We started last lecture, no we ended last lecture, with an analysis of the FTPL model and a prototype Eukinesian model where I argued that these two models can actually be seen as different interpretations of the same set of equations.
And I would like to illustrate this once again here to repeat and reintroduce to you what we did last time.
These are the three equations we have been dealing with as the linear approximation to our fiscal theory of the price level model.
And the way how we can solve this model was always somewhat technical when we wanted to derive closed expressions for the solution of the endogenous variables, but in a sequential way it is actually very easy to see how the system can be solved by just
following through these three equations here under the assumption that epsilon Ts is the fiscal shock, so epsilon Ts is something which is given and cannot be influenced by decisions of the central bank or the consumer.
So it is active fiscal policy which we have here, it can of course be influenced by decisions of the government, but if the government just exogenously sets the fiscal surpluses, the present value of fiscal surpluses,
then we see from equation 69 that with a shock occurring in period T, there is no way how on the left hand side of this equation this term may react because this was already determined in period T minus one.
So the expectation of the previous period's expectation of today's inflation rate is given, so obviously pi T is determined by the fiscal shock by virtue of equation 69.
Now if pi T is determined, then from equation 73 we see that nu T is some exogenous process, which depends on monetary policy shocks which are exogenous, so nu T is exogenous, r is just a fixed constant, pi T has already been determined from equation 69, so we see that equation 73 determines the nominal interest rate by virtue of the Taylor rule.
And therefore, if the nominal interest rate is determined by equation 73 and r is constant, then this determines next period's expected inflation rate or today's expectation of next period's inflation rate.
And, well, then we can move on to the next period because we then have a shock epsilon s, we can again say well this thing here, period T's expectation of period T plus one's inflation rate was determined already, so that thing is given and then this determines the actual inflation rate and so forth.
So, this seems a very straightforward way to see how the system determines the endogenous variables, where we more or less have pretended that we have three endogenous variables for three equations, and each of the three equations seemed to determine one variable.
So equation 69 determined pi T, equation 73 determined pi T, and equation 70 determined the expectation of next period's pi T, where basically we pretend as if the expectation of next period's inflation rate is a separate endogenous variable which has a separate way of being determined.
Well, actually, of course, there is a relationship between pi T and pi T plus one because the expectation must truly be the rational expectation of pi T.
And we have not really seen how this enters the system of equations, but it is easy to verify that the solution of the physical theory model actually indeed satisfies this requirement that the expectation of next period's pi T is just the expectation of the solution for pi T,
the conditional expectation of the solution for pi T.
So in that sense, we may say, well, that's quite a regular way to solve a model, three equations and three endogenous variables if we treat the expectation of a separate endogenous variable.
Now, then I told you that basically the same system of equations can also be interpreted in a new Keynesian way and today we will actually see a little better that this is a new Keynesian model which we have here because we will also then talk about the typical
ingredients of a new Keynesian models which you may not yet see in this set up of the equations. So we will talk about a new Keynesian-Phillips curve and a new Keynesian-IS curve which are typically the ingredients of a new Keynesian model which you currently don't see in this set of equations.
But you learn that actually the standard way of writing down a new Keynesian model with a new Keynesian-Phillips curve and a new Keynesian-IS curve collapses to a set of equations which is very similar to the set of equations.
So we come to this point.
Currently just accept that this model is basically a new Keynesian or if you don't want to buy this yet, then just say that this model is sort of the opposite of the fiscal theory model and we
just don't give it a name yet what kind of a model it is but is the opposite of the fiscal theory model in the sense that epsilon t plus s is not anymore.
And exogenous shock, so we do not have active fiscal policy anymore, but rather we assume passive fiscal policy. So we may treat epsilon t plus s as an endogenous variable.
And when epsilon t plus s is an endogenous variable, then obviously we see that this first equation here is of little value and of little relevance for the solution of the model because this first equation here just determines epsilon t plus s as an endogenous variable.
Epsilon t plus s does not occur in the other two equations, so it is rather irrelevant what kind of value epsilon t plus s takes on.
We just know from equation 69 this determines one endogenous variable which is the present value of a future surpluses and does this well by this expression on the left hand side of this equation.
But actually this is completely unimportant because we don't need this equation for anything and we don't need the epsilon t plus s for anything.
So we are left with two equations here. And if you accept our previous handling of the FDPL model, then you will say well it looks a little weird now because if we count equations and if we count endogenous variables, then it seems
taking again the expectation of tomorrow's inflation rate as a separate endogenous variable, then it seems as if in two equations we have three endogenous variables to determine.
Because we have it to determine we have pi t to determine that we have the expectation of tomorrow's it to determine.
Or if you want, you can also have a look at the model as a whole, and you would also say well it seems as if we have four endogenous variables for three equations, because the fourth endogenous variable would be the epsilon t, which
by notation still suggests something exciting as a shock but actually in this case with passive fiscal policy it would not be a shock would be an end obviously determined variable so there is no fiscal shock in the new Keynesian deportation of this model
there's just a monetary shock which is the epsilon t i shock which is built into this new process here, as you know, new is some PR one with innovation, epsilon t i, for instance.
So, whatever the way you look at this model, you may say it seems that this model cannot be uniquely determined or cannot uniquely determine all four variables since we just have three equations.
But then of course is not correct to say, since the expectation of next periods pi t depends on the solution of next periods it and that's actually the way how we solve the new Keynesian model, we just make sure that the solution of pi t is in fact uniquely determined,
and that the expectation of next periods pi t, next periods inflation rate is consistent with rational expectations.
And this can be done, as I told you, if we make the assumption that the parameter phi in the Taylor rule is greater than one, then as many people have analyzed for this Keynesian model, then there is a unique solution for pi and there is a unique solution for the expectation
of pi, even though we just have two equations here to determine it it and the expectation of next periods.
And this was where we left off last lecture. And now I want to show you how we solve this model under the assumption that phi is greater than one. So, I solve the model under the new Keynesian interpretation, where phi must be greater than one to give us a solution.
Okay, how do we do this. Well, first thing is that we look at equation 70 and 73, which imply that the expectation conditional on period t information of next periods inflation rate is equal to phi pi t plus nu t and nu t is of course an example.
So basically what happened in the equation 70 and 73 there was also also the difference between normal interest rate and real interest rate in both equations, so we could substitute this out and then we are left with just one equation in the expectation of inflation and actual inflation.
So just to show you again an equation 70 and 73 there is it minus R basically in this equation and this it minus R basically in this equation, so we can just solve for it minus R and substitute it to the other equation and then we get exactly the
equality here of the expected of the expected inflation rate with phi times pi t plus nu t.
So, that's simple to do we have equation 84 which is an expectation difference equation in pi, in pi, excuse me, pi.
So, how do we solve this equation, well you see already phi is smaller spice greater than one. So one over phi is smaller.
We will have to solve in the future, because this year relates pi t to its future values and now we have to use or may use successive substitutions by which we substitute out next periods inflation rate and substitute in periods t plus two's inflation
rate and for t plus two's inflation rate we substitute in periods t plus three's and so forth.
So that's the way to proceed here and then do this in these two lines or three lines here. This is still equation 84 just without any change, and now I use this successive substitution technique so I replace the expected value of pi t plus
one by one over phi expected value conditional on periods t plus one information of period of pi t plus two minus one over phi nu t plus one.
Perhaps I think I spoke a little bit. I do not replace the expectation because the expectation operator with respect to periods t information is still here, right, nothing has changed there.
I just replace pi t plus one by this expression here just shifted ahead in time by one period. So by pi t plus one is equal to one over phi e conditional t plus one information of pi t plus two.
This is this term here minus then one over phi nu t plus one.
And this term of course is left unchanged. So this was the first substitution we did and now you know by the law of iterated expectations, the expectation conditional on periods t information of an expectation conditional on period t plus one information is actually equal to just the
expectation conditional on period t information. So basically we can drop this term here so that we would have pull out this one over phi here, we would have one over phi squared times the expectation of pi t plus two
plus a term here which consists of the news. And then we can replace the pi t plus two again by equation 84 just shifted ahead in time by two periods.
And when we do this many times, then we arrive at an expression which would come to one over phi to the capital T.
So I've done this now, capital T times, and I would have one over phi to the capital T e t the expectation conditional on period t information of pi t plus capital T minus.
And now we have to look at what happens to this term here that's actually easy. There's always a term one over phi times nu t or one over phi squared t plus one or one over phi to the power of three new two plus t and so forth.
So I can write this as a sum of this form here, one over phi to the tau plus one times the expectation conditional on period t information of nu t plus tau.
You may wonder why do we have this expectation operator in here. I mean it's clear that it is this expectation operator which applies to this term here and all others which build up by the success of substitutions, but there was no expectations operator in here.
