HETEROSCEDASTICITYThe assumption of equal variance Var(ui) = 2, for all i, is called homoscedasticity, which means equal scatter (of the error terms ui around their mean 0)

Equivalently, this means that the dispersion of the observed values of Y around the regression line is the same across all observationsIf the above assumption of homoscedasticity does not hold, we have heteroscedasticity (unequal scatter)

Consequences of ignoring heteroscedasticity during the OLS procedureThe estimates and forecasts based on them will still be unbiased and consistentHowever, the OLS estimates are no longer the best (B in BLUE) and thus will be inefficient. Forecasts will also be inefficient

The estimated variances and covariances of the regression coefficients will be biased and inconsistent, and hence the t- and F-tests will be invalid

Testing for heteroscedasticity1. Before any formal tests, visually examine the models residuals i Graph the i or i2 separately against each explanatory variable Xj, or against i, the fitted values of the dependent variable

2. The Goldfeld-Quandt test Step 1. Arrange the data from smallto large values of the indp variable XjStep 2. Run two separate regressions,one for small values of Xj and one forlarge values of Xj, omitting d middleobservations (app. 20%), and recordthe residual sum of squares RSS foreach regression: RSS1 for smallvalues of Xj and RSS2 for large Xjs.

Step 3. Calculate the ratioF = RSS2/RSS1, which has an F distribution with d.f. = [n d 2(k+1)]/2 both in the numerator and the denominator, where n is the total # of observations, d is the # of omitted observations, and k is the # of explanatory variables.

Step4. Reject H0: All the variances i2 are equal (i.e., homoscedastic) if F > Fcr, where Fcr is found in the table of the F distribution for [n-d-2(k+1)]/2 d.f. and for a predetermined level of significance , typically 5%.

Drawbacks of the the Goldfeld-Quandt test It cannot accommodate situations where several variables jointly cause heteroscedasticityThe middle d observations are lost

3. Lagrange Multiplier (LM) tests (for large n>30)The Breusch-Pagan testStep 1. Run the regression of i2on all the explanatory variables. Inour example (CN p. 37), there isonly one explanatory variable, X1,therefore the model for the OLSestimation has the form: i2 = 0 + 1X1i + vi

Step 2. Keep the R2 from this regression. Lets call it R22 Calculate either(a) F = (R22/k)/[(1-R22)/(n-(k+1)], where k is the # of explanatory variables; the F statistic has an F distribution with d.f. = [k, n-(k+1)] Reject H0: All the variances i2 are equal (i.e., homoscedastic) if F >Fcr

or(b) LM = n R22, where LM is called the Lagrangian Multiplier (LM) statistic and has an asymptotic chi-square (2) distribution with d.f. = kReject H0: All the variances i2 are equal (i.e., homoscedastic) if LM> cr2

Drawbacks of the Breusch-Pagan test It has been shown to be sensitive to any violation of the normality assumptionThree other popular LM tests: the Glejser test; the Harvey-Godfrey test, and the Park test, are also sensitive to such violations (wont be covered in this course)

One LM test, the White test, does not depend on the normality assumption; therefore it is recommended over all the other tests

The White test Step 1.The test is based on the regr.of 2 on all the explanatory varia-bles (Xj), their squares (Xj2), andall their cross products. E.g., whenthe model contains k = 2 explanat.variables, the test is based on an estim. of the model: 2 =0+ 1X1 +2X2+3X12+4X22 + 5X1X2 + v

Step 2. Compute the statistic 2 = nR22, where n is the sample size and R22 is the unadjusted R-squared from the OLS regression in Step 1. The statistic 2 = nR22, has an asymptotic chi-square (2) distrib. with d.f. = k, where k is the # of ALL explanatory variables in the AUXILIARY model. Reject H0: All the variances i2 are equal (i.e., homoscedastic) if 2 > cr2

Estimation Procedures when H0 is rejected1. Heteroscedasticity with a known proportional factorIf it can be assumed that the error variance is proportional to the square of the indep. variable Xj2, we can correct for heteroscedasticity by dividing every term of the regression by X1i and then reestimating the model using the transformed variables. In the two-variable case, we will have to reestimate the following model (CN, p. 39): Yi/X1i = 0/X1i + 1 + ui/X1i

2. Heteroscedasticity consistent covariance matrix (HCCM) As we know, the usual OLS inference is faulty in the presence of hetero-scedasticity because in this case the estimators of variances Var(bj) are biased. Therefore, new ways have been developed for estimation of hetero-scedasticity-robust variances. The most popular is the HCCM procedure proposed by White.

The heteroscedasticity consistent covariance matrix (HCCM) procedure. Lets consider the model: Yi = 0 + 1X1i + 2X2i + ... + kXki + uiStep 1. Estimate the initial model by the OLS method. Let i denote the OLS residuals from the initial regression of Y on X1, X2, .., Xk

Step 2. Run the OLS regression of Xj (each time for a different j) on all other independent variables. Let ij denotes the ith residual from regressing Xj on all other independent variables.

Step 3. Let RSSj be the residual sum of squares from this regression: RSSj = SXjXj(1-R2).RSSj can also be calculated as RSSj = [n-(k+1)]SER2, where SER is the standard error of regression and can easily be found in the Excels OLS solution.

Step 4. The heteroscedasticity-robust variance Var(bj) can be calculated as follows: Var(bj) = ij2i2/RSSj2. The square root of Var(bj) is called the heteroscedasticity-robust standard error for bj.Example: CN, p. 44.

3. Feasible Generalized Least Squares (FGLS) methodStep 1. Compute the residuals i from the OLS of the initial regression model

Step 2. Regress i2 against a constant term and all the explanatory variables from either the Breusch-Pagan test for heteroscedasticity (e.g., when k =2: i2 = 0 + 1X1i + 2X2i + vi )or the White test for heteroscedasticity: i2 = 0 + 1X1i + 2X2i + 3X1i2 + 4X2i2 + 5X1i X2i + vi

Step 3. Estimate the original model by OLS using the weights zi = 1/i, where i2 are the predicted values of the dependent variable (the i2) in the Breusch-Pagan (or White) model. Note: the model must be estimated without a constant term.Such OLS procedure is called WLS (weighted least squares).

It may happen that the predicted values i2 of the dependent variable may not be positive, so we cannot calculate the corresponding weights zi = 1/i. If this situation arises for some observations, then we can use the original i2 and take their positive square roots.