This is frustrating me. People (e.g. the Atlanta Fed Macroblog, the St Louis Fed Economic Synopses (pdf)) still aren't getting it. What can I do to attract attention to my simple point? Think up some totally insulting inflammatory blog post title? Nope, that's not really me. I'm just going to try again. And use bold.
You can't test whether core inflation is a useful indicator for a central bank to look at just by seeing whether core inflation forecasts future total inflation (or whatever the bank is targeting). You can't test whether anything is a useful indicator for central banks to look at that way. Everything ought to look useless by that test, if the bank is doing it right. What you are testing is whether the bank is doing it right.
If a central bank is targeting (say) 2% total inflation at a (say) 2-year horizon, and if it's doing it right, then deviations of total inflation from 2% ought to be uncorrelated with anything that the bank knew 2 years ago. This means that nothing (i.e. nothing in the bank's information set) should forecast 2-year ahead total inflation, if the bank is reacting correctly to those indicators. Core should fail to forecast future total inflation. Total inflation should fail to forecast future total inflation. Trimmed mean inflation should fail to forecast future total inflation. Unemployment should fail to forecast future inflation. Everything should fail to forecast future total inflation.
This is an immediate implication of rational expectations (on the part of the central bank). The bank sets monetary policy, looking at the indicators in its information set, so that the bank's forecast of 2-year ahead total inflation is equal to the 2% target. Deviations of actual inflation from 2% are therefore forecast errors. Under rational expectations (on the part of the bank) forecast errors should be unforecastable from anything in the bank's information set. They wouldn't be rational if you could forecast errors.
When you look at correlations between core inflation and future total inflation, you are not testing whether core inflation is a useful indicator of future headline inflation that the bank should pay attention to. You are doing something quite different. If you do find a positive correlation between core inflation and future total inflation, it means one of three things: 1. the bank is not targeting total inflation; 2. The bank's target for total inflation has changed over time; 3. the bank is targeting total inflation, and the target is not changing over time, but the bank is getting it wrong, and isn't responding as strongly to core inflation as it should, and so is violating rational expectations.
Now, if you want to learn how to do it properly, go and read my old post.
Worthwhile Canadian Initiative 
In the above, I should have said that the CB reacts to the state variables, obviously!
But about rational expectations, the Michigan survey is useful because they asked people about inflation expectations for the next year as well, and at least in that survey, inflation expectations lagged behind actual inflation on the way up (pre-Volcker) and subsequently the public was over-estimating inflation on the way down. You can argue that the mean over the entire window was zero (I haven’t checked this, but it seems reasonable), nevertheless the errors were serially correlated.
It’s one thing to say that, on average, expectations will be rational ex-post, when averaged over long time periods, and quite another to say that errors wont be serially correlated, or that you have a hard bound on what the time interval will be.
I think the latter requires that people be endowed with some organ that they don’t actually have, whereas the former requirement is easier to believe. And in this sense, I view the CB as having no better predictive powers than those who answered the Michigan survey.
RSJ: we are now getting closer to being on the same page (thankfully!).
“Now maybe I am wrong, and the CB is able to forecast shocks and that CB inflation errors are not serially correlated. I am open to that. But at least a case needs to be made — it’s a non-obvious assertion that requires evidence, right? And even if this is true for a period of time, what would guarantee that it would continue to hold in the future?”
Here’s how I look at it. There are lags in monetary policy. It takes time for the central bank to get information on shocks, react to shocks, and for inflation to react to monetary policy. If those shocks were unforecastable white noise, and if those shocks had an immediate impact on inflation, there would be no way the central bank could react to them, given those lags. All it could do would be to hold monetary policy constant, and hope for the best.
By asking this question, we are implicitly assuming that there is some information the bank can usefully react to. It can’t keep inflation exactly constant, but it can do better than just holding monetary policy constant.
On your second comment, remember that when I talk about rational expectations here, I am only saying that the central bank ought to have rational expectations. I am saying nothing about other people’s expectations, or whether their expectations even matter.
Put it another way. Unless you say that the central bank should hold the interest rate constant (and nobody says that) you must implicitly be assuming there is some information the central bank can usefully react to. And if it says, like the Bank of Canada says, that it is reacting to its information in such a way to keep its forecast of inflation 2 years in the future at 2%, we can test empirically whether it is reacting to that information in such a way to make its forecast rational.
Nick, I think the main mistake in your argumentation is right in the bold paragraph of your article:
“Everything ought to look useless by that test, if the bank is doing it right.”
