Do you have evidence that states with higher government spending as a percentage of the economy have a more quickly increasing (or even a higher) HDI? It seems that efficiency of spending would influence the trajectory of your metric of interest more directly than the metric's current level (this seems obvious to me, but I can explain why in detail, if required). I can think of a few examples which would support your case, and many counterexamples, but would be happy to hear your case.

I would also think that a system's efficiency would affect the economy as well as all other indices of interest (though perhaps in varying proportion), as a healthier, happier citizenry or one with more capital goods, and better investments would grow economically as well as in other metrics.