LECTURE 11 LINEAR PROCESSES III: ASYMPTOTIC RESULTS

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1 PRIL 7, 9 where LECTURE LINER PROCESSES III: SYMPTOTIC RESULTS (Phillips ad Solo (99) ad Phillips Lecture Notes o Statioary ad Nostatioary Time Series) I this lecture, we discuss the LLN ad CLT for a liear process f t g geerated as t C (L) " t j () C (L) " t ; L j ; ad f" t g is a sequece of iid radom variables with zero mea ad ite variace. The LLN ad CLT for f t g rely oly o the LLN ad CLT for iid sequeces ad a certai decompositio of the lag polyomial C (L) : The method also works for more geeral sequeces with " t s beig idepedet but ot idetically distributed (iid) ad martigale di erece sequeces (mds). De itio Let ff t g be a icreasig sequece of - elds. The f(u t ; F t )g is a martigale if E (u t jf t ) u t with probability oe. The sequece f(" t ; F t )g is said to be a martigale di erece sequece if E (" t jf t ) with probability oe. For fu t g, the - eld F t is ofte take to be (u t ; u t ; : : :) : Suppose that f(u t ; F t )g is a martigale. The f(u t u t ; F t )g is a mds, sice u t u t u t E (u t jf t ) ; ad, therefore, E (u t u t jf t ) : Note either of the requiremets for f" t g, iid, iid or mds, is stroger tha just a WN. Lemma (SLLN for iid sequeces, White (), Corollary 3.9) Let f" t g be a sequece of idepedet radom variables such that sup t E j" t j + < for some > : The, P t " t P t E" t a:s: : Lemma (SLLN for mds, White (), Exercise 3.77) Let f(" t ; F t )g be a mds such that sup t E j" t j + < for some > : The, P t " t a:s: : Lemma 3 (CLT for iid sequeces, White (), Theorem 5.) Let f" t g be a sequece of idepedet radom variables such that E" t for all t; sup t E j" t j + < for some > ; ad for all su cietly large P t E" t > > : The, P t " t P t E" t d N (; ) : Lemma 4 (CLT for mds, White (), Corollary 5.6) Let f(" t ; F t )g be a mds such that sup t E j" t j + < for some > : Suppose that for all su cietly large P t E" t > > ; ad P t " t P t E" t p : The, P t " t P t E" t d N (; ) : Lemma 5 (CLT for strictly statioary ad ergodic mds) Let f(" t ; F t )g be a strictly statioary ad ergodic mds such that E" t < : The P t " t d N ; E" t : Beveridge ad Nelso (BN) decompositio First, we discuss a algebraic decompositio of a lag polyomial ito log-ru ad trasitory elemets. The decompositio was itroduced by Beveridge ad Nelso (98). Lemma 6 Let C (L) P L j : The (a) C (L) C() ( L) e C (L) ; where e C (L) P el j with e P hj+ c h:

2 (b) If P j j j j < ; the P e < : (c) If P j j jj < ; the P jej < : Proof. For part (a), write L j + j + + : : : + + : : : : j j j j3 jh jh+ L L L h Rearragig the terms, L j ( L) j ( L) L j : : : ( L) L h jh+ : : : ( L) hj+ C () ( L) e C (L) : c h L j

3 For part (b), ec j hj+ hj+ hj+ c h jc h j jc h j h 4 jc h j h 4 jc h j h hj+ hj+ jc h j h jc h j h jc h j h : h hj+ Next, cosider P P hj+ jc hj h : The term jc j appears i the sum oly oce, whe j. The term jc j appears i the sum twice, whe j ; : Hece, jc h j appears whe j ; ; : : : ; h ; total h times. Therefore, jc h j h j j j j hj+ j j j ; ad For part (c), ec j j j j : je j c h hj+ jc h j hj+ j j j: Notice that the assumptios P j j j j < ad P j j jj < are stroger tha iteess of the log-ru variace P j jj <. ccordig to the BN decompositio, if f t g is a liear process, the t C (L) " t C () " t ( L) e C (L) " t C () " t (e" t e" t ) ; () 3

4 where e" t e C (L) " t : Furthermore, e" t has ite variace provided that P j j j <. The rst summad o the right-had side of (), C () " t ; is the log-ru compoet, ad e" t e" t is trasiet. P The similar decompositio exists i the vector case. The trasiet compoet has ite variace if j j kc j k < : C e j C h hj+ kc j k hj+ hj+ j kc j k : kc h k h 4 kc h k h 4 The coditio i part (c) of Theorem 6, becomes P j j kc jk < i the vector case. WLLN Suppose that t C (L) " t : f" t g is a sequece of iid radom variables with E j" t j < ad E" t. C (L) satis es P j j jj < : We will show that uder these coditios t t p : The key is the BN decompositio (). Notice that P t (e" t e" t ) is a so called telescopig sum: so that (e" t e" t ) (~" ~" ) + (~" ~" ) + (~" 3 ~" ) + : : : + (~" ~" ) t e" e" ; t t C () t " t (e" e" ) : Due to P j j jj <, jc ()j < : Hece, by the WLLN for iid sequeces, C () t 4 " t p :

