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Is ols-based estimator for regression coefficient unbiased

**Problem**

Twenty people participated in a medical study. Each participant contributed three longitudinal measurements to the response variable y for the study. Thus the study had a total of 60 measurements on y. It is suspected that the longitudinal measurements from the same person are likely correlated. Without sufficient knowledge of the study background, a data analyst performed a multiple linear regression analysis using a model with four regressors and assuming that all 60 measurements are statistically independent. Thus the data analyst used OLS (i.e., ordinary least-squares) estimation for all regression coefficients.

1. Is the OLS-based estimator for each regression coefficient unbiased? Why? Provide mathematical arguments for your answer.

2. The data analyst assumed that all 60 measurements are statistically independent and used the OLS estimator to generate t test for testing significance of each regressor at the target significance level α. Is the type I error probability of this ?? test equal to the targeted ?? level. Provide mathematical arguments for your answer.

3. The data analyst assumed that all 60 measurements are statistically independent and used the OLS estimator to generate 95% confidence interval for each regression coefficient. Does the 95% confidence interval have a correct coverage probability? Provide mathematical arguments for your answer.

4. Later the data analyst was told that the assumption of statistical independence may not hold. Should the analyst adjust estimation and statistical testing? Why?

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