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How can I get cluster-robust errors (for generating confidence intervals) on a Pearson correlation coefficient?

I know that I can generate cluster-robust errors for a coefficient in a linear model. And I know that square-root of the r-squared in the linear model is equivalent to the Pearson correlation. But I don't know how to get an appropriate standard error on the Pearson correlation.

Note: this is how i'm accounting for cluster-robust errors

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Standard error alone cannot be used to generate a confidence interval for a Pearson correlation coefficient. One way to generate confidence intervals is to do a 'percentile bootstrap' (see here). Instead of bootstrapping a standard error, use the bootstrap process to generate a distribution of correlation coefficients. Then, identify the bounds the inner (eg) 95% of the data to generate a (eg) 95% confidence interval.

To account for clustered data, use a clustered bootstrap: instead of sampling-with-replacement N individuals for each bootstrap estimate, sample entire clusters. (see a practitioners guide to cluster robust inference)

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