Do "Mixed Effects Time Series Models" Exist?

Question

Suppose I have the number weekly of hospital visits in a certain ZIP code (10 years of data) - this ZIP code has a small population in general and as a result, not that many people visit the hospital within this ZIP code. If I were to fit a basic time series model (e.g. ARIMA) to this data, I might run into some difficulties because I don't have enough data.

Suppose I also have the weekly number of hospital visits at the State Level and at the Country Level. Additionally, at the yearly level - I have information on the total population and the average income of all residents within the ZIP code, the State and the Country. Furthermore, I also have information on whether a given year the President of the Country and the State Governor was a Democrat or Republican, and indicator variable for Pre-Covid/Post-Covid.

Based on this information, I had the following ideas:

• Suppose I were to use a VAR Model - in which the weekly hospital visits in the ZIP code are modelled using weekly hospital visits from the State Level and the Country Level. The idea being that perhaps I could "statistically pool" relevant and related information to improve my weekly estimates. By doing this, am I basically creating a "Mixed Effects Time Series Model"?
• On top of what I just described - suppose I decide to add information to the above model on Covid and Democrat/Republican, thus making the model similar to an ARMAX/ARIMAX. From a conceptual point-of-view, is this also similar to a "Mixed Effects Time Series Model"?

In general, I am not sure if these approaches that I have described above "statistically valid approaches" or if they fundamentally flawed and likely will result in model violations.

Thanks!

Answer 1

I am not an expert on these statistical tools, but from what I know the VAR model is useful when you have dependent multivariate time series, which might not be your case from your description. ARMA models are useful when there is a clear temporal correlation in your time series. While there might be correlation in number of hospital visits among days , this is mainly due to time trends and other variables (e.g. more people hospitalized due to flu in winter months) than a causal relationship between them.

You have hierarchical time series data and I imagine it's not easy to find the right modelling perspective.
I suggest a two-stage design approach for time-series modelling with complex hierarchical structure. This is practically described (with R code example) in a paper called Extended two-stage designs for environmental research by Sera and Gasparrini.

Their exposure are mainly environmental (temperature, air pollution etc.) but you can use it for any exogenous exposure. The idea is that you first run a ZIP code specific models from which you will get estimates. You then include these estimates as dependent variables in a model that includes meta-regressors at higher hierarchical level (such as county or country). In this larger model you can include random-effects for the ZIP code within the same county and also define lag structures. Because you might have several zero counts at ZIP code level, I suggest using a zero-inflated Poisson to model your time-series. It might not be easy to fit this to your data, but with some work on the model specifications you might find this method helpful toward your goal.

Update: sorry, I forgot to mention that the method described is particularly helpful when you have several small-area time-series, but if you don't have many you could also just run a zero-inflated Poisson model with random effects for the entire dataset.