4. The R² coefficient: how much your model really explains
Level 4: Correlation, regression and the myth of causality
Two indices that have been rising almost without pause for a decade can yield an R² of 92% when one is regressed on the other. A number that looks impeccable.
And it may mean nothing at all.
What R² measures exactly
Once you have the regression line, an obvious question arises. How much of what happens to the share each day is really explained by the sector, and how much escapes the model?
That is exactly the question answered by the coefficient of determination, known as R².
It is expressed as a percentage between 0% and 100%, and it measures what proportion of the daily variation of the dependent variable is captured by the fitted line.
The shortcut in simple regression
For a simple regression, with a single explanatory variable, there is a convenient shortcut: R² is literally the Pearson correlation coefficient squared.
If the correlation between the airline and its sector is 0.55, R² is 0.55 squared. Roughly 0.30, that is, 30%.
How to read that 30%
Interpreting that 30% correctly is the part most open to confusion. Let us start with what it does not mean:
- It does not mean that the model gets the direction of the move right on 30% of days.
- It does not mean that the slope is 30.
What it does mean is this. Of all the variation shown by the airline's daily return across the sample, about a third is explained by the sector's movements.
The remaining 70% —the larger part— corresponds to factors the line does not capture: company-specific news, changes in expectations, pure noise. That uncaptured 70% is exactly what we earlier called residuals.
High is not good, low is not bad
A high R² is not automatically good news.
Let us go back to the opening example: a broad equity index and a broad property index, both rising over a decade. Regressing one on the other using price levels can give an R² of 92%, simply because they share a trend, without any strong causal relationship between them.
It is the same problem of non-stationary series seen in the previous module, now disguised as a number that looks impeccable.
Conversely, a low R² does not automatically make a model useless.
Regressions of individual shares against the general market usually show R² values of barely 20% to 30%. And they remain hugely relevant, because they capture that share's systematic risk: a factor the market pays for and that any risk manager needs to know about.
That holds even though most of the share's daily movement responds to other causes.
What the pattern of the residuals says
Looking only at the final R² figure can also hide problems.
Plot a model's residuals over time. If you observe that they become much larger during periods of high market volatility and stay small in the calm phases, that pattern is a signal.
It indicates that the model is not well specified for every market regime. Something relevant that only shows up when the market gets choppy may be missing.
The mechanical defect
One last nuance, which connects with the next module: R² has a mechanical defect.
When more explanatory variables are added to a model, R² almost always tends to rise. Even if those additional variables carry no real information.
The reason is that the model has more degrees of freedom with which to fit the sample data.
That apparently harmless property is the gateway to overfitting in multiple regression.