Amhers College Deparmen of Economics Economics 360 Fall 0 Wednesday, December 5 Handou: Panel Daa and Unobservable Variables Preview Taking Sock: Ordinary Leas Squares (OLS) Esimaion Procedure o Sandard Ordinary Leas Squares (OLS) Premises o OLS Bias Quesion o OLS Reliabiliy Quesion Preview: Panel Examples Firs Differences and Cross Secion Fixed Effecs (Dummy Variables) Period Fixed Effecs (Dummy Variables) Random Effecs Sandard Ordinary Leas Squares (OLS) Premises Error Term Equal Variance Premise: The variance of he error erm s probabiliy disribuion for each observaion is he same; all he variances equal Var[e]: Var[e ] = Var[e ] = = Var[e T ] = Var[e] Error Term/Error Term Independence Premise: The error erms are independen: Cov[e i, e j ] = 0. Knowing he value of he error erm from one observaion does no help you predic he value of he error erm for any oher observaion. Explanaory Variable/Error Term Independence Premise: The explanaory variables, he x s, and he error erms, he e s, are no correlaed. Knowing he value of an observaion s explanaory variable does no help you predic he value of ha observaion s error erm. Taking Sock of he Ordinary Leas Squares (OLS) Esimaion Procedure OLS Bias Quesion: Is he explanaory variable/ Saisfied: Independen Violaed: Correlaed error erm independence premise saisfied or violaed? Is he OLS esimaion procedure for he value of he coefficien unbiased? OLS Reliabiliy Quesion: Are he error erm equal Saisfied Violaed variance and he error erm/error erm independence premises saisfied or violaed? Can he OLS calculaion for he coefficien s sandard error be rused? Is he OLS esimaion procedure for he value of he coefficien BLUE?
Panel Daa: Time Series and Cross Secion Daa Three Scenarios Scenario - Mah Class Panel Daa: Three college sudens are enrolled in a mah class: Jim, Peg, and Tim. A quiz is given in each week. We have weekly daa for each suden s quiz scores, SAT scores, and number of minues each suden sudied. Quiz Mah Minues Quiz Mah Minues Suden Week Score SAT Sudied Suden Week Score SAT Sudied Jim 8 70 3 Peg 3 760 7 Jim 0 70 7 Peg 3 760 3.. Jim 0 70 3 Peg 0 4 760 9 Tim 5 670 7 Tim 5 670. Tim 0 0 670 9 The 0 weeks provide daa. The 3 sudens provide daa. Scenario - Chemisry Class Panel Daa: Two sudens are enrolled in an advanced undergraduae chemisry course: Ted and Sue. A lab repor is due each week. We have weekly daa for each suden s lab score and number of minues each suden devoed o he lab. Each week a differen graduae suden grades he lab repors of he wo sudens. Scenario 3 - Sudio Ar Class Panel Daa: Three college sudens are randomly seleced from a heavily enrolled sudio ar class: Bob, Dan, and Kim. An ar projec is assigned each week. We have weekly daa for each suden s projec scores and number of minues each suden devoed o he projec. Scenario - Mah Class Panel Daa: Three college sudens are enrolled in a mah class: Jim, Peg, and Tim. A quiz is given in each week. We have weekly daa for each suden s quiz scores, SAT scores, and number of minues each suden sudied. Projec: Assess he effec of sudying on quiz scores. Quiz Score Model: MahScore i = β Cons + β Sa MahSai + β MahMins MahMinsi + ei where MahScore i = Mah quiz score for suden i in week MahSa i = Mah SAT score for suden i in week MahMins i = Minues sudied by suden i in week Noaion: Focus on he superscrips and subscrips of he variables: The superscrip i denoes he individual suden; ha is, i equals Jim, Peg, or Tim. The subscrip denoes ime; ha is, equals he week:,,, or 0. For clariy, le us apply his o each of he hree sudens For Jim, i = Jim: MahScore Jim = β Cons + β Sa MahSa Jim For Peg, i = Peg: MahScore Peg For Tim, i = Tim: MahScore Tim = β Cons + β Sa MahSa Peg = β Cons + β Sa MahSa Tim + β MahMins MahMins Jim + β MahMins MahMins Peg + β MahMins MahMins Tim A closer look a he SAT scores: Jim s SAT score equals a consan 70: MahSa Jim = 70 for =,,, 0 Peg s SAT score equals a consan 760: MahSa Peg = 760 for =,,, 0 Tim s SAT score equals a consan 670: MahSa Tim = 670 for =,,, 0 + e Peg + e Tim This allows us o simplify he noaion by dropping he ime subscrip for he MahSa variable: MahScore i = β Cons + β Sa MahSai + β MahMins MahMins i + ei
