Assessing others rationality in real time

Assessing others rationality in real time Gaetano GABALLO Ph.D.candidate University of Siena, Italy June 4, 2009 Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 1 / 13

Extending adaptive learning approach in macroeconomics Aim: exploring the potentiality of adaptive learning (Marcet and Sargent, 1989; Evans and Honkapohja 2001) in solving behavioural uncertainty in real time (so overcoming the bounded rationality hypothesis); Key idea: agents can learn about rationality of others exactly as they do for exogenous variables as long as they have some noisy perceptions (predictors) about others actual actions; aetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 2 / 13

Extending adaptive learning approach in macroeconomics Aim: exploring the potentiality of adaptive learning (Marcet and Sargent, 1989; Evans and Honkapohja 2001) in solving behavioural uncertainty in real time (so overcoming the bounded rationality hypothesis); Key idea: agents can learn about rationality of others exactly as they do for exogenous variables as long as they have some noisy perceptions (predictors) about others actual actions; Economic relevancy: towards a model of decentralized and endogenous emergence (or failure) of REE looking for an evolutive solution to the "forecasting the forecast of others" problem. aetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 2 / 13

The paper The paper proposes an adaptive learning analysis of a simple expectations coordination game between two agents having non negligible impact on the aggregate expectation, perfect understanding of the model, but noisy perception of each others simultaneous expectations; Results: a) requirements for a situation like the one entailed by common knowledge assumption (REE) can arise as asymptotic result of a decentralized adaptive learning mechanism without assuming any common knowledge; b) existence of learneable equilibria embodying observational errors and yielding excess volatility; Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 3 / 13

The model The economy is represented by y t = α 0 x t + β λ 1 Et 1 1 y t + (1 λ 1 ) Et 2 1 y t + η t where E i t 1 y t E[y t jω i t 1 ], x t is a vector of exogenous variables and η t is a i.i.d. noise. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 4 / 13

The model The economy is represented by y t = α 0 x t + β λ 1 Et 1 1 y t + (1 λ 1 ) Et 2 1 y t + η t where E i t 1 y t E[y t jω i t 1 ], x t is a vector of exogenous variables and η t is a i.i.d. noise. The unique rational expectation for the model is y t = α 0 x t (1 β) 1. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 4 / 13

The model The economy is represented by y t = α 0 x t + β λ 1 Et 1 1 y t + (1 λ 1 ) Et 2 1 y t + η t where E i t 1 y t E[y t jω i t 1 ], x t is a vector of exogenous variables and η t is a i.i.d. noise. The unique rational expectation for the model is y t = α 0 x t (1 β) 1. Assuming agents are endowed with the following loss function (and they are rational) 2 Λ i = E y t Et i 1 y t then REE emerges i both hold RE, that is E i t 1 y t = y t, i = 1, 2. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 4 / 13

The model The economy is represented by y t = α 0 x t + β λ 1 Et 1 1 y t + (1 λ 1 ) Et 2 1 y t + η t where E i t 1 y t E[y t jω i t 1 ], x t is a vector of exogenous variables and η t is a i.i.d. noise. The unique rational expectation for the model is y t = α 0 x t (1 β) 1. Assuming agents are endowed with the following loss function (and they are rational) 2 Λ i = E y t Et i 1 y t then REE emerges i both hold RE, that is E i t 1 y t = y t, i = 1, 2. RE is a best expectation i both don t doubt the other one holds the same. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 4 / 13

The model The economy is represented by y t = α 0 x t + β λ 1 Et 1 1 y t + (1 λ 1 ) Et 2 1 y t + η t where E i t 1 y t E[y t jω i t 1 ], x t is a vector of exogenous variables and η t is a i.i.d. noise. The unique rational expectation for the model is y t = α 0 x t (1 β) 1. Assuming agents are endowed with the following loss function (and they are rational) 2 Λ i = E y t Et i 1 y t then REE emerges i both hold RE, that is E i t 1 y t = y t, i = 1, 2. RE is a best expectation i both don t doubt the other one holds the same. Therefore for REE it has to be E i t 1 y t 2 Ω j t 1. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 4 / 13

Introducing Behavioral Uncertainty Let s deviate from epistemic assumption required for REE emergence assuming a general speci cation of behavioral uncertainty Et i 1 y t 62 Ω j t 1 : Et i 1 E j t 1 y t E j t 1 y t v i,t 1 Υ (µ i, δ i ) where Υ (µ i, δ i ) is a generic distribution whose mean is µ i and variance δ i. In such a case agents best expectations are Et 1 1 (y t y t ) = bb Et 2 1 y t + v 1,t 1 y t Et 2 1 (y t y t ) = bc Et 1 1 y t + v 2,t 1 y t where bb and bc are respectively solutions of min b Λ 1 and min c Λ 2. In words, (bb, bc) are coe cients for which agents expectations are optimal linear projection given available information Ω i t 1. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 5 / 13

The Dynamic System Agents expectations follow the following exogenous stochastic processes: E 1 t 1 y t = y t + b bbc 1 bbbc v 2,t 1 + b b 1 bbbc v 1,t 1 E 2 t 1 y t = y t + b bbc 1 bbbc v 1,t 1 + bc 1 bbbc v 2,t 1 Actual Law of Motion is 0 y t = y t + β @ b (λ 1 + (1 λ 1 ) bc) v 1,t 1 bbbc 1 + bc λ 1 bb + (1 λ 1 ) 1 bbbc v 2,t 1 1 A + η t Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 6 / 13

