LAD ASYMPTOTICS UNDER CONDITIONAL HETEROSKEDASTICITY WITH POSSIBLY INFINITE ERROR DENSITIES. Jin Seo Cho, Chirok Han, and Peter C. B.

Transcription

1 LAD ASYMPTOTICS UNDER CONDITIONAL HETEROSKEDASTICITY WITH POSSIBLY INFINITE ERROR DENSITIES By Ji Seo Cho, Chirok Ha, ad Peter C. B. Phillips Jue 29 COWLES FOUNDATION DISCUSSION PAPER NO. 73 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 2828 New Have, Coecticut

2 LAD Asymptotics uder Coditioal Heteroskedasticity with Possibly Ifiite Error Desities Ji Seo Cho Korea Uiversity Chirok Ha Korea Uiversity Peter C. B. Phillips Yale Uiversity, Uiversity of York Uiversity of Aucklad & Sigapore Maagemet Uiversity April, 29 Abstract Least absolute deviatios (LAD) estimatio of liear time-series models is cosidered uder coditioal heteroskedasticity ad serial correlatio. The limit theory of the LAD estimator is obtaied without assumig the fiite desity coditio for the errors that is required i stadard LAD asymptotics. The results are particularly useful i applicatio of LAD estimatio to fiacial time series data. Keywords: Asymptotic leptokurtosis, Covex fuctio, Ifiite desity, Least absolute deviatios, Media, Weak covergece. Subject Classificatios: Primary C2, Secodary G. Itroductio This ote derives asymptotics for the least absolute deviatios (LAD) estimator i a liear time series model uder coditioal heteroskedasticity ad serial correlatio. Our methods follow the approach pioeered i Kight (998, 999), who derived LAD asymptotics uder ostadard coditios that allow for possibly ifiite or zero error desity fuctio coditios. Nostadard coditios such as these, particularly a ifiite error desity at the origi, are likely to arise i

3 may empirical applicatios to fiacial data, which are well kow to be characterized with high kurtosis. I recet related work, the authors (Ha, Cho ad Phillips, 29) have developed a test for ifiite desity i stock returs ad foud i empirical applicatios that a sigificat umber of leadig compaies i U.S. idustries have asset returs with ifiite desity at the media. The preset asymptotic results assist i validatig such applicatios for fiacial data. I particular, the curret paper cosiders time-series models with predetermied variables as regressors, coditioally heterogeeous errors, ad possibly ifiite error desities, thereby expadig the rage of potetial empirical applicatios of LAD estimatio theory. Previous work i the literature has developed asymptotic theory for LAD estimatio i various time series settigs. Phillips (99) developed LAD limit theory for statioary data uder the assumptios of a fiite error desity ad exogeous regressors. Koeker ad Zhao (996) cosidered the ifluece of coditioal heteroskedasticity o the LAD estimator, also uder fiite error desity coditios. Kight (999) established LAD asymptotics for a liear model with exogeous regressors ad heteroskedastic errors uder possibly ifiite error desity coditios, but coditioal heteroskedasticity ad weakly exogeous regressors were ot cosidered. Fially, Rogers (2) geeralized the framework i Kight (998) to statioary ad itegrated time-series cotexts without hadlig coditioal heteroskedasticity. The preset ote exteds this literature by establishig a asymptotic theory for LAD estimatio with time series data that exhibits coditioal heteroskedasticity ad whose errors have possibly ifiite desity. The model, regularity coditios ad mai result are give i Sectio 2 ad Sectio 3 provides proofs. 2 Mai Results We cosider the followig liear media regressio model: y t = x tβ + ε t, Ex t sg(ε t ) = iff β = β, where the regressor vector x t R k may cotai weakly exogeous variables ad sg(x) = {x > } {x < }. The error term ε t is coditioally heteroskedastic i the sese that ε t := σ t e t, where σ t is adapted to F t, the sigma field geerated by {x t, ε t, x t, ε t 2,...}, ad the primitive shock e t is assumed to have a distributio fuctio F e (z) := P (e t z F t ) for all t ad for all z i a eighborhood of zero. We maitai the followig assumptio throughout the paper: Assumptio A. (i) σ t σ > ; (ii) (x t, σ t ) is statioary ad ergodic; (iii) σ2 t = O p (); ad (iv) E x t 4 <. 2

