A First Guide to Hypothesis Testing in Linear Regression Models. A Generic Linear Regression Model: Scalar Formulation
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1 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo A rs Gude o Hypoess Tesng n Lnear Regresson Models A Generc Lnear Regresson Model: Scalar mulaon e - populaon ( sample observaon, e scalar fmulaon of e PRE s wren as: Y β + β X + β X + L + β X + u Noe: k u k k β + β X + ( k β X + u, were X Y e - populaon value of e regressand, dependen varable; X e - populaon value of e - regress, ndependen varable; β e paral regresson coeffcen of X ; u e - populaon value of e unobservable random err erm ( Lower case k denoes e number of slope coeffcens n e PR ( Upper case K denoes e oal number of regresson coeffcens n e PR ( Terefe: K k + ECON 5*: A rs Gude o Hypoess Tesng Page of 9 5gudedoc
2 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Tess of One Coeffcen Resrcon: One Resrcon on One Coeffcen H specfes only one equaly resrcon on one coeffcen Two-Tal Tess of One Resrcon on One Coeffcen Example: H : β b versus H : β b were b s a specfed consan Use: eer a wo-al -es an -es Tes Sascs: β (ˆ β sample value sê(ˆ β [N K] ( β (ˆ β [, N K] sample value Vâr(ˆ β Noe: [ ] Decson Rules: ( $ β ( β $ (ˆ (ˆ β Reec H f [N K] / Rean H f [N K] b (ˆ β sê(ˆ β ( b (ˆ β Vâr(ˆ β β and [ [N K] ] [, N K] α / α > α wo-al p-value f Pr( > < α Pr( > < > α [, N K] p-value f α / α [, N ; α wo-al p-value f Pr( > α K] p-value f Pr( > α ; ECON 5*: A rs Gude o Hypoess Tesng Page of 9 5gudedoc
3 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo One-Tal Tess of One Resrcon on One Coeffcen Examples: H : β b ( β b versus H : β < b a lef-al es H : β b ( β b versus H : β > b a rg-al es Use: a one-al -es Tes Sasc: β (ˆ β sample value sê(ˆ β [N K] b (ˆ β sê(ˆ β Decson Rules -- lef-al -es: Reec H f Rean H f < α[n K] lef-al p-value f Pr( < < α ; α[n K] lef-al p-value f Pr( < α Decson Rules -- rg-al -es: Reec H f > α [N K] rg-al p-value f Pr( > < α ; Rean H f α [N K] rg-al p-value f Pr( > α ECON 5*: A rs Gude o Hypoess Tesng Page of 9 5gudedoc
4 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Tess of One Lnear Resrcon on Two Me Coeffcens H specfes only one lnear resrcon on wo me regresson coeffcens Two-Tal Tess of One Lnear Resrcon on Two Coeffcens Example: H : c β + c β c versus H : c β + c β c Use: eer a wo-al -es an -es Tes Sascs: c β ˆ (c (c β sê(c + c β ( [N K] [N K ] were: sê(c β ˆ Vâr(c Vâr(c c Vâr(ˆ β sample value ( c ˆ ( c β ˆ + c Vâr(ˆ β (c ˆ ˆ β + cβ c β sê(c [(c (c β + c β ] + c c Côv(ˆ β, [, N K] [, N K] Vâr(c were: ˆ ˆ Vâr(c β + c β c Vâr(ˆ β + c Vâr(ˆ β + c c Côv(ˆ β, sample value ( c β ˆ Decson Rules: Reec H f [N K] / Rean H f [N K] [(c c ] Vâr(c > α wo-al p-value f Pr( > < α Pr( > < > α [, N K] p-value f α / α [, N ; α wo-al p-value f Pr( > α K] p-value f Pr( > α ; ECON 5*: A rs Gude o Hypoess Tesng Page of 9 5gudedoc
