Applying Time Series Econometrics to Temperature and Carbon Dioxide

For an economist, what I suggest below is obvious, but I wonder if anybody has done it.

1. Does the world temperature rise over time? Check the statistical significance of a time trend.
2. Next, check for serial correlation, and re-check for whether there is a time trend.
3. Now regress temperature on carbon dioxide levels. Is there a significant relationship?
4. Now add a time trend for 1950-2009. Does it explain temperature better than carbon dioxide?
5. Now start applying some time series econometrics. I don't know that stuff really. But what we'd want to do is to detrend the variables and include lagged values.

Eyeballing the data, it's hard for me to believe there's a significant relationship. The temperature is highly variable from year to year, and the amount of average increase 1970-2000 is tiny compared to year-to-year variation. Once we allow even for first-order autocorrelation, finding an effect would be tough.

As I said, to an economist this is the obvious way to proceed. But has anybody done it? Now that we are finding out how poorly constructed the temperature series are, we'd better ask about all kinds of things.