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I'm following these notes from NYU (1) for developing the OLS estimator with the Gauss-Markov theorem. By the assumptions, we can prove that $ E(\hat\beta) = \beta$ and that $E(\epsilon\epsilon'|X) = \sigma^2I$.

Isn't that enough to do some hypothesis testing for $\hat\beta$ using Chebyshev's inequality?

Why does the text say we must assume a 6th assumption for normality in order to be able to do hypothesis testing - namely why do we need to assume that $\epsilon = N(0, \sigma^2I)$?

(1): NYU-notes on OLS in matrix form
(https://web.stanford.edu/~mrosenfe/soc_meth_proj3/matrix_OLS_NYU_notes.pdf)

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Isn't that enough to do some hypothesis testing for $\hat\beta$ using 'Chebyshev's inequality'?

Not quite.

  1. To use Chebyshev directly, you'd need to know the population $\sigma^2$.

  2. Even if we got around that (say we came up with a Chebyshev bound on a purely-sample-statistic), the bounds are going to be quite broad. We have poor control of type I error (in that our rejection rule will in practice mean that our type I error rate is much lower than what we want (and hence, our power is low), so if we could make it work it wouldn't be a very good test.

Why does the text say we must assume a 6th assumption for normality in order to be able to do hypothesis testing - namely why do we need to assume that $\epsilon = N(0, \sigma^2I)$?

It's not necessary to assume normality; there are numerous things that can be done.

However the text doesn't say that we need to assume normality. If you read what it actually says:

However, we often assume it to make hypothesis testing easier.

that's a rather weaker statement that you said. The hypothesis testing is pretty easy under this assumption.

But as mentioned, other things could be done. For example:

  • we could make some other parametric assumption, for example (but then the obvious question would be "why use an inefficient estimator of the coefficients, given you're prepared to make that parametric assumption?"). So for example I might assume that the noise term has a Laplace distribution and base my inference on that -- but then the least squares parameter estimates will be considerably less efficient than using L1 estimation.

  • we could use a test that doesn't make a parametric assumption, such as a permutation or a bootstrap test, or use a robustified test based on a parametric assumption (i.e. one which is robust to deviations from that distributional assumption). [On the other hand, if we're going to go to something robust, why not robustify our estimation as well?]

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