Why use empirical macroeconomic models when they are not policy invariant (Lucas Critique)? With high probability this question is duplicate, and I tried to find one already in the community, but I was unsuccessful. 

According to the Lucas Critique, in its more general terms, the problem with empirical macroeconomic models is that they are not invariant to policy changes, and hence we cannot take any policy conclusions out of them. 

My question then is why do we still use them? 

Any help would be appreciated.
The response to the Lucas Critique was the emergence of RBC and DSGE models. Using microeconomic foundations of macro models we can simulate how behavior changes when policy changes and only estimate "deep" structural paramateres that are not policy variant. Before microfoundations we were estimating models where the estimation included the actions of people. With microfounded models today we try to seperate the actions. This was not possible in older models as we actually did not consider how people act or react, whereas microfoundations tell you how people would react.

However this is a difficult task as once you introduce agents that actively think about their actions you must take into acount their future expectations, which are unknown. One way to deal with this are rational expectations.

A further issue is that such models predict reactions that are quicker than in the data. If agents are perfectly rational, have perfect foresight and all information, they react quickly and perfectly. The solution to this today is to add frictions that slow these reactions down. However old models that we used to estimate (think about IS-LM and especially AS-AD models) also have the huge issue that people are too stupid (only adaptive (backward looking) expectations, do not take information into account when forming expectations, don't think about what may come in the future) and this is partially a piece of the Lucas Critique. Now we have the problem that people are too smart or super-rational. Some models (see models where some fraction of agents are "Rule of Thumb Consumers") now also assume that some agents are fully rational (many indeed are) while some just are backward looking (as in the normal IS-LM or AS-AD model), as a balance of the two things to generate a more realistic picture.

As for the critique of rationality: In many experimental micro studies rationality fails. However this does not tell us how many small deviations from rationality aggregate, which is what macro is interested in. It may occur that from many different deviations in different directions that in aggregate rationality is still a good approximation.

Furthermore approaches have been developed now to deviate from rational expectations. These are very difficult to solve though. One interesting way is to assume people are rational, but do not have all the information, which is a reason most people make mistakes. These are models of information frictions. Key words: Rational Inattention (e.g. Sims) and Inattentiveness (e.g. Reis). Another related approach includes Learning Models.

To summarize: we try to have models that are immune to the Lucas Critique. These often require rational expectations to be solved, but other approaches are also being developed.