However, obviously the expectation conditional on periods t information of nu t is equal to nu t. So we can replace nu t here by expectation conditional on period t information of nu t, which we have essentially done here.
So we arrive at this expression here.
Yes.
Here we have now the expectation of nu t plus tau. And this thing can be simplified because we've made a special assumption of how the process nu t plus tau which is an exogenous error process.
How this is built up we have assumed that it is a r one so autoregressive of the first order. So we know nu t plus tau is equal to row times nu t plus tau minus one is the autoregressive part plus the innovation epsilon t plus tau i.
So this is the monetary policy shock.
And then we can replace nu t plus tau minus one by exactly the same equation here just shifted backwards in time by one period.
So when we do this, the usual way we would get this is row squared times nu t plus tau minus two plus row epsilon t plus tau minus one i plus epsilon t plus tau i.
And again doing a substitution for the nu t plus tau minus two with its autoregressive representation and doing this over and over again, we would arrive at that nu t plus tau is equal to row to the tau times nu t plus a sum over weighted epsilon terms.
The weights are row to the chase. And here we have epsilon t plus tau minus j.
The j here goes from zero to tau minus one. So if j is zero, then obviously the epsilon t plus tau is a future shock, which we do not know yet in period t so its expectation is zero.
This is also true for any other future shock up to period t plus one.
We would know in period t, of course, the shock epsilon t.
But we do not yet know the shock epsilon t plus one because the epsilon shocks here are white noise so it's not possible to predict them one period ahead in time.
Or let's see whether there's something which we can predict in these terms where the sum goes up to tau minus one.
So if j is tau minus one, then we subtract tau minus one here, which means that the tau cancels and negative times negative becomes plus, but we have epsilon t plus one.
This is actually the last term in this sum here and extra epsilon t plus one can still not be forecasted cannot be expected its expectation is zero.
So when we take the expectation of new t plus tau conditional period t information, then this sum here is zero expectation of the sub year zero.
The expectation conditional period t information of the sum is zero. So the expectation of this term would here would just be row to the tau new t because new t is known and create t.
So we find that the expectation conditional period t information of new t plus tau is row to the power of tau times new t.
So we can therefore replace in the week we can replace this rather ugly some here just by row tau to the row to the power of tau times new t.
Okay. Sorry, not the sum but the single expectation. This expectation here, we can replace by row to the power of tau times new t.
This I have done here. So my solution for pi in the new Keynesian model is now pi t is equal to one over five to the power of capital T times condition period t's conditional expectation of pi t plus capital T
minus new t which I can put out of the sum because there is no tau anymore in this expectation or the expectation relates to new t new t over five times the sum of well terms which have the form row to the tau divided by five to the tau.
Now we know that row to the tau divided by five to the tau is smaller than one because row is smaller than one by assumption and five is greater than one by assumption. So this whole thing here certainly smaller than one.
So therefore we know that for t going to infinity, this thing here can actually be written as one over one minus row divided by five.
So that's the limit of this sum here as t goes to infinity.
Moreover, as t goes to infinity phi to the power of t will go to infinity and therefore one over five to the power of t will go to zero. So this term here is asymptotically zero provided that we don't have explosive solution path for inflation.
So this thing here stays bounded.
Okay, this is why I write here, we have to rule out explosive solution paths for inflation. So in the case of hyperinflation everything may be different here but suppose we are in the economy where there is no hyperinflation and the solution path for inflation is somehow bounded or at least doesn't grow faster, inflation doesn't grow faster
than phi to the t, then this thing here will go to zero.
So, the solution converges to negative nu t over phi. This nu t over phi times the value of the sum which is one over one minus row over phi. So when we multiply the phi in here we get the very simple expression nu t over row minus phi.
That's our solution for pi t in the new Keynesian model. So pi t is just a linear transformation of the exogenous monetary policy shots of the nu t shots in this case not of the innovation, but of the nu t so nu t was supposed to be an AR one.
So inflation will also be an AR one, a scale by row minus phi.
We can also write this in terms of the structural shocks epsilon t i. Well, we know how to replace the nu t by the epsilon t i.
Trucks just use the moving average representation of the nu t. So we know that we can write the nu t as an infinite sum of a row to the power of tau epsilon t minus tau to the power of i. And this is our solution for pi t.
Why do I make the simple equation, the simple solution nu t divided by row minus phi complicated again by substituting out the nu t against the structural shocks.
Well, simply because we want to compute the impulse response and the impulse responses are defined in terms of uncorrelated shocks in terms of the epsilon shock. So we need to have a representation
in terms of the epsilon shocks rather than a representation in terms of thoughts which are autocorrelated over time.
But here it is our solution for pi t in terms of the structural shock. In this case, the monetary policy shock. So inflation depends solely on the monetary policy shock.
There is no fiscal shock, as you know, and there is no endogenous propagation mechanism, as I will emphasize a little later. So it's a simple function of the monetary policy innovations.
Now, what does this imply for the difference between, well, actually for the nominal interest rate, but I write it here as the difference between nominal interest rate and the constant real rate.
Well, we know from equation, what was it, 70, right?
That it minus RT is just the expectation conditional period t information of pi t plus one.
Now it's easy to see, right? Here we are. Write this here as pi t plus one is equal to, well, this term here, nu t plus one divided by rho minus phi.
Then we can write actually the nu t plus one as rho times nu t plus epsilon t plus one i.
And take the expectation of this conditional period t information, then we would have this is rho times nu t and the expectation of the epsilon t plus one would period t is zero. So that would vanish.
So we know that from this equation 70, it minus r is equal to this thing here. So the nominal interest rate is actually also just a transformation of the exogenous shock.
Just with different scaling factor, namely the scaling factor of the inflation rate multiplied by rho.
Otherwise, it's the same expression as we have had for inflation.
So we get this expression here when we then again substitute out the nu t and substitute in the infinite sum, the infinite weighted sum of the epsilon t minus tau i.
And this is the solution of the New Keynesian model derived under the assumption that phi was greater than one.
As an exercise, please verify that is indeed the complete solution or that the solution is consistent because what we have not yet done is that we have computed the solution for the expectation of an experience.
So in this exercise, please first answer the question which I actually did already answer. Why do we need to rule out explosive solutions for pi?
Then compute the expectation, the conditional expectation of t plus tau for taus greater than one from equation 85.
Verify that pi t plus one is equal to rho pi t plus epsilon t plus one i divided by rho minus phi and verify that equation 70 and 73 imply the same solution for it.
So basically, you shall verify that these two things are indeed the solution and give a consistent solution for the expectation of future inflation, regardless of which way we computed which equation we use to compute the expectation of a future inflation.
Inflation, it will always be the same expression that will be derived and it solves the equations, two equations, 70 and 73.
69 was, as you know, basically superfluous because it determined automatically the physical shock variable.
Now, given the two equations that we just had on the previous slide, so 86 and 87, it is easy to derive the impulse responses of the New Keynesian model because the New Keynesian
impulse responses are just the partial derivatives of pi t plus tau and of i t plus tau with respect to a shock epsilon ti.
So depending on how big tau is, we either have the instantaneous impulse response or we have the impulse response after a lag of tau period. So that's why I write here tau is greater than or equal to zero in both cases.
Now this impulse response here, this partial derivative, as this one here, is always computed for a value of epsilon ti being equal to one.
That's actually the normal way to do it. Either you shock by, that's one unit or you shock by a standard deviation.
But we will actually, in the next slides, also use different size of the shocks and the impulse responses to make a point very clear. So I emphasize this here, what I have assumed when computing these impulse responses here is that the size of the shock was epsilon ti is equal to one.
And this shock of this size then has responses, just the partial derivatives of the endogenous variables with respect to the shock. This should be very clear.
Okay. What I want to do is that I do not compute the impulse responses for a unit shock of epsilon ti, but I want to compute the impulse responses for a shock of size phi minus rho, whatever that is.
Okay.
Let's see in a minute why I do it that way. Let's suppose that phi is greater than one, for instance, phi is equal to 1.5 and rho is smaller than one, let's say it is 0.8.
Just to evaluate the impulse responses with these settings, what would happen if the shock epsilon ti is equal to phi minus rho, then obviously these multipliers here must be multiplied with the size of the shock.
So both multipliers, both impulse responses here are or have as the opponent in the denominator phi minus rho and this must be multiplied with the size of the shock which is phi minus rho.
So essentially the phi minus rho just cancels.
That cancels and as impulse response for shock, monetary policy shock of size phi minus rho, we would have negative rho to the tau for inflation and negative rho to the tau plus one for the nominal interest rate.
Right. So this makes it particularly easy to draw the impulse response because now we actually don't have to bother anymore about the size of phi.
Phi has just canceled out by setting a certain size for the monetary policy shock here and the impulse responses only depend on the size of rho and rho is equal to 0.8.
Now when we plot the zero, the impulse responses for the new Keynesian model with these parameters here, then we get exactly this.
