EDO UNIVERSITY, IYAMHO EDO STATE, NIGERIA DEPARTMENT OF ECONOMICS

EDO UNIVERSITY, IYAMHO EDO STATE, NIGERIA DEPARTMENT OF ECONOMICS ECO 4: APPLIED ECONOMETRICS INSTRUCTOR: LECTURES: OFFICE HOURS: Davd Umoru, Emal: davd.umoru@edounversy.edu.ng Alernave emal: davd.umoru@yahoo.com Tuesday, 0am pm, Lecure Classroom 5 (LC5), Moble Lne: (+34)803388844 Wednesdays, pm o 3.30pm, Offce: s Floor, MH Admnsrave Block GENERAL OVERVIEW OF LECTURE: Ths course s desgned prmarly for sudens a he posgraduae level. The am of he course s o buld upon he sudens exsng knowledge of economercs and essenally, o help he sudens develop a praccal knowledge of economercs and s applcaons o real-world economc daa. PREREQUISITES: Sudens should be famlar wh he conceps of nroducory economercs (ECO 33 & ECO 33) wh specfc knowledge of economercs mehods, Ordnary Leas Squares(OLS) mehod, sngle equaon modellng, hypohess esng, random varables and her dsrbuons, lm heorems, fndng momen funcons, and workng wh characersc funcons and also be able o solve marc algebra, ec. LEARNING OUTCOMES: The sudens a he end of hs course should be able o:. Demonsrae a sound undersandng of he economerc modelng and esmaon as well as exhb evdence-based polcy makng. Prove he Gauss Markov heorem and deec, and fnd soluons o economerc problems n he conex of esmaed regresson models. Ener polcy dalogues a he naonal and nernaonal levels, and engage n relaed polcy research o provde new soluons o exsng problems n a changng envronmen v. Oban daa ha are relevan o he saed economc problem and presen a model ha s sued o deal wh he phenomena under sudy v. Demonsrae compeence n he use of economerc packages (lke E-VIEWS) as may be needed o perform he analyss. v. Use he economerc model for analyss and predcon. Ths nvolves explorng he economc mplcaons of he emprcal resuls. v. Carry ou good qualy appled economc research wh confdence P a g e

ASSIGNMENTS: Classroom es and an Economercs erm paper wll also be gven o faclae learnng of he more challengng areas of he course. Ths wll make up he connuous assessmen of 30% of he fnal grade of every suden. GRADING: We wll assgn 0% of hs class grade o homeworks, 0% for he programmng projecs, 0% for he md-erm es and 70% for he fnal exam. The Fnal exam s comprehensve. The gradng for hs course s a combnaon of connuous assessmen and fnal examnaons. A fnal examnaon wll be wren a he end of he course and hs wll cover 70%. REFERENCE TEXTS The recommended exbooks for hs class are as saed: Tle: Appled Economercs. Auhors: Aserou, D. and Hall, S.G. Publsher: Palgrave Macmllan, New York, nd Edon Year: 007 Tle: Economerc Mehods Auhor(s): Johnson, J. and J. DNardo Publsher: McGraw Hll Inernaonal Edons, 4 h Edon Year: 008 Tle: Appled Tme Seres Modellng and Forecasng Auhor: Harrs, R.I.D and Solls, R. Publsher: John Wley & Sons, Inc., nd Edon, Year: 003 VIOLATIONS OF BASIC ASSUMPTIONS Inroducon Volaon of he basc assumpons of OLS esmaor leads o economercs Problems such as auocorrelaon, heeroscedascy and mulcollneary. AUTO-CORRELATION Auocorrelaon s he seral dependence of he successve values of he sochasc error erm. In oher words, he sochasc dsurbance n he curren perod depends on s mmedae pas values. I s ndeed, a volaon of he assumpon of he absence of auocorrelaon or seral ndependence of he OLS echnque for esmang he classcal lnear regresson model. In effec, he value aken on by he sochasc dsurbance n one perod depends on he value akes on n he prevous perod such ha: Cov( u, u ) 0,,..., n Thus, u f ( u ) e Auocorrelaon has sgnfcan occurrence n me-seres economc daa and hence n me seres economercs. Forms of Auocorrelaon P a g e

There are dfferen forms of auocorrelaon. These nclude: (a) Frs order auocorrelaon (b) Second-order auocorrelaon (c) Thrd-order auocorrelaon (d) Kh-order auorrelaon The frs-order auoregressve scheme s specfed as: U U e Where s he coeffcen of frs-order auocorrelaon, e sasfes he usual OLS assumpons, ha s, Ee ( ) 0 Ee ( ) e E( e e ) 0 s he sochasc error erm whch The second-order auoregressve scheme s specfed as: U U U e The hrd-order auoregressve scheme s specfed as: U U U 3U 3 e The kh-order auoregressve scheme s specfed as: U U U... kuk e Causes of Auo-correlaon Incdence of Inera Inera s he flucuaons n me seres varables such as employmen. So, n regressons nvolvng me seres daa, successve observaons are lkely o be nerdependen. Non-Saonary A me seres s saonary f s mean, covarance and varance do no vary wh me. Ths means ha a non-saonary me seres s a seres ha changes wh me. Non-saonary causes auocorrelaon because he mean, varance and covarance of a non-saonary seres are me varan, ha s, hey change over me Dfference-Transformaon Auocorrelaon s easly nduced by frs-dfference ransformaon of he me seres varables. In shor, seral correlaon mos ofen characerzed models regressed on he successve dfferences of he values of varables. Consder he followng level and dfference models: Y Z u Y Z where Y Y Y Z Z Z Noe ha he error erm n he level equaon s no auocorrelaed bu can be shown ha he error erm n he frs dfference form s auocorrelaed. Specfcaon Bas Specfcaon bas s n wo folds, he (a) Omed varable bas and 3 P a g e

(b) Incorrec funconal or mahemacal form bas. Omed Varable Bas Auocorrelaon wll occur when here s an omsson of an mporan explanaory varable from he regresson model. In oher words, he omsson bas akes place when varables ha are germane o he phenomenon beng suded are excluded from he model. Funconal Bas Wrong funconal form of a model nduces auocorrelaon. For example, when a lnear mahemacal form of a model s specfed nsead of a quadrac form as n he followng equaons, auocorrelaon s nduced. C Q 0 C 0 Q Thus, when a researcher fs an emprcal model wh a wrong funconal form o he avalable daa, such as, fng a lnear model when a non-lnear model s acually he mos approprae, auocorrelaon s provoked auomacally. Lag Srucures n Economcs The use of lags n economcs s a major cause of auocorrelaon. Thus, auoregressve models are mosly auo correlaed because he error erm s a reflecon of he sysemac paern due o he nfluence of he lagged erm n he model. Consder he usual auoregressve consumpon funcon: d C Y C 0 Is hs no a reflecon of u f ( u ) e.the underlyng assumpon s ha consumers expendures n he curren perod depend for mos mes on her prevous level of expendures. As were, economc agens mos ofen do no sgnfcanly change her consumpon habs. Thus, due o he presence of C, ha s, he effec of C onc, auocorrelaon s provoked. In hs very nsance, he error erm wll reflec he paern of seral dependence gven he smlar paern of consumpon. Daa Mnng Daa mnng s he nerpolaon and exrapolaon of daa. I s a smoohenng process of daa manpulaon whch dampens he flucuaon n he orgnal daa se. Such daa manpulaon s known as daa massagng whch leads o sysemac paern n he dsurbances hereby nang seral correlaon. A good example of daa mnng s averagng of quarerly daa. Consequences of Auocorrelaon on he OLS Esmaor (a) The unbasedness propery of he OLS esmaor s no affeced even n he presence of auo correlaed errors. Thus, he OLS esmaor, β, s sll unbased and hghly conssen. (b) The effcency propery of he OLS esmaor s desroyed. The varances and he sandard errors of he OLS esmaor are upwardly based. (c) Consequenly, he -values of he coeffcen esmaes are dsored and rendered unrelable. (d) Erroneous sascal decsons are made. Ths s because he use of he convenonal and F sascal ess o evaluae he sascal sgnfcance of he esmaed coeffcens of a model are no longer vald. (e) Confdence nervals of he OLS esmaes are flawed and a bes ouszed. 4 P a g e

