Problem: I had planned to use a linear regression model to model time series data in retrospect (i.e., no forecasting). However, I am wondering if this is the best option having come across a few posts - (https://www.quora.com/Is-regression-analysis-legitimate-for-time-series-data) - that regression analysis might not be legitimate for time series data. Preliminary plotting also shows a concave shape in the data, but this would still be a regression model, I think.
Question: Would anyone have any good sources to link to regarding how to build a legitimate model for time series data and the type of predictions and assumptions one is able to make?
Context: My dataset consists of 48 years and I'm using words in the context of corpus data. In the figure below you can see I've tried getting the Pearson's correlation between time and my outcome variable, fitting a quadratic line (which suggests multicollinearity in data when statistically compared to linear regression), and have also fit a 3-year moving average to the data. Somewhat frustratingly, there is less data in earlier years (pre-1976) which seems to be adding to noise in the trend.