Review from last time Time Series Analysis, Fall 2007 Professor Anna Mikusheva Paul Schrimpf, scribe October 23, 2007.

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1 Review fro last tie 4384 ie Series Analsis, Fall 007 Professor Anna Mikusheva Paul Schrif, scribe October 3, 007 Lecture 3 Unit Roots Review fro last tie Let t be a rando walk t = ρ t + ɛ t, ρ = where ɛ t is a artingale difference sequence (E[ɛ t I t] = 0, with E[ɛ It ] σ as and E ɛ 4 t t < K < hen {ɛ t } satisf a functional central liit theore, ξ (τ = [τ ] ɛt t= σ ( e showed last tie that: σ (td (t σ (t 0 σ (t t ɛ t 3/ t t 0 0 and our OLS estiator and t-statistic have non-standard distributions: d ( ρˆ ρ d t his is quite different fro the case with ρ < If x t = ρx t + ɛ t, ρ < then, xt ɛ t x 3/ t x t σ N(0, 4 ρ Ex t = 0 Ex t = σ ρ and the OLS estiate and t-stat converge to noral distributions: (ρˆ ρ N(0, ( ρ t N(0, Cite as: Anna Mikusheva, course aterials for 4384 ie Series Analsis, Fall 007 MI OenCourseare (htt://ocwitedu, Massachusetts Institute of echnolog Downloaded on [DD Month YYYY]

2 Adding a constant Adding a constant Suose we add a constant to our OLS regression of t his is equivalent to running OLS on deeaned t Let t = t ȳ hen, ρˆ = t ɛ t ( t Consider the nuerator: t ɛ t = ( t ȳɛ t = t ɛ t ȳɛ ȳɛ = 3/ t ɛ t e know that t σ, and 3/ we know that t ɛ t σ d Cobining this, we have: ( t ɛ t σ (sd (s ( ɛt σ ( Also, fro before (t e can think of this as the integral of a deeaned Brownian otion, ( (sd (s ( (t = (s (t d (s = (sd (s he liiting distributions of all statistics (ρˆ, t, etc would change In ost cases, the change is onl in relacing b one also can include a linear trend in the regression, then the liiting distribution would deend on a detrended Brownian otion Liiting Distribution Let s R return to the case without a constant he liiting distribution of the OLS esitate is (ρˆ R d his distribution is skewed and shifted to the left If we include a constant, the distribution is even ore shifted If we also include a trend, the distribution shifts et ore For exale, the 5%-tile without a constant is -05, with a constant is -69, and with a trend is -5 he 975%-tiles are 6, 04, and -8, resectivel hus, estiates of ρ have bias of order his bias is quite large in sall sales he distribution of the t-statistic is also shifted to the left and skewed, but less so than the distribution of the OLS estiate he 5%-tiles are -3, -3, -366 and of the 975%-tiles are 6, 03, and -066 for no constant, with a constant, and with a trend, resectivel Allowing Auto-correlation So far, we have assued that ɛ t are not correlated his assution is too strong for eirical work Let, t = t + v t where v t is a zero-ean stationar rocess with an MA reresentation, v t = c j ɛ j j=0 Cite as: Anna Mikusheva, course aterials for 4384 ie Series Analsis, Fall 007 MI OenCourseare (htt://ocwitedu, Massachusetts Institute of echnolog Downloaded on [DD Month YYYY]

3 Augented Dicke Fuller 3 where ɛ j is white noise Now we need to look at the liiting distributions of each of the ters we looked at before ( v t, t v t, t, etc v t is a ter we alread encountered in HAC estiation: where ω is the long-ter variance, vt N(0, ω t= ω = γj = σ c( ω j= For linearl deendent rocess, v t, we have the following central liit theore: ξ (τ = [ τ] vt t= ω ( his can be roven b using the Beveridge-Nelson decoosition All the other ters converge to things siilar to the uncorrelated case, excet with σ relaced b ω For exale, An excetion is: = ξ (t/ ω (t/ 3/ t ω (sds t v t = his leads to an extra constant in the distribution of ρˆ: t v (ω ( σ v ω ω d + σ v (ρˆ d ω σ v ω so it is iossible to just look u critical values in a table Phillis & Perron (988 suggested corrected statistics: ωˆ σˆv d (ρˆ + where ωˆ t is an estiate of the long-run variance, sa Newe-est, and σˆv is the variance of the residuals Augented Dicke Fuller Another aroach is the augented Dicke-Fuller test Suose, t = a j t j + ɛ t j= Cite as: Anna Mikusheva, course aterials for 4384 ie Series Analsis, Fall 007 MI OenCourseare (htt://ocwitedu, Massachusetts Institute of echnolog Downloaded on [DD Month YYYY]

4 Augented Dicke Fuller 4 where z = is a root, Suose we factor a(l as a j z j =0 j= with b(l = j j= β jl e can write the odel as his suggests estiating: = a j a j L j = ( Lb(L j t = β j t + ɛ t i= t = ρ t + β j t + ɛ t j i= and testing whether ρ = to see if we have a unit root his is another wa of allowing auto-correlation because we can write the odel as: a(l t =ɛ t b(l t =ɛ t t = t + v t t =b(l ɛ t where v t = b(l ɛ t is auto-correlated he nice thing about the augeented Dicke-Fuller test is that the coefficients on t j each converge to a noral distribution at rate, and the coefficient on t converges to a non-standard distribution without nuisance araeters o see this, let x t = [ t, t,, t + ] and let θ = [ρ, β,, β ] Consider: θˆ θ = (X X (X ɛ e need to noralize this so that it converges Our noralizing atrix will be Q = 0 Multiling b Q gives: ˆ Q(θ θ =(Q (X X Q (Q X ɛ he denoinator is: Q (X X Q = t t 3/ t t 3/ t X X Cite as: Anna Mikusheva, course aterials for 4384 ie Series Analsis, Fall 007 MI OenCourseare (htt://ocwitedu, Massachusetts Institute of echnolog Downloaded on [DD Month YYYY]

5 Other ests 5 Note that t 3/ t j 0 Also, we know that t ω, so ω 0 Q (X X Q 0 EX X Siilarl, ( Q X ɛ = ( t ɛ t σω d t j ɛ t N(0, E[X X]σ hus, or ωˆ (ρˆ σˆ σ d (ρˆ ω (ρˆ = d βˆ j Other ests Sargan-Bhargrava (983 look at: For a non-stationar rocess, the variance of t grows with tie, so we could when ɛ t is an ds, this converges to a distribution, converges to 0 in robabilit t t P t P t ( For a stationar rocess, this Range (Mandelbrot and coauthors You could also look at the range of t, this should blow u for non-stationar rocesses for a rando walk, we have: est Coarison (ax t in t su (λ t λ inf (λ λ he Sargan-Bhargrava (983 test has otial astotic ower against local alternatives, but has bad size distortions in sall sales, eseciall if errors are negativel autocorrelated he augented Dicke-Fuller test with the BIC used to choose lag-length has overall good size in finite sales Cite as: Anna Mikusheva, course aterials for 4384 ie Series Analsis, Fall 007 MI OenCourseare (htt://ocwitedu, Massachusetts Institute of echnolog Downloaded on [DD Month YYYY]