On the Use of White Test for Heteroskedasticity in the Presence of Autocorrelation

Transcription

1 O the Use of White Test for Heteroskedasticity i the Presece of Autocorrelatio by Juwo Seo School of Ecoomics Yosei Uiversity ad Jiook Jeog 1 School of Ecoomics Yosei Uiversity October 008 Abstract We show that the variace-covariace matrix estimator by White (1980) is ot cosistet i the presece of autocorrelatio. Thus, White s correctio is ot valid i the presece of autocorrelatio. We also show that the asymptotic properties of White test is still valid eve i the presece of autocorrelatio, after a simple decompositio of the error variace matrix.. However, the presece of autocorrelatio worses the fiite sample bias i White test. Thus, oe should be cautious whe the sample size is small ad the errors are suspected to be serially correlated. Keywords: White test, heteroskedasticity, autocorrelatio JEL classificatios: C1, C15 Ackowledgemet: We are grateful for the helpful commets from Tae H. Kim ad the participats of 008 KEA coferece at Yosei Uiversity. The authors are members of Brai Korea 1 Research Group of Yosei Uiversity. 1 Correspodig author: Professor Jiook Jeog, School of Ecoomics, Yosei Uiversity, Seoul, Korea; Tel.: ; address: jiook@yosei.ac.kr.

2 I. Itroductio Heteroskedasticity ad autocorrelatio may co-exist i a regressio model. There have bee a umber of studies o autocorrelatio tests i the presece of heteroskedasticity. Epps ad Epps (1977) show that Durbi-Watso test for autocorrelatio is robust to the existece of heteroskedasticity. Harriso ad McCabe (1975) examie the effect of heteroskedasticity o various autocorrelatio tests such as Durbi-Watso test, Geary test, ad Heshaw test. They coclude that the size ad power of the autocorrelatio tests are robust to the existece of heteroskedasticity. Cotrary to these studies, Diebold (1986) shows that the asymptotic sizes of Box-Pierce test ad Ljug-Box test are distorted i the presece of ARCH errors. Wooldridge (1991) also shows that Breusch-Godfrey LM test suffers from asymptotic size distortios uder coditioal heteroskedasticities such as ARCH. Whag (1998) fids that autocorrelatio tests for oliear regressio models are ot valid i the presece of heteroskedasticity, ad proposes a test based o kerel estimatio. O the robustess of heteroskedasticity test i the presece of autocorrelatio, however, there have bee fewer studies. Epps ad Epps (1977) show that Goldfeld- Quadt test ad Glejser test for heteroskedasticity are seriously distorted whe the errors are autocorrelated, ad suggest a correctio usig Cochrae-Orcutt trasformatio. I this paper, we will aalyze the effects of autocorrelatio o White test, oe of the most popular tests for heteroskedasticity, proposed by White (1980). More specifically, we will show that a simple decompositio of White test statistic makes White test robust to the existece of autocorrelatio. We also show that autocorrelatio i the errors could iflate the fiite sample bias i the revised White test.

3 II. White Test i the Presece of Autocorrelatio White (1980) proposes a heteroskedasticity test based o the differece betwee the variace estimator of the regressio coefficiets uder homoskedasticity ad the oe uder heteroskedasticity. Let us cosider the followig liear regressio model. Y i = X i β + ε i ( i = 1,,, ) (1) where (X i, ε i ) is a sequece of idepedet (ot ecessarily idetically distributed) radom vectors, β is a (k 1) vector of true coefficiet, ad ε i is a uobservable radom scalar satisfyig E(X i 'ε i )=0. X i ad Y i are assumed to be fully observable. The homoskedasticity hypothesis is defied as H0: E(ε i ) = E ε j for all i j, while the heteroskedasticity is defied as a violatio of H0. From the model, the distributios of least squares estimator of β, β, are as follows uder homoskedasticity ad uder heteroskedasticity, respectively. (β β ) d N(0, σ plim[ X X ] 1 ) (uder homoskedasticity) () (β β ) d N(0, plim[ X X 1 X Ω X X X 1 ] ) (uder possible heteroskedasticity) (3) i=1 where β = ( X i X i ) 1 i=1 X i Y i (X X) 1 X Y, σ is defied as E(ε i ) i i the case of homoskedasticity, ad Ω is defied as a diagoal matrix as follows. Ω = E(ε ε ) = E( ε1 ) E( ε ) E( ε3 ) E( ε ) 3 (4)