(f) Also, he resdual varance ( S ) wll under esmae he rue parameer. As a resul, he coeffcen of deermnaon, R wll be overesmaed. (g) Auocorrelaon can be subjugaed for predcons. The reason s ha an auocorrelaed me seres s probablscally predcable because fuure values depend on curren and pas values. Sascal Tess for Auocorrelaon Graphc Mehod The graphcal mehod enals plong he values of he error erms agans me. Three graphcal ools for assessng he auocorrelaon of a me seres are he me seres plo, he lagged scaer plo, and he auocorrelaon funcon. Durbn-Wason d Tes Sasc Due o Durbn and Wason (970),n her arcle led, Tesng for Seral Correlaon n Leas Squares Regresson, he d es for auocorrelaon s defned as he rao of he sum of squared dfferences n successve resduals o he sum of squared resduals. d n ( u u ) n u Gven ha, he values of he D-W sasc les beween 0 and 4, ha s, 0d 4. Thus, When d, () 0 sgnfyng absence of auocorrelaon When d 0, (0) sgnfyng perfec posve auocorrelaon When d 4, (4) sgnfyng perfec negave auocorrelaon 5 P a g e

Graphcal Represenaon of Durbn-Wason d Tes Sasc of 0 dl du 4-dU 4-dL d The Durbn-Wason emprcal value d mus be compared wh he crcal d values denoed by d U w, he upper lm for he sgnfcance level of d and d, he lower lm for he sgnfcance level of. Gven hese condons; he es s appled on he bass of he followng hypohess: d w Auocorrelaon 0 d w The applcaon of he d w sasc depends on he followng condons (a) An nercep erm mus be ncluded n he regresson model. Ths s because f he model s fed whou an nercep, resdual sum of squares [RSS] even f compued mgh no sum o zero hereby surmounng he noon of zero sum of squares. Consequenly,R mgh be negave [see Johnson and Dnardo (997)] (b) The regressors of he model mus be ruly exogenous.e. non sochasc, (c) The error process mus be generaed by he fs-order auoregressve scheme and no by hgher orders. As were, he d w s only applcable for esng he presence or oherwse of frs-order auoregressve scheme, (d) The error erm mus be normally dsrbued wh zero mean and consan varance, (e) No mssng daa pon n he enre me seres or seres of observaons (f) The regresson model mus no nclude lagged endogenous varable as explanaory varable. Thus, f he model o be fed s of he money demand ype. D D M 0 Y M The d w sasc becomes napproprae and ofen erroneously selng for he absence of auocorrelaon even f here s. The only way ou of hs emprcal problem s o ulze he Durbn-h sasc. w H : 0 ( non auocorrelaed resduals) H : 0 ( auocorrelaed resduals) L 6 P a g e

Durbn h Tes Sasc The napplcably of he d sasc n models wh lagged endogenous varables as regressors necessaed he Durbn h-sasc [see Durbn (974)]. The h-sasc s defned as: Where N s he sample sze, Var( ) s he varance of he esmaed coeffcen of he lagged D endogenous regressor, ha s, he varance of he coeffcen of n he money demand funcon and s he esmaed frs-order auocorrelaed. Correcve Measures of Auocorrelaon The remedal measures of he auocorrelaon problem exs for boh when he auocorrelaon coeffcen s known and when s unknown. Generalzed Leas Squares [GLS] Esmaor The generalzed leas squares esmaor s bascally he applcaon of OLS o he ransformed model ha fulflled he classcal OLS assumpons. However, s applcaon s faclaed when he frs-order auocorrelaon coeffcen s known. Gven he underlyng model whose error erm follows he AR scheme: Y v [0, v ], [ ] To correc he model for auocorrelaon, we lag (5.) by one-perod so ha we have: Y 0 X Mulply he resulng equaon by he auocorrelaon coeffcen o oban Y 0 X Subrac equaon (5.0) from equaon (5.9) above o oban: h X 0 where v N n[ Var( )] M (5.0) Y Y ( ) ( X X ) 0 Y X v 0 where Y Y Y 0 ( ) 0 X ( X X ) v (5.9) (5.) Cochrane-Orcu Ierave Mehod The Cochrane-Orcu erave mehod of ransformaon enals a process of convergence o he auocorrelaon coeffcen. Accordngly, he applcably of he Cochrane-Orcu 7 P a g e

ransformaon requres he model o be esmaed. Thus haven appled he OLS echnque; he esmaed model can be gven as: y 0 X To correc he model for auocorrelaon, compue he OLS resdual seres as follows: Y 0 X Usng he resdual seres, run he followng auxlary regresson or smply compue he auocorrelaon coeffcen from he varance-covarance rao as follows: Usng he esmaed, esmae he generalzed frs-dfference model. In oher words, use o ransformed model and use OLS o furher esmae he model. y X v Compue he second round OLS resdual seres, Y 0 X Compue he second-round auocorrelaon coeffcen from he varance-covarance rao as follows: Use o ransform he orgnal model and apply OLS esmaor o he model: Compue he hrd-round OLS resdual seres, Y 0 X Compue he hrd-round auocorrelaon coeffcen as follows: Use Y o ransform he orgnal model and apply OLS esmaor o he model: Y v 0 Y Y v y X v 0 Y Y 8 P a g e

y 0 X v The erave process connues unl he auocorrelaon effec s elmnaed from he esmaed se of regresson resuls. Numercal Examples Example These are regresson resuls usng OLS for observaons wh sandard errors n parenheses: Y.3 0.97Y.3X (0.3) (0.8) (0.04) DW. Tes for he presence of auocorrelaon n he dsurbances [Greene (003): 8, (Exercse 3)] Soluon Gven ha he D-W sasc canno be used o es for auocorrelaon n dynamc models, we resolve o usng he Durbn h-sasc whose es sasc s gven as: h N n[ Var( )] The auocorrelaon coeffcen can be esmaed from D-W he sasc dw dw. 0.395 Applyng he Durbb h-sasc, we have as follows: d w ( ) d w 9 P a g e

