Likelihood ratio test on Nonlinear regression done manually. Possible or typo?

#1
Dear all,
this forum is great! If I knew about it before, my studies would have been much more enjoyable.

Anyways, right now I’m looking at past exams of my advanced econometrics course; I’ve been trying for two days now to solve this exercise, advanced text books at hand and constantly on the web, only to come to the conclusion that the question probably contains a typo (but I couldn’t reach the professor yet).

It goes like this: given the NLS model

y = b/(x - c) + u (b,c parameters, y d.v., x i.v.)

with u i.i.d. and N(0,1), define the Likelihood ratio test for the null hypothesis c = 1.

Now, Maximising the likelihood function and calculating the RSS of the restricted model is easy, but I couldn’t find a way to come up with a closed form of the maximum likelihood for the unrestricted model. Is there any point I’m missing? Anyway to obtain a closed form by hand? Is there a simple way to linearize the model?

Or, as I’m suspecting by now, did the professor mean “Lagrange multiplier test”, which doesn’t need the ML of the unrestricted model.

Any ideas are much appreciated!