Using an OLS coefficient to estimate a non-linear coefficient

Question

Using OLS, I've estimated the following equation:

$y_i = \alpha_0 + \alpha_1 X_i + \epsilon_i$

I know that theoretically, the following should be true:

$y_i = a + (1-e^{-\lambda 60}) X_i$

Is there any way, having an estimate of $\alpha_1$ I can translate it to an estimate of $\lambda$?

As a follow up, if this is not possible without some difficulty, If I knew the distribution of $\alpha_1$ was a normal distribution with some mean and variance, is there a way to describe what the form of the distribution of $\lambda$ would be? I feel like it would be a log-linear distribution, but I'm not sure what the mean/variance would be.

Answer 1

Judging from your equations there is no reason for the OLS estimate of $\alpha_1$ not to be consistent and asymptotically normal. So you can use plug-in estimate for $\lambda$:

$$\hat{\lambda}=-\frac{1}{60}\log(1-\hat{\alpha}_1)$$

Using delta method it would be possible to show that this estimate is also consistent and asymptotically normal. The only caveat is that the estimate of $\alpha_1$ can lie outside the interval $[0,1]$, but this would indicate that your postulated model is incorrect.