Example Consder ha n a regresson analyss ha relaes he consumpon level of elecrcal applance o he ncome level of consumers, he economerc resuls ha follow below were obaned for he sample perod, 980-00. C 83.55 0.3I raos 0.005, R 0.99, R Adjused 0.89.864,.0683 (a) Tes for auocorrelaon (b) Commen succncly on he presence or oherwse of auocorrelaon (c) Esmae, he auocorrelaon coeffcen Soluon 5.3 The es hypohess s gven as follows: H h 0.395 [0.006] 0.395 0.0336 0.395 0.9664 0.395.73 0.395[4.66].84 H : 0 or dw Nex, we compue he d-sasc as follows: 0.009 d 0 : 0 or dw N w ( ) N.0683.864 dw 0.798 Wha follows nex s o oban he crcal values of dl and du. Thus, a K (number of explanaory varables), N 30 and 5%, dl.8 and du.57. Analyss Snce he compued d les beween 0 and, we conclude ha he esmaed se of regresson resuls for w dl he consumpon level of elecrcal applance o he ncome level of consumers s neffcen. Hence, he esmaed dsurbances are serally dependen. 0 P a g e

Esmaon of : We can esmae from he sasc. Recall ha: d w d w ( ) d w dw dw 0.798 0.399 P a g e

MULTICOLLINEARITY Mulcollneary refers o he lnear relaonshp beween he explanaory varables of a mulple regresson model (MRM). In oher words, mulcollneary occurs when he regressors of an MRM X, X,...,Xk are hghly correlaed wh each oher. By economerc nuon, lnear relaonshp ough no o exs beween he explanaory varables n a MRM. Ths s acually a desecraon of he full rank assumpon of he classcal lnear regresson model. Exreme Cases of Mulcollneary There are wo exreme scenaros of mulcollneary. These nclude perfec mulcollneary and orhogonal mulcollneary. Perfec Mulcollneary Perfec mulcollenary means ha he relaonshp beween he explanaory varables s exac n he sense ha he correlaon coeffcen beween he explanaory varables s uny, ha s, rr. Consder he log-lnear regresson of LnY on LnX and LnX. x x j LnY LnX LnX 0 where r r x x s suppose o gve he rae of charge n he mean value of as changes by a percenage Y X pon holdng X fxed. Unforunaely enough, he perfec collneary beween he regressors X and means ha canno be kep consan whle changes and vce-versa. X X X Orhogonal Mulcollneary An orhogonaly means ha explanaory varables have no relaonshp. In effec, he correlaon coeffcen beween he explanaory varables s zero. Thus, rr 0. In parcular, he explanaory varables n he mulple regresson models are no correlaed n any form. Thus, orhogonal varables are he varables whose covarance s zero. LnY LnX LnX Imperfec Mulcollneary In he pracce of appled economercs, neher of he wo exremes of orhogonaly and perfec mulcollneary exs. The mulcollneary problem ha exss n pracce les n beween he wo exremes and s called mperfec mulcollneary whch means hgh bu less han perfec mulcollneary. In hs case, he correlaon coeffcen les beween zero and uny, ha s, 0 rr. x x j Causes of Mulcollneary Dsrbued Lag Models Mulcollneary can be caused by he use of lagged varables n a mulple regresson model. Ths s because s nnae for successve values of a parcular varable o be hghly nercorrellaed. Consder he followng sysem of equaons: j 0 where r r 0 LnY LnX LnX 0 x where 0 r r x j x x j x x j P a g e

LnM LnC Y Y r D D LnM 0 3 LnY LnY LnC 0 3 3 LnI LnY LnY 0 3LnI The ncluson of pas and presen levels of ncome n he money demand, consumpon and nvesmen equaons respecvely can nduce he problem of mulcollneary as hese varables are ceranly gong o be hghly correlaed. Over-deermned Models An over-deermned regresson model s he model whose explanaory varables are more han he number of observaons. In hs ype of regresson, mulcollneary problem s severe. Consequences of Perfec Mulcollneary Indeermnacy of he OLS Esmaor: Coeffcen esmaes of he OLS esmaor are ndeermnae Emprcally, wh perfec mulcollneary such ha [ r ], mples ha he rue relaonshp beween he explanaory varables x and x s exac and as such x x.the ndeermnacy of he OLS esmaor arses because he daa marx of he explanaory varables ' ' ' ( XX) n he OLS esmaor canno be nvered. Consequenly, OLS esmaor [( X X ) X Y] breaks down. Ths shows ha n he presence of perfec mulcollneary, he varances and hence he sandard errors of he OLS esmaors are nfne and consequenly ndeermnae. xx j Consequences of Imperfec Mulcollneary Accordngly, as collneary ncreases beween any wo regressors, he varances and hence sandard errors of he OLS esmaors evenly ncreases. Thus, he VIF measures he exen by whch he varance of OLS esmaor s nflaed due o he presence of mulcollneary. Emprcally, wh mperfec mulcollneary [0 ], he followng consequences are on he OLS esmaor: (a) Large Varances and sandard errors: The varances and sandard errors of he OLS esmaor are unduly large, and so he -raos are rendered sascally nsgnfcan whch leads o a ype II error of accepng an ncorrec null hypohess null hypohess nsead of rejecng. (b) Erroneous Sascal Inferences: Msleadng sascal nferences are drawn from he es of hypohess (c) Wde confdence nervals: Confdence nervals are unduly ouszed (d) Unsable coeffcens: The OLS parameer esmaes become hghly unsable. Such nsably of coeffcens could cause a dramac change n he coeffcen sgn as he degree of mulcollneary ncreases. Tess for Mulcollneary r xx j 3 P a g e

There are several ess for deecng he problem of mulcollneary. As s, we have he formal and nformal ess for mulcollneary. The nformal ess for mulcollneary nclude: (a) Hgh R : Even n he presence of nsgnfcan -values, he overall measure of goodness-of-f, R could very hgh. (b) Low -raos (c) Wrong coeffcen sgn (d) R delee The formal sascal ess nclude he: (a) Varance Inflaon Facor [VIF] (b) Farrar-Glauber es Varance Inflaon Facor [VIF] The varance nflaon facor (VIF) quanfes he proporon by whch he varance of he OLS esmaor s nflaed. In oher words, he VIF quanfes he varance of he OLS esmaor due o mulcollneary. Compuaonally, s defned as he recprocal of he olerance ndex. Appled economercans mos ofen desre lower values of VIF, as hgher values of VIF are known o adversely affec he regresson resuls. For example, a VIF of 8 mples ha he sandard errors of he OLS esmaor are larger by a facor of 8 han would oherwse be he case, f here were no mulcollneary beween he regressors n he mulple regresson analyss. VIF( ) r j Tolerance( ) VIF r j Where rxx s he smple correlaon coeffcen beween any pars of regressors say, [ x j, x] defned by he Karl Pearson s produc momen equaon below: xx r x x Thus, as he correlaon coeffcen, 4 P a g e r xx j or N x x j ( x x j ) N x ( x ) N x j ( x j ) r xx j measures he degree of collneary beween he regressors, he VIF quanfes he speed a whch varances and covarances ncreases. For hs reason, as he collneary beween he regressors ncreases and ends o uny.e he case of