Why do you assume that the central bank is “doing it right”? Why do you assume that it and the rest of the economy is operating under “rational expectations”?
Yes, in an ideal world you would be correct to point out that iff the actions of a CB constitute a Markov process then there can be no forcasting performed on the observed time series – because this is the very definition of a Markov process.
Do you claim that we live in an ideal world?
Do you claim that CBs are employing policy action that is a Markov process?
The article you are criticising is making no such assumption: it simply states, by observing the real world, that if a CB wants to improve its targeting accuracy in this real world then it should go from headline inflation to core.
That is a very simple observation of reality.
How is your information-free statement that in essence says that a Markovian process is a Markovian process relevant to that observation and to the resulting discussion?
White Rabbit: “Why do you assume that it and the rest of the economy is operating under “rational expectations”?”
I make no assumptions whatsoever about whether the rest of the economy has rational expectations.
I am not asserting that the bank has rational expectations.
I am asserting the normative claim that the central bank ought to have rational expectations.
What this post is about is how we interpret a zero or non-zero correlation between indicator and (future) target variable.
A zero correlation does not tell us that the indicator is useless. It tells us only that the bank is responding to that indicator rationally (which might include ignoring it.
A non-zero correlation tells us only that the bank does not have rational expectations. It does not tell us that the indicator is useful.
Perhaps a simple (non-numerical) example will help.
Image a one dimensional table with a ball rolling on it. The CB wants to keep the ball at the origin. If the ball is to the left, it tilts the table so the ball slides back to zero. If the ball is to the right, it tilts the table in the opposite direction.
If we are to make a guess as to where the ball will be 2 years from now, we would guess it would be at the origin.
But, if we know that the ball will be to the left of the origin in 2 years, then we would rationally guess that 2 years + 1 day from now, it will also be to the left. Similarly, conditional on the ball being to the right of the origin in 2 years, we would rationally expect it to continue to be to the right of the origin in 2 years + 1 day.
The errors are serially correlated, even though the mean of the errors is zero.
And this would be true for any control that has “inertia” — i.e. for any situation in which the energy or effort needed to push the ball back to the origin from a displacement away from the origin goes to infinity as the time allotted to push the ball back to the origin goes to zero.
I think it’s reasonable to assume that inflation has this inertia property.
everything has this property, so you always get serial correlation of errors in any controlled process.
God, this is really frustrating. Non-stupid people keep on not getting my simple point!
Let me try a simple example:
Assume the economy is P(t+1) = aZ(t) + bM(t) + S(t+1) where P is inflation, Z is some indicator, M is monetary policy, and S is a white noise shock, and a and b are fixed parameters. In this model, Z is useful iff a is not zero.
Let the central bank follow a reaction function M(t)=-R.Z(t)
If R=a/b, then P(t+1) = S(t+1) and there is zero correlation between P(t+1) and I(t). The feds will say that Z is useless, even when it might not be.
If R is less than a/b, then there is a positive correlation between P(t+1) and I(t). The feds say Z is useful, even when it might not be.
RSJ: (my above comment was in response to WR).
I like your example.
Except: “But, if we know that the ball will be to the left of the origin in 2 years, then we would rationally guess that 2 years + 1 day from now, it will also be to the left. Similarly, conditional on the ball being to the right of the origin in 2 years, we would rationally expect it to continue to be to the right of the origin in 2 years + 1 day.”
No we wouldn’t. Not if the bank tilts the table to get the ball back to zero, on average, at a 2 years horizon. The bank of Canada does not say it will act to ensure that inflation approaches 2% asymptotically. It says 2 years.
The errors will not be AR(1). They will be MA(23). (That’s 23 months).
“Unless you say that the central bank should hold the interest rate constant (and nobody says that) you must implicitly be assuming there is some information the central bank can usefully react to.”
Even in a world where monetary policy really can control (not merely influence) inflation, a CB would recognize that there is value in interest rate stability. Large random jumps in interest rates with no smoothing would not be the optimal policy.
RSJ: Yep. You are assuming that “serial correlation” of the bank’s forecast errors means an AR process, which only dies away as time goes to infinity. But it will be an MA process, not an AR process. If it’s an MA(23) process, then the 24 month ahead expectation will be zero.
Max: agreed. That is presumably one reason why central banks try to get inflation back to target in 2 years, not next month.
RSJ: just to clarify. The shocks hitting the economy may well be an AR process. But if the bank is targeting inflation at a 24 month horizon, the bank’s forecast errors will be MA(23).