5 Next, ad provided that P j j jj < : Therefore, P je" t j > E je" tj ; E je" t j E e " t j je j E j" t E j" j < ; je j (e" e" ) p : If we assume that E" t < ; the we ca replace P j j jj < with P j j j j < ; sice P j j je" t j > Ee" t ; ad Ee" t < provided that P j j j j < holds. We ca prove similar WLLNs with f" t g beig iid or a mds by usig the correspodig LLNs for iid or mds. For example, the result holds if f" t g is iid, sup t E j" t j + < for some > ; ad P j j jj < : If f(" t ; F t )g is a mds, the result holds with sup t E j" t j + < for some > ; ad P j j j j < : So far we assumed that E t : Oe ca modify the rst assumptio so that t + C (L) " t ; where is the mea of t : I this case, uder the same set of coditios, we have P t t p : For example, R () process with mea is give by ( L) ( t ) " t : If jj < ; the the sample average of t coverges i probability to provided that the correspodig momet restrictios hold. CLT Suppose that t C (L) " t : f" t g is a sequece of iid radom variables with E j" t j < ad E" t. C (L) satis es P j j j j < : C() 6 : The BN decompositio allows us to write t t C () t " t (e" e" ) C () " t + o p () t d C () N ; N ; C () : 5

6 Here, covergece i distributio is by the CLT for iid radom variables. The approach illustrates why i the serially correlated case the asymptotic variace depeds o the log-ru variace of f t g : gai, the approach ca be exteded to the case where f" t g is iid of mds. I the vector case, suppose that f" t g is a sequece of iid k-vectors with E" t ; ad E" t " t ; a ite matrix. Let t C (L) " t ; ad P j kc j k < ; C () 6 : Sice P t " t d N (; ) ; we have that t Covergece of sample variaces t d N ; C () C () : Estimators of coe ciets i the liear regressio model ivolve secod sample momets P t t t. Here, we discuss covergece of sample secod momets whe f t g is a liear process. We assume that f t g is a scalar liear process satisfyig the same assumptios as i the previous sectio. Write t (C (L) " t ) " t j l c l " t j " t l c j" t j + c l " t j " t l l>j c j" t j + +h " t j " t j h (chage of variable l j + h, so that h ; ; : : : ) B (L)" t + where for h ; ; : : :, h B h (L) " t " t h ; h B h (L) b h;j L j +h L j : Thus, B (L) B (L) b ;j L j c jl j : b ;j L j + L j : : : : The BN decompositio of B h (L) is B h (L) B h () ( L) e B h (L) ; (3) 6

7 where eb h (L) e bh;j e bh;j L j ; lj+ lj+ b h;l c l c l+h : The BN decompositio of B h (L) is valid provided that P j j j j < : e b h;j c l c l+h lj+ lj+ lj+ l 4 c l c l+h l l c l l c l l l c l l l c l ; l 4 lj+ lj+ ad P l l c l is ite provided that P l l jc l j is ite: l c l l jc l j l l sup j j j c l+hl c l+hl c l+hl l l jc l j < ; where sup j j j < because P l l jc l j < ad therefore jc l j as l. Thus, we have t B (L)" t + B h (L) " t " t h B ()" t + h l B h () " t " t h ( L) e B (L) " t + c j h eb h (L) " t " t h h " t + u t ( L)ev t ; 7

8 where u t " t B h () " t h ; h ev t e B (L) " t + eb h (L) " t " t h : h We have We will show ext that Let F t (" t ; " t t t c j t t " t + u t (ev ev ) : u t a:s: : ; : : :) : We have that f(u t ; F t )g is a mds. Lemma 7 (White (), Theorem 3.76) Let f(u t ; F t )g be a mds. If for some r, P t E ju tj r t +r < ; the P t u t a:s: : We will verify that fu t g satis es the coditio of the above lemma. Set r : The coditio is satis ed if sup t Eu t < ; sice P t t < : Eu t 4 E t B h () " t h h 4 4 B h () : h Next, B h () h +h h h c j c j h c j j jc j; c j+h c j+h ad, by the same argumet as o page 3 of Lecture, P j j j j < implies that P j j < as well. s before, oe ca show that (ev ev ) p ; provided that P j j j j <. 8

9 Lastly, by the WLLN for iid sequeces, t " t p : Therefore, t p c j t E t : 9