The correlation coefficient is a statistical measure of the strength of a linear relationship between two variables. Its values can range from -1 to 1. A correlation coefficient of -1 describes a perfect negative, or inverse, correlation, with values in one series rising as those in the other decline, and vice versa. A coefficient of 1 shows a perfect positive correlation, or a direct relationship. A correlation coefficient of 0 means there is no linear relationship.
Correlation coefficients are used in science and finance to assess the degree of association between two variables, factors, or data sets. For instance, while high oil prices might be favorable for crude producers, correlation analysis may reveal that the relationship between oil prices and forward returns on oil stocks isn’t consistently strong.
Key Takeaways
- Correlation coefficients assess the strength of associations between data variables.
- The most common, called a Pearson correlation coefficient, measures both the strength and the direction of a linear relationship between two variables.
- Values always range from -1 (a perfectly negative correlation) to 1 (a perfectly positive correlation). Values at zero indicate no linear relationship.
- The coefficient values required to signal a meaningful association vary by application and can be calculated for statistical significance.
Unveiling the Correlation Coefficient
Different types of correlation coefficients exist, with the Pearson coefficient, or Pearson’s r, being the most common. It measures the linear relationship between two variables. It is essential to note that the Pearson coefficient cannot assess nonlinear associations between variables and cannot differentiate between dependent and independent variables.
The Pearson coefficient uses a mathematical statistics formula to evaluate how closely the data points connecting the two variables approximate the line of best fit, which can be determined through regression analysis.
The further the coefficient is from zero, whether positive or negative, the better the fit and the greater the correlation. Values of -1 and 1 describe perfect fits where all data points align in a straight line, indicating perfect correlation. Conversely, the closer the correlation coefficient is to zero, the weaker the correlation.
Assessments of correlation strength based on the correlation coefficient value vary by application. In physics and chemistry, meaningful correlation typically requires values lower than -0.9 or higher than 0.9, while in social sciences, thresholds might be as high as -0.5 and as low as 0.5. For sample-based correlation coefficients, statistical significance is determined through the p-value, calculated from the data sample’s size and the coefficient value.
Formula for the Correlation Coefficient
To calculate the Pearson correlation, determine each variable’s standard deviation as well as the covariance between them. The correlation coefficient is covariance divided by the product of the two variables’ standard deviations.
ρxy = Cov(x,y) / (σx * σy)
where:
- ρxy = Pearson product-moment correlation coefficient
- Cov(x,y) = covariance of variables x and y
- σx = standard deviation of x
- σy = standard deviation of y
Standard deviation measures data dispersion from its average. Covariance indicates whether two variables tend to move in the same direction, while the correlation coefficient measures the strength of that relationship on a normalized scale, from -1 to 1.
An alternative formula is:
r = [n × (∑(XY) - (∑X * ∑Y))] / √[(n × ∑X^2 - (∑X)^2) * (n × ∑Y^2 - (∑Y)^2)]
where:
- r = Correlation coefficient
- n = Number of observations
Correlation Statistics and Investing
The correlation coefficient is particularly beneficial in assessing and managing investment risks. Modern portfolio theory suggests diversification can reduce a portfolio’s return volatility, thereby mitigating risk. The correlation coefficient between historical returns can reveal whether adding an investment to a portfolio will enhance its diversification.
Additionally, correlation calculations are pivotal in factor investing, a strategy where portfolios are built based on factors associated with excess returns. Quantitative traders leverage historical correlations and correlation coefficients to forecast near-term changes in securities prices.
Limitations of the Pearson Correlation Coefficient
Correlation does not imply causation. The Pearson coefficient does not determine whether one correlated variable depends on the other nor what proportion of the dependent variable’s variation is due to the independent variable. This is measured by the coefficient of determination, also known as R-squared.
Moreover, the Pearson correlation coefficient does not describe the slope of the line of best fit; this can be determined using the least squares method in regression analysis. The coefficient can’t assess nonlinear associations or sampled data not subject to normal distribution. Extreme outliers can also distort the correlation.
For relationships not suited to the Pearson coefficient, nonparametric methods like Spearman’s correlation coefficient, the Kendall rank correlation coefficient, or a polychoric correlation coefficient can be used.
Finding Correlation Coefficients in Excel
To calculate correlation in Excel, simply input two data series in adjacent columns and use the built-in correlation formula.
Alternatively, for a correlation matrix across a range of data sets, Excel offers a Data Analysis plugin on the Data tab. Selecting the table of returns—ensuring columns are titled—allows for easy calculation and formatting of results.
FAQ
Are R and R² the Same?
No, R and R² are not the same. R represents the Pearson correlation coefficient, showing the strength and direction between variables. R², the coefficient of determination, measures the strength of a model.
How Do You Calculate the Correlation Coefficient?
The correlation coefficient is calculated by determining the covariance of the variables and dividing by the product of the variables’ standard deviations.
How Is the Correlation Coefficient Used in Investing?
The correlation coefficient is crucial in portfolio risk assessments and quantitative trading strategies, helping portfolio managers to limit volatility and risk.
The Bottom Line
The correlation coefficient describes how one variable moves concerning another. A positive correlation indicates a concurrent movement, with a value of 1 denoting a perfect positive correlation. A value of -1 shows a perfect negative correlation, while zero means no linear correlation exists.
Related Terms: linear relationship, negative correlation, positive correlation, covariance, R-squared, statistical significance.
References
- DataTrek Research. “Oil Prices/Energy Stock Correlations, Rate Expectations”.