This graph you have already seen, right? If you recall, we discussed exactly the same graph when we discussed the new Keynesian model.
Here the graph gives us the reaction of the new Keynesian model.
Sorry, when we discussed the fiscal theory model, I think I haven't spoken.
Here we see the graphs in the interpretation of the new Keynesian model.
We know that in a new Keynesian model where the monetary policy shock is of size phi minus rho and the parameters are chosen in this way here, the impulse responses would actually look exactly like this.
So for the inflation rate, the first impulse is just equal to one because, well, tau is equal to zero, rho to the tau is one, negative rho to the tau is negative one.
So that's the first period response of inflation to a shock, a monetary policy shock in the new Keynesian model of size phi minus rho.
And for the nominal interest rate, the response is negative 0.8. Of course, it's negative rho because of the nominal interest rate, it is rho to the tau plus one.
So if tau is zero, the impulse response is negative rho to the power of one. So there's 0.8 here.
And then the shocks become great, the responses become great, greater but smaller and absolute number. So as you know, we've discussed this last time.
The two impulse responses are just the same, except that there is a lag of one period in inflation response.
So in period two, we have 0.8 here. We have 0.64 here, which would be rho squared, an absolute value.
All the values are, of course, negative, as you can see here, negative one, negative 0.5 here, zero. So this is negative 0.64, which we have here.
We would have negative 6.4 then for inflation in period three, and here it is then 0.8 to the power of three, whatever that is.
And this is again then taken up by the inflation rate in period four and so forth. So this property we have seen in the fiscal theory model, actually we have seen exactly the same graph.
Now don't think that I or that Cochrane picked these parameter values just because we were too lazy to draw a new graph for new impulse responses.
There's a much deeper point in that, that we see exactly the same impulse responses in the new Keynesian model in response to monetary policy shock.
As we have seen in the fiscal theory model in response to a fiscal policy shock.
Because recall these impulse responses we obtained in the fiscal theory model when we shocked the fiscal policy shock.
So when we shocked the present value of surpluses by one unit.
Now we shocked the money will use the monetary policy shock and took a particular science finance role.
And then we obtained exactly the same type of impulse responses.
So this is a little worrying, and you'll see it becomes much more worrying soon, because suppose that we are just observers of an economy.
And we just observe co movements of variables responses of variables to shocks, which we cannot observe, we just don't know.
Shocks typically are unobservable shocks are things which economic traditions like to formulate and which we can estimate under some circumstances, but they are not directly observable I can go to the system of national accounts and tell them hey tell me what the
shock is and what would a shock in the present value of our future surpluses is you'd find this nowhere.
Also you'd not find some type of monetary policy shock in the system of national accounts so the shots are unobservable.
Another worrying thing is that if we have just the capacity to observe certain data like the inflation rate and the non interest rate.
We see that the pattern, which evolves in these two models in the fiscal theory model and in the new Keynesian model is exactly the same.
So when we are just given the data series on it and on it and suppose we observe such a behavior in the data.
We now know that we have no clue why the data actually develop this way.
Do they develop this way because there has been a fiscal policy shock, and we are in the fiscal theory of the price level vert, or do they develop in this way because there was a monetary policy shock and we have a new Keynesian model.
We are unable to say we have something which is called observational equivalence. So two models generate exactly the same type of data.
And we define observational equivalence in a minute, but let us first see or look at some other implication of this finding.
Now, you know that we have in our model, the Taylor, the net Taylor rules and integral part of the new Keynesian model and the Taylor rule was formulated in our case as the non interest rate will be adjusted by a central bank which conducts active monetary policy
by or in response to changes in inflation and in response to certain shocks which occur and I have sometimes say that this is maybe world market developments developments to interest rates in other countries which are not explained our model or whatever happens.
And that is whatever is not captured in the inflation rate what whatever else may motivate the central bank to adjust interest rates, that's this new tea.
So basically, the central bank in the Keynesian model, but also in the fiscal theory model we also have a Taylor rule reacts to developments which it observes in the world and markets.
For once to inflation, and also to anything else which is captured in the shock.
The focal point is always the constant real interest rate this doesn't really matter. What does matter is that there are two influences in this Taylor rule on the nominal interest rate.
Now we can replace this new tea shock this exogenous shock, of course, by the right hand side of its AR one representation assuming that new tea is an AR one.
So in this case, we would write this is art plus five it plus row times new T minus one plus excellent tea.
Now why are the results that we have obtained in the previous slide in terms of these imposter responses remarkable.
They are remarkable because at first inspection of equation 73.
One would expect that positive interest rate shock positive monetary policy tea shock, epsilon tea, I raises nominal interest rates.
But what we have seen is that if the monetary policy shock is positive five is 1.5 negative row is negative 0.8. So, this is 0.7 it's a positive monetary policy shock.
We have seen here that the instantaneous reaction of it to a monetary policy shock in period one. So, same period is negative.
So that's the remarkable finding here, even though it seems that it depends positively on the innovation.
It actually depends negatively or the reaction is negative is negative.
Why is this so well, that's easy to see because I write here at first inspection so let's do a second inspection. The second inspection is that in period tea, we don't only have the epsilon, the eye which is say positive.
And the normal interest rate which may respond to that, but there's also pi t, which is also a period tea variable.
So before inferring how a positive monetary policy shock affects the nominal interest rate.
We also have to study how does the monetary policy shock change pi tea.
Well, and how does it change pi tea, we've seen this positive interest rate shock here, lowest pi tea.
Right. This was our result here, positive interest rate shock, lowest pi tea.
The response to the same shock is negative one.
Okay.
So, now, this year is a shock of plus point seven. This is a shock of negative one, it is being multiplied by five and five is greater than one.
So, if I is negative one, if I is one point five, the product of these terms is actually negative one point five negative one point five here.
We have plus point seven here gives us as a result and effect on it of negative point eight. And that was what we saw here.
This is why the interest rate actually falls when something happens, of which people may at first inspection think well this will drive up interest rates, because there was a positive shock in this equation.
Completely untrue doesn't happen.
Positive shock here implies negative reaction here and that is what is surprising.
Now, when you read papers, which set up new Keynesian models, then you will often read something like people believe that raising interest rates, lowest inflation.
This is actually somewhat the core of the new Keynesian model that the central bank fights inflation by raising interest rates.
So, this is what the Taylor rule should express then that when inflation rises, then the normal interest rate will rise and and in response to a rising nominal interest rate inflation goes down, because the normal interest rate rate has been increased
by the central bank, more than one for one, since five is greater than one. So there is sort of some kind of excessive reaction of the central bank.
It raises the normal interest rate by more percentage points than the inflation rate has actually been.
And therefore, the central bank is able to force down inflation rates fight inflation that's the relief and the new Keynesian model.
Now here in our thought experiment we have not directly observed an increase in inflation, or we have not started, let's say, it will start with an increase in inflation but we have started with a monetary policy shock.
So suppose inflation was at some level, constant or whatever.
The central bank is induced by some events to raise the nominal interest rate.
Then what would happen to the inflation rate. Well, we know when the epsilon ti here is a positive shock which with constant inflation would actually increase the nominal interest, right.
You would simultaneously have the effect that inflation falls. So in some sense it is the desired effect inflation falls, but it doesn't fall because the normal interest rate is increased, since actually the normal interest rate will be decreased.
In fact, Taylor rule will give us a lower nominal interest rate.
So inflation falls because there was a positive epsilon ti shock which is observable.
It does not fall because the central bank raises the nominal interest rate effect the central bank doesn't raise the nominal interest rate.
It is true to say that a positive monetary policy shock lowers inflation but it is not true to say that a higher interest rate lowers inflation.
But this actually you read in many papers, which set up a new Keynesian models that people say, raising the interest rate lowers inflation, completely untrue.
Because lower inflation is the epsilon ti shock our model, at least in the new Keynesian model, actually the nominal interest rate and inflation are positively correlated.
So, increasing the nominal interest rate should go hand in hand with with increasing it.
And the interest rates and inflation are positively correlated in the new Keynesian model this is one big insight, we have now gained.
The problem is that the epsilon ti shocks are not observable. Only the interest rates are observable. So while we now know that an increase in epsilon ti would in the new Keynesian model leads to a decrease in patient.
This knowledge is good for nothing, actually, because we do not know if epsilon ti has increased. We don't observe epsilon ti ti.
The only thing we observe our interest rates and the interest rate has actually decreased.
So, we don't know what epsilon ti is and we don't really know whether new Keynesian model is true. So can we base any inference on the new Keynesian model in the setting. Well, that's a big question.
I would actually say no.
The knowledge we have about the relationship between interest rates and inflation rates in the new Keynesian model suggests actually the opposite of the type of inference people draw from new Keynesian models, at least in a suggestive way.
