Heterogeneous Expectations, Adaptive Learning, and Evolutionary Dynamics

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1 Heterogeneous Expectations, Adaptive Learning, and Evolutionary Dynamics Eran A. Guse West Virginia University November 006 (Revised December 008) Abstract This paper presents a linear self-referential macroeconomic model with the possibility of multiple equilibria where agents have the choice of using one of two forecasting models (one of minimum state variable form and the other of sunspot form) to form expectations of current and future prices. Endogenous predictor selection is modeled as an evolutionary game where individuals choose among the forecasting models based on relative performance. Some Nash solutions are not relevant as they are not stable under evolutionary or adaptive learning. Finally, it is shown that the sunspot equilibrium is fragile against temporary shocks to information costs. Key Words: Adaptive Learning; Evolutionary Dynamics; Heterogeneous Expectations; Multiple Equilibria; Rational Expectations. JEL Classi cation: C6, D84, E37 Address correspondence to: West Virginia University, College of Business and Economics, Department of Economics, Morgantown, WV , USA. Phone: + (304) Fax: + (304) eran.guse@mail.wvu.edu. This paper was written when I was a postdoctoral fellow at the Research Unit on Economic Structures and Growth (RUESG), Department of Economics, University of Helsinki. Financial support from the Academy of Finland, Yrjö Jahnsson Foundation, Bank of Finland and Nokia Group is gratefully acknowledged. I would like to thank William Branch, Shankha Chakraborty, George Evans, Van Kolpin, Seppo Honkapohja, Sunny Wong, and two anonymous referees for useful comments and suggestions. All errors are mine.

2 Introduction Since its introduction by Muth (96), Lucas (97, 973), and Sargent (973), the Rational Expectations Hypothesis (REH) has been the dominant paradigm in expectations formation and continues to play a key role in macroeconomic research. Under the REH, agents form expectations using the mathematical expectations operator conditioned upon available information. Economists typically assume that agents in an economic model posses perfect knowledge of the true stochastic process of the variables they wish to forecast. Two objections to the REH come from the bounded rationality literature. First, it may be a very strong assumption that agents know the true stochastic process of the variables they need to forecast. Alternatively, one could allow agents to form expectations from less sophisticated schemes as in Bray and Savin (986), Evans and Honkapohja (00), and Hommes and Sorger (998). Another objection to the REH is that under an environment with heterogeneous expectations, economic outcomes depend upon expectations of all participants. Heterogeneous expectations may alter the stochastic process of aggregate variables. Thus, if agents with rational expectations are to know the form of this stochastic process, then they must be able to observe the expectations of all agents in the economy. The learning literature has also discussed expectation formation schemes with heterogeneity, e.g. in Evans and Honkapohja (997), Evans, Honkapohja, and Marimon (00), Honkapohja and Mitra (006), and Giannitsarou (003). Recently, Adam (005) and Guse (005) have presented models of heterogeneous expectations where each agent uses one of several available forecasting models to form expectations of a stochastic process. One limitation in this literature is the restrictive assumption that the proportion of agents using each forecasting model is exogenously determined. If heterogeneity is exogenously determined, then many agents may be using an obviously ine cient forecasting model due to the stochastic form of the equilibrium. Agents who notice this e ciency disparity would most likely choose to switch to using a more e cient forecasting model. I propose that including a predictor choice in these types of learning models would remove the possibility of agents continually using ine cient forecasting models. Many papers have recently studied including predictor choice as an economic decision in models with expectation formation. In Evans and Ramey (99) agents choose whether or not to use a costly algorithm to update beliefs every period. This is later extended in Evans and Ramey (998) where they allow agents to pay a resource cost for the privilege of using a mechanism that directly calculates expectations. Brock and Hommes (997) use an approach they call the Adaptively Rational Equilibrium There are some rational expectation models with heterogeneous information where agents try to improve upon their information using market data and the actions or expectations of other agents (if they are observable). In the macro learning literature, the assumption that other agents expectations or actions are observable is typically not made. The works of Arthur (99), De Grauwe, DeWachter, and Embrechts (993), and Sethi (996) present numerical results for this type of research.

