Mastering Correlation Tests in Finance | CFA Level I Quantitative Methods

Today we’ll dive into the world of hypothesis tests concerning correlation. We’ll start with the parametric test based on Pearson correlation, explore the differences between parametric and non-parametric tests, and wrap up with a non-parametric test based on Spearman rank correlation.

Parametric Test: Pearson Correlation

In finance, we often want to assess the strength of the linear relationship between two variables. The correlation coefficient, bound between -1 and +1, can help us do just that:

• Near -1: significant negative correlation
• Near +1: significant positive correlation
• Near 0: no significant linear relation between X and Y

To test if there’s a significant linear relation, we design a hypothesis test with the null hypothesis that the correlation in the population is equal to zero, and the alternate hypothesis that it’s not equal to zero. We then use a t-test to evaluate our hypothesis.

The Pearson correlation coefficient (r) measures the linear relationship between two variables. Here’s the formula:

r = S_xy / (S_x * S_y)

Where:

• Sxy is the covariance between variables X an Y
• Sx and Sy are the standard deviations of variables X and Y, respectively

Parametric vs. Non-Parametric Tests

Parametric tests, like the ones we’ve discussed so far, have two common characteristics:

1. They are concerned with parameters.
2. Their validity depends on a set of assumptions.

Non-parametric tests, on the other hand, are not concerned with parameters and make minimal assumptions about the population. They are useful in three scenarios:

1. When the distributional assumptions of parametric tests are not met.
2. When data are ranks rather than values.
3. When the question doesn’t concern a parameter.

Non-Parametric Test: Spearman Rank Correlation

If we can’t assume that the variables are normally distributed, we might use the Spearman rank correlation coefficient, which applies to ranked data. To perform this test, we first convert quantitative data to pairwise ranked data, calculate the difference in ranks between the two variables, and square the differences.

Formula for Spearman Rank Correlation Coefficient:

ρ = 1 – (6 * Σd^2) / (N * (N^2 – 1))

The remaining steps of the test are similar to those for the Pearson correlation test. We apply the Spearman rank correlation coefficient to the test statistic and perform a standard t-test with N – 2 degrees of freedom. Note that a large sample size is necessary for the t-test to be reliable.

It’s unlikely that you’ll have to conduct the Spearman Rank test in the exam, but it’s essential to know that quantitative data must be converted to ranked data and understand how the correlation coefficient is calculated in this case.

Conclusion

And there you have it! You’re now a correlation-testing wizard. We’ve covered hypothesis tests concerning correlation, including parametric tests based on Pearson correlation and non-parametric tests like the Spearman rank correlation test. We also discussed the differences between parametric and non-parametric tests.

Next up, we’ll learn how to perform tests of independence using contingency tables. See you in the next lesson, and keep up the fantastic work!