4 The asymptotic behavior of the LAD estimator is well kow to deped o the distributio of e t ear zero. Specifically, as Kight (998, 999) shows, if we let ψ e (x) := F e (a x) F e (), where a is selected i a way that esures ψ e ( ) coverges to a odegeerate fuctio, the ψ e (x) turs out to be the critical compoet i determiig LAD asymptotics, as detailed below. The sequece a is determied differetly depedig o whether or ot f e () is fiite. For example, if f e () is fiite, so that F e ( ) has a fiite slope at zero, the we ca let a be, implyig that ψ e (x) = f e ( /2 x)x for some x betwee x ad by first-order Taylor expasio. As teds to, f e ( /2 x)x coverges to f e ()x, which is o-degeerate. As aother example, for some α, suppose F e (x) = F e () + λ sg(x) x α i a eighborhood of zero, the ψ e (x) = /2 λ sg(x)a α x α, so that ψ e (x) = λ sg(x) x α whe a = /2α. I this case, ote that for α < the desity f e () is ot fiite. Also, ψ e (x) icreases with respect to x, ad ψ e (x) = if ad oly if x =. It is useful i the developmet of the asymptotics to make some assumptios regardig the fuctioal shape ad properties of ψ e ( ) as well as some geeral regularity coditios to facilitate cetral limit theory. We maitai the followig assumptios. Assumptio B. For a fuctio h( ), F e (x) F e () h(x) for all x i a ope iterval V cotaiig zero such that h(x) icreases with respect to x, ad for some fiite C ad, /2 h(a x) C ( + x ) for all x R provided that >. I geeral, Assumptio B is satisfied by a wide class of desity fuctios. If the desity of e t is fiite ad cotiuous at zero, for example, the we ca let a =, ad F e (x) F e () = f e ( x)x for some x betwee x ad. If we further let f be the maximum desity i the eighborhood, the F e (x) F e () = f e ( x) x f x, so that Assumptio B follows by lettig h(x) = f x, because /2 h(a x) = f x f ( + x ). These are ot the oly desities satisfyig Assumptio B. If a desity aroud zero ca be approximated by a power desity, the it also satisfies Assumptio B. More specifically, suppose that the desity of e t is bouded by two power fuctios with the same expoet, i.e., for some α, 2 λα x α f e (x) 2λα x α o a ope iterval cotaiig zero, the we have a = /(2α), ad F e (x) F e () λ x α = h(x) o the same iterval. We ote that h( ) satisfies the last part of Assumptio B because /2 h(a x) = λ x α λ( + x ) for all x R, where the last iequality holds because α. Next we assume that ψ e ( ) coverges to some ψ e ( ) i a proper mode as follows: Assumptio C. For some ψ e ( ), there is a symmetric ad oegative sequece δ ( ) such that δ (s) is icreasig as s icreases, ψ e ( ) ψ e ( ) δ ( ), () lim sup E x t δ ( x t ) <, 3

