Transformation of confidence intervals I'm using Matlab to perform a linear regression. In order to prevent the prediction of negative values I used a box-cox-transformation of the dependent variable ($=y_t$) with $\lambda = 0.5$. 
$y^{(\lambda)} = \frac{y_t^{\lambda} - 1}{\lambda}$
After that I perform the linear regression with $y^{(\lambda)}$ as dependent variable. To get the result in my original form I transform $y^{(\lambda)}$ back into $y_t$.
$y_t = (\lambda(\frac{1}{\lambda} + y^{(\lambda)}))^{\frac{1}{\lambda}}$
My question now is, can I transform the confidence intervals in the same way as I transform my dependent variable and how do I prove it or disprove it?
 A: Let's say the transformed regression equation is $E \left[ y^{\left( \lambda \right)} |x
\right] = x \beta$ and let's call $\hat{y}^{\left( \lambda \right)} = x
\hat{\beta}$ where $\hat{\beta}$ is the estimated parameters vector. Let's call
$H$ the covariance matrix of $\hat{\beta}$. Finally let's create this new
function $g \left( \hat{y}^{\left( \lambda \right)} \right)$ such that
$$ g \left( \hat{y}^{\left( \lambda \right)}  \right) = \left( \lambda \left(
   \frac{1}{\lambda} + \hat{y}^{\left( \lambda \right)} \right)
   \right)^{\frac{1}{\lambda}}$$
Now, in order to construct the confidence interval, you need to approximate
the variance of your new estimator $g \left( \hat{y}^{\left( \lambda \right)}
\right)$. It is possible to do so using the delta method
$$ \frac{\partial g}{\partial \hat{\beta}^T} H \frac{\partial g}{\partial
   \hat{\beta}} $$
where $\frac{\partial g}{\partial \hat{\beta}}$ is a vector of derivatives of
$g$ with each of $\hat{\beta}_1, \ldots, \hat{\beta}_k$ the different
parameters you estimated and $\hat{\beta}^T$ is obviously the transpose of
$\hat{\beta}$.