Now, suppose we just do not know what the true model is, whether this is fiscal theory or new Keynesian.
Suppose that we have data on inflation, on interest rates and on surpluses.
Possibly, even though this is normally not the case, but let's say we even have data on all planned future surpluses so that we can compute the present value of surpluses.
The same argument goes through if we don't have this information. So, as the argument against what I want to show, I say it's not really lack of plausible data or lack of data which we may deplore.
We have actually everything at our disposal which one could reasonably hope for.
With the exception, of course, of the shocks. We do not observe the shocks and we do not observe the parameters of the model because those typically just not the case that we observe.
In this case, we have what I already introduced a couple of minutes ago, we have observational evidence.
The data, which we have cannot distinguish between fiscal policy shock in the fiscal theory of the price level model and monetary policy shock in the new Keynesian model.
What I claim and what I will show now in the sequel is that the model.
Well, that that we have this phenomenon of observational equivalence with the data which we may have at our disposal, even in really ideal circumstances where we also know about plant future surpluses.
The data are completely uninformative about the model which is Edward Burke and the shock, which is at work. So they tell us nothing. Right.
And that's a dangerous insight, which we have here, because what people usually do is that they say, well, assume we have a certain model assume let's say we have a new Keynesian model because new Keynesian state sort of still the main contender macroeconomic
model everybody works with it, almost everybody works with the new Keynesian model. So the people say, suppose we have a new Keynesian model.
Now we estimate somehow the parameters with Bayesian estimation or something like this and then we simulate the time series data from which we can generate from such a model and then we compare this with the actual data we have, and then people say who
looks similar. This looks similar. The model, the data we have generated from our model seem to look similar, hopefully, hopefully, or not so much but anyway, if you're if you're lucky if you're good, then the data generated by a model look similar to real world data, real world time series.
That's all fine can do so, but implicit in what people do is then the inference that they say, since the data looks similar to real world data we conclude that our model which we have assumed is true or close to the truth that we can use this model for policy analysis.
And what we now learn is that this later part of the inference is unfounded is plain wrong. There's no compelling reason to believe that if the data is generated by a particular model are similar to real world data are perhaps even exactly identical
under ideal circumstances. Let's say that they are exactly identical to the data we observe in reality.
And we still have no idea whether these data were in reality were generated by a new Keynesian model or perhaps by an observation of the equivalent fiscal theory model.
So the point I made here is, it is completely inadmissible in economics in macro economics to conclude from the fact that a certain model has features which match real world data that this model is true or close to the truth or explains anything.
Because what we actually would need to show in addition is that this is the only model which can generate this type of data patterns.
And we see here that this is not necessarily the case that may actually be the case that there is another model which generates the same patterns of data.
What I will tell you now that is that there are situations very simple situations where exactly the same patterns of the data can be generated by a completely different model named the fiscal theory model.
This does of course not say that the fiscal theory model is true.
The fiscal theory model is prone to exactly the same type of criticism. If we start out saying, well, this is the fiscal theory model and the fiscal theory model generates data which are similar to the data we observe in reality, then anybody may come and say, well, it could equally well be generated by a new Keynesian model.
We're just ignorant, currently, we have observation, which I will not show you that we do have it. We completely ignored what the true model is.
But we should be very careful to conclude that new Keynesian theory explains anything of our data, because it may be the case that the true model is a completely different model.
It may be a fiscal theory model, or it may perhaps be a different type of observationally equivalent structure.
There are other examples of macroeconomics of observationally equivalent structures to standard models, like news models.
I've worked some years over news models, and news models are very often observationally equivalent to standard type models which do not contain news and economics.
This is different.
Now, of course, one may say when we have two observationally equivalent models that one model is plausible and the other model is not plausible for some kind of extraneous reasons which we may have.
Well, if we have such reasons, then it's fine. If there are strong arguments to say that one model is implausible and the other model is plausible, then you may say, okay, then perhaps observationally equivalent is not much of a problem.
But as far as fiscal theory and new Keynesian models are concerned, I would say that both of them are plausible, or perhaps, perhaps I wouldn't even say they are equally plausible.
It depends on how one assesses the actual real world macroeconomy.
I always find this kind of fairy tale in the new Keynesian model not so convincing that some entrepreneurs are not allowed to change their prices because I just don't see this in reality that people are not allowed to change their prices.
First of all, this business about computing over an infinite horizon, the optimal prices, given that people expect they may not be allowed to change the prices in some period with some probability.
I don't know of any entrepreneur who would think that this is a very realistic way of modeling his behavior.
But be it as it may, I have my doubts about this kind of price stickiness which is built in the new Keynesian model.
Others have their doubts about active fiscal policy and say this is not how governments behave and most governments are actually responsible and they try to adjust to monetary policy sitting so they accept the primacy of monetary policy of a fiscal policy
and I would say yes, in certain periods, this was certainly the case and perhaps it is still the case for some countries or perhaps for all countries, perhaps it will be the case again in the future.
It depends on how we assess the actual situation. I personally think that in recent years we have seen quite some evidence of countries which seem to be not so much concerned about the solvency of their debt.
So they went into debt rather strongly and it seems that the European Central Bank at least has accommodated this fiscal behavior by some years on governance, for instance, but that's my personal point of view, which is why I think the fiscal theory model is not that
unplausible. It's a plausible alternative to new Keynesian model and the new Keynesian model to my view has more weaknesses than other people think it has, whatever that may be.
I conclude, we just don't know whether the new Keynesian analysis is really true.
It tells us anything about the real world because we cannot correctly distinguish between fiscal theory models and new Keynesian models since the two are observationally familiar, which I have not yet shown to you in detail.
I just gave you some clue by looking at these in parts responses here and pointing out that the same type of responses can be obtained in a fiscal theory model that we have actually discussed them already.
Now we want to go into more detail and see that indeed there is a very protracted problem of observational equivalence between the theory models and new Keynesian models.
So let us first define observational equivalence. I do this in a very general way here. Suppose we have two models which I call A and B.
And models A and B are different. They may be different in their economic structure. They may be different in the parameterization.
I mean, even if they are the same in structure as it is the case for the fiscal theory model and the Keynesian model, the parameterization may be different and then there are different models obviously.
Or they may be different in underlying shocks. Whatever is the case, I do not mind.
The only thing which is important is that along some dimension either in structure or in parameterization or in types of underlying shocks or in a combination of these things, models A and B are different.
Now, if under some specific setting, model A generates exactly the same patterns of observable data as model B does under a different setting, then model A and model B are called observationally equivalent.
So to frame it somewhat simpler, models A and B are observationally equivalent if the models are different, but they generate the same observations, the same patterns of the observations in any way exactly the same data structures coming from two different types of models.
And that is bad news for economics because then the data cannot discriminate between the two models.
And there's actually no way how we can infer from observable data which model is true.
And that is we can actually not do any kind of reliable policy analysis or any kind of reliable policy recommendation.
Because very likely if the models are different in structure or in parameterizations or in shocks, then the policy recommendations would be very different depending on what kind of model is true.
So we'll see that there is some type of correspondence between the fiscal theory of the price level model and the new Keynesian model in terms of shocks, basically, in the sense that a fiscal shock in the fiscal theory model
causes the same type of dynamics in the data as a monetary policy shock causes in the new Keynesian model.
So we cannot give any policy advice as long as we do not know whether the new Keynesian model is true or the fiscal theory model is true because
we would not know whether to recommend to the fiscal policy or whether to do monetary policy. This would depend on the body,
on the question of which is the true model.
Now, let's look at an example first before we come to a more general statement of the problem. I think the example is quite illustrative of the problem.
Suppose that some values of phi and rho are a specific parameterization of the FTPL model.
We know these two parameters occur in the FTPL model, phi in the Taylor rule and rho in the persistence of the AR1 process nu, which affects the Taylor rule.
Note now by phi nk and rho nk a parameterization of the nk model of the new Keynesian model.
With a specific property that rho nk, so the value we set for rho in the new Keynesian model, is equal to the value of phi in the fiscal theory of the price level model.
So we just take any parameterization of the fiscal theory of the price level model and what I now show in this example is actually whatever the specific parameterization of the fiscal theory model is, I can find the parameterization of the new Keynesian model, which is observationally equivalent.
And in order to do so, I need only take the new Keynesian model and set in the new Keynesian model rho nk equal to phi of the fiscal theory model.
I don't show it now for the complete model. I first showed in this example for the hypothesis actually we need another condition for the complete model.
Now suppose that we have a fiscal policy shock of unity in the fiscal theory of the price level model. So epsilon t s is equal to one.
We have computed the impulse responses for this. This was equation 83. We know the impulse response of inflation in period t plus tau to a fiscal shock and period t is negative phi to the tau.
And the impulse response of the normal interest rate in period t plus tau with respect to a unit shock in the fiscal innovation in period t is negative phi to the tau plus one.
