Lecture 6 - Testing Restrictions on the Disturbance Process (References Sections 2.7 and 2.10, Hayashi)

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1 Lecure 6 - esing Resricions on he Disurbance Process (References Secions 2.7 an 2.0, Hayashi) We have eveloe sufficien coniions for he consisency an asymoic normaliy of he OLS esimaor ha allow for coniionally heeroskeasic an serially correlae regressors coniionally heeroskeasic, serially uncorrelae isurbances. In his lecure we will consier esing he isurbances for coniional heerokeasiciy an serial correlaion. I is clear why we woul wan o es for serial correlaion in he isrubances: one of he mainaine assumions of he moel is ha here is no serial correlaion. he case for esing coniional heeroskeasiciy is less comelling since our heory rovies roceures ha o no een on wheher he isurbances are coniionally homoskeasic or coniionally heeroskeasic. Bu, such ess sill migh be useful. Uner assumions A.-A.5, if he isurbances are coniionally homoskeasic we showe las ime ha he inference roceures ha are aroriae in he classical linear regression moel wih sricly exogenous regressors an sherical isurbances can be alie for esing an inerval consrucion an hese roceures will be asymoically vali. If he isurbances are coniionally homskeasic, roceures ha ake his ino accoun may erform beer in finie samles. (Mone Carlo suies migh be helful in his regar.) If he isurbances are coniionally heeroskeasic, he OLS esimaor is no asymoically efficien. If he isurbances are coniionally heeroskeasic he FGLS esimaor will be asymoically efficien. (o aly he FGLS esimaor we nee o be able o moel he coniional heeroskeasiciy an have a consisen esimaor of he arameers of ha moel.) he mos oular nonarameric ess for serial correlaion (Box-Pierce an Lung-Box ess) rely on he assumion ha he isurbances are coniionally homoskeasic.

2 here are wo ways o aroach hese esing roblems: arameric ess an nonarameric ess. A arameric es secifies a moel for he serial correlaion or he coniional heeroskeasiciy an hen ess a se of resricions on he arameers of ha moel. For insance, we migh hink ha if he isurbances are coniionally heeroskeasic i is because he isurbances follow he firs-orer ARCH rocess: v = ε α 0 + αv In his case, coniional homoskeasiciy is he resricion ha α = 0 an so a naural way o es for coniional homoskeasiciy woul be o consruc a es of he null hyohesis, H 0 : α =0. A nonarameric es oes no require you o formulae a secific moel of he coniional heeroskeasiciy or he serial correlaion uner he alernaive hyohesis.

3 esing for Coniional Heeroskeasiciy Mos ess for coniional heeroskeasiciy in ime series regressions are arameric ess ha formulae he ossible heeroskeasiciy in erms of an ARCH-ye moel. We will come back o hese if we have ime. (hese moels are iscusse in more eail in Econ 674.) Secion 2.7 of Hayashi s exbook oulines a nonarameric es of coniional heeroskeasiciy eveloe by Whie (980). Since his es relies on he assumion of i.i. regressors, i is no aricularly useful in ime series seings an, herefore, no of much ineres o us. he iea unerlying he es is ha when regression isurbances are coniionally homoskeasic, here are (a leas) wo consisen esimaors of he marix S ha aears 2 ' in he asymoic variance formula for he OLS esimaor, where S = E ε x x ): ( an 2 S = ε x x ' s 2 S xx = s 2 x ' x, where s 2 = SSR/. So, if he isurbances are coniionally homoskeasic, he ifference beween hese wo esimaors shoul be geing small as samle size increases. In fac, he ifference will converge in robabiliy o zero if he isurbances are homoskeasic. Whie consruce a saisic base on his ifference ha converges in isribuion o a Χ 2 ranom variable if he isurbances are coniionally homoskeasic.

