Model Misspecification

Welcome to the World of Model Misspecification!

In our previous Quantitative Methods lessons, we learned how to build and interpret regression models. But what happens if the model we build is fundamentally flawed? This is where Model Misspecification comes in. Think of a regression model like a blueprint for a house. If the blueprint is wrong—maybe it’s missing a support beam or uses the wrong materials—the whole house might collapse, no matter how good the builders are.

In this chapter, we are going to learn how to spot these "blueprint errors," why they happen, and how they mess up our investment analysis. Don’t worry if this seems a bit abstract at first; we will break it down piece by piece!

1. What is Model Misspecification?

Model misspecification occurs when we violate the basic assumptions of linear regression by setting up the model incorrectly. If a model is misspecified, the regression coefficients (\(b_i\)) can become biased and inconsistent. This means our predictions will be off, and our hypothesis tests (like t-tests) won't be reliable.

The Three Major Categories of Misspecification

We generally group these errors into three buckets:

1. Functional Form: We chose the wrong mathematical relationship (e.g., using a straight line when the data is curved).
2. Explanatory Variables: We either left out something important or included something we shouldn't have.
3. Measurement Error: The data itself is "noisy" or inaccurately recorded.

Quick Review: If a model is misspecified, the error term (\(\epsilon\)) becomes correlated with the independent variables (\(X\)). This breaks a core rule of regression!

2. Omitted Variable Bias (The Missing Ingredient)

Imagine you are trying to predict the price of a house. You include the number of bedrooms, but you omit (leave out) the neighborhood's quality. Because the neighborhood usually correlates with the number of bedrooms (bigger houses are often in nicer areas), the model will "blame" the number of bedrooms for price increases that were actually caused by the neighborhood.

The Result of Omitting a Variable:

• The model is biased: The estimated coefficients are systematically too high or too low.
• The model is inconsistent: Even if we add thousands of more data points, the error won't go away.

Did you know? This is often called "lurking variable" bias in basic statistics. In the CFA curriculum, we focus on how it makes our \(b\) coefficients unreliable.

3. Functional Form Misspecification

Sometimes the "shape" of our model is just wrong. Here are the most common ways we mess this up:

A. Omitting Variable Transformations

If the relationship between \(X\) and \(Y\) is curved (exponential), but we use a standard linear regression (\(Y = b_0 + b_1X\)), our model won't fit. We might need to use the natural log of a variable to make it linear.

B. Incorrectly Pooling Data

If you take data from 1990 and data from 2024 and mash them together into one regression, you are "pooling." If the economy changed fundamentally between those times, your coefficients won't be accurate for either period. It’s like trying to find the average shoe size by measuring both toddlers and NBA players—the average won't fit anyone well!

Key Takeaway: Always check if your data comes from the same "population" or time period before combining it.

4. Endogeneity and Correlation Between Variables and Error

This sounds like a scary word, but Endogeneity just means that an independent variable (\(X\)) is correlated with the error term (\(\epsilon\)). This is a "cardinal sin" in regression. It usually happens for three reasons:

1. Omitted Variables (As discussed above)

2. Measurement Error

If your data for \(X\) is "fuzzy" or inaccurate, that inaccuracy ends up in the error term. This makes the variable (\(X\)) look less important than it actually is (this is called attenuation bias).

3. Simultaneity (The Two-Way Street)

In a standard regression, we assume \(X\) causes \(Y\). But what if \(Y\) also causes \(X\)?
Example: Do high marketing spends (\(X\)) cause high sales (\(Y\))? Or do companies with high sales (\(Y\)) simply have more money to spend on marketing (\(X\))? If the relationship goes both ways, the model is misspecified.

Summary Tip: If you see "Simultaneity," "Omitted Variables," or "Measurement Error," think: Inconsistent and Biased Coefficients.

5. Qualitative Dependent Variables

Up until now, we’ve assumed the dependent variable (\(Y\)) is a number (like stock price). But what if \(Y\) is a "Yes/No" or "Pass/Fail" outcome? We call these Qualitative Dependent Variables.

You cannot use standard Ordinary Least Squares (OLS) for these because OLS might predict a probability of 150% or -20%, which is impossible! Instead, we use these three models:

• Probit Model: Based on the normal distribution. It estimates the probability that an event will happen.
• Logit Model: Based on the logistic distribution. Similar to Probit, but uses a different mathematical "curve."
• Discriminant Analysis: This creates a "score" to sort observations into groups (e.g., "Will Bankrupt" vs. "Will Not Bankrupt"). Think of the Altman Z-score as a famous example.

Quick Review Box:
Y is a number? Use Linear Regression.
Y is Yes/No? Use Probit or Logit.

6. Summary of Key Problems and Consequences

To help you study, here is a "Cheat Sheet" of what happens when things go wrong:

Final Encouragement

Quantitative methods can feel like learning a new language. If "Endogeneity" and "Simultaneity" feel confusing, just remember the core message: The independent variable must not be "buddy-buddy" with the error term. If they are related, your results are biased and you can't trust your model to make investment decisions. You've got this!