perfec mulcollneary, he VIF approaches nfny ( ). Also, f he collneary beween he regressors, [ x, x ] s zero, he varance- nflang facor wll be equal o uny. Correcve Measures for Mulcollneary The remedal measures o be adoped f mulcollneary exss n a model depends on he followng facors, severy of he mulcollneary problem, avalably of daa, he mporance of he collnear regressors, he purpose of esmaon ec. In any case, he remodel measures are dscussed as follows: Chrs s Correcon Chrs (966:389) suggesed ha mulcollneary can be correced by ncreasng he sample sze. Ths enals brngng no he sample, more daa pons. However, he remedal measures are only vald f error of measuremen n he explanaory varables s he cause of mulcollneary. Droppng Varables Anoher measure of resolvng severe mulcollneary problem s o drop he varable ha s hghly collnear wh he ohers. However, droppng one of he collnear varables from a model may pu he economercan a he verge of commng a specfcaon error. Ths s because n lne wh economc heory, f neres rae and ncome are he key deermnans of money demand and as such mus be ncluded n he money demand funcon, droppng eher of neres rae or ncome would mean commng a specfcaon error. Under hs scenaro, he cure could be worse han he dsease. Transformaon Transformaon of varables o s generally regularly useful as a way ou of he mulcollneary problem. Thus, nsead of runnng he regresson n he orgnal varables hemselves, wha becomes desre s o run mulple regresson model on he ransformed daa marx on he varables under specfcaon. Poolng Daa A combnaon of boh cross secon and me seres daa can help o resolve he problem of mulcollneary. Poolng has he followng lmaons: (a) Pooled observaons do suffer from he seral correlaon problem, (b) Poolng observaons n dfferen me perods do erroneously assume sably n he casual relaonshp across me raher han varaon across sub-perods (c) Poolng canno dsngush beween varaons across me and across seconals. For example, an ncluson of a dummy varable akes no care he dfferen slopes or baselne values raher han he dfferen slopes wh varous perods. Mulcollneary In he Presence of Economc Forecasng I has been assered ha he problem of mulcollneary s harmless f he objecve for esmang a model s o forecas he values of he endogenous varable only [Chrs (966): 390, Greene (003)]. In hs case, he values of he collnear varables can be ncluded n he model whle gnorng he consequence. Ths can only be successful provded he economercan s ceran ha he correlaon paern ha exss beween he explanaory varables wll reman he same hroughou he predcon perod. 5 P a g e

HETEROSKEDASTICITY Heeroskedascy means ha he varance of he sochasc dsurbance erm (u) s no consan (he same) for all values of he explanaory varables. Ths s because he varance of he sochasc dsurbance s no longer gven by a fne consan and hus would end o change wh an ncreasng range of values of he explanaory varables hereby makng mpossble o be aken ou of summaon. Thus, he homoskedasc varance covarance marx s gven by : E u u 0 0 u 0 0 u 0 0 u 0 0 0 0 ' (, j ) 0 u 0 u I However, gven he presence of heeroskedascy, Var( u ) E( u ) u u In effec, he heeroskedasc varance-covarance marx s gven by: 0 0 u ' E( u, u j ) 0 u 0 0 0 u 3 6 P a g e I u The subscrp denoe he fac ha he varances of each sochasc dsurbance are all dfferen. The occurrence of heeroskedascy s found n boh me seres and cross secon daa bu more ofen encounered and severe wh cross secon daa. Ths s because he assumpon of consan varance over he heerogeneous uns may be raher unrealsc. Causes of Heeroskedascy (a) Oulers Problem Oulyng observaons are he roo cause of heeroskedascy. An ouler s an observaon ha s eher excessvely small or excessvely large n relaon o oher observaons n he sample. The able below llusraes a scenaro of an ouler Y X 38 80.6 680. 800

(b) In oher words, he oulyng observaon exhbs huge dfference from ohers n he sample. In effec, he populaon of he oulyng observaon s dfferen from he populaon of he oher sample observaons Omed Varable Bas The omsson of key explanaory varables from a regresson model causes heeroskedascy. For example, consder he followng model of consumpon expendure: d C r I T C F W Y u 0 3 4 5 6 7 where C s consumpon exp endues, r s neres rae, I s nflaon rae T s aseof he consumer, F s fashon, W s weaher condon d Y s dsposable ncome In lne wh economc heory, ncome s he mos crucal deermnan of consumpon expendures. Thus, f ncome s omed from he model, he omed varable bas would have been nduced. Ths n urn aracs heeroscedascy. (c) Error Learnng Facors/Models As people learn everyday from her pas msakes, her errors of behavor become smaller and smaller over me and as such canno be relavely consan. (d) Wrong Funconal Form Specfcaon error or ncorrec funconal form of a regresson model causes heeroskedascy. Ths occurs when a model s beng regressed wh a pool of level and log varables a he same me d LnC r LnY u 0 d where C s consumpon exp endues, r s neres rae, Y s dsposable ncome (e) Erroneous daa ransformaon Incorrec daa ransformaon s anoher cause of heeroskedascy. I occurs when a regresson model s beng regressed wh a pool of rao and frs-dfference se f daa a he same me (f) Skewness Skewness n he dsrbuon of he explanaory varables causes heeroskedascy. For example, he dsrbuon of ncome and wealh s mos ofen unequal bu skewed n such a way ha he bulk of ncome and wealh s owed by a few ndvduals a he op. Thus, whle he spendng behavor or he expendure profle of a cross-secon of famles wh low ncome may exhb smlar paern n addon o beng relavely sable, such expendure profle of he cross-secon of he rch famles wh hgh ncome could be dfferen and hghly volale Consequences of Heeroskedascy To nvesgae he effecs of heeroskedascy on he OLS esmaor, s varance and sandard errors, becomes desrous ha we rever o marx specfcaon of he classcal lnear regresson model [CLRM]. 7 P a g e

X ' Y where Var[ / X ] I,,..., N Gven havar( u ) I, he general form of he heeroskedasc varance-covarance marx can hen be descrbed as: Var[ / X] Where s a posve defne marx such ha Dag. Thus, he varance-covarance [V-C] marx of he OLS esmaor Var Cov( ) E[( )( )'] wll be gven by: E [( X ' X ) X ' ][( X ' X ) X ' ]' [( X ' X ) X ' ][( X ' X ) X '] [( X ' X ) X '] E( ')[( X ' X ) X ] ( X ' X ) [ X ' X ] ( X ' X ) Var Cov ( ) ( X ' X ) Ths porrays he fac ha wh heeroskedascy, (a) The varances and sandard errors of he OLS esmaor are no longer effcen, no even asympocally. In oher words, no even n large samples. Thus, he mnmum varance propery of he OLS esmaor s los. (b) The varances and sandard errors are overesmaed by he OLS esmaor hereby geng he sandard errors of he esmaed coeffcens dsored by beng over boosed. (c) Sascal es of sgnfcance are rendered nvald. As were, he valdy of he convenonal formulae for and f es sascs becomes mpared (d) Sascal nferences are erroneous. Consequenly, wh heeroskedascy, here s a hgher rsk of commng ype error whch enals rejecng a correc null hypohess nsead of accepng and also here s he lkelhood of commng a ype II error whch has o do wh he accepance of an ncorrec null hypohess raher han rejecng. (e) Confdence nerval of he esmaed coeffcens becomes nordnaely wde. In oher words, confdence nervals are overly ouszed (f) In general, he hypohess-esng procedures on he bass of he OLS esmaes are conamnaed and spurous. (g) Heeroskedascy does no desroy he unbasedness propery of he OLS esmaor. As a maer of emprcal fac, remans unbased.,, 3,..., N E / X 0sll holds. Consequenly, he OLS esmaor Sascal Tess for Heeroskedascy There are numerous ess for deecng he presence or oherwse of he problem of heeroskedascy. These nclude he nformal and formal echnques. The formal mehods for deecng he presence or oherwse of heeroskedascy are mehods ha sugges ha he economercan has some apror nformaon se abou he rue paern of 8 P a g e

