1. Diagnostic (Misspeci cation) Tests: Testing the Assumptions

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1 Business School, Brunel Universiy MSc. EC5501/5509 Modelling Financial Decisions and Markes/Inroducion o Quaniaive Mehods Prof. Menelaos Karanasos (Room SS269, el ) Lecure Noes 6 1. Diagnosic (Misspeci caion) ess: esing he Assumpions and Consequences of heir Failure Having compleed a discussion of he classical normal linear regression model, i seems naural o review each of he model s assumpions in urn. I is imporan o es he validiy of he assumpions which underlie our model because when one or more assumpions fail our inference migh be misleading. he properies of he leas squares esimaor depend on he assumpions of he CLRM. he derivaion of he and F ess also depends on hese assumpions. Much of economerics is abou esing wheher hese assumpions hold. If hey do no, hen he model should be respeci ed and perhaps esimaed by a di eren mehod, depending on he exac naure of he misspeci caion. he various diagnosic ess will be presened in he conex of he classical muliple linear regression model: 1.1. No Serial Correlaion y = 0 x + " ; = 1; :::; or y = x x 3 + ::: + k x k + " : (1) When he error erms from di eren ime periods (or cross-secion observaions) are correlaed, we say ha he error erm is serially correlaed. Serial correlaion occurs in ime-series sudies when he errors associaed wih observaions in a given ime period carry over ino fuure ime periods. For example, if we are predicing he growh of sock dividends, an overesimae in one year is likely o lead o overesimaes in succeeding years. Serial correlaion (also called auocorrelaion) in he residuals means ha hey conain informaion, which should iself be modelled. Firs order serial correlaion arises when he residuals in one ime period are correlaed direcly wih he residuals in he ensuing ime period; second-order serial correlaion refers o correlaion

2 beween residuals wo periods apar, hird-order serial correlaion refers o correlaion beween residuals hree periods apar, and so on. (Wih annual daa he mos common form of serial correlaion is rs order, wih quarerly daa here is usually fourh order serial correlaion.) A paern where successive residuals end o have he same sign indicaes posiive rs order serial correlaion, whereas a paern where successive residuals end o have opposie signs indicaes negaive rs order serial correlaion. (Observe ha we need o use he words end o because no all he residuals can have he same sign - hey mus sum o zero.) he assumpion of no serial correlaion can be expressed as E (" " s ) = 0; 6= s; or V (y =x ) E ("" 0 ) = 2 I: On he oher hand, he presence of serial correlaion can be expressed as E (" " s ) 6= 0; 6= s; or V (y =x ) E ("" 0 ) = 2 ; where he o -diagonal elemens of are no all equal o zero (he diagonal elemens are all equal o one) esing for rs order serial correlaion: he Durbin-Wason saisic (DW) he DW saisic is a es for he deecion of rs order serial correlaion in he residuals. I is prined ou by every economeric package, and is compued as: DW = P (b" b" 1 ) 2 =2 P b" 2 1 =2 = 2 (1 b) ; where b is he rs order auocorrelaion coe cien of he residuals, i.e. i is he coe cien in he regression: b" = b" 1 + u ; 1 < < 1; u iid (0; 2 u). he null hypohesis is no serial correlaion (H 0 : = 0) : he DW saisic will lie in he 0 o 4 range, wih a value near 2 indicaing no rs order serial correlaion. he abulaed values of he DW saisic have one peculiar feaure. For any signi cance level, sample size and number of regressors, wo values of he saisic are abulaed: hese are usually referred o as d L (lower value) and d U (upper value). Posiive serial correlaion is associaed wih DW values below 2 (in his 2

