Linear Models

Question 1

What four assumptions must be met to use linear models?

Answer 1

1. Linearity and additivity
   
   • ​The expected value of dependent variable is a straight-line function of each independent variable, holding the others fixed​
   • The slope of that line does not depend on the values of the other variables
   • The effects of different independent variables on the expected value of the dependent variable are additive
2. Statistical independence of the errors
3. Homoscedasticity: costant variance of the error
   
   • ​Versus time (in case of time series)
   • Versus the predictions
   • Versus any independent variable
4. Normality of the error distribution

Question 2

When do you want to use a:

• Generalized Linear Model
• Panel Regression Model

And what characterizes them?

Answer 2

GLM

GLMs are appropriate when the outcome is not normal (gaussian), e.g. if outcomes are binary

A GLM is characterized by 3 components:

• random: associated with the dependent variable and its probability distribution
• systematic: identifies the selected covariates through a linear predictor
• link function: identifies the function of E[Y] such that it is equal to the systematic component

PLM

PLMs are appropriate when the assumption of independence does not hold, e.g. repeated measures of the same subject.

2 Possible approaches are possible:

• Fixed Effects (FE)
  
  • Explore the relationship between predictor and outcome variables within a subject
• Random Effects (RE) (error component)
  
  • Sssume that the variation across subjects is random and uncorrelated with the predictors
  • To be used when differences across subjects have some influence on the dependent variable