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I am trying to understand some basic stuff about New Keynesian models and monetary policy. Despite that almost every single paper on monetary policy talks about monetary policy shocks, I still don’t understand what a monetary policy shock is. So my main question is this:

What is a monetary policy shock?

Now, in order to avoid uninformative answers of the sort: "it is a shock to the rule the central bank follows", I will try to explain my thoughts so to make clear what exactly I don't understand.

I see that many New Keynesian (NK) models incorporate some variant of a Taylor rule of the form: \begin{equation} i_t= \delta \pi_t + \kappa y_t + u_t \tag{1} \end{equation} where $i_t$ is the nominal interest rate, $\pi_t$ is the inflation rate, $y_t$ is the output gap, $\delta, \kappa$ are some parameters chosen by the central bank and $u_t$ is a random process.

Is the random variable $u_t$ the monetary policy shock (MPS)? Mathematically speaking it seems to me that a MPS in a NK model is just a random variable (that appears in the rule that the central bank follows) which is exogenous in the sense that one does not solve for it using other “endogenous” variables. Instead a MPS is determined by some specific stochastic process specified by the modeler. Also, this random variable appears to have the property that its conditional expectation conditional on available information up to time t is zero. Is an exogenous random variable “attached” to a Taylor rule, which is zero in expectation conditional on available information a type of MPS? And whose information set? Does it have to satisfy other properties in order to call it a (monetary policy) shock mathematically speaking?

In any case, I don’t quite understand why a central bank rule such as the above “needs” a random term like $u_t$ in the first place. What does the existence of $u_t$ mean?

1) Does it imply, for example that the central bank observes inflation and output gap and sets the nominal interest rate according to: \begin{equation*} \widehat{i_t} = \delta \pi_t + \kappa y_t \tag{2} \end{equation*}

but for some reason the actual interest rate, (the one that agents in the model observe at the end of the period say) is given by (1), i.e. it is “disturbed” by some random force despite the central bank’s best efforts to control it? If so, what are these “forces”? Where do they come from? And are we seriously to believe that movements in real variables in actual economies are (at least partly) just responses to movements in a variable like $u_t$? In any case this would seem to me to go against standard stories one hears about monetary policy by central banks in which it is purposeful changes in the nominal interest rate by the central bank that have impact on other real magnitudes in actual economies, not some random movements (of interest rates) that nobody controls.

2) Maybe the term $u_t$ captures the influence on the decision of the central bank about $i_t$ of variables other than $\pi_t$ and $y_t$. If so why aren’t these other variables modeled explicitly? Is the term $u_t$ really random only for agents inside the model other than the central bank, i.e. is $u_t$ under the direct control of the central bank but to other agents it just seems as an additional term to rule (2) for which the best they can say is that it is on average zero and in any case unpredictable given their available information? But if the term $u_t$ is really under the central bank’s control, how does the central bank decide its value? Given the importance the literature gives to MPS it seems bizarre that something so important is left unmodeled.

Moreover, after one linearizes and solves a standard three equation NK model typically one then gets IRFs by assuming a onetime increase in $u_t$. This is done to obtain the impact of a change in $u_t$ on other endogenous variables. This gives me the impression that changes in $u_t$ are really thought to be very important in explaining business cycles but I don’t get why this is the case, given that I don’t understand what the variable $u_t$ represents in the real world. I guess we could take a step back and ask what happens if we solve the model without assuming the existence of a variable like $u_t$. Suppose in other words that we solve a NK model assuming no “exogenous shocks” of any kind. I guess that then we would just have a steady state solution, i.e. a solution where nothing really happens. If so, then “exogenous shocks” (interpreted as “random forces” that “disturb” otherwise constant variables), would be a way to generate movements around the steady state which seems to be an improvement given that we would like to obtain a solution that resembles what we see in the data, and I am pretty sure that in the data variables such as output, interest rates and inflation are rarely constant. But then one faces the question: what kind of shocks? Surely one cannot just begin to throw shocks around (or can he?). So, why MPS? Why a random term on a Taylor rule? What is the connection with reality? And I think now we have come back to my initial question: what is a monetary policy shock? And why is it there?

I know I have asked way too many questions here, but really all of them can be summarized into my main question. The rest are just to explain what I don't understand.

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