Behavioral learning equilibria for the New Keynesian model


 Πηνελόπεια Παυλόπουλος
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1 Behavioral learning equilibria for the New Keynesian model Cars Hommes a, Mei Zhu b a CeNDEF, School of Economics, University of Amsterdam and Tinbergen Institute, Netherlands b Institute for Advanced Research & School of Economics, Shanghai University of Finance and Economics, and the Key Laboratory of Mathematical Economics(SUFE), Ministry of Education, Shanghai , China August, 205 Abstract We generalize the concept of behavioral learning equilibrium (BLE) to a general high dimensional linear system and apply it to the standard New Keynesian model. Boundedly rational agents learn to use a simple AR() forecasting rule with parameters consistent with the observed mean and autocorrelation of past data of inflation and output gap. Boundedly rational agents do not fully recognizing the more complex structure of the economy, but learn to use an optimal simple AR() rule. We find that BLE exists, under general stationarity conditions, typically with near unit root autocorrelation parameters. BLE thus exhibit persistence and volatility amplification with much higher persistence in inflation and output gap against a background of relatively small changes in fundamental factors. Furthermore, the ACFs of output gap and inflation at the BLE are similar to those in empirical work. In a boundedly rational world, coordination of individual expectations on an aggregate outcome described by our simple, parsimonious BLE seems more likely. We also consider monetary policy under BLE for different Taylor interest rate rules and study whether inflation and/or ouput gap targeting can stabilize BLE. addresses:
2 Keywords: Bounded rationality; Behavioral learning equilibriium; Adaptive learning; New Keynesian macromodel; Inflation persistence 2
3 Introduction Rational Expectations Equilibrium (REE) requires that economic agents subjective probability distributions coincide with the objective distribution that is determined, in part, by their subjective beliefs. There is a vast literature that studies the drawbacks of REE. Some of these drawbacks include the fact that REE requires an unrealistic degree of computational power and perfect information on the part of agents. Alternatively, the adaptive learning literature (see, e.g., Evans and Honkapohja (200, 20) and Bullard (2006) for extensive surveys and references) replaces rational expectations with beliefs that come from an econometric forecasting model with parameters updated using observed time series. A large part of this literature involves studying under which conditions learning will converge to the rational expectations equilibrium. When the perceived law of motion (PLM) of agents is correctly specified, convergence of adaptive learning to an REE can occur. However, in general the PLM will be misspecified. As shown in White (994), an economic model or a probability model is only a more or less crude approximation to whatever might be the true relationships among the observed data and consequently it is necessary to view economic and/or probability models as misspecified to some greater or lesser degree. Whenever agents have misspecified PLMs a reasonable learning process may settle down to some sort of misspecification equilibrium. In the existing literature, different types of misspecification equilibria have been proposed, such as a Restricted Perceptions Equilibrium (RPE) where the forecasting model is underparameterized (Sargent, 99; Evans and Honkapohja, 200) and a Stochastic Consistent Expectations Equilibrium (SCEE) (Hommes and Sorger, 998; Hommes et al., 203), where agents learn the optimal parameters of a simple, parsimonious AR() rule. A SCEE is a very natural misspecification equilibrium, where agents in the economy do not know the actual law of motion or even recognize all of the explanatory variables, but prefer a parsimonious forecasting model. The economy is too complex to fully understand and therefore, as a first order approximation, agents forecast the state of the economy by simple autoregressive models. In the simplest model applying this idea, agents run an univariate AR() regression to generate outofsample forecasts of the state of the economy. In Hommes and Zhu (204) we provide the firstorder SCEE with an intuitive behavioral interpretation and therefore, we refer to them as a Behavioral Learn Branch (2006) provides a stimulating survey discussing the connection between these types of misspecification equilibria. 3