3 case he alernaive hypohesis is H 1 : > 0 ). he decision rule is as follows: If: 0 < DW < d {z L ; } d L < DW < d {z U ; } d U < DW < 2 {z } hen: Rejec H 0 ; No Conclusion; Accep H 0 : Negaive serial correlaion is associaed wih DW values above 2 (i.e. he alernaive hypohesis is H 1 : < 0 ). In his case we subrac he value of he DW from 4 and proceed as follows: If: 0 < 4 DW < d {z L ; d } L < 4 DW < d {z U ; d } U < 4 DW < 2 {z } hen: Rejec H 0 ; No Conclusion; Accep H 0 : Noe: if R 2 > DW; hen we are probably dealing wih a spurious regression. ha means ha despie he apparen saisical signi cance of he explanaory variables here is no underlying relaionship beween he dependen and explanaory variables. ime series can appear highly correlaed because of common rends and no because one a ecs he oher (spurious correlaion). In he presence of a lagged endogenous variable in he model he DW saisic becomes unreliable. Insead we can use Durbin s h saisic: s h = 1 DW 2 ; 1 s 2 l where s 2 l is he esimaed variance of he coe cien of he lagged dependen variable. Since Durbin has shown ha he h saisic is approximaely normally disribued, he es for rs order serial correlaion can be done direcly by using he normal disribuion able. o conclude, he drawbacks of he DW saisic are: (i) i can only es for rs order serial correlaion in he residuals, (ii) i is no reliable in he presence of lagged endogenous variables, (iii) no decision can be reached when i lies in he d L o d U range. As a resul, i is beer o adop he esing procedure given below. 3

4 esing for rs and higher order serial correlaion We es by running he following (auxiliary) regression: b" = x 2 + ::: + k x k + H 0 : i = 0; i = 1; :::; N H 1 : i 6= 0; N i b" i + u ; (2) i=1 where b" are he residuals of eq.(1). I is no di cul o see ha model (2) is he unresriced model, while model (1) is he resriced one. We can es he above hypoheses individually, using ess; or joinly using an F es : =N F es = bu 2 b" 2 bu 2 = ( k N) F (N; k N) : 4

5 Alernaively, we can es he join signi cance of he s by using he asympoic Lagrange-muliplier es (LM es) : he operaional version of he es is carried ou by obaining he produc of he number of observaions ( ) and he coe cien of deerminaion (R 2 ) of he auxiliary regression (2): LM es = R 2 2 (N) : Decision rule: if he es saisic is smaller han is criical value we canno rejec he null Wha are he e ecs of serial correlaion? Exacly wha e ec serial correlaion has on he properies of he OLS esimaor depends on why i arises: If he residuals are serially correlaed because of serial correlaion in he disurbances (he unsysemaic par), hen he OLS esimaor b remains unbiased and consisen bu ceases o have minimum variance. In paricular, OLS produces biased esimaes of he sandard errors of he coe ciens; his renders hypohesis esing unreliable. In his case consisen esimaors of he sandard errors can be obained by appropriaely ransforming he variables and hen esimae he ransformed model wih OLS (his esimaion procedure is called feasible or esimaed Generalized Leas Squares (GLS)); noe ha Micro provides consisen sandard errors on reques. he mos usual cause of serial correlaion in he residuals is he omission of relevan variables as regressors (in oher words, he cause of residual serial correlaion lies in he sysemaic par of he regression). In his case he coe cien esimaes hemselves will be biased and inconsisen. he mos obvious candidaes for he omied variables which produce serial correlaion are lagged values of he regressors already included and of he dependen variable iself. herefore, he appropriae way o deal wih he problem of serial correlaion is o respecify he model by including hese lagged variables. 5

6 1.2. Homoscedasiciy here are occasions in economeric modeling when he assumpion of consan variance, or homoscedasiciy, will be unreasonable. For example, consider a cross secion sudy of family income and expendiures. I seems plausible o expec ha low-income individuals would spend a a raher seady rae, while he spending paerns of high-income families would be relaively volaile. his suggess ha in a model where expendiures are he dependen variable, error variances associaed wih high-income families would be greaer han heir low-income counerpars. In oher words, heeroscedasiciy is presen in he model. he assumpion of homoscedasiciy can be expressed as E " 2 = 2 ; for all ; or V (y =x ) E ("" 0 ) = 2 I: On he oher hand, heeroscedasiciy can be expressed as V (y =x ) E ("" 0 ) = 2 ; where he diagonal elemens of are no all equal o one (he o -diagonal elemens are all equal o zero) esing for heeroscedasiciy here are various ypes of ess depending on he naure of heeroscedasiciy. In wha follows we are going o examine wo of he mos commonly used ess. he Rese-ype es involves he esimaion of he following (auxiliary) regression: b" 2 = by 2 + u ; (3a) H 0 : 1 = 0; H 1 : 1 6= 0; i.e. we regress he squared residuals of model (1) on a consan and on he squared ed values of model (1). Under he assumpion of homoscedasiciy, he slope coe cien of eq. (3a) is zero. We can es he saisical signi cance of 1 by using a es; or an F es (in his case he F es F (1; 2) and is given by he square of he es), or an LM es (in his case i follows a 2 (1) disribuion): 6

