香港大学：《计量经济学》（英文版） Chapter 11 Heteroskedasticity

CHAPTER 11 HETEROSKEDASTICITY Chapter 1l Heteroskedasticity 11.1 Whites test for heteroskedasticity For a model with het eroskedasticity E(b)=B and Var (b)=(X'XXQX(X'X) ag We may express X9X=∑ uppose that Tx1→Q, a finite and nonsingular matrix. Then XD=7可x can be estimated consistent ly by 1-∑ (Note that b 2, B even in the presence of heteroskedasticity. If there is no heteroskedasticity(of 02),n-lX'QX is consistently estimated either by 8(n-1X'X)where 2=n-1(y-Xb(y-X'b)as Vn. Thus, comparing Vn and 2(n-X'X)provides an indicator of heteroskedasticity. When there is no heteroskedas- ticity, Vn-02(n-1X'X)2,0 Otherwise, Vn-02(n-lX'X)#0

CHAPTER 11 HETEROSKEDASTICITY 1 Chapter 11 Heteroskedasticity 11.1 White’s test for heteroskedasticity For a model with heteroskedasticity, yi = X ′ iβ + εi , we have E (b) = β and V ar (b) = (X ′X) −1 X ′ΩX (X ′X) −1 where Ω = diag
σ 2 1 , · · · , σ2 n . We may express X ′ΩX = n i=1 σ 2 i xix ′ i . Suppose that n −1X ′X = n −1xix ′ i → Q, a finite and nonsingular matrix. Then n −1X ′ΩX = n −1σ 2 i xix ′ i can be estimated consistently by Vˆ n = n −1n i=1 e 2 i xix ′ i where ei = yi − x ′ i b. (Note that b p→ β even in the presence of heteroskedasticity.) If there is no heteroskedasticity (σ 2 1 = · · · = σ 2 n ), n−1X′ΩX is consistently estimated either by σˆ 2 (n −1X′X) where σˆ 2 = n −1 (y − X′ b) (y − X′ b) as Vˆ n. Thus, comparing Vˆ n and σ 2 (n −1X′X) provides an indicator of heteroskedasticity. When there is no heteroskedas￾ticity, Vˆ n − σ 2 (n −1X′X) p→ 0. Otherwise, Vˆ n − σ 2 (n −1X′X) p
0.

HAPTER 11 HETEROSKEDASTIO The test stat ist ic White suggests is WHEnd ba Bid ba E . E y, is the 1 x K(K+ 1)/2 vect or cont aining the element of the lower triangle of the matrix ayIn 亚=n 业 Under the null of no heteroskedasticity, WH→xk(K (K: no of regressors) rk 1 ha-D b,o is he vec--rized f rm-fV-a"(nxX rk 2 The limi-ing dis-ribu-i-n-fwh depends-n he number-f regress-rs in he 11.2 Lagrange multiplier test for heteroskedasticity Breusch and Pagan(1979)"A Simple Test for Het eroskedasticity and Random Coefficient Variation. Econometrica X6+ Et N didn 0.anm' d=h(zla)(the first element of Zt is one) Ho: an=.=ap=0(no heteroskedasticity The lm test is EB 6, E3 Stft

CHAPTER 11 HETEROSKEDASTICITY 2 The test statistic White suggests is WH = nD  b, σˆ 2  Bˆ−1D  b, σˆ 2  , where D  b, σˆ 2  = n −1Ψ ′ i  e 2 i − σˆ 2  Bˆ = n −1 e 2 i − σˆ 2   Ψi − Ψˆ ′  Ψi − Ψˆ  Ψi is the 1 × K (K + 1) /2 vector containing the element of the lower triangle of the matrix xix ′ i , Ψ =ˆ n −1n i=1 Ψi Under the null of no heteroskedasticity, W H d→ χ 2 K(K+1)/2 (K : no of regressors). Remark 1 Note that D  b, σˆ 2  is the vectorized form of Vˆ − σˆ 2 (n −1X′X). Remark 2 The limiting distribution of WH 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 = X ′ tβ + εt εt ∼ iidN  0, σ2 t  σ 2 t = h (Z ′ tα) (the first element of Zt is one) H0 : α2 = · · · = αp = 0 (no heteroskedasticity) The LM test is LM = 1 2 Ztft ′ ZtZ ′ t −1 Ztft 

