆβ = 2.44, β = and ê(ˆ ) = in he formula for ( ˆβ ): s β β (ˆ β ) = = = sê(ˆ β ) = The wo-ail criical value of he [N 2] disribuion a he 5 percen significance level (a α =.5) is α / 2[N 2] =. 25[72] = Decision Rule: If > α [N 2] rejec H a significance level α; / 2 / 2[N If α 2] reain H a significance level α. Inference: = >.9935 =. 25 [72] rejec H a significance level α =.5 Rejec H : β = in favour of H : β a he 5 percen significance level. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 23 of 49 pages

24 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tes 2: Tes he proposiion ha a -pound increase in weigh i is associaed wih an increase in average price i of dollar. Use he 5 percen significance level (α =.5). Null and Alernaive Hypoheses H : β = H : β a wo-sided alernaive hypohesis. The feasible es saisic is he -saisic for ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) Compue he sample value of ( ˆβ ) under he null hypohesis H: Se ˆβ = 2.44, β = and ê(ˆ ) = in he formula for ( ˆβ ): s β β (ˆ β ) = = = 2.77 sê(ˆ β ) = The wo-ail criical value of he [N 2] disribuion a he 5 percen significance level (a α =.5) is α / 2[N 2] =. 25[72] = Decision Rule: If > α [N 2] rejec H a significance level α; / 2 / 2[N If α 2] reain H a significance level α. Inference: = 2.77 >.9935 =. 25 [72] rejec H a significance level α =.5 Rejec H : β = in favour of H : β a he 5 percen significance level. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 24 of 49 pages

25 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tes 3: Tes he proposiion ha a -pound increase in weigh i is associaed wih an increase in average price i of 2 dollars. Perform he es a he 5 percen significance level (α =.5). Null and Alernaive Hypoheses H : β = 2 H : β 2 a wo-sided alernaive hypohesis. The feasible es saisic is he -saisic for ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) Compue he sample value of ( ˆβ ) under he null hypohesis H: Se ˆβ = 2.44, β = 2 and ê(ˆ ) = in he formula for ( ˆβ ): s β β (ˆ β ) = = =.7 sê(ˆ β ) = The wo-ail criical value of he [N 2] disribuion a he 5 percen significance level (a α =.5) is α / 2[N 2] =. 25[72] = Decision Rule: If > α [N 2] rejec H a significance level α; / 2 / 2[N If α 2] reain H a significance level α. Inference: =.7 <.9935 =. 25 [72] reain H a significance level α =.5 Reain H : β = 2 agains H : β 2 a he 5 percen significance level. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 25 of 49 pages

26 ECONOMICS 35* -- NOTE 8 M.G. Abbo How o perform all hree of hese wo-ail hypohesis ess a once Tes : H : β = versus H : β Tes 2: H : β = versus H : β Tes 3: H : β = 2 versus H : β 2 Compue he wo-sided 95 percen confidence inerval for β. = + α [N 2]sê(ˆ ) = = = U 2 β = [N 2]sê( ˆ α ) = = =.293 L 2 β Resul: The wo-sided 95% confidence inerval for β is [.293, 2.795]. Decision Rule: If he hypohesized value of β lies ouside he wo-sided 95 percen confidence inerval for β, rejec he null hypohesis H a he 5 percen significance level. If he hypohesized value of β lies inside he wo-sided 95 percen confidence inerval for β, reain he null hypohesis H a he 5 percen significance level. Tes : Since he value lies ouside he wo-sided 95 percen confidence inerval for β, rejec H : β = in favour of H : β a he 5 percen significance level. Tes 2: Since he value lies ouside he wo-sided 95 percen confidence inerval for β, rejec H : β = in favour of H : β a he 5 percen significance level. Tes 3: Since he value 2 lies inside he wo-sided 95 percen confidence inerval for β, reain H : β = 2 agains H : β 2 a he 5 percen significance level. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 26 of 49 pages

