ACE 564 Spring Lecture 9. Violations of Basic Assumptions II: Heteroskedasticity. by Professor Scott H. Irwin

Transcription

1 ACE 564 Spring 006 Lecure 9 Violaions of Basic Assumpions II: Heeroskedasiciy by Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Heeroskedasic Errors, Chaper 5 in Learning and Pracicing Economerics Kennedy. Inroducion, Secion 8.; Consequences of Violaion, Secion 8.; Heeroskedasiciy, Secion 8.3 in A Guide o Economerics ACE 564, Universiy of Illinois a Urbana-Champaign 9-

2 The Naure of Heeroskedasiciy In he firs par of his course, we inroduced he linear economic model o explain he relaionship beween food expendiure and income, y = β + β x This linear economic model predics ha food expendiure for a given level of income will be he same for all households We specified a saisical model by recognizing ha acual expendiure for a given level of income will no be he same for all households, y = β+ βx + e =,..., T where, y is he dependen variable x is he independen, or explanaory, variable e is he error, or disurbance, erm T is he number of households in he sample ACE 564, Universiy of Illinois a Urbana-Champaign 9-

3 Before re-saing he esimaion resuls for he sample of 40 households, consider he following quesions: Will i be easier o predic food expendiure for low-income or high-income households? Who will have more choices regarding food expendiure? Esimaes for Food Expendiure Daa b = b = 0.33 ˆ σ = var( ˆ ) b = cov( ˆ b, b ) = 0.34 var( ˆ b ) = Based on his informaion and he assumpion ha he saisical model is correcly specified, we can esimae he disribuions of e and y as, e ~ N(0, ) y ~ N( x, ) We also can esimae he sampling disribuions of b and b as, b ~ N(7.38,6.0669) b ~ N (0.33, 0.003) ACE 564, Universiy of Illinois a Urbana-Champaign 9-3

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5 The scaer plo suggess ha daa poins have a endency o deviae more and more from he esimaed mean funcion (line) as income increases Anoher way o say his is ha he leas squares residuals, e = y β β x ˆ increase in absolue value as income grows Since he observable leas squares residuals eˆ are proxies for he rue, unobservable errors, e = y β β x he sample evidence suggess ha he unobservable, rue errors also increase in absolue value as income grows Since he spread of errors is conrolled by he variance of he error erm, we are suggesing ha he variance of he error erm increases as income grows ACE 564, Universiy of Illinois a Urbana-Champaign 9-5

6 One of he assumpions of he classical linear regression model is ha he variance of he (populaion) error erm is consan, E( e ) = σ Termed homoskedasiciy When his assumpion is violaed, he variance of he (populaion) error erm is no consan E( e ) = σ Subscrip on variance indicaes ha i is differen for differen levels of he independen variables Termed heeroskedasiciy ACE 564, Universiy of Illinois a Urbana-Champaign 9-6

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9 Why Does Heeroskedasiciy Arise? Cross-secional daa Scale effecs As income grows, people have more discreion over heir consumpion choices Larger firms have more flexibiliy in producion and invesmen plans han small firms Time-series daa Learning curves As people learn, errors (hopefully!) become smaller Forecas errors in new fuures markes Improvemens in echnology New seed variey ha is more droughresisan ACE 564, Universiy of Illinois a Urbana-Champaign 9-9

10 Improvemens in daa collecion echniques Correlaion in "shocks" o economic sysems Large shocks end o be followed by large shocks and vice versa ACE 564, Universiy of Illinois a Urbana-Champaign 9-0

11 Consequences of Heeroskedasiciy for he Leas Squares Esimaors If heeroskedasiciy is presen, hen, The leas squares esimaor is sill a linear and unbiased esimaor, bu no longer he bes linear unbiased esimaor (no longer BLUE) The sandard errors usually compued for he leas squares esimaor are incorrec, and hence, confidence inervals and hypohesis ess ha use hese sandard errors may be misleading To explore hese issues, i is mos sraighforward o coninue working wih he simple linear regression model, y = β + β x + e where all assumpions are he same as before excep he variance of he regression is assumed o be heeroskedasic, var( e ) = E( e ) = σ ACE 564, Universiy of Illinois a Urbana-Champaign 9-

