Session 2 - Multivariate Models

What does unbiasedness of OLS mean?

• E(β̂) = β

• On average, estimates equal true parameter

Why is β̂ random and β constant?

• β̂ depends on sample

• β is population value

When is an estimator unbiased?

If across inifinite samples, mean of β̂ = β

What does bias mean?

Difference between E(β̂) and β

What ensures unbiased OLS?

SLR.1–SLR.4

What is omitted variable bias?

Leaving out relevant variable correlated with regressor

• Example: True model RET = β0 + β1SAT + β2EXP + u

• Estimated model RET = β0 + β1SAT + v

• If SAT and EXP correlated → β̂1 biased

Why does OVB occur?

• Because omitted variable is in the error term

• If it’s correlated with regressor → E(u|x) ≠ 0

• Violates zero conditional mean (SLR.4)

Formula for OVB bias

Bias = δ̃1β2

When is no OVB?

β2 = 0 or δ̃1 = 0

General form of multivariate regression

y = β₀ + β₁x₁ + β₂×₂ + … + β_kx_k + u

Key advantage of multivariate regression

• In univariate: β₁ may capture both x₁’s effect + effects of omitted correlated vars

• In multivariate: model separates effects → β₁ reflects x₁’s impact only

• Hence gives a more genuine ceteris paribus effect

How to interpret β1 in multivariate regression?

Effect of x1 on y, holding other x’s fixed (cetris paribus)

What is a dummy variable?

• Variable = 1 if condition true, 0 if false

• Lets regression compare descriptive variables

• Changes the intercept between groups

Do irrelevant regressors cause bias?

No, but increase variance

Key rule for variable selection

• Include only theoretically relevant variables

Dummy-only regression: intercept meaning

Intercept = mean of group with dummy=0

Population Regression Function (PRF)

E[Y|X]=β₀+β₁x₁

(or multivariate: E[Y|X]=β₀+β₁X₁+…+β_kX_k). The theoretical linear relation of Y on X; OLS estimates this.