Step 1 — Restate the setup

We have an OLS regression with design matrix $X$ and response $y$. The estimator variance is $\operatorname{Var}(\hat\beta)=\sigma^{2}(X^{T}X)^{-1}$. All features and the target are multiplied by 2, producing a new dataset $(X',y')$ and a new OLS estimator $\hat\beta'$. Find a constant $c$ such that $\operatorname{Var}(\hat\beta')=c\,\operatorname{Var}(\hat\beta)$.

Step 2 — Assumptions & givens

Linear model: $y = X\beta + \varepsilon$ with $\varepsilon\sim N(0,\sigma^{2}I)$.
$X$ has full column rank.
Scaling factor $a=2$.
New design matrix $X' = aX$ and new response $y' = ay$.

Step 3 — Method & formulas

Use the variance formula for OLS: $$\operatorname{Var}(\hat\beta)=\sigma^{2}(X^{T}X)^{-1}.$$ For the scaled data we need the new error variance $\sigma'^{2}$ and the new information matrix $X'^{T}X'$.

Step 4 — Derivation

$X' = 2X \Rightarrow X'^{T}X' = (2X)^{T}(2X)=4X^{T}X,\qquad (X'^{T}X')^{-1}= \frac{1}{4}(X^{T}X)^{-1}.$ Since $y' = 2y = 2X\beta + 2\varepsilon = X'\beta + 2\varepsilon$, the new error term is $\varepsilon' = 2\varepsilon$ with variance $$\sigma'^{2}= \operatorname{Var}(2\varepsilon)=4\sigma^{2}.$$ Thus $$\operatorname{Var}(\hat\beta') = \sigma'^{2}(X'^{T}X')^{-1}= (4\sigma^{2})\Big(\tfrac{1}{4}(X^{T}X)^{-1}\Big)= \sigma^{2}(X^{T}X)^{-1}.$$

Step 5 — Plug in & compute

Comparing with the original variance, $$\operatorname{Var}(\hat\beta') = \operatorname{Var}(\hat\beta).$$ Hence the constant $c$ satisfying $\operatorname{Var}(\hat\beta') = c\,\operatorname{Var}(\hat\beta)$ is $$c = 1.$$

Step 6 — Sanity checks

Scaling both $X$ and $y$ by the same factor leaves the OLS coefficients unchanged (as the hint notes) and the increase in error variance ($4\sigma^{2}$) exactly cancels the increase in the information matrix ($4X^{T}X$), so the variance remains unchanged—consistent with $c=1$.

Final answer

c = 1

81. Double Data Variance

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