Correlation after stacking a dataset twice

Question

An OLS fit of share price $Y$ on news sentiment $X$ gives $Y=156.3+10.21X$ with correlation $0.78$ . You append an exact copy of every data point and refit. By how much does the correlation coefficient go up?

Short answer

$0$ . Duplicating every point leaves all sample means, variances and covariances unchanged, so the correlation (and fitted line) stay the same.

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Lessons behind this question

Maximum Likelihood Estimation: Score and Log-Likelihood
Covariance and Correlation
Simple Linear Regression: OLS Estimators and Geometry

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