OK, I’m not making any assumptions about the shocks — there is no reason to believe that the shocks are an AR or MA process.
Without a central bank, if there were only shocks — the price level would be undefined. Therefore you cannot say anything about the shocks.
What I am saying is that even though you know that the expectation (e.g. mean) of inflation in 2 years will be 2%, nevertheless you also know that the probability of inflation being exactly 2% is zero.
Inflation will always be above or below 2%. And conditional on inflation being above target, you also know that the next day, it is likely to remain above target. And conditional on inflation being below target, the next day it is likely to remain below target — with probability 1.
With probability 1, the errors, or difference between actual and target inflation — are going to be serially correlated over any non-zero interval of time.
“Without a central bank, if there were only shocks — the price level would be undefined”
RSJ, seriously, was there no price level in the US before 1913?
Why do you so frequently undermine your own agrument with a blatantly false statement that doesn’t appear to be essential to your point anyway?
Nick,
I don’t think these guys are missing your point, I think that you’re missing theirs. Seems to me you have an entirely different underlying model to theirs.
In particular, I think the difference is that the engineers are assuming that the mapping from control variable to objective is itself noisy.
As far as I can tell your viewing the world through the lens of MV=PY with shocks to V. Thus, in your view if a velocity shock hits the bank tries to offset it. In the absence of a further shock to V they are always able to do this, if they’re doing it right. Further, if they’re doing it right they should optimally predict V (to the extent this is possible) and so only miss if the change in V is truly a shock. Is that accurate?
Thus, only unpredictable shocks to V cause them to miss and you get your result.
The engineers probably have in mind something like PY = MV + noise, in this case even if the CB does all it can in offsetting a velocity shock, it is not the case that in the absence of a new shock to V it hits the target with probability one.
If we take V as the inverse of the demand for real balances, as it is supposed to be, and not simply PY/M, then I think the engineers are correct here.
“P(t+1) = aZ(t) + bM(t) + S(t+1)”. Actually you probably meant P(t) = aZ(t) + bM(t) + S(t+1)
Look very closely at what you’ve formulated here Nick. Your system determines its response at time t by looking at input from time t+1. Your system is not causal. In fact it is acausal. At t=0
You have violated causality.
In the real world, you can only definitely know things that happened now, or in the past. You can’t know the future. You can guess at the future, but not know it definitively. That’s causality. Now if you wish to use a non-zero correlation between core and total inflation, Bayes Rule is a good place to start. But in order to usefully construct a system that looks into the future, you have to start using probabilistic methods like Bayes Rule.
It may be simple, but when it comes to control systems the math is deceptively complex. There is a great deal of complexity in simple-looking equations and your first instinct is often wrong. You have to derive then analyze.
RSJ: let me re-state, to maybe resolve our differences here.
Assume the shocks to inflation when the central bank is optimally targeting 2% inflation at a 24 month horizon, so that E[P(t+24)/I(t)]=2%, are an MA(23) process. (I assert that they will be, or rather, I assert that they will be an MA process of up to 23 months). In math, P(t) = 2% + shock(t) where shock(t) is MA(23).
The expectation at time t of inflation at time t+23 will (almost certainly) not equal 2%. But the expectation at time t of inflation at time t+24 will nevertheless equal 2%.
(My statement above does not contradict what you were saying, if I re-interpret you correctly.)
Adam: I don’t really see the difference.
The way I look at it is this: assuming the bank targets 2% inflation at a 24 month horizon, then it sets E[P(t+24)/I(t)]= 2%, where I(t) includes the instrument, and the vector of indicators, plus lagged values of indicators and instrument. Then I impose the standard bit of econometrics that P(t+24)=E[P(t+24)/I(t)]+S(t+24)=2%+S(t+24) where S(t+24) must be unforecastable (uncorrelated, orthogonal) with respect to I(t). And S(t) must be an MA(23) process (or less than 23).
If the bank expects P(t+24) to be above 2% it should tighten monetary policy, and if less than 2%, it should loosen monetary policy.
If monetary policy has no effect on P(t+24) then this won’t work. If the bank has zero relevant information this won’t work. If the distributed lag structure is ugly enough to cause ever-increasing oscillations in the monetary policy instrument this won’t work. Under those circumstances, inflation targeting at t+24 is impossible. Otherwise, it’s just econometrics.
I normally tell the story in a New Keynesian way, assuming monetary policy instrument is R(t) rather than M(t). But it makes no difference to the econometrics.
