I highly doubt patent numbers are very correlated with anything we care about. For example, its harder to get software patents, so what this might say is that mid sized metros have worse tech industries. O would expect relative economic or productivity growth to be much more in tune with a cities importance.
People in big cities have already figured out that patents are a time suck.
I'd also be curious, why the cutoff at 2012. Do the results change if they go to 2020? 2024? Whenever I see a cutoff like that it makes me suspicious that that date was chosen because another date would not product the results they wanted.