12 When exploring he sampling properies of he leas squares esimaor of he slope parameer in he simple linear regression model, we noed ha he esimaor could be wrien as, b T = β + we = where w = T = ( x x) ( x x) The mean is derived by aking he expecaion of b, T Eb ( ) = E β + we = T Eb ( ) E( β ) wee ( ) = + Eb ( ) = = β This shows ha he mean of he sampling disribuion of b is β, he populaion slope parameer, even when he variance of he error erm is heeroskedasic ACE 564, Universiy of Illinois a Urbana-Champaign 9-

13 We can derive he variance of b as follows, T T var( b) = var β + we = var we = = T T T var( ) var( ) scov(, s) = = s= b = w e + ww e e s T var( b ) = w var( e) = var( b ) T = w σ = If we noe ha, w ( x x) = T ( x x) = and T ( ) T x x = w T = ( x x) = = ACE 564, Universiy of Illinois a Urbana-Champaign 9-3

14 Then, var( b ) T ( x x) σ = = T ( x x) = This can be compared he sampling variance in he homoskedasic case, var( b ) σ = T ( x x) = Conclusions: Leas squares esimaor of sampling variance is no longer minimum variance, or bes Leas squares is no longer BLUE If we use he original leas squares formula when heeroskedasiciy is presen, our esimae of he sandard error will be biased ACE 564, Universiy of Illinois a Urbana-Champaign 9-4

15 In general, we canno predic he direcion of he bias in he leas squares esimaor of he sampling variance of b The naure of he bias depends on he relaionship beween he variance of he regression and he independen variable Using leas squares in he presence of heeroskedasiciy will make confidence inerval esimaes and hypohesis es resuls misleading ACE 564, Universiy of Illinois a Urbana-Champaign 9-5

16 Whie's Approximae Esimaor for he Sampling Variance of Leas Squares Esimaors One approach o he problem of heeroskedasiciy is o seek "correc" sandard error esimaes for leas squares parameer esimaes Whie has developed once such mehod, based on replacing σ wih e ˆ in he formula for sandard error The argumen is ha large variances are likely o lead o large esimaed squared residuals Because of his approximaion, Whie sandard error esimaes are valid only "large" sample sizes Someimes Whie sandard errors are called "heeroskedasic-consisen variance-covariance esimaes" Mos economeric packages have commands or opions o compue Whie sandard errors ACE 564, Universiy of Illinois a Urbana-Champaign 9-6

17 To derive he Whie esimaor, recall ha he formula for he sampling variance of he leas squares slope esimaor wih non-consan variance is, var( b ) Whie's esimaor is, var ˆ ( b ) w T ( x x) σ = = T ( x x) = T ( x ˆ x) e = = T ( x x) = Leas squares wih Whie sandard error esimaes is sill no BLUE Insead, Whie sandard error esimaor is consisen In very large samples, Whie sandard error esimaor converges owards he rue esimaor ACE 564, Universiy of Illinois a Urbana-Champaign 9-7

18 X=Weekly Income Y=Food Expendiure Residuals eha x-bar (x-xbar) numerm Whie's var(b) Whie's se(b) ACE 564, Universiy of Illinois a Urbana-Champaign 9-8

19 Applying Whie s sampling variance esimaor, we obain he following resuls for he food expendiure daa, yˆ = x R = 0.37 (4.90) (0.0705) (Whie s.e.) These can be compared o he incorrec leas squares sampling variance esimaes, yˆ = x R = 0.37 (4.0080) (0.0553) (LS s.e.) In his case, ignoring heeroskedasiciy and using incorrec sandard errors ends o oversae he precision of esimaion; we end o ge confidence inervals ha are narrower han hey should be. We can consruc wo corresponding 95% confidence inervals for β Whie: b ± se( ˆ ) c b = 0.33 ±.04(0.0705) = [0.0896, ] Incorrec: b ± se( ˆ ) c b = 0.33 ±.04(0.0553) = [0.04, 0.344] ACE 564, Universiy of Illinois a Urbana-Champaign 9-9

20 Whie s esimaor helps overcome he problem of drawing incorrec inferences wih leas squares in he presence of heeroskedasiciy However, we can go furher and ask if i is possible o obain an esimaor for he regression parameers ha is superior o leas squares or leas squares wih Whie sandard errors In oher words, is here a BLUE esimaor in he presence of heeroskedasiciy? ACE 564, Universiy of Illinois a Urbana-Champaign 9-0

21 Generalized Leas Squares When σ is Known In he case of heeroskedasic variances of he error erm, we would like o have an esimaor ha: Places more weigh on observaions drawn from error disribuions wih lower variances, and Places less weigh on observaions drawn from error disribuions wih higher variances Leas squares esimaors do no mee his crierion Places same weigh, or "influence," on each observaion, regardless of he disurbance populaion from which i was drawn An esimaor known as generalized leas squares does mee he above crierion Also known as weighed leas squares ACE 564, Universiy of Illinois a Urbana-Champaign 9-

22 Sar wih he following regression model, y = β + β x + e where all classical assumpions hold excep he variance of he regression is assumed o be heeroskedasic, var( e ) = E( e ) = σ Assume ha all he heeroskedasic variances σ are known Nex, divide hrough he original model by σ as follows, y x e = β + β + σ σ σ σ which can be re-wrien as, where y = β x + β x + e * * * *,, y x e y =, x =, x =, e = σ σ σ σ * * * *,, ACE 564, Universiy of Illinois a Urbana-Champaign 9-

23 The effec of ransforming he model on he variance of he error erm is e e = E e = E σ var( * * ) ( ) = σ ( ) Ee = σ σ = The variance of he error in he ransformed model is a consan, and herefore, homoskedasic If we apply leas squares o he ransformed model, he leas squares esimaors will once again be BLUE Basic idea of generalized leas squares is o ransform he variables so ha he classical linear regression assumpions hold ACE 564, Universiy of Illinois a Urbana-Champaign 9-3

24 We can gain furher insigh abou he GLS (generalized leas squares) esimaor by noing ha he objecive of GLS is o minimize he sum of ransformed squared errors, min WSSE e T T * e = = σ = = This clearly shows ha he squared errors are weighed by he reciprocal of σ When σ is small, he observaion conains more informaion, and he observaion is weighed more heavily When σ is large, he observaion conains less informaion, and he observaion receives less weigh This shows how GLS akes advanage of heeroskedasiciy o improve parameer esimaion Imporan o emphasize ha GLS is BLUE when is assumed o be known σ ACE 564, Universiy of Illinois a Urbana-Champaign 9-4

25 Generalized Leas Squares When σ is Unknown: Proporional Heeroskedasiciy Normal siuaion is ha σ is unknown Wih GLS, an esimaion problem is creaed because we only have T sample observaions and T differen error variances plus he inercep and slope o esimae More parameers han observaions! Bu, economericians are a clever bunch (or hey like o hink so!) and have developed ways of geing around his problem The soluion is o make furher assumpions regarding he process generaing error variances Typically, assume error variance is a proporional o x, he square of x, or some oher simple funcional form, such as quadraic or absolue value ACE 564, Universiy of Illinois a Urbana-Champaign 9-5

26 Generalized Leas Squares: Heeroskedasiciy is Proporional o x Sar wih he following regression model, y = β + β x + e where all classical assumpions hold excep he variance of he regression is assumed o be heeroskedasic, var( e ) = E( e ) = σ = σ x Implies ha he variance of he h error erm is given by a posiive unknown consan ( σ ) muliplied by he posiive variable x A high levels of x, error variance will be high A low levels of x, error variance will be low Our earlier observaion of he leas squares residuals for he food expendiure problem is consisen wih his model; error variance increases as income increases ACE 564, Universiy of Illinois a Urbana-Champaign 9-6

