As a result of recent research, a new stochastic methodology of assessing causality was developed. Its application to instrumental measurements of temperature (T) and atmospheric carbon dioxide concentration ([CO2]) over the last seven decades provided evidence for a unidirectional, potentially causal link between T as the cause and [CO2] as the effect. Here, I refine and extend this methodology and apply it to both paleoclimatic proxy data and instrumental data of T and [CO2]. Several proxy series, extending over the Phanerozoic or parts of it, gradually improving in accuracy and temporal resolution up to the modern period of accurate records, are compiled, paired, and analyzed. The extensive analyses made converge to the single inference that change in temperature leads, and that in carbon dioxide concentration lags. This conclusion is valid for both proxy and instrumental data in all time scales and time spans. The time scales examined begin from annual and decadal for the modern period (instrumental data) and the last two millennia (proxy data), and reach one million years for the most sparse time series for the Phanerozoic. The type of causality appears to be unidirectional, T→[CO2], as in earlier studies. The time lags found depend on the time span and time scale and are of the same order of magnitude as the latter. These results contradict the conventional wisdom, according to which the temperature rise is caused by [CO2] increase.