Sometimes in macro models, something causes instability. For example, when the Taylor principle is not satisfied or you hit the lower bound under some conditions, you get instability.

But, say, I estimate the Taylor rule i_t = \alpha + \beta_1 \pi + \beta_2 y_t + e_t for economy A (say, with OLS or GMM), and I find that \beta_1 <1, which is a violation of the Taylor principle. But there is no instability in the data for economy A. Actually, output and inflation are stable. Then I guess there is only one reason why instability didnât happen as the theory predicts, which is, there were no shocks to spur the instability?

John Cochrane humorously once said, âwe all know what happens when you hit the lower bound, nothingâ.

I guess the correct way to think about it is that if you believe the theory and âtake it seriouslyâ, then you should, sort of, investigate why the instability didnât happen as the theory predicts.

I find the literature on indeterminacy hard to distill into simple messages. For example, we know that in regime-switching contexts, there may be determinacy as long as people assign a sufficiently high probability to return to a normal regime. See https://www.jstor.org/stable/30035014
For that reason I donât assign too much weight to estimates in one particular regime/sample.

Does this statement also apply to other parameters in the model? Or only parameters in equations not derived from FOCs. Maybe even parameters in policy functions derived from FOCs may have this regime-switching behavior.

Also, thanks for the paper. I have not read all of it yet, but it seems the point they make is that instability is not always due to indeterminacy. I guess we can also say stability is not always due to determinacy (the point in my question).

Instability is not the same as indeterminacy. The point of the paper is that there are can be periods of both instability or indeterminacy that are still consistent with overall stability and determinacy as long as there is a high enough probability to move to a stable and determinate regime at some point.

It is good to remind that the violation of the Taylor principle does not necessarily means instability or indeterminacy in the model, but maybe an alternative regime. If you consider a Government in the model, an underlying assumption of the Taylor principle is that fiscal policy is backing up any increase of nominal rates, providing (today or future, at least in expectations) surpluses to stabilize public debt. This is automatic when you consider lump-sum taxes, for example.

Now, imagine that Government surpluses fluctuates without this gravity pushing it toward debt stability, say s_t = s^* + \epsilon_t and that monetary policy increases nominal rates above inflation (Taylor principle). This increases real debt service r_t b_{t-1}, requiring more real debt issue b_t. Since there is no expected (or current) increase in surpluses E_ts_{t+j} to compensate that new debt, real debt keeps increasing, violating transversality conditions. So, actually, in this case you need the opposite of the Taylor Principle, \beta_1 < 1. Fiscal policy is not accomodating monetary policy, therefore monetary policy should accomodate fiscal policy. This makes much sense when you think about emerging markets.

For more details see John Cochrane manuscript about the Fiscal Theory of Price Level and the Leeper and Leith (2016) paper.

Yes, the intuition is that inflation is responsible to pay some of the nominal debt, since Government is not expected to do it through surpluses. It will run in dynare, as long as you write the debt equation in real terms (making inflation appear) and setting \beta_1 < 1. For example, see what happens after a deficit shock \epsilon_t< 0 in the following NK model: