HETEROSCEDASTICITY q The assumption of equal variance Varui

HETEROSCEDASTICITY q. The assumption of equal variance Var(ui) = σ2, for all i, is called homoscedasticity, which means “equal scatter” (of the error terms ui around their mean 0) 1

2

§ Equivalently, this means that the dispersion of the observed values of Y around the regression line is the same across all observations q. If the above assumption of homoscedasticity does not hold, we have heteroscedasticity (unequal scatter) 3

4

q. Consequences of ignoring heteroscedasticity during the OLS procedure § The estimates and forecasts based on them will still be unbiased and consistent § However, the OLS estimates are no longer the best (B in BLUE) and thus will be inefficient. Forecasts will also be inefficient 5

§ The estimated variances and covariances of the regression coefficients will be biased and inconsistent, and hence the t- and F -tests will be invalid 6

q. Testing for heteroscedasticity q 1. Before any formal tests, visually examine the model’s residuals ûi § Graph the ûi or ûi 2 separately against each explanatory variable Xj, or against Ŷi, the fitted values of the dependent variable 7

8

q 2. The Goldfeld-Quandt test Step 1. Arrange the data from small to large values of the indp variable Xj Step 2. Run two separate regressions, one for small values of Xj and one for large values of Xj, omitting d middle observations (app. 20%), and record the residual sum of squares RSS for each regression: RSS 1 for small values of Xj and RSS 2 for large Xj’s. 9

Step 3. Calculate the ratio F = RSS 2/RSS 1, 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. 10

• Step 4. Reject H 0: All the variances σi 2 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 α, typically 5%. 11

q Drawbacks of the Goldfeld-Quandt test § It cannot accommodate situations where several variables jointly cause heteroscedasticity §The middle d observations are lost 12

q 3. Lagrange Multiplier (LM) tests (for large n>30) § The Breusch-Pagan test Step 1. Run the regression of ûi 2 on all the explanatory variables. In our example (CN p. 37), there is only one explanatory variable, X 1, therefore the model for the OLS estimation has the form: ûi 2 = α 0 + α 1 X 1 i + vi 13

Step 2. Keep the R 2 from this regression. Let’s call it Rû 22 Calculate either § (a) F = (Rû 22/k)/[(1 -Rû 22)/(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 H 0: All the variances σi 2 are equal (i. e. , homoscedastic) if F >Fcr 14

or § (b) LM = n Rû 22, where LM is called the Lagrangian Multiplier (LM) statistic and has an asymptotic chisquare (χ2) distribution with d. f. = k Reject H 0: All the variances σi 2 are equal (i. e. , homoscedastic) if LM> χcr 2 15

q Drawbacks of the Breusch- Pagan test § It has been shown to be sensitive to any violation of the normality assumption q. Three other popular LM tests: the Glejser test; the Harvey-Godfrey test, and the Park test, are also sensitive to such violations (won’t be covered in this course) 16

q. One LM test, the White test, does not depend on the normality assumption; therefore it is recommended over all the other tests 17

§ The White test Step 1. The test is based on the regr. of û 2 on all the explanatory variables (Xj), their squares (Xj 2), and all their cross products. E. g. , when the model contains k = 2 explanat. variables, the test is based on an estim. of the model: û 2 =β 0+ β 1 X 1 +β 2 X 2+β 3 X 12+β 4 X 22 + β 5 X 1 X 2 + v 18

Step 2. Compute the statistic χ2 = n. Rû 22, where n is the sample size and Rû 22 is the unadjusted R-squared from the OLS regression in Step 1. The statistic χ2 = n. Rû 22, has an asymptotic chi-square (χ2) distrib. with d. f. = k, where k is the # of ALL explanatory variables in the AUXILIARY model. Reject H 0: All the variances σi 2 are equal (i. e. , homoscedastic) if χ2 > χcr 2 19

q. Estimation Procedures when H 0 is rejected • 1. Heteroscedasticity with a known proportional factor § If it can be assumed that the error variance is proportional to the square of the indep. variable Xj 2, we can correct for heteroscedasticity by dividing every term of the regression by X 1 i 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/X 1 i = β 0/X 1 i + β 1 + ui/X 1 i 20

• 2. Heteroscedasticity consistent covariance matrix (HCCM) § As we know, the usual OLS inference is faulty in the presence of heteroscedasticity because in this case the estimators of variances Var(bj) are biased. Therefore, new ways have been developed for estimation of heteroscedasticity-robust variances. § The most popular is the HCCM procedure proposed by White. 21

q. The heteroscedasticity consistent covariance matrix (HCCM) procedure. § Let’s consider the model: Yi = β 0 + β 1 X 1 i + β 2 X 2 i +. . . + βk. Xki + ui • Step 1. Estimate the initial model by the OLS method. Let ûi denote the OLS residuals from the initial regression of Y on X 1, X 2, . . , Xk 22

• Step 2. Run the OLS regression of Xj (each time for a different j) on all other independent variables. Let ŵij denotes the ith residual from regressing Xj on all other independent variables. 23

• Step 3. Let RSSj be the residual sum of squares from this regression: RSSj = SXj. Xj(1 -R 2). § RSSj can also be calculated as RSSj = [n-(k+1)]SER 2, where SER is the standard error of regression and can easily be found in the Excel’s OLS solution. 24

• Step 4. The heteroscedasticityrobust variance Var(bj) can be calculated as follows: Var(bj) = Σŵij 2ûi 2/RSSj 2. § The square root of Var(bj) is called the heteroscedasticity-robust standard error for bj. § Example: CN, p. 44. 25

• 3. Feasible Generalized Least Squares (FGLS) method Step 1. Compute the residuals ûi from the OLS of the initial regression model 26

• Step 2. Regress ûi 2 against a constant term and all the explanatory variables from either the Breusch-Pagan test for heteroscedasticity (e. g. , when k =2: § ûi 2 = α 0 + α 1 X 1 i + α 2 X 2 i + vi ) or the White test for heteroscedasticity: § ûi 2 = α 0 + α 1 X 1 i + α 2 X 2 i + α 3 X 1 i 2 + α 4 X 2 i 2 + α 5 X 1 i X 2 i + vi 27

• Step 3. Estimate the original model by OLS using the weights zi = 1/σi, where σi 2 are the predicted values of the dependent variable (the ûi 2) 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). 28

§ It may happen that the predicted values σi 2 of the dependent variable may not be positive, so we cannot calculate the corresponding weights zi = 1/σi. If this situation arises for some observations, then we can use the original ûi 2 and take their positive square roots. 29