# Unconditional and conditional models

I don't know if the question is worded weirdly, but I'm having difficulties understanding its logic. I have the solution, but if possible, can someone explain the reason behind it?

We have two models (assume random sampling): $$E[y]=\alpha$$ and $$E[y|x]=\alpha$$

How can we estimate for both models, $$\alpha$$ consistently and as efficiently as possible;

And from using these estimates, how can we test the hypothesis $$H_0: \alpha= 0$$

The solution given is: For model 1, which is a regression of y on a constant, the only consistent estimator is the analog estimator $$\hat{\alpha}=\bar{y}$$. Model 2 is a regression on x and assuming that x is not constant with a very tight regression function. The regression parameters (here only one) can be efficiently (as the true conditional density is unknown) estimated through GLS:

• Interesting question. You might find it helpful to contemplate the simplest situation where the two models differ: namely, $x$ can take on only two distinct values (which might as well be $0$ and $1$). In this case you have a bunch of $y$ observations associated with $x=0$ and another bunch associated with $x=1.$ What do you do when the former set is much more scattered than the latter? Answer: construct a weighted mean with weights in inverse proportion to the group variances.
– whuber
Mar 17, 2021 at 18:19
• I am sorry, I am still not following. Is $x=1$ in the second model for the GLS to look like that? Mar 17, 2021 at 19:05
• @MaybelineLee you have to think beyond regression and look into likelihood. You would need the EM algorithm to estimate the MLE in whuber's example. Mar 17, 2021 at 19:41
• @Adam The quotation does not seem to be pushing for MLE, as evidenced by its reference to "nonparametric regression."
– whuber
Mar 17, 2021 at 19:52
• @whuber I would think GLS is a rather a parametric regression routine, and a convenient off-the-shelf way of estimating the covariance structure (via EM). But is the question how do we go to the next step and construct the hypothesis test? Mar 17, 2021 at 21:05