Chapter 11 Heteroscedasticity: What Happens if the Error Variance is Nonconstant ve NITIATI
Chapter 11 Heteroscedasticity: What Happens if the Error Variance is Nonconstant
11.1 The Nature of Heteroscedasticity Homoscedasticity: equal variance Heteroscedasticity: unequal variance HEteroscedasticity is usually found in cross-sectional data VITIA
11.1 The Nature of Heteroscedasticity • Homoscedasticity：equal variance. • Heteroscedasticity：unequal variance. Heteroscedasticity is usually found in cross-sectional data
11.2 Consequences of Heteroscedasticity 1. OLS estimators are still linear 2. They are still unbiased 3. But they no longer have minimunm variance. 4. The usual formulas to estimate the variances of ols estimators are generally biased 5. The bias arises from the fact that o, namely, e2/d is no longer an unbiasedestimator of 02KESaU 6. The usual confidence intervals and hypothesis tests based on t and f distributions are unreliable
11.2 Consequences of Heteroscedasticity • 1. OLS estimators are still linear. • 2. They are still unbiased. • 3. But they no longer have minimunm variance. • 4. The usual formulas to estimate the variances of OLS estimators are generally biased. • 5. The bias arises from the fact that , namely, , is no longer an unbiased estimator of . • 6. The usual confidence intervals and hypothesis tests based on t and F distributions are unreliable. 2 e /d.f. 2 i 2
In short, in the presence of heteroscedasticity, the usual hypothesis-testing routine is not reliable, raising the possibility of drawing misleading conclusions Heteroscedasticity is potentially a serious problem, for it might destroy the whole edifice of the standard, and so routinely used OLS estimation and hypothesis-testing procedure. A
• In short, in the presence of heteroscedasticity, the usual hypothesis-testing routine is not reliable, raising the possibility of drawing misleading conclusions. • Heteroscedasticity is potentially a serious problem, for it might destroy the whole edifice of the standard, and so routinely used, OLS estimation and hypothesis-testing procedure
11.3 Detection of heteroscedasticity How Do We know When There is a Heteroscedasticity Problem? 1. Nature of the Problem In cross-sectional data involving heterogeneous units, heteroscedasticity a may be the rule rather than / the exception NITIATI
11.3 Detection of Heteroscedasticity: How Do We Know When There is a Heteroscedasticity Problem? • 1. Nature of the Problem In cross-sectional data involving heterogeneous units, heteroscedasticity may be the rule rather than the exception
2. Graphical Examination of Residuals These residuals can be plotted against the observation to which they belong or against one or more of the explanatory variables or against y, the estimated emean value of y For example, we can plot erFs the residuals squared, against sales NITIATI
• 2. Graphical Examination of Residuals • These residuals can be plotted against the observation to which they belong or against one or more of the explanatory variables or against , the estimated mean value of . For example, we can plot , the residuals squared, againstsales. i ˆ i ˆ 2 i e
3. Park Test we can regress o 2 on one or more of the X variables ho2=B2+B2nX;+v;(14) Park suggests using ei as proxies for u he2=B2+B2hX;+V1(11.5) A
• 3. Park Test. • we can regress on one or more of the X variables. (11.4) • Park suggests using ei as proxies for ui . (11.5) 2 2 i i 2 i lnσ = B + B lnX + v 2 2 i i 2 i lne = B + B lnX + v 2 σ i
(1)Run the original regression despite the heteroscedasticity problem (2)Obtain the residuals ei, square them, and take their logs 3)Regress ei2 against each X variable. Alternatively, run the regression against yGu, the estimated VITIA . (4)Null hypothesis: B2=0;
• (1) Run the original regression despite the heteroscedasticity problem • (2) Obtain the residuals ei , square them, and take their logs . • (3)Regress ei2 against each X variable. Alternatively, run the regression against , the estimated Y. • (4 )Null hypothesis: B2=0; i ˆ
(5)Serious problem with the Park test, the error term v may itself be heteroscedastic VITIA
• (5) Serious problem with the Park test, the error term vi may itself be heteroscedastic!
4. Glejser Test (Obtaining residuals e, from the original model, (2)Regressing the absolute values of ei, el,on the X variable lei l=B+B2Xi+v (11.7) eil =B+B2x+V (1.8) e|=B1+B2(x)+v(19 (3)Null hypotheses: here is no heteroscedasticity, that is: B2=0 ITIA If this hypothesis is rejected, there is probably evidence of heteroscedasticity
4.Glejser Test (1)Obtaining residuals ei from the original model, (2)Regressing the absolute values of ei , |ei | , on the X variable . |ei | =B1+B2Xi +vi (11.7) |ei | =B1+B2 +vi (11.8) |ei | =B1+B2 +vi (11.9) (3)Null hypotheses: There is no heteroscedasticity, thatis : B2=0. If this hypothesis is rejected, there is probably evidence of heteroscedasticity. X i 1 Xi