1Spring 02 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

2Spring 02 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.

3Spring 02 Tests of Heteroscedasticity Lagrange Multiplier Tests Goldfeld-Quant Test White’s Test

4Spring 02 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 ESS 1 and ESS 2 Calculate

5Spring 02 Problem Salvatore – Data on income and consumption

6Spring 02 Problem

7Spring 02 Problem Regression on the whole sample: Regressions on the first twelve and last twelve observations:

8Spring 02 To Correct for Heteroscedasticity To correct for heteroscedasticity of the form Var(  i )=CX 2, 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

9Spring 02 Problem

10Spring 02 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  i is correlated with  i-1, then we say there is first order serial correlation. Serial correlation may be positive or negative. E(  i,  i-1 )>0 E(  i,  i-1 )<0

11Spring 02 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  hat is biased downward, leading to higher t stats With negative serial correlation, the standard errors of  hat is biased upward, leading to lower t stats

12Spring 02 Durbin-Watson Statistic 0 d L d U 2 4-d U 4-d L 4 +SC inconcl no serial correlation inconcl -SC

13Spring 02 Problem Data 9-4 shows corporate profits and sales in billions of dollars for the manufacturing sector of the U.S. from 1974 to Estimate the equation Profits =  1 +  2 Sales + e Test for first-order serial correlation.

14Spring 02 Problem OLS Estimate of Profit as a function of Sales:

15Spring 02 Problem Test for serial correlation SPSS

16Spring 02 Correcting for Serial Correlation We assume: Where u t is distributed normally with a zero mean and constant variance. Follow a Durbin Procedure

17Spring 02 Correcting for Serial Correlation

18Spring 02 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 .

19Spring 02 Correcting for Serial Correlation Create new independent and dependent variables by the following process: Estimate the following equation:

20Spring 02 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

21Spring 02 Problem Begin by regressing Profit (  ) on Profit lagged one period, Sales, and Sales lagged one period. The estimated coefficient on the lagged dependent variable is .

22Spring 02 Problem  =.49

23Spring 02 Problem Then generate the transformed (starred) variables. Run regression on transformed variables Profit*= Sales* Profit = Sales With no serial correlation