27 GLS ransformaion begins by dividing boh sides of he regression model by he square roo of x (we will see momenarily why we use he square roo), y x e = β + β + x x x x which can be re-wrien as, y = β x + β x + e * * * *,, where * y * * x * e y =, x, =, x, = = x, e = x x x x We can show ha he error erm for he ransformed model is homoskedasic, The effec of ransforming he model on he variance of he error erm is, e = = x * * var( e ) E( e ) E ACE 564, Universiy of Illinois a Urbana-Champaign 9-7

28 = x Ee ( ) = x x σ = σ The variance of he error in he ransformed model is a consan, and herefore, homoskedasic The esimaed model will be of he form, y = bx + bx * * * * ˆ,, which is a muliple regression model wihou an inercep * b is he esimae of β b is he esimae of β Usual problem in inerpreing R in a model wihou an inercep ACE 564, Universiy of Illinois a Urbana-Champaign 9-8

29 Some imporan poins o noe: By assuming ha error variance is a muliplicaive funcion of x, we have solved he problem of having T+ parameers o esimae and only T observaions; now only have wo parameers o esimae The ransformed model is linear in he unknown parameers β and β, he original parameers ha we are ineresed in esimaing Transformaion of variables wih GLS should be viewed as a device for convering a heeroskedasic error model ino a homoskedasic error model, no as somehing ha changes he meaning of he coefficiens The ransformed error erm will reain he properies Ee ( ) = 0 and zero correlaion beween differen observaions, cov( e, e ) = 0 for s As a consequence, we can apply leas squares o he ransformed variables, y, x, and x, o obain he bes linear unbiased esimaors for β and β s ACE 564, Universiy of Illinois a Urbana-Champaign 9-9

30 The ransformed model saisfies he condiions of he Gauss-Markov Theorem, and he leas squares esimaors defined in erms of he ransformed variables are BLUE The esimaor obained in his way is also called he weighed leas squares esimaor (WLS) ACE 564, Universiy of Illinois a Urbana-Champaign 9-30

31 GLS Transformaion of he Food Expendiure Daa y x sqr(x) y* x* x* ACE 564, Universiy of Illinois a Urbana-Champaign 9-3

32 GLS Regression Resuls in Excel for he Food Expendiure Daa SUMMARY OUTPUT Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 40 ANOVA df SS MS F Significance F Regression Residual Toal Coefficiens Sandard Error Sa P-value Lower 95% Upper 95% Inercep 0 #N/A #N/A #N/A #N/A #N/A X Variable X Variable ACE 564, Universiy of Illinois a Urbana-Champaign 9-3

33 The GLS esimaion resuls are, yˆ = x (3.57) (0.0489) (GLS s.e.) These can be compared o he Whie and incorrec leas squares esimaes, yˆ = x R = 0.37 (4.90) (0.069) (Whie s.e.) yˆ = x R = 0.37 (4.0080) (0.0553) (LS s.e.) Since GLS is a beer esimaion procedure han leas squares or leas squares wih Whie sandard errors, we expec he GLS sandard errors o be lower ACE 564, Universiy of Illinois a Urbana-Champaign 9-33

34 The smaller sandard errors have he advanage of producing narrower more informaive confidence inervals 95% confidence inerval esimaes for β, GLS: [0.56, 0.354] Whie: [0.0896, ] LS: [0.04, 0.344] GLS wih proporional heeroskedasiciy is BLUE, assuming he form of he proporional relaionship is known ACE 564, Universiy of Illinois a Urbana-Champaign 9-34

35 Generalized Leas Squares: Heeroskedasiciy is Proporional x Sar wih he following regression model, y = β + β x + e where all classical assumpions hold excep he variance of he regression is assumed o be heeroskedasic, var( e ) = E( e ) = σ = σ x Implies ha he variance of he h error erm is given by a posiive unknown consan ( σ ) muliplied by he posiive variable x A high levels of A low levels of x, error variance will be high x, error variance will be low ACE 564, Universiy of Illinois a Urbana-Champaign 9-35

36 GLS ransformaion begins by dividing boh sides of he regression model by x (we will see momenarily why we do no use he square roo in his case), y x e = β + β + x x x x which can be re-wrien as, where y = β x + β x + e * * * *,, y x e y =, x =, x = =, e = * * * *,, x x x x We can show ha he error erm for he ransformed model is homoskedasic, The effec of ransforming he model on he variance of he error erm is, e e = E e = E x var( * * ) ( ) = x Ee ( ) ACE 564, Universiy of Illinois a Urbana-Champaign 9-36

