Problems in Regression Analysis

1. Problems in Regression Analysis • Heteroscedasticity • Violation of the constancy of the variance of the errors. • Cross-sectional data • Serial Correlation • Violation of uncorrelated error terms • Time-series data

2. Heteroscedasticity • The OLS model assumes homoscedasticity, i.e., the variance of the errors is constant. In some regressions, especially in cross-sectional studies, this assumption may be violated. • When heteroscedasticity is present, OLS estimation puts more weight on the observations which have large error variances than on those with small error variances. • The OLS estimates are unbiased but they are inefficient but have larger than minimum variance.

3. Tests of Heteroscedasticity • Lagrange Multiplier Tests • Goldfeld-Quant Test • White’s Test

4. Goldfeld-Quant Test • Order the data by the magnitude of the independent variable, X, which is thouth to be related to the error variance. • Omit the middle d observations. (d might be 1/5 of the total sample size) • Fit two separate regressions; one for the low values, another for the high values • Calculate ESS1 and ESS2 • Calculate

5. Problem • Salvatore – Data on income and consumption

6. Problem

7. Problem Regression on the whole sample: Regressions on the first twelve and last twelve observations:

8. To Correct for Heteroscedasticity • To correct for heteroscedasticity of the form Var(ei)=CX2, where C is a nonzero constant, transform the variables by dividing through by the problematic variable. • In the two variable case, • The transformed error term is now homoscedastic

9. Problem

10. Serial Correlation • This is the problem which arises in OLS estimation when the errors are not independent. • The error term in one period is correlated with error terms in previous periods. • If ei is correlated with ei-1, then we say there is first order serial correlation. • Serial correlation may be positive or negative. • E(ei,ei-1)>0 • E(ei,ei-1)<0

11. Serial Correlation • If serial correlation is present, the OLS estimates are still unbiased and consistent, but the standard errors are biased, leading to incorrect statistical tests and biased confidence intervals. • With positive serial correlation, the standard errors of bhat is biased downward, leading to higher t stats • With negative serial correlation, the standard errors of bhat is biased upward, leading to lower t stats

12. Durbin-Watson Statistic 0 dL dU 2 4-dU 4-dL 4 +SC inconcl no serial correlation inconcl -SC

13. Problem • Data 9-4 shows corporate profits and sales in billions of dollars for the manufacturing sector of the U.S. from 1974 to 1994. • Estimate the equation Profits = b1+b2Sales + e • Test for first-order serial correlation.

14. Problem • OLS Estimate of Profit as a function of Sales:

15. Problem • Test for serial correlation • SPSS

16. Correcting for Serial Correlation • We assume: • Where ut is distributed normally with a zero mean and constant variance. • Follow a Durbin Procedure

17. Correcting for Serial Correlation

18. Correcting for Serial Correlation • Move the lagged dependent variable term to the right-hand side and estimate the equation using OLS. The estimated coefficient on the lagged dependent variable is r.

19. Correcting for Serial Correlation • Create new independent and dependent variables by the following process: • Estimate the following equation:

20. Correcting for Serial Correlation • The estimates of the slope coefficients are the same (but corrected for serial correlation) as in the original equation. • The constant of the regression on the transformed variables is

21. Problem • Begin by regressing Profit (p) on Profit lagged one period, Sales, and Sales lagged one period. • The estimated coefficient on the lagged dependent variable is r.

22. Problem • r = .49

23. Problem • Then generate the transformed (starred) variables. • Run regression on transformed variables • Profit*=.167+.042 Sales* • Profit = .327 +.027 Sales • With no serial correlation