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1、Econometrics王维国王维国东北财经大学东北财经大学计量经济学计量经济学东北财经大学数量经济系本本章内容章内容第一节 异方差的性质第二节 异方差的后果第三节 异方差的检验第四节 异方差的补救措施第八讲第八讲 异方差异方差东北财经大学数量经济系定义：对于不同的观测点，随机扰动项ui的方差不同。用公式表示为：注意这样一个事实，在给定X的条件下：异方差问题多存在于横截面数据中。第一节第一节 异方差异方差的性质的性质东北财经大学数量经济系1. OLS估计量仍然是线性无偏的。2. OLS估计量不再具有最小方差性。3. 样本方差不再是总体方差的无偏估计量。4. OLS估计量的方差的估计量是有偏的。
2、5. t检验和F检验失效。第二节第二节 异方差异方差的后果的后果东北财经大学数量经济系q检验回归模型中是否存在异方差问题，也就是检验对于不同的观测点i，随机扰动项ui的方差是否相同。q然而，我们很少能够知道整个总体的信息。更一般地，我们仅仅知道一个样本。也就是说我们仅有与给定变量X值相对应的单独的一个Y值。而根据单独的这个Y值无法确定对应于给定X值的Y的条件分布的方差。第三节第三节 异方差异方差的检验的检验东北财经大学数量经济系一、一、帕克检验（帕克检验（1 1）q由图形法可知，如果存在异方差问题，那么异方差 可能与一个或多个解释变量系统相关。利用这一原理，帕克作 对一个或多个解释变量的回归。
3、例如在双变量模型中，我们可以得到如下的回归模型：这就是我们通常所说的帕克检验。注意，在帕克检验中模型的函数形式是不唯一的。东北财经大学数量经济系一、一、帕克检验（帕克检验（2 2）在实际应用中，帕克建议用 来代替 ，得到下面的回归模型：注意：在帕克检验的方程中误差项也可能存在异方差问题。东北财经大学数量经济系q帕克检验的步骤： 1.作普通最小二乘回归，不考虑异方差问题。 2.从原始回归方程中求得残差序列ei ,并求其平方,再取对数形式。 3.利用原始模型中的一个解释变量做回归；如果有多个解释变量，则对每个解释变量作上面形式的回归，或者作被解释变量估计值的回归。 4.检验零假设:B2=0。 5.
4、如果拒绝零假设，则原模型中存在异方差问题；如果接受零假设，则表示为同方差。一、一、帕克检验（帕克检验（3 3）东北财经大学数量经济系从原始模型中获得残差ei之后，格莱舍尔建议做残差的绝对值|ei|对Xi的回归分析。函数形式如下:q在每种情形下，若都接受零假设：B2=0，则表示不存在异方差；如果有一种情况拒绝零假设，则表明可能存在着异方差。二、二、 GlejserGlejser检验检验东北财经大学数量经济系假定有如下回归模型：q怀特检验步骤：1.首先用OLS估计回归方程，获得残差ei。2.作辅助回归：三、三、怀特检验（怀特检验（1 1）东北财经大学数量经济系3.求辅助回归方程的R2值。在零假设:
5、不存在异方差下，怀特证明了R2值与样本容量n的乘积服从 分布:q自由度等于辅助回归方程中解释变量的个数,不包括截距项。4.如果从辅助回归方程中计算得到的统计量值大于所选显著水平下分布的临界值，则拒绝零假设，表示存在异方差。如果计算的统计量的值小于临界值，则不能拒绝零假设。三、三、怀特检验（怀特检验（2 2）东北财经大学数量经济系 异方差的存在并不破坏普通最小二乘估计量的无偏性，但是估计量却不是有效的，即使对大样本也是如此。缺乏有效性就使得通常的假设检验值不可靠。因此，如果怀疑存在异方差或者已经检测到存在异方差，那么我们就要积极地寻求补救措施，最常用的方法是加权最小二乘法。第四节第四节 异方差异
6、方差的补救措施的补救措施东北财经大学数量经济系考虑双变量回归模型： 假设误差方差 已知，对模型作如下变换： 令 ，把它称为变换后的误差项。加权最小二乘法加权最小二乘法 WLSWLS（1 1）东北财经大学数量经济系q则 满足同方差性，从而可以按常规的方法进行回归分析。在实际估计回归模型时，将Y和X的每个观察值都除以已知 ，然后再对这些变换后的数据进行OLS回归，由此获得的估计量就称为加权最小二乘估计量， 为权数。这种估计方法就称为加权最小二乘法。加权最小二乘法加权最小二乘法 WLSWLS（2 2）东北财经大学数量经济系q 为未知的情况情形1：误差方差与Xi成比例：作如下变换：加权最小二乘法加权最
7、小二乘法 WLSWLS（3 3）东北财经大学数量经济系情形2：误差方差与Xi2成比例：作如下变换：加权最小二乘法加权最小二乘法 WLSWLS（4 4）东北财经大学数量经济系. .xiyi0Homoscedastic pattern of errorsThe scattered points spread out quite equally 东北财经大学数量经济系The variance of Yi increases as family income, X1i, increases.Heteroscedasticity Case.x1ix11x12Yif(Yi)expenditurex13.i
8、ncomeVar(i) = E(i2)= i2 东北财经大学数量经济系Heteroscedastic pattern of errors.xiyi0The scattered points spread out quite unequally Small i associated with small value of Xilarge i associated with large value of Xi东北财经大学数量经济系Definition of Heteroscedasticity:Two-variable regression: Yi = 0 + 1 X1i + i 1 = = wi
9、 Yi = wi (0 + 1Xi + i) xy x2= 1 = 1 + wi i E(1) = 1 unbiasedif 12 = 22 = 32 = i.e., homoscedasticity 2 2 xi2Var ( 1) = xi2 if 12 22 32 i.e., heteroscedasticity= i i2 2= i i2 x2( xi2)2Var ( 1) = E (1 - 1)2 = E (wi i)2= E (w12 12 + w22 22 + . + 2w1w2 1 2 + )= w12 12 + w22 22 + .+ 0 + .= i i2 wi2Var( i
10、 i) = E( i i2) = i i2 2 2 2Refer to lecture notesSupplement #03A 东北财经大学数量经济系Consequences of heteroscedasticity1. OLS estimators are still linear and unbiased2. Var ( i )s are not minimum. = not the best = not efficiency = notnot BLUE4. 2 = is biased, E(E( 2 2) ) 2 2 i2 n-k-1 5. t and F statistics ar