4 White (1980) suggests the followig statistic, based o the differece betwee () ad (3), for testig H0: E(ε i ) = E ε j i j. W = D B 1 D d χ ( K(K+1) where D = 1 i=1 Ψ i [e i σ ], B = 1 [ ) (5) i=1 e i σ ] (Ψ i Ψ ) (Ψ i Ψ ), e i s are the residuals from the least squares estimatio of (1), σ = 1 (e e ), Ψ i is the 1 K(K+1) row vector of the elemets i the lower triagle of X X, ad Ψ is the sample mea of Ψ i. D is a estimator for the differece betwee variaces of () ad (3), which are respectively estimated i terms of cosistecy as follows. σ X X 1 σ E[ X X ] 1 X X 1 1 e ix i X i i=1 X X a. s 0 (uder homoskedasticity) (6) 1 E[ X X ] 1 E X Ω X E[ X X ] 1 a. s 0 (uder possible heteroskedasticity) (7) White (1980) also shows that the White test statistic, W, is asymptotically equivalet to (R ), where R is computed from the followig auxiliary regressio. K K e i = α 0 + j=1 k=j α s X ij X ik + v i (8) There are a umber of studies o the performace of White test. Dastoor (1994) shows that the cosistecy of White test statistic is robust to the form of heteroskedasticity. small sample bias. However, it is well kow that White test suffers distortios from I this respect, MacKio ad White (1985) evaluate the small sample performace of White test i compariso to some alterative heteroskedasticity tests. Cribari-Neto ad Zarkos (1999), Godfrey ad Orme (1999), ad Jeog ad Lee (1999) apply bootstrap methods to improve the fiite sample properties of White test. 4

5 Let us ow geeralize the model (1) by itroducig autocorrelated errors. Allowig autocorrelatio, the variace-covariace matrix of ε becomes: Ω = E(ε ε ) = E( ε1) E( εε 1 ) E( εε 1 3) E( εε 1 ) E( εε 1) E( ε) E( εε 3) E( εε ) E( εε 3 1) E( εε 3 ) E( ε3) E( εε 3 ) E( εε 1) E( εε ) E( εε 3) E( ε ) (9) We assume that the structure of autocorrelatio does ot violate the regularity coditios of White test. To aalyze the effect of autocorrelatio o White test, let us defie the followig matrices, ω ad ω. ω ε ε ε ε (10) ω eˆ eˆ eˆ eˆ (11) We ca decompose Ω ito two matrices, Ω (1) ad Ω (). Ω = Ω (1) + Ω () (1) For example, if the error term is a radom walk or ay other ostatioary process, the assumptio of bouded variace i White s setup is violated. We do ot cosider such cases because the whole practice of variace estimatio or heteroskedasticity testig is rather meaigless with such irregularities. I other words, we cosider the autocorrelatios that guaratee the cosistecy of the least squares estimatio. 5

6 where Ω (1) E(ω), ad Ω () Ω Ω (1). Please ote that Ω (1) is a diagoal matrix with the error variace i each diagoal term, ad Ω () is a collectio of off-diagoal terms, i.e. the covariaces betwee errors, of Ω. Sice the least squares estimator of β, β, is still cosistet with autocorrelatio, the followig result is immediate from White (1980): 1 i=1 e i X i X i 1 i=1 E[ε i X i X i ] a. s 0 (13) Thus, X X 1 1 e ix i X i i=1 X X 1 E[ X X ] 1 E[ X Ω (1) X Please ote that Ω (1) appears i (14) i the place of Ω i (7). ]E[ X X ] 1 We ca summarize the fidigs i the followig propositio. a.s 0 (14) Propositio 1: White s variace-covariace matrix estimator, W vvvvvv (ββ ) = 1 XX XX 1 1 ee ii=1 ii XX ii XX ii of autocorrelatio. XX XX 1 does ot guaratee cosistecy i the presece It seems somewhat obvious that (β β ) has differet limitig distributio with autocorrelated disturbace, as autocorrelatio chages the distributio of X i ε i. However, with the above assumptios, the distributio of (β β ) still remais ormal while its variace is differet. To see this, let us rewrite var(β ) i terms of Ω (1) ad Ω (). var β = plim { X X 1 [ X ω X ] X X 1 } + plim{ X X 1 X [ ε ε ω X ] X X 1 } = E X X 1 E X Ω (1) X E X X 1 + E X X 1 E[ X Ω () X Sice (3) is still satisfied regardless of autocorrelatio structure, ]E X X 1 (15) (β β ) d N(0, plim( X X 6 ) 1 ( X ΩX )( X X ) 1 )