3 Dynamics (A.R.E.D.) to examine predictor decision. Using a simple cobweb model, they conclude that when the set of predictors is a stable predictor, rational expectations, and an unstable predictor, naive expectations, the dynamics of the system may not settle down to an equilibrium. However, this result may disappear when the set of available predictors increases as shown by Branch (00). Finally, Sethi and Franke (995) consider a model where agents have the choice between using a costless adaptive expectations rule or paying a cost to use rational expectations. Predictor choice is then dictated via an evolutionary process. 3 The predictor choice literature has focused mainly on deterministic models and not stochastic models of learning. A reason for this hole in the literature is that prior to papers like Adam (005) and Guse (005), there were no learning models studied with multiple available forecasting models. 4 This paper uses the Taylor (977) model, a linear self-referential stochastic macroeconomic model with the possibility of multiple equilibria, as a prototype where agents form expectations using adaptive learning and predictor choice. Agents must pay an ad hoc cost of k 0 to use a possibly overspeci ed forecasting model that uses lagged data and (possibly) sunspots (referred to as the AR() forecasting model). Agents also have the opportunity to use a "free", however possibly misspeci ed, forecasting model with a form corresponding to the minimum state variable (MSV) rational expectations equilibrium (REE). Heterogeneity is endogenized in an evolutionary game where agents tend to use the forecasting models that award the highest payo s. The concept of Evolutionary E-stability is introduced as a situation where a Nash equilibrium 5 is stable against a small "mutation" in the population s distribution of forecasting model use and the resulting stochastic equilibrium de ned by the Nash equilibrium (and in the neighborhood of this equilibrium) is also stable under learning. It turns out that under the game with a simple ad hoc cost to using the AR() forecasting model, the only two possible outcomes are: all the agents using the MSV forecasting model and learning the MSV REE or all of the agents using the AR() forecasting model and learning the AR() REE. Finally, I show that the fragility of sunspots as discussed in Evans (989) can be further discussed in my model with heterogeneous expectations. If there is a temporary shock that su ciently increases the cost of acquiring sunspot data, then agents would tend to abandon using the sunspot (AR()) forecasting model forever. Using the technique described in the paper, one can test if a stochastic equilibrium 3 Branch and McGough (008) compare the ARED dynamics of the model in Brock and Hommes (997) with the model under evolutionary dynamics as discussed in Sethi and Franke (995). They nd that the cobweb model with heterogeneous expectations under evolutionary dynamics generates instability as it did under the ARED in Brock and Hommes (997). 4 Branch and Evans (006) examine multiple available (misspeci ed) forecasting models in a cobweb model. They endogenize heterogeneity using a multinomial logit and show that intrinsic heterogeneity will exist in the neoclassical limit if the correctly speci ed forecasting model is not available. This approach di ers from mine as I endogenize heterogeneity using an evolutionary approach with the possible availability of a correctly speci ed forecasting model. 5 A Nash equilibrium is de ned in terms of a vector describing the proportion of agents using each forecasting model and the parameters describing the resulting stochastic equilibrium. This is formally described below. 3

4 is resistant to a small adjustment in the distribution of heterogeneity under evolutionary and adaptive learning. A stochastic equilibrium is commonly considered to be relevant if it is stable under adaptive learning, or expectationally stable (E-stable). However, some models produce multiple equilibria where more than one solution may be E-stable. A natural question would be: If a model has multiple equilibria (of di erent forms) and agents can choose a forecasting model (to form expectations) based on past performance, what solution, or solutions, would be stable under adaptive and evolutionary learning? Furthermore, would the resulting (stable) Nash solution, under the predictor choice model, involve homogeneous or heterogeneous expectations? Stability under evolutionary and adaptive learning can be thought of as a "stronger" necessary condition for the relevance of a stochastic equilibrium (under homogeneous or heterogeneous expectations) than that of stability only under adaptive learning. The Model and E-stability The model I consider is a self-referential linear stochastic macroeconomic model with the possibility of multiple rational expectations equilibria, as presented in Taylor (977) and discussed in the learning literature, 6 e.g. Evans and Honkapohja (00), Heinemann (000), and Guse (005). It is a linear stochastic model with real balance e ects consisting of four parts: aggregate demand, aggregate supply, money demand, and a xed money supply. The reduced form is as follows: 7 y t = + E t y t E t y t+ + vt () where y t is the price level at time t, Et denotes a not necessarily rational expectation at time t, and v t is a linear combination of iid shocks where v t N 0; v. Under homogeneous expectations, there are two possible Rational Expectations Equilibria (REE): the minimum state variable (MSV) solution: and the "bubble" or sunspot solution: 8 y t = + v t () y t = + y t + c v t + d t + v t (3) 6 Although the model is ad hoc and not derived based on microfoundations, it continues to be a workhorse to study learning in a model with the possibility of multiple equilibria. 7 The Sargent and Wallace (975) "ad hoc" model has a similar form. However, as the model is always determinate, there exists only one Nash equilibrium where all agents will use the MSV forecasting model. A more interesting outcome occurs under an indeterminate model as multiple Nash solutions may exist. 8 This solution will exist only if the model is indeterminate, see Blanchard and Kahn (980). 4

5 where t N 0; is an extraneous or sunspot variable and c and d can take on any value. 9 For the majority of the discussion below, I impose c = d = 0 as it greatly simpli es the mathematics. I will refer to an equilibrium of the second form as the AR() solution. Assume that agents have the choice of using one of two forecasting models. One forecasting model will take the MSV REE form and the other will take the AR() REE form. Agents will recursively estimate the parameters of their forecasting model to form expectations of y t and y t+. Suppose that a proportion of agents have a perceived law of motion (PLM) of the following form and the remaining ( ) agents have a PLM of the form 0 P LM : y t = a + v t (4) P LM : y t = a + b y t + v t (5) Under these assumptions, the actual law of motion (ALM) is obtained by plugging the average expectations of y t and y t+ into the reduced form of the model to obtain: y t = ( )a b + ( ) ( b ) b y t + v t. (6) The mapping from the PLM s to each type of agent s "projected" ALM is as follows: T (; ) = T (; ) T (; ) 0 B ( )a b ( )b ( b ) ( )a b ( ) ( b ) b C A (7) where = (a ; a ; b ) 0. If b 6= 0, then the MSV forecasting model is underparameterized and the T-map T (; ) is obtained from a linear projection (commonly referred to as a "projected T-map") under the assumption that (6) is a stationary process. Guse (005) refers to a stochastic process de ned by a xed point of the T-mapping and equation (6) as a "mixed expectations equilibrium" (MEE). 9 Evans and McGough (005) refer to the REE in equation (3) as the general form (GF) sunspot representation. One could also consider the common factor (CF) representation as well (equation () with a sunspot), however, I do not discuss this as it complicates the mathematics without providing new results. 0 When agents include sunspots in their PLM, the results found in the paper are slightly di erent, however, the main conclusion that heterogeneity can not exist in the game discussed below remains. The results with sunspots are available upon request. A projected T-map is a mapping from a possibly misspeci ed PLM to the ALM obtained via a linear projection (the "projected ALM"). See Evans and Honkapohja (00) chapter 3 and Guse (008) for more details on how to deal with misspeci ed PLM s. The equilibria are referred to as "mixed" as they are generated from more than one PLM. Branch and McGough (004) refer to such an equilibrium as a Heterogeneous Expectations Equilibrium (HEE). 5