4 ing Equilibrium (BLE). Although it is possible for some agents to use more sophisticated models, one may argue that these practices are neither straightforward nor widespread. A simple BLE seems more likely as a description of aggregate behavior, because a large population of individual agents may coordinate their expectations more easily to learn a simple, parsimonious behavioral equilibrium. In Hommes and Zhu (204), we formalize the concept of BLE in the simplest class of models one can think of: a onedimensional linear stochastic model driven by an exogenous linear stochastic AR() process. Agents do not recognize, however, that the economy is driven by an exogenous AR() process y t, but simply forecast the state of the economy x t using an univariate AR() rule. The parameters of the AR() forecasting rule are not free, but fixed (or learned over time) according to the observed sample average and firstorder sample autocorrelation. Within this simple, but general, class of models Hommes and Zhu (204)we are able to fully characterize the existence and multiplicity of BLE. They also study the stability of BLE under a simple adaptive learning scheme Sample Autocorrelation Leaning (SAClearning). and provide simple and intuitive stability conditions 2. SAClearning also has a simple behavioral interpretation, with agents guestimating the sample average and firstorder persistence (i.e. autocorrelation) from observed time series. Although the class of models is simple, Hommes and Zhu (204) consider two important standard applications: an asset pricing model driven by autocorrelated dividends and the New Keynesian Philips curve with inflation driven by autocorrelated output gap (or marginal costs). However, as shown in Fuhrer (2009), although the skeleton model of the New Keynesian Philips curve with AR() driving variable provides important insights into some of the structural sources of inflation persistence, the model leaves implicit the determination of real output and the role of monetary policy in influencing output and inflation. In theory, both the systematic component of monetary policy and the nature of the transmission of policy through the real side can have significant effects on the dynamic properties of inflation. Hence in this paper we extend the BLE to a standard twodimensional dynamic stochastic general equilibrium (DSGE) modelthe New Keynesian model and study the role of monetary policy. Agents are boundedly rational. They do not know the exact form of the actual law of motion (2.) because of cognitive limitations or simply prefer a parsimonious prediction 2 In earlier work, learning of SCEE has been studied only by numerical simulations, see, for example, Hommes and Rosser (200) and Tuinstra (2003). 4
5 rule. They only use a simple, parsimonious forecasting model where agents perceived law of motion (PLM) is a simple univariate AR() process for each variable to be forecasted. As shown in Enders(200, p.8485), coefficient uncertainty increases as the model becomes more complex, and hence it could be that an estimated AR() model forecasts an real ARMA(2,) process better than an estimated ARMA(2,) model. Numerous empirical studies also show that overly parsimonious models with little parameters uncertainty can provide better forecasts than models consistent with the actual datagenerating complex process (e.g. Nelson, 972; Stock and Watson, 2007; Clark and West, 2007). Chung and Xiao (204) also indicate by simulations the simple AR() model is more likely to prevail in reality because of limited information restriction when they study predictions in a two dimensional New Keynesian model. In view of these, similar to Hommes and Zhu (204), agents are assumed to use a simple AR() model for each of the variables to make prediction. The same two consistency requirements are imposed upon BLE to pin down to two parameters of the forecasting model: for each endogenous variable observed sample average and firstorder sample autocorrelations match the corresponding parameters of the forecasting rule. In this paper we analytically prove the existence of BLE in highdimensional linear systems, in particular the New Keynesian model and provide stability conditions of BLE under SAC learning. One of our main findings is the persistence and volatility amplification along BLE, that is, BLE exhibit much higher persistence in inflation and output gap against a background of relatively small changes in fundamental factors. In particular, persistence and volatility