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An assumption for linear regression confidence intervals is that the variance is the same for the dependent variable for whatever of the independent variable.

If in practice the variance is different, what does this imply for the ability of the model to function as a prediction mechanism?

I have a few questions that come immediately to mind:

Does this mean that the confidence intervals will be wider in some places than they need to be?

Does it mean the model will be inaccurate for some inputs?

Are linear regression models nearly-always inaccurate, and so the last point might be a minor problem compared to the general inaccuracy?

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    $\begingroup$ If, in practice, the variance is different then use weighted least squares! See here. $\endgroup$ Mar 31 '15 at 14:55
  • $\begingroup$ @TrynnaDoStat Thanks for the quick and useful suggestion! I found this useful: biostat.jhsph.edu/~iruczins/teaching/jf/ch5.pdf I'd still be interested to answers regarding the way normal linear regression would be affected by changes in variance as in the question -- mainly so I can decide whether normal linear regression can be used as a quick approximation. $\endgroup$
    – Tom
    Mar 31 '15 at 15:31
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If the variance of the dependent variable $Y$ is not constant, this is called heteroscedasticity. Its effect is usually not severe unless the heteroscadacitity is quite strong. Even if the heteroscedasticity is strong, if it's the only assumption violated then the least squares estimates will still be unbiased and will still be Normally distributed. However, the estimate for the variance of the least squares estimates will be inaccurate and consequently so will the confidence intervals. We cannot say if, in general, these incorrect confidence intervals are wider or narrower then they should be.

If, in practice, the variance is quite heteroscedastic then weighted least squares is preferred.

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  • $\begingroup$ Thank you very much for this answer -- very clear and informative. I marked it as the answer because it answers my core question about the nature of the confidence intervals. $\endgroup$
    – Tom
    Mar 31 '15 at 16:44

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