Asymptotc Propertes of the Jarque-Bera est for Normalty n General Autoregressons wth a Determnstc erm Carlos Caceres Nuffeld College, Unversty of Oxford May 2006 Abstract he am of ths paper s to analyse the asymptotc propertes of the Jarque-Bera test for normalty from a tme seres perspectve. For ths purpose, we study the characterstcs of ths test when appled to a frst order autoregressve model wth and wthout a constant term, and wth ndependent, dentcally and normally dstrbuted errors. It wll be proved that, n such cases, the Jarque-Bera test for normalty s parameter ndependent. hs means that ths normalty test can be appled to any frst order autoregressve process wth and wthout ntercept, regardless of whether the latter contans a statonary, a unt or an explosve root.
Contents Introducton. he Jarque-Bera test for Normalty........................2 Assumptons.................................... 4 2 Autoregressve process wth no determnstc terms 4 3 Autoregressve process wth determnstc terms 9 4 Concluson 24 A Appendx 24 References 25
Introducton An mportant ms-specfcaton test s the normalty test due to Jarque and Bera 980). Many econometrc models and results are based on the fact that some varables follow a normal dstrbuton. In partcular the maxmum lkelhood estmator concdes wth the OLS estmator when normalty s assumed. hus the mportance of havng a theoretcal test for normalty that can be used n tme seres analyss. In ths paper, the asymptotc propertes of the Jarque-Bera test for normalty are analysed from a tme seres perspectve. Some studes n the past have analysed the valdty of the Jarque-Bera test for normalty when dealng wth statonary autoregressve processes. Notably the contrbuton of Lutkepohl and Shneder 989). Other authors have expanded ths ssue to accommodate for more general models. Klan and Demroglu 2000) proved the valdty of the Jarque-Bera normalty test for Vector-Error Correcton VEC) models and for unrestrcted Vector Autoregresson VAR) models wth possbly ntegrated or contegrated varables. Yet, theoretcal results regardng the valdty of the Jarque-Bera test for normalty n the general case are not avalable. Here the general case refers to all three, statonary, margnally stable and explosve cases together. It s found n ths paper that the Jarque-Bera test for normalty s ndeed vald n the general case, when appled to a frst order autoregressve process. In other words, ths normalty test s ndependent of the values of the parameters of nterest, and s therefore applcable regardless of whether we have a statonary, a unt or an explosve root. hs last statement s also true f a constant term s ncluded n the autoregressve model. In ths sense, the Jarque-Bera test for normalty shares the parameter ndependence property.e. ts valdty s ndependent of the values of the characterstc roots) found n the lkelhood based methods used for order determnaton n general vector autoregressons, see Nelsen 200). However, note that ths s not a general property of all ms-specfcaton tests. In fact, Whte s test for heteroskedastcty s only vald when the characterstc roots are non-explosve, see Caceres and Nelsen 2006b). Smlarly, ths drawback s shared by correlograms based on the Yule-Walls equatons, but not by correlograms based on correlatons, see Nelsen 2006).. he Jarque-Bera test for Normalty As we sad earler, our workhorse n ths paper s a frst order autoregressve model or AR) model). hus, lets start by defnng the latter n a more formal way. Let X 0, X,..., X ) be a one-dmensonal tme seres satsfyng the frst order autoregressve equaton: X t = αx t ε t for t =,..., ) where ε t ) s a sequence of..d. normally dstrbuted random shocks and α R represents the parameter of nterest.e. the parameter space). Note that equaton ) represents an autoregressve model wthout determnstc terms.
