What kind of econometric models would be appropriate to determine the presence of a causal relationship? I am conducting some econometric research on the impact of austerity on birth rates in the UK.
I would be using publicly available data from the UK government covering the number of births per year, local government spending (which will be used as a proxy for austerity) and demographic data (age breakdown, population, educational attainment, unemployment among other measures). I have this data for the years 2004-2018 (the period of austerity covered 2010-18) and it is collected for 203 local authorities (districts of local government).
I also have access to a dataset with the simulated average amount lost per person due to austerity (as calculated by this paper http://shura.shu.ac.uk/15883 ) but this is only available for the years 2013 and 2016.
I am thinking of using a fixed effects model, difference in difference regression or vector autoregressive model to determine if there is a causal relationship between austerity imposed by the government and birth rates. This is because there are  unobserved heterogeneity issues as there is a very limited amount of data available meaning I am unable to used a traditional OLS regression.
I am thinking of following a method similar to the one used in this paper https://academic.oup.com/jpubhealth/article/38/3/417/2239829.
However, I am very new to these types of models and I am unsure if I am on the right tracks with my thinking.
I would appreciate any feedback or suggestions anyone has!

A good exposition in this direction is a recent review from Huenermund-Bareinboim
Causal Inference and Data Fusion in Econometrics. So probably we would need to build structural equation models to test the causal assumptions.
This might also be helpful from implementation perspective, R's causaleffect package's vignette Identifying Causal Effects with the R Package causaleffect.
