Learning in Macroeconomic Models Wouter J. Den Haan London School of Economics � 2011 by Wouter J. Den Haan c August 28, 2011
Intro Simple No Feedback Recursive LS With Feedback Topics Overview • A bit of history of economic thought • How expectations are formed can matter in the long run • Seignorage model • Learning without feedback • Learning with feedback • Adaptive learning • Least-squares learning • Bayesian versus least-squares learning • Decision theoretic foundation of Adam & Marcet
Intro Simple No Feedback Recursive LS With Feedback Topics Overview continued Topics • Learning & PEA • Learning & sun spots
Intro Simple No Feedback Recursive LS With Feedback Topics Why are expectations important? • Most economic problems have intertemporal consequences • = ⇒ future matters • Moreover, future is uncertain • Characteristics/behavior other agents can also be uncertain • = ⇒ expectations can also matter in one-period problems
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought • adaptive expectations: � � � E t [ x t + 1 ] = � x t − � E t − 1 [ x t ] + ω E t − 1 [ x t ] • very popular until the 70s
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought problematic features of adaptive expectations: • agents can be systematically wrong • agents are completely passive: � � • � E t x t + j , j ≥ 1 only changes (at best) when x t changes • = ⇒ Pigou cycles are not possible • = ⇒ model predictions under estimate speed of adjustment (e.g. for disinflation policies)
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought problematic features of adaptive expectations: • adaptive expectations about x t + 1 � = adaptive expecations about ∆ x t + 1 • (e.g. price level versus inflation) • why wouldn’t (some) agents use existing models to form expectations? • expectations matter but still no role for randomness (of future realizations) • so no reason for buffer stock savings • no role for (model) uncertainty either
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought rational expectations became popular because: • agents are no longer passive machines, but forward looking • i.e., agents think through what could be consequences of their own actions and those of others (in particular government) • consistency between model predictions and of agents being described • randomness of future events become important � � c − γ � = ( E t [ c t + 1 ]) − γ • e.g., E t t + 1
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought problematic features of rational expectations • agents have to know complete model • make correct predictions about all possible realizations • on and off the equilibrium path • costs of forming expecations are ignored • how agents get rational expectations is not explained
Intro Simple No Feedback Recursive LS With Feedback Topics History of economic thought problematic features of rational expectations • makes analysis more complex • behavior this period depends on behavior tomorrow for all possible realizations • = ⇒ we have to solve for policy functions , not just simulate the economy
Intro Simple No Feedback Recursive LS With Feedback Topics Expectations matter • Simple example to show that how expectations are formed can matter in the long run • See Adam, Evans, & Honkapohja (2006) for a more elaborate analysis
Intro Simple No Feedback Recursive LS With Feedback Topics Model • Overlapping generations • Agents live for 2 periods • Agents save by holding money • No random shocks
Intro Simple No Feedback Recursive LS With Feedback Topics Model c 1, t , c 2, t ln c 1, t + ln c 2, t max s.t. P t c 2, t ≤ 1 + ( 2 − c 1, t ) P e t + 1 no randomness = ⇒ we can work with expected value of variables instead of expected utility
Intro Simple No Feedback Recursive LS With Feedback Topics Agent’s behavior First-order condition: 1 P t 1 1 1 = = P e π e c 1, t c 2, t c 2, t t + 1 t + 1 Solution for consumption: c 1, t = 1 + π e t + 1 /2 Solution for real money balance (=savings): m t = 2 − c 1, t = 1 − π e t + 1 /2
Intro Simple No Feedback Recursive LS With Feedback Topics Money supply s M t = M
