Identifying Relationships Between Variables

There is an infinite number of relationships around us— some more obvious, some less so—and finance is no exception. One might want to verify some of the many relationships in the market, such as that interest rates and bond prices move in opposite directions, or that inflation affects the prices of publicly traded securities.

We don't have to go very far or use complex statistics to shed some light on whether there is a relationship—a correlation—between a set of variables. In this publication we will have a look at how this can be done using the various types of correlation coefficients, or even just by looking at a specific type of chart: the scatter plot.

1. Visualising the Correlation

The beauty of statistics is that before we even have to do any calculations, we can often simply take a look at a chart to get a general idea of our data. This is no different when it comes to correlation. We can use a specific type of chart—the scatter plot—to visualize two, or even three, variables. Visualizing more than three variables is usually very difficult and can even be counterproductive.

Reading a scatter plot might seem daunting for first-timers, but it's actually fairly simple. When looking for a correlation, we focus on how the points on the plot cluster together and whether they follow a trend or a direction.

Despite the fairly simple construction of scatter plots, they can convey a surprisingly large amount of insight about our data. In fact, they don’t just show whether a correlation exists—they can also indicate the direction of the correlation (positive or inverse), its strength, and its form (linear or nonlinear). In some cases, we can even spot obvious outliers before examining the dataset more closely.

All this information can help us make important decisions and assumptions—for example, choosing the appropriate method to measure correlation, or deciding whether a correlation may exist at all—thus saving significant time and effort in our analysis.

2. Quantifying the Correlation

While scatter plots are great for a first assessment of whether there is a trend or not in our data, to actually confirm and quantify empirically our observations, we must go a step further. A great tool to quantify our observations is the covariance coefficient. This coefficient helps us measure how two variables move together.

$$s_{XY}=\dfrac{\sum_{i=1}^{n}{(X_i-\overline{X})(Y_i-\overline{Y})}}{n-1}$$

In the equation above, we can see that this covariance coefficient is nothing else but the product of the deviations from the mean of the two variables, divided by $n-1$ in the denominator. When looking at this equation, we could say that the covariance coefficient is a measure of the joint variability of the two variables—which we also saw in the previous publication about Dispersion as a Measure of Risk.

The only downside of the covariance coefficient is that it yields a result in squared units. For example, when calculating with variables with a unit of euros (€) we get the covariance coefficient with a unit of euros squared (€²), which is quite hard to interpret.

2.1. Pearson’s Correlation Coefficient

To help us interpret the covariance coefficient in a more intuitive way, we can use the correlation coefficient. The most popular one is the Pearson’s Correlation Coefficient, also called Pearson’s correlation for short. This coefficient standardizes the covariance coefficient, making it easier to read and interpret the correlation between the two variables. We can compute it using the following equation:

$$r_{XY}=\dfrac{\sum{(X_i-\overline{X})(Y_i-\overline{Y})}}{\sqrt{\sum(X_i-\overline{X})^2(Y_i-\overline{Y})^2}}$$

By a quick look at it, we notice that this equation can be simplified to a fraction with the covariance coefficient in the numerator and the product of the standard deviations of the two variables in the denominator.

$$r_{XY}=\dfrac{s_{XY}}{s_x\cdot s_Y}$$

The result of this equation is a coefficient without a unit of measure, which indicates the intensity and direction of the correlation between the two compared variables.

2.2. Properties of Pearson’s Correlation

To interpret the coefficient of correlation, we can follow a simple set of rules. The correlation ranges from -1 to 1, with -1 being the highest negative (inverse) correlation and 1 being the highest positive (direct) correlation. The size of the correlation coefficient dictates the intensity of the correlation, while its sign tells us the direction. A correlation coefficient of 0 tells us there is no correlation at all between the two variables.

2.3. Limitations of Pearson’s Correlation

Pearson’s correlation coefficient is not always an ideal or reliable tool. If there is a nonlinear relationship between variables, the coefficient might be low even if a strong relationship exists. Pearson’s correlation is also sensitive to outliers, which can distort results.

To address these limitations, other methods can be used. The Spearman’s Rank Correlation Coefficient is more robust to outliers and suitable for monotonic relationships, while partial correlation can account for the influence of a third variable.

3. Correlation is not causation

It is important to note—especially in today’s world, where access to statistics is easy—that correlation does not imply causation. This common logical error is part of the post hoc family of fallacies. It’s easy to assume that if there is a correlation between two variables, one must cause the other.

A spurious correlation happens when two seemingly related variables are not related at all. This may occur when a third variable influences both, or when the correlation is purely coincidental.

For example a dataset might show a strong positive correlation between global gold prices and the number of new technology patents filed in a given year. The apparent relationship might simply reflect the influence of a third factor, such as overall global economic growth, which can simultaneously increase investment demand for gold and funding for research.

For this reason, correlation should not be taken as the whole story of our data. It instead should be examined in combination with domain knowledge (e.g., finance) and tested with other methods to confirm any underlying relationships.

4. Conclusion

Correlation analysis is a powerful first step in identifying potential relationships between financial variables. Visual tools like scatter plots provide an immediate sense of whether a trend might exist, while statistical measures such as covariance and Pearson’s correlation quantify the relationship’s strength and direction. However, the method’s limitations—sensitivity to outliers, inability to detect nonlinear patterns, and the risk of confusing correlation with causation—mean that results should always be interpreted with caution. A careful approach that combines statistical analysis with domain specific expertise and additional testing will offer us the most reliable insights.