Regression with an Evaporating Logarithmic Trend

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1 Regressio with a Evaporatig Logarithmic Tred Peter C. B. Phillips Cowles Foudatio, Yale Uiversity, Uiversity of Aucklad & Uiversity of York ad Yixiao Su Departmet of Ecoomics Yale Uiversity October 5, 00 Abstract Liear regressio o a itercept ad a evaporatig logarithmic tred is show to be asymptotically colliear to the secod order. Cosistecy results for least squares are give, rates of covergece are obtaied ad asymptotic ormality is established for short memory errors. JEL Classificatio: C Key words ad phrases: Asymptotic theory, asymptotic expasios, colliear regressors, liear process, liear regressio, logarithmic factors, logarithmic itegral. Phillips thaks the NSF for research support uder Grat Nos. SBR ad SES Su thaks the Cowles Foudatio for support uder a Cowles Prize Fellowship. The paper was typed by the authors i SW.5. Computig was doe i GAUSS.
2 . Problem Part A The time series X t is geerated by the model X t α + β l t + u t, t,..., where α ad β are ukow parameters whose least squares regressio estimates are deoted by α ad β, respectively. The error u t i is assumed to be iid 0, σ with fiite fourth momet.. Show that α ad β are strogly cosistet for α ad β as.. Fid the asymptotic distributio of α ad β. Part B Suppose that u t i is the liear process u t c i ε t i, i0 with j c j <, j0 ad where ε t is iid 0, σ with fiite fourth momet. Explai how you would modify your derivatios i Part A to allow for such a error process i the regressio model.
3 . Solutio Part A Let z t. The l t l t β β α α z t u t, u t β l t β. We start with β ad fid a asymptotic represetatio of the compoets / l t ad / l t that appear i z t. Usig Euler summatio which justifies 3 below ad partial itegratio which justifies 4 below we obtai a asymptotic series represetatio as follows: l t m dx + O l x 3 k! l k k! l k + m! l m+ dx + O x 4 k m k k! l k k + m! l m+ dx + O. 5 x To show that 5 is a valid asymptotic series, we ca igore the O term ad by a further applicatio of partial itegratio to the secod term we see that the remaider is proportioal to R m+ l m+ x dx Let L α for some α 0, ad write But L l m+ x dx L l m+ x dx l L L It follows from 68 that l m+ x dx + L l m+ l m+ + m + l m+ x dx O α + L l m+ x dx l L l m+ dx. 6 x l m+ dx. 7 x l m+ x dx l L R m+, 8 R m+ l m+ + m + l m+ x dx l m+ + m + l L R m+ + O α 3 L l m+ + m + l m+ x dx + m + l L R m+
4 so that ad thus R m+ O l t m k showig that 5 is a valid asymptotic series. I a similar way we may establish that l t m k l m+, 9 k! l k + O l m+, 0 l dx + O x k! l k + O l m+ is a valid asymptotic series. Combiig 0 ad, we get l t l t l x dx l x dx + O 3 l + l l 4 + O l 5 4 { l + l + } l 3 + O l 4 l 4 + O l 5 a.s., 5 observig that we have to go to the third order terms i expasio 4 to avoid degeeracy. We ca obtai higher order asymptotic expasios by takig the above process to further terms, leadig to the followig explicit expressio to order / l 0 l l + 56 l l l l l 6 + O l. 6 It is immediately apparet from the size of the coefficiets i 6 that very large values of are required before these approximatios ca be expected to work well. Of course, such approximatios are hardly ecessary sice is ameable to direct calculatio or to direct approximatio usig the logarithmic itegrals give i 3 ad above. The latter ca be evaluated from the followig well kow series represetatio of the logarithmic itegral e.g., Gradshtey ad Ryzhik, 967, 8.3., p. 96 liy y 0 l x dx γ + l l y + 4 k l y k, for y >, 7 kk!
