Understanding the Power of Inverse Correlation
An inverse correlation, also termed as negative correlation, defines a situation where two variables move in opposite directions. When one variable exhibits a high value, the corresponding variable typically shows a low value and vice versa. Statistically, this relationship is quantified using the correlation coefficient “r,” which ranges from -1 to 0, with an ‘r’ of -1 signifying a perfect inverse correlation.
Key Takeaways
- Inverse (or negative) correlation occurs when one variable increases while the other decreases.
- A strong negative correlation does not imply causation between the variables.
- The relationship between two variables can fluctuate over time.
Visualizing Inverse Correlation
A scatter diagram, which plots data points concerning two variables on the x and y axes, is an excellent tool to visualize correlation. A strong inverse correlation would appear as a distinct downward trend on such a graph.
Image by Sabrina Jiang
Determining Inverse Correlation Using Calculations
To calculate correlation accurately, one might use Pearson’s r. When r is below 0, it indicates an inverse correlation. Here’s a breakdown of calculating Pearson’s r:
Given this dataset with seven observations on variables X and Y:
- X: 55, 37, 100, 40, 23, 66, 88
- Y: 91, 60, 70, 83, 75, 76, 30
Let’s follow these steps to derive r:
- Determine SUM(X), SUM(Y), and SUM(X, Y):
$$\text{SUM}(X) = 55 + 37 + 100 + 40 + 23 + 66 + 88 = 409$$
$$\text{SUM}(Y) = 91 + 60 + 70 + 83 + 75 + 76 + 30 = 485$$
$$\text{SUM}(X, Y) = (55 \times 91) + (37 \times 60) + … + (88 \times 30) = 26,926$$
- Calculate SUM(X^2^) and SUM(Y^2^):
$$\text{SUM}(X^2) = (55^2) + (37^2) + (100^2) + … + (88^2) = 28,623$$
$$\text{SUM}(Y^2) = (91^2) + (60^2) + (70^2) + … + (30^2) = 35,971$$
- Use the following formula to find Pearson’s r:
$$r = \frac{n \times (\text{SUM}(X, Y) - (\text{SUM}(X) \times \text{SUM}(Y))}{\sqrt{(n \times \text{SUM}(X^2) - \text{SUM}(X)^2) \times (n \times \text{SUM}(Y^2) - \text{SUM}(Y)^2)}}$$
Substituting the values:
[(7 \times 26,926 - (409 \times 485)) / sqrt{((7 \times 28,623 - 409^2) \times (7 \times 35,971 - 485^2))}]
This calculation yields an r of -0.42, revealing an inverse correlation.
Insights from Inverse Correlation
Inverse correlation signifies that when one variable rises, the other tends to fall. This relationship can be instrumental in financial contexts, such as understanding how stock and bond markets may move in opposite directions.
In financial markets, an exemplary case of inverse correlation is seen between the U.S. dollar and gold. As the U.S. dollar weakens, gold prices often rise and vice versa.
Leveraging Inverse Correlation in Portfolio Diversification
Inverse correlation is valuable for portfolio diversification. If two asset returns are negatively correlated, blending them can reduce overall risk.
Limitations of Inverse Correlation
- Correlation vs. Causation: An inverse correlation does not prove causality. Two variables might strongly correlate negatively, yet this alone does not indicate one causes the other.
- Dynamic Relationships: Especially in time series data, correlations can change. Variables may have inverse correlations in some periods and show positive correlations in others, making future projections risky.
Related Terms: positive correlation, correlation coefficient, statistical significance, linear relationship, diversification.
References
- Federal Reserve Bank of St. Louis. “Trade Weighted U.S. Dollar Index vs. Gold Fixing Price”.