7 W E[ X X ] 1 E X Ω (1) X E[ X X 1 plim [W var β ] = E X X ] 1 a.s 0 (16) 1 X E[ Ω () X ]E X X 1 0 (17) (17) shows that W is ot a cosistet estimator of var β uless Ω () is a ull matrix, which meas o autocorrelatio. Propositio 1 states that the so-called White s Correctio is o loger valid i the presece of autocorrelatio. However, what Propositio 1 also states is that White test ca be doe eve i the presece of autocorrelatio if we appropriately isolate Ω () from Ω (1). To see this poit, let us cosider var( β ) uder homoskedastic autocorrelatio ad uder heteroskedastic autocorrelatio. With autocorrelatio, from (), (3), ad (1), var(β ) = plim X X 1 X Ω X X X 1 = E X X 1 E X Ω (1) X E X X 1 + E X X 1 E X Ω () X E X X 1 (18) Particularly, uder homoskedastic autocorrelatio, var(β ) = plim X X = E X X 1 X Ω X 1 E σ X X X X 1 E X X 1 + E X X 1 X E Ω () X E X X 1 ] (19) Sice the secod terms i (18) ad (19) are idetical, it does ot affect the differece betwee (18) ad (19). I other words, the ull hypothesis of homoskedasticity i the respect of variace ca be simplified as: H 0 : E X X 1 E X Ω (1) X E X X 1 = E X X 1 E σ X X E X X 1 (0) 7

8 Now, the questio is how to estimate both the sides of the homoskedasticity hypothesis i (0). The followig corollaries are immediate from (16) ad the cosistecy of β. Corollary 1: Eve i the presece of autocorrelatio, the White's variacecovariace matrix estimator i Propositio 1 is a cosistet estimator for 1 EE XX XX 1 XX EE ΩΩ (1) XX EE XX XX 1. Corollary : Eve i the presece of autocorrelatio, σ, is a cosistet estimator for σσ. 3 Corollary 1 says White variace-covariace matrix estimator multiplied by is cosistet estimator to estimate the left-had side of ull hypothesis (0), ad corollary says that the usual variace estimator used i White test statistic is good for the estimatio of the right-had side of ull hypothesis (0), eve i the presece of autocorrelatio. After substitutig these estimators, Propositio summarizes the fidigs o White test i the presece of autocorrelatio. Propositio : Eve i the presece of autocorrelatio, WW = DD BB 1 DD AA χ ( KK(KK+1) ) (1) Proof: By Corollaries 1-, ad Theorem i White (1980), with the Liapouov cetral limit theorem replaced by the cetral limit theorem by Serflig (1968). Propositio implies that the usual White test ca be directly used for heteroskedasticity test eve whe the errors are autocorrelated. The robustess of 3 Eve though there exists autocorrelatio problem i the error terms, the cosistecy of β is maitaied uder the regularity coditios that White (1980) postulates, which meas plim σ = σ 8

9 White test to autocorrelatio could be quite useful for the researchers dealig with time-series data, as they are usually ot sure about the ature of serial errors. Moreover, it ca be applied to every autocorrelatio of ukow form, as we did ot impose ay restrictios o Ω (). III. Bias Aalysis As was briefly metioed i sectio II, the small sample bias of White test has bee poited out by may authors. The small sample bias is of course caused by the differece betwee the estimated residuals ad the uobserved errors. Chesher ad Jewitt (1987) derive the fiite sample bias of White test, ad show that the bias could be quite cosiderable eve i large samples, depedig o the desig matrix. Cribari- Neto, Ferrari ad Codeiro (000) propose a complicated bias correctio procedure. I this sectio, we will derive the fiite sample bias of White test i the presece of autocorrelatio based o fixed regressors, ad show that autocorrelatio makes the bias more severe. β Whe the errors are ot autocorrelated, the bias i White's variace-covariace matrix estimator is as follows. Bias i var( β )= (X X) 1 X E(e e ) X(X X) 1 (X X) 1 X Ω X (X X) 1 = (X X) 1 X {MΩ M Ω} d X(X X) 1 = (X X) 1 X {HΩ[H I]} d X (X X) 1 () where e = Y Xβ = Mε, M I X(X X) 1 X ad H X(X X) 1 X. To derive the bias i var (β ) i the presece of autocorrelatio, let us defie M (i) as the i th row of M. The, e i = M (i) ε, ad E(e i ) = M (i) Ω M (i). Or, 9

