R-Squared for Stock Return Models

Business Context

At NorthBridge Capital, a quant analyst is comparing two linear regression models for predicting monthly stock returns. The team wants to know why a model with more predictors can show a higher $R^2$ but still be a worse choice once model complexity is considered.

Problem Statement

You are given summary statistics for two regression models fit on the same 60 months of stock return data. Model A uses market excess return as the only predictor. Model B adds size, value, momentum, and short-term reversal factors. Explain the difference between $R^2$ and adjusted $R^2$, compute both adjusted values, and determine which model is preferable based on goodness-of-fit after penalizing unnecessary predictors.

Given Data

Assume both models include an intercept.

Requirements

1. Define $R^2$ and adjusted $R^2$ in plain language.
2. Compute adjusted $R^2$ for Model A.
3. Compute adjusted $R^2$ for Model B.
4. Compare the two models and explain why adjusted $R^2$ can decrease when new predictors are added.
5. State which metric is more appropriate for comparing these two stock return models.

Assumptions

• Both models are estimated on the same response variable and same 60 monthly observations.
• Predictors were added without changing the sample period.
• The goal is in-sample explanatory comparison, not causal inference.
• Ignore out-of-sample validation for this exercise, though you should mention it in interpretation.