Intro Simple No Feedback Recursive LS With Feedback Topics Equilibrium Equilibrium in period t implies M = M t � � 1 − π e M = P t t + 1 /2 M P t = 1 − π e t + 1 /2
Intro Simple No Feedback Recursive LS With Feedback Topics Equilibrium Combining with equilibrium in period t − 1 gives 1 − π e P t t /2 π t = = 1 − π e P t − 1 t + 1 /2 Thus: π e t & π e t + 1 = ⇒ money demand = ⇒ actual inflation π t
Intro Simple No Feedback Recursive LS With Feedback Topics Rational expectations solution Optimizing behavior & equilibrium: P t = T ( π e t , π e t + 1 ) P t − 1 Rational expectations equilibrium (REE): π e π t = t = ⇒ π t = T ( π t , π t + 1 ) = ⇒ 3 − 2 π t + 1 = π t π t + 1 = R ( π t )
Intro Simple No Feedback Recursive LS With Feedback Topics Multiple steady states • There are two solutions to π = 3 − 2 π = ⇒ there are two steady states • π = 1 (no inflation) and perfect consumption smoothing • π = 2 (high inflation) and no consumption smoothing at all • Initial value for π t not given, but given an initial condition the time path is fully determined
Intro Simple No Feedback Recursive LS With Feedback Topics Rational expectations and stability 2 1.8 1.6 π t +1 1.4 1.2 1 45 o 0.8 1 1.2 1.4 1.6 1.8 2 π t
Intro Simple No Feedback Recursive LS With Feedback Topics Rational expectations and stability π 1 : value in period 1 < π 1 1 : divergence = 1 : economy stays at low-inflation steady state π 1 π 1 > 1 : convergence to high-inflation steady state
Intro Simple No Feedback Recursive LS With Feedback Topics Alternative expecations • Suppose that t + 1 = 1 2 π t − 1 + 1 π e 2 π e t • still the same two steady states, but • π = 1 is stable • π = 2 is not stable
Intro Simple No Feedback Recursive LS With Feedback Topics Adaptive expectations and stability 1.1 1.05 1 0.95 0.9 π t 0.85 Initial conditions: π e 1 = 1 . 5, π e 2 = 1 . 5 0.8 0.75 0.7 0.65 0 5 10 15 time
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback Setup: 1 Agents know the complete model, except they do not know dgp exogenous processes 2 Agents use observations to update beliefs 3 Exogenous processes do not depend on beliefs = ⇒ no feedback from learning to behavior of variable being forecasted
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback & convergence • If agents can learn the dgp of the exogenous processes, then you typically converge to REE • They may not learn the correct dgp if • Agents use limited amount of data • Agents use misspecified time series process
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback - Example • Consider the following asset pricing model P t = E t [ β ( P t + 1 + D t + 1 )] • If → ∞ β t + j D t + j = 0 lim j − then � � ∞ β j D t + j ∑ P t = E t j = 1
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback - Example • Suppose that D t = ρ D t − 1 + ε t , ε t ∼ N ( 0, σ 2 ) (1) • REE: D t P t = 1 − βρ (note that P t could be negative so P t is like a deviation from steady state level)
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback - Example • Suppose that agents do not know value of ρ • Approach here: • If period t belief = � ρ t , then D t P t = 1 − β � ρ t • Agents ignore that their beliefs may change, � � � � = E t D t + j • i.e., � P t + j E t is assumed to equal 1 − β � ρ t + j � � 1 ρ t E t D t + j 1 − β �
Intro Simple No Feedback Recursive LS With Feedback Topics Learning without feedback - Example How to learn about ρ ? • Least squares learning using { D t } T t = 1 & correct dgp • Least squares learning using { D t } T t = 1 & incorrect dgp • Least squares learning using { D t } T T & correct dgp t = T − ¯ • Least squares learning using { D t } T T & incorrect dgp t = T − ¯ • Bayesian updating (also called rational learning) • Lots of other possibilities
Intro Simple No Feedback Recursive LS With Feedback Topics Convergence again • Suppose that the true dgp is given by D t = ρ t D t − 1 + ε t � � ρ t ∈ ρ low , ρ high � ρ high w.p. p ( ρ t ) ρ t + 1 = ρ low w.p. 1 − p ( ρ t ) • Suppose that agents think the true dgp is given by D t = ρ D t − 1 + ε t • = ⇒ Agents will never learn (see homework for importance of sample used to estimate ρ )
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