And provded that ths s the true model the resduals are gven by: wth ˆα α) defned as: ˆε t = ε t ˆα α)x t 2) ˆα α) = t= X t ε t t= X2 t 3) Smlarly, we can defned a second model n whch X 0, X,..., X ) s a one-dmensonal tme seres satsfyng the frst order autoregressve equaton: X t = αx t µδ t ε t for t =,..., 4) where ε t ) s a sequence of..d. normally dstrbuted random shocks, δ t s a determnstc term and α, µ R represent the parameters of nterest. he above s an autoregressve model wth a determnstc term. he latter represents the second type of models that we are gong to use when analysng the asymptotc propertes of the Jarque-Bera test for normalty. In ths paper we consder the partcular example where δ t =. hen, n that partcular case the model presented n equaton 4) becomes: X t = αx t µ ε t for t =,..., 5) Equaton 5) s thus satsfed by an autoregressve process wth an ntercept. provded that ths s the true model the resduals are gven by: Agan, ˆε t = ε t ˆα α)x t ˆµ µ) 6) wth the vector ˆα α, ˆµ µ) defned as: ˆα α ˆµ µ = t= X2 t t= X t t= X t t= X t ε t t= ε t 7) he Jarque-Bera normalty test s based on the dea of analyzng the asymptotc propertes of the followng two statstcs: And, ˆK 3 = [ t= ˆε t ˆε) 3 3/2 = S 3 t= ˆε t ˆε) 2 S 2 3/2 8) ˆK 4 = [ t= ˆε t ˆε) 4 2 3 = S 2 4 3S 2 t= ˆε t ˆε) 2 2 9) S 2 2
Where: S 2 = t= ˆε t ˆε) 2 = t= ˆε2 t S 3 = t= ˆε t ˆε) 3 = t= ˆε3 t 3 S 4 = t= ˆε t ˆε) 4 ) 2 t= ˆε t t= ˆε2 t ) ) t= ˆε t 2 ) 3 t= ˆε t And, S 4 3S 2 2 = t= ˆε4 t 4 6 t= ˆε t t= ˆε t ) 4 3 ) t= ˆε2 t ) t= ˆε3 t ) 2 2 ) ) 2 t= ˆε2 t t= ˆε t hen, under the null.e. the ε t s are normally dstrbuted) we have the followng results: ˆK 2 3 6 d χ 2 ) ˆK 2 4 24 d χ 2 ) 0) ) ˆK2 3 ˆK 4 2 d χ 2 6 24 2) hs means that under the null hypothess of normalty, the above three statstcs should have a ch-square as lmtng dstrbuton. Now, as mentoned prevously, t wll be shown n ths paper that Jarque and Bera s method for testng for normalty works well n the general case. Agan, the general case denotes all three statonary, unt root and explosve cases together. Addtonally, ths result also holds regardless of whether we nclude an ntercept or not n the autoregressve model. hus ths paper s organsed n the followng way: the valdty of the Jarque-Bera test for Normalty when dealng wth a frst order autoregressve process wthout determnstc terms, whch satsfes equaton ), wll be presented n secton 2. hs s gong to be shown explctly n heorem 2.. hen, heorem 3. n secton 3 wll reveal that the Jarque-Bera test for normalty s stll vald when appled to a general frst order autoregressve process wth a determnstc term n ths case an ntercept). Followng the outlne just descrbed, t s worth emphaszng here that for each of the two cases studed n ths paper.e. autoregressve process wth and wthout determnstc terms) a dfferent dstrbuton theory s requred dependng on the values of the parameters of nterest α and µ. Nevertheless, the lmtng dstrbutons of the Jarque-Bera statstcs n equaton 0) are ndependent of the values of these parameters. he statonary case s based on the applcaton of the Law of Large Numbers LLN) and the Central lmt heorem CL) to statonary ergodc mxng) autoregressve processes and ergodc martngale dfferences 3
respectvely. he results for an autoregressve process wth a unt root are consequences of the Functonal Central Lmt heorem FCL), the Contnuous Mappng heorem and a result provded by Caceres and Nelsen 2006a). Last but not least, the results for an autoregressve process wth an explosve root are based on a dfferent theory from the prevous two. As ponted out by Nelsen 2005), ths theory s based on the use of strong consstency arguments rather than weak consstency and weak convergence arguments used n non-explosve tme seres. Furthermore, t wll be ponted out that the order of magntude of the process X t ) t N vares accordng to the values of the parameters α and µ. Before proceedng to