And that holds for all periods tau, starting from tau equal to zero.
Now suppose that we have a monetary policy shock of size epsilon ti is equal to phi and chi minus rho and chi in the new Keynesian model.
That was the case we have already analyzed. This was the case we had when we looked at the impulse responses here, right? Because there we have said that the size of the fiscal shock policy shock is phi minus rho.
Now I just index the parameters phi and chi and rho and chi, okay? So we say that the monetary policy shock has a size phi and chi k minus rho and chi in the new Keynesian model.
We have computed the expressions for the impulse responses in equation 88. And now know from 88 that in all periods tau greater than or equal to zero, we have that the impulse response with respect to a shock of size phi and chi minus rho and k is the partial of phi t plus tau with respect to epsilon ti multiplied by the size of the shock.
So this would be phi and chi minus rho and chi times this partial derivative, which is rho and k to the tau divided by rho and chi minus phi and k.
Well, obviously, this thing here divided by this thing here is negative one. So we get this is equal to negative rho and k to the tau.
But we have set rho and k equal to the phi of the fiscal theory model. So this is then the same as negative phi to the tau.
Okay, which tells us that the impulse response of inflation in the new Keynesian model to a monetary policy shock of this size here is exactly the same as the impulse response of inflation in the fiscal theory model in response to the fiscal shock of size one.
It's exactly the same thing. It's negative phi to the tau. So it's exactly the same thing we found in the fiscal theory model.
And the same thing holds for the interest rate. You know, the two impulse responses are closely related. So the impulse response of the nominal interest rate is the partial of the nominal interest rate with respect to the monetary policy innovation
multiplied by the size of the shock, phi and k minus rho and k. So phi and k minus rho and k times the partial, which we have here, again, resides in canceling out this term here, just using negative one.
So this is negative rho and k to the tau plus one. And since rho and k is equal to phi, this is negative phi to the tau plus one. And therefore, that is the same impulse response as in the fiscal theory model.
With respect here to fiscal theory shock, whereas here it is with respect to a monetary policy shock of a different size though.
But this isn't really helpful because we don't observe the shocks. We don't know what the shocks are. We don't know what size they have. We just see that the impulse responses and therefore also the co-movement between inflation and interest rate are just the same for the fiscal theory model and a suitably chosen
Eukensian model. So when somebody comes up and presents to you in a seminar, a fiscal theory model, saying, well, this fiscal theory model
generates these type of impulse responses and look, they are exactly as the impulse responses, which I can compute from some type of analysis, which I can estimate from some type of analysis with real world data,
then you may immediately raise your hand and say, well, I can give you a new Keynesian model with a completely different economic message, which would generate exactly the same impulse responses.
So you just don't know what you're talking about. Perhaps you can say it a little bit more politely, but that's the essence of what you should say in such essay.
So we can, in this setting, always construct a new Keynesian model for any given fiscal theory model, which is observationally equivalent.
And obviously we can turn the thing around for any new Keynesian model of this particular structure of these simple three equations we have discussed at the beginning of today's lecture.
We would also be able to construct a fiscal theory model, which has completely different economic mechanisms because it has active fiscal policy and passive monetary whereas the Keynesian model has active monetary and passive fiscal policy.
But the data generated by these models would be the same, absolutely the same.
Now here comes an exercise for you.
We have shown that fiscal policy shock in the fiscal theory of the price level model is under those parameter settings observationally equivalent to a monetary policy shock in the new Keynesian model.
So we have shown that for any fiscal theory model, we can find a suitably parameterized Keynesian model, which is observationally equivalent.
Now turn the thing around. I told you already that's possible. So find a parameterization for the new Keynesian model such that a monetary policy shock in the fiscal theory of the price level model is observationally equivalent to a monetary policy shock in the new Keynesian model.
Actually, that's not quite what I had in mind. So sorry, the exercise here is a little differently post.
We had this cross correlation or this cross correspondence between fiscal policy shock in the fiscal theory model and monetary policy shock in the new Keynesian model.
So there was a correspondence between the two active parts of the models.
Now what I ask you in Part A of this exercise is that you find a parameterization for the new Keynesian model, such that a monetary policy shock in the fiscal theory model, so passive monetary policy, is observationally equivalent to a monetary policy shock in the new Keynesian model.
So that would say passive policy here can, under some parameters, generate the same effects as active policy here with the same variable, namely with the constraint.
You may be surprised that I now ask you to show the equivalence or the observational equivalence between monetary policy shocks in one model and monetary policy shock in the other model.
After having shown to you in the previous example that sort of the shocks switched roles in the sense that what a fiscal policy shock did in the FTP model was what a monetary policy shock did in the
new Keynesian model. So you may have expected, and actually not quite unreasonably so, that I would ask you in the exercise show that monetary policies shock in the fiscal theory model
can have observationally equivalent shocks to a suitably parameterized new Keynesian model in which we have a fiscal policy shock.
But that would not make sense, because in the new Keynesian model we don't have any fiscal shocks. At the fiscal shock term, this epsilon t was endorsed as one was the shock.
So we cannot pose this question, we can just pose the question of whether there is also observational equivalence between monetary policy shocks and that's what question A is about.
Now, question B, when you found this parameterization for exercise Part A, is this parameterization compatible with the observational equivalence between fiscal policy shock in the FTP model and monetary policy shock in the NK model.
So the question I ask here is, suppose you find a parameterization which solves exercise A, do you under this parameterization still have the observational equivalence of monetary policy shocks and fiscal policy shocks in the Keynesian model and fiscal theory model.
So, is there a parameterization such that whatever shock occurs in the fiscal theory of the price level model, there's always an observationally equivalent explanation in the new Keynesian model.
I know that this exercise is a little demanding and I would like to discuss it next week in the interactive part of the session.
But I would love to hear your solutions of this exercise as far as you get. I hope you can solve completely. That would be nice.
So please prepare it at home and present your solution next week in the interactive part and if you were unable to solve it well then I will show you how to solve it.
So that is not actually the point of the interactive part and I think you learn much more from it by working on the model yourself so please give it a serious try to solve problems A and B in this exercise.
Some remarks on that.
So the new Keynesian model does not have an exogenous fiscal policy shock so no epsilon TS, it does imply exactly the same values of the epsilon 2S TS by 69.
69 was the equation which determined unexpected inflation.
This equation would in the new Keynesian interpretation of the model, give exactly the same values for epsilon TS, as we would have as exogenous shocks in the fiscal theory interpretation of the model.
The only difference is that in the new Keynesian interpretation of the model epsilon TS is endogenous and not an exogenous shock.
Why is this so? Well, just have a look at the new Keynesian model. I told you in the new Keynesian model, this thing here is actually, well, superfluous, we don't really need it, it just determines the epsilon T plus 1S as an endogenous variable.
But what we know is that high and high, so inflation rate and non interest rate, are determined in these two equations in an observationally equivalent way to the fiscal theory model if we have suitable parameterization and suitable size of the shock.
So we know in these two equations here, there are certain data being generated which match exactly the time series which are generated in the fiscal theory model.
So this means that this pi here and the expectation of pi in the new Keynesian model is actually the same as the pi and the expectation of pi in the fiscal theory model, and therefore, epsilons must also be the same.
The epsilon S's, the surplus shocks, must also be the same. And so this is the reason for my remark here that the epsilon TS series are actually also indistinguishable.
Even if we knew them, they would be indistinguishable, we would still not be in the position to say is this epsilon TS now endogenously determined or is it exotlessly determined.
Obviously the parameterizations of the two models are drastically different in terms of the phi because in the new Keynesian model, we have a phi, which is greater than one, and in the fiscal theory model, we have a phi, which is smaller than one, more or less just by convention.
It is because the stability condition for the fiscal theory model is that phi is smaller than one absolute value, whereas the stability condition for the new Keynesian model is phi is greater than one in the new Keynesian model.
There is this little distinction here in the sense that we could also solve the fiscal theory of the price model level for phi greater than one.
This would only violate stability, so it would result in the inflation rates, which are not stationary, which would explode and thereby violate stability.
But that may actually happen, I mean we have had economies with exploding inflation rates, at least for a number of periods, rather short in Germany in 1922 and 23 very long in the bubble, for instance, or in Latin America.
So actually the fiscal theory model may be useful even for the case of phi greater than one. Mathematically, we can still solve the model, the fiscal theory model for the case of phi greater than one.
We can not really solve the new Keynesian model, or well no, that's also quite true. We can solve the new Keynesian model for phi smaller than one, but then the solution would not be unique.
We would actually have infinitely many solutions, we would have sunspot solutions and these type of things, so that's usually not wanted, but in the fiscal theory model we can actually derive a unique solution.
Even in the case where phi is greater than one, we have the stability condition is violated, the model would still make sense in the sense that gives us a unique solution.