10 E(ω ) = M ΩM ' 0 0 (1) (1) 0 M ΩM ' () () M ΩM ' (3) (3) (3) Ad, E(ω ) Ω = M ΩM Ω (1) (1) M ΩM Ω () () M ΩM Ω (3) (3) 33 Ω () = D Ω () (4) where M D Usig (3) (5), ΩM ' Ω 0 0 (1) (1) 11 0 M ΩM ' Ω () () M ΩM ' Ω (3) (3) 33 = diag(h Ω[H I]) (5) Bias i var (β )= (X X) 1 X E(ω ) X(X X) 1 (X X) 1 X Ω X (X X) 1 = (X X) 1 X {D Ω () } d X (X X) 1 (6) Equatio (6) shows that the bias due to autocorrelatio still remais i the variace estimator eve if we ca successfully separate the covariace effects, (X X) 1 X Ω () X (X X) 1. To see this, let us cosider: 10

11 M Ω M = M ΩM ' M ΩM ' M ΩM ' M ΩM ' M ΩM M ΩM ' M ΩM ' M ΩM ' M ΩM ' (1) (1) (1) () (1) (3) () (1) () () () (3) (3) (1) (3) () (3) (3) = E(ω )+ μ where μ ij = M (i)ω M (j) if i j 0 if i = j Thus, Bias i var (β ) = (X X) 1 X {D Ω () } d X (X X) 1 =(X X) 1 X {M Ω M μ}x (X X) 1 (X X) 1 {X ΩX }(X X) 1 = (X X) 1 X {HΩ (1) [H I]} X (X X) 1 +(X X) 1 X {HΩ () [H I]} X (X X) 1 (X X) 1 {X μ X} (X X) 1 (7) Compared to (), (7) has two more terms: (X X) 1 X {HΩ () [H I]} X (X X) 1 ad (X X) 1 {X μ X} (X X) 1. These two terms represet the additioal bias from autocorrelatio. The additioal bias i var (β ) due to autocorrelatio weakes the small sample properties of White test. Therefore, as autocorrelatio does ot affect the properties of White test i large samples, it may deteriorate the small sample properties of White test. IV. Mote Carlo Simulatio To verify the fidigs i the sectios II ad III, we perform a Mote Carlo simulatio. The simulatio model is as follows. y i = β 0 + β 1 x i + β z i + ε i (8) 11

12 ε i = z i v i + u i (9) u i = ρu i 1 + ( 1 ρ )η i 4 (30) The regressio coefficiets (β 0, β 1, β ) are fixed at 1, x i ad z i are idepedet radom picks from a uiform distributio ad the stadard ormal distributio, respectively. The white oise error, v i ad η i are draw from the stadard ormal distributio. To remove the iitial value effect i AR(1) process, some observatios i the begiig have bee deleted. Note that the variace of the regressio error, ε i, is: var(ε i ) = z i σ v + σ u (31) as σ η = σ u = 1. cotrolled by σ v Accordig to the structure, the magitude of heteroskedasticity is 5 ad the magitude of autocorrelatio is cotrolled by the value of ρ. 6 To idetify the small sample properties, we experimet with various sample sizes from 30 to 1,000. The omial size of White test is set to 5%. (1) Size Table 1 presets the empirical size of White test, with various levels of autocorrelatio. First, whe there exists o autocorrelatio(ρ = 0), the size of White test is accurate as a whole. I additio, the size becomes more accurate as the sample size icreases, eve whe autocorrelatio exists. The size distortio dimiishes as the sample size grows. This cofirms Propositio i sectio II. Secod, whe sample size is small, White test teds to uder-reject the true ull hypothesis, as the degree of autocorrelatio icreases. This result is related to the bias aalysis i sectio III. 4 By multiplyig 1 ρ oto the white oise error (η), we ca make σ u ot to be affected by the value of ρ. 5 Homoskedasticity is imposed by settig σ v = 0. 6 We try (-0.9, , -0.5, 0, ) for ρ, ad (0, 0., 0.4, 0.6, 0.8, 1) for σ v. 1

