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I have 2 questions I was stuck on about.

What would you say linear regression assumes?

a. The relationship between X and Y is a straight line.

b. The residuals are normally distributed.

c. The residuals are homoscedastic.

d. Both homoscedastic and normally distributed residuals.

Models need to be validated:

a. In-sample

b. Out-of-sample

c. Both in-sample and out-of-sample

For the first one, I think its b). For the second one, I think its c, but need some conceptual explanation to them both and why this is the case.

I came to the conclusion of the 2nd one as c) since the model in an anova has considered weights accordingly. Is this correct?

Shane Gurtin
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You don't need to be fitting a straight line in order to be doing linear regression; see this answer.

Linear regression using ordinary least squares assumes homoscedasticity.

See this Wikipedia article about the Gauss–Markov theorem. That theorem assumes

  • The errors have expected value $0$, and
  • The are homoscedastic (in particular, they have finite variance), and
  • They are uncorrelated (not necessarily independent).

Identical distribution is not assumed.

The conclusion of the theorem is that the least-squares estimators of the parameters have the smallest variance of all estimators of those parameters that are linear combinations of the observed response variables.

So it appears that homoscedasticity is the only thing among your multiple choices that is correct.

Community
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Michael Hardy
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  • so you're saying for the first question the answer is c? How does the Gauss markov relate to the linear regression. Also, for the second question, wouldn't it be c) as well, both in-sample and out of sample? – Shane Gurtin Apr 07 '17 at 06:18
  • @ShaneGurtin : (c) is the only one that I can identify with the information I've got, but I wonder if you've given us the whole context. How Gauss–Markov relates to linear regression is just what I explained above. – Michael Hardy Apr 07 '17 at 17:22
  • I think the answer to the one you answered is d, not c. Wouldn't you agree? – Shane Gurtin Apr 07 '17 at 20:41
  • @ShaneGurtin : In some contexts, in fact quite often, one assumes normally distributed errors. However, linear regression can reasonably be done in some contexts where the errors are not normally distributed, and it's still linear regression. I would be more careful about the words "error" and "residual". A residual is an observable estimate of an unobservable error. One has $\text{error}_i = y_i - (\alpha + \beta x_i)$ and $\text{residual}_i = y_i - (\widehat\alpha + \widehat\beta x_i). \qquad$ – Michael Hardy Apr 07 '17 at 22:53