Gauss–Markov Theorem: OLS, BLUE, Assumptions, and GLS

All flashcards

What does the regression model $y=Xβ+ε$ represent?

Observed outcomes $y$ equal the systematic component $Xβ$ plus unobserved errors $ε$.

How does OLS estimate the coefficient vector when $X$ has full column rank?

It minimizes squared residuals, giving $\hat β_{OLS}=(X^TX)^{-1}X^Ty$.

What does BLUE mean?

Best linear unbiased estimator: minimum variance among linear unbiased estimators under the theorem’s assumptions.

What is homoscedasticity?

All errors have the same finite variance, $Var(ε_i)=σ^2$.

How do errors differ from residuals?

Errors are unobserved model disturbances; residuals are observed outcomes minus fitted outcomes.

Quiz questions

1. When $X$ has full column rank, which expression gives the ordinary least squares coefficient estimate?Answer: $(X^TX)^{-1}X^Ty$
2. Within which class does the Gauss–Markov theorem say OLS has minimum variance?Answer: Linear unbiased estimators
3. For a linear estimator $Cy$ to be unbiased for every coefficient vector, which condition must hold?Answer: $CX=I$
4. When the error covariance matrix is non-spherical and known, which method uses that covariance structure?Answer: Generalized least squares
5. What is the consequence of perfect multicollinearity for the usual OLS coefficient formula?Answer: $X^TX$ is singular, so the coefficient vector is not uniquely computable.