Determinant:” “P(t+1) = aZ(t) + bM(t) + S(t+1)”. Actually you probably meant P(t) = aZ(t) + bM(t) + S(t+1)”
No, I meant “P(t+1) = aZ(t) + bM(t) + S(t+1)”. Monetary policy affects inflation with a lag. Some of the stuff the bank observes (Z) affects inflation with a lag, so the bank can counteract it. And other stuff S affects inflation too quickly for the bank to counteract it.
Then you canonically have a non-causal and therefore unstable system. You may try to implement it with Bayesian methods and core inflation ought to make a good prior for Bayesian estimation, but your equation is not a well-behaved system.
Which bring me to my next point. The Fed’s study is extremely suggestive of using core inflation as a Bayesian prior, surely as an economist you would agree?
Determinant: You lost me. Are you telling me that causes can’t have lagged effects? Are you telling me that lags always mean the system is unstable? Regardless of the central bank reaction function?
My economists intuition tells me that core is probably a useful indicator for the central bank to look at. In the sense that it would be a good Bayesian prior of (say) 2-year ahead inflation if the central bank didn’t react strongly when core changes. But I want empirical evidence to back up my intuition. And the Fed’s study is no help, because the Fed is probably reacting to core, as well as to total, so I can’t tell what would happen if the Fed didn’t respond to core. The empirical work I did do for Canada, 10 years ago, using the method outlined here, did suggest that core might be useful, IIRC, but my data series was too short to get strong results. And that was for Canada, not the US.
Adam P,
I was referring to the steady state being defined. Of course, there are always prices. But prior to 1913, if you looked at the price level, it veered up and down and followed no observable AR pattern. Have you ever read “This time it’s different”? They have long term price data, and while there is a general bias towards inflation over deflation, there doesn’t seem to be any obvious characterization of the inflation rate as m + MA(N) for some N as low as 24 months. The “m” is what is undefined.
Which addresses my point to Nick..
Suppose, without a central bank, that the price level is m + MA(infinity), or m + MA(very large number). Basically you cannot determine m with any confidence. But the shocks are i.i.d.
With a central bank, inflation is mean-reverting, due to the interventions in response to (and as you would argue, in anticipation of) the shocks. As the CB shocks the economy in the opposite direction, it can reduce the effective period and cause the errors to be a MA(23) process.
However, the terms are not i.i.d., because they include interventions in response to the shocks. The interventions are not distributed independently of the shocks. Now your conclusions about S no longer follow.
Try this definition of causality:
Causal System:
The output at any time t depend on the past or current input and for the output depencency it depends on the past output only.
Or put it another way, for a non-causal the excitation of the system starts before t=0. Normally we say that if y(0) != 0 the system is not causal.
Second, you never express something as y(t+T). y is the dependent variable. Lags are expressed in the independent side of the equation.
Let’s rephrase.
P(t) = aZ(t-1) + bM(t-1) + S(t)
“Monetary policy affects inflation with a lag.”
Further, you always want P(t), current inflation, to be K%. So
K% = aZ(t-1) + bM(t-1) +S(t) where S(t) is a random variable representing the shock.
Rearranging, you want K%-S(t) = aZ(t-1) + bM(t-1). You want the indicator and monetary policy at t-1 to anticipate the shock at T. Remember definition 1? This contradicts it since S(t) is a random variable. You want your indicator and monetary policy together to anticipate a random variable.
Rearranging further, you said Z(t) is useful if and only if it is not zero. You also defined monetary policy as M(t)=-RZ(t)
So, K%-S(t) = aZ(t-1) – b(-RZ(t-1))
K%-S(t) = (a+Rb)Z(t-1)
Which means the Indicator has to indicate S(t) = K – (a+Rb)Z(t-1). Alternatively, Z(t) = S(t+1)/(K-a-Rb)
This is the causality problem right here. Your indicator isn’t causal, it isn’t reacting to any past or present physical inputs, it is anticipating future shocks. Now as I said you can estimate to determine you indicator, but your posts always seem to want it be behave perfectly. You can’t physically realize this indicator perfectly. You can use approximations and estimates, Bayes Rule being an excellent place to start, but us engineers have a horrible time when you economists just assume that the indicator will be perfect and then get angry when it isn’t. We say “of course it’s not perfect, what do you expect when you have a look-ahead system”
Determinant:
1. Your P(t) = aZ(t-1) + bM(t-1) + S(t) is exactly the same as my
P(t+1) = aZ(t) + bM(t) + S(t+1)
2. Since the central bank does not know S(t+1) at t, because S is iid, the best it can do is set M(t)=-(a/b)S(t). If it does this, P(t) = S(t), which is the best it can do to keep inflation as close as possible to the target (assumed 0%).
