My question is about confidence intervals of the slopes estimated in a multivariate regression.
I would like to clear up something that is probably a fault somewhere in my understanding. As I understand things:
1) $95\%$ confidence interval gives the region, around an estimate of y, for which we are $95\%$ confident that the true y lies, for a given x.
--> the narrower this interval, the more confident we are that the estimate lies closer to the true values - i.e. the better our estimate.
2) a higher t-stat (and therefore lower p-value) for a coefficient means we are more confident with the estimated value of that coefficient.
3) The width of the confidence interval is directly proportional the t-stat, (equation 3.2). A higher t-stat will produce a wider confidence interval.
3 means that 1 & 2 contradict each other. The two figs below show examples of how the confidence interval looks for a given t-stat, for two different explanatory variables of the same regression.
Fig 1: variable with low t-stat, high p-val, but 'narrow' confidence band
Fig 2: variable with high t-stat, low p-val, but 'wide' confidence band
Can anyone clear this up? What intuition can we apply that explains why estimates with higher confidence (t-stat/p-value) have a wider confidence interval? Or am I misunderstanding?