heeroskedascy. In effec, he economercan s ask s o conduc he regresson analyss on he assumed paern of heeroskedascy. Glejser Tes Due o Glejser (969), he Glejser es s a formal es for heeroskedascy ha regresses he absolue values of he esmaed resduals on varous powers of he explanaory varable of he model. The es s based on he followng hypohess. H : are homoskedasc 0 H : are heeroskedasc Spearman s Rank Correlaon Tes [SRCT] The SRCT sasc s a deecve measure of heerokedascy ha ranks he values of he explanaory varable and he esmaed regresson resduals eher n ascendng or n descendng order of magnude whou regard for he sgns of he resduals. Gven he followng regresson model o be esmaed: Y 0 X u Wha follows nex s o: (a) F he regresson o he daa on Y and X and (b) Generae he resduals u. (c) Dsregardng he sgn of he esmaed resdual, ha s akng only he absolue value of he esmaed resdual, we rank u and X eher n a descendng or n an ascendng order. (d) Nex, s he compuaon of he Spearman rank correlaon coeffcen (SRCC). The es sasc s gven as: k d r X, 6 nn Where d s he dfference beween he values of correspondng pars of X and, n s he number of observaons n he sample, ha s, he number of ndvdual uns beng ranked. The es s based on he followng hypohess. H : are homoskedasc 0 H : are heeroskedasc Decson rule: k If s low, accep H, he error valances s homoskedasc rx, 0 k If rx, s hgh, accep H, he error varance s heeroskdasc Alernavely, k r n If r X,, accep H and rejec 0 H r k r n If r X,, accep r and rejec H k.96 If rx, z, accep H0 n and rejec H H 0 9 P a g e

k.96 If rx, z, accep and rejec n Where r s he compued SRCT sasc,.96 n s he sandard normal z crcal value. Here, he sandard error of he SRCT sasc s assumed o obey he sandard normal dsrbuon n hs regard. The crcal -value can be obaned as and s he level of sgnfcance and n s he degree of freedom. Goldfeld-Quand Tes Ths s a formal es for heeroskedascy due o Goldfeld and Quand (97). By defnon, he G-Q es s a fundamenal F-es sasc ha enals he orderng of he se of observaon n accordance o he magnude of he values of he explanaory varable and hereafer dvdes he se of observaons no hree pars such ha he frs and hrd halves are equal. The mddle half whch s made up of one-quarer of he oal number of observaons n he sample ( n 4) s excluded from he es. Thus, f n 6, mples ha he 4 mddle observaons n he ordered se mus be omed or deleed and he balance observaons dvded no wo equal halves of 6 observaons each. Havng asceraned hs dvson, separae error varances or resdual varances are esmaed from he OLS regresson of he wo equal halves of 6 observaons each. Worhy of noe s he fac ha he orderng of he daa se s deermned on he bass of an ascendng order of he values of he explanaory varable. In sum, he G-Q mehod requres he followng seps: Sep : Rankng he daa pons on he regressor, X sarng wh he leas observaon Sep : Dvde he ordered se of observaons no equal groups each of n f observaons haven omed he cenral group of observaons f. The cenral group of observaons o be excluded from he heeroskedascy es should be abou one quarer of he oal number of observaons n. For example, f n 3, abou 3/4 = 8 daa pons mus be excluded and he remanng dvded no wo equal halves of n and n observaons, where n and n. Sep 3: Fng he OLS regresson on separae bass o he wo groups of observaons each, and oban he resdual sums of squares RSS, and RSS wh he degree of freedom gven by nk and nk respecvely, and k s he number of parameers o be esmaed. Sep 4: Compue he F- rao as follows: The es sasc s gven as: Fsasc H H0 %, n 0 P a g e

SSE u n k u n u k u u u u uu ' n k ( Y Y) n SSE n k n u k k F sasc u u uu ' n k ( Y Y) n k ( Y Y) n k ( Y Y) n k Where s he error varance, oherwse known as he resdual varance, s he sum of squared resduals, n s he number of observaons n he sample [sample sze] and k s he number of esmaed regresson coeffcens. The es s based on he hypohess: H : are homoskedasc H : are heeroskedasc Decson Rule: If S S, accep H, he error varance s homoskedasc P a g e 0 0 If S S, accep H, he error varance s heeroskedasc The G-Q es rao herefore obeys he F-dsrbuon wh nk and n k degress of freedom. Sgnfcan F ndcaes presence of heeroscedascy and vce versa.

If F, accep and rejec calculaed Fcrcal 0 If F F, rejec and accep The assocaed crcal F-value s obaned as. The decson rule s o accep he homoskedascy assumpon f Fcal Fcr and rejec f Fcal Fcr. In whch case, f he boh he numeraor and he denomnaor are equal we accep. Correcve Measures for Heeroskedascy Transformaon based on he Paern of Heeroskedascy In economerc leraure, dfferen assumpons abou he error erm have been made and hs warrans he ype of daa ransformaon n order o elmnae heeroskedascy from an emprcal model. Logarhmc Transformaon [LT] Gven he unknown naure of heeroskedascy, a logarhmc ransformaon [esmang he orgnal model n log] s also applcable n resolvng he problem of heeroskedascy. In hs case, he ransformaon of he orgnal model becomes: LnY LnX Mers of Logarhmc Transformaon In general, logarhmc ransformaon helps o reduce f no oal elmnaon of he problem of heeroskedascy. Ths s evden n he followng facs. Logarhmc ransformaon: Compresses he scales n whch he varables are measured, hereby reducng a enfold dfference beween wo values a wo-fold dfference. For example, he number 0 s en mes larger han, bu Ln(0) gves 4.787 whch s jus abou wce as large as Ln() whch s equal o.485 Yelds drec elasces. For example, he slope coeffcen of a logarhmc ransformed model measure he elascy of he dependen varable wh respec o he regressor n queson. In parcular, measures he percenage change n he dependen varable due o a percenage change n he explanaory varable. Logarhmc Transformaon Problems Log ransformaon s no applcable f some of he observaons [daa pons] for boh he dependen and he explanaory varables are zero or negave The problem of spurous correlaon wll be encounered beween he raos of he ransformed varables even when he orgnal varables are uncorrelaed or random. The convenonal F and ess for model robusness are only vald n large samples gven ha he varances are unknown and are esmaed from any of he ransformaon procedures. In he mulple regresson model [MRM], model wh more han one regressors, s dffcul o asceran on apror basc whch of he regressors o be chosen for daa ransformaon Example Consder he daa below: H H calculaed H crcal 0 H F %, n k, n k H 0 P a g e