5 ad δ ( ) coverges uiformly to zero o every compact eighborhood of zero. Assumptio C slightly differs from coditios i the previous literature. First, primitive coditios are provided i Assumptio C for the covergece of ψ e ( ) to ψ e ( ) such that Ψ e (x) := x ψe (s)ds is covex with respect to x, as proved i Lemma 5 below, whereas Kight (999) assumes this directly. Secod, we allow for x t to be stochastic, differetly from Kight (999). May distributio fuctios satisfy Assumptio C. For example, uiformly fiite desities o a eighborhood of zero satisfy the coditios. For this kid of desity, we have a = /2, ad ψ e (x) = f e ( /2 x)x = f e ()x + f e ( /2 x) f e ()x = ψ e (x) + δ (x) for some x betwee x ad zero. Thus, if we let δ (x) = sup x x f e ( /2 x) f e () x ad E x t 2 <, the δ ( ) trivially satisfies all the coditios i Assumptio C. Power desities are aother example satisfyig Assumptio C. We further illustrate the use of Assumptio C by cosiderig oe of the examples give i Kight (998, p. 76). Suppose F e (x) F e () = λx l( x ) i a eighborhood of zero so that a is /2 (l )/2, ad for every x ad for large eough such that a x is i the eighborhood. We the have ψ e (x) = /2 λa xl(a ) + l( x ) = λx + 2λx(l l l 2 + l( x )). l Lettig ψ e (x) = λx yields ψ e (x) ψ e (x) 2λ x (l l + + x + x ) l ( ) l l + 2λ( + x + x 2 ), l because l( x ) x for x ad > l( x ) > x for x >. We may deote the right had side (RHS) of the fial iequality by δ (x), which is a icreasig fuctio of x ad coverges to zero uiformly o every compact set as. The other coditios stated i Assumptio C trivially hold if E x t 3 <. As the fial regularity coditio, it is coveiet to impose the followig high level cetral limit theorem (CLT). Assumptio D. For some positive defiite A, /2 x t sg(ε t ) ζ N(, A). If the media of ε t coditioal o x t is zero, the Assumptio D follows by the Lideberg coditio ad A = plim x tx t i this case. I order to preset our mai results, we briefly explai some techical details adapted maily from Kight (998). First, ote that for ay a, the cetered ad rescaled LAD estimator ˆθ := a (ˆβ β ) miimizes Z (θ) := a ( ) ε t a x tθ ε t. 4

6 For every x, we have x y x = y sg(x) + 2 y I(x s) I(x )ds, as show by Kight (998, 999) ad Rogers (2), where I( ) deotes the idicator fuctio. Thus, (2) Z (θ) = θ x t sg(ε t ) + 2a = Z () (θ) + Z (2) (θ), say. a x I(ε t s) I(ε t ) ds Here, Z () (θ) θ ζ by Assumptio D, ad (3) Z (2) (θ) = 2 /2 I(ε t a s) I(ε t ) ds, which coverges i probability to a oradom quatity as show by Lemmas 2, 4 ad 5 i the followig sectio. Deotig this limit by τ(θ), the weak limit of Z (θ) has the form Z (θ) = θ ζ + τ(θ). Importatly, both Z (θ) ad its limit Z (θ) are covex fuctios of θ, as proved i Lemma 5. Hece, if Z (θ) is miimized at a uique poit uder some regularity coditios for the properly chose a, we ca ivoke a argmi cotiuous mappig theorem to obtai the limit distributio of a (ˆβ β ) without establishig stochastic equicotiuity (e.g., Geyer, 996). The followig LAD asymptotic theory is established for liear models with weakly exogeous regressors ad coditioally heteroskedastic disturbaces. Theorem. Uder Assumptios A D, a (ˆβ β ) argmi θ R k{ θ ζ + τ(θ)} provided that the limit fuctio Z (θ) = θ ζ + τ(θ) is uiquely miimized almost surely. Theorem is the mai result of the curret paper ad exteds the rage of potetial applicatios of Kight s (998, 999) limit theory to our time-series framework that allows for coditioal heterogeeity ad serial depedece. Also, our limit theory differs from earlier work i the way that τ(θ) is deployed. To see this more clearly let (4) D t := I(ε t a s) I(ε t ) ds, so Z (2) (θ) = (2/) D t. We ca split D t ito the sum of E(D t F t ) ad D t E(D t F t ). Lemma 2 i the ext sectio shows that the secod term is asymptotically egligible, whereas for the first term, we have (5) E(D t F t ) = Ψ t (x tθ), 5