The Dynamic System Agents expectations follow the following exogenous stochastic processes: E 1 t 1 y t = y t + b bbc 1 bbbc v 2,t 1 + b b 1 bbbc v 1,t 1 E 2 t 1 y t = y t + b bbc 1 bbbc v 1,t 1 + bc 1 bbbc v 2,t 1 Actual Law of Motion is 0 y t = y t + β @ b (λ 1 + (1 λ 1 ) bc) v 1,t 1 bbbc 1 + bc λ 1 bb + (1 λ 1 ) 1 bbbc v 2,t 1 1 A + η t De nitions Rational Expectations Equilibrium (REE) is a sequence of fy t g such that b b, bc = (0, 0). Behavioral Sunspots Equilibrium (BSE) is a sequence of fy t g such that b b, bc 6= (0, 0). Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 6 / 13

Existence results (analytic proof.) Proposition The unique REE is an equilibrium of the system at hand in all the parameters space. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 7 / 13

Existence results (analytic proof.) Proposition The unique REE is an equilibrium of the system at hand in all the parameters space. Proposition In the symmetric case (equal impact and equal observational errors variance), two BSE exist provided β 2 [1, 2). Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 7 / 13

Existence results (analytic proof.) Proposition The unique REE is an equilibrium of the system at hand in all the parameters space. Proposition In the symmetric case (equal impact and equal observational errors variance), two BSE exist provided β 2 [1, 2). Notice: Here we not deal with economic relevance of BSE, the issue is referred to a di erent paper. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 7 / 13

T-map Figure: T-map for the case of two simmetrical agents (λ = 0.5 and ε 1 = ε 2 ) and no correlation between observational errors ρ v = 0. Curves are obtained for di erent values of β, respectively: 0.8 normal, 1 dotted, 1.08 dashed, 1.4 dotted-dashed. "BSE" denotes the high behavioral Sunspot Equilibrium and "bse" the low one. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 8 / 13

Existence and Learneability of REE The concept of learneability refers to the nature, stable or unstable of the learning dynamics under a recursive least square algorithm around the equilibria computed above. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 9 / 13

Existence and Learneability of REE The concept of learneability refers to the nature, stable or unstable of the learning dynamics under a recursive least square algorithm around the equilibria computed above. Let s suppose institutional forecasters estimate respectively bb and bc in each point in time with a standard ordinary least square regression. They form expectations according to the rule E 1 t 1 (y t y t ) = b t 1 E 2 t 1 y t + v 1,t 1 y t E 2 t 1 (y t y t ) = c t 1 E 1 t 1 y t + v 2,t 1 y t De nition An equilibrium (bb, bc) is locally learnable under recursive least square (RLS) algorithm if and only if there exist some neighborhood =(bb, bc) of (bb, bc) such that for each initial condition (b 0, c 0 ) 2 =(bb, bc) the estimates converge almost surely a.s. a.s. to the equilibrium, that is lim t! b t 1 = bb and lim t! c t 1 = bc. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 9 / 13

Learneability results (analytic proof.) Proposition REE (0, 0) is learneable i β 1 p 1 4λ(1 λ)(1 ρ 2 v ) 2λ(1 λ)(1 ρ 2 v ) Figure: The grey area denotes REE learneability region in the whole parameter space x λ (1 λ) 1 ρ 2 v. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 10 / 13

Learneability results (analytic proof.) Proposition REE (0, 0) is learneable i β 1 p 1 4λ(1 λ)(1 ρ 2 v ) 2λ(1 λ)(1 ρ 2 v ) Figure: The grey area denotes REE learneability region in the whole parameter space x λ (1 λ) 1 ρ 2 v. Proposition In the symmetric case, whenever BSE exist, the high one is always learneable whereas the low one it is never. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 10 / 13

Real time simulation b(t), c(t) b(t), c(t) y(t), E1(t), E2(t) y(t), E1(t), E2(t) 1.78 0 5 0 a) RLS convergence to REE 0 50 100 150 200 250 300 5 0 50 100 150 200 250 300 1.78 0 b) RLS convergence to BSE 0 50 100 150 200 250 300 5 0 5 0 50 100 150 200 250 300 t Figure: Two cases of RLS convergence up to 300 periods with λ 1 = 0.5, x t = 0, β = 1.08. Initial conditions are set equal for both at REE value b 0 = c 0 = 0. The two are runned for di erent series of v i,t 1 and η t i.i.d. centred normally distributed schoks with unitary variance. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 11 / 13

Conclusions The paper shows how adaptive learning can work in a truly behavioral uncertainty problem; Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 12 / 13

Conclusions The paper shows how adaptive learning can work in a truly behavioral uncertainty problem; Interactive learning generates very robust convergence to REE in real time and without assuming any common knowledge property; Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 12 / 13

Conclusions The paper shows how adaptive learning can work in a truly behavioral uncertainty problem; Interactive learning generates very robust convergence to REE in real time and without assuming any common knowledge property; The paper characterizes a new type of non rational equilibria tagged behavioural sunspots equilibria (BSE) entailing excess volatility regimes; Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 12 / 13

Conclusions The paper shows how adaptive learning can work in a truly behavioral uncertainty problem; Interactive learning generates very robust convergence to REE in real time and without assuming any common knowledge property; The paper characterizes a new type of non rational equilibria tagged behavioural sunspots equilibria (BSE) entailing excess volatility regimes; The latter arises provided β 2 [1, 2) and takes shape as a coordination failure. Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 12 / 13

The end Thanks for your attention Gaetano GABALLO Ph.D.candidate University of Siena, Italy Assessing () others rationality in real time June 4, 2009 13 / 13