Can I have multiple individuals for each time point in ARIMA?

Question

I am a novice learner of ARIMA modeling. I have a dataset which includes many cities that are followed up regularly for several years. Each city has its own record (household income, and well-being index) for each time point. Then I sort it by year (so it's a long-form data, a single record for each city at each time point).

If I want to use ARIMA to check the relationship between household income (covariate) and well-being (outcome) over time, is it okay to perform ARIMA by using this data set? Or I must generate two "summary" variables called "average well-being of all cities in a given year" and "average income of all cities in a given year" (which is the mean of the whole sample for each time point) before performing ARIMA? I am confused because most tutorials online only include one subject (rather than multiple cities) for each time point. Thanks!

Answer 1

ARIMA is for a univariate time series. You would look at how well-being changes over time, given the previous values of well-being. There is no opportunity to include covariates in an ARIMA model.

For regression, ARIMA models are therefore not the answer. However ARIMA models can have covariates added, these models are called ARIMAX (Autoregressive integrated moving average with exogenous predictors) models. You can also have regression models with time series errors, dynamic generalised linear models. I would have a look at ARIMAX models online as I don't think they are massively more complicated than the standard ARIMA models.

Forgetting regression and talking about your confusion about having multiple cities, ARIMA models are univariate and so not appropriate for multiple cities either. The extension to multivariate time series is called VARIMA, where the V stands for vector and is for when you can put your multiple time series into a vector.

Both of the above are extensions to the standard ARIMA model and, therefore, are somewhat more difficult than standard ARIMA models. Combining both ideas is significantly more difficult still. Your idea for averaging over all cities might be a good idea, although obviously not ideal. You could then attempt an ARIMAX model given your averages. You can always look at just averaged well-being on its own first and fit an ARIMA model if you are completely new to ARIMA and want to make sure you are comfortable with the ARIMA model first.