are much higher under BLE than under REE. We also study the role of monetary policy under BLE and SAClearning and study how inflation and output gap targeting can stabilize BLE. Related literature Our behavioral learning equilibrium concept is closely related to Exuberance Equilibria (EE) in Bullard et al. (2008), where agents perceived law of motion is misspecified. However, because of difficulty of computation, in Bullard et al. (2008) there are only numerical results on the exuberance equilibria while here we can theoretically show the existence of BLE and stability under learning, especially in the same typical New Keynesian model. Another closely related misspecification equilibrium is Limited Information Learning Equilibrium (LILE) defined in Chuang and Xiao (204), which is defined by the leastsquares projection of variables on the past information of the actual law of motion 5
6 equal to that in the perceived law of motion. Different from the LILE, our general Behavioral Learning Equilibrium is defined by all the means and firstorder autocorrelations of each variable of the actual law of motion consistent with those corresponding in the perceived law of motion. As shown above, the BLE makes coordination of individual expectations on such an aggregate outcome more likely. We further study the effects of monetary policy under the more plausible BLE. The concept of natural expectations in Fuster et al. (200) and Fuster et al. (20, 202) is a similar misspecification concept to our BLE, where agents use simple, misspecified models, e.g., linear autoregressive models, for their perceived law of motion. Natural expectations, however, do not pin down the parameters of the forecasting model through consistency requirements as for a restricted perceptions equilibrium nor do they allow the agents to learn an optimal misspecified model through empirical observations. The paper is organized as follows. Section 2 introduces the main concepts of BLE and Section 3 generalizes to a general ndimensional linear system. Section 4 introduces the New Keynesian model and studies BLE and their stability for different specifications of the Taylor rule. Finally, Section 5 concludes. 2 Main concepts In Hommes and Zhu (204), we discuss BLE in the simplest setting, a onedimensional linear stochastic model driven by an exogenous linear stochastic AR() process. In this paper we generalize the BLE to ndimensional (linear) stochastic models driven by exogenous linear stochastic AR() processes of multiple shocks. Suppose that the law of motion of an economic system is given by the stochastic difference equation x t = F(x e t+, u t, v t ), (2.) where x t is a ndimensional vector denoted by [x,x 2,,x n ] describing the state of the system (e.g. output gap and inflation) at date t and x e t+ is the expected value of x at date t +. This denotation highlights that expectations may not be rational. Here F is a continuous ndimensional vector function, v t is an i.i.d. ndimensional vector noise process with mean zero and finite absolute moments 3, where the variance is denoted by 3 The condition on finite absolute moments is required to obtain convergence results under SAClearning. 6
7 Σ v, and u t is an exogenous ndimensional shock, assumed to follow a stochastic AR() process u t =a+ρu t +ε t, (2.2) where ε t is another uncorrelated ndimensional i.i.d. noise process with mean zero and finite absolute moments, with variance Σ ε, and uncorrelated with v t and ρ is a n n diagonal matrix with n elements denoted by ρ i [0,),i =,2,,n. The mean of the stationary process u t is u = a(i ρ), the variance is Σ u = Σ ε (I ρ 2 ) and the kthorder correlation coefficient of u t is ρ k, where I is a n n identity matrix. See for example, Hamilton (994). Agents are boundedly rational and do not know the exact form of the actual law of motion (2.). They only use a simple, parsimonious forecasting model where agents perceived law of motion (PLM) is a simple univariate AR() process for each variable to be forecasted. As shown in Enders (200, p.8485), coefficient uncertainty increases as the model becomes more complex, and hence it could be that an estimated AR() model forecasts an real ARMA(2,) process better than an estimated ARMA(2,) model. Numerous empirical studies also show that overly parsimonious models with little parameters uncertainty can provide better forecasts than models consistent with the actual datagenerating complex process (e.g. Nelson, 972; Stock and Watson, 2007; Clark and West, 2007).Thus agents perceived law of