古扎拉蒂计量经济学第四版讲义Ch7 Heteroscedasticity

第七章 异方差 Heteroscedasticity1、异方差的实质异方差和自相关是一对，分别检测误差项的方差和协方差，涉及的方法都是GLS 或EGLS 。

同方差的假定如下表示：()221,2,,i E i n εσ== 11.1.1异方差则表示为()22i i E εσ=11.1.22、存在异方差的OLS 估计首先举一个两变量回归模型的例子：异方差下2β的OLS 估计量与同方差假定下的公式(3.1.6)相同，但是它的方差现在由下式给出：()()()22222var ii i x x b x x σ − = −∑∑11.2.2这显然与同方差下的公式3.3.1不同。

()()222var ib x x σ=−∑ (3.3.1) (11.2.3)Proof for 11.2.2.:从一元回归中已知，()()21i i nii x x k x x =−=−∑()2122i i i i i i i b k y k x k ββεβε==++=+∑∑∑()()()()()2222222222211221212112222221122var 22i i n n n n n n n n b E b E k E k k k k k k k E k k k βεεεεεεεεεεε−−=−==++++++=+++∑这是因为无序列相关的假定，误差项交互项乘积的期望等于0。

由于i k 已知，而且()22ii Eεσ=，()()()()()()()222222211222222222211222222222var n n n n i i i i i i i i b k E k E k E k k k k x x x x x x x x εεεσσσσσσ=+++=+++=−−==−−∑∑∑∑∑可以证明在异方差情况下，2b 估计量仍然是线性的和无偏的；同理，不管误差项是否同方差还是异方差，2b 估计量都是一致的估计量；进一步，2b 是asymptotically normally distributed 。

这里关于一元回归在异方差出现下OLS 估计量2b 的特性可以完全推广到多元回归的情况。

但是，在异方差下，2b 虽然是线性、无偏和一致的，却不是有效的和最优的，不具有无偏估计量族中的最小方差。

3、广义最小二乘Generalized Least Squares (GLS)1）广义最小二乘（GLS ） 还是首先回到简单回归模型12i i i y x ββε=++ or ()10201i i i ii y x x x ββε=++=Now assume that the heteroscedasticity variance2i σ are known .012i i i ii i i i y x xεββσσσσ=++11.3.3 For ease of exposition we write as102i i i i y x x ββε∗∗∗∗∗∗=++ 11.3.4where the starred, or transformed, variables are the original variables divided by (the known) i σ.What’s the purpose of transforming the original model?()()2222221var 11i i i i i ii iE E knownεεεσσσσσ∗ ==← == 11.3.5Therefore, the variance of the transformed disturbance term is now homoscedastic.Since we are still retaining the other assumptions of the CLRM, the finding suggest that if we apply OLS to the transformed model 11.3.3 it will produce estimators that are BLUE.This procedure of transforming the original variables in such a way that the transformed variables satisfy the assumptions of the classical model and then applying OLS to them is known as the method of generalized least squares (GLS). In short, GLS is OLS on the transformed variables that satisfy the standard least-squares assumptions. The estimators thus obtained are known as GLS estimators , and it is these estimators that are BLUE.GLS 的估计程序如下：First, we write down the SRF of 11.3.3012ii iii i i i y x x e b b σσσσ∗∗ =++or102i i i i y b x b x e ∗∗∗∗∗∗=++11.3.6Now, to obtain the GLS estimators, we minimize()22102i i i i e y b x b x ∗∗∗∗∗∗=−+∑∑ 11.3.7The actual mechanics of minimizing 11.3.7 follow the standard calculus techniques. The GLS estimator of 2b ∗is()()()()()()()222ii iii ii iii ii iw w x y w x w y bw w x w x ∗−=−∑∑∑∑∑∑∑ 11.3.8 and its variance is given by()()()()222var iii ii iwb w w x w x ∗=−∑∑∑∑ 11.3.9where 21/i i w σ=.2）加权最小二乘（WLS ） Weighted Least Squares (WLS)以简单回归为例。

The unweighted least squares method minimizes2212()iiie y b b x =−−∑∑ 1The weighted least squares minimizes2212()i iiii w e w y bb x ∗∗=−−∑∑2where 12,b b ∗∗are the weighted least-squares estimators 。

In the case of heteroscedasticity, 21i i w σ=, which are inversely proportional to the variance ofi ε or i y .Differencing 2 with respect to 12,b b ∗∗, we obtain()()212121222()12()i i i i i i ii i i i w e w y b b x bw ew y b b x x b ∗∗∗∗∗∗∂=−−−∂∂=−−−∂∑∑∑∑Setting the preceding expressions equal to zero, we obtain the following two normal equations12212i iii ii i ii ii iw y b w b w xw x y b w x b w x∗∗∗∗=+=+∑∑∑∑∑∑Solving these equations simultaneously, we obtain12b y b x ∗∗∗∗=−()()()()()()()222ii iii ii iii ii iw w x y w x w y bw w x w x ∗−=−∑∑∑∑∑∑∑ where /i i i y w y w =∑∑ and /i i i x w x w ∗=∑∑.3）OLS, GLS and WLS总的说来，WLS 只是GLS 的一个特例，但是，在异方差的背景下，GLS 和WLS 术语可以互换；以后我们会讨论GLS 估计的其它特例。

如果确实存在异方差和序列相关性，则通过GLS 这些违背被有效地消除了；如果不存在异方差和序列相关，则GLS 等价于OLS 。

4）矩阵描述GLS 和EGLSTo take into account heteroscedasticity variances (the elements on the main diagonal of 'εε) and autocorrelations in the error terms (the elements off the main diagonal of 'εε), assume that()2'E σ=εεVwhere V is a known n*n matrix. Therefore, if the model is=+y X βεwhere ()0E =ε and ()2var cov σ−=εV . In case 2σ is unknown , which is typically thecase, V then represents the assumed structure of variances and covariances among the random errorsi ε.Under the stated condition of the variance-covariance of the error terms, it can be shown that()1gls 11''−−−=b X V X X V ygls b is known as the generalized least-squares (GLS) estimator of β.It can also be shown that()()1gls21var cov 'σ−−−=bX V XIt can be proved that glsb is the best linear unbiased estimator (BLUE) of β.The real problem in practice is that we do not know 2σ as well as the true variances and covariances (i.e., the structure of the V matrix). As a solution, we can use the method of estimated (or feasible) generalized least squares (EGLS).For EGLS, we first estimate our model by OLS disregarding the problems of heteroscedasticity and/or autocorrelation. We obtaine the residuals from this estimated model and form the(estimated) variance-covariance matrix of the error term, V. It can be shown that EGLS estimators are consisten t estimators of GLS. Symbolically,()111egls''−−−=bX VX X Vy () ()11egls2var cov 'σ−−−=bX V Xwhere Vis an estimate of V .4、异方差下使用OLS 估计的结果假如我们不使用GLS 方法，而是继续使用OLS 方法，我们分别考虑异方差和不考虑异方差两种情况来分析置信区间和假设检验可能出现的不同情况。