7 (his is opional) he (auoregressive condiional heeroscedasiciy es) ARCH-es involves he esimaion of he following (auxiliary) regression: b" 2 = b" b" ::: + p b" 2 p + u (3b) p = 0 + i b" 2 i + u ; i=1 H 0 : i = 0; i = 1; 2; :::; p H 1 : i 6= 0: We can use individual ess or we can es he join signi cance of he i s wih an F es and/or an LM es : =p F es = bv 2 bu 2 LM es = R 2 2 (p) ; bu 2 = ( 1 p) F (p; 1 p) ; where bv is he residual series obained from a regression of b" on a consan Wha are he e ecs of heeroscedasiciy? When he residuals are heeroscedasic he OLS esimaor b remains unbiased and consisen bu ceases o have minimum variance. In paricular, OLS produces biased esimaes of he sandard errors of he coe ciens; his renders hypohesis esing unreliable. In his case consisen esimaors of he sandard errors can be obained by appropriaely ransforming he variables and hen esimae he ransformed model wih OLS (his esimaion procedure is called feasible or esimaed Generalized Leas Squares (GLS)); noe ha Micro provides consisen sandard errors on reques Lineariy he assumpion of lineariy can be expressed as follows: E (y =x ) = x x 3 + ::: + k x k : 7

8 his speci caion is no as limiing as i migh seem, because he linear regression model can be applied o a more general class of equaions ha are inherenly linear. Inherenly linear models can be expressed in a form ha is linear in he parameers by a ransformaion of he variables. Inherenly nonlinear models, on he oher hand, canno be ransformed o he linear form. he non-lineariies of ineres here are he ones which canno be accommodaed ino a linear condiional mean afer ransformaion. One of he mos common ways o es he lineariy assumpion is o use he Rese-ype es. his esing procedure involves he esimaion of he following (auxiliary) regression: b" = x 2 + ::: + k x k + by 2 + u ; (4) H 0 : = 0; H 1 : 6= 0: I is no di cul o show ha (4) is he unresriced model, whereas model (1) is he resriced one. We can now es he saisical signi cance of by using he es =, or he F and/or LM ess : F es = esimae of sandard error of b b" 2 bu 2 LM es = R 2 2 (1) : bu 2 =1 = ( k 1) F (1; k 1) ; he failure of lineariy has major consequences for our esimaion. In paricular, when lineariy does no hold he OLS esimaors are biased and inconsisen. In oher words esimaion and esing resuls are invalid and we need o respecify our model Normaliy he assumpion of normaliy can be expressed as follows: " N 0; 2 ; or (y =x ) N 0 x ; 2 : 8