motion (PLM) is assumed to be the simplest VAR model with minimum parameters, i.e. a VAR() process x t =α+β(x t α)+δ t, (2.3) where α is a vector denoted by [α,α 2,,α n ], β is a diagonal matrix 4 denoted by β β 2 0 with β i (,) and δ t is a white noise process; α is the unconditional mean of x t, β i is the firstorder correlation coefficient of variable x i. Given 0 0 β n the perceived law of motion (2.3), the 2period ahead forecasting rule for x t+ that minimizes the meansquared forecasting error is x e t+ =α+β2 2 (x t α). (2.4) 4 Chung and Xiao (204) also indicate by simulations the simple AR() model is more likely to prevail in reality because of limited information restriction when they study predictions in a two dimensional New Keynesian model. 7
8 Combining the expectations (2.4) and the law of motion of the economy (2.), we obtain the implied actual law of motion (ALM) with u t an AR() process as in (2.2). x t =F(α+β 2 (x t α), u t, v t ), (2.5) Behavioral Learning Equilibrium (BLE) Similar to Hommes and Zhu (204), the concept of BLE is defined as follows. Definition 2. A vector (µ,α,β), where µ is a probability measure and α and β are vectors with β i (,) (i =,2,,n), is called a behavioral learning equilibrium (BLE) if the following three conditions are satisfied: S The probability measure µ is a nondegenerate invariant measure for the stochastic difference equation (2.5); S2 The stationary stochastic process defined by (2.5) with the invariant measure µ has unconditional mean α, that is, the mean of x i is α i, (i =,2,,n). S3 Each element x i for the stationary stochastic process of x defined by (2.5) with the invariant measure µ has unconditional firstorder autocorrelation coefficient β i, (i =,2,,n). That is to say, a BLE is characterized by two natural observable consistency requirements: the unconditional mean and the unconditional firstorder autocorrelation coefficient generated by the actual (unknown) stochastic process (2.5) coincide with the corresponding statistics for the perceived linear AR() process (2.3), as given by the parameters α and β. This means that in a BLE agents correctly perceive the two simplest and most important statistics: the mean and firstorder autocorrelation (i.e., persistence) of the state of the economy, without fully understanding its structure and recognizing all explanatory variables and cross correlations. Sample autocorrelation learning In the above definition of BLE, agents beliefs are described by the linear forecasting rule(2.4)withfixedparametersα k andβ k. However, theparametersα k andβ k areusually unknown. In the adaptive learning literature, it is common to assume that agents behave 8
9 like econometricians using time series observations to estimate the parameters as additional observations become available. Following Hommes and Sorger (998), we assume that agents use sample autocorrelation learning (SAClearning) to learn the parameters α i and β i, i =,2,,n. That is, for any finite set of observations x i,0,x i,,,x i,t, the sample average is given by α i,t = t+ and the firstorder sample autocorrelation coefficient is given by t x i,k, (2.6) k=0 β i,t = t k=0 (x i,k α i,t )(x i,k+ α i,t ) t k=0 (x i,k α i,t ) 2. (2.7) Hence α i,t and β i,t are updated over time as new information arrives. It is easy to check that, independently of the choice of the initial values (x i,0,α i,0,β i,0 ), it always holds that β i, = 2, and that the firstorder sample autocorrelation β i,t [,] for all t. As shown in Hommes and Zhu (204), define R i,t = t+ t (x i,k α i,t ) 2, k=0 then the SAClearning is equivalent to the following recursive dynamical system α i,t = α i,t + t+ (x i,t α i,t ), β i,t = β i,t + [ t+ R i,t (x i,t α i,t ) ( x i,t + x i,0 t+ t2 +3t+ ) α (t+) 2 i,t (t+) 2x i,t t ] t+ β i,t (x i,t α i,t ) 2, R i,t = R i,t + [ t ] t+ t+ (x i,t α i,t ) 2 R i,t. (2.8) The actual law of motion under SAClearning is therefore given by with α i,t,β i,t as in (2.8) and u t as in (2.2). x t =F(α t +βt (x 2 t α t ), u t, v t ), (2.9) In Hommes and Zhu (204), f is a onedimensional function. In this paper F may be a general ndimensional vector function. We focus on the simple case where F is linear, so that BLE can be computed analytically. 9
