The setting

Consider the simple regression model:

\begin{equation} y_i=\hat{\beta}_0 + \hat{\beta}_1 x_i + \hat{\epsilon}_i \end{equation}

Now, suppose I want to apply a linear transformation to the response, say:

\begin{equation} u_i = b_0 + b_1 y_i \end{equation}

The model with the transformed response would be

\begin{equation} u_i = \hat{\gamma}_0 + \hat{\gamma}_1x_i + \hat{\epsilon}_i, \textrm{with} \\ \hat{\gamma}_0 = b_0 + b_1 \hat{\beta}_0 \\ \hat{\gamma}_1 = b_1\hat{\beta}_1 \end{equation}

The problem

So far, so good. Now let's say I have a response in liters, and I want to transform to milliliters. \begin{equation} 1ml = 0.001 l, \textrm{ i.e.} \\ u_i = 0.001 y_i \end{equation}

So I think, "Ah, that's a linear transformation. I can just apply the formula from above." But when I do, I end up with

\begin{equation} \hat{\gamma}_0 = 0.001\hat{\beta}_0, \textrm{and} \\ \hat{\gamma}_1 = 0.001\hat{\beta}_1 \end{equation} which is obviously wrong. The right transformation would be:

\begin{equation} \hat{\gamma}_0 = 1000\hat{\beta}_0, \textrm{and} \\ \hat{\gamma}_1 = 1000\hat{\beta}_1 \end{equation}

So I suppose, the transformation formula would have to go: \begin{equation} u_i = 1000y_i \end{equation}

The question

How can I systematically get it right, when starting from something like $1ml = 0.001l$?


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