Chapter 11 Autocorrelation
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1 Chaper Aocorrelaio Oe of he basic assmpio i liear regressio model is ha he radom error compoes or disrbaces are ideically ad idepedely disribed So i he model y = Xβ +, i is assmed ha σ if s = E (, s) = if s ie, he correlaio bewee he sccessive disrbaces is zero I his assmpio, whe E (, s) = σ, s= is violaed, ie, he variace of disrbace erm does o remais cosa, he problem of heeroskedasiciy arises Whe E (, s) =, s is violaed, ie, he variace of disrbace erm remais cosa hogh he sccessive disrbace erms are correlaed, he sch problem is ermed as problem of aocorrelaio Whe aocorrelaio is prese, some or all off diagoal elemes i E( ') are ozero Someimes he sdy ad explaaory variables have a aral seqece order over ime, ie, he daa is colleced wih respec o ime Sch daa is ermed as ime series daa The disrbace erms i ime series daa are serially correlaed The aocovariace a lag s is defied as γ = E (, ); s=, ±, ±, s s A zero lag, we have cosa variace, ie, γ E ( ) = = σ The aocorrelaio coefficie a lag s is defied as E( s) γ s ρs = = ; s =, ±, ±, Var( ) Var( ) γ s Assme ρ s ad γ s are symmerical i s, ie, hese coefficies are cosa over ime ad deped oly o legh of lag s The aocorrelaio bewee he sccessive erms ( ad ), ( 3 ad ),,( ad ) gives he aocorrelaio of order oe, ie, ρ Similarly, he aocorrelaio Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr
2 bewee he sccessive erms ( 3 ad ),( 4 ad )( ad ) gives he aocorrelaio of order wo, ie, ρ Sorce of aocorrelaio Some of he possible reasos for he irodcio of aocorrelaio i he daa are as follows: Carryover of effec, aleas i par, is a impora sorce of aocorrelaio For example, he mohly daa o expedire o hosehold is ifleced by he expedire of precedig moh The aocorrelaio is prese i cross-secio daa as well as ime series daa I he cross-secio daa, he eighborig is ed o be similar wih respec o he characerisic der sdy I ime series daa, he ime is he facor ha prodces aocorrelaio Wheever some orderig of samplig is is prese, he aocorrelaio may arise Aoher sorce of aocorrelaio is he effec of deleio of some variables I regressio modelig, i is o possible o iclde all he variables i he model There ca be varios reasos for his, eg, some variable may be qaliaive, someimes direc observaios may o be available o he variable ec The joi effec of sch deleed variables gives rise o aocorrelaio i he daa 3 The misspecificaio of he form of relaioship ca also irodce aocorrelaio i he daa I is assmed ha he form of relaioship bewee sdy ad explaaory variables is liear If here are log or expoeial erms prese i he model so ha he lieariy of he model is qesioable he his also gives rise o aocorrelaio i he daa 4 The differece bewee he observed ad re vales of variable is called measreme error or errors i-variable The presece of measreme errors o he depede variable may also irodce he aocorrelaio i he daa Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr
3 Srcre of disrbace erm: Cosider he siaio where he disrbaces are aocorrelaed, γ γ γ γ γ γ E( ') = γ γ γ ρ ρ ρ ρ γ = ρ ρ ρ ρ ρ ρ = σ ρ ρ Observe ha ow here are ( + k) parameers- β, β,, β, σ, ρ, ρ,, ρ These ( + k) parameers are k o be esimaed o he basis of available observaios Sice he mber of parameers are more ha he mber of observaios, so he siaio is o good from he saisical poi of view I order o hadle he siaio, some special form ad he srcre of he disrbace erm is eeded o be assmed so ha he mber of parameers i he covariace marix of disrbace erm ca be redced The followig srcres are poplar i aocorrelaio: Aoregressive (AR) process Movig average (MA) process 3 Joi aoregressio movig average (ARMA) process Esimaio der he firs order aoregressive process: Cosider a simple liear regressio model y = β + β X +, =,,, Assme ' s follow a firs order aoregressive scheme defied as i = ρ + where ρ <, E( ) =, Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 3