27 ECONOMICS 35* -- NOTE 8 M.G. Abbo Deermining Criical Values for One-Tail Tess CASE A Lef-Tail Tes Problem: Deermine he criical values for he following one-ail -es: H : β = b or β b = H : β < b or β b < where b is a specified consan a lef-sided alernaive hypohesis. a one-ail es, specifically a lef-ail es.. The appropriae es saisic is he -saisic for ˆβ, he OLS esimae of he slope coefficien β: ˆ β β β (ˆ β ) = =. Vâr(ˆ β ) sê(ˆ β) 2. The sample value of he es saisic ( ˆβ ) is calculaed by evaluaing ( ˆβ ) under he null hypohesis H. This involves evaluaing ( ˆβ ) using he equaliy form of he null hypohesis H, which is β = b. Seing β = b in he expression for ( ˆβ ) yields he sample value of he es saisic ( ˆβ ) under H: b (ˆ β ) =. sê(ˆ β ) Noe: The null hypohesis for a lef-ail es is someimes wrien as H : β b raher han H : β = b. Bu he compuaional procedure for esing H : β b agains H : β < b is exacly he same as he procedure for esing H : β = b agains H : β < b. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 27 of 49 pages

28 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion: Why do we use he equaliy form of he null hypohesis, β = b, o calculae he sample value of he es saisic? Answer: A wo-par answer: A es ha akes he null hypohesis as H : β = b is he mos favorable o he null hypohesis, and hence is he leas favorable o he alernaive hypohesis H : β < b. This means ha, a any chosen significance level α, if we rejec H : β = b in favor of he alernaive hypohesis H : β < b, hen we would also rejec H : β = b + c in favor of H : β < b + c, where c > is any posiive consan. Calculaing he value of ( ˆβ ) for all values of β > b would be exremely edious!! (How would you know when you're done?) Moreover, is unnecessary. 3. The null disribuion of ( ˆβ ) is [N 2], he -disribuion wih N 2 degrees of freedom. In oher words, if he null hypohesis H is rue i.e., if β = b, hen b (ˆ β ) = ~ [N 2] under H : β = b. sê(ˆ β ) ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 28 of 49 pages

29 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion : Wha values of ˆβ and (ˆ β ) would lead us o rejec H agains H? H : β = b or β b = H : β < b or β b < a lef-sided alernaive hypohesis. Answer: Examine he numeraor of he calculaed -saisic: Remember: (ˆ β ) = b sê(ˆ β ) ˆβ = he esimaed value of β b = he hypohesized value of β ê(ˆ β ) > ( ê(ˆ β ) is always a posiive number) s s We would rejec H agains H if ˆβ is much less han b, if he esimaed value of β is much less han b, he hypohesized value of β. More specifically, we would rejec H agains H in he following case: ˆβ << b ˆβ b << b (ˆ β ) = << sê(ˆ β ) Values of ˆβ much less han b imply large negaive values of ( ˆβ ). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 29 of 49 pages

30 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion 2: How much less han zero does he value of ( ˆβ ) have o be for us o rejec H in favour of H? Answer: I depends on he significance level we choose for he es and he null disribuion of our es saisic. Le α = he chosen significance level for he es (e.g.,.,.5, or.) = he probabiliy of making a Type I error. Because only large negaive values of favour he alernaive hypohesis, here is only one criical value a lower criical value of he [N 2]-disribuion ha delineaes a lower or lef-ail rejecion area equal o α. α [N 2] = he lower α criical value of he [N 2]-disribuion. Implicaions: If H : β = b is rue, hen he following wo probabiliy saemens hold. () ( β ) α[n 2] ) = α Pr (ˆ () (2) ( β ) < α [N 2] ) = α where Pr (ˆ (2) (ˆ β ) = he calculaed sample value of he -saisic under H; α [N 2] = he lower α-level criical value of he [N 2]-disribuion; α = he significance level for he es; α = he confidence level for he es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 3 of 49 pages

31 ECONOMICS 35* -- NOTE 8 M.G. Abbo Deermine he Rejecion and Nonrejecion Regions -- Lef-Tail Tes The non-rejecion region for ( ˆβ ) is defined by he inequaliy in probabiliy saemen () above. () ( β ) α[n 2] ) = α Pr (ˆ () I consiss of all values of ( ) such ha ˆβ (ˆ β ) α[n 2] non-rejecion region under H: β = b. The rejecion region for ( ˆβ ) is he se of values defined by he inequaliy in probabiliy saemen (2) above. (2) ( β ) < α [N 2] ) = α Pr (ˆ (2) I consiss of all values of ( NOTE: ) such ha ˆβ (ˆ β ) < α[n 2] rejecion region under H: β = b is a one-ail lef-ail rejecion region. ) For a lef-ail es, he rejecion region ( ˆβ ) < α[n 2] consiss only of he lower lef-hand ail of he -disribuion wih N 2 degrees of freedom. 2) This lef-ail rejecion region conains unexpecedly small values of ( ˆβ ) under H i.e., values ha we would only expec o obain wih small probabiliy α if he null hypohesis H : β = b is rue. 3) The lef ail area under he [N 2]-disribuion in his lower ail equals he significance level α. This area is called he lower α-level (or lower α percen) ail area of he [N 2]-disribuion. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 3 of 49 pages

