Jae Kim Monash University. Abstract

Transcription

1 ias Correed oosrap Inferene for Regression Models wih Auoorrelaed Errors ae Kim Monash Universiy Absra A boosrap bias orreion mehod is applied o saisial inferene in he regression model wih auoorrelaed errors. I is found ha his mehod subsanially redues small sample size disorions relaive o alernaive mehods proposed in he lieraure. I would like o hank Professors re Inder, Max King, an Kivie, and Mihael Veall for useful ommens. Ciaion: Kim, ae, (25) "ias Correed oosrap Inferene for Regression Models wih Auoorrelaed Errors." Eonomis ullein, Vol. 3, No. 44 pp. 8 Submied: May 3, 25. Aeped: Sepember 2, 25. URL: hp:// 5C27A.pdf

2 . Inroduion This paper is onerned wih saisial inferene for he oeffiiens of regression model when he error erm is auoorrelaed. Pas sudies have repored size disorion where he onvenional -es based on a sandard feasible GLS (FGLS) ehnique over-rejes he rue null hypohesis in small samples. This size disorion is pariularly severe, when he daa is rended and auoorrelaion (AR) of error erm is high (see Park and Mihell, 98; King and Giles, 984; Veall, 986; Kwok and Veall, 988; Rayner 99; and Rilsone, 993). A brief review of his lieraure also appears in Li and Maddala (996, Seion 3.3). Pas sudies noed ha he problem is aused mainly by downward bias in he esimaion of he AR oeffiien. In response o his, Kwok and Veall (988) used bias-orreed esimaors for he AR() oeffiien based on he jakknife and boosrap mehods. Rayner (99) ondued a boosrap es proedure where he jakknife is used o esimae he AR() oeffiien, while Rilsone (993) onsidered ieraed boosrap onfidene inervals. Alhough hese auhors repored some improvemens, serious size disorions sill remain espeially when he error erm is highly auoorrelaed. This paper proposes an improved boosrap proedure when saisial inferene is ondued for he regression model wih AR() errors. I is disin from he pas sudies on he following poins. Firs, bias-orreion is ondued in wo sages of he boosrap. Tha is, following Kilian (998), pseudo-daa ses of he boosrap are generaed using a bias-orreed esimaor for he AR() oeffiien, and hen biasorreion is again given o he AR() oeffiien esimae obained from he pseudodaa ses. For his purpose, he bias-orreed esimaors based on he boosrap and jakknife mehods are used. Seondly, he FGSL esimaes for regression oeffiiens are re-alulaed using he bias-orreed esimae for he AR() oeffiien, again in wo sages of he boosrap. As a resul, bias-orreion is implemened o esimaion of he regression oeffiiens as well as o he AR() oeffiien. The hird poin is relaed o he way in whih boosrap inferene is arried ou. In view of he resul obained by van Giersbergen and Kivie (22), aenion is paid o wha hey referred o as he es saisi approah as opposed o he onfidene inerval approah. Wih he former, resampling is ondued using he resried parameer esimaors and residuals under he null hypohesis. Oher hybrid approahes, suh as he one adoped by Rayner (99), an show undesirable small sample properies, aording o van Giersbergen and Kivie (22). Mone Carlo simulaions are ondued o ompare size and overage probabiliies of alernaive bias-orreed boosrap inferene based on he es saisi and onfidene inerval approahes. I is found ha he bias-orreed boosrap es based on he es saisi approah subsanially improves size properies, regardless of wheher biasorreion is ondued using he jakknife or boosrap. I provides empirial size values fairly lose o he nominal one for nearly all ases onsidered. This improvemen is pariularly srong when he daa is rended and he error erm is highly auoorrelaed. The nex seion presens he mehods of parameer esimaion and bias-orreion. Seion 3 provides he deails of he bias-orreed boosrap inferene. Seion 4 presens Mone Carlo resuls, and Seion 5 onludes he paper.

