Nonlinear regression model linear in some parameters In a set of lecture notes that I stumbled upon online, the author discusses a nonlinear regression model, which is linear in some parameters, like this
$$
y = \theta_1 + \theta_2\exp \left( {\theta_3x} \right) + \varepsilon
$$
and suggests to first perform a nonlinear regression to estimate $\theta_3$, and then use OLS to estimate the other two parameters. I am not very experienced in regression modeling, so please forgive me if I ask a very simple question, but I don't understand how you can do this consistently without also estimating the other 2 parameters? Apparently, there is some trick involved, which I don't understand...
Another question is what happens to the ability to do inferences based on parameter estimates to the complete model if you do the regression in two stages like that?
Many thanks!
 A: OLS for the model $y = \theta_1 + \theta_2 u + \varepsilon$ can be understood as a formula to convert an array of data $(u_i, y_i)$ into least squares (LS) parameter estimates $(\hat{\theta_1}, \hat{\theta_2})$.  Associated with any parameter estimates (not necessarily the least squares ones) is a sum of squares of residuals.  Let's call this $\text{SS}(\hat{\theta_1}, \hat{\theta_2})$.  
Now, given data $((x_i,y_i),i=1,\ldots,n)$, fix a value of $\theta_3$ and let $u_i = \exp(\theta_3 x_i)$.  The composite function given by
$$\lambda: \theta_3 \rightarrow ((u_i,y_i))\xrightarrow{LS} (\hat{\theta_1}, \hat{\theta_2}) \rightarrow \text{SS}(\hat{\theta_1}, \hat{\theta_2})$$
is the residual sum of squares when the LS estimates $(\hat{\theta_1}, \hat{\theta_2})$ are computed with the third parameter set to $\theta_3$.  The least squares solution $\hat{\theta_3}$ is obtained by minimizing $\lambda(\theta_3)$.  (The minimization can be done in any way that is appropriate and efficient; it need not even be iterative when direct solutions are available.)  Once that is obtained, we take $(\hat{\theta_1}, \hat{\theta_2})$ to be the OLS estimates associated with this value of $\theta_3$.  This is a considerable simplification because the minimization is a one-variable problem, not a three-variable problem.