32 ECONOMICS 35* -- NOTE 8 M.G. Abbo Rejecion and Nonrejecion Regions for a Lef-Tail Tes lef-ail rejecion region nonrejecion region area = α area = α α Pr( < α ) = α Pr( α ) = α ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 32 of 49 pages

33 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 5: Apply he Decision Rule and Sae Inference Sep 5: Apply he decision rule of he es and sae he inference, or conclusion, implied by he sample value of he es saisic.. Formulaion of he Decision Rule for a Lef-Tail Tes Formulaion : Deermine if he sample value of he es saisic lies in he rejecion or nonrejecion region a he chosen significance level α. Decision Rule for a Lef-Tail Tes -- Formulaion. If he sample value of he es saisic lies in he lef-ail rejecion region a he chosen significance level, hen rejec he null hypohesis H. For a lef-ail es: < α [N 2] rejec H a significance level α. Rejec H in favour of H a significance level α if < α [N 2] meaning lies in he lower lef-ail rejecion area. Inference: Rejec H in favour of H a significance level α. 2. If he sample value of he es saisic lies in he nonrejecion region a he chosen significance level, hen reain (do no rejec) he null hypohesis H. For a lef-ail es: α [N 2] reain H a significance level α. Reain H agains H a significance level α if α [N 2] meaning lies in he nonrejecion area. Inference: Reain H agains H a significance level α. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 33 of 49 pages

34 ECONOMICS 35* -- NOTE 8 M.G. Abbo CASE 2 A Righ-Tail Tes Problem: Deermine he criical values for he following one-ail -es: H : β = b or β b = where b is a specified consan H : β > b or β b > a righ-sided alernaive hypohesis. a one-ail es, specifically a righ-ail es. NOTE: This one-ail es is a righ-ail es because, as we will see, he rejecion region for he es consiss of he upper righ-hand ail of he appropriae -disribuion.. Again, he appropriae es saisic is he -saisic for ˆβ, he OLS esimae of he slope coefficien β: ˆ β β β (ˆ β ) = =. Vâr(ˆ β ) sê(ˆ β) 2. The sample value of he es saisic ( ˆβ ) is calculaed by evaluaing ( ˆβ ) under he null hypohesis H. This involves evaluaing ( ˆβ ) using he equaliy form of he null hypohesis H, which is β = b. Seing β = b in he expression for ( ˆβ ) yields he sample value of he es saisic ( ˆβ ) under H: b (ˆ β ) =. sê(ˆ β ) Noe: The null hypohesis for a righ-ail es can be wrien as H : β b raher han H : β = b. Bu he compuaional procedure for esing H : β b agains H : β > b is exacly he same as he procedure for esing H : β = b agains H : β > b. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 34 of 49 pages

35 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion: Why do we use he equaliy form of he null hypohesis, β = b, o calculae he sample value of he es saisic? Answer: A es ha akes he null hypohesis as H : β = b is he mos favorable o he null hypohesis, and hence is he leas favorable o he alernaive hypohesis H : β > b. This means ha, a any chosen significance level α, if we rejec H : β = b in favor of he alernaive hypohesis H : β > b, hen we would also rejec H : β = b c in favor of H : β > b c, where c > is any posiive consan. 3. The null disribuion of ( ˆβ ) is [N 2], he -disribuion wih N 2 degrees of freedom. In oher words, if he null hypohesis H is rue i.e., if β = b, hen b (ˆ β ) = ~ [N 2] under H : β = b. sê(ˆ β ) ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 35 of 49 pages

36 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion : Wha values of ˆβ and (ˆ β ) would lead us o rejec H agains H? H : β = b or β b = H : β > b or β b > a righ-sided alernaive hypohesis. Answer: Examine he numeraor of he calculaed -saisic: Remember: b ( ) =. sê( ) ˆβ = he esimaed value of β b = he hypohesized value of β ê(ˆ β ) > ( ê(ˆ β ) is always a posiive number) s s We would rejec H agains H if ˆβ is much greaer han b, if he esimaed value of β is much greaer han b, he hypohesized value of β. More specifically, we would rejec H agains H in he following case: ˆβ >> b ˆβ b >> b (ˆ β ) = >> sê(ˆ β ) Values of ˆβ much greaer han b imply large posiive values of ( ˆβ ). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 36 of 49 pages

