1. Expectations Adaptive Expectations Rational Expectations Modeling Economic Shocks

2. Letz 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. Problem: Errors are systematic and repeated.

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

6. 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)

7. 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)

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

9. 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 <  measures the degree of persistence.

10. Where  is a random “white noise” shock with mean zero:and constant variance.  = 1  permanent shock to z, “random walk”  = 0  purely temporary shock, no persistence. 0 <  < 1  temporary but persistent Examples: Macroeconomic data: GDP, Money Supply, ect.

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

12. Monetary Policy: 2004 - 2008

13. 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:

14. Notice the effect on z t depends on the value of  which measures the amount of persistence for the shock .   purely temporary   temporary but persistent

15.   permanent

16. Let’s use  for the shock to z. Comparison of adaptive expectations (AE with  ) and rational expectations (RE) of z. Actual value of z is in red, expected values for z are in blue. Adaptive ExpectationsRational Expectations

17. Rational expectations (RE) is the statistical forecast of future variables given all current information available at time t (Info t ) Notice since z t is known at time t: With RE, the errors in expectations are random and average to zero: When    “Random Walk” or “Martingale”

18. 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.

19. 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.