Expectations

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Expectations

• Adaptive Expectations
• Rational Expectations
• Modeling Economic Shocks

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• Let zt value of variable z at time t,
• zet1 expectation of zt1 at time t.
• Perfect Foresight
• Adaptive Expectations
• where is the speed of adjustment of
  expectations.
• Problem Errors are systematic and repeated.

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• Rational Expectations The expectation of zt1
  at time t given all currently available
  information. (Statistical conditional expected
  value)

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Notes about Statistical Expectations

• Let X random variable
• f(x) Pr (X x) probability density of X
• The expected value of X is
• (discrete)
• (continuous)

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• 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) Eg(X) Eh(X)
• EXY E(X)E(Y) COV (X,Y)

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• Let X and Y be random variables.
• The conditional expectation of X given Y y is
  given by
• where

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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
• where 0 lt r lt 1 measures the degree of
  persistence.

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• Where e is a random white noise shock with mean
  zero and constant variance.
• r 1 ? permanent shock to z, random walk
• r 0 ? purely temporary shock, no persistence.
• 0 lt r lt 1 ? temporary but persistent
• Examples Macroeconomic data GDP, Money
  Supply, ect.

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Figure 3.2 Percentage Deviations from Trend in
Real GDP from 1947--2003
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Monetary Policy 2004 - 2008
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Numerical Example

• Consider t 20 periods
• There is a one-time shock to et in period 1 where
  e1 10 and et 0 for all other time periods

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• Notice the effect on zt depends on the value of r
  which measures the amount of persistence for the
  shock e.
• r 0 ? purely temporary
• r 0.80 ? temporary but
• persistent

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• r 1 ? permanent

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• Lets use r 0.80 for the shock to z.
• Comparison of adaptive expectations (AE with
  a0.5) and rational expectations (RE) of z.
  Actual value of z is in red, expected values for
  z are in blue.
• Adaptive Expectations Rational Expectations

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• Rational expectations (RE) is the statistical
  forecast of future variables given all current
  information available at time t (Infot)
• Notice since zt is known at time t
• With RE, the errors in expectations are random
  and average to zero
• When r 1, ? Random Walk
• or Martingale

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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
• ?
• 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.

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• 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.