11、e unreliableunreliable.Y = 0+ 1 X + Cannot be min.SEE = RSS = i23. Var ( 1) = instead of Var( 1)= i2 x2 2x2Two-variable case东北财经大学数量经济系Detection of heteroscedasticity1. Graphical method :plot the estimated residual ( i ) or squared (i 2 ) against the predicted dependent Variable (Yi) or any independ
12、ent variable(Xi).Observe the graph whether there is a systematic pattern as:Y 2Yes, heteroscedasticity exists东北财经大学数量经济系Detection of heteroscedasticity: Graphical method Y 2yesY 2yesY2yesY2yesY 2yesY 2no heteroscedasticity东北财经大学数量经济系Yes, Yes, heteroscedasticityheteroscedasticityno no heteroscedastic
13、ityheteroscedasticityYes, Yes, heteroscedasticityheteroscedasticityYes, Yes, heteroscedasticityheteroscedasticity东北财经大学数量经济系Park test procedures:1. Run OLS on regression: Yi = 1 + 2 Xi + i , obtain i2. Take square and take log : ln ( i2)4. Use t-test to test H0 : 2* = 0 (Homoscedasticity)If t* tc =
14、reject H0 = heteroscedasticity existsIf t* not reject H0 = homoscedasticity3. Run OLS on regression: ln ( i2) = 1* + 2* ln Xi + viStatistical test: (i) Park testH0 : No heteroscedasticity exists i.e., Var( i ) = 2 (homoscedasticity)H1 : Yes, heteroscedasticity exists i.e., Var( i ) = i2Suspected var
15、iablethat causes heteroscedasticity东北财经大学数量经济系Example: Studenmund (2006), Equation 10.21 (Table 10.1), pp.370-1Park TestPracticalIn EVIEWSProcedure 1:PCON: petroleum consumption in the ith stateREG: motor vehicle registrationTAX: the gasoline tax rateMay misleading东北财经大学数量经济系Graphical detectionGraph
16、ical detection东北财经大学数量经济系Procedure 2: Obtain the residuals, take square and take log 东北财经大学数量经济系Scatter plotHorizontal variable东北财经大学数量经济系东北财经大学数量经济系东北财经大学数量经济系Procedure 3 & 4Refers to Studenmund (2006), Eq.(10.23), pp.373If | t | tc = reject H0 = heteroscedasticity 东北财经大学数量经济系(ii) Whites general he
17、teroscedasticity test (no cross-term) (Breusch-Pagan test, or LM test)(4) Compare the W and 2df (where the df is #(q) of regressors in (2)if W 2df = reject the HoH0 : homoscedasticity Var ( i ) = 2H1 : heteroscedasticity Var ( i ) = i i2 2(3) Compute LM=W= n R2Or F= R2u / q(1 - R2u) / n-kif F* Fcdf
18、= reject the HoTest procedures:(1) Run OLS on regression: Yi = 0 + 1X1i + 2X2i +.+ qXqi + i , obtain the residuals, i(2) Run the auxiliary regression: i2 = 0 + 1 X1i + 2X2i + +qXqi + vi东北财经大学数量经济系Yi = 0 + 1X1i + 2X2i + 3X3i + i东北财经大学数量经济系WW=BPG test for a linear modellinear modelPCON=0+1REG+2Tax+The
19、 WW-statistic indicates thatthe heteroscedasticity is existed.FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 Decision rule: WW 2df = reject the Ho东北财经大学数量经济系FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 Decision rule: WW 2df = reject the HoWW=The BPG test for a transformed log
20、-log model:log(PCON)=0+1log(REG)+2log(Tax)+The WW-statistic indicates that the heteroscedasticity is still existed.Therefore, a double-log transformation may notnot necessarilynecessarily remedy theHeterocsedasticity.东北财经大学数量经济系(4) Compare the W and 2df (where the df is # of regressors in (2)if W 2d