the presentaton of the mentoned theorems, t s mportant at ths pont to ntroduce the man assumptons that are used for the rest of the paper, unless stated otherwse..2 Assumptons hroughout the entre paper, the followng two assumptons are used: Assumpton.. ε t ) t N s a sequence of..d. normally dstrbuted random varables wth mean zero and varance one. Assumpton.2. X 0 = 0 Note that assumpton. could be modfed so that varance equal to one could be replaced by constant varance σ 2. hs s due to the scale nvarance property of ths partcular problem and smplfes the presentaton thereafter. In other words, the dstrbutons of the sgnfcant statstcs n equaton 0) are unaffected when the varance σ 2 s scaled to one. Addtonally, t s worth mentonng that most of the results presented n ths paper stll hold f we further relax assumpton.. Notably f..d.-ness s replaced by lettng ε t ) t N be a martngale dfference sequence wth respect to an ncreasng sequence of σ-felds F t ) t N where F t denotes the so called natural fltraton). hs s due to the fact that as mentoned earler) most of the theorems used here: Law of Large Numbers, CL, Functonal Central Lmt heorem FCL), etc, are applcable to martngale dfferences. Assumpton.2 s used for smplcty only. he latter could be effectvely replaced by the assumpton that X 0 s fxed and the man results would be unchanged. However the presentaton would become more cumbersome due to the extra term. It s therefore convenent to use assumpton.2 henceforth. 2 Autoregressve process wth no determnstc terms he characterstcs of ths test for normalty have been consdered prevously by a number of researchers n both, the statonary case and the unt root case c.f. Lutkepohl and Shneder 989), and Klan and Demroglu 2000), respectvely). However, no results have been prevously gven n the case of a general autoregressve process,.e. an autorgressve process that can have statonary, margnally stable and explosve roots. 4
In ths secton, the asymptotc propertes of the Jarque-Bera test when appled to a general frst order autoregresve process wthout determnstc terms are analysed. herefore the fndngs presented n ths secton are ground-breakng n the sense that t allows for the possblty of havng an explosve root n the model. he man result of ths secton s that of the valdty of the Jarque-Bera normalty test when dealng wth a frst order autoregressve process wthout determnstc terms, whch satsfes equaton ). hs pont s presented n the followng theorem: heorem 2.. Let ˆK3 and ˆK 4 be defned as n equatons 8) and 9), where ˆε t ) t [[, s gven by equaton 2). If assumptons. and.2 are satsfed, then ˆK 3 2 d χ 2 ), ˆK 4 2 d ˆK2 χ 2 ) and 3 6 24 6 ˆK ) 4 2 d χ 2 2) ) 24 In order to establsh the proof for the above theorem, the followng two lemmas are requred. Lemma 2.. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let X t ) t N be the autoregressve process satsfyng equaton ) and X 0 = 0. hen, for any fxed postve ntegers k and p, we have t= X t = O p ) f α < Xt t= = Op ) f α = α k t= X t = O p ) f α > And, t= X t ε p t m p ) = O p ) f α < t= Xt εt p m p ) = O p ) f α = α k t= X t ε t p m p ) = O p ) f α > where: m p = E [ε t p. Addtonally, wth ˆα α) gven by equaton 3), we have ˆα α) = Op ) f α < ˆα α) = O p ) f α = α ˆα α) = O p ) f α > 5
Proof of lemma 2.. : In the case where α < we obtan the correspondng results for Lemma 2. above from Lemmas 2. and 2.2 n Caceres and Nelsen 2006b). Also, the case α = s dealt wth by those authors n ther Lemmas 3.2, 3.3 and 3.4. Fnally, Caceres and Nelsen 2006b) proved the above results for α > n ther Lemmas 4. and 4.2. Lemma 2.2. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let the resduals ˆε t ) t [[, be defned as n equaton 2). hen, for α and for any postve nteger k, we have And, ˆε k t = t= ˆε k t = t= ) ε k t O p t= t= ε k t O p ) Also, for α > and for any postve nteger k, we have ˆε k t = t= ) ε k t O p t= f k s even 2) f k s odd 3) 4) Proof of lemma 2.2. : Usng the bnomal rule, we have that for any postve nteger k, ˆε k t = t= = = = [ε t ˆα α)x t k t= k ) ˆα α) X t ε t k t= =0 k ε k t ) ˆα α) t= = t= k [ k ε k t ) ) ˆα α) t= = t= k [ k ) ) m k ˆα α) = t= X t X t ε k t X t ˆα α )k ε k t m k ) t= X k t usng the conventon m 0 =. 6
