The Consequences of Violating Linear Regression Assumptions. How violating assumptions can impact prediction and inference
Recently, a friend learning linear regression asked me what happens when assumptions like multicollinearity are violated. Despite being a former statistics student, I could only give him general answers like “you won’t be able to trust the estimates of your model.” Unsatisfied with my response, I decided to create a real-world example, via simulation, to show what can happen to prediction and inference when certain assumptions are violated.
Suppose researchers are interested in understanding what drives the price of a house. Let’s pretend that housing prices are determined by just two variables: the size and age of the house. While age holds a negative, linear relationship with price, the size of the house has a positive, quadratic (non-linear) relationship with price. Mathematically, we can model this relationship like so:
Priceᵢ = β₀ + β₁sqftᵢ + β₂sqftᵢ² − β₃*age_yearsᵢ + eᵢ
where Price is the price of a house in thousands of dollars, sqft is the square footage of a house in thousands, and age_years the age of the house in years. The residuals _e _are normally distributed with mean 0 and variance σₑ_². _Let’s call this the true model since it accounts for everything that drives housing prices (excluding residuals). Since researchers don’t have a crystal ball telling them what the true model is, they test out a few linear regression models. Here’s what they came up with, in no particular order:
(1)_ Priceᵢ = β₀ + β₁sqftᵢ + β₂sqftᵢ² − β₃*age_yearsᵢ + eᵢ_
(2)_ Priceᵢ = β₀ + β₁sqftᵢ + β₂sqftᵢ² − β₃age_yearsᵢ − β₄age_monthsᵢ + eᵢ_
(3)_ Priceᵢ = β₀ + β₁sqftᵢ − β₂age_yearsᵢ + eᵢ_
(4)_ Priceᵢ = β₀ − β₁*age_yearsᵢ + eᵢ_
The researchers were smart and nailed the true model (Model 1), but the other models (Models 2, 3, and 4) violate certain OLS assumptions. Lastly, let’s say that there were 10K researchers who conducted the same study. Each took 50 independent observations from the population of houses and fit the above models to the data. By examining the results of these 10K models, we can see how these different models behave.
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