3 Apply his o each of he hree sudens For Jim, i = Jim: MahScore Jim = β Cons + β Sa MahSa Jim + β MahMins MahMins Jim For Peg, i = Peg: MahScore Peg = β Cons + β Sa MahSa Peg + β MahMins MahMins Peg For Tim, i = Tim: MahScore Tim Theory: β Sa = β Cons + β Sa MahSa Tim + β MahMins MahMins Tim Mah SAT scores represen a cross secion fixed effec. Each suden s MahSa does no vary across ime. : Higher mah SAT scores a suden s quiz score β MahMins : Sudying more a suden s quiz score Ordinary Leas Squares (OLS) Esimaion Procedure We begin by using he ordinary leas squares (OLS) esimaion procedure o esimae he parameers of he model: Dependen Variable: MATHSCORE Mehod: Panel Leas Squares Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Coefficien Sd. Error -Saisic Prob. MATHSAT 0.793 0.0833 4.6589 0.0003 MATHMINS 0.43906 0.4739.0305 0.0544 C -73.54375 8.05883-4.07454 0.0004 Inerpreaion: EsMahScore = + MahSa + MahMins + e Peg + e Tim b Sa = : We esimae ha a 00 poin increase in a suden s mah SAT score increases his/her quiz score by an esimaed poins. b MahMins = : We esimae ha a 0 minue increase in sudying increases a suden s quiz score by poins. Jim: MahSa Jim = 70 EsMahScore Jim = 73.54 +.8MahSa +.43MahMins = 73.54 +.8 70 +.43MahMins = 73.54 + 84.96 +.43MahMins =.4 +.43MahMins Peg: MahSa Peg = 760 EsMahScore Peg = 73.54 +.8MahSa +.43MahMins = 73.54 +.8 760 +.43MahMins = 73.54 + 89.68 +.43MahMins = 6.4 +.43MahMins Tim: MahSa Tim = 670 EsMahScore Tim = 73.54 +.8MahSa +.43MahMins = 73.54 +.8 670 +.43MahMins = 73.54 + 79.06 +.43MahMins = 5.5 +.43MahMins EsMahScore 6.4.4 5.5 Peg Jim Tim Slope =.43 MahMins
4 Unobserved Variable: Wha if privacy concerns did no permi he release of suden SAT daa? Wha are he ramificaions of omiing MahSa from he regression model? MahScore i = β Cons + β Sa MahSai + β MahMins MahMins i + ei = β Cons + β MahMins MahMins i + β Sa MahSai + e i = β Cons + β MahMins MahMins i + (β Sa MahSai + e i ) = β Cons + β MahMins MahMins i + εi where εi = β Sa MahSai + e i Dependen Variable: MATHSCORE Mehod: Panel Leas Squares Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Coefficien Sd. Error -Saisic Prob. MATHMINS.0735 0.0404 4.98467 0.0000 C 0.594368 3.75463 0.58303 0.8754 Inerpreaion: EsMahScore = + MahMins b MahMins = : We esimae ha a 0 minue increase in sudying increases a suden s quiz score by poins. Quesion: Migh a serious economeric problem exis when using he ordinary leas squares (OLS) esimaion procedure o esimae his model? OLS Bias Quesion: Is he explanaory variable/error erm independence premise saisfied or violaed? Quesion: Is he model s error erm, ε i, correlaed wih is explanaory variable, MahMinsi : MahScore i = β Cons + β MahMins MahMinsi + εi where εi = β Sa MahSai + e i Quesion: Do high school sudens who receive high SAT mah scores end o sudy more or less han hose sudens who receive low scores? Answer: Quesion: Would you expec MahSa and MahMins o be correlaed? Answer: Quesion: Would his cause he ordinary leas squares (OLS) esimaion procedure for he MahMins coefficien o be biased? Answer: MahSa i and MahMins i Correlaed MahSa i up εi = β Sa MahSai + e i β Sa MahMins i εi. MahMins i and Error Term εi Correlaed OLS esimaion procedure for he value of he MahMins is biased