6 De nition For the above model, a Mixed Expectations Equilibrium: is a stochastic process such that y t = ( )a b + ( ) ( b ) b y t + v t (8) j = T j ; : Similar to a REE, the coe cients in a MEE are such that each PLM is consistent with its associated projected ALM. I will refer to such an equilibrium as j = (a ; a ; b ), which implies the stochastic process in equation (8) given j. The MEE can thus be solved using the method of undetermined coe cients or similar methods. The MSV MEE is the stochastic process de ned by equation (8) and () = (a ; a ; b )0 where a = a =, b = 0. (9) If the model is indeterminate, then the AR() MEE the stochastic process de ned by equation (8) and () = (a ; a ; b )0 where a =, a = ( ), b = ( ). (0) Under the standard homogeneous expectations setup, the Taylor (977) model is determinate when and indeterminate when >. Guse (005) shows that under this heterogeneous expectations setup, the model is determinate when and indeterminate when <. When the model is determinate, under this heterogeneous expectations setup, the projected T-map T (; ) can not be obtained under the AR() MEE since jb j > and thus the MSV MEE is the only solution when. One may wish to examine the local stability properties of each MEE when all agents are learning the parameters of their speci c forecasting model (PLM). The condition for an equilibrium to be locally stable under a simple learning rule, such as ordinary least squares, is known as Expectational Stability, or E-stability. De nition An equilibrium,, is said to be E-stable if the following ordinary di erential equation: 3 d = T (; ) d is stable when evaluated at. It is well known that (Evans and Honkapohja (00), Marcet and Sargent (989)) under fairly general conditions, E-stability governs convergence under least squares and closely related learning rules. 3 denotes "notional" or "arti cal" time. 6

7 Guse (005) presents the E-stability conditions for a xed. It turns out that when one MEE is E-stable, the other is not. The MSV MEE is E-stable if > and the AR() MEE is E-stable if <. Note that when >, E-stability is determined by. In this case, if changes, the two equilibria may exchange stability.. The Evolutionary Game Under an MEE, for any given (0; ), it may be that some agents are using an obviously misspeci ed, or ine cient, forecasting model. Therefore, suppose that agents have the ability to change their forecasting model (PLM) if they believe that the other is doing a better job at predicting future prices. In this case, agents may act like econometricians and test the performance of the available forecasting models. The choice of using di erent forecasting models can be set up in a game in normal form: G = (I; S; ) where I is the set of players, S is its pure-strategy space, and is the pure-strategy payo function. Assume that there exists a continuum of agents that each form price expectations using information from one of two forecasting agencies, 4 i.e. the player set is I = f[0; ] ; A ; A g. Agency (A ) will choose = (a ) and Agency (A ) will choose = (a ; b ) 0. At the beginning of each period, every agent will choose which agency they will use to form expectations of prices and then each agency will use all the information available to choose j. After expectations are made, prices will be realized from equation (6). The set of strategies available to each agent is ST a = fj f; gg and the set of strategies available to each forecasting agency is ST j = f j R q j g where q j = dim ( j ). Note that agents can only use one forecasting model at a time and thus are restricted to using pure strategies. Later, learning about which forecasting model to choose will be dictated by a predictor choice mechanism and learning about which j to use (given the forecasting model choice) will be dictated by an adaptive learning algorithm. The pure strategy pro le will be denoted as a vector representing the proportion of agents using the MSV forecasting model () and each j. The pure strategy pro le, p S (S de ned below) is thus represented as p = ; 0 ; 0 0. As [0; ], R, and R, the pure strategy space is S = [0; ] ; R 3. Next, consider the payo s for playing each strategy. According to Elliot and Timmerman (008), the most common loss function used for forecasting is the mean 4 I am assuming homogeneous expectations in a forecasting model as the pure strategy pro le can not be represented if agents using the same forecasting model have heterogeneous expectations. One could use a genetic algorithm setup to avoid this problem as in Arifovic (994). The bene t of using the evolutionary game approach is that the size of the population is in nite whereas one must restrict the population size using the genetic algorithm due to computational restrictions. 7

8 squared error (MSE). If we assume that forecasting agencies have rational expectations 5, then each MSE can be obtained by supposing a quadratic loss function for forecast errors: h MSE j (p) = E zt;j 0 i j () where zt;j 0 j is the forecast of y t+ obtained using forecasting model j given the pure strategy pro le, p. Since the agents in this model are concerned with predictability, they would like to use the forecasting model (and its parameter values) that gives the smallest MSE. 6 If forecasting agencies do not have rational expectations and must learn the coe cients of their forecasting models (PLM s), then each MSE can not be obtained via equation (). In this case, forecasting agents will update their (estimated) MSE each period in the following manner: MSE j (p t ) = MSE j (p t ) + jt y t z 0 j;t j;t MSE j;t (p t ) () where j;t is the learned coe cient values associated with forecasting model j at time t, z j;t is the matrix of regressors used for forecasting model j, p t is the pure strategy pro le at time t, and jt is a small xed or decreasing gain. It is important to note that when discussing the learning dynamics to an E-stable MSV MEE, the MSV PLM is misspeci ed on the transition path, but the MSE s of the two forecasting models will converge if!. When there is convergence to the AR() MEE, it turns out that MSE ; > MSE ; so the estimated MSE for the MSV forecasting model will typically be larger than the estimated MSE for the AR() forecasting model. Next, assume that there is a xed cost associated with using each forecasting model based on informational and operational costs. As some information may be more costly to obtain and/or use than other information, informational and operational 7 costs may vary between the forecasting models. Finally, using MSE can present a problem when the temporary equilibrium de ned by equation (6) and the pure strategy pro le p may be non-stationary. As estimating a mean (the MSV forecasting model) will do a very poor job tracking a non-stationary process, I will assume that agents strictly prefer to use the AR() forecasting model over the MSV forecasting model when y t is non-stationary. 8 5 This assumption is made to nd the Nash Equilibria of the game and to determine evolutionary stability of these equilibria. 6 One could consider other properties as well, however, predictive performance of the estimated model (not just estimators) is very natural. 7 The operational costs are minimal if one uses a computer to run regressions. However, one could assume that agents must run their regressions without a computer. In this case, expectations would be easier to form using less variables. Alternatively, one could think of these regressions as being used in some sort of thought process. The thought processes using more variables would require more time to calculate expectations. 8 When y t is non-stationary, each MSE is unde ned. Therefore, making this assumption about the payo for non-stationary processes will de ne the pure-strategy payo function over the entire pure strategy space. The goal of this paper is to show the stability properties of the model under adaptive and evolutionary learning. The non-continuous payo function will thus restrict the ALM to be stationary y t+ 8

9 Therefore, given p, if the ALM is stationary, then the payo function for using strategy (j; j ), j : (S; R q j )! R, is the following: j (p) = MSE j (p) K j where K j 0 represents the xed cost for using forecasting model j. Without loss of generality, the cost of using the MSV forecasting model will be normalized to zero, so that K = 0 and K = k 0. 9 If the ALM is non-stationary, then the payo functions are represented as follows: (p) = > (p) =. The pure-strategy payo function of the game, : S! R is thus the following:. The Nash Equilibria (p) = ( (p) ; (p)). The Nash equilibria can be solved using backward induction. Assume that the forecasting agencies have rational expectations and wish to maximize the payo for their clients. They will do this by choosing the j that maximizes j (p) conditional on. For a given, the best replies of and are thus de ned by the xed points of the T-mapping described above by equation (7). I will refer to these forecasting models de ned by the coe cients of an MEE as "optimal" forecasting models. De nition 3 For a given [0; ] and j = (a ; a ; b )0, the optimal MSV forecasting model is E t y t = a and the optimal AR() forecasting model is E t y t = a + b y t As optimal forecasting models minimize MSE j (p), each optimal forecasting model is, therefore, h a best-reply for a given. Note that if the model is determinate (if i), ; the optimal forecasting models are de ned by. If the model is h indeterminate ( 0; ), then there are two possible MEE meaning that there are two possibilities for the optimal forecasting models: the optimal forecasting models may be de ned by or by. If the optimal forecasting models are de ned by, then they are equivalent and agents have "homogeneous" expectations. If the optimal forecasts are de ned by, then some agents are using a misspeci ed forecasting model as Guse (005) shows that the MEE must be stationary to be stable under adaptive learning. 9 This can be done since agents will only consider di erences in payo s and not the di erences in the estimated parameter values. 9

10 (the MSV PLM), however, it is the optimal forecast in its class of forecasting models. Finally, let ^p j = ; j and 0 LP ^p j = ^p j ; ^p j represent the pure-strategy payo function of the game when the forecasting agencies are using optimal forecasting models corresponding to the j th MEE. Next, I wish to consider the best-replies for ST a under optimal forecasting models. First, I de ne the forecasting model best-reply correspondence: De nition 4 A forecasting model best-reply correspondence B j i :! ST a is the mapping from [0; ] to the non-empty ( nite) set B j i () = h ST a : h ^p j g ^p j 8g ST a. For a given, some individuals may not be following their (forecasting model) best reply. Therefore, I also express the combined forecasting model, best-reply set (under j j ), De nition 5 The combined forecasting model, best-reply set (under j j ), BRj [0; ], is the set where all players simultaneously play their forecasting model best-reply under optimal forecasting models: 8 < BR j = : [0; ] : ^p j = ^p j if (0; ) ^p j ^p j if = ^p j ^p j if = 0 Given the set BR j, a Nash equilibrium for the game is de ned as: De nition 6 x; j (x) S is a Nash Equilibrium if j (x) de nes the coe cients of an MEE and x BR j. Under a Nash equilibrium, x will be referred to as a population equilibrium and j (x) will be referred to as a forecasting parameter equilibrium. Since agents are restricted to using pure strategies, a Nash equilibrium where x (0; ) is de ned as a genetically polymorphic population in Maynard Smith (974). In this type of population, there are appropriate proportions of agents using each forecasting model to make up x. To alleviate any confusion, a Nash equilibrium where x (0; ) will be referred to as a heterogeneous Nash equilibrium rather than a mixed Nash equilibrium, ; and a Nash equilibrium where x = (0; ) will be referred to as a homogeneous Nash equilibrium. 0 The superscript, LP, stands for "linear projection" as the coe cients of the MEE are obtained via a linear projection. I will only be concerned with the proportions of agents that use each forecasting model and not with which agents are using each forecasting model. Therefore, I will consider a heterogeneous Nash solution as one solution rather than a continuum of solutions. A mixed Nash equilibrium here would be where each agent uses A with probability and A with probability ( ). A heterogeneous Nash equilibrium is where there is a proportion of agents,, who always use A and the others, ( ), always use A. 0 9 = ;

