I would like an explanation on the statement in bold below. At first glance, I'd think that a weak instrumental variable would yield a even bigger standard error estimate.
"When instruments are weak, however, two serious problems emerge for two-stage least squares. First is a problem of bias. Even though two-stage least squares coefficient estimates are consistent — so that they almost certainly approach the true value as the sample size approaches infinity — the estimates are always biased in finite samples. When the instrumental variable is weak, this bias can be large, even in very large samples. Second, when an instrumental variable is weak, two-stage least squares’ estimated standard errors become far too small. Thus, when instruments are weak, confidence intervals computed for two-stage least squares estimates can be very misleading because their mid-point is biased and their width is too narrow, which undermines hypothesis tests based on two-stage least squares."
Murray, Michael P.. Avoiding Invalid Instruments and Coping with Weak Instruments. Journal of Economic Perspectives, v. 20, pp. 111-132.