5 Quesion: Wha can we do? Firs Differences Dummy Variable/Fixed Effecs Firs Differences Approach Focus on he firs suden, Jim: For week : MahScore Jim = β Cons + β Sa MahSa Jim + β MahMins MahMins Jim For week : MahScore Jim = β Cons + β Sa MahSa Jim + β MahMins MahMins Jim Subrac he wo equaions: MahScore Jim MahScore Jim = β MahMins MahMins Jim β MahMins MahMins Jim = β MahMins (MahMins Jim MahMins Jim ) The firs wo erms are he righ hand side, β Cons and β Sa MahSa Jim, subrac ou. Using similar logic for Jones and Smih: MahScore Peg MahScore Peg = β MahMins (MahMins Peg MahScore Tim MahScore Tim Generalize: Firs Difference Model MahScore i MahScorei = β MahMins (MahMins Tim MahMins Peg ) + (epeg + (ejim e Peg ) MahMins Tim ) + (etim e Tim ) = β MahMins (MahMinsi MahMinsi ) + (ei ei ) Now, generae wo new variables in EViews: DifMahScore = MahScore MahScore( ) DifMahMins = MahMins MahMins( ) Dependen Variable: DIFMATHSCORE Mehod: Panel Leas Squares Periods included: 9 Cross-secions included: 3 Toal panel (balanced) observaions: 7 Coefficien Sd. Error -Saisic Prob. DIFMATHMINS 0.6500 0.6398.59463 0.568 Inerpreaion: EsDifMahScore = DifMahMins e Jim e Jim ) b MahMins = : We esimae ha a 0 minue increase in sudying increases a suden s quiz score by poins. Firs Differences Criical Assumpion: For each suden (cross-secion) he unobserved (and hence omied) variable mus equal he value in each week (ime period). Tha is, from week o week: MahSa Jim. MahSa Peg. MahSa Tim. Quesion: Why are hese condiions criical?
6 Dummy Variable/Fixed Effecs Approach Again begin by focusing on Jim: Because MahSa is consan for Jim, we can fold he MahSa erm ino he consan. MahScore Jim = β Cons + β Sa MahSa Jim + β MahMins MahMins Jim MahSa Jim = 70 = β Cons + β Sa 70 + β MahMins MahMins Jim Le α Jim Cons = β Cons + β Sa 70 = α Jim Cons Using he same logic for Jones and Smih: MahScore Peg = α Peg Cons MahScore Tim = α Tim Cons + β MahMins MahMins Jim + β MahMins MahMins Peg + β MahMins MahMins Tim + e Peg + e Tim Now, inroduce hree dummy variables: DumJim i = if he suden is Jim (if i = Jim); 0 oherwise DumPeg i = if he suden is Peg (if i = Peg); 0 oherwise DumTim i = if he suden is Tim (if i = Tim); 0 oherwise The following model describes he hree equaions for Jim, Peg, and Tim concisely: MahScore i = α Jim Cons DumJimi + α Peg Cons DumPegi + α Tim Cons DumTimi + β MahMins MahMins i + ei For Jim, i = Jim: DumJim i = DumPeg i = 0 DumTim i = 0 MahScore i = α Jim Cons For Peg, i = Peg: DumJim i = 0 DumPeg i = DumTim i = 0 MahScore i = αpeg Cons For Tim, i = Tim DumJim i = 0 DumPeg i = 0 DumTim i = MahScore i = αtim Cons + β MahMins MahMinsi + ei + β MahMins MahMinsi + ei + β MahMins MahMinsi + ei Dependen Variable: MATHSCORE Mehod: Panel Leas Squares Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Coefficien Sd. Error -Saisic Prob. MATHMINS 0.37305 0.377.489 0.698 DUMJIM.8633 3.94885 3.0047 0.0058 DUMPEG 9.09 5.40950 3.53049 0.006 DUMTIM 7.5354 3.6458.0630 0.049 Inerpreaion: EsMahScore = DumJim + DumPeg + DumTim + MahMins Jim: Peg: Tim: EsMahScore = + MahMins EsMahScore = + MahMins EsMahScore = + MahMins
EViews makes i easy for us o do his: Click on MahScore and hen while holding he <Crl> key down, click on MahMins. Double click he highlighed area. Click Open Equaion. Click he Panel Opions ab. In he Effecs specificaion box, selec Fixed from he Cross-secion drop down box. Click OK. EsMahScore 9.0.86 7.5 7 Peg Jim Tim Slope =.33 Dependen Variable: MATHSCORE Mehod: Panel Leas Squares Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Coefficien Sd. Error -Saisic Prob. MATHMINS 0.37305 0.377.489 0.698 C.893 4.84080 3.06678 0.0050 Effecs Specificaion Cross-secion fixed (dummy variables) MahMins Inerpreaion: b MahMins = : We esimae ha a 0 minue increase in sudying increases a suden s quiz score by poins. Quesion: Wha does he esimae of he consan,.83, represen? Answer: To obain he inerceps: CROSS Effec Click View. -0.966005 Click Fixed/Random Effecs 6.73785 Click Cross-secion Effecs 3-5.307779 The inercep for each group equals he consan (C) from he regression prinou plus he effec value from he able: Inercep for Jim: s crossid.83 = Inercep for Peg: nd crossid.83 = Inercep for Tim: 3 rd crossid.83 = Quesion: How are hese inerceps relaed o he dummy variable inerceps? Dummy Variable/Fixed Effecs Criical Assumpion: For each suden (cross-secion) he unobserved (and hence omied) variable mus equal he value in each week (ime period). Tha is, from week o week: MahSa Jim. MahSa Peg. MahSa Tim.
