Question

1. If the errors in the CLR model are not normally distributed, although the OLS estimator is no longer BLUE, it is still unbiased.
2. In the CLR model, β^OLS is biased if explanatory variables are endogenous.
3. The value of R² in a multiple regression cannot be high if all the estimates of the regression coefficients are shown to be insignificantly different from zero based on individual t tests.
4. Suppose the CNLR applies to a simple linear regression y = β₁ + X₂β₂ + ε, and you obtained the OLS results 0.73 with standard error 0.2 from a data set. Since β₂^OLS is unbiased, the sampling distribution of β₂^OLS is distributed around 0.73 with standard error 0.2.
5. In the CLR model, y = X₁β₁ + X₂β₂ + ε (population) and y = X₁β₁^OLS + X₂β₂^OLS + e (sample), β₂^OLS can be obtained when the residuals from a regression of y on X₁ alone are regressed on the set of residuals obtained when each column of X₂ is regressed on X₁.

Answer 1

Statement 1 is true as normality plays no role in unbiasedness.Normality is required only to derive distribution of OLS estimators.

Statement 2 is true that OLS estimator are biased if explanatory variables are endogeneous.

Statement 3 is False as in case of muticollinearity R² can be high even when coefficient are insignificant

Statement 4 is False as sampling distribution of β₂^OLS  is distributed around its expected value i.e β₂ and not a particular sample value of β₂^OLS

Statement 5 is correct. Because in this process we are purifying Y and X₂ of the influence of X_1.Here residuals indicate their purified value on which simple regression model can be applied.