37 = x x σ = σ The variance of he error in he ransformed model is a consan, and herefore, homoskedasic If we apply leas squares o he ransformed model, he leas squares esimaors will once again be BLUE The esimaed model is, y = b x + b x * * * * ˆ,, Muliple regression model wihou an inercep * b is he esimae of β b is he esimae of β y In pracice, simply regress x on and a consan, x bu reverse he inerpreaion of he parameer esimaes! ACE 564, Universiy of Illinois a Urbana-Champaign 9-37

38 Imporan poins: The GLS ransformaion "works" (generaes BLUE esimaors) because i is a funcion of known x values GLS ransformaion generalizes o he muliple variable regression case Simply ransform all variables as in he wovariable case Esimaion and inerpreaion is he same as in wo-variable case Heeroskedasiciy can be a muliplicaive funcion of more han one independen variable ACE 564, Universiy of Illinois a Urbana-Champaign 9-38

39 A final cauion regarding GLS If assumed form of heeroskedasiciy is correc, GLS will produce BLUE parameer esimaors Whie s correcion is less efficien han GLS If assumed form of heeroskedasiciy is incorrec, GLS will produce biased parameer esimaes and biased sandard error esimaes A form of specificaion error If here is "subsanial" uncerainy abou form of heeroskedasiciy, probably beer o use Whie sandard error correcion, as parameer esimaors are unbiased and consisen ACE 564, Universiy of Illinois a Urbana-Champaign 9-39

40 An Inroducion o Esimaed Generalized Leas Squares: A Sample wih a Heeroskedasic Pariion In each of he preceding examples of generalized leas squares, he ransformaion eliminaed he need o direcly esimae he changing variance parameer There may be siuaions where such ransformaions canno be applied Such siuaions requires he use of esimaed generalized leas squares (EGLS) The Problem Need o model he supply of whea in an area of Ausralia Informaion on supply response o price is imporan for governmen policy purposes Governmen pays a guaraneed price for whea and needs o know how much whea will be produced a he guaraneed price ACE 564, Universiy of Illinois a Urbana-Champaign 9-40

41 The Economic Model Producion economics eaches us ha he quaniy of whea supplied will depend on he producion echnology of he firm, price expecaions of he firm, and weaher condiions We can depic his economic model as, Quaniy = f (Price, Technology, Weaher) The Saisical Model Choice of saisical model will depend on he ype of daa available Cross-secion Adjus prices for differences in ransporaion and inpu coss Technology is likely o be similar across farmers Weaher will be similar for farmers in a he same region ACE 564, Universiy of Illinois a Urbana-Champaign 9-4

42 Time-series Prices will vary across ime Producion echnology will presumably improve hrough ime Weaher condiions will vary from year-o-year Time-series is available for 6 years of aggregae quaniy supplied and price Use a simple linear ime-rend variable o proxy changes in producion echnology Effec of weaher is argued o be random and included in he error erm ACE 564, Universiy of Illinois a Urbana-Champaign 9-4

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44 The saisical model is specified as, where y = β+ βx, + β3x3, + e =,...,6 y is he quaniy of whea produced in year x, is he price of whea guaraneed in year x 3, is a rend variable o capure changes in producion echnology and i akes on values,..,6 e is he random error erm ha accouns for random influences on whea producion, including weaher To complee he model, as before, we need o specify he saisical assumpions for he error erm We could assume he error erm is homoskedasic Bu, we have addiional informaion ha suggess his is no correc ACE 564, Universiy of Illinois a Urbana-Champaign 9-44

45 We know ha afer year 3 new whea varieies were inroduced ha were less suscepible o variaions in weaher condiions Mean yields did no change Yield variabiliy did change This is modeled as follows, Ee ( ) = 0 =,..,6 var( e ) = E( e ) = σ =,..,3 var( e ) = E( e ) = σ = 4,..,6 cov( e, e ) = 0, s where s We furher assume ha, s σ > σ ACE 564, Universiy of Illinois a Urbana-Champaign 9-45

