One aspect of government purchases the current crisis has highlighted is how volatile they can be. Quite obviously, they are influenced by politics, to the point of complete reversal between massive spending and severe belt-tightening within months as in the US and the UK. But there could also be a more systematic component that is linked to the business cycle. After all, the government may be trying to improve the welfare of its constituents and for example substitute public consumption for lacking private consumption, or the same for investment.
Ruediger Bachmann and Jinhui Bai look at this using an augmented real business cycle model. They claim that 25-40% of the variance of public consumption can be accounted for by shocks to total factor productivity once implementation lags and costs of public consumption, as well as taste shocks to public vs. private consumption. I am no particular fan of taste shocks, as they are the symptoms of a modeler who is giving up on trying to explain something and simply equates the error term in the Euler equation to a shock. Then much is driven by how this shock is calibrated, in this case to match a four year electoral cycle and some data moments. When I think about shocks in this context, I think indeed about who is in power to decide on public expenditures. But that is not completely exogenous. Indeed, the state of the economy has an impact on who gets elected or reelected. And this can be calibrated without trying to match the data moments one is trying to explain.