For α <, we can rewrte the above equalty as ˆε k t = k ) ) [ k [ ε k t ) ˆα α) t= t= = t= k ) ) [ k [ ) m k ˆα α) Xt = t= X t ε k t m k ) Smlarly, for α =, we can wrte ths as ˆε k t = t= k ) ) [ k ε k t ) [ ˆα α) t= k = = ) k ) m k ) [ [ ˆα α) t= t= ) Xt Xt ) ε k t m k ) hus, usng the results from Lemma 2. correspondng to the cases α < and α =, and combned wth the fact that m k = E ε t k ) = 0 for k even, we recover the results for α gven by equatons 2) and 3) of Lemma 2.2. Fnally, for α >, we have that ˆε k t = t= k ) [α ε k t ) ˆα α) [ α t= = k ) [α ) m k ˆα α) [ α = t= t= X t X t ε k t m k ) Hence, from the results n Lemma 2. for α >, we obtan the correspondng result for Lemma 2.2. Corollary 2.. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let the resduals ˆε t ) t [[, be defned as n equaton 2). hen, for α and for any postve nteger k, we have ˆε k t m k ) = t= ) ε k t m k ) O p t= f k s even 5) And, t= ˆε k t = ε k k t m k ˆα α) t= t= X t O p ) f k s odd 6) 7
Also, for α > and for any postve nteger k, we have ˆε k t m k ) = t= ) ε k t m k ) O p t= 7) Now, havng establshed Lemmas 2. and 2.2 and Corollary 2., we are n the poston of formally presentng the proof of heorem 2.. Proof of theorem 2.. : Lets consder agan the terms ˆK 3 and ˆK 4 defned n equatons 8) and 9), where ˆε t ) t [[, s gven by equaton 2). hen, usng the results presented n Lemma 2.2 and Corollary 2., combned wth the Law of Large Numbers and the Central Lmt heorem CL), we obtan that: And S 2 = S3 = S4 3S 2 2 ) = herefore, combnng the above results we now obtan that: t= ) ε 2 P t O p 8) t= ) ε 3 d t 3 ε t op ) N0, 6) 9) ε 4 t 6 ε 2 t 3 ) d o p ) N0, 24) 20) t= ˆK ) 3 2 6 = 2 S3 d 3 χ 2 ) 2) S 2 6 ˆK 4 2 24 = [ S4 3S ) 2 2 2 d 4 χ 2 ) 22) S 2 24 [ Furthermore, let Z t = 6 ε 3. t 3 ε t ), 24 ε 4 t 6 ε 2 t 3) hen, we have: ) ) 0 0 EZ t ) = and EZ 0 t Z t) = 0 And, from the multvarate CL we obtan that: t= Z t [ d 0 N 0 hus, from the above results we conclude that ) 0, 0 ) 23) 8
ˆK2 3 6 ˆK 4 2 24 ) d χ 2 2) 24) hs completes the proof. Hence, the results presented n ths secton prove the valdty of the Jarque-Bera normalty test when appled to a general frst order autoregressve process wthout determnstc terms. hs s qute an mportant result n the sense that the asymptotc propertes of the Jarque- Bera test for normalty are ndependent of the values of the parameter α. hus, the use of ths test s not restraned to statonary autoregressons. It can also be used when dealng wth margnally stable or even explosve autoregressve processes. 3 Autoregressve process wth determnstc terms In ths secton, the asymptotc propertes of the Jarque-Bera test for a general frst order autoregresve process wth a constant, whch satsfes equaton 5), are analysed. Once agan, the fndngs presented n ths secton are nnovatve n the sense that t allows for the possblty of havng an explosve root n the model. he man result of ths secton s that of the valdty of the Jarque-Bera normalty test when dealng wth a general frst order autoregressve process wth a constant term. hs pont s presented n the followng theorem: heorem 3.. Let ˆK3 and ˆK 4 be defned as n equatons 8) and 9), where ˆε t ) t [[, s gven by equaton 6). If assumptons. and.2 are satsfed, then ˆK 3 2 d χ 2 ), ˆK 4 2 d ˆK2 χ 2 ) and 3 6 24 6 ˆK ) 4 2 d χ 2 2) 25) 24 In order to establsh the proof for heorem 3., the followng two lemmas are requred. Lemma 3.. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let X t ) t N be the autoregressve process satsfyng equaton 5) and X 0 = 0. hen, for any fxed postve ntegers k and p, we have t= X t = O p ) f α < and µ R) t= Xt = Op ) f α = and µ 0) Xt t= = Op ) f α = and µ = 0), or α = and µ R) α k t= X t = O p ) f α > and µ R) 9