Tes for he presence or oherwse of heeroskedascy usng he spearman rank coeffcen es sascs. Y 4 6 3 6 X 0 6 4 8 Soluon Frs: We sae he hypohess: H0 : e arehomoskedasc H : e areheeroskedasc Secondly: We would esmae an SRM whch s of he followng specfcaon Y 0 X e Y X X YX ỳ y-ỳ 0 00 0. -0. 4 44 48 3. 0.8 6 6 56 96 5. 0.8 3 4 96 4 4. -. 6 8 34 08 6. -0. 70,00 34 0 Esmaed Regresson lne: Y 0 0 Y X X YX (00) 70(34) 5(00) (70).8 N X X N YX Y X NX X 5(34) (70) 5(00) (70) 0.5.8 0.5X when X 0, Y.8 0.5(0). when X, Y.8 0.5() 3. when X 6, Y.8 0.5(6) 5. when X 4, Y.8 0.5(4) 4. when X 8, Y.8 0.5(8) 6. 3 P a g e

Y Y e Y Y X. -0. 0.5-0.5 0.5 4 3. 0.8 3 3.5 -.5.5 6 5. 0.8 4 6 4 3.5 0.5 0.5 3 4. -. 5 4 3 5-4 6 6. -0. 8 5.5 3.5.5 9.0 r X re d rx re d Now, we can apply he spearman rank coeffcen es sasc 6d rxe. n n 6(9) r xe. 5(4) 0.95 0.05 where d s he dfference beween he values of correspondng pars of X and e observaons, n s he number of observaons n he sample. Solvng es: +=.5 3+4 = 3.5 Decson Rule: We can evaluae he z and crcal values as follows:.96 z 5.96 r Xe, 4.96 0.98 0.05 0.98 0.05 5 rxe. (0.05) 0.05 Gven ha Snce r xe. s on he low sde, we conclude ha he error varances are homoskedasc. So we would accep and rejec H H0 Example 4.5 Gven: 4 P a g e

Y X 0 0 4 6 6 8 8 8 0 30 4 8 6 0 30 6 0 6 8 6 8 8 8 6 6 Tes for heeroskedascy usng he Goldfeld-Quand es sasc a boh he 5% and 5 sgnfcance levels. Soluon 4.5 Dae Orderng Y X 0 0 6 0 8 6-4 = = 6 4 6 8 6 6 6 8 8 8 Delee x 6 = 4 8 8 4 Hypohess: 6 8 8 8 0 30 0 30 6-4 = = 6 4 6 6 H0 : resdual varance s homoskedasc H : resdual varance s heeroskedasc F sasc e e 5 P a g e

Solvng frs half where n Y Y n Y X X YX ỳ (y -ỳ) 0 0 400 00 3.34.6 6 0 400 30 3.34 7.08 484 64 4.6 5. 8 484 396 4.6 3.99 4 6 676 364 6. 4.4 8 6 676 468 6. 3.6 88 36 30,0 45.36 The underlyng model can be specfes hus: Esmaed Regresson lne: Y 6 P a g e where To ge Y we procede as follows : b b 0 b b 0 Y b0 b X Y X X YX NX X 88(30) 36(0) 6(30) (36) 4.4 N YX Y X NX X 6(0) 88(36) 6(30) (36) 0.46 4.4 0.46X when X 0, Y 4.4 0.46[0] 3.34 when X 0, Y 4.4 0.46[0] 3.34 when X, Y 4.4 0.46[] 4.6 when X, Y 4.4 0.46[] 4.6 when X 6, Y 4.4 0.46[6] 6.0 when X 6, Y 4.4 0.46[6] 6.0

Solvng hrd half Also, n n 45.36 4.34 Y Y n The underlyng model can be specfes hus: Esmaed Regresson lne: Y where b To ge Y we procede as follows : Y X X YX ỳ Y - ỳ (y- ỳ) 6 8 0 0 6 8 8 30 30 4 6 784 784 900 900 764 3844 448 784 300 300 504 60 5.05 5.05 5.5 5.5 8.7.87 0.95.95-5.5-5.5-6.7 3.3 0.90 67.70 30.36 30.36 39.3 9.67 0 0 8976 3948 0 78.43 b b b 0 0 Y b0 b X Y X X YX NX X 0(8976) 0(3948) 6(8976) (0) 8.6 N YX Y X NX X 6(3948) 0(0) 6(8976) (0) 0.3 8.60.3X 7 P a g e

when X when X when X when X when X when X 8, Y 8.6 0.3[8] 5.05 8, Y 4.4 0.46[8] 5.05 30, Y 4.4 0.46[30] 5.5 30, Y 4.4 0.46[30] 5.5 4, Y 4.4 0.46[4] 8.7 6, Y 4.4 0.46[6].87 n 78.43 4 69.6 F sasc e e 69.6.34 6.4 Decson Rule: A F F. Snce we accep he null hypohess and %,[ n k, n k] 5%,[4,4] 6.39 rejec he alernave. Alernavely, F compued (6.4) F crcal (6.39), we accep H0 and conclude ha he error varance are homoskedasc. 8 P a g e

IDENTIFICATION PROBLEM Idenfcaon n economercs has o do wh beng able o solve for unque values of he parameers of he srucural model from he coeffcens of he reduced-form of he model. In effec, denfcaon s concerned wh he possbly of obanng meanngful esmaes of he srucural parameers from he reduced form coeffcens such ha here mus be no oher equaon n he model ha can be formed by algebrac manpulaon of some oher equaons whn he model whch conans he same varables as he funcon n queson. The denfcaon problem hus occurs because dfferen ses of srucural coeffcens are compued from he same sample daa. In oher words, a gven reduced form equaon s found o be compable wh dfferen srucural equaons hereby makng dffcul o dsenangle he parcular hypohess ha s beng esed emprcally. As were, he denfcaon problem s a mahemacal problem assocaed wh smulaneous equaon sysems. I s herefore a problem of model specfcaon and no of model esmaon. Types of Idenfcaon In economerc modelng, wo ypes of denfcaon are dscernble. These are: (a) Under-denfed equaon (b) Idenfed equaon (b.) Exacly (jus) denfed equaon (b.) Over denfed equaon Under-denfcaon An under-denfed equaon s an equaon whose coeffcens canno be esmaed. Indeed, an equaon s under-denfed f s sascal form s no unque. Idenfed Equaon A sysem s denfed f all of s equaons are denfed. An denfed equaon could eher be exacly denfed or over-denfed. Exacly (Jus) Idenfcaon An equaon s exacly or jus denfed f only one se of srucural coeffcen esmaes can be compued from he reduced-form coeffcens. Over Idenfcaon An equaon s over denfed f more han one se of srucural coeffcen can be compued from he coeffcens of he reduced form equaon. In sum, a model (sysem of equaons) s denfed f all he equaons n he model are denfed. Idenfcaon Resrcons The denfyng resrcon enals he placemen of resrcons on he varables of a smulaneous equaons model usng economc heory and exraneous nformaon o solve he denfcaon problem of he smulaneous equaons. These resrcons can ake a varey of forms such as: (a) Use of exraneous esmaes of parameers, (b) Knowledge of exac relaonshp among parameers, 9 P a g e

