which will distributed as FNi-k Na-k under the null hypothesis of homoscedasticity 2N XN-k and by the null assumption that o=of=a2 2.1.2 The Breush-Pagan Test The Goldfeld-Quandt test has been found to be reasonably powerful when we are able to identify correctly the variable to use in the sample separation. This requirement does limit its generality, however. Breush-Pagan(1979)assume a border class of heteroscedasticity defined by 02=02h(ao +4 a1), where zi is a(p x 1)vector of exogenous variables. This model is homoscedastic if a1=0. Breush and Pagen consider the general estimation equation 元2 where e: represent the i-th OlS residual and a-= >ise/N. The null hy- pothesis a1=0 can be tested if the Ei are normally distributed. Let SSR denote the sum of squares obtained in an Ols estimation of Letw=,=∑1/N,amd=a+a1, Then SSR=∑1(,-m2) Breush and Pagan shows, if a1=0, then SsR 2.1.3 White's general est White address the case where nothing is known about the structure of the het- eroscedasticity other than the heteroscedastic variance a? are uniformly bounded It would be desirable to be able to test a general hypothesis of the form for all i Hi Not Ho If there is no heteroscedasticity (under Ho), then s2(X'X)will give a consistent estimator of variance 6, where if there is, then it will not(see Ch. 8 sec. 1). Whitewhich will distributed as FN1−k,N2−k under the null hypothesis of homoscedasticity since e 0 1 e1 σ2 ∼ χ 2 N1−k and by the null assumption that σ 2 = σ 2 1 = σ 2 2 . 2.1.2 The Breush-Pagan Test The Goldfeld-Quandt test has been found to be reasonably powerful when we are able to identify correctly the variable to use in the sample separation. This requirement does limit its generality, however. Breush-Pagan (1979) assume a border class of heteroscedasticity defined by σ 2 = σ 2h(α0 + z 0 iα1), where zi is a (p × 1)vector of exogenous variables. This model is homoscedastic if α1 = 0. Breush and Pagen consider the general estimation equation eˆ 2 i σ¯ 2 = α0 + z 0 iα1 + vi , where eˆi represent the i − th OLS residual and σ¯ 2 = PN i=1 eˆ 2 i /N. The null hy￾pothesis α1 = 0 can be tested if the εi are normally distributed. Let SSR denote the sum of squares obtained in an OLS estimation of eˆi σ¯ 2 = α0 + z 0 iα1 + vi . (Let yi = eˆ 2 i σ¯ 2 , y¯ = PN i=1 yi/N, and yˆi = αˆ0 + z 0 iαˆ1. Then SSR = PN i=1(yˆi − y¯) 2 .) Breush and Pagan shows, if α1 = 0, then 1 2 SSR ∼ χ 2 p . 2.1.3 White’s General Test White address the case where nothing is known about the structure of the het￾eroscedasticity other than the heteroscedastic variance σ 2 i are uniformly bounded. It would be desirable to be able to test a general hypothesis of the form: H0 : σ 2 i = σ 2 for all i, H1 : Not H0. If there is no heteroscedasticity (under H0), then s 2 (X0X) will give a consistent estimator of variance βˆ, where if there is, then it will not (see Ch. 8 sec.1). White 4