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How do I test if a non-linear least squares regression best fits a given set of data in R? More specifically I'm interested in getting estimates for a regression like this: $$y=\beta_0+\beta_1(x_1-a)+u$$
(for the variables $\beta_0$,$\beta_1$ and $a$). How would I code this?
If (as your question seems to say) that you have three parameters to estimate ($\beta_0,\beta_1,a$) then this model is not identifiable.
\begin{eqnarray} y&=&\beta_0+\beta_1(x_1-a)+u\\ &=&(\beta_0-\beta_1 a)+\beta_1 x_1+u\\ &=&\beta_0^*+\beta_1 x_1+u \end{eqnarray}
Now consider that there's some best estimate of slope $\hat{\beta}_1$, and a corresponding best intercept $\hat{\beta}_0^*$ (this doesn't rely on how they're estimated - it doesn't matter).
Then in your formulation the intercept (given the slope) has two free parameters ($\beta_0$ and $a$), so that for a given $\hat{\beta}_0^*$ there's an infinite set of pairs ($\hat{\beta}_0$ and $\hat a$) which will give that best fitting value of intercept. So you simply can't tell which pair it might be -- they will all give identical fits.
As it stands, there's nothing to be done here.