I know, that’s why I reformulated it to follow conventional notation. It makes analysis easier and it’s what I’m used to. Independent variables on the right, dependent variables on the left, lags shown on the right not the left.
Right, so the best control a central bank can achieve is total inflation equal to the random shock. Not the K% target you enunciated. Yet somehow a central bank is supposed to analyze all information and react in such a way that this is achieved. How that is supposed to happen with variables available right now is the question that many in this thread, including myself, are having trouble with.
Again, calibrating the parameters in your indicator variable Z(t) = S(t+1)/(K-a-Rb) with Bayesian methods related to core inflation input seems a good a place as any to start, but I don`t see how you can justify the strenuous objection you outlined in your OP based on the system you have here.
I don’t get this. What Nick is saying ought to be totally uncontroversial in principle. The amount by which the CB will miss will be unpredictable, conditional on the information available to the CB 24 months ahead of the measurement date. If this is not true, then either 1) they aren’t really trying or 2) they don’t have the means to affect it. I argued above that there is plenty of evidence for 1. I.e. policy changes are serially autocorrelated (like ratings) which is strong evidence that they are not taking all available information into account; and secondly that the CB shouldn’t really be trying to totally neutralize shocks because of inertial effects: if they correct shocks back to zero in finite time, then they will overshoot. If inflation is a second order system, they should aim to damp shocks like a critically damped spring, i.e. asymptotically. If the system was first, instead of second order, they could kill shocks arbitrarily fast by moving rates arbitrarily much, in which case they wouldn’t need 24 months. The reason they don’t do this is that they they know that higher order terms can cause insane oscillations, as you point out, Nick. And if they know about higher order terms, then they can’t seriously mean 24 months, either. Asymptotically, at the highest rate possible, is the only coherent policy.
But to argue for 2), they don’t have the means, doesn’t make any sense to me. If they really wanted to kill every shock with a 24 month lag, then (ignoring the ZIRB) they are fully empowered to do so without predictable error, by moving rates arbitrarily much. And even adding uncertainty in the knowledge of the control function is not an excuse to miss in predictable ways.
Nick, I re-read the macroblog post (again) and it’s more clear to me that they are not necessarily using core as a predictor. Maybe they are but all I’m getting is that they are stating that core provides better “controllability” than looking at headline. They do not explicitly state they are using core as a predictive indicator to formulate current monetary policy, only that it validates using core as the sole feedback measurement (even though they use different but similar method) ex post.
Now if the Feds are using some Bayesian Witchsmeller mumbo jumbo to forecast future core based on… core… within a closed loop system well, I’ll send over the ghost of Nyquist to slap them upside the head. More likely, though, they are looking backwards, not forwards.
Nick, I’m sure you are tired of repeating yourself but please correct me if I am wrong here. Your argument rests on the following assumptions:
1. The CB behaves rationally (nothing wrong with that)
2. At time t, the CB is able to optimally exploit an information set so as to set its policy in accordance with achieving an inflation target at t+h (2% at the 2 year horizon in Canada)
So, if the CB is not able to respond optimally to the information set there may be deviations from target that one can forecast. Now, it is not hard to imagine that the CB’s response to its information set is imperfect (despite rational behavior), especially given the uncertainty inherent in unobservable measures such as the output gap that the CB is known to follow. Furthermore, your argument only holds for the horizon at which the CB wishes to achieve the target. Does that mean that even if your case holds, core could still be useful for forecasting total inflation at other horizons? Say, 12 or 36 months rather than 24 months.
K,
So if mean{X{t}} = 0, then X{t} is i.i.d.? There are no serial correlations?
Mik: that’s very close.
“Does that mean that even if your case holds, core could still be useful for forecasting total inflation at other horizons? Say, 12 or 36 months rather than 24 months.”
If h=24, then core (or anything else) could (perhaps) still be useful in forecasting total inflation at less than 24 months ahead, but it could not be useful in forecasting inflation at more than 24 months ahead.
Here’s the explanation of why it would not be useful at (say) 36 months: remember that the bank’s information set at t=12 includes core at t=0 (assuming the bank does not forget old data). And since the bank at t=12 is targeting total inflation at t=36, inflation at t=36 must be unforecastable from the bank’s information set at t=12, which includes core at t=0.