8 Scenario - Chemisry Class Panel Daa: Period Fixed Effecs Two sudens are enrolled in an advanced undergraduae chemisry course. A lab repor is due each week. We have weekly daa for each suden s lab score and number of minues each suden devoed o he lab. Each week a differen graduae suden grades he lab repors of he wo sudens. In he firs week, boh Ted s and Sue s lab repors are graded by one graduae suden. In he second week, Ted s and Sue s repors are graded by a second graduae suden, ec... Projec: Assess he effec of ime devoed o lab on he lab repor score. We begin by formulaing a model: LabScore i = β Cons GraderGenerosiyi LabMins i + ei where LabScore i = Chemisry lab score for suden i in week LabMins i = Minues devoed o lab by suden i in week GraderGenerosiy i = Generosiy of he grader for suden i in week Applying o each individual suden: LabScore Ted = β Cons GraderGenerosiy Ted = β Cons GraderGenerosiy Sue + e Ted Since he grader is he same for each week, we can drop he suden superscrip in GraderGenerosiy: LabScore Ted = β Cons GraderGenerosiy + e Ted = β Cons GraderGenerosiy and hen apply his o each week: Week ( = ): LabScore Ted = β Cons GraderGenerosiy + e Ted = β Cons GraderGenerosiy Week ( = ): LabScore Ted = β Cons GraderGenerosiy + e Ted = β Cons GraderGenerosiy. Week 0 ( = 0): LabScore Ted 0 = β Cons GraderGenerosiy 0 0 + eted 0 0 = β Cons GraderGenerosiy 0 0 0 Unobserved Variable: The explanaory variable GraderGenerosiy is unobservable. I mus be omied from he regression. Wha are he ramificaions of omiing MahSa from he regression model? LabScore i = β Cons GraderGenerosiy LabMinsi + ei = β Cons LabMins i GraderGenerosiy + ei = β Cons LabMins i + (β GG GraderGenerosiy + ei ) = β Cons LabMins i + εi where εi = β GG GraderGenerosiy + ei
9 Ordinary Leas Squares (OLS) Pooled Regression Dependen Variable: LabScore Explanaory Variable: LabMins Dependen Variable: LABSCORE Mehod: Panel Leas Squares Sample: 0 Periods included: 0 Cross-secions included: Toal panel (balanced) observaions: 0 Coefficien Sd. Error -Saisic Prob. LABMINS 0.538 0.7404.948883 0.0086 C 5.6796.5803 4.8696 0.0006 Inerpreaion: EsLabScore = + LabMins b LabMins = : We esimae ha an addiional 0 minue of ime devoed o he lab increases he lab score by poins. Quesion: Bu migh here be a serious economeric problem wih using he ordinary leas squares (OLS) esimaion procedure o esimae his model? LabScore i = β Cons LabMinsi + εi OLS Bias Quesion: Is he explanaory variable/error erm independence premise saisfied or violaed? Quesion: Would his cause he ordinary leas squares (OLS) esimaion procedure for he LabMins coefficien o be biased? Answer: where εi = β GG GraderGenerosiy + ei Grader unusually generous ã é ε i up LabMinsi é ã ε i and LabMinsi Now we can move on he he coefficien reliabiliy quesion: OLS Reliabiliy Quesion: Are he error erm equal variance and he error erm/error erm independence premises saisfied or violaed? Quesion: Since each week s lab repor is graded by a differen graduae suden would you expec some graduae sudens o be more demanding han ohers? Answer: Quesion: In each week, would you expec error erms for he wo sudens o be correlaed? Answer: Quesion: Does i appear ha he ordinary leas square calculaion for he coefficien s sandard error can be rused and ha he esimaion procedure for he coefficien value he bes linear unbiased esimaion procedure (BLUE)? Answer: OLS esimaion procedure for coefficien value is
0 The error erms are. Bu we can hink of he residuals as he. Period Fixed Effecs (Dummy Variables) Begin by focusing on Week. For boh Ted and Sue, we can fold he consan and he grader generosiy erm ino a new consan for Week : Week ( = ): LabScore Ted = β Cons GraderGenerosiy + e Ted LabScore Ted = α + e Ted = β Cons GraderGenerosiy = α NB: The new consan erm, α, for boh Ted and Sue is idenical because. Using he same logic for each oher week: Week ( = ): LabScore Ted = α + e Ted = α. Week 0 ( = 0): LabScore Ted 0 = α 0 0 + eted 0 0 = α 0 0 0 For each week, we have folded he generosiy of he grader ino he consan. In each week he consan is idenical for boh sudens because he same graduae suden grades boh lab repors. We now have en new consans for each week, one consan for each of he en weeks. Period fixed effecs esimaes he values of parameers.
