CHAPTER 11 HETEROSKEDASTICITY 1 Chapter 11 Heteroskedasticity 11.1 White’s test for heteroskedasticity Foramodelwithheteroskedasticity, yi = Xprimeiβ + εi, wehave E (b) = β and V ar(b) = (XprimeX)?1 Xprime?X (XprimeX)?1 where ? = diagbracketleftbigσ21,··· ,σ2nbracketrightbig. Wemayexpress Xprime?X = nsummationdisplay i=1 σ2ixixprimei. Supposethat n?1XprimeX = n?1 summationdisplay xixprimei → Q, afiniteandnonsingularmatrix. Then n?1Xprime?X = n?1 summationdisplay σ2ixixprimei canbeestimatedconsistentlyby ?Vn = n?1 nsummationdisplay i=1 e2ixixprimei where ei = yi ?xprimeib. (Notethat b p→ β eveninthepresenceofheteroskedasticity.) Ifthereisnoheteroskedasticity (σ21 = ··· = σ2n), n?1Xprime?X isconsistentlyestimated eitherby?σ2 (n?1XprimeX)where?σ2 = n?1 (y ?Xprimeb)(y ?Xprimeb)as ?Vn.Thus,comparing ?Vn and σ2 (n?1XprimeX) providesanindicatorofheteroskedasticity. Whenthereisnoheteroskedas- ticity, ?Vn ?σ2 (n?1XprimeX) p→ 0. Otherwise, ?Vn ?σ2 (n?1XprimeX) pnotarrowright0. CHAPTER 11 HETEROSKEDASTICITY 2 The test statistic White suggests is WH =nDparenleftbigb,?σ2parenrightbig ?B?1Dparenleftbigb,?σ2parenrightbig, where Dparenleftbigb,?σ2parenrightbig = n?1 summationdisplay Ψprimeiparenleftbige2i ? ?σ2parenrightbig ?B = n?1summationdisplayparenleftbige2i ? ?σ2parenrightbigparenleftBigΨi ? ?ΨparenrightBigprimeparenleftBigΨi ? ?ΨparenrightBig Ψi is the 1 ×K(K+ 1)/2 vector containing the element of the lower triangle of the matrixxixprimei, ?Ψ =n?1 nsummationdisplay i=1 Ψi Under the null of no heteroskedasticity, WH d→χ2K(K+1)/2 (K : no of regressors). Remark 1 Note that Dparenleftbigb,?σ2parenrightbig is the vectorized form of ?V ? ?σ2 (n?1XprimeX). Remark 2 The limiting distribution ofWH depends on the number of regressors in the model. 11.2 Lagrange multiplier test for heteroskedasticity Breusch and Pagan (1979) “A Simple Test for Heteroskedasticity and Random Coefficient Variation.” Econometrica yt =Xprimetβ+εt εt ～iidNparenleftbig0,σ2tparenrightbig σ2t =h(Zprimetα) (the first element ofZt is one) H0 :α2 = ··· =αp = 0 (no heteroskedasticity) TheLM test is LM = 12 parenleftBigsummationdisplay Ztft parenrightBigprimeparenleftBigsummationdisplay ZtZprimet parenrightBig?1parenleftBigsummationdisplay Ztft parenrightBig CHAPTER 11 HETEROSKEDASTICITY 3 where ft = e 2 ?σ2 ?1. eand ?σ2 are obtained by OLS. Asn→∞, LM d→χ2p?1. Remark 1 The LM test is independent of the functional formh(·). Remark 2 We need to specify exogenous variableZt to apply the LM test. 11.3 GLS Suppose thatVar(εi|Xi) =σ2i. The GLS estimator is obtained by regressing Py= ? ?? y1/ √σ 1. .. yn/√σn ? ?? onPx= ? ?? x1/ √σ 1. .. xn/√σn ? ?? This gives the GLS estimator ?βGLS = bracketleftBigg nsummationdisplay i=1 parenleftbigg 1 σ2i parenrightbigg xixprimei bracketrightBigg?1bracketleftBigg nsummationdisplay i=1 parenleftbigg 1 σ2i parenrightbigg xiyi bracketrightBigg . Ifσ2i =xprimeiα,we may write ε2i =σ2i +νi where νi =ε2i ?Eparenleftbigε2i|xiparenrightbig. Replacingε2i withe2i,we have an approximate relation e2i =xprimeiα+ν?i. Running OLS on this equation, we can obtain ?αand ?σi =xprimei?α.The feasible GLS estimator is obtained plugging ?σi into the formula of GLS. We may use other models for heteroskedasticity. Examples are: σ2i = (xprimeiα)2 σ2i = exp(xprimeiα) ... CHAPTER 11 HETEROSKEDASTICITY 4 11.4 Autoregressive conditional heteroskedasticity (ARCH) Consider yt = βprimeXt +εt εt = ut radicalBig α0 +α1ε2t?1, ut ～ iidN (0,1) This is the ARCH (1) model. Since E (εt|εt?1) = radicalBig α0 +α1ε2t?1E(ut) = 0, Var(εt|εt?1) = Eparenleftbigε2t|εt?1parenrightbig = parenleftbigα0 + α1ε2t?1parenrightbigE (ut)2 = α0 +α1ε2t?1 Thus, εt is conditionally heteroskedastic with respect to εt?1. The unconditional variance of εt is Var(εt) = EparenleftbigEparenleftbigε2t|εt?1parenrightbigparenrightbig = α0 +α1Eparenleftbigε2t?1parenrightbig = α0 +α1Var(εt?1). If the unconditional variance does not change over time, Var(εt) = Var(εt?1) = α01?α 1 , |α| < 1. Thus, the model obeys the condition of the classical linear regression model. Various generalizationof theARCH (1) model is available in the literature: ARCH (p), GARCH (1,1), etc.