5 where γ is Euler s costat. I the preset case we have ad l x dx l dx li li, l x l + l x dx l + [li li ]. l Observe that β β is the same as the error i the OLS estimator of β i the regressio X t z t β + u t, where {u t } is a martigale differece sequece with respect to the atural filtratio F t. The persistet excitatio coditio 5 holds i this regressio ad so β a.s. β ad β is strogly cosistet. Now observe that α α u t β l t β o a.s. + o a.s. O a.s. l o a.s., ad so α a.s. α. Next, tur to the asymptotic distributio of α ad β. We have β β z t u t z t z t u t s l + o, s. Let y t z t u t /s, the y, y 3,..., y are idepedet radom variables with zero meas ad variaces that sum to σ. To apply the Liapouoff cetral limit theorem for y t, we eed to show that E y t 3 0, as. But ad so E y t 3 z t 3 E u t 3 y t s 3 Cost. l 4 3/ o, z t u t s N0, σ. 5
6 Hece Note that l β β z t u t s + o p N0, σ. 8 α α u t β l t β O p / O p / l l O p / l. Thus, the term β β l t domiates the asymptotics of α ad we deduce that l α α l β β + o p d N0, σ. 9 I view of 8 ad 9, we have the followig joit asymptotics: / l α α / l β β N0, Σ with Σ σ. 0 Remarks. The limit distributio 0 is sigular ad the compoets / l α α ad / l β β are perfectly egatively correlated as. Observe that the rate of covergece of α exceeds that of β, by virtue of the fact that the sigal from the itercept is stroger tha the sigal from the evaporatig logarithmic regressor l t.. Result 0 gives first order asymptotics. As is apparet from 6, asymptotic series expasios i the preset model ivolve factors of i cotrast to the l more usual ad, correspodigly, they deliver oly very slow improvemets i the first order asymptotics. The asymptotic variace σ l 4 ad higher order approximatios based o 6 are therefore poor approximatios to the variace of β eve for quite large. To illustrate, for values of the sample size [0, 0 4 ] ad for σ, Fig. provides graphs of the exact variace of β, the asymptotic variace, the twoterm ad threeterm series approximatios to the variace based o the first few terms of 6, ad direct calculatio of the logarithmic itegral represetatios of the variace obtaied from 3 ad 7. Apparetly, oly the latter are adequate for sample sizes i this rage. 3. The theory developed here is part of a geeral theory of regressio o slowly varyig regressors, a subject that has recetly bee studied i Phillips
7 Figure : σ / t z t ad Asymptotic ad Itegral Approximatios 7
8 Part B Usig the PhillipsSolo 99 device, we have u t CLε t Cε t + ε t ε t, where ε t CLε t, CL u0 C u L u, C u su+ c s. The, we ca write It follows that z t u t z t Cε t + z t ε t z t ε t C z t ε t + z t+ z t ε t + z ε z ε. β β C z t ε t z t + z t+ z t ε t + z ε z ε. Note that the first term satisfies the limit theory give i the earlier part. If we prove that the last three terms coverge strogly to zero, the we are doe with the strog cosistecy of β. To obtai the asymptotic distributio, we eed to cosider / β β. We prove that the last three terms, so ormalized, are o p, ad the the asymptotic distributio is determied by the first term. The followig three results are give first. a z ε z t o a.s ad z ε z t / o p. b Proof. These are immediate sice z t ad z t /. z ε z t o a.s ad z ε z t / o p. Proof. Note that z is bouded ad for ay δ > 0 z ε P > δ z E ε δ O, so z P ε > δ < ad the first result follows. The secod follows z t z because P ε > δ O z t / 0. 8
9 c z t+ z t ε t z t o a.s ad z t+ z t ε t z t / o p Proof. Note that { ε t } has fiite fourth momet because ε t 4 : E ε 4 t /4 t j0 t C t j ε j 4 C t j ε j 4 <. j0 Here we have employed the fact that t j0 C t j <, which follows from the assumptio that j0 j c j <. Next, E ε t E ε 4 t + ε s ε t s, s<t Therefore, for ay δ > 0, Sice P O + z t+ z t ε t z t > δ s, s<t E z t+ z t ε t 4 4 δ 4 E [ [ E ε 4 s ] / [ E ε 4 t ] / O. z t+ z t ε t ] 4 δ 4 z t+ z t O 4 δ 4 4 δ 4. E z t+ z t ε t z t+ z t l t + l t l + t [l t][l t + ] < Cost. t <, we have z t+ z t ε t P > δ O l 8 9
10 ad so z t+ z t ε t / o a.s. Similarly, z t+ z t ε t P / > δ E [ z t+ z t ε t ] z t δ E z t+ z t ε t + s>t z t+ z t z s+ z s E ε t ε s z t δ E z t+ z t ε t + s>t + z t z s+ z s E ε t / E ε s / δ Cost. z t+ z t z t δ l Cost. δ 0. The last equality follows as z t+ z t O /t Ol. Hece / z t+ z t ε t op. Combiig results a, b, ad c, we have β β o a.s., ad the / β β / C z t ε t + o p d N0, C σ. With, ad u t / a.s. 0 e.g. Phillips ad Solo, 99, the previous argumets ca ow be repeated to obtai the strog cosistecy ad the asymptotic distributio of α. Specifically / l α α / l β β 3. Refereces N0, Σ with Σ C σ Gradshtey I. S. ad I. M. Ryzhik 965. Tables of Itegrals, Series ad Products. New York: Academic Press. Phillips, P. C. B Regressio with Slowly Varyig Regressors, Cowles Foudatio Discussio Paper #30, Yale Uiversity. Phillips, P. C. B. ad V. Solo 99. Asymptotics for Liear Processes, Aals of Statistics 0,
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