OLS Variance After Scaling the Data

This quant interview question is about understanding how linear regression behaves when you rescale all the data. It focuses on the relationship between the design matrix, the error term, and the resulting distribution of the estimated coefficients. In quant prep, this sits at the intersection of probability, statistics, and linear algebra, which are all core to quantitative finance interviews.

It trains your grasp of OLS variance, scaling properties, and how model uncertainty depends on data geometry rather than arbitrary units. You need to be comfortable with matrix expressions for variance and how transformations of inputs and noise affect them in regression.

This matters for quant interviews because real-world data often require rescaling, normalization, or changes of units. Interviewers use this to test if you can reason rigorously about model stability, estimator precision, and invariance properties without relying on rote formulas.

What it tests

The variance of OLS coefficient estimates is governed by how both the scale of the predictors (the design matrix) and the noise variance interact in the formula $\text{Var}(\hat{\beta}_{OLS}) = \sigma^2 (\bm{X}^T \bm{X})^{-1}$. When all data points are scaled by a constant, the design matrix and the error variance both scale, but in a way that these effects exactly offset each other in the variance formula. This is because scaling the predictors by $a$ multiplies $\bm{X}^T \bm{X}$ by $a^2$, and if the errors are scaled by $a$ as well, $\sigma^2$ becomes $a^2\sigma^2$. The inverse in the formula then divides by $a^2$, perfectly canceling the scaling of $\sigma^2$. This invariance reflects the fact that OLS regression is equivariant to linear scaling: the relative geometry of the data, not its absolute scale, determines the estimator's precision.