Spearman Test - rileywheadon/ffa-framework GitHub Wiki

The Spearman Test is used to identify autocorrelation in a time series $y_{t}$. A significant lag is a number $i$ such that the correlation between $y_{t}$ and $y_{t-i}$ is statistically significant. The least insignificant lag is the largest $i$ such that all $j < i$ are significant lags.

• Null hypothesis: The least insignificant lag is $0$.
• Alternative hypothesis: The least insignificant lag is greater than $0$.

To carry out the Spearman test, we use the following procedure:

1. Compute Spearman's correlation coefficient $\rho_{i}$ for $y_{t}$ and $y_{t-i}$ for all $0 \leq i < n$.
2. Determine the $p$-value $p_{i}$ for each correlation coefficient $\rho _{i}$.
3. Iterate through $p_{i}$ to find the largest $i$ such that $p_{j} \leq \alpha$ for all $j \leq i$.
4. The value of $i$ found in (3) is the least insignificant lag at confidence level $\alpha$.

Remark: To compute the $p$-value of a correlation coefficient $\rho _{i}$, first compute:

$$ t_{i}= \rho_{i} \sqrt{\frac{n-2}{1 - \rho _{i}^2}} $$

Then, the test statistic $t_{i}$ has the $t$-distribution with $n-2$ degrees of freedom.

For more information, see the Wikipedia pages on Autocorrelation and Spearman's Correlation Coefficient.