10 3 Main results in a ndimensional linear framework Assume that a reduced form model is a ndimensional linear stochastic process x t, driven by an exogenous AR() process u t. More precisely, the actual law of motion of the economy is given by 5 x t = F(x e t+, u t, v t ) = b 0 +b x e t+ +b 2u t +v t, (3.) u t = a+ρu t +ε t, (3.2) where ρ is a diagonal matrix as before, and b i (i =,2) is a matrix. 3. Rational expectations equilibrium Under the assumption that agents are rational, assume the perceived law of motion (PLM) corresponding to the minimum state variable REE of the model x t = ξ +ηu t +v t. (3.3) Assuming that shocks u t are observable when forecasting x t+ the onestep ahead forecast is E t x t+ = ξ +ηa+ηρu t, (3.4) and the corresponding actual law of motion is x t = b 0 +b (ξ +ηa+ηρu t )+b 2 u t +v t. (3.5) The rational expectations equilibrium (REE) is the fixed point of ξ = b 0 +b ξ +b ηa, (3.6) η = b ηρ+b 2. (3.7) A straightforward computation (see Appendix A) shows that the mean of the REE x satisfies x = (I b ) [b 0 +b 2 (I ρ) a]. (3.8) 5 In fact, as shown in Section 4 for the alternative case with lagged Taylor rule, our results on BLE still hold for the more general model including the term of lagged x t in the right side of the equation (3.). 0
11 In the special case of ρ with ρ i = ρ j = ρ for i,j =,2,,n and if (I b ) exists, the rational expectation equilibrium x t satisfies 6 x t = (I b ) b 0 +(I b ) b (I ρb ) b 2 a+(i ρb ) b 2 u t +v t. (3.9) Thus its unconditional mean is x = E(x t ) = ( ρ) (I b ) [b 0 ( ρ)+b 2 a]. (3.0) Its covariance is Σ x = E(x t x )(x t x ) = ( ρ 2 ) (I ρb ) b 2 Σ ε [(I ρb ) b 2 ] +ΣΣ v. (3.) Σ v Furthermore, the firstorder autocovariance is, Σ x x = E(x t x )(x t x ) = ρ( ρ 2 ) (I ρb ) b 2 Σ ε [(I ρb ) b 2 ].(3.2) Therefore the firstorder autocorrelation of the ielement x i of x is the ith diagonal element of matrix Σ x x divided by the corresponding ith diagonal element of matrix Σ x. Note that in the special case Σ v = 0, the firstorder autocorrelation of the ielement x i of x is equal to ρ, that is, when there is no exogenous noise v t in (3.) and ρ i = ρ j, the persistence of the ith variable x i in the REE coincides exactly with the persistence of the exogenous driving force u i,t. 3.2 Existence of BLE Now assume that agents are boundedly rational and do not believe or do not recognize that the economy is driven by an exogenous AR() process u t, but use a simple univariate linear rule to forecast the state x t of the economy. Given that agents perceived law of motion is an AR() process (2.3), the actual law of motion becomes x t = b 0 +b [α+β 2 (x t α)]+b 2 u t +v t, (3.3) with u t given in (3.2). The mean of x t in (3.3), denoted by x, is computed as x = (I b β 2 ) [b 0 +b α b β 2 α+b 2 (I ρ) a]. (3.4) 6 Note ρ is a matrix while ρ is a scalar number.
12 Imposing the first consistency requirement of a BLE on the mean, i.e. x = α, and solving for α yields α = (I b ) [b 0 +b 2 (I ρ) a]. (3.5) Comparing with (3.8), we conclude that in a BLE the unconditional mean α coincides with the REE mean. That is to say, in a BLE the state of the economy x t fluctuates on average around its RE fundamental value x. Consider the second consistency requirement of a BLE on the firstorder autocorrelation coefficient matrix β of the PLM. Assume that the ALM (3.3) is a stationary process. We denote the firstorder autocorrelation coefficient Corr(x i,t,x i,t ) of the ith element of the ALM(3.3)by G i (β,β 2,,β n ) 7. FromtheALM (3.3)andthe theory ofstationary linear time series, G i (β,β 2,,β n ) is an analytic function with respect to β i and other model parameters 8. The second consistency requirement of BLE concerning the firstorder autocorrelation coefficient β yields G(β) = β. (3.6) G(β) maps [,] n into [,] n. For each G i, given (β,,β i,β i+,,β n ), G i (β i ) is a continuous function from [,] to [,]. Define H i (β i ) := G i (β i ) β i. Because H i ( ) = G i ( ) ( ) 0, H i () = G i () 0, (3.7) given(β,,β i,β i+,,β n ), thereexists β i(β,,β i,β i+,,β n ) [,]such that G i (βi ) = β i. We conclude: Proposition In the case that 0 < ρ i < and all the eigenvalues of b lie inside the unit circle, there exists at least one behavioral learning equilibrium (BLE) (α,β ) for the economic system (3.3) with α = (I b ) [b 0 +b 2 (I ρ) a] = x and G(β ) = β. 7 G i just depends only on β, not α. 8 For example, refer to the expression (3.9) in Hommes and Zhu (204) for the special case n =. In Section 4 we consider the twodimensional New Keynesian model and will compute the complicated expressionsofg (β,β 2 ) andg 2 (β,β 2 ). Followingthe similarideas, inthegeneralndimensionalsystem, it can be to calculate G i but the expressions are very complicated. 2
