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The Blind Target Shooter

Summary:
This is an attempt by someone who is not very good at math to understand the alleged "indeterminacy problem" of central banks using market signals of expected inflation to help them target inflation (or whatever). [I used to do rifle shooting at school. "If you can't group you can't shoot". But if you could group, and you had a spotter to tell you where your shots were hitting, and you adjusted the sights accordingly, you could hit the bull. Except for changing winds etc. So here's a parable:] Let the deviation from target at time t, Y(t), depend on the instrument setting X(t), and an exogenous random variable S(t), according to Y(t) = X(t) + S(t). If the target shooter can see the wind S(t), and adjust his aim X(t) according to a reaction function X(t) = -S(t), he can hit the target

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This is an attempt by someone who is not very good at math to understand the alleged "indeterminacy problem" of central banks using market signals of expected inflation to help them target inflation (or whatever).

[I used to do rifle shooting at school. "If you can't group you can't shoot". But if you could group, and you had a spotter to tell you where your shots were hitting, and you adjusted the sights accordingly, you could hit the bull. Except for changing winds etc. So here's a parable:]

Let the deviation from target at time t, Y(t), depend on the instrument setting X(t), and an exogenous random variable S(t), according to Y(t) = X(t) + S(t).

If the target shooter can see the wind S(t), and adjust his aim X(t) according to a reaction function X(t) = -S(t), he can hit the target perfectly, so his misses Y(t)=0 for all t.

Suppose he cannot see the wind S(t), but he can see where his shots hit (or miss) Y(t).

If he is firing continuously (it's a machine gun), and his shots hit instantly, and he sees his misses instantly, and he can adjust his aim instantly, and if the windspeed never jumps discontinuously, then by adjusting his aim in response to his misses, he can make his misses arbitrarily small, but not always precisely zero.

[Mathematicians tell me that what happens "in the limit" isn't always the same as what happens "at the limit"

To see this, suppose he has a reaction function X(t) = -bY(t), so his misses are Y(t) = -bY(t) + S(t) = [1/(1+b)]S(t). By making the adjustment parameter b arbitrarily large, he can make his misses arbitrarily small.

But he cannot make his misses precisely zero (unless the wind never changes [edit: I should have said "never changes unpredictably"]). Proof is by contradiction: if he never misses, then he never adjusts his sights (because misses are the only information he can respond to), so if the wind changes he will miss. Which is a contradiction.

It makes no difference if the target shooter cannot see his misses, but uses a second person, a spotter with a telescope, to tell him where his shots hit (if there are no communication lags between spotter and shooter). The misses can be made arbitrarily small, but not always precisely zero.

It makes no difference if the spotter can also see the wind S(t), and the shooter's aim X(t), and so knows exactly what the miss will be, but can only communicate the misses Y(t) and cannot communicate the wind S(t). The misses can be made arbitrarily small, but not always precisely zero.

Now let us make a small change in the assumptions. Assume the misses Y(t) can only be observed with a 2-year lag. But everything else happens instantly, and the model is still a continuous time machine gun. There is now a big difference between a spotter who can see the wind, and can communicate his continuous forecast of what the 2-year ahead observations of the misses Y(t) will be, vs a different spotter who cannot see the wind. By listening to the forecasts of the spotter who can see the wind, and adjusting his aim accordingly, the target shooter can make his misses arbitrarily small, but not always precisely zero.

If the spotter can see the wind only imperfectly, the misses cannot be made arbitrarily small of course. But if the target shooter cannot see the wind, and the spotter can see the wind imperfectly, the target shooter should still respond to the spotter's forecasts. And the difference between the misses when the spotter communicates only his forecasts, vs the misses when the spotter communicates his imperfect observation of the wind, can be made arbitrarily small, but not always precisely zero.

The target shooter is a central bank trying to hit in inflation (or whatever) target at a 2-year ahead horizon with variable "headwinds". (It won't observe the misses until 2 years later.) The spotter is a market forecast of 2-year ahead inflation. Set aside the question of whether the market forecast is better or worse than the central bank's own internal forecast (who can see the wind better), which depends on market participant's incentives to collect and process information on headwinds, and the fact that that incentive will be reduced if the central bank is targeting inflation. Those are good and relevant questions, but not what I'm trying to get my head straight on in this post.

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