56546,应用计量经济学原书im

1、Chapter 10Pure HeteroskedasticityPure heteroskedasticityreferred to as heteroskedasticityoccurs when Classical Assumption V is violated in a correctly specified equation.Classical Assumption V assumes homoskedasticity: a constantWith heteroskedasticity, error term variance is not constant and depend

2、s on the observation:Pure Heteroskedasticity (continued)Heteroskedasticity often occurs in data sets in which there is a large disparity between largest and smallest observed values of the dependent variable.Simplest way to visualize pure heteroskedasticity is to picture a world in which observation

3、s of error term can be grouped into two distributions: “wide” and “narrow.”Both groups can be centered on zero but one has a larger variance (Figure 10.1).Pure Heteroskedasticity (continued)Pure Heteroskedasticity (continued)Heteroskedasticity can take on many complex forms.One model has the varianc

4、e of the error term related to an exogenous variable Zi.For a typical regression equation:The variance of the error term might be equal to:Z is called a proportionality factor.Pure Heteroskedasticity (continued)Pure Heteroskedasticity (continued)Pure Heteroskedasticity (continued)Heteroskedasticity

5、can occur in a time-series model with a significant amount of change in the dependent variable.It can also occur in any model, time series or cross-sectional, where the quality of data collection changes dramatically.As measurement errors decrease in size, so should the variance of the error term.He

6、teroskedasticity caused by an error in specification is referred to as impure heteroskedasticity.The Consequences of HeteroskedasticityThere are three major consequences of heteroskedasticity: 1. Pure heteroskedasticity does not cause bias in the coefficient estimates. 2. Heteroskedasticity typicall

7、y causes OLS to no longer be the minimum variance estimator (of all the linear unbiased estimators). 3. Heteroskedasticity causes the OLS estimates of the standard errors to be biased, leading to unreliable hypothesis testing and confidence intervals.Testing for HeteroskedasticityThere are many test

8、s for heteroskedasticity; two popular: 1. Breusch-Pagan test 2. White testBefore testing for heteroskedasticity, start with asking: 1. Are there any obvious specification errors? 2. Are there any early warning signs of heteroskedasticity? 3. Does a graph of the residuals show any evidence of heteros

9、kedasticity?Testing for Heteroskedasticity (continued)Testing for Heteroskedasticity (continued)The Breusch-Pagan TestBreusch-Pagan test investigates whether the squared residuals can be explained by possible proportionality factors.It has three steps: Step 1: Obtain the residuals from the estimated

10、 equation. For an equation with two independent variables:The Breusch-Pagan Test (continued) Step 2: Use the squared residuals as the dependent variable in an auxiliary equation. Step 3: Test the overall significance of Equation 10.7 with a chi-square test.H0: 1 = 2 = 0HA: H0 is falseThe test statis

11、tic is NR2 and the degrees of freedom is equal to the number of slope coefficients in auxiliary.The Breusch-Pagan Test (continued)Example: Woodys restaurant locationAuxiliary equation:R2 = 0.0441, N = 33 so chi-square statistic = 1.4555-percent critical value (3 degrees of freedom) = 7.81We cant rej

12、ect the null hypothesis.There is no evidence of heteroskedasticity.The White TestThe White test investigates whether the squared residuals can be explained by the equations independent variables, their squares, and their cross-products.It has three steps: Step 1: Obtain the residuals from the estima

13、te equation. For an equation with two independent variables:The White Test (continued) Step 2: Use the squared residuals as the dependent variable in an auxiliary equation with each X from the original equation, the square of each X, and the product of each X times every other X as explanatory varia

14、bles.The White Test (continued) Step 3: Test the overall significance of Equation 10.9 with a chi-square test.H0: 1 = 2= 3= 4= 5 = 0HA: H0 is falseThe test statistic is NR2 and the degrees of freedom is equal to the number of slope coefficients in auxiliary.A weakness of the White test is that as th

15、e number of explanatory variables in original regression rises, the number of right hand variables in the White test auxiliary goes up much faster.Remedies for HeteroskedasticityIf heteroskedasticity is found, the first thing to do is examine the equation carefully for specification errors.If there

16、are no obvious specification errors, the heteroskedasticity is probably pure in nature and one of the following remedies should be considered.Heteroskedasticity-CorrectedStandard ErrorsHeteroskedasticity-corrected (HC) standard errors are standard errors calculated specifically to avoid consequences

17、 of heteroskedasticity.HC standard errors are biased but are generally more accurate than uncorrected standard errors in large sample.HC standard errors can be used in t-tests and other hypothesis tests.Redefining the VariablesRedefining variables can help avoid heteroskedasticity.Be careful! Redefi

18、ning variables is a functional form specification change.In some cases, the only redefinition needed is to switch from a linear to double-log functional form.In some situations, it might be necessary to rethink project in terms of its underlying theory.Redefining the Variables (continued)Example: To

19、tal expenditures by city governments where:EXPi = expenditures of the ith cityPOPi = population of the ith cityINCi = e of the ith cityWAGEi = average wage of the ith cityThe transformed equation: Redefining the Variables (continued)Redefining the Variables (continued)A More Complete ExampleExample:

20、 Demand for gasoline by state where: PCONi= petroleum consumption in the ith state (trillions of BTUs) REGi = motor vehicle registrations in the ith state (thousands) PRICEi= the price of gasoline in the ith state (cents per gallon) i= a classical error termA More Complete Example (continued)Using data from 2005 we can estimate:The equation seems to have no problems.Coefficients are significant in the hypothesized direction. The overall equation is statistically significant.A More Complete Example (co