9 If he assumpion of normaliy does no hold, hen he OLS esimaor b remains he Bes Linear Unbiased Esimaor (BLUE), i.e. i has he minimum variance among all linear unbiased esimaors. I remains consisen, bu is no he maximum likelihood esimaor which can only be de ned if a paricular disribuion is speci ed for y : However, wihou normaliy one canno use he sandard formulae for he and F disribuions o perform saisical ess. Forunaely, he cenral-limi heorem provides a raional for using sandard saisical ess as approximaely correc for reasonably large sample sizes. Before we proceed wih he normaliy ess le us specify he null hypohesis. he null is ha he skewness ( 3 ) and kurosis ( 4 ) coe ciens of he condiional disribuion of y (or, equivalenly, of he disribuion of " ) are 0 and 3, respecively: H 0 : 3 = 0; 4 = 3; (if 3 < 0 hen f (y =x ) is skewed o he lef) (if 4 > 3 hen f (y =x ) is lepokuric) (his is opional) he above assumpions can be esed joinly using he Jarque- Bera es (JB) which follows asympoically a chi-square disribuion: JB es = 6 b (b 4 3) 2 2 (2) ; 2 3 where b 3 = 4 1 3=2 b" 3 1 = b" 2 5 ; 2 3 and b 4 = b" 4 1 = b" 2 5 : Noe ha he JB es is sensiive o ouliers. he above assumpions can also be esed individually, using he asympoic disribuions of b 3 and b 4 : H 0 : 3 = 0; r es saisic = 6 b 3 N (0; 1) : H 0 : 4 = 3; r es saisic = 24 (b 4 3) N (0; 1) : 9

10 1.5. No Perfec Mulicollineariy: Rank () = k If here is an exac linear relaionship among he righ-hand side variables of our model, hen we say ha we have he problem of perfec mulicollineariy: rank () < k; and so ( 0 ) is no inverible, and as a consequence esimaion of he model is no feasible. Mulicollineariy arises when wo or more variables (or combinaions of variables) are highly (bu no perfecly) correlaed wih each oher. In his case he esimaed coe ciens b remain unbiased bu heir sandard errors ge very large. Generally, worrying abou mulicollineariy does more damage han mulicollineariy iself. 10

11 1.6. Parameer ime Invariance Wihou loss of generaliy we are going o presen and es he above assumpion in he conex of he following bivariae linear model: y = x + " ; = 1; 2; :::; : (5) Under he null hypohesis he parameers and 2 remain consan hroughou he sample period. Under he alernaive hypohesis and 2 change beween wo speci ed ime periods: y = x + u 1 ; u 1 iid 0; 2 1 ; = 1; 2; :::; 1; (6a) y = x + u 2 ; u 2 iid 0; 2 2 ; = 1 + 1; 1 + 2; :::; : (6b) I is easy o see ha he number of observaions of he rs subsample is 1; whereas he number of observaions of he second subsample is 2 = 1: Below we presen hree ess depending on wheher ; 2 ; or boh may change. esing Variance Equaliy (H 0 : 2 1 = 2 2) : s 2 1 F ( 1 k; 2 k) ; P bu 2 1 s 2 2 P bu 2 2 where s 2 1 = ; 1 k s2 2 = ; and k is he number of coe ciens we esimae 2 k in eq. (5) (noe ha in his case k = 1). he larger variance should be used as he numeraor. his is he Goldfeld-Quand Variance Raio es, for heeroscedasiciy of a very speci c ype; i is sensiive o he failure of he normaliy assumpion. esing Coe cien Equaliy (H 0 : = ) condiional on variance equaliy: 1 =k F es = =1 1 =1 b" 2 bu =1 bu 2 1 = 1+1 bu 2 2 = 1+1 bu 2 2 = ( 2k) F (k; 2k) : Noe ha in he conex of model (5) k = 1: he unresriced model comprises of eq. (6a) and (6b), while he resriced model is given by eq. (5). he above is Chow s es for coe cien equaliy and i can be used when 1 > k and 2 > k: 11

12 esing Predicive Failure (i.e. wheher he model of he rs subsample predics he second subsample): 1 = 2 F es = b" 2 =1 1 bu 2 1 =1 =1 bu 2 1 = ( 1 k) F ( 2; 1 k) : he above es is paricularly useful when he observaions of he 2nd subsample do no allow us o esimae he model. In his case eq. (5) is he resriced model, whereas eq. (6a) is he unresriced one. Observe ha he all hree ess above assume ha you know he poin a which he parameers change, i.e. you know when he srucural break occurs. If you do no, hen Micro provides he CUSUM and CUSUMSQ plos o es for srucural sabiliy. If he plos cross he wo lines denoing he 95% con dence inerval bounds, hey indicae ha here has been a srucural shif. 12