09 Techniques: Regression & Qualitative Variables

Question 1

What are Probit & Logit models and how do they differ?

Answer 1

• types of generalized linear models
• modeling the relationship between predictor variables and a categorical outcome
• have versions for binary, ordinal, or multinomial categorical outcomes

Real difference is theoretical: different link functions

• > logistic regression uses a logit link function
• > probit regression uses an inverse normal link function

Question 2

How many columns does a model with 1 qualitative regressor have?

Answer 2

d₁ = number of levels of the regressor

Question 3

How many columns does a model with 2 qualitative regressors have?

Answer 3

d₁ + d₂ + 1

d₁ = number of levels of the first regressor
d₂ = number of levels of the second regressor

Question 4

How many columns does a model with 2 qualitative regressors with interactions have?

Answer 4

d₁ ⋅ d₂

d₁ = number of levels of the first regressor
d₂ = number of levels of the second regressor

Question 5

What are the steps to build data matrix X for a model with two qualitative regressors with interactions?

Answer 5

a) 1 column for constant
b) for the first regressor, make (d₁-1) columns.
- if d₁ = 2: make 1 column. code one level as 1, one level as -1
- if d₁ = 3: make 2 columns. code one level as 1, one level as 0, and the third as -1. switch 1 and 0 for the second column.
c) repeat step b) for the second regressor
d) (d₁-1) ⋅ (d₂-1) columns for interactions, where the relevant columns are multiplied element by element

Question 6

Find Y* as a linear combination of vector X!

β = (β₁; β₂; β₃)ᵀ
X = (X₁;X₂;X₃)ᵀ

Answer 6

Y* = β₁ X₁ + β₂ X₂ + ; β₃ X₃ + ε

Question 7

Interpret β₁ :

Y* = β₁ X₁ + β₂ X₂ + ; β₃ X₃ + ε

Answer 7

If X₁ changes by one unit, we expect Y* to change by β₁, ceteris paribus.