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      Question

      Consider a standard Ordinary Least Squares (OLS)

      regression model: Yi=β0+β1X1i+ϵi. If the errors (ϵi) are serially correlated (autocorrelation) but all other Gauss-Markov assumptions hold, which of the following properties of the OLS estimators (β^0,β^1) is compromised?
      A Unbiasedness: The expected value of the estimator remains equal to the true parameter value. Correct Answer Incorrect Answer
      B Efficiency: The estimator has the minimum variance among all linear unbiased estimators (BLUE). Correct Answer Incorrect Answer
      C Consistency: The estimator converges to the true parameter value as the sample size increases. Correct Answer Incorrect Answer
      D Normality: The sampling distribution of the estimator is asymptotically normal. Correct Answer Incorrect Answer

      Solution

      Solution: The Gauss-Markov Theorem states that under certain assumptions, OLS is the Best Linear Unbiased Estimator (BLUE). The assumption violated here is the No Autocorrelation/Serial Correlation assumption (Cov(ϵi,ϵj)=0 for i=j).

      • Impact of Autocorrelation: The OLS estimator (β^) remains Unbiased (A) and Consistent (B). However, the formulas for the standard errors will be biased, and the OLS estimator is no longer the most efficient among the class of linear unbiased estimators.
       
      • The Compromised Property: Since the OLS estimator is no longer the minimum variance estimator, its Efficiency (C) is lost. This often leads to underestimated standard errors, causing t-statistics to be too large, and potentially leading to incorrect conclusions about the statistical significance of the coefficients.

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