11 Since there are two possible MEE, MSV and AR(), I solve for the Nash equilibria associated with each MEE. When both forecasting agencies use an optimal forecasting model de ned by the MSV MEE so that = de ned by equation (9), it turns out that 8 [0; ], MSE = MSE = v. The Nash equilibrium under the MSV MEE is given in the following proposition. Proposition Under the MSV MEE, the Nash (population) equilibrium under the game de ned above is =. The proof is given in appendix B. In this paper, the focus is to combine stability analysis of evolutionary learning and adaptive learning. Therefore, when discussing the AR() MEE, I will assume that > so that the AR() MEE may be learnable under some [0; ]. If the forecasting agencies use optimal forecasting models de ned by the AR() MEE given by equation (0) so that =, then there may be several Nash equilibria as shown in the following proposition. Proposition In the game de ned above, under the AR() MEE 3 where >, If k = 0, then there are two population Nash equilibria: = 0 and = =. If 0 < k k, then the h three population equilibria are: = 0, = and = where k = ( ) v, 0; and ; are given in appendix C. If k > k, then, under the AR() MEE, there is one population equilibrium: = ;. The proof is given in appendix C..3 Evolutionary Stability One may study whether a Nash equilibrium is stable when disturbed by a group (or groups) of individuals experimenting with a strategy other than their own under the Nash solution. In evolutionary games, such individuals are commonly referred to as "mutants". Maynard Smith and Price (973) and Maynard Smith (974) introduced the concept of evolutionary stable strategies (or evolutionary stable populations) in player games. The original de nition of a Evolutionary Stable Strategy (ESS) given by Maynard Smith is "An ESS is a strategy such that, if all members of a population adopt it, then no mutant strategy could invade the population under the in uence of natural selection" (Maynard Smith, 974). In this paper, I will show that an evolutionary stable Nash solution (under optimal forecasting models) may become unstable when forecasting agents have to learn their j. First, I extend Maynard Smith s de nition of an evolutionary stable strategy to a game such that agents only choose their optimal forecasting model. 3 For (0:5; ), it turns out that if k [0; k ], then the population Nash equilibrium is = 0. If k > k, then the population Nash equilibrium is. Proposition 7 infers that these Nash solutions will not be stable under evolutionary dynamics and adaptive learning since <.

12 De nition 7 ; j ( ) is an ESS if for every alternative population state 0 [0; ], 0 6=, there exists some " y (0; ) such that ^p j (") ^p j (") 7 0 if 0? (3) holds for all " (0; " y ) where ^p j (") = " 0 + ( ") ; j (" 0 + ( ") ). " y is chosen such that for any 0 6=, f ^p j ("), is at least weakly monotonic within the range of " for all f ST a. If ; j ( ) is evolutionary stable, 4 then (under optimal forecasting models) the payo to individuals using forecasting model where the use has decreased should be larger that the payo to individuals using the forecasting model where the use has increased due to the mutation. Under "natural selection", agents will tend to the forecasting models awarding higher payo s and thus returning to the evolutionary stable population. Therefore, this de nition pertaining to the above game is consistent with the original de nition in Maynard Smith (974). Consider the population Nash equilibrium under the MSV MEE, = 0. Proposition 3 shows that the only Nash equilibrium under the MSV MEE is evolutionary stable. Proposition 3 The Nash (population) equilibrium under the MSV MEE described in the game above, =, is an evolutionary stable strategy. The proof is straightforward. Since both available optimal forecasting models are equivalent, forecasting agent A is equivalent to A and thus all agents are using the MSV forecasting model ( = ). Any "mutation" will result in absolutely no change to since forecasting agents will both still be using the equivalent of the MSV optimal forecasting model. (Figure about Here) Next, consider the Nash equilibria under the AR() MEE. The following proposition gives the evolutionary stable strategies for the AR() MEE. Proposition 4 In the game de ned above, under the AR() MEE where > : If 0 k < k, then 0; (0) and ; ( ) are ESS and ; ( ) is not an ESS. If k k, then ; ( ) is an ESS. The proof is given in appendix D. Figure represents ^p ^p for di erent values of k and a given v and >. The curve in this gure shifts up as the cost of using the AR() forecasting model increases. Each horizontal line in Figure represents ^p ^p = 0 for a given k in a speci ed range. For 0 k < k, inequality (3) holds for only 0; (0) and ; ( ), so these Nash equilibria are ESS s 4 Note that the only potential strategies that can be evolutionary stable are the Nash Equilibria.

