The cetral limit theorem for Studet s distributio Problem 03.6.1 Karim M. Abadir ad Ja R. Magus Ecoometric Theory, 19, 1195 (003)
Z Ecoometric Theory, 19, 003, 1195 1198+ Prited i the Uited States of America+ DOI: 10+10170S06646660319613 PROBLEMS AND SOLUTIONS PROBLEMS 03.6.1. The Cetral Limit Theorem for Studet s Distributio Karim Abadir Uiversity of York, UK Ja Magus Tilburg Uiversity, The Netherlads Let x 1,+++,x be a radom sample from Studet s t~! distributio, where R + Ivestigate whether z : ( i 1 x i 0l is asymptotically N~0,1! for a suitable choice of l + 03.6.. Ubiasedess of the OLS Estimator with Radom Regressors Michael Jasso UC Berkeley Cosider the liear regressio model y Xb u, where X is a k matrix of radom regressors, u is a -vector of error terms, ad b is a k-vector of parameters+ Suppose X has full colum rak with probability oe+ It is a stadard textbook claim that the ordiary least squares ~OLS! estimator b ~X ' X! 1 X ' y of b is ubiased if E~u6X! a+s+ 0, where a+s+ sigifies almost sure equality+ Specifically, it is claimed that ubiasedess follows from the law of iterated expectatios ad the relatio E~ b6x Z! a+s+ b ~X ' X! 1 X ' E~u6X!+ As it turs out, this argumet is flawed+ ~a! Show by example that E~u6X! a+s+ 0 does ot imply existece of E~ b!+ Z ~b! Provide stroger coditios uder which E~ b! Z exists ~ad equals b!+ SOLUTIONS 0.6.1. Oblique Projectors 1 Solutio Götz Trekler ~the poser of the problem! Uiversity of Dortmud, Germay It is well kow that a oblique projector P ca be writte as U P I r K 0 0 U*, 003 Cambridge Uiversity Press 066-4666003 $1+00 1195
Solutio Ecoometric Theory, 0, 161 163 (004)
Ecoometric Theory, 0, 004, 161 164+ Prited i the Uited States of America+ DOI: 10+10170S06646660406089 PROBLEMS AND SOLUTIONS SOLUTIONS 03.6.1 The Cetral Limit Theorem for Studet s Distributio Solutio Karim Abadir Uiversity of York, UK Ja Magus ~the poser of the problem! Tilburg Uiversity, The Netherlads Cosider the Lideberg Feller cetral limit theorem ~CLT!, which we state as follows+ Let $x % be a sequece of idepedet radom variables with meas $m % ad ozero variaces $s % ~both existig!, ad c+d+f+s $F %+ Defie l 0byl ( i 1 s i +The, Lideberg s coditio e lim u m i r` (i 1 df i ~u! 0, e 0, 6u m i 6 l l is equivalet to z : ( i 1 ~x i m i! a N~0,1! ad lim max l Pr 6x i m i e 6 0, r` 1 i l where the latter limit is kow as the uiform asymptotic egligibility ~u+a++! coditio+ Oe ca usually iterpret l as the variace of the umerator of z + We shall see, however, that there are cases where asymptotic ormality holds i spite of $x % havig ifiite variaces+ Let $x % be a radom sample from Studet s t~!+ For, o l exists that ca lead to z : ( i 1 x i 0l a N~0,1!+ This is because the tails of the desity of t~! decay at a rate of u 1 ad the stable limit theorem tells us that a oormal stable law arises if the tails of the p+d+f+ of x i decay at a rate of u a where a 3; e+g+, see Loève ~1977, 5! or Hoffma-Jørgese ~1994, 5+5!+ For example, for 1, the average of stadard Cauchy variates is stadard Cauchy too, so that there exists o l achievig asymptotic ormality of z + For, both the mea ad the variace exist, ad the Lideberg Feller CLT applies, with l var~x i!+ The iterestig part is, where we will show that asymptotic ormality of z holds, i spite of var~x i! beig ifiite, ad we will derive the appropriate l + We will require the additioal assump- 004 Cambridge Uiversity Press 066-4666004 $1+00 161
16 PROBLEMS AND SOLUTIONS tio that l r ` as r `+ I the stadard CLT, this assumptio was uecessary, as it followed from l var~x i!+ We will see subsequetly that l ca be iterpreted i terms of trucated variaces for + To prove the asymptotic ormality of z, we eed to show that the characteristic fuctio w~t! : E~exp~itx i!! satisfies w lim log t r` l t (1) for some choice of l, with l r ` as r `+ Because the sequece $x % is i+i+d+, the uiform asymptotic egligibility coditio lim r` max Pr 6x e i 6 1 i l 0 is satisfied for all e 0, thus implyig lim r` w t l 1 0+ This allows us to take the leadig term of the logarithmic expasio of the left-had side of ~1! as w lim log t lim r` l w t r` l 1 t lim u r` 6u6 l e l df~u!, l r `, where the liear term i t drops out because the sequece $x % is cetered aroud zero+ Asymptotic stadard-ormality obtais if we ca fid the appropriate l that makes the latter limit equal to 1 for all e 0+ Notice that this limit is the complemet of Lideberg s coditio, where ( i 1 is replaced by because $x % is a i+i+d+ sequece+ From Studet s t~! desity, cm u cm c du log~m1 c c! M8 1 u M1 c 30 sih 1 ~c! c M1 c
teds to ifiity as c r `+ We eed to solve 1 lim r` l 6u6 l e sih 1 ~l u e M! df~u! lim r` l where we have dropped c M1 c r that is domiated by sih 1 ~c! r `+ By usig the logarithmic represetatio of the latter ad simplifyig, log~l! 1 lim r` l PROBLEMS AND SOLUTIONS 163 is solved by l M log~! or ay other fuctio that is asymptotically equivalet to it ~such as M log~! M!+ Therefore, z 1 ( x i a M log~! i 1 N~0,1!+ REFERENCES Hoffma-Jørgese, J+ ~1994! Probability with a View toward Statistics, vol+ I+ Chapma ad Hall+ Loève, M+ ~1977! Probability Theory I, 4th ed+ Spriger-Verlag+ 03.6.. Ubiasedess of the OLS Estimator with Radom Regressors Solutio Michael Jasso ~the poser of the problem! Uiversity of Califoria, Berkeley ~a! Suppose 1 ad let X ad u be idepedet stadard ormal variates+ The X is ozero with probability oe ad E~u6X! a+s+ 0, but E6b6 Z ` because the distributio of bz b u0x is Cauchy+ ~b! The matrix X has full colum rak with probability oe if ad oly if Pr@l mi ~X ' X! 0# 1, (1) where l mi ~{! deotes the miimal eigevalue of the argumet+ E~ b! Z exists if ~ad oly if! E6c ' ~ bz b!6 ` for ay k-vector c with c ' c 1+ I the sequel, let c be a arbitrary k-vector with uit legth+ Now, 6c ' ~ bz b!6 6c ' ~X ' X! 1 X ' u6 Mc ' ~X ' X! 1 cmu ' u Ml 1 mi ~X ' X!Mu ' u, where the first iequality uses the Cauchy Schwarz iequality ad the secod iequality uses Magus ad Neudecker ~1988!, Theorem 11+4+ If X ad u are idepedet, E~u6X! a+s+ 0, ad E @l 10 mi ~X ' X!# `, (3)