Ordinary least squares vs. generalized linear model assumptions

Introduction: Moving Beyond the "Standard" Model

Welcome! If you’ve spent any time studying statistics, you’ve likely met the Ordinary Least Squares (OLS) regression model. It’s the "bread and butter" of predictive modeling. However, in the real world of actuarial science—where we deal with insurance claims, frequency of accidents, and mortality rates—OLS often falls short.

In this chapter, we are going to explore why we sometimes need to trade in our rigid OLS model for a more flexible Generalized Linear Model (GLM). We will look at the specific assumptions that separate the two. Don't worry if this seems a bit abstract at first; once you see the patterns, it becomes much clearer!

1. The Starting Point: OLS Assumptions

Before we can understand how GLMs are different, we have to remember what OLS expects from our data. You can remember the core OLS assumptions using the mnemonic LINE:

L – Linearity: The relationship between the predictors and the response is a straight line.
I – Independence: The observations (data points) are independent of each other.
N – Normality: The error terms (residuals) follow a Normal (Gaussian) distribution.
E – Equal Variance: The variance of the errors is constant (this is a fancy word called homoscedasticity).

Analogy: Imagine OLS is like a standard "Medium" sized t-shirt. It fits a lot of people reasonably well, but if you are much taller or shorter, it’s going to look weird. OLS works great for "Normal" data, but insurance data is often "weird."

Key Takeaway:

OLS assumes your target variable is continuous, can be negative or positive, and has a consistent "spread" (variance) regardless of the predicted value.

2. Why OLS Fails in Actuarial Work

In the "Linear Models" section of Exam SRM, you'll find that insurance data frequently breaks the OLS rules. Consider these two common scenarios:

1. Claim Counts: You can't have -2 accidents. OLS might predict a negative number, which makes no sense! Furthermore, counts are discrete integers (0, 1, 2...), not continuous values.
2. Claim Severity: Big insurance claims (like a house burning down) usually have much higher variance than small claims (like a broken windshield). OLS assumes the variance is the same for both, which is often wrong.

3. The GLM Solution: A Flexible Framework

A Generalized Linear Model (GLM) relaxes the strict rules of OLS. Instead of forcing everything into a "Normal" box, a GLM consists of three specific components:

A. The Random Component (The Distribution)

In OLS, we assume the response variable \(Y\) follows a Normal distribution. In a GLM, we assume \(Y\) follows a distribution from the Exponential Family. This family includes:
- Normal (for standard continuous data)
- Poisson (for counting how many times something happens)
- Binomial (for Yes/No outcomes)
- Gamma/Inverse Gaussian (for positive, skewed insurance claim amounts)

B. The Systematic Component (The Linear Predictor)

This is the part that looks familiar! It’s the combination of our explanatory variables (\(X\)) and their coefficients (\(\beta\)). We call this eta (\(\eta\)):
\(\eta = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + ... + \beta_p X_p\)

C. The Link Function

This is the "bridge." It connects the Mean of our data (\(\mu\)) to the Linear Predictor (\(\eta\)).
The formula looks like this: \(g(\mu) = \eta\).

In OLS, the link is "identity," meaning we assume \(\mu = \eta\). In GLMs, we can use a "Log" link (\(ln(\mu) = \eta\)) to ensure our predictions stay positive!

Did you know?

OLS is actually just a specific type of GLM! If you choose a Normal distribution and an Identity link function, your GLM becomes an OLS model.

4. OLS vs. GLM: Side-by-Side Comparison

Let's break down the major differences in their assumptions so you can easily spot them on the exam:

1. Distribution of the Response Variable
- OLS: Must be Normal.
- GLM: Can be any distribution from the Exponential Family (Poisson, Binomial, Gamma, etc.).

2. The Variance Relationship
- OLS: Variance is constant and does not depend on the mean (Homoscedasticity).
- GLM: Variance is a function of the mean. For example, in a Poisson distribution, as the mean increases, the variance also increases. This is much more realistic for insurance data!

3. The Relationship Transformation
- OLS: Assumes \(\mu = \beta X\) (a direct, linear relationship).
- GLM: Assumes \(g(\mu) = \beta X\). The Link Function \(g(.)\) allows the relationship to be non-linear while the parameters remain linear.

Quick Review: Common Mistakes

- Don't say that GLMs assume the data is non-linear. The parameters (\(\beta\)) are still linear! It's the link to the mean that can be non-linear.
- Don't assume GLM residuals must be normal. We care about the distribution of the response variable, not just the "errors" in the same way OLS does.

5. Summary of Assumptions for the Exam

When you are sitting for Exam SRM, remember these three pillars of GLM assumptions vs. OLS:

1. Flexibility: GLMs allow for non-normal distributions (Exponential Family).
2. Non-Constant Variance: GLMs allow the variance to change along with the mean (Function of the mean).
3. The Link: GLMs use a link function to relate the mean to the linear predictors, which helps keep predictions within valid ranges (like ensuring claim counts aren't negative).

Encouraging Note: If the math behind link functions feels heavy, just remember the purpose: we are just finding a way to make sure our model's "math" matches the "reality" of the data we are studying. You've got this!

Key Takeaway:

The primary advantage of a GLM over OLS is that it removes the "straightjacket" of Normality and Constant Variance, allowing us to model real-world risks more accurately.