More Regression

Question 1

What is required to make predictions using a regression model?

Answer 1

Gauss-Markov assumptions should be met

Question 2

What is a dummy variable?

Answer 2

A variable which summarises a category with discrete options e.g. D could be a dummy variable for sex with male corresponding to D = 0 and female to D = 1, in which case

Question 3

How could regression be used for a model where there is a mean male wage and a mean female wage?

Answer 3

If w_m = α_m + ε_m and likewise for females, a dummy variable set to 0 for males would make it so that w = Dα_f + (1 - D)α_m + Dε_f + (1 - D)ε_m = α_m + D(α_f - α_m) + ε_m + D(ε_f - ε_m) which can be written α + Dβ + u with the intercept being the male wage and the slope being the difference in means

Question 4

What is functional form?

Answer 4

The method to go beyond a linear model so that logs can be used with CRM

Question 5

What is given by the slope in the relationships Y = α + βX + ε, ln(Y) = α + βX + ε, and ln(Y) = α + ln(β)X + ε?

Answer 5

The (constant) average change to Y from each unit of X
The % change in Y associated with each additional unit of X regardless of the initial level (x_i + 1 –> y_i + 100β) based on the approximation for a small change in a log, called the semi-elasticity of Y wrt X
The constant % change to Y from a 1% increase in x, called the constant elasticity of Y wrt X (so β = 1 means 1% increase in X increases Y by 1%)

Question 6

When is a log transformation used?

Answer 6

When it turns a non-linear relationship to a linear relationship

Question 7

How can the total sum of squares of a regression model be broken down?

Answer 7

Σ(Y - Ȳ)² = Σ(Y - Ŷ)² + Σ(Ŷ - Ȳ)²
Total sum of squares TSS = Error sum of squares ESS + Regression sum of squares RSS
To prove this, use Y = a + bX + e and Ȳ = a + bX̄ to get an expression for TSS which will include Σ2b(X - X̄)e which is assumed to equal zero for OLS
Using b(X - X̄) = Ŷ - Ȳ leaves the required relation

Question 8

What is R²?

Answer 8

R² = RSS/TSS = 1 - ESS/TSS gives the % variation in Y that is explained by the model

Question 9

What is multiple regression?

Answer 9

When there is more than one random variable on the RHS of the regression (multiple regressors)
A systematic difference in residuals of two groups after bivariate regression would suggest there should be another regressor

Question 10

What is the non-parametric method for predicting Y conditional on X₁ and X₂?

Answer 10

Calculate a mean wage for every possible paired value of Xs and use that as the CEF E(Y|X₁, X₂) for distinct X pairs

Question 11

What is the parametric method for multiple regression?

Answer 11

Use the OLS condition to get FOCs for Ŷ = a + bX₁ + cX₂
∂SSR/∂a = -2Σe = 0
∂SSR/∂b = -2ΣeX₁ = 0
∂SSR/∂c = -2ΣeX₂ = 0
Substituting e = Y - Ŷ gives a = Ȳ - bX̄_{1 - cX̄2}

Question 12

If using a dummy variable for X₂ where 1 is true, what can be said about the CEF?

Answer 12

E(Y|X₁, X₂ = 0) = a + bX₁ and E(Y|X₁, X₂ = 1) = a + bX₁ + c so c is the difference between the latter and the former (the intercept of the CEF shifts but the slope doesn’t change)

Question 13

What is the Frisch Waugh Lovell Theorem?

Answer 13

A method used to estimate residuals when trying to estimate a model with two regressors but only interested in one
Regress Y on X₁ to leave residuals e_Y and regress X₂ on X₁ to get residuals e_x, then regress Y residuals on X residuals and the coefficient on the latter will be the same as c from the multiple regression

Question 14

How does the Frisch Waugh Lovell Theorem work?

Answer 14

Regressing Y on X₁ and X₂ on X₁ partials out the affects of other independent variables (X₁), since residuals in a linear regression represent the left-over variation in the dependent variable after accounting for one independent variable