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!

In general, causality is a function of your data, not necessarily your analysis. That's not to say you can't run bad analysis, but if your data wasn't collected in a way to allow for casual analysis, no analysis will overcome that.
In the case of econometrics, the general way causal relationships are found is through a "natural experiment" where something happened in the world, essentially at random, that can be used to split the data into a control and experimental group. The "difference of differences" is by far the most common way of performing this analysis. If you have two groups (in your case, local authorities) you would need to determine the random "event" that split them into a control and experimental group.
So really, the question you need to ask yourself is "what is my natural experiment?" If you can identify that, the analysis should be fairly straight forward to identify.