And, t= X t ε p t m p ) = O p ) f α < and µ R) Xt t= p εt m p ) = O p ) f α = and µ 0) t= Xt εt p m p ) = O p ) f α = and µ = 0), or α = and µ R) α k t= X t ε t p m p ) = O p ) f α > and µ R) where: m p = E [ε p t. Addtonally, wth ˆα α, ˆµ µ) gven by equaton 7), we have that for α < and µ R), ˆα α) O p ) = ˆµ µ) O p ) for α = and µ 0), 3/2 ˆα α) O p ) = ˆµ µ) O p ) for α = and µ = 0), and for α = and µ R), ˆα α) O p ) = ˆµ µ) O p ) for α > and µ R), α ˆα α) ˆµ µ) = O p ) O p ) An nterestng pont to note s that the presence of the determnstc term µ only affect the order of the above processes n the case when α =, whch also depends on the value of µ. However, n the statonary and explosve cases, the presence of a constant term and ts value) does not affect the order of the correspondng processes. Proof of lemma 3.. : In what follows, unless stated otherwse, assume that µ 0. Addtonally, t s worth mentonng that ths proof s dvded nto four parts, correspondng to the four cases α <, α =, α = and α >. Frst of all, consder the statonary case when α <. And, consder the sequence X t ) t N 0
defned as X t = =0 α ε t µ α = Z t µ α where Z t = α ε t =0 hus, usng the bnomal rule, we obtan that for any postve nteger k Xt ) k = [ ) k µ Z t α t= t= = k ) j k µ Z j t j α t= j=0 µ k ) ) [ k j k µ = α j α And, from Lemma 2. we have that for any postve nteger j, j= t= Z j t Z j t = O p ) t= whch mples that Xt ) k = O p ) Now, consder X t ) t N satsfyng equaton 5). For α < ths s gven by whch can also be wrtten as t= t ) α X t = α t ε t µ α t X 0 α =0 X t = X t α t X 0 X 0) And, from Caceres and Nelsen 2006b, Lemma 2.), we have that Xt k = t= Xt ) k o p ) t= hus, Xt k = O p ) t=
Smlarly, for any postve ntegers p and k, consder Xt ) k ε p t m p ) t= = = = t= t= [ µ Z t k j=0 µ α α ) k ε p t m p ) ) j k µ Z j j α t ε p t m p ) ε p t m p ) t= k ) ) [ k j k µ j α j=0 Zt ε j p t m p ) t= From Lemma 2. we have that for any postve nteger j, Zt ε j p t m p ) = O p ) t= whch mples that Xt ) k ε p t m p ) = O p ) Agan, from Caceres and Nelsen 2006b, Lemma 2.2), we have t= t= X k t ε p t m p ) = Xt ) k ε p t m p ) o p ) t= hus, herefore, for α <, we have ˆα α) ˆµ µ) = t= X2 t Xt ε k p t m p ) = O p ) t= t= X t t= X t t= X t ε t t= ε t = O p ) O p ) Secondly, consder the case when α =. Hence, the sequence X t ) t N satsfyng equaton 5) s now gven by X t = t ε µt = Z t µt where Z t = = t = ε 2
Agan, usng the bnomal rule we obtan that Xt k = t= = µ k [Z t µt k t= k t k t= j= [ k )µ k j j t k j Z j t t= t= Z k t herefore, t= = µk k Xt k ) ) [ j k t k µ k j j t= j= j ) j ) [ k t Zt t= t= ) k Zt Now, let P k ) = t= tk. Note that P k ) s a polynomal of order O k ). See Lemma A. n the appendx. Hence, µ k k t= t k = µk k P k ) = O) And, from Caceres and Nelsen 2006a, heorem 3), we have that t= j ) j t Zt = O p ) and t= Zt = O p ) hus, t= Xt Smlarly, usng the bnomal rule we can also wrte t= Xt ε p t m p ) = µk k t t= k j j= = O p ) ε p t m p ) ) µ k j From Johansen 2006, heorem B.2) we have that ) [ j ) [ k t= t t= ) k Zt ε p t m p ) j Zt ) j ε p t m p ) t= t ε p t m p ) = O p ) 3
And, from Caceres and Nelsen 2006a, heorem ), we have that t= j ) j t Zt ε p t m p ) = O p ) and t= Zt ε p t m p ) = O p ) hus, herefore, for α =, we have 3/2 ˆα α) ˆµ µ) = t= 3 t= X2 t Xt ε p t m p ) = O p ) 2 t= X t 2 t= X t 3/2 t= X t ε t t= ε t = O p ) O p ) Now, consder the case when α =. Hence, the sequence X t ) t N satsfyng equaton 5) s now gven by t X t = ) t ) ε µ γ t = Z t µ γ t where Z t = ) t = t ) ε and γ t = = Agan, usng the bnomal rule, we obtan that { f t s odd 0 f t s even Xt k = t= = µ k [Z t µ γ t k t= k γ t t= j= [ k )µ k j j γ t Z j t t= t= Z k t herefore, = t= Xt µ k ) ) [ k j k γ t µ k j j t= j= Now, from the defnton of γ t, we have that t= ) j Zt γ t t= Zt γ t 2 t= µ γ t = o p ) t= 4