Whereas the new Keynesian model would not give us a unique solution in the opposite case of the case which is stability, so in the case of phi smaller than one or equal to one, we would not have a unique solution anymore. So that's slightly different there.
Okay, and I think I stop here for the break. Let's meet again in 10 minutes, please.
And then I will come to estimation and see whether we can actually discriminate, distinguish the fiscal theory model from the new Keynesian model by just estimating phi.
So one idea may be to say, well, if we just estimate the Taylor rule and we get a phi greater than one, then obviously we know this is a new Keynesian model.
And if we estimate the Taylor rule with a phi smaller than one, then we know that is fiscal theory model. As I just said, this is not quite compelling as an argument because the fiscal theory model is also defined, well defined for phi greater than one, but okay.
Then we would have explosive inflation and that one would also see from the data, the new Keynesian model would not have explosive inflation for phi greater than one, so would still allow us perhaps to distinguish between fiscal theory and new Keynesian model.
So the question is, can econometric estimation help us to distinguish between two models which in theory, as we have now seen, are observationally equivalent.
So I stop for the break here and see you back in ten minutes.
Good. I hope you are all back. Let's now come to estimation and pose the question whether we can distinguish between the two observationally equivalent models by econometric techniques and in this case by discriminating between the two models
by the value of phi, which normally should be greater than one in the Keynesian model and smaller than one in the fiscal theory of the price level model.
So the easiest way to do that would be that we say, well, we estimate the Taylor rule. We have data on the nominal interest rate, possibly data even on real rate, but we take it as a constant here.
We have data on inflation and this would be an error term. So we would just regress the nominal interest rate on a constant and on inflation and then see whether the phi is greater than one or whether the phi is smaller than one.
The question is, would this allow us to distinguish between the fiscal theory model and the new Keynesian model, supposing that either model has generated the data. So either one or the other has generated the data.
And then, I mean, it seems to be a natural question why couldn't we just estimate the phi and then tell what model is correct and which model is not correct.
Now it will turn out that this is not possible to do this kind of distinction, but perhaps not so easy to see why it doesn't work.
Let's look first at the fiscal theory model.
Under the fiscal theory model, so supposing that the data has been generated by fiscal theory of the price level model.
We would get a consistent estimate of phi and this estimate would have the property asymptotically since it's just a consistency property and that phi is smaller than one absolute value.
Why is that? So, well, we know what the solution for the fiscal theory model is. We have computed this in equation 77. It is displayed here again for your convenience.
Phi t is actually the difference between two infinite sums or can be represented as the difference of two infinite sums. One phi, one sum is a weighted sum of lagged new terms.
So weighted by phi to the tau and then lagged new terms. So new t minus one minus tau goes here from zero to infinity. So very clearly pi t does not depend on new t.
Right. Does not depend on new t because here we only have lagged values at least a leg of one period, possibly much more than most of the terms much more than that. So this term here is completely independent of new t.
And this year is an infinite sum of the epsilon shocks, the epsilon s shocks, so the surplus shocks, while the epsilon s shocks are completely independent of the epsilon i shocks, the monetary policy shocks which propel this new term.
So therefore we know the pi t, if the fiscal theory model is true, then this regressor pi t will be independent of the error new t in this regression.
And you know from your econometrics classes that we get a consistent estimate if we know that there's independence between the regressor and the error term.
However, if there is correlation between the regressor and the error term, then the coefficient estimate of the regressor will be biased even asymptotically. So the estimate will not be consistent.
In the fiscal theory model, there is apparently no such problem because from the solution of the fiscal theory model we can infer that pi t does not correlate with new t and therefore the estimate is consistent.
And then the new Keynesian model, the solution for pi t was new t divided by rho minus phi. This we derived in the beginning of this lecture. So very easy solution for pi t, just a linear transformation as I had pointed out of new t by a factor of 1 over rho minus phi.
So pi t and new t are now perfectly correlated because pi t is just a linear transformation of new t. So we have a problem here in our regression equation if the data were generated from the new Keynesian model because pi t here is actually perfectly correlated with the error term.
So what consequence would that have? Well, the regression equation is I t is equal to constant plus phi times pi t plus new t.
But that is then the same thing as constant r plus phi pi t plus rho minus phi times pi t because we can solve this equation here for new t.
And instead of new t, we can also write rho minus phi times pi t. So phi times pi t, this is the regressor, the error term is rho minus phi times pi t.
When then obviously as you see the term negative phi times pi t cancels against the term plus phi times pi t here so that we are left with the regression being I t is equal to r plus rho times pi t.
Which shows you that if the data were generated by a new Keynesian model, then in estimating the Taylor equation, we wouldn't estimate the phi t, we would estimate rho.
Right. We thought we estimate phi, because we can write the equation this way, but paying tribute to the perfect correlation between pi and the error term in the new Keynesian model, we realize that the parameter we actually estimate is rho.
Well, and you know that in the observational equivalence result we have discussed, there was always this parameterization set in such a way that we get observational equivalence when we associate the phi of the fiscal theory model with the rho of the new Keynesian model.
Do you see that the two of them switch places also in the econometric center, which of course follows directly from the theoretical structure.
So there is no help in rely on econometrics actually I mean we might have known this from the outset on because since the data are completely uninformative about the underlying model, given that they are observationally equivalent to models.
Econometrics cannot have any useful function to discriminate between them as suggestive as the thought was that we just estimate the Taylor equation and find out what size I as it doesn't work that way.
We would come up with an estimate of phi, which is always smaller than one, because the row here is smaller than one, provided of course that the new t is indeed an AR1 process with parameter rho.
But if this is the case, we would come up with an estimate of what we think is phi, but which is not necessarily phi would come up with an estimated coefficient which is smaller than one.
And if we then concluded hey obviously the phi coefficient in the Taylor rule is smaller than one, so the underlying model is ftpl that would also be wrong.
Right, because we have to realize that the new Keynesian model would also produce an estimation coefficient, a regression coefficient in this model which is smaller than one, just that this would not be an estimate of phi.
It would be an estimate of rho.
So, estimation is of no help here. The observational equivalence destroys any possibility to infer from the data which of the two competing models underlies the data generating mechanism or is the underlying data generating mechanism.
Observational equivalence really means we just don't know and we cannot know there is no technique to infer from the data what the true model is.
I'll give you here a third way to see this observational equivalence. And this is probably the most general way because so far I have only given you actually specific examples of where you can easily see observational equivalence.
First, by looking at the impulse responses of phi and nominal interest rate and showing that the impulse responses are exactly the same, then by looking at estimates of the Taylor coefficient in the Taylor rule.
Well, okay, that also didn't work, but you may think perhaps there are other ways to estimate something. So if you're still not convinced that we have observational equivalence between the Keynesian model and the theory model, look at this third way to see that this observational equivalence.
So suppose that we have data, as many as nobody ever has, but this also makes our argument stronger. Suppose we have data for the inflation rate for the nominal interest rate for surpluses and for bond sales or debt, however you want to determine,
for all periods from minus negative infinity to infinity. So we have all the data we may just wish to have. And I claim we are still not able to distinguish between the Keynesian model and the fiscal theory model.
Suppose we have such data which are for given parameters, r, so real interest rate factor, rho, the autoregressive coefficient in the new series, phi, the Taylor coefficient, and some series of shocks, nu t, a solution to the model.
So in this case, I do not assume that nu t is an AR1 process. I just say we have some shocks which are given to us, like the parameters are given. We have r, rho, and phi, that these are certain given parameters and there are shocks, somehow some type of shocks.
All of these we do not know, right? We ignore the values of r, rho, and phi and we ignore the shocks, but we know they are there and we know that the data which we have have been generated by a model which was parameterized with these parameters and subjected to these shocks, all of which we do not know.
So the data are actually a solution to this type of model here, which is the familiar way of writing our model, pi t plus one minus e t of pi t plus one is equal to negative epsilon t plus one s, and epsilon t plus one s is, as you know, these are the innovations, the shocks, to the present value of surpluses.
And then here we have the Fischer equation. Again, at here we have the Taylor rule in the usual way. So these are the equations we know 69, 70, and 73.
Okay, now suppose we change the parameter phi.
So currently we have pi given at some value which we do not know. And suppose we change it from, let's say, a value which is an absolute value smaller than one to some phi tilde, which is greater than one.
So we would move from some fiscal theory type model to some new kind of type model by changing one of those parameters.
What can we do? We can say, well, then we change the shock.
The shock is also something which is unobservable. Let us define a new shock as nu t is equal to it minus r minus phi tilde times pi t.
Of course, we don't really know the size of the shocks since we don't know phi tilde, right? This is not known to us. But in principle, we can say, well, suppose the shock is exactly of that size, it minus r minus phi tilde times pi t.
If this is the case, then equation 73 is automatically satisfied because just plug nu t in when it is defined as this.