4 esing for Serial Correlaion Parameric an nonarameric ess are wiely use in ime series o es for serial correlaion in he regression isurbances. he nex maor secion of he course will be concerne wih ime series moels an one of he alicaions of ime series will be o arameerize serial correlaion in regression moels: ha will allow o es for serial correlaion an, if i aears o be resen, aly he FGLS esimaor an es roceures. Our focus oay will be on nonarameric ess. he avanage of hese ess is ha hey o no require us o formulae a secific moel of serial correlaion. he isavanages are ha ) hey will rely on he assumion of coniionally homoskeasic isurbances an 2) if we fin evience of serial correlaion, hen wha o we o? Noe: ess for serial correlaion in he resence of coniional heeroskeasiciy o exis. Le s begin by assuming ha {z } is a saionary an ergoic rocess wih finie variance (i.e., i is covariance saionary,oo). hen he -h auocovariance of he rocess is: cov( z, z µ = γ for all an, where µ = E z ) ) = E[( z z )( z µ z )] z ( he samle -h auocovariance is efine by: γ = ( z z )( z = + z ), where z is he samle mean of z,,z Corresoning o hese are he -h auocorrelaions an he samle -h auocorrelaions: an ρ ρ = = corr ( z, z ) = cov( z, z ) / var( z ) = γ / γ γ / γ 0 0 We have alreay claime as a fac ha uner he assumion ha z is a saionary an ergoic rocess, he samle auocovariances an auocorrelaions are consisen esimaors of he oulaion auocovariances an auocorrelaions.so, if z is a serially uncorrelae rocess, he samle auocorrelaions will converge in robabiliy o zero for all > 0. (Noe: Since he auocovariance an auocorrelaion funcions are symmeric aroun 0, we only have o consier > 0.)

5 Assume ha {z µ z } is a coniionally homoskeasic m..s. ha is: where z = µ z + ε E(ε ε -, ) = 0 an E(ε 2 ε -, ) = σ 2 for all. hen, for any osiive ineger, ρ N(0, I ), ρ = [ ρ... ρ ]' an, since uncorrelae normally isribue ranom variables are ineenen ranom variables, { nρ } is asymoically an i.i.. N(0,) sequence. o es for firs-orer serial correlaion: τ = ρ (/ ) N(0,) o es for -h orer serial correlaion: τ = ρ (/ ) N(0,) o es H 0 : ρ =0,ρ 2 =0,,ρ =0 2 Q = ρ Χ 2 ( ) which is he Box-Pierce Q saisic. (his resul follows from he fac ha he asymoically i.i.. N(0,).) nρ s are he Lung-Box Q saisic, Q LB, Q LB = ρ ( + 2) 2 is asymoically equivalen o he B-P Q saisic (i.e., Q-Q LB 0 an Q Χ 2 ( ) ), bu seems o work a lile beer in racice. LB

6 hese ess can be alie o any saionary ime series o es for serial correlaion. We are inerese in alying hem o es for serial correlaion in regression isurbances. hese isurbances are unobservable! I woul seem naural o consier wheher we can aly hese ess using he resiuals ( ε ' s) from he fie regression in lace of he unobservable regression isurbances (ε s). Le an le ~ ρ ~ / ~ = γ γ 0, where ~ γ = ε ε, for = 0,,2, + / = γ γ 0, γ = ε ε + ρ for = 0,,2, Recall, we have alreay moifie our assumions A.-A.5 o resric he isurbances o be coniionally homoskeasic. If, in aiion, he regressors are sricly exogenous, ρ N(0, I ), ρ = [ ρ... ρ ]' In oher wors, he limiing isribuion of he an Q saisics we formulae above o no een on wheher we consruc he samle auocorrelaions using he ρ ~ s or he ρ s.

7 Suose he regressors are reeermine bu no sricly exogenous. In aricular, suose: an E(ε ε -,ε -2,,x,x -, ) =0 for all. E(ε 2 ε -,ε -2,,x,x -, ) =σ 2 In his case, he limiing isribuion of ρ N(0, I Φ), ρ = [ ρ... ρ ]' where Φ is a x marix whose i-h elemen is: Φ i = E(x ε -i ) E(x x ) - E(x ε - )/σ 2 his leas o he moifie Box-Pierce Q saisic Q MBP = ρ'( Φ ) I ρ where Φ is he consisen esimaor of Φ, given by equaion (2.0.9) in Hayashi. Uner he null hyohesis of no serial correlaion, Q MBP Χ 2 ()