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Expectations
Adaptive Expectations
Rational Expectations
Modeling Economic Shocks
• Let zt = value of variable z at time t,
zet+1 = expectation of zt+1 at time t.
zte1  zt 1
• Perfect Foresight:
• Adaptive Expectations
zte1  zt  (1   ) zte
where 0    1 is the “speed” of adjustment of
expectations.
• Problem: Errors are systematic and repeated.
• Rational Expectations: The expectation of zt+1 at time t
given all currently available information. (Statistical
“conditional” expected value):
zte1  E{zt given info avaialble at time t}
 E{zt 1 informatio n at time t}
 Et {zt 1}
Notes about Statistical
Expectations
• Let X = random variable
• f(x) = Pr (X = x) = probability density of X
• The expected value of X is
E ( X )   X  Pr( X  x)   X  f ( x)
x
E ( X )   xf ( x)dx
x
(discrete)
x
(continuous)
• Properties of Expected Value: For X and Y
random variables and b constant:
E(b) = b
E(bX) = bE(X)
E{ g(X) + h(X) } = E{g(X)} + E{h(X)}
E{XY} = E(X)E(Y) + COV (X,Y)
• Let X and Y be random variables.
• The conditional expectation of X given Y = y is
given by
E ( X Y  y)  x x  Pr( X  x Y  y)
where
Pr( X  x Y  y)  Pr( X  x, Y  y)
Pr(Y  y )
Modeling Economic Shocks
• Many economic variables exhibit persistence:
*
If z is above (below) trend today, it
will likely be above (below) trend
tomorrow.
• One way to model the idea of persistence of
shocks is by an autoregressive (AR) process:
zt 1  rzt   t 1
where 0 < r < 1 measures the degree of
persistence.
• Where  is a random “white noise” shock with
mean zero: Et  t 1  0 and constant
variance.
 r = 1  permanent shock to z, “random walk”
r = 0  purely temporary shock, no persistence.
0 < r < 1  temporary but persistent
Examples: Macroeconomic data: GDP, Money
Supply, ect.
Figure 3.2 Percentage Deviations from Trend in
Real GDP from 1947--2003
Monetary Policy: 2004 - 2008
Numerical Example
• Consider t = 20 periods
• There is a one-time shock to t in period 1 where 1 = 10 and t = 0
for all other time periods:
10
e
t
10
5
0
0
0
0
5
10
t
15
20
20
• Notice the effect on zt depends on the value of r which measures
the amount of persistence for the shock .
22
20
r0
purely temporary
15
z
t
10
5
0
0
0
5
0
r  0.80 
10
15
t
20
20
22
20
temporary but
persistent
15
z
t
10
5
0
0
0
0
5
10
t
15
20
20
r  1  permanent
22
20
15
z
t
10
5
0
0
0
0
5
10
t
15
20
20
• Let’s use r  0.80 for the shock to z.
• Comparison of adaptive expectations (AE with 0.5)
and rational expectations (RE) of z. Actual value of z is
in red, expected values for z are in blue.
25
25
20
20
z
t
z
t
AEz
t
0
REz
t
10
0
0
0
0
5
10
t
15
20
20
Adaptive Expectations
10
0
0
0
5
10
t
15
20
20
Rational Expectations
• Rational expectations (RE) is the statistical
forecast of future variables given all current
information available at time t (Infot)  E{zt 1 info t }
• Notice since zt is known at time t:
Et zt 1  Et rzt  Et  t 1  rzt
• With RE, the errors in expectations are random
and average to zero:
Error  zt 1  Et zt 1   t 1
• When r  1, Et zt 1  zt
 “Random Walk”
or “Martingale”
Application: Theory of Efficient
Markets
• If investors in stock markets have rational
expectations, then the value of the stock market (z)
should follow a random walk:

zt 1  zt   t 1
Et zt 1  zt
• Why? RE says that investors buy and sell based
upon all information publicly available. I.e., the
current stock price already reflects current public
information.
• Implications:
(i) Only unpredictable events cause stock
market fluctuations.
(ii) Market fluctuations cannot be
systematically forecasted. Best to
“follow” the market, cannot
systematically “beat” the market.