What is the time-lagged correlation between historical $\ce{CO2}$ records from ice cores and global temperature records?

Is there a time-lagged correlation with CO2 levels preceding temperature? Does the time lag change from time to time?

• Studies of the Vostok ice core show that at the "beginning of the deglaciations, the CO$_2$ increase either was in phase or lagged by less than ~1000 years with respect to the Antarctic temperature, whereas it clearly lagged behind the temperature at the onset of the glaciations".$^{[1]}$ Recent warming is followed by carbon dioxide levels with only a 5 months delay.$^{[2]}$ The time lag has been used to argue that the current rise in CO$_2$ is a result of warming and not a cause. While it is generally agreed that variations before the industrial age are mostly timed by astronomical forcing,$^{[3]}$ a main part of current warming is found to be timed by anthropogenic releases of CO$_2$, having a much closer time relation not observed in the past (thus returning the argument to the importance of human CO$_2$ emissions). Analysis of carbon isotopes in atmospheric CO$_2$ shows that the recent observed CO$_2$ increase cannot have come from the oceans, volcanoes, or the biosphere, and thus is not a response to rising temperatures as would be required if the same processes creating past lags were active now.$^{[4]}$