5 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo One-Tal Tess of One Lnear Resrcon on Two Coeffcens Examples: H : c β + c β c vs H : c β + c β < c a lef-al es H : c β + c β c vs H : c β + c β > c a rg-al es Use: a one-al -es Tes Sasc: c β ˆ (c (c β sê(c + c β ( [N K] [N K ] were sê(c β ˆ Vâr(c Vâr(c c Vâr(ˆ β sample value ( c ˆ Decson Rules -- lef-al -es: + c Vâr(ˆ β (c ˆ ˆ β + cβ c β sê(c + c c Côv(ˆ β, Reec H f Rean H f < α[n K] lef-al p-value f Pr( < < α ; α[n K] lef-al p-value f Pr( < α Decson Rules -- rg-al -es: Reec H f > α [N K] rg-al p-value f Pr( > < α ; Rean H f α [N K] rg-al p-value f Pr( > α ECON 5*: A rs Gude o Hypoess Tesng Page 5 of 9 5gudedoc
6 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Tess of Two Me Lnear Coeffcen Resrcons H specfes wo me lnear coeffcen resrcons Example: H : β β and β β 5 H : β β and/ β β 5 Use: a general -es; only an -es can be used o es only wo me coeffcen resrcons Tes Sascs: Eer of e followng wo general -sascs (RSS RSS (df RSS df df (RSS RSS (K K RSS (N K (R U R R (df df (R U R R (K K ( R df ( R (N K U U df N df f RSS ; df N K df f RSS K Null dsrbuon: [df df, df ] [K K, N K] Sample value of -sasc under H Decson Rules: Reec H f > α [df df, df] α [K K, N K] p-value f Pr( > < α Rean H f α [df df, df] α [K K, N K] p-value f Pr( > α ECON 5*: A rs Gude o Hypoess Tesng Page 6 of 9 5gudedoc
7 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Tess of Two Lnear Coeffcen Resrcons: Example Example: H : β β and β β 5 H : β β and/ β β 5 Unresrced model s gven by PRE (: Y β + β X + β X + β X + β X + β X + u ( 5 5 Resrced model s gven by PRE (: se β β and β 5 β n PRE (: Y β + β X + β X + β X + β X + β X + u ( Y β + β X + β X + β X + β X + β X + Y β + β X + β (X + X + β (X + X + u ( u OLS esmaon of ( yelds e unresrced SRE (*: Y + X + X + X + X + X + û (* 5 5 N RSS û û T û w df N K N 6 OLS esmaon of ( yelds e resrced SRE (*: Y + (* β + βx + β(x + X + β(x + X5 u β β and β5 β RSS N u u T u w df N K N Subsue values of RSS, RSS, df and df no fmula f general -sasc: (RSS RSS (df RSS df df (RSS RSS (K K RSS (N K Apply decson rule ECON 5*: A rs Gude o Hypoess Tesng Page 7 of 9 5gudedoc
8 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Example of Tess of Two Lnear Coeffcen Resrcons: Saa Commands Example: H : β β and β β 5 H : β β and/ β β 5 Unresrced model s gven by PRE (: Y β + β X + β X + β X + β X + β X + u ( 5 5 Esmae unresrced model gven by PRE ( by OLS usng regress command: regress y x x x x x5 Perfm on -es of H versus H usng es command: es x x es x x5, accumulae es x - x es x - x5, accumulae ECON 5*: A rs Gude o Hypoess Tesng Page 8 of 9 5gudedoc
9 ECON 5* -- A rs Gude o Hypoess Tesng MG Abbo Example of Tess of One Lnear Coeffcen Resrcon: Saa Commands Perfm ndvdual wo-al -es of H : β β versus H : β β usng lncom command: lncom x - x lncom _b[x] - _b[x] Perfm ndvdual -es of H : β β versus H : β β usng es command: es x x es x - x How s sasc produced by lncom command relaed o sasc produced by es command? lncom command compues: ˆ ˆ ˆ ˆ ( ˆ ˆ ( β β ( β β β [ N 6 β ] under H sê( sê( es command compues: ( [ ˆ ˆ ] ˆ ˆ ˆ ˆ ( β β ( β β β [, N 6 β ] Vâr( Vâr( under H Relaonsp beween and ess: ey yeld dencal nferences because ( ˆ ( ( β β ( β ˆ ˆ and [, N 6] [ N ] α α 6 p-value f wo-al p-value f ECON 5*: A rs Gude o Hypoess Tesng Page 9 of 9 5gudedoc
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