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Heterogeneous Gain Learning and Long Swings in Asset Prices
Blake LeBaron!International Business School!
Brandeis University!www.brandeis.edu/~blebaron!
Microfoundations for Modern Macroeconomics
Center on Capitalism and Society
Columbia University
November, 2010
Long Swings: S&P Price/Dividend Ratio!
Gain Parameters!
ft = ft−1 + gj (xt − ft−1 )
Outline!
Model! Structure! Time series! Wealth distributions!
Learning dynamics! Summary!
Connections!
Adaptive learning! Agent-based markets!
Securities! Equity !
Risky dividend (Weekly U.S. Data)! Annual growth = 2%, std. = 12%! Macro shock!
Fixed supply (1 share)! Risk free!
Infinite supply! Constant interest: 0% per year!
Agents!
Myopic CRRA,!
Consumption!
Budget!
Equity fraction = α t ,i =Et ,i (rt+1 )− rf + (1 / 2)σ t ,i
2
γσ t ,i2 + t ,i
Ct ,i = λWt ,i
PtSt ,i + Bt ,i = (1− λ)(PtSt−1,i + Bt−1,i (1+ rf ))
γ = 3.5
Forecast Families! Adaptive: !
Adaptive expectations (log returns)! Fundamental:!
Log(P/D) regressions! Recursive least squares!
Short AR (autogressive) !! AR(3) on lagged returns! Recursive least squares!
Buy and hold! Low gain return mean and variance estimates!
Forecast Families!
Adaptive
Low Gain
High Gain
Fundamental
Low Gain
High Gain
Short AR
Low Gain
High Gain
Buy and Hold
Low Gain
High Gain
Variance Forecasts (Adaptive)!
σ̂ t , j2 = σ̂ t−1, j
2 + gj ,σ (et , j2 − σ̂ t−1, j
2 )
et , j2 = (rt − ft−1
j )2
gj ,σ = variance gain
Gain Levels 5 Discrete Values!
Low Gain: 50-45 years! All Gain: 50-1 year!
Discrete half-life: [50,18, 7,2.5,1] years
Forecast Selection!Agents
Forecasts
Learning and Wealth!
Active! Recursive parameter estimation! Active forecast rule selection!
Passive! Wealth moves to more successful
strategies!
Low Gain (long memory)!
All Gain!
Price/Dividend Ratios!
Annual Return Regressions R(t+1) on log(P/D(t))!
Series! Coefficient! R-squared!All Gain! -0.41! 0.19!
(0.03)!Low Gain! -0.01! 0.00!
(0.05)!S&P ! -0.13! 0.03!
(0.06)!
Weekly Return Density!
Weekly Return Autocorrelations!
Outline!
Model! Structure! Time series! Wealth distributions!
Learning dynamics! Summary!
Wealth Time Series!
Return Gain Wealth Distributions!
Variance Gain Wealth Distributions!
Utility Surfaces Annual Certainty Equivalent Returns!
Outline!
Model! Structure! Time series! Wealth distributions!
Learning dynamics! Summary!
Fundamental Portfolio Strategies by Gain!
Variance Forecasts!
Variance versus Log(P/D)!
Variance Forecast Comparisons!
High Gain Only Experiment!
Gain half-lives: [1-5] years!
High Gain Only!
Outline!
Model! Structure! Time series! Wealth distributions!
Learning dynamics! Summary!
Empirical Features!
Large and persistent fundamental deviations!
Fat tailed return series! Persistent volatility! Low gain convergence!
Necessary Components!
Multiple time scales! Risk and return! Active and passive learning!
Return Forecasts!
P/D regressions do not time market well! Difficult regressions, unstable learning! Is 50 year half-life long enough?!
Adaptive forecasts control large wealth share!
Short AR behaves predictably!
Variance Forecasts!
Forecast structure! Time series features give short gain
forecasts an edge! Range of gains survive!
Persistence in beliefs about risk!
Future!
Other forecast rules?! Estimation/validation! Fat tails/robust filtering! Macro dynamics!
Extra Figures/tables!
These tables may be useful for discussions!
Summary Statistics!Statistic S&P (1871-2008) All Gain Simulation Return mean (real) 7.9% 8.7% Return std. 0.17 0.25 P/E mean 15.3 15.2 P/E std. 6.0 4.2 P/E correlation (t,t+1) 0.68 0.64
Wealth over Utility Gain!
Forecast Comparisons!
Strategy Fractions!