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The Statement of the Problem:

When the predictor variable is so coded that $\bar X = 0$ and the normal error regression model applies, are $b_0$ and $b_1$ independent? Are the joint confidence intervals for $\beta _0$ and $\beta _ 1$ then independent?

Where I Am:

I know that, in this case, the covariance between $b_0$ and $b_0$ is

$$ \sigma \{ b_0, b_1 \} = -\bar X\sigma ^2\{b_1 \}. $$

So, if $\bar X = 0$, then the covariance would be $0$. Of course, one can't conclude independence from this (though, the converse is obviously true). And, I certainly can't conclude that they're definitely NOT indepedent. Basically, I'm wondering if it's even possible to establish independence either way here -- in which case, this is somewhat of a "trick" question, which would be annoying. Any help here would be appreciated.

thisisourconcerndude
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    What can you say about the independence (or lack thereof) of uncorrelated bivariate normal variables? – whuber Oct 26 '15 at 22:54
  • @whuber That they are independent, sure. I realize that these are both normal, but how do I know that these are $jointly$ normal? – thisisourconcerndude Oct 26 '15 at 22:57
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    Because both are linear combinations of the same jointly normal random deviations, as per the assumptions. If that's *not* your assumption, you should be explicit about just what you are assuming. – whuber Oct 26 '15 at 22:58
  • @whuber Ah, yes! I see. Thank you. And, since the standard deviations are independent of one another, so must be the joint confidence intervals. Is that right? – thisisourconcerndude Oct 26 '15 at 23:16
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    I'm not sure what you mean by joint confidence intervals being "independent." Even though the coefficient estimates $\hat\beta_0$ and $\hat\beta_1$ might be independent, a joint CI for $(\beta_0,\beta_1)$ will still not be the same thing as the product of separate CIs for $\beta_0$ and $\beta_1$. – whuber Oct 27 '15 at 00:13
  • @whuber Yeah, I'm not really sure how to interpret the question, either. The context is that of constructing Bonferroni joint confidence intervals which involves computing separate confidence intervals and the probability of the union of the events in which both of these are correct. But, even after playing around with the calculations a bit under the assumption of independence, nothing is popping out to me. Oh, well. Thanks for all of the help! – thisisourconcerndude Oct 27 '15 at 12:28
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    When two random variables are independent and you construct a $1-\alpha$ confidence interval for each, then it follows immediately from the definition of independence (and the fact that [measurable functions of independent variables are independent](http://stats.stackexchange.com/questions/94872)) that the Cartesian product of the intervals is a joint $(1-\alpha)^2\approx 1 - 2\alpha$ confidence interval (when $\alpha$ is small). – whuber Oct 27 '15 at 13:05

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