In this post, I have two sets of plots of the phase between centered sunspot numbers and centered temperature anomalies, where the centering is done by subtracting the 128 month linearly filtered data from the original time series. Also plotted is the phase divided by the frequency – or tau. I apologize to anyone I confused by my earlier uncentered plots.
If the slope of the phase plot is constant, then there is a constant lag between the two plots. The sign of the slope gives which series precedes the other. In my plots a negative slope indicates that the second series lags behind the first. (See Introduction to Statistical Time Series, W.A. Fuller, 1976, Wiley, pp. 152-153 and pp.308-324.)
The first set of plots is of the average of the daily sunspot numbers, averaged over each month, from WDC-SILSO at the Royal Observatory of Belgium, Brussels, http://sidc.oma.be/silso/datafiles, where the data is the previous data from the site (the data changed in July of 2015) and of the temperature anomaly from Berkeley Earth of monthly global average temperature data for land and ocean, http://berkeleyearth.org/data. The temperature series goes from 1851 to 2014. There appears to be a lag of about four or five months from the sunspot incidence to changes in the temperature anomaly.
The second set of plots uses the current sunspot data from Belgium and the land temperature anomaly from Berkeley Earth. The Berkeley Earth data goes back to 1750, the Belgian data to 1749. There does not appears to be a lag from the sunspot incidence to the temperature anomaly.
The confidence intervals should be taken with a grain of salt, since the sunspot series is not a stationary time series.