And, from Lemma 2., we have that t= Zt = O p ) For the mddle terms, assume that s even.e. we can wrte = 2n), then t= γ t Zt ) j = = = 2n t= n r= n r= ) j Zt γ t ) j Z2r γ 2r ) j Z2r = O p ) from Lemma 2. and, f s odd.e. = 2n ), then t= γ t Zt ) j = = = 2n t= [ 2n t= 2n t= = O p ) ) j Zt γ t ) j ) j Zt Z2n γ t γ 2n γ t Zt ) j as γ 2n = 0 hus, t= Smlarly, usng the bnomal rule, we can wrte Xt = O p ) t= Xt ε p t m p ) = µ k j= t= ) k µ k j j γ t ε p t m p ) ) [ k j t= t= Zt ε p t m p ) γ t Zt ) j ε p t m p ) 5
From Lemma 2. we have that t= Zt ε p t m p ) = O p ) Furthermore, usng exactly the same argument as above.e. consderng separately the two cases, when s even and then when s odd), we obtan that t= γ t Zt ) j ε p t m p ) = O p ) and γ t ε p t m p ) = O p ) t= hus, herefore, for α =, we have ˆα α) ˆµ µ) = t= 2 t= X2 t Xt ε p t m p ) = O p ) 3/2 t= X t 3/2 t= X t t= X t ε t t= ε t = O p ) O p ) Fnally, consder the case when α >. Hence, the sequence X t ) t N satsfyng equaton 5) s now gven by t X t = α ε t µ =0 ) α t = Z t µ α Hence, usng the bnomal rule, we obtan α k Xt k = α k t= = j= t= [ Z t µ ) α t α ) α t k α t where Z t = α ε t Zt µ α k α t ) k α α t= t= k ) ) [ j k µ α j α t ) ) k j j Zt j α α t= =0 6
Frst, note that usng the bnomal rule once agan, we have that µ α k α t α t= µ k = α k ) k r α t ) r α r t= r=0 ) [ k µ k ) k = α k r α r ) t α r t= r= t= ) [ k µ k ) ) k α = α k r r α r α r r= ) [ k µ k ) = α k ) ) α k r r k) α k α k α r α r α k r= hus µ α α t= Also, from Lemma 2., we have that α t ) ) ) k µ α k k ) µ = o) α α k α α k 26) α k Zt k = t= Zt α t= = O p ) 27) Smlarly, usng the bnomal rule once more, we have that [ α j α t ) ) k j j Zt α t= [ j ) ) k j j = α j ) j r α r ) t Zt r α t= r=0 j = )j j Zt ) j ) j r j α j α r α α r ) t Zt α j ) t Zt j r) α α t= r= t= t= j ) j ) j r j = r α α r ) t Zt j α j ) t Zt o j r) α α p ) r= t= t= j 7
And, from the rangle and Jensen s nequaltes, we have ) k j E α r ) t Zt ) α r t E Z t α α t= t= }{{} =O) k j c.f. La and We 983)) K j α r α r t where K j s a constant t= ) α r K j α r Kj ) α r whch mples that herefore, t= α j [ t= α r ) t Zt α j = O p ) α t ) ) k j j Zt = O p ) α Hence, k j= ) k µ j α ) [ j α j α t ) ) k j j Zt = O p ) 28) hus, combnng equatons 26), 27) and 28), we obtan that t= α α k Xt k = O p ) 29) t= Smlarly, usng the bnomal rule, we can wrte α k Xt ε k p t m p ) = t= ) µ α k α t ε p t m p ) α k j= k j t= ) µ α t= Zt α ) [ j α j α t ) j Zt t= α ε p t m p ) j ε p t m p ) 8
Frst, usng the bnomal rule once more, we have that ) µ α k α t ε p t m p ) α t= ) µ α k k = r α r ) t ε p t m p ) α r t= r=0 ) [ k µ k ) k ) = α k ε p k r t m p ) α r ) t ε p α r α k r) t m p ) t= r= t= ) [ k k ) µ k ) k r = α r ) t ε p α r α k r) t m p ) o p ) r= And, from Caceres and Nelsen 2006b, Lemma 4.2), we know that hus, Also, from Lemma 2., we have t= α r ) t ε p t m p ) = O p ) t= ) µ α k α t ε p t m p ) = O p ) 30) α t= t= Zt α And, from the bnomal rule, we can wrte α j [ t= = )j α j j = r= t= j r α t ) j Zt t= Zt α ) ) j r α j r) α j ) t Zt α α j ε p t m p ) j ε p t m p ) t= t= α r ) t Zt α j j ε p t m p ) ε p t m p ) = O p ) 3) j α j ) t Zt ε p t m p ) α j ε p t m p ) r= ) j ) j r r α j r) t= α r ) t Zt α j ε p t m p ) o p ) Now, from the rangle nequalty, and then the Cauchy-Schwarz and Jensen s nequaltes, 9
we obtan that j E α r ) t Zt ε p α t m p ) t= { } k j α r t Z t E α ε p t m p t= [ ) α r t E Z t 2k j) /2 α E ε p t m p 2 t= K j α r α r t where K j s a constant t= ) K α r j α r ) Kj α r whch mples that herefore, t= α j [ t= α r ) t Zt α α t ) j Zt j ε p t m p ) = O p ) α j ε p t m p ) = O p ) Hence, k ) ) [ j k µ α j j α j= t= α t ) j Zt α j ε p t m p ) = O p ) 32) hus, combnng equatons 30), 3) and 32), we obtan that α k Xt ε k p t m p ) = O p ) 33) t= Fnally, from equatons 29) and 33), we have that, for α >, α ˆα α) ˆµ µ) = = α 2 t= X2 t α α t= X t α 2 t= X2 t o p ) o p ) t= X t α t= X t ε t t= ε t α t= X t ε t t= ε t O p ) = O p ) Now, when µ = 0 we recover the results for t= Xk t and t= Xk t ε p t m p ) drectly from Lemma 2.. herefore, the proof of Lemma 3. s completed by nsertng these last 20