13 and ; ( ) is not an ESS. 5 For k > k, as inequality (3) holds for the Nash equilibrium ; ( ), it is an ESS..4 Evolutionary E-stability Finally, I wish to consider whether an evolutionary Nash equilibrium continues to be stable when forecasting agents learn their j. After a mutation, as j is a function of, a mutation to will change j. Therefore, a boundedly rational agent may have to re-learn j after a mutation to (; ). A natural question to ask is "will agents be able to coordinate to the new MEE after a mutation to (; )?" When agents are boundedly rational, for the Nash solution to be stable, it must be that the population,, is stable (under optimal forecasting models) when disturbed by a mutation and every j is stable under learning along the transition from the mutation to the Nash equilibrium. I call this combination of stability evolutionary E-stability. De nition 8 A Nash equilibrium, ; j ( ), is evolutionary E-stable if for every 0 [0; ], 0 6=, ; j ( ) is an evolutionary stable strategy under optimal forecasting models and j (" 0 + ( ") ) is E-stable for all " (0; " y ). Under the above de nition, a Nash Equilibrium is evolutionary E-stable if the Nash solution (under the optimal forecasting model) is locally stable against a mutation to and every MEE in the relevant neighborhood is locally stable under learning. If an MEE is not E-stable, then it will not be possible for agents to coordinate on the MEE when learning. Therefore, an E-unstable MEE should not be considered a relevant equilibrium under a ESS ; j ( ) S. Furthermore, if the MEE for a population state near is not E-stable, then the payo functions along the transition from the mutation to the Nash equilibrium will not converge to the payo functions under the optimal forecasting models. In the example below, I show that not all ESS s in the game described above will have a corresponding E-stable MEE. Evolutionary E-stability can thus be used to remove some evolutionary stable strategies (under optimal forecasting models) as possible xed points in the above endogenous predictor choice model. The most interesting candidates for evolutionary E-stability are the heterogeneous Nash solutions where (0; ). If a heterogeneous Nash solution was evolutionary E-stable, then it would be possible for heterogeneous expectations to exist in the long run under a selection mechanism such as the imitation dynamics (described below). The following proposition states that heterogeneous Nash solutions can never be evolutionary E-stable under the above game. Proposition 5 In the two strategy game above, for any k 0, there does not exist an evolutionary E-stable Nash equilibrium ; j ( ) where (0; ). 5 When k = 0, there are only two population solutions under the AR() MEE, = 0 and = = =. In this case, 0; (0) is the ESS. When k = k, the two Nash equilibria are = = 0 and = where only ; ( ) is an ESS. 3

14 The proof for this proposition is given in appendix E. This proposition rules out the evolutionary stable strategy (under optimal forecasting models): ; ( ) and thus rules out heterogeneous expectations in the long run under learning. 6 The two remaining evolutionary stable strategies are ; () and 0; (0). The resulting stochastic equilibrium under each homogeneous Nash equilibrium can be referred to as the REE as agents are only using one forecasting model. First, consider the Nash equilibrium under the MSV REE where =. Proposition 6 shows that the MSV REE is an evolutionary E-stable strategy. Proposition 6 For the above two strategy game, ; () is Evolutionary E-stable. The proof is given in appendix F. The MSV REE thus cannot be ruled out as a relevant equilibrium as it is always evolutionary E-stable. Next, consider the Nash equilibrium under the AR() REE where = 0. Recall that this Nash equilibrium can exist only if jb j < and k k. Therefore, this solution is not always evolutionary E-stable as shown in Proposition 7. Proposition 7 For the above two strategy game, if 0 k < k and > : 0; (0) is Evolutionary E-stable The proof is given in appendix G. In the indeterminate case, it turns out that for [0; ], b is at its maximum absolute value when = 0. As MSE =, it is b minimized when = 0. For an individual to be indi erent between the two strategies, it must be that (p) = (p). k is the cost for using the AR() strategy such that 0; = 0;. Under a cost of k, any increase in will thus lead to ^p > ^p. If k k, then the only evolutionary E-stable strategy is ; (). Therefore, although either REE in the above model may be E-stable, only the MSV solution is guaranteed to be evolutionary E-stable for any k 0. In this section I showed that in the Taylor (977) model, there may be more evolutionary stable strategies than there are evolutionary E-stable strategies. Furthermore, the only potential evolutionary E-stable strategies are those with homogeneous expectations. It turns out that the only solution that is guaranteed to be evolutionary E-stable is the MSV REE. 6 This result where heterogeneity disappears in the long run is similar in spirit to Giannitsarou (003) and Honkapohja and Mitra (006). Branch and Evans (006) show that heterogeneity can exist in the long run in a similar model, however, all available forecasting models are underparameterized. Future research will ask whether equilibria with (0; ) can arise in other applications of the above approach. Was this result a general feature when PLM s corresponding to all REE are available as forecasting model choices or does this result appear only when one PLM nests all others? 4