“Now, it is not hard to imagine that the CB’s response to its information set is imperfect (despite rational behavior), especially given the uncertainty inherent in unobservable measures such as the output gap that the CB is known to follow.”
Strictly speaking, when we test whether the bank has been responding rationally to its information, we should be using real time data (i.e. unrevised data). Measures like the output gap are often revised, and Simon van Norden has shown that revised measures of the output gap often differ a lot from real time measures of the output gap. If we used final revised data, it would be a bit unfair, because that final revised data was not available to the bank at the time.
(There is also the problem inherent in any test of rational expectations when we use the whole data series for test for correlations, when the bank has a smaller data set, and the structure of the economy may be changing over time. Strictly speaking we should use some sort of rolling sample method to do this test of whether the bank is responding rationally.)
Which is unrealistic in any event. I much prefer a Bayesian interpretation whereby the probability expressed the given confidence that an outcome will occur given a specified prior, rather than frequency.
Bayesian interpretations are both much more intuitive and much more realistic.
C’mon, admit the Central Bank isn’t pefect and be Bayesian. Become One with the Bayesians. :mad cackle:
Determinant: let me give a good Bayesian gloss. The central bank, as a good bayesian, looks back over its past performance looking for systematic mistakes, so it can adjust its decision rule accordingly. It will look for non-zero correlations, and when it finds some it will adjust the weight it places on those indicators.
I appreciate the response. My output gap example was not a good one. As you point out, the CB’s optimal response can only be seen in terms of what is available to it at the time. I guess more broadly speaking I meant it is probably unrealistic to view monetary policy decisions as optimal responses to an information set, despite the best efforts of a CB to react optimally to the information at hand.
All of the core inflation measures currently used at the BoC have been shown to have good forecasting properties for total inflation. On the other hand, simple contemporaneous correlations with the policy rate vary considerably depending on which core measure is used. While simplistic, in this framework your conclusions could range from 1) They should pay more attention to core to 2) They are paying too much attention to core. Depends on which core measure is used!
RSJ: I’m not saying anything about X(t). It doesn’t matter what X(t) is, since the relevant process that we are discussing is E_t[X(T)] which is a martingale. If you calculate an expectation E_t[X(T)] based on all available information at time t, then you can’t improve that calculation by taking into the account some random variable whose value is in the information set at the time of the expectation. It’s obvious, but nevertheless has a name in probability theory: the law of iterated expectations.
K,
No, the E_t(X_t) is not a martingale, and I challenge you to prove that it is. You will find that you are assuming your conclusion…
but, RSJ, for T > t, E_t[X(T)] is a martingale and that is what K asserted.
“for T > t, E_t[X(T)] is a martingale and that is what K asserted.”
I guess the question is what empirical evidence, not just assertion, is there for the “stochastic” process modelling interest rate adjustment by a CB to be a martingale ?
Sure, we can assert anything, but based on what evidence ?
Aha! I now finally understand K’s comment @04.47am (with help from Adam)! Holding T constant, as t increases the expectation of P(T) at time t will follow a martingale. Immediate implication of the law of iterated projections/expectations. Today’s expectation of tomorrow’s expectation of the day after tomorrow’s rainfall = today’s expectation of the day after tomorrow’s rainfall. Not an empirical assertion at all. Doesn’t even assume the CB is targeting inflation. Damn, that took me a while. (my brain was thinking that T=t+h, where h is a constant, rather than T being a constant.)
Nick:
Do you mean Doob martingale construction (E_t[X(T)]) ? I am not sure how helpful the construction is by itself here as it can be made from any arbitrary random variable with bounded expectations.
Nick: Yup. But the martingale property of conditional expectations was not my central point; it’s just a nice way to think about the problem. The central point was the (bleedingly obvious) fact that you can’t improve on the expected value of a process by taking into account information that was already available to you when you first computed the expectation. This is also the result of iterated projection. I.e. when we say that the CB could have done a better job of computing the expected future value of inflation, we mean given all the information they had then. But that information obviously includes the value of core inflation (and every other bit of the state of the world), so adding core inflation to the information shouldn’t improve their estimate. It’s seems like a silly point when you break it down, but for some reason it’s not obvious on the face of it, which is why your post was clearly worth writing.
vjk: I just Googled “Doob martingale”. First time I had heard the term. I think that’s what I mean.
K: Yes. My whole point does seem bleedingly obvious, almost trivial, when you state it like that. Which is why I get frustrated when people don’t get it!
Everyone: if you still don’t get it, then re-read K’s comment above (@11.44) slowly and carefully.