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My statistics professor mentioned in the class that we can generally model the regression relationship as $y = f(x) + \epsilon$ where $E(\epsilon|x) = 0$. He told us that $\epsilon$ is not independent of $x$ and a way to show that is through $(\epsilon|x)\sim N(0,x)$

Does anyone know why is this the case? I thought that random error is always independent of X.

RofyRafy
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  • I don't understand your claim. By $N(0,x)$ do you mean the distribution with mean $0$ and standard deviation (or variance) $x$? But that's not a standard notion for error. – lulu Sep 28 '21 at 14:07
  • Yes, I was curious too. The instructor told me that error is not independent of x and a way to prove that is by setting the error to follow a gaussian distribution of mean 0 and variance = x. – RofyRafy Sep 28 '21 at 14:14
  • Perhaps you misunderstood. Perhaps the instructor was asserting that it was possible that the error was dependent on the value. That is certainly true. But that's a very different claim that saying than the error is always dependent on the value, let alone that the error must always follow some explicit form. – lulu Sep 28 '21 at 14:16
  • I guess I must have misunderstood. Nonetheless, how should I give the counterexample so that there are I can say that it is possible for the error to be dependent on x? My instructor told me that I can think of showing how there are times that the error can follow a gaussian distribution and have a variance of x. I still can't wrap my mind around how it actually work – RofyRafy Sep 28 '21 at 14:26
  • Oh, the example you gave is fine. My objection was not that it was an impossible form for an error, just that it certainly wasn't a universal form for the error. – lulu Sep 28 '21 at 14:34
  • I see, noted with thanks. One more doubt that I have (if you don't mind). If I can say that there are times that epsilon is not independent of x s.t. $E(\epsilon | x) = 0$, does it means that the error is not always independent of x? – RofyRafy Sep 28 '21 at 14:40
  • Of course there are situations in which the error depends on $x$ and situations in which the error is independent of $x$. For an unbiased estimator one has that the error has expectation $0$, but you did not specify that your estimator was unbiased. – lulu Sep 28 '21 at 14:43

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