Pearson Correlation — Narrow in Meaning, Wide in Misuse
Pearson linear correlation coefficient is commonly used to describe both the direction and strength of association between two variables. However, it is often misapplied to situations where the relationship is not linear, despite the coefficient being explicitly designed to measure only linear relationships.
Moreover, the coefficient is frequently misinterpreted as a measure of causal impact, especially when assumed, without justification, that one variable causes the other.
This dual misuse—treating Pearson correlation coefficient as both a general measure of dependence and as a proxy for causality—is widespread but fundamentally flawed.
Remark:
Pearson linear correlation coefficient is often used inappropriately:
- It is applied as a measure of any type of correlation, not just linear ones.
- It is mistakenly treated as an indicator of causality
When the Relationship Is Linear: What Pearson Correlation Tells Us
Let \(X\) and \(Y\) be random variables. Define
\[Y=aX+b+\varepsilon _Y,\]
where \(a\in \mathbb{R}-\{0\}\), \(b\in \mathbb{R}\) , \(\varepsilon _Y\) is either a constant (\(\varepsilon _Y\in \mathbb{R}\)) or a random variable (\(\varepsilon _Y:\mathrm{ }\Omega \rightarrow \mathbb{R}\)).
Assuming additionally that:
\[Cov\left( X,\mathrm{ }\varepsilon _Y \right) =0,\]
we obtain the Pearson correlation coefficient as:
\[Corr\left( X,Y \right) =\frac{Cov(X,Y)}{\sqrt{Var\left( X \right) Var(Y)}}=\frac{aVar(X)}{\sqrt{a^2\left[ Var\left( X \right) \right] ^2+Var\left( X \right) Var(\varepsilon _Y)}}.\]
If \(\varepsilon _Y\) is a constant (i.e. \(Var\left( \varepsilon _Y \right) =0\)) then:
If \(\varepsilon _Y\) is a random variable (i.e. \(Var\left( \varepsilon _Y \right) \ne 0\)), then:
\[Corr\left( X,Y \right) \in (-1,1).\]
Thus, we can see that if the relation between Y and X is deterministically linear (\(\mathrm{i}.\mathrm{e}.\mathrm{ }Y=aX+b)\), then \(Corr\left( X,Y \right) \in \{-1,1\}\). In contrast, if the relationship is linear but includes an additive random disturbance (\(Y=aX+b+\varepsilon _Y\)), then \(Corr\left( X,Y \right) \in (-1,1)\).
Remark:
Pearson linear correlation coefficient is exclusively intended for gauging:
- The strength of a linear relationship between random variables (measured by the absolute value of the coefficient)
- The direction of this relationship (indicated by the sign of the coefficient)
A greater strength corresponds to less randomness in the linear association between the variables.
Why Pearson Correlation Cannot Reveal Causal Direction
Let \(X\) and \(Y\) be random variables. Define
\[X=cY+d+\varepsilon _X,\]
where \(c\in \mathbb{R}-\{0\}\), \(d\in \mathbb{R}\), \(\varepsilon _X\) is either a constant (\(\varepsilon _X\in \mathbb{R}\)) or a random variable (\(\varepsilon _X\in \mathbb{R}\)).
Assume that one of the two equations — either \(Y=aX+b+\varepsilon _Y\) or \(X=cY+d+\varepsilon _X\) —represents the true causal relationship between \(X\) and \(Y\), though we do not know which one. Additionally, suppose that \(sgn\left( a \right) =sgn\left( c \right) \in \{-1,1\}\), and that both error terms are negligible: \(Var(\varepsilon _Y)\approx 0\), and \(Var(\varepsilon _X)\approx 0\). Under these assumptions, for both formulations we have:
\[Corr\left( X,Y \right) \approx \pm 1.\]
This indicates that the association between the variables is very strong in both cases. However, can we determine, based solely on \(Corr\left( X,Y \right) \), which of the two directions reflects the true causal relationship?
No—we cannot. The Pearson correlation coefficient does not allow us to distinguish between causal directions, nor does it provide evidence of causality in either case.
The situation described above may, for example, concern the relationship between operating profit (\(EBIT\)) and sales value (\(S\)). Regardless of whether \(Y\mathrm{ }=\mathrm{ }S\) and \(X\mathrm{ }=\mathrm{ }EBIT\), or vice versa, the above equations are mathematically sound. Moreover, the Pearson correlation between these variables is very strong. However, it is known that over the selected time period, it is \(S\) that determines \(EBIT\), not the other way around — \(S\) is the cause of \(EBIT\).
Remark:
Pearson linear correlation coefficient does not generally allow for testing hypotheses about assumed linear causality, and it does not address non-linear causal relationships. It does not indicate causality per se.
We can verify the reasoning just outlined in the following presentation.
As we slide sigma to the right, we increasingly disturb the linear relationship between \(X\) and \(Y\) with random noise. The grey dots—representing paired observations of the variables—become more loosely scattered around the green line, and Pearson’s \(\rho\) decreases.
Sliding sigma in the opposite direction reduces this randomness: the grey dots cluster more tightly around the green line, and Pearson’s \(\rho\) increases. Eventually, when \(\sigma\) equals zero—meaning there is no randomness in the relationship between \(X\) and \(Y\) —all the dots lie exactly on the green line, and Pearson’s rho reaches 1.
However, the fact that all grey dots lie exactly on the green straight line does not imply that \(X\) causes \(Y\) (or vice versa) in a linear manner. It merely indicates a perfect linear co-movement or trade-off between the variables. It does not denote a causal connection.
Takeaways
- Pearson correlation measures only linear dependence, not general association.
- It tells us how strongly and in what direction two variables co-vary—not whether one causes the other.
- A high correlation does not imply causality, nor can it reveal causal direction.
- When used outside its scope—especially for non-linear or causal analysis—Pearson correlation coefficient is easily misinterpreted.
- Use it only when linearity is assumed and justified.
For a related deep dive, see: When Perfect Correlation Meets a Pearson Shrug — Nonlinear Dependence Woes
If you want to go beyond correlation and measure portfolio risk directly, try my free tool: VaRCalc — Multi-Method Value at Risk Calculator.
