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I've got an easy question concerning residual analysis. So when I compute a QQ-Plot with standardized residuals $\widehat{d}$ on the y-axis and I observe normal distributed standardized residuals, why can I assume that the error term $u$ is normal distributed? I'd think that if $\widehat{d}$ looks normal distributed I just could assume that the standardized error term $d$ should be normal distributed.

So why can we assume that $u\sim N$ when we just observe that $\widehat{d}\approx N$. By the way can we assume it?

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could you please mention how you calculate $u$ and $\hat d$ – suncoolsu Jul 12 '11 at 16:41
Hi. you can't calculate $u$ (except you're observing the whole population). So I'd say I calculate $\widehat{u}=y-\widehat{y}$ and $\widehat{d}=\widehat{u}/\widehat{\sigma^2}$. I guess I've got an idea of the answer, but I can not show it. What I know is, that we often assume that $u\sim N(0,\sigma^2)$, so when we divide $u$ by $\sigma^2$ I assume that we divide the first 2 moments of the distribution as well $u/\sigma^2 \sim N(0/\sigma^2,\sigma^2/\sigma^2)$ where $u/\sigma^2=d$ ist the std.error term which is (from above) $d\sim N(0,1)$. But I'm not sure if this makes sense (That's why I ask) – MarkDollar Jul 12 '11 at 19:46
... Thus when $u\sim N(0,\sigma^2)$ we assume automatically that $d\sim N(0,1)$. Is this correct? And where can I find a proof that this is correct? – MarkDollar Jul 13 '11 at 8:09
In practical situations we estimate $\sigma^2$, then $\hat d$ is no longer gaussian. Instead, it is a $t$. But a $t$ with higher than 10 is not very different from a gaussian distribution. – suncoolsu Jul 13 '11 at 14:31
@suncoolsu: I absolutely don't understand what you mean :) Please clarify and give a reference. Thank you! – MarkDollar Jul 13 '11 at 17:27

2 Answers

up vote 1 down vote

This inference is no different from any other inference we make. We assume a default (you could call it a 'null'). In this case, it's that the underlying distribution is Gaussian. We examine the data to see if they are inconsistent with our default hypothesis. If the qq-plot of our residuals looks sufficiently Gaussian for your satisfaction, then we stick with that assumption. In truth, no matter how non-normal our residuals appear, they could still have come from an underlying Gaussian distribution, but at some point, we just don't believe it anymore. Another way to phrase this is that we don't assume they're Gaussian because the qq-plot looks Gaussian, rather we don't stop assuming they're Gaussian because the qq-plot doen't look sufficiently non-Gaussian.

Some people have trouble with this line of reasoning; which is perfectly fine. You might be interested in checking out the Bayesian approach to statistics.

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up vote 0 down vote

The $u$s are unobserved and the $\hat{d}$s are just estimates of them.

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