results nto the correspondng terms of ˆα α, ˆµ µ) gven by equaton 7). Lemma 3.2. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let the resduals ˆε t ) t [[, be defned as n equaton 6). hen, for α and µ R) and for any postve nteger k, we have And, ˆε k t = t= ˆε k t = t= ) ε k t O p t= t= ε k t O p ) f k s even 34) f k s odd 35) Addtonally, for α > and µ R) and for any postve nteger k, we have ˆε k t = t= ) ε k t O p t= 36) Proof of lemma 3.2. : hs proof s based on the same argument as that used to prove Lemma 2.2. Usng the trnomal rule, we have that for any postve nteger k, t= =0 ˆε k t = [ε t ˆα α)x t ˆµ µ) k t= = k r k r ) r [ˆα α)x t r ˆµ µ) ε k r t where t= r=0 =0 = k r k [ ε k t r ) r ˆα α) r ˆµ µ) t= r= =0 t= k [ k ) ˆα α ˆµ µ) Xt k = ε k t t= k r= r =0 k r= k r t= r k [ r ) r ˆα α) r ˆµ µ) =0 [ ) r m k r ˆα α) r ˆµ µ) t= t= X r t Xt ε r k r t k r = ) Xt r ε k r t m k r k! k r)!r )!! 2
usng the conventon m 0 =. Hence, the proof of Lemma 3.2 s completed by multplyng the four terms, t= Xk t, t= Xk t ε p t m p ), ˆα α) and ˆµ µ), by ther correspondng weghts as presented n Lemma 3., accordng to the dfferent values of α and µ, and by fnally nsertng them nto the above expresson for t= ˆεk t. Corollary 3.. Let ε t ) t N be a sequence of..d. random varables satsfyng assumpton. and let the resduals ˆε t ) t [[, be defned as n equaton 6). hen, for α and µ R) and for any postve nteger k, we have ˆε k t m k ) = t= ) ε k t m k ) O p t= f k s even 37) And, t= ˆε k t = ε k t [ k m k ˆα α) t= ) X t ˆµ µ) O p Addtonally, for α > and µ R) and for any postve nteger k, we have ˆε k t m k ) = t= t= ) ε k t m k ) O p t= f k s odd 38) 39) Now, havng establshed Lemmas 3. and 3.2 and Corollary 3., we are n the poston of formally presentng the proof of heorem 3.. Proof of theorem 3.. : Lets consder agan the terms ˆK 3 and ˆK 4 defned n equatons 8) and 9), where ˆε t ) t [[, s gven by equaton 6). hen, usng the results presented n Lemma 3.2 and Corollary 3., combned wth the Law of Large Numbers and the Central Lmt heorem CL), we obtan that: And S 2 = S3 = S4 3S 2 2 ) = herefore, combnng the above results we now obtan that: t= ) ε 2 P t O p 40) t= ) ε 3 d t 3 ε t op ) N0, 6) 4) ε 4 t 6 ε 2 t 3 ) d o p ) N0, 24) 42) t= 22
ˆK ) 3 2 6 = 2 S3 d 3 χ 2 ) 43) S 2 6 ˆK 4 2 24 = [ S4 3S ) 2 2 2 d 4 χ 2 ) 44) S 2 24 Fnally, usng the multvarate CL exactly as n heorem 2., we conclude that ˆK2 3 6 ˆK ) 4 2 d χ 2 2) 45) 24 whch completes the proof. Hence, the results presented n ths secton prove the valdty of the Jarque-Bera normalty test, when the latter s appled to a general frst order autoregressve process wth a constant. In other words, t was shown that the Jarque-Bera test for normalty s ndependent of whether the process s statonary or not and whether the latter ncludes a determnstc term: a constant n ths partcular case. It s worth mentonng that, when a dfferent type of determnstc term s ncluded e.g. a lnear trend nstead of a constant term) n equaton 4), a dfferent order would be certanly obtaned for some of the terms such as t= Xk t and t= Xk t ε p t m p ), where k and p are postve ntegers. In fact, f we consder a model that satsfes the autoregressve equaton hen, the process X t ) t N can be wrtten as and t X t = α ε t =0 X t = X t = αx t µ γ t ε t µ γ α ) α t ε j j= ) ) α t γ t for α 46) α α µ γ ) t γ 2 2 t2 for α = 47) Hence, dfferent orders than those presented n Lemma 3. would be obtaned at least n the non-explosve case) for the terms t= Xk t and t= Xk t ε p t m p ). However, t s not clear whether ths wll affect the asymptotc propertes of the Jarque-Bera statstcs exhbted n equaton 0). Strctly speakng, the entre analyss presented n ths secton would then have to be carred out agan n order to prove the valdty of ths normalty test when the process X t ) t N s gven by equatons 46) and 47). 23