3 2. Parameer Esimaion and ias-correion The simple regression model wih an AR() error erm is onsidered, whih an be wrien as y = β + β x + u, =, 2, 3, n, () u = ρ u - + e, ρ <, (2) where e ~ iid N(,) and x is non-random. For FGLS esimaion of equaion (), he Cohrane-Oru ieraive proedure is used wih he firs observaions adjused based on he Prais-Winsen ransformaion. The FGLS esimaors for β, β, and ρ are denoed as ˆ β, ˆβ and ρˆ, while ê s indiae he assoiaed enred residuals from (2). To orre for he downward bias assoiaed wih ρˆ, he bias-orreed esimaors based on he boosrap and jakknife mehods are used. The former an be alulaed in hree sages: (i) Le u ˆ where e is a random draw from { ê }. Generae.5 = e /( ρ) ˆ = u e u + ê wih replaemen. (ii) Generae pseudo-daa as ˆ ˆ n y = β + βx + u. Using { y, x } =, alulae he boosrap FGLS esimaor ˆρ for ρ. (iii) Repea Sages (i) and (ii) imes o obain he boosrap disribuion ˆ ρ ( ). { i } = ρ, where e is a random draw from { } i If ρˆ <, he boosrap bias-orreed esimaor for ρ is alulaed as ˆ ρ = ˆ ρ ias( ˆ) ρ, (3) where ias( ρˆ ) = ρ - ρˆ and ρ is he sample mean of { ˆ ( i )} i= ρ. When ρˆ >, is se o.99 or.99 following he saionariy orreion proposed by Kilian (998). I has been found o be highly effeive for he bias-orreed boosrap of AR ime series (see, for example, erkowiz and Kilian, 2). The impa of his adjusmen is asympoially negligible, beause i effeively shrinks he bias esimae ias( ρˆ ). No bias-orreion is given o ρˆ when ρˆ, also following Kilian (998). As an alernaive o boosrap bias-orreion, he half-sample jakknife of Quenoulle (949) is used. I akes he following form: ˆ ρ 2 ˆ ( ˆ ˆ = ρ ρ + ρ 2 ) / 2, (4) where ˆρ and ˆρ 2 are he esimaors for ρ on eah half of he sample. If ρˆ >, Fisher s ransformaion F(ρ) =.5 log[(+ρ)/(-ρ)] is used so ha ρˆ is appropriaely bounded, following Kwok and Veall (988). Tha is, F( ρˆ ) an be jakknifed o obain F ˆ ( ˆ ρ) = 2 F( ˆ ρ).5[ F( ˆ ρ ˆ ) + F( ρ)], and he bounded esimae ρˆ an be obained by aking he inverse of F ˆ ( ˆ ρ ). ρˆ 2

