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This is an example,
or rather, a story,

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that's supposed to
give us some insight

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and some intuition
about what the law

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of iterated expectations
really means.

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Suppose that you work for
a forecasting company.

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And suppose that
you make forecasts

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by calculating expected
values of the quantities

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that you want to forecast.

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And of course, when you
calculate an expected value,

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you always use whatever
information you have.

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So here we have the
beginning of the year.

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And you're working
for a company that's

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trying to forecast the sales
during the month of February.

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That's a random
variable, capital

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X. You're sitting in your office
in the beginning of the year.

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What is going to
be your forecast?

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It's going to be the expected
value of the random variable X.

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So this is a forecast that you
make at this point in time.

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Now, time goes by,
and we're sitting now

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in the beginning of February
or the end of January.

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At that time, you obtain
some new information,

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which is the value, little y,
of a random variable, capital

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Y. What should your
new forecast be?

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Well, once you have this
information in your hands,

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your new forecast should
be the expected value

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of x, given the specific
available information

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that you have.

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So this is the
revised forecast as

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calculated at the
end of January.

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But if you're sitting here
in the beginning of the year

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and you ask yourself, what
is the revised forecast going

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to be, your answer
would be, I don't

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know what it's going to be.

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It's random.

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It depends on what capital
Y would end up being.

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My revised forecast is a random
variable, the expected value

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of X given Y, which will take
this particular numerical value

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if it turns out the
random variable Y

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takes a specific
value, little y.

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So this is the
forecast calculated

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at this point in time.

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This is the forecast
viewed at the beginning

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of the year, at which
time we do not know yet

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the value of the
revised forecast.

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Now, what does the law
of iterated expectations

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tell us in this case?

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It tells us that the expected
value of the revised forecast

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is the same as the
original forecast.

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What does this mean
in practical terms?

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It means that given
today's forecast,

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the original
forecast, you do not

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expect the next
forecast, the revised

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one, to be higher or lower.

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It could be either
higher or lower.

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But on the average,
you expect the revision

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of the forecast going
from this one to that one,

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the revised one, that
revision on the average,

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to be equal to 0.

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You do not expect forecasts
to be revised either upwards

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or downwards on the average.

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Of course, this is not what
happens always in real life.

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So suppose that capital X, the
quantity you're forecasting,

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is the cost of some big project.

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And your original budget
or original forecast,

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expected value of X,
is what you expect

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the cost of the project to be.

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Well, from experience
with real life,

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we kind of know that
budgets or cost estimates

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tend to be revised upwards
more often than downwards.

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Does this real life
fact contradict

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the law of iterated
expectations?

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Well, not really.

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What is going on here is that
real life forecasts are not

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really honestly calculated
expected values.

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But maybe they're calculated
with some implicit or hidden

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biases so that the
forecasts that are given

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are actually not
the expected values.

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So there's no contradiction
between this mathematical fact

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and possible life experiences.