Duplication Summary

Effects of duplicating every data point in linear regression:

Answer

  • Coefficients: $\hat{\beta}$ remains unchanged. Duplication does not affect the OLS estimates.

  • Variance of estimators: It is halved, $\mathrm{Var}^*(\hat{\beta}) = \tfrac{1}{2}\sigma^2(X'X)^{-1}$, due to the doubled information matrix.

  • Estimated variance: Approximately halved (not exact due to how the sample variance of the errors is calculated).

  • p-values: Become smaller. This due to $t_{\text{dup}} \approx \sqrt{2}\,t_{\text{orig}}$, giving a false impression of greater precision.

  • Confidence intervals: Narrower by a factor of $\approx\tfrac{1}{\sqrt{2}}$.

  • R-squared: Remains unchanged since both RSS and TSS double proportionally.

Notes and comments

Comment 1: These apparent improvements are spurious. Duplicating data violates independence and adds no new information. The true variances, p-values, and confidence intervals therefore remain unchanged.

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