7 where Ψ t (x tθ) := ψ t (s)ds ad ψ t (s) := F t (a s) F t (). Further, Lemma 4 i the ext sectio shows that whe ψ t (s) ψ t (s) poitwise for each t, the probability limit of (5) equals the probability limit of ψ t (s)ds. Thus, essetially we have (6) Z (2) (θ) = 2 Ψ t (θ) + o p (), where Ψ t (θ) := ψ t (s)ds. The first term o the right side of (6) should obey the ergodic theorem, ad we have ψ t (s) = σ t ψ e (σ t s), where ψ e (s) is the limit of ψ e (s) := F e (a s) F e (). Cosequetly, τ(θ) which is defied as the probability limit of 2 Ψ t(θ), is i fact the probability limit of 2 σ tψ e (σ t x tθ). I this expressio we ote that the stochastic regressors x t ad the coditioal heteroskedasticity process σ t iteract so that σ t plays the role of a stadardizig factor, which differs from previous limit theory ad distiguishes the limit form of τ(θ) i the preset paper. We ow aalyze two special examples usig Theorem to show explicity how the results i Kight (998, 999) are modified i the presece of coditioal heteroskedasticity ad weakly exogeous regressors. Example (Fiite desity). Let < f e () <. The for a =, we have ψ e (s) f e ()s by first-order Taylor expasio, which implies that ψ t (s) f t ()s with f t () = σ t f e (). Thus Ψ t (x) = f t () x sds = f 2 t()x 2, ad Z (2) (θ) = 2 2 f t()(θ x t ) 2 + o p () p τ(θ) = θ Bθ, where B := plim f t ()x t x t. Thus, Z (θ) θ ζ + θ Bθ, ad this weak limit is miimized almost surely by (2B) ζ N(, 4 B AB ). Therefore, this expressio gives the limit distributio of the LAD estimator accordig to Theorem. Further, we ote that f t ()x t x t = ft e ()σ t x t x t, implyig that B = f e () plim σ t x t x t, ad thereby showig how σ t ad x t ifluece B. Example (Power desity). If e t has a power desity, so that for some α, f e (x) = λα x α i a eighborhood of zero, the we have ψ e (s) = λ sg(s) s α = ψ e (s) by lettig a = /2α. Therefore, Ψ e (x) = λ(α + ) x α+, ad this implies that Ψ t (x) = λ(α + ) σ α t x α+ by virtue of the fact that Ψ t (x) = σ t Ψ e (σt x). From this, for each θ, 2 Ψ t (x tθ) = 2λ α + If we further suppose that x t is a scalar, the τ(θ) = 2λh α α + θ α+, σ α t x tθ α+ τ(θ). where h α := plim 6 σ α t x t α+,

8 so that ζ = 2λh α sg(θ ) θ α whe θ miimizes θ ζ + τ(θ). Thus, sg(ζ) = sg(θ ), ad we obtai tha = (2λh α ) /α sg(ζ) ζ /α. This result is very close to Example i Kight (998) except that his λ is replaced with our λh α due to the ifluece of σ t i τ(θ). 3 Proofs For the proof of Theorem, the most difficult part is associated with (2) for which we eed to prove a law of large umbers (LLN) for Z (2) (θ). We start by writig Z (2) (θ) = 2 D t usig a chage of variables, where D t is defied i (4), ad decompose this sum as follows (7) D t = Ψ t (x tθ) + Ψ t (x tθ) Ψ t (x tθ) + D t Ψ t (x tθ). The proof proceeds by showig that the secod ad third terms o the right side of (7) are egligible i probability, as verified i Lemmas 4 ad 2 below. The, Lemma 5 below shows that the first term o the right side obeys the ergodic theorem. These lemmas are itegrated ito the overall proof of the theorem to aid readability i what follows. We begi with the third term o the right side of (7), otig that Ψ t (x tθ) = E(D t F t ). Lemma 2. Uder Assumptios A(i,ii) ad B, if E x t 3 <, the D t E(D t F t ) p. Proof. If we let ξ t = D t E(D t F t ), the E(ξ t ) =, so that it suffices to show that the variace of ξ t goes to zero. We have var(ξ t F t ) = E(D 2 t F t ) E(D t F t ) 2 E(D 2 t F t ), where E(D 2 t F t ) = = = E I(ε t a (s r)) I(ε t a (s )) I(ε t a (r )) + I(ε t ) F t drds ψ t (s r) ψ t (s ) ψ t (r ) drds ψ t (s r)drds 2 θ x t ψ t (s )ds. Therefore, if x tθ >, the ψ t (s r) ψ t (s) over the domai of the secod itegral, so that E(Dt F 2 t ) ψ t (s)drds + = θ x t Ψ t (x tθ). 7

