1. The effects of an interest rate shock are highly fiscal policy dependent. In Cochrane's model, fiscal policy is non-ricardian, but the effect is the same if fiscal policy is active as described by Leeper (1991). If fiscal policy is passive, then conventional wisdom holds so that inflation reacts negatively to an interest rate shock.

2. Cochrane's result only happens when monetary policy moves from one interest rate peg to another. So, even in a fiscal dominant regime, inflation doesn't jump with interest rates if the central bank follows an interest rate rule (that must violate the Taylor principle).

Here is a comparison between a fiscal dominant (without an interest rate peg) and monetary dominant regime during a nominal interest rate shock:

As you can see, inflation eventually rises in the fiscal dominant regime, but this is simply due to the persistence of the shock. The initial effects are, in fact, more severe than in the standard model. As stated before, Cochrane's result prevails when the central bank switches from a lower peg to a higher peg and vice versa, but switching from peg to peg as Cochrane's paper implies is hardly realistic, so it's safe to say that conventional wisdom would hold is most empirical cases, even under fiscal dominance. Perhaps the only situation in which Cochrane's inflation jumping result (to be replicated shortly) would occur is if a country like Japan decided to switch from what is effectively an interest rate peg regime to an inflation targeting regime with active monetary policy (this remains un-modeled due to the limitations of modelling software).

Inflation |

Nominal Interest Rate |

So, under a fiscal dominant regime, raising the nominal interest rate once and for all results in this equilibrium, but allow the interest rate to follow a Taylor rule and this equilibrium is replaced by one with a harsher reduction in inflation than than the standard monetary dominant model.

Appendix:

Here is the model I used above in case you'd like to check it:

The government budget constraint is expressed in real terms and the real money supply is assume constant and equal to one, so seigniorage is simply expressed as the rate of inflation, $ \pi_t $:

$$ (1) \: b_t + \tau_t = (1 + \rho)b_{t-1} - \pi_t $$

$ b_t $ is the real bond supply, $ \tau_t $ is the lump sum tax levied by the government, and $ \rho $ is the constant real interest rate. The Fisher relation follows; relating the nominal interest rate to expected inflation:

$$ (2) \: i_t = \rho + E_t \pi_{t+1} $$

Fiscal policy is simply a function of the current stock of government bonds and can be adjusted by changing $ \phi^f $ between $ \phi^f > \rho $ for passive fiscal policy and $ \phi^f < \rho $ for active fiscal policy.

$$ (3) \: \tau_t = \phi^f b_t $$

Monetary policy can similarly be adjusted between passive ($ \phi^\pi \leq 1 $) and active ($\phi^\pi > 1 $) regimes. $ v_t $ is a shock term that follows an AR(1) process.

$$ (4) \: i_t = \rho + \phi^\pi \pi_t + v_t $$

$$ (5) \: v_t = \rho^v v_{t-1} + \epsilon_t $$

$ \rho^v $ is the persistence of the monetary policy shock and $ \epsilon_t $ is white noise.

During the deterministic simulation, the interest rate feedback rule is removed and the nominal interest rate is pegged exogenously.

Updates:

The response to the interest rate shock seems to depend on whether or not the shock is expected. There is strangely always a drop in inflation on the period when agents get news of the shock, but inflation does end up jumping when it comes into effect.

The stochastic simulation becomes a bit more informative when $ \phi^\pi $ is set to zero as there is just an initial drop in inflation in response to the shock and then a subsequent jump as inflation meets expected inflation.

Updates:

The response to the interest rate shock seems to depend on whether or not the shock is expected. There is strangely always a drop in inflation on the period when agents get news of the shock, but inflation does end up jumping when it comes into effect.

The stochastic simulation becomes a bit more informative when $ \phi^\pi $ is set to zero as there is just an initial drop in inflation in response to the shock and then a subsequent jump as inflation meets expected inflation.