古扎拉蒂计量经济学第四版讲义Ch7 Heteroscedasticity

第七章 异方差 Heteroscedasticity

1、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=

11.1.2

2、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：

()()()2

222

2var i

i 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。

()()

2

22

var i

b x x σ=

−∑ (3.3.1) (11.2.3)

Proof for 11.2.2.:

从一元回归中已知，

()

()

2

1

i i n

i

i x x k x x =−=

−∑

()2122i i i i i i i b k y k x k ββεβε==++=+∑∑∑

()()()

()()

2

2

2222222221122121211222222

1122var 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()22

i

i E

εσ

=，

()()()()()

()()2222

222112222222222

11222

22222

2var 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 i

i y x x x ββε=++=

Now assume that the heteroscedasticity variance

2i σ are known .

012i i i i

i i i i y x x

εββσσσσ

=++

11.3.3 For ease of exposition we write as

102i i i i y x x ββε∗∗∗∗∗∗=++ 11.3.4

where the starred, or transformed, variables are the original variables divided by (the known) i σ.

What’s the purpose of transforming the original model?

()()2

222221

var 1

1

i i i i i i

i i

E E known

εεεσσσσσ∗ ==← == 11.3.5

Therefore, 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 11.3.3

012i

i i

i

i i i i y x x e b b σσσσ∗∗ =++

or

102i i i i y b x b x e ∗∗∗∗∗

∗=++

11.3.6

Now, to obtain the GLS estimators, we minimize

()2

2102i i i i e y b x b x ∗∗∗∗∗∗=−+∑∑ 11.3.7

The actual mechanics of minimizing 11.3.7 follow the standard calculus techniques. The GLS estimator of 2b ∗

is

()()()()()()()

2

22i

i i

i

i i

i i

i

i i

i i

w w x y w x w y b

w w x w x ∗−=

−∑∑∑∑∑∑∑ 11.3.8 and its variance is given by

()()()()

2

2

2var i

i

i i

i i

w

b w w x w x ∗=

−∑∑∑∑ 11.3.9

where 2

1/i i w σ=.

2）加权最小二乘（WLS ） Weighted Least Squares (WLS)