21、f = reject the HoH0 : homoscedasticity Var ( i ) = 2H1 : heteroscedasticity Var ( i ) = i i2 2(3) Compute W (or LM) = n R2Test procedures:(1) Run OLS on regression: Yi = 0 + 1 X1i + 2 X2i + i , obtain the residuals, i(2) Run the auxiliary regression: i2 = 0 + 1 X1i + 2 X2i + 3 X21i + 4 X22i + 5 X X1
22、i 1i X X2i2i + vi(ii) Whites generalgeneral heteroscedasticity test (with cross-termswith cross-terms) (The White Test)(The White Test)Cross-termCross-term东北财经大学数量经济系东北财经大学数量经济系W=White test for a linear modellinear modelPCON=0+1REG+2Tax+The WW-statistic indicates thatthe heteroscedasticity is existe
23、d.FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 Decision rule: W 2df = reject the Ho东北财经大学数量经济系FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 Decision rule: W 2df = reject the HoW=The White test for a transformed log-log model:log(PCON)=0+1log(REG)+2log(Tax)+The WW-statistic i
24、ndicates that the heteroscedasticity is still existed.Therefore, a double-log transformation may notnot necessarilynecessarily remedy theHeterocsedasticity.东北财经大学数量经济系Another example 8.4 (Wooldridge(2003), pp.258)The White test for a linear modelPCON=0+1REG+2Tax+The test statistic indicates heterosc
25、edasticity is existed.2(0.05, 9) = 16.922(0.10, 9) = 14.68 W =Decision rule: W 2df = reject the H0东北财经大学数量经济系Testing the log-log modelThe White test for a log-log modelThe test statistic indicates heteroscedasticity is not existedUsing the log-log transformationin some cases may remedy the heterosce
26、dasticity, (But not necessaryBut not necessary).2(0.05, 9) = 16.922(0.10, 9) = 14.68 Decision rule: WW not reject H0W=东北财经大学数量经济系RemedyRemedy :Weighted Least SquaresWeighted Least Squares (WLSWLS)Suppose : Yi = 0 + 1 X1i + 2 X2i + iE(i) = 0, E(i j )= 0 i jVqr (i2) = i2 = 2 f (Z ZX2i) = 2Z Zi2If all
27、Z Zi i = 1 (or any constant), homoscedasticity returns. But Z Zi i can be any value, and it is the proportionality factor.In the case of 2 was known :To correct the heteroscedasticityTransform the regression: Yi 1 X1i X2i i =0 + 1 + 2 +Z Zi i Z Zi i Z Zi i Z Zi i Z Zi i= Y Y* * = = 0 0 X X0 0* * + +
28、 1 1 X X1 1* * + + 2 2 X X2 2* + * + i i* *If Var(i2)=2Z Zi iThen each term divided by Z Zi i东北财经大学数量经济系In the transformed equation where(i) E ( ) = E (i) = 0iZ Zi iZ Zi i1(ii) E ( )2 = E (i2) = Z Zi i2 22 = 2iZ Zi iZ Zi i2 211Z Zi i2 2(iii) E ( ) = E ( i j ) = 0ijZ Zi iZ Zj jZ Zi iZ Zj j1These thre
29、e results satisfy the assumptions of classical OLSThese three results satisfy the assumptions of classical OLS. .Why the WLS transformation can remove the Why the WLS transformation can remove the heteroscedasticityheteroscedasticity? ?东北财经大学数量经济系Therefore, we might expect i2 = Zi2 2Zi2 = X2i2 =Zi =
30、X2iThese plots suggest variance is increasing proportional to X2i2. The scattered plots spreading out as nonlinear patternnonlinear pattern.= Yi* = 1 X0* + 2 X1* + 3 + *Now this becomes the interceptcoefficientIf the residuals plot against X2i are as following :X2i+0-X22Yi 1 X1i X2i iX2i X2i X2i X2i
