Can I use an ACF to test for a feedback control system?

The broader question is: How can I test whether a variable is feeding back in a system, to influence future values of itself? The application is in a control system: Specifically a performance measurement framework (PMF), where managers monitor measures in monthly reports, in order to keep the system stable.

If PMFs alert managers to unfavourable changes in measures (variables) being monitored, one would expect these managers to make decisions aimed at limiting or reversing the unfavourable change. For example, as part of PMFs, police managers monitor crime volumes in order to identify emerging patterns, so that they can direct resources to prevent escalation of the problem.

In a feedback control system, managers may accept a regular seasonal cycle in a variable, such as crime. So, the presence of statistically significant seasonal components in the time series may not indicate whether or not the PMF is effective. However, if managers are unable to stop these variables from trending, this suggests that the PMF is not acting as an effective feedback control mechanism. So, if the ACF indicates an integrated series, it follows that the PMF is not acting as an effective feedback control mechanism.

If PMFs are effective as feedback control mechanisms, we would expect to see serial correlation in the variables being monitored. Specifically, the PACF of the time series for the variables, would include one or more negative coefficients at a lag of 1-3 months. This period of time allows for the measurement, reporting, management decision and subsequent implementation of the decision.

So, all I need to do is examine the lags of the PACF. If they are negative in months 1-3, the PMF may be acting as a feedback control mechanism; otherwise it is not?

Have I missed something? This seems too simple.
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