厦门大学：《高级经济计量学》讲义 第六章 异方差 Heteroskedasticity

Chapter5异方差 Heteroskedasticity

Chapter 5 异方差 Heteroskedasticity

1. Recall the assumption for the cmlrm var(;)=σ= const.i=1l,…,n (Homoskedasticity) 2. Counterexamples 1)rich family and poor family expenditures 2)large company and small company sales There exists heteroskedasticity in lots of econometric problems

1. Recall the assumption for the CMLRM: (Homoskedasticity) 2. Counterexamples 1) rich family and poor family expenditures; 2) large company and small company sales. There exists heteroskedasticity in lots of econometric problems. 2 var( ) const. 1, , i  = = =i n

3. What happens if there is heteroske dasticity in an econometric problem? 1) The Ols estimators are maybe not blues( they are not efficient 2) The hypothesis tests for the parameters do not hold good though they is very important and so on

3. What happens if there is heteroske￾dasticity in an econometric problem? 1) The OLS estimators are maybe not Blues (they are not efficient ). 2) The hypothesis tests for the parameters do not hold good though they is very important. and so on

4. Tests for Heteroskedasticity 1) Goldfeld-Quandt Test Given a sample with size n (1) Sort it by the order of an independent variable and then portion it into three parts Sub-sample 1 with size n1 Sub-sample 2 with size n2 Sub-sample 3 with size n3 1 tn<n

4. Tests for Heteroskedasticity 1) Goldfeld-Quandt Test: Given a sample with size n (1) Sort it by the order of an independent variable and then portion it into three parts. Sub-sample 1 with size n1 Sub-sample 2 with size n2 Sub-sample 3 with size n3 1 3 n n n + 

(2)Estimate the regression equation with sub-sample 1 and 3 respectively (3)let k-1 (4)Test H0:=G3wH1:≠σ

(2) Estimate the regression equation with sub-sample 1 and 3 respectively. (3) let (4) Test 2 2 1 1 1 ˆ 1 e n k  = − −  2 2 3 3 3 ˆ 1 e n k  = − −  2 2 2 2 0 1 3 1 1 3 H H : vs :     = 

testing statistic F F(n3-k-1,n1-k-1) Given a significant level a and if the critical value is Fothen reject ho when F> Fa 2)Breusch-Pagan test 3)White test

testing statistic: Given a significant level and if the critical value is , then reject when 2) Breusch-Pagan test 3) White test 2 3 2 3 1 1 ˆ ( 1, 1) ˆ F F n k n k   = − − − −  F H0 F F  

5. Heteroskedasticity Correction Suppose X=b+b1X1+…+bXk+E;(1) and var(e=of=of(x), x1=(X1,11…k),=1,…,m,andf>0 Then var vara f(X f(X) x)0/X )=a2,i=1…,n

5. Heteroskedasticity Correction 1) Suppose (1) and Then Y = b b X b X + i i k ki i 0 1 1 + + +  2 2 1 1 var( ) ( ), ( , , ), 1, , , and 0 i i i i i i ki f X X X i n f    = = = =  X X 2 2 1 1 var( ) var( ) ( ) ( ) 1 ( ) , 1, , ( ) i i i i i i f f f i n f     = = = = X X X X

Let Xn,20 f(X,) f(x (1)becomes V=6Zi0+b,Z1i+ +bi Zki+8i (2) where v=Y/5(X, ) ' =6/f(X,) The variance of the error term in(2) is constant

Let (1) becomes (2) where The variance of the error term in (2) is constant. 0 1 1 , ( ) ( ) ji ji i i i Z X Z f f = = X X * V b Z b Z b Z i i i k ki i 0 0 1 1 = + + + +  * ( ) , ( ) V Y f f i i i i i i = = X X  

2 WLS weighted least squares)Method (加权最小二乘法) If we give a weight to the residual squared then the weighted sum of residual squared is ∑We2=∑W-(h+hX+…+bX) Applying the Ols procedure to it, we obtain the Wls estimator For example the variances of the error terms let, given

2) WLS (weighted least squares) Method (加权最小二乘法） If we give a weight to the residual squared then the weighted sum of residual squared is Applying the OLS procedure to it, we obtain the WLS estimator. For example, given the variances of the error terms, let 2 i e 2 2 0 1 1 [ ( )]   W e W Y b b X b X i i i i i k ki = − + + + 2 1 i i W  =

and we can obtain the parameter estimators Which are blues Homework P109eX6.7

and we can obtain the parameter estimators which are BLUEs. Homework: P109 ex6.7