(c) Knowledge of he relave varances of dsurbances, (d) Knowledge of zero correlaon beween dsurbances n dfferen equaons, (e) Zero resrcons, akng he form of specfcaon ha ceran srucural parameers are zero,.e., ha ceran endogenous varables and exogenous varables do no appear n ceran equaons. Formal Rules for Idenfcaon (a) Order condon for denfcaon (b) Rank condon for denfcaon Order Condon The order condon saes ha for an equaon o be denfed, he oal number of varables excluded from bu ncluded n oher equaon of he model mus be a leas as grea (mus be equal o or greaer han) as he number of equaon of he model less one. The order condon s whch s a NECESSARY condon for denfcaon s ndeed based on he counng rule of he varables ncluded and excluded from he parcular equaon ha s beng denfed. Mahemacally, he order condon s gven by: Q Q > E Where Q s he oal number of varables n he model Q s he oal number of varables n he parcular equaon ha s beng denfed E s he oal number of endogenous varables (number of equaons) n he model Illusraon : Consder he followng smple verson of he Keynesan ncome deermnaon model: C Y T 0 I I r T 0 Y 0 3 Y C I G In commenng on he denfcaon saus of he above sysem of equaons, we noe he followng: (a) There are four (4) endogenous varables, namely C, I, T and Y (b) There are hree (3) predcve varables namely, r, I and G. Applyng he order condon o he frs and second equaons (he consumpon and nvesmen equaons), we have; Q = 6, Q = 3 E = 4 6-3 = 4-3 = 3 We herefore conclude ha he consumpon and nvesmen equaons are exacly denfcaon (jus denfed). Applyng he same order condon o he hrd equaon (he ax equaon), we have: Q =, 30 P a g e

Q=6 E = 4, 6 - > 4-3 = 3 The ax equaon s over denfed. Therefore, s possble o frufully esmae he srucural parameers of he model from he reduced-form equaon. In shor, he srucural parameers can be rereved from he reduced from coeffcens. Illusraon: Consder he model: C Y 0 I Y I 0 Y C I G Usng he order condon: Equaon () Q = 5, Q =, E = 3, 5 > 3 3 > over denfed Q = 5 Q = E = 3 Therefore, 5 > 3 3 > The consumpon funcon s over denfed Equaon () Q = 5 Q = 3 E = 3 Therefore, 5 3 = 3 The nvesmen funcon s exacly (jus) denfed. Equaon Degrees of over denfcaon 0 L = Gven ha L L6, use ILS esmaor o esmae model. Rank Condon The rank condon saes ha n a sysem of K equaons, a parcular equaon s denfed f and only f s possble o consruc a leas one non-zero deermnan of order ( K ) from he coeffcens of he varables excluded from ha parcular equaon bu conaned n he oher equaons of he model. Ths condon s called he rank condon because refers o he rank of he marx of parameers of excluded varables and he rank of a marx s he order of he larges nonzero deermnan whch can be formed from he marx. 3 P a g e

In economerc analyss, he relevan marx s he sub marx of he coeffcen of he excluded varables. I s a SUFFICIENT creron for he denfcaon. When an equaon s suffcenly denfed, s necessarly denfed bu he converse s no he case. In effec, he common use of he order condon for denfcaon s no jusfed because s only a necessary condon for denfcaon. Thus an equaon mgh be necessarly denfed bu no suffcenly denfed. Tha s, even f he order condon s sasfed for a parcular equaon, may happen ha very equaon s no denfed. Illusraon: Consders he srucural Keynesan model gven below: C l l Y 0 I m my m 0 Y C I G Ths model could be re-wren n he form 0 Y l l Y o 0 I m my m I o 0 Y C I G Ignorng he random dsurbance he able of Ps of he model becomes varables. Equaons C I Y I G Equaon C - 0 l 0 0 Equaon I 0 - m m 0 Equaon Y - 0 Snce we are denfyng equaon (), he consumpon funcon, we srke ou he frs row n he able of srucural parameers as follows. Table of srucural parameers Equaons C I Y I G Equaon C - 0 l 0 0 Equaon I 0 - m m 0 Equaon Y - 0 Tables of parameers of Excluded Varables 3 P a g e I G - o o

Formng he deermnan(s) of order (M ) x (M ) ha s (3 ) by (3 ) = x, we have ha: m 0 0 m m 0 0 0 0 3 m m 0 0 0 m 0 We are able o form 3 non zero deermnans of order, he consumpon funcon of he model s denfed. Implcaons of Idenfcaon (a) If an equaon (model) s under-denfed s mpossble o esmae s parameers wh any economerc echnque. (b) If an equaon (model), s coeffcens can be esmaed. The suable esmaon echnque s asceraned by denfcaon saus,.e. exacly denfed or over-denfed. (c) If an equaon s exacly denfed, he approprae economerc echnque o be used for s esmaon s he ILS (d) If an equaon s over denfed, he approprae economerc echnque o be used for esmang s he SLS, 3SLS, ML ec. 33 P a g e

GAUSS-MARKOV THEOREM The Gauss Markov Theorem [GMT] saes ha he OLS esmaor provdes he bes, lnear and unbased [BLU] esmaor. In oher words, n he class of lnear and unbased esmaors, he OLS esmaor s he mos effcen esmaor. Dervaon of he OLS Esmaor Usng Marx Approach Y Xa Y X a Y Y Y X a ' ( Y X a)'( Y X a) Y ' Y ay ' X a' X ' Y a' X ' X a () () Observe ha ay ' X and a' X ' Y are boh scalars and hus equal o her ranspose ' Y ' Y ay ' X a' X ' Y a' X ' X a ( ' ) X ' Y a X ' X a ( ' ) Seng 0 a X ' Y X ' X 0 X ' Y X ' X ( X ' X ) ' X Y Y X X X3 0 Y X X X 3 Y 3 X 3 X 3 X 33 3 where Y s an ( n) column vecor of endogeneous varables X s an ( n k) marx of exogeneous varables s a ( k ) column vecor of populaon parameer s an ( n) column vecor of sochasc dsurbances 34 P a g e

Combnng he wn assumpons of homoskedascy and absence of auocorrelaon, ha s, Var E n ( ) ( ) (,,..., ) Cov E n ' ' (, j ) (, j ) 0 (,,..., ) We now derve he varance covarance marx as follows : 35 P a g e E 0 0 0 0 0 0 0 0 0 0 ' (, j) 0 0 I Proof of Unbasedness a X X X Y ( ' ) ' () Recall ha Y Xa () Subsung equaon () no () a X X X Xa Varance Covarance ( ' ) '[ ] (3) ( X ' X ) X ' Xa ( X ' X ) X ' (4) ( X ' X) a X X X ( X ' X) ( ' ) ' (5) ( X ' X ) X ' (6) ( ) [ ( ' ) ' ] (7) E a E a X X X a X X X E ( ' ) ' ( ) (8) E( a) a (9) Var Cov( a) E[( aa)( aa)'] (0) E X X X X X X [( ' ) ' ][( ' ) ' ]' () [( X ' X ) X ' ][( X ' X ) X '] () [( X ' X ) X '] E( ')[( X ' X ) X ] (3) ( X ' X ) [ X ' X ] I( X ' X ) (4) ( ) ( ' ) (5) Var Cov a X X