Saisical sofware makes i easy o compue hese esimaes: Geing Sared in EViews Click on MahScore and hen while holding he <Crl> key down, click on MahMins. Double click he highlighed area. Click he Panel Opions ab. In he Effecs specificaion box, selec Fixed from he Period drop down box. Click OK. Period Fixed Effecs (FE) Dependen Variable: LabScore Explanaory Variable(s): Esimae SE -Saisic Prob LabMins 0.36678 0.6859 3.33509 0.0 Cons 63.6684 8.434896 7.500607 0.0000 Number of Observaions 0 Cross Secions Periods 0 Esimaed Equaion: EsLabScore = 63.7 +.37LabMins Inerpreaion of Esimaes: b LabMins =.37: A 0 minue increase devoed o he lab increases a suden s lab score by 3.7 poins. Quesion: Wha does he esimae of he consan, 63.7, represen? Answer: The average of he weekly ime consans. Saisical sofware allows us o obain he esimaes of each week s consan. Geing Sared in EViews Firs, esimae he cross-secion fixed effecs. Click View. Click Fixed/Random Effecs Click Period Effecs Period ID Fixed Effec 3.738 4.466844 3 4.94860 4.805060 5 4.0843 6 3.553 7.44860 8 4.8904 9 5.84780 0 4.83755 The period fixed effecs sugges ha he graduae suden who graded he lab repors for week 9 was he oughes grader and he graduae suden who graded for week 8 was he mos generous. Period Dummy Variable/Fixed Effecs Criical Assumpion: For each week (ime period) he omied variable mus equal he same value for each suden (cross secion).
Scenario 3 - Sudio Ar Class Panel Daa: Random Effecs Approach Three college sudens are randomly seleced from a heavily enrolled sudio ar class: Bob, Dan, and Kim. An ar projec is assigned each week. We have weekly daa for each suden s projec scores and number of minues each suden devoed o he projec. Projec: Assess he effec of ime devoed o projec on he projec score. We begin by formulaing a model: ArScore i = β Cons + β AIQ ArIQi + β ArMins ArMins i + ei where ArScore i = Projec score for suden i in week ArIQ i = Innae arisic alen for suden i ArMins i = Minues devoed o projec by suden i in week The variable ArIQ represens innae arisic alen. ArIQ is an absrac concep. ArIQ is no numerically observable. Neverheless, we do know ha differen sudens possess differen quaniies of innae arisic alen. Quesion: In general wha does he variable ArIQ represen? Answer:. Mean[ArIQ] ArIQ i Since our hree sudens were seleced randomly, define a random variable, v i, o equal he amoun by which a suden s innae arisic alen deviaes from he mean: v i = ArIQ i = Mean[ArIQ] Hence, ArIQ i = Mean[ArIQ] + v i where v i is a random variable. Nex, le us incorporae our specificaion of ArIQ i o he model: ArScore i = β Cons + β AIQ ArIQi + β ArMins ArMins i + ei = β Cons + β AIQ (Mean[ArIQ] + v i ) + β ArMins ArMins i + ei = β Cons + β AIQ Mean[ArIQ] + β AIQ v i + β ArMins ArMins i + ei Le α Cons = β Cons + β AIQ Mean[ArIQ] Fold he Mean[ArIQ] erm ino he consan. = α Cons + β AIQ v i + β ArMins ArMins i + ei Rearranging erms = α Cons + β ArMins ArMins i + β AIQ vi + e i Fold he v i erm ino he error erm = α Cons + β ArMins ArMins i + εi where ε i = β AIQ vi + e i
