Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R). Assertion (A) : The presence of heteroscedasticity problem in regression analysis implies that the least square estimators are still unbiased but inefficient. Reason (R) : The estimates of the variances are also unbiased. In the light of the above statements, choose the most appropriate answer from the options given below :
Assertion (A) Analysis: Heteroscedasticity and Estimator Properties
The statement asserts that heteroscedasticity in regression leads to unbiased yet inefficient least squares (LS) estimators. Heteroscedasticity implies that the variance of the error term is not constant across observations.
- Unbiasedness: While heteroscedasticity violates the constant variance assumption of the Gauss-Markov theorem, the LS estimators ($\hat{\beta}_{LS}$) remain unbiased. This means their expected value equals the true parameter values (E[$\hat{\beta}_{LS}$] = $\beta$).
- Inefficiency: However, the minimum variance property of BLUE (Best Linear Unbiased Estimator) is lost. Consequently, the LS estimators are no longer the most efficient among linear unbiased estimators, meaning they are inefficient.
Therefore, Assertion (A) is correct.
Reason (R) Analysis: Variance Estimation Bias
The statement claims that the estimates of the variances are also unbiased under heteroscedasticity. This is incorrect. When heteroscedasticity is present, the standard formulas used to calculate the variance-covariance matrix of the LS estimators are misspecified.
The resulting estimates of the variances (standard errors) of the coefficients are biased and inconsistent. Special methods are needed to obtain correct standard error estimates in the presence of heteroscedasticity (e.g., robust standard errors).
Therefore, Reason (R) is incorrect.
Conclusion
Assertion (A) correctly describes the properties of LS estimators under heteroscedasticity (unbiased but inefficient). Reason (R) incorrectly states that variance estimates remain unbiased.
Thus, the appropriate choice is that (A) is correct and (R) is incorrect.
Important Questions from Correlation and Regression
If r is the coefficient of correlation between x and y, then what is the correlation coefficient between (3x + 4) and (-3y + 3)?
Which of the following statements is/are correct in respect of regression coefficients?
1. It measures the degree of linear relationship between two variables
2. It gives the value by which one variable changes for a unit change in the other variable.
Select the correct answer using the code given below.If two lines of regression are x + 4y + 1 = 0 and 4x + 9y + 7 = 0, then what is the value of x when y = -3 ?
A bivariate data set contains only two points (-1, 1) and (3, 2). What will be the line of regression of y on x ?
If two regression lines between height (x) and weight (y) are 4y – 15x + 410 = 0 and 30x – 2y – 825 = 0, then what will be the correlation coefficient between height and weight?