HAPTER 11 HETEROSKEDASTICIT ty( e and d are obtained by ols. As n LM fXp Remark1~LM→- nd>pxndx-~×fum-b-malf-mhe Remark2W×mx- sp>cify x7-gx- us waria×2t-pply×LM-8 11. 3 GLS Suppose that var B OriEy oi. The GLS estimator is obt ained by regressing /voi P This gives the gls estimat or n77 2 If oy x a, we may write En y a Xv h Replacing el with ei, we have an approximate relation e y c Xv Running Ols on this equation, we can obtain deand d y a, de The feasible GLS estimator is obtained plugging d into the formula of GLs We may use other models for heteroskedasticity. Examples are

CHAPTER 11 HETEROSKEDASTICITY 3 where ft = e 2 σˆ 2 − 1. e and σˆ 2 are obtained by OLS. As n → ∞, LM d→ χ 2 p−1 . Remark 1 The LM test is independent of the functional form h (·). Remark 2 We need to specify exogenous variable Zt to apply the LM test. 11.3 GLS Suppose that V ar (εi |Xi) = σ 2 i . The GLS estimator is obtained by regressing Py =    y1/ √ σ1 . . . yn/ √ σn    on P x =    x1/ √ σ1 . . . xn/ √ σn    This gives the GLS estimator βˆ GLS = n i=1  1 σ 2 i  xix ′ i −1 n i=1  1 σ 2 i  xiyi  . If σ 2 i = x ′ iα, we may write ε 2 i = σ 2 i + νi where νi = ε 2 i − E  ε 2 i |xi  . Replacing ε 2 i with e 2 i , we have an approximate relation e 2 i = x ′ iα + ν ∗ i . Running OLS on this equation, we can obtain αˆ and σˆi = x ′ iα. ˆ The feasible GLS estimator is obtained plugging σˆi into the formula of GLS. We may use other models for heteroskedasticity. Examples are: σ 2 i = (x ′ iα) 2 σ 2 i = exp (x ′ iα) . . .

CHAPTER 11 HETEROSKEDASTICITY 11. 4 Autoregressi ve conditional heteroskedasticity (ARCH) This is the ARCH (1) mode Since 0)t|)+-1) a0+a1)21(u)=0 v120)-1)=02)=1)=(ao+a1)2-1)^(ur) Thus, )t is conditionally heteroskedast ic with respect to )t-1 The unconditional variance of )t is v12()=^(()2-1)=a0+a1^02-1 +a1y12() If the unconditional variance does not change over time 业120)=亚12()+-1 Thus, the model obeys the condition of the classical linear regression model Various generalization of the ARC H (1)model is available in the literature: ARCH(P) GARCH (1,1),etc

CHAPTER 11 HETEROSKEDASTICITY 4 11.4 Autoregressive conditional heteroskedasticity (ARCH) Consider yt = β ′Xt + εt εt = ut  α0 + α1ε 2 t−1 , ut ∼ iidN (0, 1) This is the ARCH (1) model. Since E (εt |εt−1) =  α0 + α1ε 2 t−1E (ut) = 0, V ar (εt |εt−1) = E  ε 2 t |εt−1  =  α0 + α1ε 2 t−1  E (ut) 2 = α0 + α1ε 2 t−1 Thus, εt is conditionally heteroskedastic with respect to εt−1. The unconditional variance of εt is V ar (εt) = E  E  ε 2 t |εt−1  = α0 + α1E  ε 2 t−1  = α0 + α1V ar (εt−1). If the unconditional variance does not change over time, V ar (εt) = V ar (εt−1) = α0 1 − α1 , |α| < 1. Thus, the model obeys the condition of the classical linear regression model. Various generalization of the ARCH (1) model is available in the literature: ARCH (p), GARCH (1, 1), etc