On the causality between the labor income share and the size of governments

One puzzling feature of national account data in recent years has been a decline in the labor income share across most economies. This is not limited to the last recession but has been happening by and large since the 1970s. Why this is occurring is an important research question, and what the consequences are as well.

François Facchini, Mickael Melki and Andrew Pickering claim that this decrease has lead to a reduction in the size of the government. For this, they build a small two-sector model from which they obtain this positive relationship. Then they run some linear regressions to confirm this. But have they really? With the same model, I can obtain a reverse causation or even both variables being jointly function of others. It all depends on what I am assuming to be exogenous. Invert the regression equation, and you cannot reject the reverse causality either (I suppose, I have not done it). So all they have shown is that there is a correlation, nothing more. Claiming causation here is misleading. And if anything, I would have assumed that the causality runs from government size (which is set by political processes and policy) to the labor income share (which responds to market forces and policy).