46 Wih he previous assumpions we can wrie our model in he following pariioned forma, y = β + β x + β x + e var( e ) = σ =,...,3, 3 3, y = β + β x + β x + e var( e ) = σ = 4,...,6, 3 3, We can ransform his pariioned model so ha he wo pariions have he same error variance as follows, y x x e = β + β + β + =,...,3 σ σ σ σ σ, 3, 3 y x x e = β + β + β + = 4,...,6 σ σ σ σ σ, 3, 3 Using he same logic as we have followed before, i can be proven ha he error variance in boh pariions is he same (equals one) Transformed model is homoskedasic ACE 564, Universiy of Illinois a Urbana-Champaign 9-46

47 We can wrie he ransformed model using one equaion, where y = β x + β x + β x + e * * * * *,, 3 3, y x x e y =, x =, x =, x =, e = σ σ σ σ σ * * *, * 3, *,, 3, i i i i i i= when =,..3 and i= when =4,,6 Providing σ and σ are known, he ransformed model provides a se of new ransformed variables o which we can apply LS o obain he bes linear unbiased esimaors for ( β, β and β 3). Like before, he complee process of ransforming variables, hen applying leas squares o he ransformed variables, is called generalized leas squares or weighed leas squares. However, we canno simply apply leas squares as before because he ransformed model depends on he unknown parameers σ and σ ACE 564, Universiy of Illinois a Urbana-Champaign 9-47

48 We now move ino a new esimaion mehod: esimaed generalized leas squares (EGLS) In his case, EGLS is a hree-sep process Sep : Apply LS o he firs half of he sample o obain an esimae of σ SUMMARY OUTPUT Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 3 ANOVA df SS MS F Significance F Regression Residual Toal CoefficiensSandard Error Sa P-value Lower 95% Upper 95% Inercep X Variable X Variable ACE 564, Universiy of Illinois a Urbana-Champaign 9-48

49 Apply LS o he second half of he sample o obain an esimae of σ SUMMARY OUTPUT Regression Saisics Muliple R R Square Adjused R Square Sandard Error Observaions 3 ANOVA df SS MS F Significance F Regression E-06 Residual Toal CoefficiensSandard Error Sa P-value Lower 95% Upper 95% Inercep E X Variable X Variable Resuls: ˆ σ = and ˆ σ = ACE 564, Universiy of Illinois a Urbana-Champaign 9-49

50 Sep : Transform he original observaions using he square roo of he variance esimaes obained in sep q p q* in* p* * ACE 564, Universiy of Illinois a Urbana-Champaign 9-50

51 Sep 3: Apply LS o he enire se of 6 ransformed observaions SUMMARY OUTPUT Regression Saisics Muliple R R Square Adjused R Square 0.95 Sandard Error.035 Observaions 6 ANOVA df SS MS F Significance F Regression Residual Toal Coefficiens Sandard Error Sa P-value Lower 95% Upper 95% Inercep #N/A #N/A #N/A #N/A #N/A X Variable X Variable X Variable ACE 564, Universiy of Illinois a Urbana-Champaign 9-5

52 EGLS resuls: yˆ = x + 3.8x, 3, (.8) (8.9) (0.8) ( s. e.) For comparison, LS resuls: yˆ = x, x3, R = (3.) (7.4) (.4) ( se..) Parameer esimaes are lile changed beween LS and EGLS Sandard error esimaes are subsanially lower wih EGLS Imporan poins: EGLS does no produce BLUE esimaors EGLS only produces consisen esimaors ACE 564, Universiy of Illinois a Urbana-Champaign 9-5