9 O the other had, if x tθ <, the ψ t (s r) ψ t (s) for all s ad r betwee ad x tθ, so that ψ t (s r)dr ψ t (s)dr = θ x t ψ t (s) for x tθ s, implyig that E(D 2 t F t ) θ x t Ψ t (x tθ) 2 θ x t Ψ t (x tθ) = θ x t Ψ t (x tθ). Thus, we have E(D 2 t F t ) θ x t Ψ t (x tθ) regardless of the value of x tθ, ad exploitig this iequality yields that ( ) var ξ t = (8) 2 E 3/2 E E(ξ 2 t F t ) 2 θ x t Ψ t (x tθ) E E(Dt F 2 t ) = 3/2 E θ x t σ t Ψ e (σ t x tθ), where the secod iequality follows from the fact that E(D 2 t F t ) θ x t Ψ t (x tθ). The fial equality of (8) holds by the fact that Ψ t (x tθ) = σ t Ψ e (σ t x tθ). Now Lemma 3 below shows that ψ e (x) C( + x ) for all x, ad thus (9) Ψ e (x) C x ( + s)ds = C( x + 2 x 2 ) C( x + x 2 ), implyig that θ x t σ t Ψ e (σ t x tθ) C( θ x t 2 +σ θ x t 3 ). Hece, the right side of (8) is bouded by C 3/2 (E θ x t 2 + σ E θ x t 3 ), which is O( /2 ) by Assumptio A(i,ii) ad the fact that E x t 3 < by assumptio. Lemma 3. Uder Assumptio B, for some C <, ψ e (x) C( + x ) for all x R. Proof. First, we suppose that the fuctio h( ) satisfies Assumptio B o V = (c, c 2 ) for some c < ad c 2 >. Secod, we let M := max{, /2h(c ), /2h(c 2 )} <. The, F e (x) F e () Mh(x) for every x i V obviously; ad if x is ot a elemet of V, the Mh(c ) /2 ad Mh(c 2 ) /2 whereas F e (x) F e () /2, so that F e (x) F e () /2 Mh(c j ) Mh(x), j =, 2. Thus, for some M <, we ca coclude that F e (x) F e () Mh(x) for all x R. Third, therefore, we have ψ e (x) M /2 h(a x) MC ( + x ) for all x R, where the last iequality holds by the fial coditio i Assumptio B. The desired result ow follows by lettig C = MC <. Next, we show that the secod term o the right side of (7) coverges to zero. Lemma 4. Uder Assumptios A ad C, := Ψ t (x tθ) Ψ t (x tθ) p. 8