37 ECONOMICS 35* -- NOTE 8 M.G. Abbo Quesion 2: How much greaer han zero does he value of ( ˆβ ) have o be for us o rejec H in favour of H? Answer: I depends on he significance level we choose for he es and he null disribuion of our es saisic. Le α = he chosen significance level for he es (e.g.,.,.5, or.) = he probabiliy of making a Type I error. Because only large posiive values of favour he alernaive hypohesis, here is only one criical value an upper criical value of he [N 2]-disribuion ha delineaes an upper or righ-ail rejecion area equal o α. α [N 2] = he upper α criical value of he [N 2]-disribuion. Implicaions: If H : β = b is rue, hen he following wo probabiliy saemens hold. () ( β ) α[n 2] ) = α Pr (ˆ () (2) ( β ) > α [N 2] ) = α where Pr (ˆ (2) (ˆ β ) = he calculaed sample value of he -saisic under H ; α [N 2] = he upper α-level criical value of he [N 2]-disribuion; α = he significance level for he es; α = he confidence level for he es. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 37 of 49 pages

38 ECONOMICS 35* -- NOTE 8 M.G. Abbo Deermine he Rejecion and Nonrejecion Regions Righ-Tail Tes The non-rejecion region for ( ˆβ ) is he se of values defined by he inequaliy in probabiliy saemen () above: () ( β ) α[n 2] ) = α Pr (ˆ () I consiss of all values of ( ) such ha ˆβ (ˆ β ) α[n 2] non-rejecion region under H: β = b. The rejecion region for ( ˆβ ) is he se of values defined by he inequaliy in probabiliy saemen (2) above: (2) ( β ) > α [N 2] ) = α Pr (ˆ (2) I consiss of all values of ( NOTE: ) such ha ˆβ (ˆ β ) > α[n 2] rejecion region under H: β = b is a one-ail righ-ail rejecion region. ) For a righ-ail es, he rejecion region ( ˆβ ) > α[n 2] consiss only of he upper righ-hand ail of he -disribuion wih N 2 degrees of freedom. 2) This righ-ail rejecion region conains unexpecedly large values of (β ˆ ) under H i.e., values ha we would only expec o obain wih small probabiliy α if he null hypohesis H : β = b is rue. 3) The righ ail area under he [N 2]-disribuion in his upper ail equals he significance level α. This area is called he upper α-level (or upper α percen) ail area of he [N 2]-disribuion. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 38 of 49 pages

39 ECONOMICS 35* -- NOTE 8 M.G. Abbo Rejecion and Nonrejecion Regions for a Righ-Tail Tes nonrejecion region righ-ail rejecion region area = α area = α α Pr( α ) = α Pr( > α ) = α ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 39 of 49 pages

40 ECONOMICS 35* -- NOTE 8 M.G. Abbo STEP 5: Apply he Decision Rule and Sae Inference Sep 5: Apply he decision rule of he es and sae he inference, or conclusion, implied by he sample value of he es saisic.. Formulaion of he Decision Rule for a Righ-Tail Tes Formulaion : Deermine if he sample value of he es saisic lies in he rejecion or nonrejecion region a he chosen significance level α. Decision Rule for a Righ-Tail Tes -- Formulaion. If he sample value of he es saisic lies in he righ-ail rejecion region a he chosen significance level, hen rejec he null hypohesis H. For a righ-ail es: > α [N 2] rejec H a significance level α. Rejec H in favour of H a significance level α if > α [N 2] meaning lies in he upper righ-ail rejecion area. Inference: Rejec H in favour of H a significance level α. 2. If he sample value of he es saisic lies in he nonrejecion region a he chosen significance level, hen reain (do no rejec) he null hypohesis H. For a righ-ail es: α [N 2] reain H a significance level α. Reain H agains H a significance level α if α [N 2] meaning lies in he nonrejecion area. Inference: Reain H agains H a significance level α. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 4 of 49 pages

41 ECONOMICS 35* -- NOTE 8 M.G. Abbo Examples of One-Tail Hypohesis Tess The Model: DATA: auo.da (a Saa-forma daa file) MODEL: price i = β + β weigh i + u i (i =,..., N). regress price weigh Source SS df MS Number of obs = F(, 72) = Model Prob > F =. Residual R-squared = Adj R-squared =.282 Toal Roo MSE = price Coef. Sd. Err. P> [95% Conf. Inerval] weigh _cons N = 74 N 2 = 74 2 = 72 ˆβ = 2.44 ê(ˆ β ) = s α =.5 α/2 = α / 2[N 2] =. [72] = α [N 2] =. [72] =.6663 ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 4 of 49 pages