4 σ if s E(, s) = + = if s for all =,,, where ρ is he firs order aocorrelaio bewee ad,,,, = Now = ρ + = ρ( + ) + = = + ρ + ρ + = r= r ρ r E ( ) = E ( ) = E( ) + ρ E( ) + ρ E( ) + 4 = + ρ + ρ + σ s 4 ' ( ) ( are serially idepede) σ = = ρ ( ) σ for all i E Similarly, I geeral, ( ρ ρ ) ( ρ ρ ) E( ) = E E = ρσ { ρ ( ρ )}{ ρ } = E ( ) = ρe + ρ + ( ) = ρσ E( ) = ρσ s s ρ ρ ρ ρ ρ ρ E( ') =Ω= σ ρ ρ ρ 3 ρ ρ ρ 3 Noe ha he disribio erms are o more idepede ad ospherical E ( ') σ I The disrbace are Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 4
5 Coseqeces of aocorrelaed disrbaces: Cosider he model wih firs order aoregressive disrbaces y = X β + k k = ρ +, =,,, wih assmpios E ( ) =, E ( ') = Ω σ if s E( ), E( s) = = + = if s where Ω is a posiive defiie marix The ordiary leas sqares esimaor of β is b= ( X ' X) X ' y = X X X Xβ + ( ' ) '( ) b β = ( X ' X) X ' Eb ( β ) = So OLSE remais biased der aocorrelaed disrbaces The covariace marix of b is Vb ( ) = Eb ( β)( b β)' = ( X ' X) X ' E( ') X( X ' X) = ( X ' X) X ' ΩX( X ' X) σ ( X ' X) The residal vecor is Sice so e = y Xb = Hy = H e' e = y ' Hy = ' H Eee ( ' ) = E ( ' ) E ' X( X' X) X' = σ r( X ' X ) X ' ΩX s ee =, so ' σ E( s ) = r( X ' X ) X ' ΩX, s is a biased esimaor of σ I fac, s has dowward bias Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 5
6 Applicaio of OLS fails i case of aocorrelaio i he daa ad leads o serios coseqeces as overly opimisic view from arrow cofidece ierval R sal -raio ad F raio ess provide misleadig resls predicio may have large variaces Sice disrbaces are ospherical, so geeralized leas sqares esimae of β yields more efficie esimaes ha OLSE The GLSE of β is ˆ β = ( ' Ω ) ' Ω E( ˆ β) = β V ˆ X X X X X y ( β) = σ ( ' Ω ) The GLSE is bes liear biased esimaor of β Drbi Waso es: The Drbi-Waso (D-W) es is sed for esig he hypohesis of lack of firs order aocorrelaio i he disrbace erm The ll hypohesis is H : ρ = Use OLS o esimae β i y = Xβ + ad obai residal vecor e = y Xb = Hy where b= X X X y H = I X X X X ( ' ) ', ( ' ) ' The D-W es saisic is d = = ( e e ) = e e e ee = = = e e e = = = = + Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 6
7 For large, d + r d ( r) where r is he sample aocorrelaio coefficie from residals based o OLSE ad ca be regarded as he regressio coefficie of e o e Here posiive aocorrelaio of egaive aocorrelaio of e s d < e s d > zero aocorrelaio of e s d As < r <, so if < r<, he < d< 4 ad if < r<, he < d< So d lies bewee ad 4 Sice e depeds o X, so for differe daa ses, differe vales of d are obaied So he samplig disribio of d depeds o X Coseqely exac criical vales of d cao be ablaed owig o heir depedece o X Drbi ad Waso herefore obaied wo saisics d ad d sch ha d< d< d ad heir samplig disribios do o deped po X Cosiderig he disribio of d ad d, hey ablaed he criical vales as d L ad d U respecively They prepared he ables of criical vales for 5< < ad k 5 Now ables are available for 6 < < ad k The es procedre is as follows: H : Nare of H Rejec H whe Reai H whe The es is icoclsive whe H : d < dl d > du dl < d < du H : d > d L d < (4 d U ) ( 4 du) < d < (4 dl) H : d < dl du < d < (4 du) dl < d < du d > (4 dl) or (4 du) < d < (4 dl) Vales of d L ad d U are obaied from ables Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 7