4 Le ρˆ { ρˆ, ρˆ }. The regression oeffiiens are re-alulaed using ρˆ, in order o obain he bias-orreed FGLS esimaors ˆβ and ˆβ for β and β. Le se( ˆβ ) denoe he sandard error esimaor for ˆβ, and { ê } he enred residuals alulaed from (2) using ρˆ. ased on hese, we an obain T = ( ˆβ -β)/se( ˆβ ), whih is a bias-orreed version of he onvenional -saisi T = ( ˆβ -β)/se( ˆβ ). 3. ias-correed oosrap The bias-orreed boosrap is used o es for H : β =β agains H : β β. Firs, he resampling sheme based on he onfidene region approah (see van Giersbergen and Kivie; 22) is presened in hree sages:.5 (i) Le u ˆ = e /( ρ ) where e is a random draw from { ê }. Generae u ˆ = ρ u + e, where e is a random draw from { ê } wih replaemen. (ii) Generae pseudo-daa as ˆ ˆ n y = β + β x + u. Using { y, x } =, alulae he boosrap FGLS esimaors ˆ β, ˆβ, and ˆρ for β, β, and ρ. The bias-orreed version of ˆρ is obained using eiher (3) or (4), and denoed as ˆρ. For boosrap bias-orreion, ias( ρˆ ) is used as an approximaion o ias( ˆρ ) following Kilian (998). Using ˆρ, he boosrap FGLS esimaors for β and β are realulaed and denoed as ˆβ and ˆβ. Le se( ˆβ ) denoe he boosrap ounerpar of se( ˆβ ). (iii) Repea Sages (i) and (ii) 2 imes o obain he boosrap disribuion ˆ ˆ ˆ = ( β ( i) β ) / se( β, i. { } 2 z i ) i= From he above boosrap disribuion, (-2α)% bias-orreed boosrap onfidene inerval based on he perenile- mehod (see Efron and Tibshirani, 993) an be onsrued as CI = [ ˆ ˆ ( ), ˆ ( ˆ β z se β β z se β ) ], where z τ is he τ h pereniles of { } 2 i i β CI. α z =. The deision rule is o reje H a he 2α level of signifiane if The resampling sheme based on he es saisi approah (van Giersbergen and Kivie; 22) is similar, bu somewha differen from he onfidene region approah. In addiion o he unresried esimaion and bias-orreion given in Seion 2, he resried esimaion under H and he assoiaed bias-orreion should be ondued. r Le ˆ β, ˆ r β = β and ρˆ r denoe he resried FGLS esimaors for β, β, and ρ under r H. Their bias-orreed versions are denoed as ˆβ, ˆ β r = β, and ρˆ r r, while { ê } is he resried version of { ê }. Noe ha hese boosrap or jakknife bias-orreions are ondued following (3) or (4), bu based on he resried regression. In Sage (i), u r r s are generaed using ρˆ and { ê } insead of ρˆ and { ê }. In Sage (ii), he pseudo daa se is generaed as ˆ r y = β + βx + u. Again, in he ase of boosrap bias- α 3

5 orreion, ias( ρˆ ) is used as an approximaion o ias( ˆρ ). In Sage (iii), he 2 boosrap disribuion of ineres is{ ˆ ˆ m i = ( β ( i) β ) / se( β, i) }. The deision rule i= is o reje H a he 2α level of signifiane, if T [m α, m -α ] where m τ is he τ h pereniles of { } 2 i i m =. 4. Mone Carlo Resuls Mone Carlo simulaions are ondued o ompare size properies assoiaed wih CI, CI, T, T and T. CI and CI are CI s alulaed using ρˆ and ρˆ respeively; while T and T are he ess based on T using ρˆ and ρˆ. The experimenal design loosely follows ha of Rayner (99). The sample sizes n simulaed are 2, 6 and, wih ρ {,.3,.6,.9,.95} and β =β =β =. The number of Mone Carlo rials is se o ; and he numbers of boosrap repliaions and 2 are se o 5 and 2 respeively. Noe ha x is fixed over Mone Carlo rials. The levels of signifiane 2α onsidered are.5 and.. However, only he resuls assoiaed wih he former are repored, beause hose assoiaed wih he laer are found o be qualiaively similar. The simulaion resuls are presened in Table. We firs onsider wo x designs simulaed by Rayner (99), whih are ime rend and GNP daa of Maddala and Rao (973). As seen in previous sudies, he onvenional es T fails dramaially, showing grossly inflaed size values as he value of ρ inreases. This is eviden for boh x designs. CI and CI show muh beer size properies han T, bu hey suffer from serious size disorions when he value of ρ is high and he sample size is small. T shows desirable size properies exep when n = 2 and ρ is high. On he oher hand, T performs well when n = 2 for nearly all ρ values. u is size values end o be lower han 5% when he sample size is larger. Two addiional x designs, labelled DGP and DGP2, are simulaed, whose deails are given a he boom of Table. The DGP is a saionary AR() ime series wih no linear rend, and he DGP2 is an AR() ime series wih linear rend. For DGP, as migh be expeed, all CI s and T s show desirable size properies for all n and ρ, exep for he onvenional es T when n = 2. For DGP2, CI and CI show inflaed size values when he value of ρ is high and he sample size is small. In onras, boh T and T show highly desirable size properies. Noe ha T ends o over-esimae 5% slighly, espeially when he sample size is small and he value of ρ is high; while T ends o under-esimae 5% slighly. As expeed, he onvenional es T shows grossly inflaed size values for he DGP2. As a final noe, he Hildreh-Lu grid searh mehod was onsidered as an alernaive mehod of FGLS esimaion o he Cohrane-Oru. However, he size properies of alernaive bias-orreed boosrap ess are found o be insensiive o he hoie of esimaion mehod, and he deails are no repored here. 4