42 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tes A Lef-Tail Tes: Tes he proposiion ha weigh i has a negaive effec on price i. Perform he es a he 5 percen significance level (α =.5). Null and Alernaive Hypoheses H : β = H : β < a one-sided lef-sided alernaive hypohesis. The feasible es saisic is he -saisic for ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) Compue he sample value of ( ˆβ ) under he null hypohesis H. Se ˆβ = 2.44, β = and ê(ˆ ) = in he formula for ( ˆβ ): s β β (ˆ β ) = = = sê(ˆ β ) = The one-ail criical value of he [N 2] disribuion a he 5 percen significance level (a α =.5) is α [N 2] =. 5[72] = Decision Rule -- Lef-Tail Tes: If < α[n 2] rejec H a significance level α; If α[n 2] reain H a significance level α. Inference: = >.6663 =. 5 [72] reain H a significance level α =.5 Reain H : β = agains H : β < a he 5 percen significance level. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 42 of 49 pages

43 ECONOMICS 35* -- NOTE 8 M.G. Abbo Tes 2 A Righ-Tail Tes: Tes he proposiion ha weigh i has a posiive effec on price i. Perform he es a he 5 percen significance level (α =.5). Null and Alernaive Hypoheses H : β = H : β > a one-sided righ-sided alernaive hypohesis. The feasible es saisic is he -saisic for ˆβ : ˆ β β β (ˆ β ) = = ~ [N 2]. Vâr(ˆ β ) sê(ˆ β) Compue he sample value of ( ˆβ ) under he null hypohesis H. Se ˆβ = 2.44, β = and ê(ˆ ) = in he formula for ( ˆβ ): s β β (ˆ β ) = = = sê(ˆ β ) = The one-ail criical value of he [N 2] disribuion a he 5 percen significance level (a α =.5) is α [N 2] =. 5[72] = Decision Rule -- Righ-Tail Tes: If > α [N 2] rejec H a significance level α; If α [N 2] reain H a significance level α. Inference: = >.6663 =. 5 [72] rejec H a significance level α =.5 Rejec H : β = agains H : β > a he 5 percen significance level. The sample evidence favours he alernaive hypohesis H : β >. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 43 of 49 pages

44 ECONOMICS 35* -- NOTE 8 M.G. Abbo Inerpreaion of he Decision Rules An hypohesis es can lead o only wo possible decisions: eiher () a decision o rejec he null hypohesis H agains he alernaive hypohesis H, in which case he sample evidence favours H over H ; or (2) a decision o reain (no o rejec) he null hypohesis H agains he alernaive hypohesis H, in which case he sample evidence favours H over H. Poins o remember in inerpreing hese alernaive decisions. ) An hypohesis es can never be inerpreed as eiher proving or disproving he ruh of he null hypohesis. Reason: The decision o rejec or reain H on he basis of sample evidence is always subjec -- explicily or implicily -- o some uncerainy, or margin of saisical error. Tha is, here is always some non-zero probabiliy of commiing a Type I or Type II error. 2) A decision o reain (no o rejec) he null hypohesis H should no be inerpreed o mean ha we accep H, or ha H is rue. Reason: Saying we accep H implies ha we are concluding ha he null hypohesis is rue, bu such a conclusion is incorrec. Nonrejecion (or reenion) of H means only ha he sample daa provide insufficien evidence o rejec H ; i does no mean ha H is rue beyond any doub. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 44 of 49 pages

45 ECONOMICS 35* -- NOTE 8 M.G. Abbo Explanaion: see example in Gujarai (23, p. 34). We obain an esimae of he slope coefficien of ˆβ =.59 and a corresponding esimaed sandard error of ê(ˆ β ) =.357. s Firs, we perform a wo-ail es of H : β =.5 agains H : β.5. The sample value of he -saisic under H is calculaed as: β.59.5 (ˆ β ) = = =.25. sê(ˆ β ).357 Bu he sample value.25 is clearly insignifican a, say, he 5% significance level (α =.5). Suppose we accep H and conclude ha he rue value of β is.5. Nex, we perform a wo-ail es of H : β =.48 agains H : β.48. The sample value of he -saisic under his H is calculaed as: β (ˆ β ) = = =.82. sê(ˆ β ).357 Bu he sample value.82 is also clearly insignifican a he 5% significance level (α =.5). Do we now accep his H and conclude ha he rue value of β is.48? Quesion: Do we accep eiher or boh of hese hypohesized values of β as he rue value? The correc answer is NO. We do no know he exac rue value of β. Conclusion: All we can legiimaely conclude from hese wo hypohesis ess is ha he sample evidence is consisen or compaible wih boh he null hypoheses we have esed. Bu he ess provide no reason o conclude ha eiher hypohesized value,.5 or.48, is he rue value of β. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 45 of 49 pages