3 We can use he ordinary leas squares esimaion procedure o esimae he exercise coefficien, β ArMins : Dependen Variable: ARTSCORE Mehod: Panel Leas Squares Sample: 0 Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Coefficien Sd. Error -Saisic Prob. ARTMINS 0.403843 0.345388.694 0.5 C 40.5786 6.35054 6.38875 0.0000 Inerpreaion: EsArScore = + ArMins b MahMins = : We esimae ha a 0 minue increase devoed o an ar projec increases a suden s score by poins. Quesion: Migh a serious economeric problem exis when using he ordinary leas squares (OLS) esimaion procedure o esimae his model? OLS Bias Quesion: Is he explanaory variable/error erm independence premise saisfied or violaed? Quesion: How are ArIQ i and ArMins i relaed? ArIQ i = Mean[ArIQ] + v i ArIQ i How are ArIQ i and ArMins i relaed? v i up ε i = β AIQ vi + e i β AIQ > 0 ε i Posiively Correlaed Independen Negaively Correlaed ArMins i up ArMinsi no affeced ArMinsi down ArMins i and εi ArMinsi and εi ArMinsi and εi NB: For purposes of illusraion we shall assume ha he unobserved (and hence omied) variable, ArIQ, and he included variable, ArMins, are no correlaed. Hence, he ordinary leas squares esimaion procedure for he coefficien value is unbiased and we can now move on o he OLS reliabiliy quesion: OLS Reliabiliy Quesion: Are he error erm equal variance and he error erm/error erm independence premises saisfied or violaed? The error erm, ε i, in his model is ineresing; i has wo componens: The firs, β AIQ v i, reflecs innae arisic alen of each randomly seleced suden: o Bob s deviaion from he innae arisic alen mean: v Bob o Dan s deviaion from he innae arisic alen mean: v Dan o Kim s deviaion from he innae arisic alen mean: v Kim Since Bob, Dan, and Kim were seleced randomly, v Bob, v Dan, and v Kim are random variables. The second, e i, represening he random influences of each suden s weekly quiz.
4 I is insrucive o illusrae he error erms: ε i = εi = εi = Individual Week β AIQ v i + e i Individual Week β AIQ vi + e i Individual Week β AIQ vi + e i Bob β AIQ v Bob + e Bob Dan β AIQ v Dan + e Dan Kim β AIQ v Kim + e Kim Bob β AIQ v Bob + e Bob Dan β AIQ v Dan + e Dan Kim β AIQ v Kim + e Kim......... Bob 0 β AIQ v Bob + e Bob 0 Dan 0 β AIQ v Dan + e Dan 0 Kim 0 β AIQ v Kim + e Kim 0 Each of Bob s error erms has a common erm, β AIQ v Bob. Similarly, each of Dan s error erms has and common error erm, β AIQ v Dan, and each of Kim s error erms has a common erm, β AIQ v Kim. Consequenly, he error erms are no independen. Quesion: Does i appear ha he ordinary leas squares (OLS) calculaion for he coefficien s sandard error can be rused and ha he esimaion procedure for he coefficien value he bes linear unbiased esimaion procedure (BLUE)? Answer: The random effecs esimaion procedure explois his error erm paern o calculae beer esimaes: Click on ArScore and hen while holding he <Crl> key down, click on ArMins. Double click he highlighed area. Click Open Equaion. Click he Panel Opions ab. In he Effecs specificaion box, selec Random from he Cross-secion drop down box. Click OK. Dependen Variable: ArScore Explanaory Variable: ArMins Dependen Variable: ARTSCORE Mehod: Panel EGLS (Cross-secion random effecs) Sample: 0 Periods included: 0 Cross-secions included: 3 Toal panel (balanced) observaions: 30 Swamy and Arora esimaor of componen variances Coefficien Sd. Error -Saisic Prob. ARTMINS 0.8360 0.67838 4.845507 0.0000 C 33.353 8.68743 3.834459 0.0007 The inuiion behind all his is ha we can exploi he addiional informaion abou he error erms o improve he esimaion procedure. Addiional informaion is a good hing. Random Effecs Criical Assumpion: The unobserved (and hence omied) variable and he included variable are.