1. Fundamentals of regression analysis 2
Obid A.Khakimov

2. OLS Estimation: Hetroscedasticity
If variance of residuals is constant then
Our equation collapses to original variance
Formula.

3. Consequences: The regression coefficients are unbiased
The usual formula for coefficient variances is wrong
The OLS estimation is BLU but not efficient.
t-test and F test are not valid.

4. Method of Generalized Least Squares

5. Method of Generalized Least Squares

6. Hetroscedasticity: Detection.
Graphical Method
Park test
White’s general Hetroscedasticity test
Breush-Pagan-Godfrey Test

7. Park test If coefficient beta is statistically different from zero,
Is not known and we use
If coefficient beta is statistically different from zero,
it indicates that Hetroscedasticity is present.

8. Goldfeld-Quandt Test
Order your sample from lowest to highest. Omit your central
your central observation and divide your sample into two samples.
2. Run two regressions for two samples and obtain RSS1 and RSS2.
RSS1 represents RSS from small X sample.
3. Each RSS has following degrees of freedom
Calculate
Follows F distribution with df of num and denom equal to

9. Breush-Pagan-Godfrey Test
If you reject your Null hypothesis then there is
Hetroscedasticity. .

10. Breush-Pagan-Godfrey Test
Step1. Estimate original regression model and get
residuals .
Step2. Obtain
Step3. Construct
Step4. Estimate the following regression model.
Step5. Obtain
m- is number of parameters of Step 4 regression

11. White’s general Hetroscedasticity test
Step1. Estimate original regression model and get
residuals .
Step2. Estimate
If you reject your Null hypothesis then there is
Hetroscedasticity. .

12. Remedial Measures Weighted Least squares
White’s Hetroscedasticity consistent variance and standard errors.
Transformations according to Hetroscedasticity pattern.

13. LM test score
and assume that
Regress each element of X2 onto all elements of X1 and collect residual in r matrix
Then form u*R
Then run regression 1 on ur

14. Autocorrelation reasons:
Inertia.
Specification Bias: Omitted relevant variables.
Specification bias: Incorrect functional form.
Cobweb phenomenon.
Lags
Data manipulation.
Data Transformation.
Non-stationary

15. Consequences: The regression coefficients are unbiased
The usual formula for coefficient variances is wrong
The OLS estimation is BLU but not efficient.
t-test and F test are not valid.

16. Detection: Breusch-Godfrey
There is no kth order serial correlation
Test Statistic
Where n- number of observations, p-number of residual lag variables.

17. Generalized Least Square
If the value of rho is known
If the value of rho is not known

18. Cochrane-Orcutt procedure :
First estimate original regression and obtain residuals
After runing AR(1) regression obtain the value of Rho and run GLS regression
Using GLS coefficients obtain new residuals and obtain new value of Rho
Continue the process until you get convergence in coefficients.

19. Endogeneity
1. Omission of relevant variables

20. What to do ?
If omitted variable does not relate to other included independent variables then OLS estimator still BLUE
Proxy variables
Use other methods other than OLS

21. Measurement error

22. Measurement error: independent variable

23. Endogenous regressors and bias
Bias. Single equation (OLS) estimators will be biased if one or more regressors is endogenous (jointly dependent).
Consistency. Indirect Least Squares, Instrumental Variables or Two Stage Least Squares. 3