15 3 The Imitation Dynamics and Evolutionary E-stability The imitation dynamics will highlight the role of forecasting model selection in real time. 7 Following Weibull (995), a discrete version of the imitation dynamics is the following: 8! e (p t) t+ = t e (p t) + ( t ) e (p t) t (4) Following Kandori et. al (993), assume that agents may not react instantaneously to their environment and they are myopic when they do react. The terms inside the bracket of equation (4) represent the degree of imitation from a payo disparity. More individuals will tend to imitate the best forecasting models as the payo disparities increase. The term of equation (4) represents non-instantaneous reaction to a payo disparity where = 0 corresponds to pure imitation (imitating another agent regardless of payo ) and! corresponds to pure best-reply adaptation at all interior population states (all reviewing agents switch to best replies). Therefore, will determine the speed of convergence to a Nash population equilibrium under real-time dynamics. Next, as Marimon and McGratton (995) show an isomorphism between adaptive learning and evolutionary learning, I will assume that, when not using optimal forecasting models, forecasting agency j will adjust their estimates of the parameter jt using recursive least squares (RLS) as discussed in Evans and Honkapohja (00) and given in appendix H. Forecasting agents will also adjust their MSE s following equation (). I will separately examine the dynamics for the model under di erent ranges of. In one case, is contained in a range where the MSV MEE is E-stable for all while in a second case, is contained in a range where E-stability of either MEE depends on the value of. In each case, I will present the imitation dynamics when agents use optimal forecasting models. Finally, I present the full system of learning in a situation where the economy may su er from a temporary shock to k. 3. Dynamics Under the Region of MSV Strong E-stability When <, the MSV REE is always the E-stable solution even if all the agents were to use the AR() forecasting model. Evans (985) shows that the MSV REE is strongly E-stable in this region as it is also stable against a small dependence of irrelevant variables. 9 In this case, either forecasting model will deliver the same MSE of v at the MEE. By assuming that agents prefer parsimony or a "sunspot" information cost 7 The imitation (replicator) dynamics are commonly used in evolutionary game theory and have been used in Weibull (995), Sethi and Franke (995), and recently by Branch and McGough (008). One could also discuss dynamics under the Multinomial Logit or similar selection mechanisms. 8 There are many alternative ways one may express the imitation dynamics. See Weibull (995, chapter 4) for details. 9 Evans (985) uses a stricter stability condition called iterative E-stability which replaces the E-stability di erential equation with a di erence equation. See Evans and Honkapohja (00, pp. 4-43) for a discussion on iterative E-stability. 5

16 exists, it turns out that ^p > ^p for all. When <, gure shows that the imitation dynamics (under optimal forecasting models) will direct the entire population to using the MSV forecasting model. Under real-time learning, if initial estimates of and are su ciently close to the MEE values, then the estimates of the parameters will converge to the MEE values and the replicator dynamics will direct the population to use the MSV forecasting model. 30 (Figure about Here) 3. Dynamics Under Region of MSV Weak E-stability When >, under homogeneous expectations, the MSV REE is E-stable only when all agents are using the MSV forecasting model. Evans (985) shows that the MSV REE and AR() REE are weakly E-stable in this range as neither equilibrium is stable against a small dependence of irrelevant variables. Under the heterogeneous framework described above, the E-stable MEE is determined by the value of. If > = then there are enough agents using the MSV forecasting model causing the MSV MEE to be E-stable. However, if <, then the AR() MEE is the E-stable equilibrium. When agents use optimal forecasting models, E-stability can be used to choose the relevant (E-stable) equilibrium to consider. When > the MSV MEE is E-stable and the AR() MEE is E-unstable. The MSE s of each forecasting model are the same at the MSV h MEE and thus by assuming k > 0, it will be that ^p > ^p for all i. ; In this range, the imitation dynamics will tend to move the entire population to use the MSV forecasting model. When <, the AR() MEE is E-stable and the MSE s are not the same. The previous section on evolutionary stability under the AR() MEE shows that ^p > ^p if > and ^p < ^p if <. (Figure 3 about Here) (Figure 4 about Here) Figure 3 presents the imitation dynamics under E-stable optimal forecasting models. The black thin line represents t = t, the dark black curves represent the imitation dynamics under the E-stable MEE, and the gray curves represent the imitation dynamics under the E-unstable MEE. If is located to the left of, then the imitation dynamics will direct the entire population to use the AR() forecasting model. However, if is located to the right of, then the imitation dynamics will direct the entire population to use the MSV forecasting model. Although the AR() MEE is E-stable when < <, 30 Real-time simulations in 3. and 3. are not given but are available upon request. Section 3.3 provides a simulation using real-time learning and the imitation dynamics. 6

17 the imitation dynamics are directing the population away from using the costly AR() forecasting model. Once >, the imitation dynamics are working with the E-stable MSV MEE instead of the E-unstable AR() MEE. Therefore, the dynamics "jump" to the MSV dynamics at = and direct the entire population to use the MSV forecasting model. Figure 4 presents convergence properties under optimal forecasting models and the imitation dynamics for some k and initial. The model converges to the AR() REE if k [0; k ) and <. 3;3 3.3 The Fragility of Sunspots Revisited Evans (989) studied several examples of models (including the Taylor (977) model) with multiple equilibria for strong E-stability. He showed that in these models, the sunspot or "bubble" solutions were always either E-unstable or only weakly E-stable. In his conclusion, he suggested that "... if a rational bubble solution is weakly E-stable, it should be possible to observe it under certain experimental conditions. However, it will be fragile in the sense that it will be eliminated by some small changes in experimental design, for example in the information provided about apparently extraneous variables." It turns out that, for the Taylor (977) model, George Evans statement can be expressed in the above context using simulations as shown in Figure 5. (Figure 5 about Here) To simulate "experimental" data, I suppose that, in every time period, 5% of all agents are "mutants" and choose their forecasting model by random and not based on past performance. Forecasting agent A learns y t using P LM, but forecasting agent A learns y t using the following PLM: 33 y t = a + b y t + d t + v t where t N 0; v = 0:5, = 0:065, 0 = 0:, = 0: and k = 0:. is an extraneous or "sunspot" variable. I let = 3, =, Initial estimates of the law of motion are set to their MEE values with d ;0 = 0. The initial and k are located inside the area where agents tend to use the AR() (sunspot) forecasting model as shown in gure 4. In gure 5, agents learn the AR() MEE and stays near = 0. At time t = 3333, I assume that k increases to k = 0:305 as there is a temporary structural shock to the economy where it becomes very expensive to acquire the sunspot data. Under 3 Note that the line from k = 0 to k = k represents for a given k. 3 Under real-time learning, cannot be used to determine the long run value of. However, as the initial is further away from, the long run result under optimal forecasting models shown by Figure 4 is more likely to be the long run result under real-time learning and the imitation dynamics. Real-time examples where the long run result is di erent than what is shown in Figure 4 are available upon request. 33 I assume that c = 0 in the general sunspot form to simplify the learning algorithm. 7