10 Proof. We have = σ tξ t, where ξ t := σ t x ψ e (s) ψ e (s)ds. By Assumptios A ad C, we also have ξ t σ t x δ (s)ds σ t x tθ δ (σ t x tθ ) σ t x tθ δ (σ x tθ ) = σ t η t, say, where the secod iequality follows from the fact that δ ( ) σ t x tθ over the domai of the itegral by virtue of the mootoicity coditio of δ ( ) i Assumptio C, so that η t = η t I(η t M) + η t I(η t > M). By the statioarity of x t ad (), the secod term i the RHS ca be made as small as desired by icreasig M. For the same M, the first term coverges to zero because its mea coverges to zero by the domiated covergece. Fially, the asymptotic limit of the first compoet o the right side of (7) is established by ergodicity as follows: Lemma 5. Uder Assumptios A C, 2 Ψ t(x tθ) p τ(θ), where τ( ) is a oradom covex fuctio. Proof. Whe (x t, σ t ) is statioary ad ergodic, Ψ t (x tθ) = σ t Ψ e (σ t x tθ) is also statioary ad ergodic, ad the covergece of Ψ t(x tθ) follows from the ergodic theorem. We ow cotiue with the proof of the theorem. The covexity of τ( ) follows from the covexity of Ψ t ( ), which, i tur, is implied by the covexity of Ψ e (x) := x ψe (s)ds with respect to x. Thus, we focus o the latter. Note that Ψ e (x) = a a x F e (s) F e () ds, by chage of variables. For otatioal simplicity, let H (s) := F e (s) F e () ad H(x) := x H (s)ds ad the H( ) is covex, which follows by showig that for ay z, z 2 ad λ,, λh(z ) + ( λ)h(z 2 ) H(λz + ( λ)z 2 ). If we let z < z 2 without loss of geerality ad that H := H(z ), H 2 := H(z 2 ), ad z λ := λz + ( λ)z 2, the λh + ( λ)h 2 = H + ( λ)(h 2 H ) = H( z λ ) + ( λ)(h 2 H ) H( z λ ) H = H( z λ ) + δ(λ), say. The proof is completed if for all λ,, δ(λ). We ote that δ (λ) = (H 2 H ) + H ( z λ )(z 2 z ), 9

11 which is a decreasig fuctio i λ because z λ decreases i λ ad H ( ) is icreasig. Further, δ () = (H 2 H ) + H (z 2 )(z 2 z ) by the mea-value theorem ad the mootoicity of H (z) i z. Similarly, we obtai that δ () = (H 2 H ) + H (z )(z 2 z ). Therefore, δ(λ) mi{δ(), δ()} =, yieldig that H( ) is covex. This proves the covexity of Ψ e ( ), so that its weighted average τ (θ) := Ψ t(x tθ) ad its limit τ(θ) must be covex i θ as well. Proof of Theorem. By (7) ad Lemmas 2, 4 ad 5, Z (2) (θ) p τ(θ) uder Assumptios A C, where Z (2) (θ) is defied i (2). Assumptio D also implies that Z () (θ) θ ζ, where Z () (θ) is defied i (2). Thus, Z (θ) θ ζ + τ(θ) for every θ, where the limit is covex by Lemma 5 ad is uiquely miimized almost surely by suppositio. The desired result follows from Geyer (996) i the same maer as Kight (998, 999). Refereces Geyer, C. J. (996) O the Asymptotics of Covex Stochastic Optimizatio. Upublished Mauscript. Ha, C., J. S. Cho, & P. C. B. Phillips (29) Ifiite Desity at the Media ad the Typical Shape of Stock Retur Distributios. Upublished Mauscript. Kight, K. (998) Limitig distributios for L regressio estimators uder geeral coditios. Aals of Statistics 26, (999) Asymptotics for L -estimators of regressio parameters uder heteroscedasticity. Caadia Joural of Statistics 27, Koeker, R., & Q. Zhao (996) Coditioal quatile estimatio ad iferece for ARCH models. Ecoometric Theory 2, Phillips, P. C. B. (99) A shortcut to LAD estimator asymptotics. Ecoometric Theory 7, Rogers, A. J. (2) Least absolute deviatios regressio uder ostadard coditios. Ecoometric Theory 7,