# Multilevel modelling and cointegration, can the two methods be combined?

I am an epidemiologist, but not a statistician. I have a good enough handle on longitudinal multi-level models to push through the material and publish a paper, but my knowledge of all possible methods to use in different scenarios is lacking. I am looking for suggestions on how to approach this problem.

Scenario: 500 health care providers, and connected to each are two weekly measures. Metric 1 is a measure of wait times, and metric 2 is a measure of how their patients access the HC system (both to themselves, and to other providers). There are plenty of missing weekly measures, and some providers may have measured sporadically every week or so for a few months, while others measured almost every week for 10 years. There are contextual effects as well at multiple levels to consider --> related to geography, provider characteristics, demographics of the patient population, as well as clinic characteristics, etc.

What we know: From a multi-level model analysis, we know that when metric 1 declines over a year, so does metric 2. When M1 is stable, M2 is stable. When M1 increases, M2 also increases. This is after much adjustment and consideration for confounding.

What we don't know: Which metric leads the other? Does an increase in Metric 1, cause an increase in Metric 2? Or is is the other way around.

Why do we want to know this: Because if we know which one leads the other, we can create an intervention to improve one metric that should bring the other along with it. We have hypothesis as to what is the most likely causal path, but nothing to back this up as of yet, statistically.

What method am I considering: Something from econometrics, such as co-integration analysis. Hopefully this will tell me which metric is leading and which is following. This may also tell us what lag to expect (e.g. if you do manage to improve Metric 1 substantially, dont expect changes in Metric 2 for about 6-9 months, or something like that).

My concerns with this method: I have only seen this done on high level aggregated statistics. I could potentially do this for one overall trend aggregating all providers together into one time series, but considering that each provider has different confounding factors to consider, I would ideally want to run 500 separate analysis and then stratify/adjust. Is there a way to do this in a mixed effect model?

Or, is there another method I should consider? Or, am I overthinking this?