46 ECONOMICS 35* -- NOTE 8 M.G. Abbo 3) A decision o rejec H on he basis of given sample daa does no imply ha we mus accep he alernaive hypohesis H, or ha H mus be rue, or ha H does in fac conain he ruh. Reason: An hypohesis es is designed only o assess he probable empirical validiy of he null hypohesis H ; i is no designed o es he alernaive hypohesis H. Regardless of wheher a es oucome for some paricular sample daa indicaes rejecion or nonrejecion of H, he se of alernaive possibiliies specified by he alernaive hypohesis H may or may no conain he ruh. I is quie possible for a es of some null hypohesis H agains some alernaive hypohesis H o indicae rejecion of H when H is false ha is, when he alernaive possibiliies specified by H do no conain he ruh. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 46 of 49 pages

47 ECONOMICS 35* -- NOTE 8 M.G. Abbo Formulaion 2 of he Decision Rule: he p-value Rule Formulaion 2: Deermine if he p-value for, he calculaed sample value of he es saisic, is smaller or larger han he chosen significance level α. Definiion: The p-value (or probabiliy value) associaed wih he calculaed sample value of he es saisic is defined as he lowes significance level a which he null hypohesis H can be rejeced, given he calculaed sample value of he es saisic. Inerpreaion The p-value is he probabiliy of obaining a sample value of he es saisic as exreme as he one we compued if he null hypohesis H is rue. P-values serve as inverse measures of he srengh of evidence agains he null hypohesis H. Small p-values p-values close o zero consiue srong evidence agains he null hypohesis H. Large p-values p-values close o one provide only weak evidence agains he null hypohesis H. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 47 of 49 pages

48 ECONOMICS 35* -- NOTE 8 M.G. Abbo Examples of p-values for common ypes of hypohesis ess For a wo-ail -es, le he calculaed sample value of he -saisic for a given null hypohesis be. Then he p-value associaed wih he sample value is he probabiliy ha he null disribuion of he es saisic akes a value greaer han he absolue value of, where he absolue value of is denoed as. Tha is, wo-ail p-value for = Pr( > ) = Pr( > ) + Pr( < ) = 2Pr( > ) if > = Pr( < ) + Pr( > ) = 2Pr( < ) if < Remember: he -disribuion is symmeric abou is mean of zero. For a one-ail -es, le he calculaed sample value of he -saisic for a given null hypohesis be. Then he p-value associaed wih he sample value is depends on wheher he es is a righ-ail or lef-ail es. () For a righ-ail -es, he p-value associaed wih he sample value is he probabiliy ha he null disribuion of he es saisic akes a value greaer han he calculaed sample value i.e., righ-ail p-value for = Pr( ). > (2) For a lef-ail -es, he p-value associaed wih he sample value is he probabiliy ha he null disribuion of he es saisic akes a value less han he calculaed sample value i.e., lef-ail p-value for = Pr( < ). ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 48 of 49 pages

49 ECONOMICS 35* -- NOTE 8 M.G. Abbo For an F-es, le he calculaed sample value of he F-saisic for a given null hypohesis be F. Then he p-value associaed wih he sample value F is he probabiliy ha he null disribuion of he es saisic akes a value greaer han he calculaed sample value F i.e., p-value for F = Pr( F F ). > Noe ha he F-disribuion is defined only over non-negaive values ha are greaer han or equal o zero. Decision Rule -- Formulaion 2. If he p-value for he calculaed sample value of he es saisic is less han he chosen significance level α, rejec he null hypohesis a significance level α. p-value < α rejec H a significance level α. 2. If he p-value for he calculaed sample value of he es saisic is greaer han or equal o he chosen significance level α, reain (i.e., do no rejec) he null hypohesis a significance level α. p-value α reain H a significance level α. ECON 35* -- Noe 8: Hypohesis Tesing in he CNLRM 35noe8.doc Page 49 of 49 pages