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Unformatted text preview: CHAPTER 13 HETEROSCEDASTICITY: WHAT HAPPENS IF THE ERROR VARIANCE IS NONCONSTANT? QUESTIONS 13.1. Heteroscedasticity means that the variance of the error term in a regression model does not remain constant between observations. ( a ) The OLS estimators are still unbiased but they are no longer efficient. ( b ) and ( c ) Since the estimated standard errors of OLS estimators may be biased, the resulting t ratios are likely to be biased too. As a result, the usual confidence intervals, hypothesis testing procedure, etc. are likely to be of questionable value. 13.2. ( a ) False . The OLS estimators are still unbiased; only they are no longer efficient. ( b ) True . Since the estimated standard errors are likely to be biased, the t ratios will be biased too. ( c ) False . Sometimes OLS overestimates the variances of OLS estimators and sometimes it underestimates them. ( d ) Uncertain . It may or may not. Sometimes a systematic pattern in the residuals may reflect specification bias, such as omission of a relevant variable, or wrong functional form, etc. ( e ) True . Since the true heteroscedastic variances are not directly observable, one cannot test for heteroscedasticity directly without making some assumptions. 13.3. ( a ) Yes, because of the diversity of firms included in the Fortune 500 list. ( b ) Probably. ( c ) Probably not. In time series data, it is often not easy to isolate the effects of autocorrelation and heteroscedasticity. ( d ) Yes, because of vast differences in per capita income data of developed and developing countries. 110 ( e ) Yes. Although the U.S. and Canadian inflation rates are similar, the Latin American countries exhibit wide swings in the inflation rate. 13.4. By giving unequal weights, WLS discounts extreme observations. The estimators thus obtained are BLUE. Note that WLS is a specific application of GLS, the method of generalized least squares. 13.5. ( a ) This is a visual method, which is often a good starting point to find out if one or more assumptions of the classical linear regression model (CLRM) are fulfilled. ( b ) and ( c ) These two tests formalize the graphical method by making suitable assumptions(s) about the explanatory variable(s) that might be the cause of heteroscedasticity. PROBLEMS 13.6. Let i i i u X B B Y + + = 2 1 . Now divide this equation through by 2 i X to obtain: i i i i i v X B X B X Y + + = 1 1 2 2 1 2 , where 2 i i i X u v = The error term i v is homoscedastic. Use the regression-through-the-origin procedure to estimate the parameters of the transformed model. 13.7. ( a ) Perhaps heteroscedasticity is present in the data. ( b ) ) (GNP ) var( 2 2 i i u σ = . ( c ) The coefficients of the original and transformed models are about the same, although the standard errors of the coefficients in the transformed model seem to be somewhat lower, perhaps suggesting that the authors have succeeded in reducing the severity of heteroscedasticity....
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