Heteroskedasticity The Problem: Classical assumption of homoskedasticity is violated: V(ei) - not constant When does this happen? Time Series Yields Prices Cross Section consumption patterns firm size - performance (revenue/sales)

Food Expenditures versus Income (Before Taxes) Data Source: 1996 Statistics Canada expenditure survey

Heteroskedasticity - Consequences 1) The OLS estimators: Estimators: Linear unbiased Not best (minimum variance) 2) More Problems: OLS variance estimate is incorrect: 3) True variance:

Ignoring Heteroskedasticity - Using OLS 1) 2) V(b2) - is not correct 3) Hypothesis tests, confidence intervals, F-tests all invalid 3) Using V(b2)* - correct variance estimator is not best

1) Graphical Methods - subjective (residual plots) DIAGNOSTICS 1) Graphical Methods - subjective (residual plots) > focus on the OLS residuals > is there a pattern ? METHOD Plot residuals Are there patterns? 2) Empirical Tests: Park Test Goldfeld Quandt Test Breusch-Pagan Test White Test

1) Graphical Methods: Income and Expenditures

PARK 2 STAGE TEST THE TEST R. E. Park Estimation with Heteroscedastic Error Terms, Econometrica, Vol. 34, No. 4. (Oct., 1966), p. 888. Park proposed the following general hypothesis: The variance of the disturbance increases as the independent variable increases. THE TEST In order to test this hypothesis he proposed the following specific model:

if  0 => heteroscedastic PARK 2 STAGE TEST STEP 1 Run a regression and compute the squared OLS residuals STEP 2 Run the regression equation above: i.e if  0 => heteroscedastic if  = 0 => homoscedastic

Issues with the Park Test 1 ) Functional Form 2) What happens if vi is Heteroscedastic ? 3) Power of the Test (Type II error) Accept Ho:  = 0 (homoscedastic) ?? => Perhaps choose   0.10 (significance)

Park Test: Example with Income and Food Expenditures Data: Statistics Canada 1996 Expenditure Survey 1000 observations were selected at random and then the data were trimmed so that income was a minimum of $5000 and a maximum of $300,000. This left 938 observations. Two versions of the Park Test were run: 1) Regress natural log of the squared residuals on the natural log of income before taxes. 2) Regress the squared residuals on income . predict err, residuals . gen e2 = err^2 . gen le2 = ln(e2) . gen libt = ln(ibt)

Park Test: Example with Income and Food Expenditures . regress le2 libt Source | SS df MS Number of obs = 938 -------------+------------------------------ F( 1, 936) = 4.87 Model | 23.0256613 1 23.0256613 Prob > F = 0.0275 Residual | 4424.28071 936 4.72679563 R-squared = 0.0052 -------------+------------------------------ Adj R-squared = 0.0041 Total | 4447.30637 937 4.74632484 Root MSE = 2.1741 ------------------------------------------------------------------------------ le2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- libt | .22007 .0997099 2.21 0.028 .0243892 .4157509 _cons | 12.28305 1.010301 12.16 0.000 10.30033 14.26577 Conclusion ?

Park Test: Example with Income and Food Expenditures . regress e2 ibt Source | SS df MS Number of obs = 938 -------------+------------------------------ F( 1, 936) = 35.88 Model | 1.4201e+16 1 1.4201e+16 Prob > F = 0.0000 Residual | 3.7050e+17 936 3.9583e+14 R-squared = 0.0369 -------------+------------------------------ Adj R-squared = 0.0359 Total | 3.8470e+17 937 4.1056e+14 Root MSE = 2.0e+07 ------------------------------------------------------------------------------ e2 | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- ibt | 158.3614 26.4392 5.99 0.000 106.4744 210.2483 _cons | 3221097 1055723 3.05 0.002 1149239 5292955 Conclusion ?

White Test (1980) Test statistic: STEP 1: STEP 2: Test H0: Regress OLS residuals on regressors + squares + X-products STEP 2: Test H0: Ho: all slope coefficients = 0 (Jointly) e.g. Homoscedastic residuals Test statistic: