t-Statistic After Scaling the Data

This question is about how basic linear regression outputs behave when you rescale the underlying data, and what that implies for standard hypothesis tests. It focuses on the stability of a key diagnostic quantity when the same relationship is viewed under a different numerical scale, a common situation in real-world quant data preprocessing and feature engineering during quant prep for interviews.

It trains your understanding of t-statistics, standard errors, and parameter estimates in linear models, and how they transform under scaling. You need a firm grasp of regression assumptions, variance structure, and the difference between absolute values and scale-free measures of significance, which is central to robust quant interviews preparation.

This matters for quant interviews because quants constantly rescale, normalize, and transform data. Interviewers want to see that you know which inferential quantities are invariant, ensuring your conclusions are sound regardless of how the data is represented.

What it tests

In linear regression, the $t$-statistic for a coefficient tests whether the estimated effect is significantly different from zero by comparing the estimate to its standard error. The key structural property is that scaling all data points by a constant factor does not change the relative relationship between variables, nor does it affect the variance of the residuals or the standard errors of the coefficient estimates. This invariance arises because both the numerator (the coefficient estimate) and the denominator (its standard error) scale proportionally, or, in some cases, remain unchanged, depending on whether the scaling is applied to the predictors, the response, or both. The $t$-statistic thus remains unchanged under uniform scaling, reflecting the fact that statistical significance is a function of relative, not absolute, magnitudes. This principle holds because the regression model is linear and the error structure is homoscedastic and independent.