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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: enter image description here

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Mar 17, 2021 at 18:19
  • $\begingroup$ I am sorry, I am still not following. Is $x=1$ in the second model for the GLS to look like that? $\endgroup$ Mar 17, 2021 at 19:05
  • $\begingroup$ @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. $\endgroup$
    – AdamO
    Mar 17, 2021 at 19:41
  • $\begingroup$ @Adam The quotation does not seem to be pushing for MLE, as evidenced by its reference to "nonparametric regression." $\endgroup$
    – whuber
    Mar 17, 2021 at 19:52
  • $\begingroup$ @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? $\endgroup$
    – AdamO
    Mar 17, 2021 at 21:05

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