6 5. Conluding Remarks This paper finds ha he bias-orreed boosrap subsanially improves size disorions of he saisial es in he regression model wih auoorrelaed errors. The bias-orreed boosrap based on he es saisi approah is found o provide superior size properies o ha based on he onfidene region approah, espeially when he sample size is small. Wih he es saisi approah, he bias-orreed boosrap provides empirial size values fairly lose o he nominal one even when he value of he AR oeffiien is high. The exen of improvemen is muh higher han hose repored by pas sudies suh as Rayner (99) and Rilsone (993). oh he boosrap and jakknife are found o be effeive for bias-orreion, bu he resuls sugges ha he boosrap be preferred as a means of bias-orreion when he sample size is more han moderaely large. 5

7 Referenes erkowiz,., and L. Kilian (2) Reen Developmens in oosrapping Time Series, Eonomeri Reviews 9, -48. Efron,., and R.. Tibshirani. (993) An Inroduion o he oosrap, Chapman & Hall: New York. Kilian, L. (998) Small Sample Confidene Inervals for Impulse Response Funions The Review of Eonomis and Saisis 8, King, M.L., and D.E.A. Giles (984) Auoorrelaion Pre-Tesing in he Linear Model: Esimaion, Tesing and Prediion ournal of Eonomeris 25, Kwok,., and M. Veall (988) The kknife and Rgression wih AR() Erors, Eonomis Leers 26, Li, H., and G.S. Maddala (996) oosrapping Time Series Models, Eonomeri Reviews 5, Maddala, G.S., and A.S. Rao (973) Tess for Serial Correlaion in Regression Models wih Lagged Dependen Variables and Serially Correlaed Errors, Eonomeria 4, Park, R.E., and.m. Mihell (98) Esimaing he Auoorrelaed Model wih Trended Daa, ournal of Eonomeris 3, Quenouille, M.H. (949) Approximae Tess of Correlaion in Time Series ournal of he Royal Saisial Soiey Series, Rayner, R.K. (99) Resampling Mehods for Tess in Regression Models wih Auoorrelaed Errors Eonomis Leers 36, Rilsone, P. (993) Some Improvemens for oosrapping Regression Esimaors under Firs-Order Serial Correlaion Eonomis Leers 42, van Giersbergen, N.P.A., and.f. Kivie (22) How o Implemen he oosrap in Sai or Sable Dynami Regression Models: Tes Saisi versus Confidene Region Approah ournal of Eonomeris 8, Veall, M. (986) oosrapping Regression Esimaors under Firs-Order Serial Correlaion, Eonomis Leers 2,

8 Table. Perenage of rejeion of he null hypohesis (The level of signifiane 2α =.5) x ρ n=2 n=6 n= CI CI T T T CI CI T T T CI CI T T T GNP DGP DGP CI : boosrap onfidene inerval, bias-orreion based on he jakknife; CI : boosrap onfidene inerval, bias-orreion based on he boosrap; T : es based on he bias-orreed boosrap, boosrap bias-orreion; T : es based on he bias-orreed boosrap, bias-orreion based on he jakknife; T: es based on FGLS esimaion; DGP: x = +.5 x - + v, v ~iidn(,), DGP2: x = x - + v, v ~iidn(,). No resuls are repored for GNP when n =, beause he sample size of he GNP series given in Maddala and Rao (973) is less han.