Proof that slope coefficient is unbiased

In this set of videos, we prove that the usual estimate for the slope of a simple linear regression is unbiased. It is useful to see this proof and think through the steps. However, your comfort with these videos will depend on your math and stats background (probability experience is helpful). From the perspective of applied data analysis, it's most important that the intuitive idea of unbiasedness makes sense to you and that you understand how we would design a simulation to illustrate the unbiasedness of this estimator.

You can also view all the videos in this section at the YouTube playlist linked here.

The total length of these videos is 15 minutes.

Part 1

Question 1: What does it mean for an estimate to be unbiased? Choose one answer.

• The estimate is equal to zero.

• On average, the estimate is equal to the truth.

• The estimate was chosen without bias from the person.

Part 2

Question 2: Which expressions are equivalent?

1. Σ (Xᵢ − X̄)Yᵢ

2. Σ (Xᵢ − X̄)X̄

3. Σ (Yᵢ − Ȳ)Xᵢ

4. Σ (Xᵢ − X̄)(Yᵢ − Ȳ)

5. Σ (Xᵢ − X̄)Ȳ - Σ (Xᵢ − X̄)Yᵢ

6. Σ (Xᵢ − X̄)Yᵢ - Σ (Xᵢ − X̄)Ȳ

Part 3

Question 3: What does linear regression assume about the x and y values?

• The y and x values are random.

• The y values are fixed values while the x values are random.

• The y and x values are fixed values in the model.

• The y values are random values while the x values are fixed.

Part 4

Question 4: What are some necessary facts that we used to prove the estimator for β₁ is unbiased?

• Ε (Yᵢ | X) = β₀ + β₁Xᵢ

• Σ(Ε(Xᵢ)) = Ε(Σ(Xᵢ))

• Σ(Xᵢ - X̄) = 0

• Σ(Xᵢ - X̄)² = Σ(Xᵢ - X̄)Xᵢ

• Ε(Σ(Xᵢ - X̄)|X) is a constant

And that is all.

During this tutorial you learned:

• More about unbiased estimates

• How to prove that the estimated slope (β1) is unbiased

Terms and concepts:

unbiasedness, expectation