8 Limiaios of D-W es If d falls i he icoclsive zoe, he o coclsive iferece ca be draw This zoe becomes fairly larger for low degrees of freedom Oe solio is o rejec H if he es is icoclsive A beer solios is o modify he es as Rejec H whe Accep H whe d< d U d d U This es gives saisfacory solio whe vales of x i s chage slowly, eg, price, expedire ec The D-W es is o applicable whe iercep erm is abse i he model I sch a case, oe ca se aoher criical vales, say d M i place of d L The ables for criical vales d M are available 3 The es is o valid whe lagged depede variables appear as explaaory variables For example, y = β y + β y + + β y + β x + + β x +, r r r+ k k, r = ρ + I sch case, Drbi s h es is sed which is give as follows Drbi s h-es Apply OLS o y = βy + βy + + βry r+ βr+ x + + βkxk, r+, = ρ + ad fid OLSE b of β Le is variace be Var( b ) ad is esimaor is Var ( b ) The he Dbi s h - saisic is h= r Var ( b ) which is asympoically disribed as N (,) ad r = = = ee e Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 8
9 This es is applicable whe is large Whe followig es procedre ca be adoped Irodce a ew variable o = ρ + The Var ( b ) <, he es breaks dow I sch cases, he e = δρ + y Now apply OLS o his model ad es H A : δ = verss HAδ sig -es I H A is acceped he accep H : ρ = If H A : δ = is rejeced, he rejec H : ρ = 4 If H : ρ = is rejeced by D-W es, i does o ecessarily mea he presece of firs order aocorrelaio i he disrbaces I cold happe becase of oher reasos also, eg, disribio may follows higher order AR process some impora variables are omied dyamics of model is misspecified fcioal erm of model is icorrec Esimaio procedres wih aocorrelaed errors whe aocorrelaio coefficie is kow Cosider he esimaio of regressio coefficie der firs order aoregressive disrbaces ad aocorrelaio coefficie is kow The model is y = Xβ +, = ρ + ad assme ha E = E = ψ σ I E = E = σ I ) ( ), ( '), ( ), ( ') The OLSE of β is biased b o, i geeral, efficie ad esimae of σ is biased So we se geeralized leas sqares esimaio procedre ad GLSE of β is where ˆ β = ( ' ψ ) ' ψ X X X y Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 9
10 ψ ρ ρ ρ ρ + ρ + ρ = + ρ ρ ρ To employ his, we proceed as follows: Fid a marix P sch ha PP ' = ψ I his case ρ ρ ρ P = ρ Trasform he variables as y* = Py, X* = PX, * = P Sch rasformaio yields ρ y ρ ρ x ρ x k y ρ y ρ x ρx xk ρx k y* = y3 ρ y, X* ρ x3 ρx x3k ρx = k y ρ y ρ x ρx,, x ρx Noe ha he firs observaio is reaed differely he oher observaios For firs observaio, ( ρ ) ( ) ( ) = ρ ' β + ρ whereas for oher observaios where y x ( ) y = ρy = x ρx )' β + ( ρ ; =,3,, ' x is a row vecor of X Also, ρ ad ( ρ) hese wo errors o be correlaed ad homoscedasic have same properies So we expec Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr
11 If firs colm of X is a vecor of oes, he firs colm of X * is o cosa Is firs eleme is ρ Now employ OLSE wih observaios y * ad X *, he he OLSE of β is β * = ( X *' X*) X *' y*, is covariace marix is Var( ˆ β) = σ ( X *' X *) ad is esimaor is = σ ( X ' ψ X) where ˆ V( β) = ˆ σ ( X ' ψ X) y X ˆ y X ˆ k ( β)' ψ ( β) ˆ σ = Esimaio procedres wih aocorrelaed errors whe aocorrelaio coefficie is kow Several procedre have bee sggesed o esimae he regressio coefficies whe aocorrelaio coefficie is kow The feasible GLSE of β is ˆ ( ' ˆ ) ' ˆ β F = X Ω X X Ω y where ˆ Ω is he Ψ marix wih ρ replaced by is esimaor ˆρ Use of sample correlaio coefficie Mos commo mehod is o se he sample correlaio coefficie r as he aral esimaor of ρ The sample correlaio ca be esimaed sig he residals i place of disrbaces as r = = = ee e where e = y xb, =,,, ad b is OLSE of β ' Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr
12 Two modificaios are sggesed for r which ca be sed i place of r k r* = r is he Theil s esimaor d r ** = for large where d is he Drbi Waso saisic for H : ρ = Drbi procedre: I Drbi procedre, he model is expressed as y ρy = β ( ρ) + β( x ρx ) +, =,3,, y = β ( ρ) + ρ y + β x ρβ x + = β + ρy + βx + β x +, =,3,, (*) * * * * where β = β ( ρ ), β = ρβ Now r regressio sig OLS o model (*) ad esimae r * as he esimaed coefficie of Aoher possibiliy is ha sice ρ (,), so search for a siable ρ which has smaller error sm of sqares 3 Cochra Orc procedre: This procedre ilizes P marix defied while esimaig β whe ρ is kow I has followig seps: (i) Apply OLS o y = β + βx + ad obai residal vecor e y (ii) (iii) Esimae ρ by r = = = ee e Noe ha r is a cosise esimaor of ρ Replace ρ by r is y ρy = β ( ρ) + β( x ρx ) + ad apply OLS o rasformed model y ry = β + β( x rx ) + disrbace erm * * ad obai esimaors of β ad β as ˆ β * ad ˆ β respecively This is Cochra Orc procedre Sice wo sccessive applicaios of OLS are ivolved, so i is also called as wo-sep procedre Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr
13 This applicaio ca be repeaed i he procedre as follows: * (I) P ˆ β ad ˆ β i origial model (II) Calclae he residal sm of sqares (III) Calclae ρ by r = = = ee e ad sbsie i is he model (IV) y ρy = β ( ρ) + β( x ρx ) + ad agai obai he rasformed model Apply OLS o his model ad calclae he regressio coefficies This procedre is repeaed il covergece is achieved, ie, ierae he process ill he wo sccessive esimaes are early same so ha sabiliy of esimaor is achieved This is a ieraive procedre ad is merically coverge procedre Sch esimaes are asympoically efficie ad here is a loss of oe observaio 4 Hildreh-L procedre or Grid-search procedre: The Hilreh-L procedre has followig seps: (i) Apply OLS o ( y ρy ) = β ( ρ) + β( x ρx ) +, =,3,, (ii) (iii) sig differe vales of ρ( ρ ) sch as ρ = ±,, Calclae residal sm of sqares i each case Selec ha vale of ρ for which residal sm of sqares is smalles Sppose we ge ρ = 4 Now choose a fier grid For example, choose ρ sch ha 3 < ρ < 5 ad cosider ρ = 3, 3,, 49 ad pick p ha ρ wih smalles residal sm of sqares Sch ieraio ca be repeaed il a siable vale of ρ correspodig o miimm residal sm of sqares is obaied The seleced fial vale of ρ ca be sed ad for rasformig he model as i he case of Cochara-Orc procedre The esimaors obaied wih his procedre are as efficie as obaied by Cochra-Orc procedre ad here is a loss of oe observaio Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 3
14 5 Prais-Wiso procedre This is also a ieraive procedre based o wo sep rasformaio (i) Esimae ρ by ˆ ρ = = = 3 ee e where e s are residals based o OLSE (ii) Replace ρ by ˆρ is he model as i Cochra-Orc procedre ( ˆ ρ ) ( ) ( ) ( ) = ˆ ρ β + β ˆ ρ ˆ + ρ y x y ˆ ρy = ( ˆ ρβ ) + β( x ˆ ρx ) + ( ˆ ρ ), =,3,, (iii) Use OLS for esimaig he parameers The esimaors obaied wih his procedre are asympoically as efficie as bes liear biased esimaors There is o loss of ay observaio (6) Maximm likelihood procedre Assmig ha y ~ N( Xβσψ, ), he likelihood fcio for β, ρ ad σ is L= y X y X ψ ( πσ ) exp ( )' ( ) β ψ β σ Igorig he cosa ad sig ψ =, he log-likelihood is ρ l L= l L( βσ,, ρ) = lσ + l( ρ) ( y Xβ) ' ψ ( y Xβ) σ The maximm likelihood esimaors of β, ρ ad σ ca be obaied by solvig he ormal eqaios l L l L l L =, =, = β ρ σ There ormal eqaios r o o be oliear i parameers ad ca o be easily solved Oe solio is o - firs derive he maximm likelihood esimaor of σ - Sbsie i back io he likelihood fcio ad obai he likelihood fcio as he fcio of β ad ρ - Maximize his likelihood fcio wih respec o β ad ρ Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 4
15 Ths l L σ σ σ = + ( y Xβ)' ψ ( y Xβ) = is he esimaor of σ Sbsiig σ = β ψ β σ ˆ i place of ˆ ( y X )' ( y X ) σ i he log-likelihood fcio yields L = L β ρ = y Xβ ψ y Xβ + ρ = l {( y Xβ) ' ψ ( y Xβ) } l( ρ ) k + l * l *(, ) l ( ) ' ( ) l( ) = k where k = l ( )' ( ) l y Xβ ψ y X ( ) ρ β Maximizaio of l L * is eqivale o miimizig he fcio ( y Xβ)' ψ ( y Xβ) ( ρ ) Usig opimizaio echiqes of o-liear regressio, his fcio ca be miimized ad esimaes of β ad ρ ca be obaied If is large ad ρ is o oo close o oe, he he erm ( ) / ρ is egligible ad he esimaes of β will be same as obaied by oliear leas sqares esimaio Regressio Aalysis Chaper Aocorrelaio Shalabh, IIT Kapr 5
Chapter 9 Autocorrelation
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