31、 X2i = 0 + 1 + 2 +Hence, the transformed equation becomesWhere * *i i satisfies the assumptions of classical OLS东北财经大学数量经济系Example: Studenmund (2006), Eq. 10.24, pp.374C.V.=0.3392东北财经大学数量经济系The correction is built in EVIEWSUse the weight (1/REG)(1/REG)to remedy theheteroscedasticity东北财经大学数量经济系Refers
32、 to Studenmund (2006), Eq.(10.28), pp.376OLS resultWLS result东北财经大学数量经济系FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 W=W=Decision rule: WW not reject HoNow, after the reformulationNow, after the reformulationThe test statistic indicates thatthe heteroscedasticity is not existed.东北财经大学数量
33、经济系Alternative remedy of heteroscedasticity: reformulate with per capitaper capita 东北财经大学数量经济系W=Now, after the reformulationNow, after the reformulationThe test statistic indicates thatthe heteroscedasticity is not existed.FC(0.05, 5, 44) = 2.452(0.05, 5) = 11.072(0.10, 5) = 9.24 Decision rule: W no
34、t reject Ho东北财经大学数量经济系WW not reject Ho东北财经大学数量经济系This plot suggests a variance is increasing proportional to X2i. The scattered plots spreading out as a linear patternlinear patternIf the residuals plot against X2i are as following :X2i+0-X22Therefore, we might expect i2 = Zi 2hi2 = X2i = hi = X2i=
35、Yi* = 1 X0* + 2 X1* + 3 X2* + * = 0 + 1 + 2 +Yi 1 X1i X2i iX2i X2i X2i X2i X2iThe transformed equation is东北财经大学数量经济系Transformation: divided by the squared root term -“sqrt(sqrt(X X) )” = 0 + 1 + 2 +Yi 1 X1i X2i iX2i X2i X2i X2i X2i东北财经大学数量经济系calculate: the C.V. = 0.369282(0.05, 5) = 11.072(0.10, 5)
36、= 9.24 W not reject HoCompare to the transformationdivided by the REG, the CV ofThat one is smaller.东北财经大学数量经济系Simple OLS result :R&D = 192.99 + 0.0319 Sales SEE = 2759 (0.194) (3.830) C.V. = 0.9026Example: Gujarati (1995), Table 11.5, pp.388东北财经大学数量经济系White Test for heteroscedasticity2(0.05, 2)= 5.
37、99142(0.10, 2)= 4.60517W=东北财经大学数量经济系Observe the shape pattern of residuals: linear or nonlinear?linear or nonlinear? 东北财经大学数量经济系( ) = -246.67 + 0.036 XiYiXi1Xi(-0.64) (5.17)=(1) R&Di = -246.67 + 0.036 Sales SEE = 7.25 (-0.64) (5.17) C.V. = 0.8195C.V. = 0.81951.Transformation equations: ( ) = 0 + 1Yi
38、Xi1 Xi Xi Xi(2) R&D = -243.49 + 0.0369 Sales SEE = 0.021(-1.79) (5.52) C.V. = 0.7467C.V. = 0.74672.Compare the C.V. To determine which weight is appropriated东北财经大学数量经济系Transformation: divided by the squared root term -“sqrt(sqrt(X X) )”1Xi0Xi+1YiXi=东北财经大学数量经济系1Xi0Xi+1YiXi=calculate: the C.V. = 0.819
39、5东北财经大学数量经济系After transformation by sqrt(xsqrt(x) ), , the W-statistic indicates there is no heteroscedasticity2(0.05, 2)= 5.99142(0.10, 2)= 4.60517W not reject Ho东北财经大学数量经济系After transformation by sqrt(Xi), residuals still spread out东北财经大学数量经济系Transformation: divided by the suspected variable (Xi)1
40、Xi0+ 1YiXi=东北财经大学数量经济系1Xi0+ 1YiXi=Calculate the C.V. = 0.7467东北财经大学数量经济系After transformation divided by the suspected X, the W-statistic indicates there is no heterosecedasticity2(0.05, 2)= 5.99142(0.10, 2)= 4.60517W not reject Ho东北财经大学数量经济系After transformation divided by Xi, residuals spread out more stable