13 <Table 1> The Empirical Size of White test (umber of replicatios=50000) ρρ (omial size = 5%) () Power Tables -4 preset the empirical power of White test for =50, =100, ad =500, respectively. As metioed before, higher value for σ v implies more severe heteroskedasticity <Table > The Empirical Power of White test (=50, umber of replicatio=50000) ρρ σσ vv

14 <Table 3> The Empirical Power of White test (=100, umber of replicatio=50000) ρρ σσ vv <Table 4> The Empirical Power of White test (=500, umber of replicatio=50000) 14

15 ρρ σσ vv As show i Tables -4, the effects of autocorrelatio o the power of White test are egligible. We observe a bit of power improvemets by autocorrelatio i small samples, although the improvemets are ot strikigly sigificat. To see if the simulatio results deped o the particular autocorrelatio structure we impose, we try some alterative autocorrelatio patters. Table 5 reports the empirical size of White test whe the errors follow AR() process. 7 As see i Table 5, our coclusio from AR(1) process does ot chage much. The results from the other autocorrelatio structures are similar to Table 5. 8 <Table 5> The Empirical Size of White test for AR() Errors (umber of replicatios=50000) 7 We set ρ 1 = ρ = ρ for simplicity, which are desiged to satisfy statioary coditio of the regressio model. 8 To coserve the space, we did ot report the results. 15

16 ρρ V. Coclusio First, we show that the variace-covariace matrix estimator by White (1980) is ot cosistet i the presece of autocorrelatio. Thus, White s correctio is ot valid i the presece of autocorrelatio. Secod, we show that the asymptotic properties of White test is still valid eve i the presece of autocorrelatio. A simple decompositio of the error variace matrix makes the usual White test feasible eve with autocorrelatio. Third, we show that the presece of autocorrelatio worses the fiite sample bias i White test. Thus, oe should be cautious whe the sample size is small ad the errors are suspected to be serially correlated. We also preset a Mote Carlo simulatio to verify our fidigs. 16

17 Referece Chesher. A ad I. Jewitt(1987) The bias of heteroskedasticity cosistet covariace matrix estimator, Ecoometrica, 55(5) 117- Dastoor, N. K. (1994), A ote o the Eicker-White heteroskedasticity-cosistet covariace matrix estimator, Joural of Quatitative Ecoomics. 10(1), 47-5 Epps T. W. ad M. L. Epps (1977), The robustess of some stadard tests for autocorrelatio ad heteroskedasticity whe both problems are preset, Ecoometrica. 45, Flachaire. E. (005), More efficiet tests robust to heteroskedasticity of ukow form, Ecoometric reviews, 4(), Fracisco C.N ad S. G. Zarkos(1999) Bootstrap methods for heteroskedastic regressio models: evidece o estimatio ad testig, Eoometric reviews. 18(), 11-8 Fracisco C. N., S. L. P. Ferrari ad G. M. Codeiro(000), Improved heteroskedasticitycosistet covariace matrix estimators, Biometrika. 87, Godfrey L. G. ad C. D. Orme(1999), The robustess, reliability ad power of heteroskedasticity tests, Ecoometric reviews, 18(), Godfrey L. G., C. D. Orme, ad J. M. C. Satos Silva(006), Simulatio-based tests for heteroskedasticity i liear regressio models: Some further results, Ecoometrics joural. 9(1), Harriso M. J. ad B. P. M. McCabe(1975), Autocorrelatio with heteroskedasticity: A ote o the robustess of the Durbi- Watso, Geary ad Heshaw tests, Biometrika. 6, Jeog. J ad Lee K (1999), Bootstrapped White s test for heteroskedasticity i regressio models, Ecoomics letters 63, MacKio, J. G. ad H. White (1985), Some heteroskedasticity-cosistet covariace matrix estimators with improved fiite sample properties, Joural of Ecoometrics 9, Seflig, R.J. (1968), Cotributios to Cetral Limit Theory for Depedet Variables, The Aals of Mathematical Statistics 39, Whag. Y. (1998), A test of autocorrelatio i the presece of heteroskedasticity of ukow form, Ecoometric Theory,14,

18 White. H (1980), A heteroskedasticity-cosistet covariace matrix estimator ad a direct test for heteroskedasticity, Ecoometrica 48,