Sept 2005 Paper Q14 Part iii)a)

[AngelFace]

I'd be really unhappy if this sort of question came up in the exam - it's mathematically incorrect. From the way that the question was asked, and the way the solution is drawn up, it is effectively assumed that the following holds:
E[log X] = log (E[X])

Which, since X is not constant, it doesn't! You could only possibly answer this question by assuming N of the form suggested, rather than E[N], and Y of the form suggested, rather than E[Y], which I guess is what the Revision Note solution was trying for?

 

[John Lee]

It's done this way but it's dodgy!

From the way that the question was asked, and the way the solution is drawn up, it is effectively assumed that the following holds:
E[log X] = log (E[X])

It's a very good question.

If we assume that Yi is normally distributed (which is necessary to do a test on beta) with additive errors ei then this implies that Ni is lognormally distributed with multiplicative errors which contradicts the additive structure asssumed in part (ii) to obtain the least squates estimates.

So there is no direct comparison - looking in Freund it simply says "it is customary to estimate" it this way. Implying that this is what's done in practice even though it's not quite sound.

However, the easiest way to resolve this is to say that the errors are small and make naff all difference what structure they have....