Ch5 Relaxing the Assumptions of the Classical Model

1. Ch5 Relaxing the Assumptions of the Classical Model • 1. Multicollinearity: What Happens if the Regressors Are Correlated? • 2.Heteroscedasticity: What Happens if the Error Variance Is Nonconstant? • 3. Autocorrelation: What Happens if the Error Terms Are correlated?

2. 1.Multicollinearity • Perfect Collinearity • Multicollinearity: two or more variables are highly(but not perfectly) correlated with each other. • The easiest way to test multicollinearity is to examine the standard errors of the coefficients. • Reasonable method to relieve multicollinearity is to drop some highly correlated variables.

3. 1.Test of Multicollinearity • A relatively high in an equation with few significant t statistics; • Relatively high simple correlations between one or more pairs of explanatory variables; • But the above criterion is not very applicable for time series data. And it can not test multicollinearity that arises because three or four variables are related to each other, either.

4. 2.Heteroscedasticity: • Impact of heteroscedasticity on parameter estimators; • Corrections for heteroscedasticity; • Tests for Corrections for heteroscedasticity.

5. 2.Impact of Heteroscedasticity • Existence of heteroscedasticity would make OLS parameter estimators not efficient, although these estimators are still unbiased and consistent ; • Often occurs when dealing with cross-sectional data.

6. 2.Correction of Heteroscedasticity • Known Variance • Unknown Variance (Error Variance Varies Directly with an Independent Variable)

7. Known Variance Two-variable regression model:

8. Known Variance Multiple linear regression model: let Because: Therefore: WLS is BLUE

9. Unknown Variance: Let Because: Therefore: WLS is BLUE

10. 2.Tests of Heteroscedasticity • Informal Test Method: • Observe the residuals to see whether estimated variances • differ from observation to observation. • Formal Test Methods: • Goldfeld-Quandt Test • Breusch-Pagan Test and The White Test

11. Goldfeld-Quandt Test • Steps: • Order the data by the magnitude of Independent Variable; • Omit the middle d observations; • Fit two separate regression, the first for the smaller X and the second for the larger X; • Calculate the residual sum of squares of each regression: RSS1 and RSS2; • Assuming the error process is normally distributed, then RSS2/RSS1~F((N-d-2k)/2, (N-d-2k)/2) .

12. Breusch-Pagan Test • Steps: • First calculate the least-squares residuals and use these residuals to estimate: ; • Run the followingregression: • If the error term is normally distributed and the null hypothesis is valid, then:

13. White Test • Steps: • First calculate the least-squares residuals and use these residuals to estimate: ; • Run the following regression: • When the null hypothesis is valid, then:

14. 3.Serial Correlation: • Impact of serial correlation on OLS estimators; • Corrections for serial correlation; • Tests for serial correlation.

15. Serial Correlation • Serial Correlation often occurs in time-series studies; • Fist-order serial correlation; • Positive serial correlation;

16. Impact of Serial Correlation • Serial correlation will not affect the unbiasedness or consistency of the OLS estimators, but it does not affect their efficiency.

17. Correction of Serial Correlation • The model with serial correlated error terms usually is described as: • Formula for first-order serial-correlation coefficient

18. Correction of Serial Correlation When is known:Generalized Differencing

19. Methods for estimating • The Cochrane-Orcutt Procedure; • The Hildreth-Lu Procedure.

20. Cochrane-Orcutt Procedure Steps: • Using OLS to estimate the original model: • Using the residuals from the above equation to perform the regression: • Using the estimated value of to perform the generalized differencing transformation process and yield new parameters. • Substituting these revised parameters into the original equation and obtaining the new estimated residuals:

21. Cochrane-Orcutt Procedure Steps: • Using these second-round residuals to run the regression and obtain new estimate of ; • The above iterative process can be carried on many times until the new estimates of differ from the old ones by less than 0.01 or 0.005.

22. The Hildreth-Lu Procedure Steps: • Specifying a set of grid values for ; • For each value of , estimating the transformed equation: • Selecting the equation with the lowest sum-of-squared residuals as the best equation; • The above procedure can be continued with new grid values chosen in the neighborhood of the value that is first selected until the desired accuracy is attained.

23. Test of Serial Correlation • Durbin-Watson Test • Test statistic is: • DW~[0,4], value near 2 indicating no first-order serial correlation. Positive serial correlation is associated with DW values below 2, and negative serial correlation is associated with DW values above 2.

24. Range of the DW Statistic Value of DW Result 4-dl<DW<4 Reject the null hypothesis; negative serial 4-du<DW<4-dl Result indeterminate 2<DW<4-du Accept null hypothesis du<DW<2 Accept null hypothesis dl<DW<du Result indeterminate 0<DW<dl Reject null hypothesis; positive serial