53 Deecing Heeroskedasiciy Economic heory may be of limied guidance as o wheher we should expec heeroskedasiciy in a paricular applied research problem Neverheless, always ry o use economic heory, or oher non-sample informaion, as he firs means of deecing his problem Wihou srong a priori informaion, all one can do is esimae he regression model using LS assuming here is no heeroskedasiciy and hen conduc a "pos-morem" analysis on he esimaed residuals If evidence of heeroskedasiciy is found, hen GLS or EGLS can be applied as a correcion Mone Carlo sudies sugges his "wo-sep" esimaor is reasonably efficien, if he overall model specificaion is correc ACE 564, Universiy of Illinois a Urbana-Champaign 9-53

54 Graphical Mehods Plo LS residuals and examine wheher hey have any paern Plos may use raw residuals or squared residuals Generae plos agains each independen variable If he errors are homoskedasic, here should be no paern of any kind If a paern is deeced, hen a more formal invesigaion is warraned ACE 564, Universiy of Illinois a Urbana-Champaign 9-54

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58 Formal Tess for Heeroskedasiciy There are a large number of ess for heeroskedasiciy Kennedy discusses hree widely used ess in Ch.8 Goldfield-Quand es Breusch-Pagan es Whie es Goldfield-Quand Tes Basic idea is o compare esimaed error variance from he high variance par of he observaions o he low variance par of he observaions Operaionally, i akes he form of an F-es Assumes ha heeroskedasiciy is posiively relaed o one of he x s ACE 564, Universiy of Illinois a Urbana-Champaign 9-58

59 . Null hypohesis H H : σ 0 : σ σ > σ. Tes saisic where ˆ GQ = σ ~ F( T K, T K) ˆ σ T is he number of observaions in he firs pariion of he sample K is he number of parameers in he model esimaed for he firs pariion T is he number of observaions in he second pariion of he sample K is he number of parameers in he model esimaed for he second pariion ACE 564, Universiy of Illinois a Urbana-Champaign 9-59

60 Given his seup i is no necessary o have he same number of observaions in he wo daa pariions Always pu he variance expeced o be he larges in he numeraor 3. Rejecion region Rejec he null hypohesis if, GQ > F ( T K, T K ) α ACE 564, Universiy of Illinois a Urbana-Champaign 9-60

61 GQ es applied o whea supply daa We firs make sure ha daa are ordered by ime Apply LS o he firs half of he sample o obain $σ Apply LS o he second half of he sample o obain $σ. Null hypohesis. Tes saisic H H : σ σ 0 : σ > σ ˆ GQ = σ = =. σˆ ACE 564, Universiy of Illinois a Urbana-Champaign 9-6

62 3. Rejecion Region Rejec he null hypohesis if, GQ > F ( T K, T K ) α If α = 0.05, hen, F T K T K F0.05(3 3,3 3) = F (0,0) =.98 α (, ) = 0.05 Rejec if GQ Decision Since.>.98 we rejec he null hypohesis and conclude ha he error variance is larger in he firs half of he sample han in he second half of he sample New varieies of whea did reduce he variance of yields ACE 564, Universiy of Illinois a Urbana-Champaign 9-6

63 GQ es applied o food expendiure daa We firs make sure ha daa are ordered by he magniude of x, household income Apply LS o he second half of he sample (high income) o obain σ ˆ Apply LS o he firs half of he sample (low income) o obain σ. Null hypohesis. Tes saisic H H ˆ : σ σ 0 : σ > σ ˆ GQ = σ = = 3.35 σˆ.377 ACE 564, Universiy of Illinois a Urbana-Champaign 9-63

64 3. Rejecion Region Rejec he null hypohesis if, GQ > F ( T K, T K ) α If α = 0.05, hen, F T K T K F0.05(0,0 ) = F (8,8) =. α (, ) = 0.05 Rejec if GQ. 4. Decision Since 3.35>. we rejec he null hypohesis and conclude ha he error variance is larger in he higher income half of he sample han in he lower income half of he sample ACE 564, Universiy of Illinois a Urbana-Champaign 9-64

65 The power of he GQ es can be improved by leaving ou some of he cenral observaions in he sample The abiliy of he es o deec heeroskedasiciy is sharpened if cenral observaions are omied Mone Carlo sudies suggess he middle 0-5% of observaions should be deleed o maximize he effeciveness of he GQ es ACE 564, Universiy of Illinois a Urbana-Champaign 9-65