Question

Consider the simple regression model: Y_i = β₀ + β₁X_i + e

(a) Explain how the Ordinary Least Squares (OLS) estimator formulas for β₀ and β₁are derived.

(b) Under the Classical Linear Regression Model assumptions, the ordinary least squares estimator, OLS estimators are the “Best Linear Unbiased Estimators (B.L.U.E.).” Explain.

(c) Other things equal, the standard error of will decline as the sample size increases. Explain the importance of this.

Answer 1

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(a)

(b) The definition of “best” refers to the minimum variance or the narrowest sampling distribution. More specifically, when your model satisfies the assumptions, OLS coefficient estimates follow the tightest possible sampling distribution of unbiased estimates compared to other linear estimation methods.

The Best in BLUE refers to the sampling distribution with the minimum variance. That’s the tightest possible distribution of all unbiased linear estimation methods!

(c)  

Stand error is defined as standard deviation devided by square root of sample size.

se=σ√n

Therefore, as sample size increases, the standard error decreases.

This can also be intuitively understood based on the fact that as sample increases the coefficient converges to a mean vlaue which reduces the possible standard error.