Cost shifting is a phenomena observed in medicare insurance where cost incurred on medicare patients were allocated to non-medicare patients by hospitals to gain higher reimbursement.
There are several points that have to be clarified/taken into account. Are we talking about private patients or charity cases? If it is the former, what conditions/services are we talking about? If it is the latter, what type of facility are we talking about and what is the reimbursement system applicable. Usually, medicare/public insurance pay less compared to privately insurance so there is an incentive to bill patients as private patients. If we know the streams of payments, we can compare the rates applied to each stream.
Cost shifting only reasonably be detected using population data where you can control for confounding factors. Such an analysis would attempt to demonstrate that the number of services have stayed consistent in the target population but have increased in the alternate population (or they have stayed consistent but cost went up).
By the way, cost shifting can also occur within the same population, by shifting costs to different services.
There are plenty of papers in which cost shifting is analyzed in the US (Medicare vs commercial/private insurance).
I have gone through cost shifting papers in medicare. But developing countries like India do not have universal insurance coverage yet. There is one public insurance scheme initiated for poor population while majority of middle class pays out of pocket. In such a case it is difficult to capture rise in OOP expenditure due to this public insurance scheme(with conservation reimbursement rates). I am unaware of any robust methodology to capture this.
If you don't have data on both populations, I can't see how you would examine cost shifting behaviors. Of course, you could model such behavior, but it is likely to be way off base.