Consder anoher lnear esmaora a ady Subsung for a and Y a a X X X DXa D (6) ( ' ) ' (7) a DXa X X X D ( ' ) ' (8) a a DXa [( X ' X ) X ' D] (9) Takng he exp eced value of equaon (9) E a E a DXa X X X D ( ) [( ' ) ' ] (0) E a a DXa X X X D E E a a DXa [( ' ) ' ] ( ) () ( ) () For a obeunbased DX 0 ( ) a (3) Varance Covarance d ( ) Var Cov d E a a a a ( ) [( )( )'] (4) E [( X ' X ) X ' D] [( X ' X ) X ' D] ' (5) E [( X ' X ) X ' D] [( X ' X ) X ' D '] ' (6) [( X ' X ) X ' D] E( ')[( X ' X ) X D '] (7) [( X ' X ) X ' D] I[( X ' X ) X D ' [( X ' X ) X ' D][ ( X ' X ) X D '] (9) ( ' ) ' ( ' ) X X X X X X ] (8) ( X ' X ) X ' D ' ( X ' X ) X D DD ' (30) ( ) ( ' ) ' (3) Var Cov a X X DD 36 P a g e

( ) ( ) Var Cov a Var Cov a Thus, he OLS esmaor, a s bes, lnear and unbased ( BLU ) In sum, he OLS esmaor formulae n marx are gven by : a0 a a ( X ' X ) X ' Y X ' Y Y Y X YX n X X X ' Y X X X X X XX X ' n k R a' X ' Y ny /( k ) where ' Y ' Y a' X ' Y a' X ' Y ny Y ' Y ny F Y ' Y a' X ' Y /( n k) 37 P a g e

0 0 a0 Var Cov( a) ( X ' X ) 0 a 0 0 0 a ( 0) Var a S ( ) Var a S ( ) Var a S a0 a0 a0 a a a a a a 38 P a g e

SIMULTANEOUS EQUATIONS MODELLING A smulaneous equaons model (SEM) s a sysem of equaons n whch he dependen varables n some equaons are explanaory varables n oher equaons and hereby feedng-off shocks o each oher. Thus, a SEM s a sysem of equaons represenng a se of relaonshps among varables and hereby relang he jon dependence of varables. The feedback effec of he SEM can be demonsraed usng he srucural model: Y d d Y d X u () 0 Y Y X u 0 where Y and Yare muually dependen varables, X and X are he exogenous varable u and u are he sochasc dsurbance erms. If u ncreases by a gven proporon, wll auomacally ncreasey. The ncrease n Y wll n urn cause Y o ncrease. Ths feedback effec beween he wo srucural equaons s conemporaneous and ndeed connuous, an ndcaon ha he endogenous varables Y and Y are jonly dependen. The correlaon s ha an ncrease n u ncreases Y whch n urn ncreases Y. So u and Y are posvely correlaed. () Sngle Equaon Model The SEM represens only one relaonshp among varables The SEM has only one equaon. The esmaon mehod s manly OLS In he SEM, only he parameers of he sngle equaon can be esmaed In he SEM, here s a sngle dependen varable and one or more explanaory varables Smulaneous Equaon Model The SEM represens more han one relaonshp among varables The SEM has more han one equaon. The OLS esmaon echnque canno be appled o esmae he SEM In he SEM, more han one parameers can be esmaed smulaneously In he SEM, here are more han one dependen varables and more han one explanaory varables Specfcaon: A SEM has hese specfcaons: (a) Reduced form specfcaon (b) Srucural form specfcaon The srucural model s a complee sysem of equaon, whch descrbes he srucure of he relaonshp beween economc varables such ha he endogenous varables are expressed as funcon of oher endogenous varables, predeermned varables and sochasc dsurbances. The regressors of srucural equaons correlaed wh sochasc dsurnaces. The srucural specfcaon of a smulaneous equaons model can be gven as: Y d d Y d X u 0 Y Y X u 0 Where Y and Y are he muually dependen varables, and u are he sochasc dsurbance erms X s an exogenous varable and u and 39 P a g e

The reduced-form model (RFM) s ha model n whch he endogenous varables are expressed as an explc funcon of only he exogenous and predeermned varables. In oher words, he RFM expresses an endogenous varable solely n erms of he predeermned varable and he sochasc dsurbances. The RFM ha corresponds o he above srucural model s gven by: Y X e 40 P a g e 0 Y 0 X e Smulaneous-Equaon Bas: Endogeney Smulaneous-equaon bas s an endogeney problem whch enals reverse causaon beween he explanaory and he dependen varables of a model. Thus, smulaney bas s a loop of causaly beween he dependen varables and he regressors of a model. I occurs when a varable on he rgh-hand sde of he causal nferenal model and he varable on he lef-hand sde of he same model nfluence each oher a he same me. In effec, boh he endogenous and he explanaory varables are relaed o each oher. Accordngly, by sequencng he causaly beween he dependen and ndependen varables of a model, endogeney s nduced. Endogeney refers o he correlaon beween he endogenous explanaory varable, ha s, he endogenous regressor and he random error erm. Cause of Smulaney Bas/Endogeney Endogeney can arse as a resul of: (a) Measuremen error (b) Omed varable bas Measuremen Error: Measuremen error n he endogenous explanaory varable causes smulaneous equaon bas. Omed Varable Bas The omsson of key explanaory varables from a regresson model causes smulaneous equaon bas. If he correc model ha explans he varaon n Y, ha s, he model o be esmaed n mean devaon was: y ax ax a3x3 u If he model s correcly specfed, he unbasedness propery would be sasfed n ha he regresson coeffcens would be unbased esmaors of he populaon parameers and as such E( a ) a, E( a ) a and E( a ) a 3 3 If by gnorance or carelessness, he economercan msakenly omed x and x 3 from he model wh he ms-specfed equaon gven as: y b x Then Eb ( ) Proof: Applyng OLS o he ms-specfed equaon, he slope coeffcen of he SRM wll be gven

as: b yx x The normal equaons for he correcly specfed model are gven as: yx a x a x x a x x (4) yx xx xx3 a a a 3 x x x Dvdng eqn (4) by x, 3 3 yx a x x a x a x x 3 3 yx where b, he slope coeffcen of he SRM of Y on X x xx x 3 x n whch X (5) (6) s omed s he slope coeffcen of he SRM of he omed varable x on x and s denoed by b,. e. x b x e xx s he slope coeffcen of he SRM of he omed varable x on x and s denoed by b,. e. x b x e 3 3 3 3 Subsung hese facs no eqn (6), b a a b a b 3 3 Obvouly, he coeffcen of he ncluded varable x n he ms specfed equaon has pcked up he ceoffcen of he omed varable, x ha was correlaed wh x Takng expecaons of eqn (7), E( b ) a ab ab 3 3 Thus, E( b ) a Omed var able bas [ E( b ) a ] a b a b 3 3 (7) In effec, he regresson coeffcen b n he ncorrec model specfcaon dffer from he regresson coeffcen of he correc model specfcaon. Thus, b s a based esmaor of and he bas s equal o b 3b3. Overall, he omsson of a key varable from he regresson model leads o based esmaes of he parameers of he ncluded varables. Consequences of Omed Varable Bas (a) Esmaed coeffcens are posvely based (b) Esmaed varances are based 4 P a g e