4 Concluson he objectve of ths paper was to analyse the asymptotc propertes of the Jarque-Bera test for normalty from a tme seres perspectve. In partcular, the man nterest was to study the valdty of ths test when the latter s appled to a frst order autoregressve process wthout determnstc terms, such as X t ) t N defned n equaton ), or wth a constant term, such as X t ) t N defned n equaton 5). In fact, t was proved that the Jarque-Bera normalty test s theoretcally vald n the general case when appled to a frst order autoregressve process. hs means that the Jarque- Bera test for normalty can be used when dealng wth a frst order autoregressve process ndependently of whether the latter contans a statonary, a unt or an explosve root. hs result s true regardless of whether a constant term s ncluded or not n the autoregressve model. Note that when dfferent types of determnstc terms are ncluded n the autoregressve model other than the constant term), then the analyss presented n ths paper would have to be repeated and modfed accordngly. But the fact that the order of the process X t ) t N and that of the sum of ts powers) dffers n some cases when dfferent determnstc terms are added to the model, does not necessarly mply that the Jarque-Bera statstcs would not satsfy the asymptotc results presented n equaton 0). hs s certanly an area of further research. Smlarly, ths analyss could also be generalsed to the case where the Jarque-Bera normalty test s appled to a general vector autorgressve VAR) model wth determnstc terms. A Appendx Lemma A.. For any postve nteger k, let P k ) = t= tk. hen, P k ) s a polynomal of order O k ) Proof of lemma A.. : Note that P k ) s also known as the Power Sum, and from the Faulhaber s formula c.f. Conway and Guy 996)), we can wrte P k ) = k ) δ r,k k r= k r ) B k r r where δ r,k s the Kronecker delta and B s the -th Bernoull number. hs way of wrtng P k ) shows that the latter s a polynomal of order O k ). In fact, the proof of the valdty of the Jarque-Bera normalty test only requres k 4. 24
For those cases the polynomals P k ) are gven by: P ) = ) 2 P 2 ) = ) 2 3 ) P 3 ) = 2 ) 2 4 P 4 ) = ) 3 3 2 2 6 6 5 ) References [ Caceres, C. and Nelsen, B. 2006a). Convergence to Stochastc Integrals ncludng Nonlnear Multvarate Functons. Dscusson paper, Nuffeld College, Oxford. [2 Caceres, C. and Nelsen, B. 2006b). Asymptotc Propertes of Whte s est for Heteroskedastcty. Dscusson paper, Nuffeld College, Oxford. [3 Conway, J.H. and Guy, R. K. 996). he Book of Numbers, Sprnger-Verlag, New York. [4 Jarque, C.M. and Bera, A.K. 980). Effcent tests for normalty, homoscedastcty and seral ndependence of regresson resduals. Economc Letters 6, 255-259. [5 Johansen, S. 996). Lkelhhod-based nference n contegrated vector autoregressve models, 2nd prnt, Oxford Unversty Press. [6 Klan, L. and Demroglu, U. 2000). Resdual-Based ests for Normalty n Autoregressons: Asymptotc heory and Smulaton Evdence. Journal of Busness and Economc Statstcs 8, 40-50. [7 La,.L. and We, C.Z. 983). Asympotc propertes of general autoregressve models and strong consstency of least-squares estmates of ther parameters. Journal of Multvarate Analyss 3, -23. [8 Lutkepohl, H. and Schneder, W. 989). estng for Nonnormalty of Autoregressve me Seres. Computatonal Statstcs Quarterly 2, 5-68. [9 Nelsen, B. 200). Order determnaton n general vector autoregressons. Dscusson paper, Nuffeld College, Oxford. [0 Nelsen, B. 2005). Strong consstency results for least squares estmators n general vector autoregressons wth determnstc terms. Econometrc heory 2, 534-56. [ Nelsen, B. 2006). Correlograms for non-statonary autoregressons. Journal of the Royal Statstcal Socety, seres B forthcomng). 25