I posted the short comment below in reply to a blog post by Richard Telford
The first thing to do before making any stats tests is to consider how the physical mechanism may work. If the relatively high solar activity of the C20th did have an effect on temperature then it was by warming the oceans. Water has a high heat capacity, so Ocean Heat Content (OHC) (and thence SST) wouldn’t be expected to react instantaneously to solar variation. So we need to low pass filter or integrate solar data to mimic the thermal inertia of the oceans.
Sunspot number (SSN) is a good proxy for total solar irradiance (TSI). Over the period of record since 1749, the average sunspot number is 40. This is also the number found when averaging the SSN over a period where SST didn’t vary much. Integrating SSN as a cumulative total departing from this ‘equilibrium’ value generates a curve which shows a slight fall from 1880 to 1934, and then a rise to 2003. When combined with a suitably phased ~60 year sinusoid representing the Atlantic multidecadal oscillation (AMO), something close to the 5 year smoothed surface temperature record is obtained.
I took this idea a little further with by combining AMO, integrated SSN, SOI and LnCO2 (at a suitable value), producing a match between model and HADSST with an R^2 of ~0.8 for MONTHLY values since 1875 and R^2=~0.9. for more accurate data since 1960
I can make the spreadsheet available to interested parties.
One of the many problems with discerning the solar effect on surface temperature is that the solar derived energy building up in the Pacific Warm Pool (PWP) between El Nino events is hidden from surface temperature datasets. Big El Nino’s tend to occur at solar minimum, and so are in anticorrelation to the solar signal on SST. This flattens the apparent magnitude of the solar effect.
There are on average three el nino events per solar cycle, so by smoothing the temperature data at 40 months, and detrending the temperature data to match the unintegrated solar data, a better solar-surface temperature correlation appears, though of small magnitude and varying phase due to the ENSO effect outlined above.