>>10994678When academics refer to "low" correlation or "moderate" or "high" correlation, they use those terms descriptively - low could be around 0.2-0.3 or lower, moderate around 0.3-0.7, and high anything above.
But you can test for the significance of the correlation coefficient (be it Pearson or Spearman rank). We do not know the true population correlation so we rely on the theoretical asymptotic convergence of a statistic (called sample correlation) to the t-distribution. The hypothesis is that the coefficient is zero, you get a p-value as usual.
If you have a single variable regression, then your t-value for the regressor coefficient will be equal to the t-value associated with the correlation coefficient. This means that, if you use unadjusted standard error estimators, the regression coefficient will have an identical p-value to the p-value for the correlation coefficient. Your p-value will likely be higher if you, for whatever reason, use adjusted SE estimators such as Huber-White heteroskedasticity robust SE or Newey-West heteroskedasticity and autocorrelation robust SE.
So, to answer your questions directly:
>can x and y be uncorrelated and still beta be significant?If by "uncorrelated" we mean an insignificant correlation coefficient and if you use unadjusted standard errors, then no.
>Can x and y be correlated and beta is not significant?Same as above - no.
>what are the thresholds for "correlated" and "uncorrelated", is ist |Pearson's r| >0.5 and |r|<=0.5?Descriptive thresholds can be whatever you want them to be. Inference-based thresholds depend on your significance level.
