I always think about the error term in a linear regression model as a random variable, with some distribution and a variance. So if the error terms come from this random variable, why do we say that they have a constant variance?
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3$\begingroup$ You might need to expand this a bit to explain what the apparent contradiction is supposed to be. $\endgroup$– Scortchi ♦Commented Feb 16, 2014 at 18:25
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$\begingroup$ It is one random variable isn't it? If the error terms come from this one random variable of course they will have the same variance. $\endgroup$– kanbholdCommented Feb 16, 2014 at 18:31
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4$\begingroup$ "Error terms" is in the plural: they are a realization of one multivariate random variable, if you like. If you don't like that point of view, then you must view them as being realizations of multiple separate random variables (which might or might not have any properties in common). $\endgroup$– whuber ♦Commented Feb 16, 2014 at 18:36
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$\begingroup$ If you're asking whether it's redundant to say of a model both that the error terms are realizations of e.g. a Gaussian random variable with mean zero & variance $\sigma^2$, and that they have constant variance, then yes it is. Constant variance is emphasized because in some models the variance is a function of the mean, or of a predictor, & therefore not constant. $\endgroup$– Scortchi ♦Commented Feb 16, 2014 at 18:41
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$\begingroup$ It may help you to read my answer here: What does having constant variance in a linear regression model mean?, which addresses this issue. $\endgroup$– gung - Reinstate MonicaCommented Feb 17, 2014 at 14:56
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1 Answer
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The error term ($\epsilon_i$) is indeed a random variable. The normality assumption holds if it has Normal distribution - $\epsilon_i$ ~ $N(\mu,\sigma)$. You are right when you say:
I always think about the error term in a linear regression model as a random variable, with some distribution and a variance
The assumption of constant variance (aka homoscedasticity) holds if the dispersion of the residuals is homogeneous along the range of values in $X$ or $Y$. This pattern of dispersion can vary.
So if the error terms come from this random variable, why do we say that they have a constant variance?
One error observation alone does not have variance. The variances come from subsets of groups of error observations. For a better comprehension, look into this picture, borrowed from @caracal's answer here.
It also helps looking to some plots which illustrates the opposite of homoscedasticity (non constant variance).
answered Feb 16, 2014 at 22:13
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$\begingroup$ This answer appears to miss the point: how can the $\epsilon_i$ be considered "a" random variable when they are heteroscedastic? $\endgroup$– whuber ♦Commented Feb 16, 2014 at 22:16
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2$\begingroup$ "One error observation alone does not have variance." True, but it can be thought of as having been drawn from a distribution that does have a variance. $\endgroup$ Commented Feb 17, 2014 at 14:59