And then you see that equation 73 is satisfied with phi tilde here and phi tilde there. These two terms cancel, the r's cancel, and then it is equal to it.
So automatically equation 73 is satisfied when you think that series of shocks, which the model is being subjected to, is defined as this one here.
Now, this means that the data we have for it are still a solution to this model.
Even though the value of phi has changed, equation 73 is still satisfied. For given values of pi t and r, we know that the same values of it as the ones we have observed here satisfy equation 73.
So, it does not change. The whole solution for it is the same from minus, for negative infinity to plus infinity.
Now, since the solution for it is unchanged, it is also true that equation 70 here holds with an unchanged inflation expectation because it is not changed, r is constant, so apparently the expectation of inflation needs not to be changed due to our change in the phi tilde.
Our change from phi to phi tilde does not require a change in the expectation of inflation, provided that the it is unchanged. The it is unchanged if we change the shock in the appropriate way.
Now, when we have an unchanged inflation expectation, and this thing here is still the same, right, all these terms here, the S's, the R's, the B's can still be the same, then pi t will also be the same.
So, equation 69 then implies that the pi series also needs not to be changed. And therefore, we would find that with different value of phi, change from the initial value of phi to phi tilde, change from a value which is smaller than one to a value which is greater than one, does not require us to have any different time series.
Actually, exactly the same time series, our solution of the problem where phi tilde is the parameter rather than phi is the parameter of the Taylor Rule, just provided that, well, we have different shock series inserted in the model.
We had some arbitrary shocks initially, we have changed them, and these shocks are also arbitrary. So by changing the shocks in a specific way, very specific way, we can ensure that these data here can also be a solution to a new Keynesian model when they were initially a solution to a fiscal theory model.
So in that sense, this is complete observational curvelets.
Yeah, so the observed data are then still an equilibrium solution to the model, even though the value of phi has changed significantly changed, namely from smaller than one to greater than one.
And the way we used in order to ensure this result is simply that we change the unobserved process.
So this technically speaking means that phi and the series of shocks new are not separately identified.
If you have dealt with identification problems and econometrics, then you know what what identification means.
We cannot separately identify in this setup of equations, which may be a fiscal theory equation or fiscal theory model or a Keynesian model, we cannot separately identify the phi and the shocks do.
And therefore, we know that we have different model structures which are observationally equivalent since the same data series are actually solutions to different models.
Now, this is kind of a depressing result.
And actually I know that Cochrane in about 2011 or 12 so some some seven years before I wrote the book, I'm teaching about here thought that the situation was just hopeless.
There was nothing we can do about it.
But actually there's a little bit we can do about this problem. And here is what are perhaps the economic implications of those problems of observational equivalence and non identifiability or no, no separate identifiability that I've just pointed out.
The problem actually lies in the Taylor rule. See the Taylor rule describes the monetary policy of the central bank as saying that the nominal interest rate is set in response to current inflation and exogenous shocks.
With observational equivalence, we cannot infer from the data why the central bank changes interest rates.
We see that there are changes in interest rates that we see from the data, but we have no idea actually whether the central bank changes the interest rates because there are changes in inflation multiplied by a certain size of the parameter phi.
Or whether the central bank changes the interest rates because there are certain shocks, new tea. This we cannot observe. We do not look into their minds.
Central bankers don't publish the minutes of their governing council meetings.
And even if we had them, we wouldn't know whether it is true what they're right there or whether they would give us the information of why they change interest rates. So we just don't know.
And in the model, we cannot distinguish whether a change in the interest rates is due to some kind of endogeneity, some type of response to an endogenous model variable like the inflation rate, or whether it is caused by outside influence by something exogenous.
So it may be an endogenous response to inflation, which operates then via phi, and then phi needs a certain size or maybe something exogenous and perhaps an unmotivated decision, at least in terms of endogenous quantities.
It is unmotivated, some unmotivated decision to peg the interest rate to a new level under the influence of this new tea here.
From the data, which we have, it is undecidable what the central bank is, why the central bank is doing what it does. So we can only see which effects its actions have.
So we can only see what does, what kind of consequences does it have if the central bank changes the interest rate.
Why the central bank changes the interest rate? That is unanswerable in these models, as long as we have observational equivalents.
But we can still look at what kind of effect does change in interest rates have, because this is observable.
So, we may say it doesn't really matter whether the Taylor principle holds. The Taylor principle recall was the requirement that phi is greater than one.
So many, many papers have been written about this question whether phi is greater than one and that it should be greater than one, whether we can estimate greater than one, all these kind of things.
It's a key ingredient of the New Keynesian model that they always assume that phi is greater than one, because otherwise, as I said, the solution of the New Keynesian model would not be unique.
So we can now say with our knowledge of observational equivalents, we don't really care.
We just don't care about this question which the New Keynesians care about so much, whether phi is greater than one or not.
If phi is not greater than one, then it may be that interest rate decisions by the central bank are motivated by the behavior, say, of other central banks, which have observational equivalent effects, or any other developments which have observational equivalent effect.
We may say we're not interested in why the central bank reacts in a specific way to developments either outside of the model, so on markets which are extraneous to our model, or to endogenous variables like the inflation rate.
We're specifically not interested in. What we are interested in is what does a change in the nominal interest rate accomplish in our economy.
So the key question is not to what extent interest rates rise, either due to the endogenous influence of phi t, or due to the exogenous influence of nu t.
I mean, there may be combinations of them, right? I mean, the governing council of the ECB may have endogenous reasons plus exogenous reasons to come to a certain change in interest rates.
We may say that is not so much an important question how strongly the governing council was influenced by either phi t or nu t.
And therefore, it's not such an interesting question and not such an important question of whether phi is greater or smaller than one.
We know due to non-identification, we just cannot answer this question.
If we tried to, then any such interpretation which, for instance, I suppose that phi is greater than one or any corresponding econometric evidence is uncompelling unless we can establish that the underlying model is not observationally equivalent to the same model with an FDPL interpretation
or better, unless we can establish that there is no model at all, no different model, which is observationally equivalent to the model we are currently analyzing.
And that can hardly ever be proven that there is no model at all, which is observationally equivalent to the model we study. So that's actually quite a high challenge to show this.
So it is completely pointless to ask the questions, which you can't often ask.
I don't know if you are aware of the philosopher Ludwig Wittgenstein, a German philosopher who worked at Cambridge University and who is his doctoral dissertation and Tractatus Logico Philosophicus concluded with a remark, where of one cannot speak, there of one must be silent.
So the conclusion here is about all these type of questions why central banks behave in certain ways and if they actually behave in such ways, we may perhaps be just be forced to be silent.
The key question then is, if the central bank raises interest rates, what happens to inflation and to output?
Therefore, to give you an example, we can indeed answer the question if an increase in interest rates fights inflation, so depressed inflation, lowest inflation.
This question we can answer from both models, actually.
So we would not ask why interest rates are changed, but we would ask what sort of effect a change in interest rate has on inflation.
So we would actually not be interested anymore in the impulse responses that we have discussed earlier, which was the partial of pi t plus tau with respect to some shock.
But we would be interested in the partial of pi t plus tau with respect to the observable variable nominal interest rate in period t.
That is not exactly what we take to be an impulse response usually because the impulse response would, or by its definition, it assumes that there are shocks to which the system is subject, and that we can study the effect of these shocks.
Here we see we cannot actually do that so well because of the observational equivalence problem.
But the observational equivalence problem may not be that much of a hurdle. If we just look at the effects, an observable variable has on some other variable, another observable and endogenous variable like the inflation rate.
So we may ask what is this partial derivative here.
Now, following on this idea, suppose that we have a given series of nominal interest rates. Again, all the data available at our disposal, which we might just want to have from negative infinity to plus infinity.
So we know all the decisions of the center bank, regardless of how they are motivated, what kind of reasoning was behind these decisions. This is not interesting, but we just have the interest rate decisions.
Now, whatever the value of phi is, we can then satisfy the Taylor rule by, as I have already explained, assuming that the shock is nu t as being equal to it minus r minus phi by t.
In this case, the Fisher equation determines the inflation expectation. So this is the Fisher equation here. And equation 69, the unexpected inflation equation, implies then that it plus one is equal to nominal rate minus r minus the fiscal theory shock or the fiscal policy shock.
So solving the system this way here gives us pi t plus one depends on the nominal interest rate in the previous period.
Well, on the real rate, but this is constant and on the fiscal shock of the same period. So we have t plus one.
This is not so surprising, and it's actually not currently in our main interest. Our main interest is what happens if the center bank raises the nominal rate of interest, it.
And here we see a rise in interest rates now leads to rise in inflation.
Quite the contrary of what New Keynesian modelers typically try to work out where rise in interest rates leads inflation to decrease.
