Empirically checking efficiency and equity I am aware that because of externalities or public goods (or for other reasons) there can be no allocative efficiency on the market. I am also aware that policy makers' subjective judgement of the situation on the market (if market outcomes are distributed "fair") can be considered as equity in economics. 

I wanted to empirically check if government spending can potentially increase efficiency or equity (as I think it should). I took labor productivity (a percentage of GDP) as a measure for efficiency and Gini index in disposable income as a measure for equity. While I am quite convinced with my measure of equity, as Gini tells me about the inequality of income distribution - I am not entirely sure about labor productivity. I am curious if I could choose better statistics to measure either efficiency or equity? 

In my results I got that government spending on environment is positively correlated with increase in efficiency in only about 30% cases and in equity in about 50% cases. This seems a little bit counter-intuitive as environment seemed like a textbook example where there is a negative externality, so the firms maximizing their profits based on private costs tend to overproduce - and that the government intervention in that case is justified. So that I thought after the intervention, there should be increase in allocative efficiency, right? But it is a case in only around 30% of cases. What could be a reason for that? 

Similarly, government spending on health. To my contention, it should increase both, equity and efficiency - but from what I've got it seems to be true in about 20% of cases. Why would that be a case? 

Is it because of the measures I chose? Or my logic was simply wrong and government spending on environment causes efficiency to decrease and on health cases both, efficiency and equity, to decrease?
Several things:


It is not clear that labor productivity as a share of GDP tracks allocative efficiency.  For example, it could rise simply because the returns to capital (capital's share of GDP) are falling, which would most obviously be a consequence of allocative inefficiency through over-investment in capital inputs.  There are potentially many other things that would confound this as an "efficiency" metric.
Gini Index isn't the only measure of inequality, and in fact it doesn't do a very good job of capturing differences in economies at the extremes of their income distributions. 
You may have the causal arrow backwards: the most widely accepted model I'm aware of (which is not without controversy, mind you) is that of the Environmental Kuznets Curve, which assumes that wage growth (not labor's share of GDP) drives environmental improvement, not the other way around. 
Because "equity" in the probability-distribution sense is attainable both by improving and destroying quality of life (e.g., everyone earns zero dollars), it's not clear that you should expect a positive correlation between this and "environmental spending".  In fact to first-order, your 50% split is consistent with what I've just pointed out.