Formation of Expectations
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Imagine a worker who has arrived at the end of some
period t - 1,
noticed that in the past the price level has moved jerkily up and down without (we
assume for the moment) any overall trend or drift.
Then he might adopt an error correction method for making adaptive expectations.
guess about the future price level would be composed of two parts.
One is the actual price level when make your forecast,
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and the other is a term that adjusts for your error in the previous forecast.
Formally, we can write equation (8) below
The first term is the actual price level at t - 1,
and the second is an adjustment factor λ times the error you made in forecasting the
price level in t - 1.
Typically, we imagine that λ lies between 0 and 1.
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We now want to see that if expectations are adjusted by an error correction
mechanism such as the one given in equation above,
the implication is that this period's expected price t-1Pt, depends solely on the
history of past prices.
This makes it exogenous to this period's actual price determination.
It is useful here to perform a Koyck transformation.
Changing time subscripts appropriately, we see that equation also describes
how expectations were formed about periods t - 1, t - 2, and so on.
For example, at the end of t - 2 the worker works out this equation for the
expected price for period t - 1:
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We can take this last equation, multiply it by lamda and add to earlier
equation In symbols
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We are now in a position to investigate the consequences of adaptive
expectations for the relationship between Pt and t-1 Pt
For a start, suppose that the price level had been constant for a long time at Po.
Then, suppose that at the beginning of a certain time T (subscript t is a
variable, indicating
some time period; subscript T is like the proper name
of a particular period), the price level jumps up to P1 and stays there
indefinitely.
At the beginning of T, all the terms on the right-hand side of equation (9) are
equal to Po, so the expected price for period T is given by Po, that is, T-1 PT =
Po:
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Here we have the extreme Keynesian case during the first period after the
price shift.
Once T is over, however, expectations are formed by equation (9) with
t set equal to T + 1. Hence, the first term on the right-hand side for period
T + 1 is P1 and not Po:
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The size of each step depends on the parameter λ and on the time elapsed
since the price level jump.
This process continues indefinitely, with the remaining error becoming smaller
and smaller
In equation (8), the second term on the right diminishes over time to make the
difference (Pt - t-l Pt) arbitrarily small.
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So adaptive expectations can make sense in an economic environment
in which the price level moves up and down in a fairly random fashion, with
the possibility of somewhat more permanent shifts in the background.
This is the basic assumption about the economic environment in any static
equilibrium model of the economy.
We analyze the consequences for changes in output and the price level of
generally unanticipated exogenous disturbances that can be interpreted as
randomly distributed before they actually occur.
Adaptive expectations seem appropriate in this environment.
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Let us get closer to a definition of rational expectations.
We posited an expectations function t-1 Pt = p( Pt). Rational expectations
are usually presented the other way around as
That is, the realized price level (or whatever variable you are predicting)
equals the predicted price level, plus a stochastic error term with mean zero.
Your prediction in (19) is said to be unbiased. If the price level is random,
that is, a nonzero єt, is a possibility, with a given distribution across possible
values
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The Law of Iterated Expectations
An implication of the unbiasedness of rational expectations is that the expected
value of all prediction errors is zero.
difference between the realization of a variable and its forecast, say,(Pt, t-1Pt),
equals a stochastic term with mean zero.
Even though some error is inevitable, it is as likely to be positive as negative..
Therefore, no one can say in advance what sign the error will have.
Unbiasedness implies that the expected prediction error is zero for all
future periods.
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For example, your forecast of the difference (Pt+1 – t-1Pt+1),
the difference between the actual value and the prediction two periods earlier,
is also a random variable with mean zero and a given variance.
Typically, the variance of the errors increases as the forecasting horizon
lengthens, but the expected value is always zero.
Rational expectations applies not only to variables such as the price level
or real output, but also to the predictions themselves.
Consider the relationship between the realization of the price level at t + 1 and
the forecasts made one and two periods in advance. If the forecasts are
unbiased, then
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so that
Here є 1t+1 and є 2t+1 are two independent "white noise" error terms. If we take
expectations through equation as of time t - 1, then the two error terms
drop out and the term t-1Pt+1 is unchanged. We are left with
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The left-hand expression in above eqn.t-1(tPt+1), is the expectation, made in
period t - 1, of the prediction that will be made at time t about t + 1.
The right·hand expression, t-1Pt+1 , is the actual prediction in t - 1 of Pt+ 1
So, equation (22) says that the expected forecast equals the current
forecast, a relationship known as the law of iterated expectations.
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Those who form their expectations rationally conform to the law of iterated
expectations because they use all available information and incorporate
all "news" as it comes in
If forecasts are fully rational, then the prediction error in one period is already
taken into account in forming the next period's prediction.
Therefore, no one can look at the forecasting errors and find that one error
helps to predict the next.
An econometrician would say that the errors are serially uncorrelated or that
the expected value of the product of two errors is zero.
With rational expectations, the expected error is always zero, and the·
errors are not linked in any way by serial correlation.
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In contrast, under adaptive expectations, we get serial correlation of forecast
errors, implying a violation of the law of iterated expectations.
The adaptive adjustment formula given in equation (8) is
For the rest of this section, a superscript" a" will be used to denote an adaptive
expectation.
At time T, just after the price increase, the forecast error equals one, as for 'rational
expectations, but in period T + 1 the error is λ and at T
+ 2 the error is λ
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and so on:
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(26c)
Correspondingly, the forecasting errors subsequent to the initial surprise at
time T are predictably different from zero.
For example, if we take expectations through equation (26c) as of time T. we
obtain
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From this equation it follows immediately that the law of iterated
expectations is broken:
and forecasts evolve in a predictable way.
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The contrast between adaptive and rational expectations is shown in
Figure 11-5, which reproduces the paths of the price level and adaptive
expectations, with the addition of the path of rational expectations.
Adaptive and rational expectations are the same until T, and
then they
both entail a one-unit prediction error at T corresponding to
the genuine price shock.
The two paths diverge subsequently as the rational
expectation leaps up to match the new price level, and the
adaptive expectation adjusts slowly toward it.
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The Pros and Cons of Rational Expectations
Rational expectations have great appeal to economists, for this hypothesis
comes closest to our vision of homo economicus, a person of thoroughgoing
rationality in pursuit of his or her maximum expected utility.
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If we propose that people and firms are very acute in choosing what to do in
anyone period,
then they ought to be equally acute in allowing for the future.
Each individual has an interest in devoting at least some time and effort to
making good predictions, as more foreknowledge cannot leave you worse off It
will usually allow you to make better decisions,
and even if adaptive expectations, say, lead to optimal actions, then with
rational expectations you could simply
reproduce that behavior.
Or if your rational expectations are more pessimistic than your adaptive ones,
given that the rational expectations are unbiased predictions of what will
happen, you might at least have a chance to mitigate the bad events to come.
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