T-stat After Data Duplication

This quant interview question is about how a simple linear regression metric behaves when you artificially inflate the dataset without adding new information. It probes whether you really understand what drives significance tests in regression, beyond just memorizing formulas. In quant prep, this kind of problem tests your intuition for how statistical quantities scale with sample size and how fragile naive interpretations of "more data" can be.

It trains your grasp of t-statistics, variance of estimators, and sampling distributions in a linear model. It reinforces how standard errors respond to changes in sample size and what truly constitutes independent information. For quant interviews, this is core statistical thinking, not plug-and-chug.

This matters in quant interviews because real-world quant research often involves reweighting, resampling, and overlapping data. Misreading t-stats under such manipulations leads to overfitting, false discoveries, and bad trading decisions, so strong intuition here is essential.

What it tests

In linear regression, the precision of coefficient estimates is fundamentally governed by sample size: as the number of observations increases, the variance of the estimated coefficients decreases. Specifically, if you double the dataset by exact duplication, the variance of the estimator for a coefficient is halved, because each data point is now counted twice, but the underlying information content (not the diversity) is not increased. The standard error, being the square root of the variance, decreases by a factor of $1/\sqrt{2}$. This relationship arises because the variance of the mean of $n$ i.i.d. samples is $\sigma^2/n$, and the same scaling applies to the variance of regression coefficients, which are linear combinations of the data. Thus, the $t$-statistic, which is the ratio of the coefficient estimate to its standard error, scales up by $\sqrt{2}$ when the dataset is duplicated, even though the estimate itself does not change.