Basically, though, what you need to bear in mind is that these [employment and unemployment measures] are imperfect measures, subject to a fair bit of noise. When the trend in the labor market is very strong in either direction, the measures move together. But when you have the kind of scene we have now — the employment situation is drifting down, but not plunging — occasional mixed signals are likely. No big deal.
The basic story is that things are sort of stabilizing — but they’re definitely not improving yet.
There are two issues. First, is it possible that a decline in the unemployment rate is not unambiguously good news? The answer is yes -- if unemployed people drop out of the labor force, then they are no longer counted as unemployed, and the unemployment rate can go down. As Krugman notes earlier in his post, that is what happened in July, but U-6, the broader measure of the unemployment rate that does not drop discouraged workers, also fell, so it may not be that it was discouraged workers who dropped out. It could have been people leaving voluntarily for any number of reasons. Second, Krugman is right that things are not improving yet. Getting worse less rapidly is not the same thing as improving.