Please see the screenshot of the book Financial Time Series Analysis by Tsay. Given the formula for $v$.
$$\boldsymbol{v} = \boldsymbol{K} (\boldsymbol{y} - \mu_{1 | 0} 1^T)$$
It below says they're related by a unit Jacobian, but I cannot seem to figure this out. The change of variable formula is given as $f_v(v) = f_y(g^{-1}(v)) \frac{d g^{-1}(v)}{dv}$ where $f$ represents the pdf. The Jacobian never is equal to the identity matrix so I am unsure exactly what the author refers to here.
The Jacobian matrix of the transformation $v=f(y)$ is the matrix whose $(i,j)$ entry is $\frac{\partial v_i}{\partial y_j}$. You can check that when $f(y) = K(y - \mu 1_T)$ this Jacobian matrix is precisely $K$ (e.g., write what $v_i$ is in terms of $y_1, \ldots, y_T$ and take the derivative with respect to $y_j$).
Confusingly, "Jacobian" is often used to refer to this matrix ("Jacobian matrix") as well as its determinant ("Jacobian determinant"). The determinant of $K$ is $1$. This is the quantity that appears in the change of variables formula.