18 this value for k, the AR() REE is not evolutionary E-stable and thus the imitation dynamics will tend to move the population away from using the "bubble" forecasting model. In gure 5, quickly increases from values near zero to values near and the agents begin to learn the MSV MEE values. At time t = 6667, the cost of the sunspot returns to k = 0:, but the agents choose to continue using the MSV forecasting model. The agents will not choose to return to using the AR() forecasting model without a rather large shock to expectations. This reluctance to return to using the (sunspot) AR() forecasting model after a temporary information shock demonstrates further the fragility of sunspots due to heterogeneous expectations Conclusion This paper considers the heterogeneous expectations model discussed in Guse (005) and extends it further by endogenizing heterogeneity using evolutionary game theory. Evolutionary and adaptive learning are combined so agents not only attempt to learn a stochastic process, but they also learn which forecasting model will be "best" to learn the stochastic process. In such a framework, where forecasting agents have rational expectations, a Nash equilibrium of heterogeneous expectations will continue only if this Nash equilibrium is an evolutionary stable equilibrium (stable under an evolutionary process). If not, heterogeneity can only exist through some random mutation process. A necessary condition for a Nash equilibrium to be relevant is that it must be stable under an evolutionary process and every resulting stochastic equilibrium must be stable under an adaptive learning process along the transition from a mutated state back to the Nash equilibrium. This paper shows that, in the above model with an ad hoc cost to using the AR() forecasting model, heterogeneity can not be sustained under the combined evolutionaryadaptive learning dynamics. In these equilibria, either the minimum state variable (MSV) or the sunspot forecasting model is always superior to the other, and the superior forecasting model is used by all agents in the long run. It turns out that if the cost of using the AR() forecasting model is su ciently large, then the parsimonious forecasting model becomes the unambiguously preferred forecasting model. Under real-time dynamics, a temporary shock to the cost of using a sunspot (AR()) forecasting model may lead to all agents using the MSV forecasting model and continue using it even after the end of the temporary shock. Such a result supports Evans (989), Evans and Honkapohja (00), and Adam, Evans, and Honkapohja (006) who suggest that bubble or sunspot solutions are fragile to small changes to information or the economic environment while the MSV REE is robust to such changes. 34 Evans and Honkapohja (00, Ch. 0) and Adam, Evans, and Honkapohja (006) also discuss fragility of sunspots to information assumptions. They nd that sunspot equilibria are more "fragile" as agents have less information when making forecasts of the endogenous variable. This conclusion is similar in spirit to mine. 8

19 Appendix A. MSE for the rst PLM Calculation of the MSE for both of the PLM s MSE = E(y a ) = E (y E (y)) = var (y) = v b Under the MSV MEE, the MSE from the MSV forecasting model is: Under the AR() MEE values, the MSE is the following: Note that the denominator is negative when MSE = v (5) MSE = ( ) v ( ) : (6) > however, this is the condition for non-stationarity of the AR() MEE. Therefore, MSE (under the AR() MEE) is well de ned for all values of such that the AR() MEE is stationary. MSE for the second PLM MSE = E(y a b y t ) = v (7) The mean square error for the will always be v as long as y follows a stationary process. This means that the MSE MSE for all E-stable stationary values of and. This intuitively makes sense because the AR() predictor is always e cient while the MSV predictor is e cient only when b = 0. Appendix B. Proof of Proposition The MSE s under each optimal forecasting model are derived in Appendix A. Under the MSV MEE, it turns out that MSE = MSE. If k = 0, agents are indi erent in which optimal forecasting model they use (^ ^p = ^ ^p for all (0; ]), so the (Nash) population equilibria consists of [0; ], the set of all possible combinations of heterogeneous expectations. However, since the forecasting models under this condition 9

20 are equivalent, all agents are e ectively using the MSV forecasting model so that =. If k > 0, the AR() optimal forecasting model is strictly dominated under the MSV MEE (^ ^p > ^ ^p for all (0; ]) and thus the (Nash) population equilibrium is = where all of the agents choose to use the MSV forecasting model. Appendix C. Proof of Proposition For (0; ) to be a Nash solution, it must be that () = () or MSE + MSE + k = 0 Using the MEE values of the MSE found in appendix A, one obtains for (interior) Nash solutions: ( ) v ( ) + v + k = 0 (8) Equation (8) can be simpli ed to v ( ) v + k ( ) + v + k = 0 Using the quadratic formula, one obtains: = v + k p ( v + k) k v The solutions are thus = ( ) v k v p ( v + k) k = ( ) v k + p ( v + k) k v where h is decreasing in k and is increasing in k. The range of all Nash solutions is 0; as can not be smaller than zero and the payo functions de ned above rule out non-stationary MEE. For k = 0, these solutions are identical where = = = Next, as is decreasing in k, there exists a k such that = 0 so that is not a Nash solution for k > k. Solving = 0 for k gives the following: k = ( ) v h Next, note that ; for k k, so the AR() MEE is a stationary process under for any k 0. 0