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Expectations

Expectations. Adaptive Expectations Rational Expectations Modeling Economic Shocks. Let z t = value of variable z at time t, z e t+1 = expectation of z t+1 at time t. Perfect Foresight : Adaptive Expectations where is the “speed” of adjustment of expectations.

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Expectations

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  1. Expectations Adaptive Expectations Rational Expectations Modeling Economic Shocks

  2. Let zt = value of variable z at time t, zet+1 = expectation of zt+1 at time t. • Perfect Foresight: • Adaptive Expectations where is the “speed” of adjustment of expectations. • Problem: Errors are systematic and repeated.

  3. Rational Expectations: The expectation of zt+1 at time t given all currently available information. (Statistical “conditional” expected value):

  4. 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)

  5. 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)

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

  7. 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 < r < 1 measures the degree of persistence.

  8. 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 < r < 1  temporary but persistent Examples: Macroeconomic data: GDP, Money Supply, ect.

  9. Figure 3.2 Percentage Deviations from Trend in Real GDP from 1947--2003

  10. Monetary Policy: 2004 - 2008

  11. 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:

  12. Notice the effect on ztdepends 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

  13. r = 1  permanent

  14. Let’s use r = 0.80 for the shock to z. • Comparison of adaptive expectations (AE with a=0.5) and rational expectations (RE) of z. Actual value of z is in red, expected values for z are in blue. Adaptive Expectations Rational Expectations

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

  16. 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.

  17. 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.

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