But here we would see the rise in interest rate clearly precedes the reaction of inflation. So it should really be causal for pi t because it happens one period earlier.
And we have when we see when we raise interest rates in period t, then period t plus one inflation rate will be greater. So inflation will go up in the succeeding period.
So interest rates, nominal interest rates and inflation rates would be positively correlated quite to the contrary of what New Keynesian analysis typically wants to suggest.
This is true under both models. It is true under the New Keynesian model, and it is true under the fiscal theory model. So we have no problem with with observational equivalence in this case because the two observational equivalent
models yield the same conclusion. When we raise interest rates, then inflation will also rise.
So that may actually suggest that our model is not very good because the central bankers, at least all believe that when they raise interest rates inflation rate will go down.
We come to this problem a little later whether the belief of central bankers there is true.
But here the first important point is to say observational equivalence is now not an issue because the Fisher equation from which this result here was derived.
The Fisher equation holds in both types of models in the fiscal theory and in the New Keynesian model. So regardless which model is true, we always would have this so called Fisherian equilibrium solution where equilibrium, nominal interest rates and inflation are positively correlated and not negatively correlated.
So it is a widespread misperception that New Keynesian models with five greater than one in the Taylor route imply that central banks aggressively combat inflation.
That's not clear because it seems that increasing nominal interest rate increases inflation.
Rather, in the New Keynesian model, higher interest rates imply that inflation will rise and the same is true for the simple fiscal theory model. We will later come to modifications of the fiscal theory model which imply that there is a negative correlation at least after some initial period.
But in the simple version of the model we study here, we have the result that the New Keynesian model implies higher inflation after raising interest rates.
Now, in some sense, I think it may be fortunate that we have the observational equivalence results because due to the observational equivalence, we are forced to look at the effect interest rates have on inflation.
Which in this case is a positive impact, rather than look at the negative impact of something which is first unobservable and second unidentifiable, namely the monetary policy shock epsilon t i, which also suggests the wrong sign, which suggests that inflation moves opposite to interest rates.
Now, this conclusion that tightening monetary policy reduces inflation obtains in the New Keynesian model only if we look at the unobservable and unidentifiable monetary policy shock. So only from this shock comes this result, only when
And it's not even fair to say that interest rates behave like the shock behaves, only if there is some initial impulse which may at unchanged
conditions raise interest rates. Only in this particular situation would we find that inflation decreases in response to the positive shock in epsilon t i, but interest rates would already move the opposite way of inflation there.
Now, if we look at the effects of changes in interest rates, rather than looking at these unobservable quantities epsilon i, we see that tightening monetary policy raises inflation.
Or conversely, loser monetary policy should make inflation go down.
Now let's turn to empirics, to real world data.
And what do you see here is the development of the ECB, of the ECB's main refinancing.
Right.
I introduced the concepts of the main refinancing rate and other key interest rates of the European Central Bank in chapter one of this lecture.
We see here in blue the ECB's main refinancing rate, so a nominal interest rate used as policy instrument and the harmonized consumer price index inflation in red.
And what we see is the following. In the years up to the financial crisis, inflation was more or less at the level which the European Central Bank came for, namely slightly above or slightly below 2%.
Actually, the European Central Bank aims at slightly below 2%, so here we were just about right. This year's 2007, this is when the financial crisis began.
At the end of 2008, there was already some movement, then we see that inflation went up.
But in principle, we see here a rather stable situation of inflation, somewhat higher than the ECB actually deserves, but perhaps not such a big problem for macroeconomic stability.
And then in the Great Recession, in the financial crisis, something really surprising happens.
First, inflation goes up to 4%, so it almost doubles or exactly doubles in peak. Right. And then it suddenly drops down in 2009, approximately.
Right. So both of this happened during the crisis.
But we see that there are two very different aspects of the crisis.
One aspect of the crisis where people apparently did not reduce their demand for goods and services, so that inflation went even the beginning of the crisis or the run up to the crisis from 2007 to 2008 inflation went up.
And then inflation crashed in the recession of 2008 and 2009 to very low levels.
And what is really interesting is that the European Central Bank had without clear motivation, I would say, increase the main refinancing rate on some, I would say, two years before inflation started rising.
Right. So this increase in the interest rate is very similar actually to the rise in inflation we've had here.
It was prior to the crisis. Right. This increase in the main refinancing rate. So we see here something which is matched very well by our model that interest rate and inflation are positively correlated with certain lag.
Okay. But then also came out of our theoretical analysis when we looked at the impulse responses, inflation lags behind interest rate decisions.
But here there's certainly strong positive correlation over a certain number of lags.
And then came the big recession. The big recession was basically in 2009 or starting perhaps in 2008 to 2009 in a recession aggregate demand drops.
And when aggregate demand drops, then obviously pressure on prices is relieved. Prices may not rise as much as they used to. They may actually fall if the fall in demand is high enough.
We see that harmonized consumer price inflation fell even to negative levels in 2009. Right. This is the trough of the depression which we see here.
So very, very low interest, very, very low inflation and actually slight deflation and this European Central Bank was mirroring this development with its key interest rate also lowering the interest rate by very much.
You may say, okay, see inflation picked up.
So perhaps the board of the European Central Bank would say, see, we have the negative correlation which we think is really at work between interest rate and inflation, because here we can really see that lowering the interest rate to this level here
results with a certain lag in the increase of inflation.
Almost immediately actually when we are here with the interest rate inflation reverses its trend, it goes up here.
Not so clear why the European Central Bank after it may have driven up inflation with this big decrease of interest rates, why they then decided to decrease interest rates even further.
But here we see it again. Here is decrease of interest rates.
And there is a decrease of inflation which follows suit with a certain time lag.
So here's again a positive correlation between interest rates and the development of inflation.
As such, it seems that the model is mixed in terms of what would we expect how higher interest rates work, how they influence inflation.
We have, however, to see that there is a huge fluctuation going on here in customer prices.
So quite probably it is the case that we have, we know it is the case that we have a very severe recession, actually one of the worst recession, which ever has been witnessed in the Eurozone.
Perhaps the worst recession if we just take post-war economic history, the worst recession and prices even sank. So we had negative inflation here.
Perhaps what, sorry, what we see in the next slides is that there are just an oscillations before the economy moves back to equilibrium.
Perhaps we may think that this here is not actually the consequence of lowering interest rates here, but rather to make up for the huge loss of interest rates or the huge decrease in interest rates that we have seen in 2009.
And then the inflation rates somewhat, fluctuations perhaps become smaller and die out over time.
And that is, we cannot decide this from this picture, but that would be a different story to the story that the decrease in interest rates here may have increased the inflation rate.
But even if that is so, if sort of the narrative of the European Central Bank were correct for this rather short period of time, clearly here it becomes incorrect again.
The European Central Bank has said, well, we decrease interest rates, hoping that inflation would rise, but inflation did exactly the opposite, inflation also fell quite a bit so, again also in negative territory.
Since approximately 2015, there was little change in the main refinancing, right? It is essentially at zero, right?
And even though there was no change here, suddenly inflation recovers and then fluctuates here at a level which will, well, this goes here until 2020.
If I had the complete data already, which would go quite big into negative territory again.
So given this discussion, it is not really clear whether the ECB main refinancing rate is positively or negatively correlated with inflation, but it is certainly clear that there is not a unanimous positive correlation or that it is clear that higher interest rates will lead to lower inflation.
So what I have done here is that I have actually looked at Eurozone data from 1999, so from the inception of the Eurozone to 2019, and computed the correlation between the ECB's main refinancing rate and the harmonized consumer price index on monthly data.
So in this first correlation, I compute just the contemporaneous correlation between inflation rate and nominal interest rate, and I come to the conclusion that this is 0.58, so clearly a strongly positive correlation, which we have here.
Now, one may say the contemporaneous correlation is not the most important correlation because possibly the inflation rate reacts just with the lag as we have had it in the model.
Then, of course, the empirical problem is what is the appropriate lag? Well, and I've tested various lags here. I tested a lag of a quarter, so three months.
I tested a lag of six months. I tested a lag of nine months, and I tested a lag of 12 months, and always computed the correlation between Pi and the previous interest rate.
Eventually, the interest rate was leading the inflation rate by one year, which I think is quite a bit.
You see, the correlation decreases somewhat. Actually, almost linearly, it does decrease down to 0.40, but it stays safely in positive territory.
So the Fisherian property that there's positive correlation between inflation and interest rates is very well visible in the data.
And actually, these are these type of data, right? These are the data of which I have computed the correlation. So the correlation analysis perhaps more clearly than the graphic tells me there's positive correlation between interest rate and the inflation, right?
And not as you can see, the theory typically assumes it is that, at least with lag, there's a negative correlation. We don't see it in the data.
Okay, here I would like to stop for today and give you the opportunity to ask questions and
possibly interaction if you like. So if you have a question, then please raise your hand.