13 3.3 Stability under SAClearning In this subsection we study the stability of BLE under SAClearning. The ALM of the economy under SAClearning is given by x t = b 0 +b [α t +β 2 t (x t α t )]+b 2 u t +v t, u t = a+ρu t +ε t. (3.8) with α t,β t updated based upon realized sample average and sample autocorrelation as in (2.8). Appendix B shows that the Estability principle applies and that the stability under SAClearning is determined by the associated ordinary differential equation (ODE) 9 dα dτ = x(α,β) α = (I b β 2 ) [b 0 +b α b β 2 α+b 2 (I ρ) a] α, (3.9) dβ dτ = G(β) β, where x(α,β) is the mean given by (3.4) and G(β) the firstorder autocorrelation. A BLE (α,β ) corresponds to a fixed point of the ODE (3.9). Moreover, a BLE (α,β ) is locally stable under SAClearning, if it is a stable fixed point of the ODE (3.9). A straightforward computation shows that the eigenvalues of the Jacobian matrix of (3.9) are given by (I b (β ) 2 ) (b I) (the coefficient of α in the first ODE) and DG i (β ). Since, by assumption, the first eigenvalue is always less than 0. Hence, the local stability of a BLE (α,β ) under SAClearning only depends on the slope DG i (β ): Proposition 2 A BLE (α,β ) is locally stable under SAClearning if for each i, DG i (β ) <, where G i (β) is the firstorder autocorrelation of variable x i. Proof. See Appendix B. Recall from Subsection 3.2 that G i ( ) and G i (), so that at least one BLE exists. If the BLE is unique, then by continuity of G i and DF i it must be that at the unique intersection point DF i (βi ) and, according to proposition 2, the unique BLE is (locally) stable under SAClearning. 0 Numerical simulations suggest that a unique BLE is even globally stable under SAClearning. In the next section, as an application we study BLE in a twodimensional New Keynesian model. 9 See Evans and Honkapohja (200) for discussion and a mathematical treatment of Estability. 0 The only exception is a hairline case where the graph of G i is tangent to the diagonal at its unique fixed point βi and G i (β i ) =. In such a hairline case, the BLE may also be locally stable under SAClearning, but stability does not follow directly from the Estability principle. 3
14 4 Application: a New Keynesian model 4. A baseline model Now we develop our results within the framework of a standard New Keynesian model along the lines of Woodford (2003). Consider a simple version given by y t = yt+ e ϕ(i t πt+ e )+u y,t, π t = λπt+ e +γy t +u π,t, (4.) where y t is the aggregate output gap, π t is the inflation rate, y e t+ and π e t+ are expected output gap and expected inflation. This denotation highlights that expectations may not be rational. Following Bullard and Mitra (2002) and Bullard et al. (2008) we will study the NKmodel (4.) with adaptive learning. The terms u y,t,u π,t are stochastic shocks and are assumed to follow AR() processes u y,t = ρ u y,t +ε,t, (4.2) u π,t = ρ 2 u π,t +ε 2,t, (4.3) where ρ i [0,) and ε i,t (i =,2) are two uncorrelated i.i.d. stochastic processes with zero mean and finite absolute moments with corresponding variances σ 2 i. Thefirstequationin(4.)isanIScurvethatdescribesthedemandsideoftheeconomy. In an economy of rational or boundedly rational agents, it is a linear approximation to a representative agent s Euler equation. The parameter ϕ > 0 is related to the elasticity of intertemporal substitution in consumption of a representative household. The second equation in(4.) is the New Keynesian Phillips curve which describes the aggregate supply relation. This is obtained by averaging each firm s pricing decisions, and the parameter γ is related to the degree of price stickiness in the economy and the parameter λ [0,) is the discount factor of a representative household. We supplement the equations in (4.) with a policy rule, which represents the behavior of the monetary authority in setting the interest rate. In this paper we assume a Taylortype policy rule setting the nominal interest rate i t = φ π π t +φ y y t, (4.4) where i t is the deviation of the nominal interest rate from the value that is consistent with the inflation at target and output at potential and the parameters φ π,φ y are assumed to Here [y e t+,π e t+] = x e t+. 4
15 positive with φ π > measuring the response of i t to the deviation of inflation and output from long run steady states. Substituting the Taylortype policy rule in equation (4.4) into the equations in (4.) and writing the model in matrix form gives x t = Bx e t+ +Cu t, u t = ρu t +ε t, wherex t = [y t,π t ],u t = [u y,t,u π,t ],ε t = [ε,t,ε 2,t ],B = +γϕφ π+ϕφ y (4.5) ϕ( λφ π) γ γϕ+λ(+ϕφ y ) C = ϕφ π +γϕφ π+ϕφ y, ρ = ρ 0. γ +ϕφ y 0 ρ 2 Before turning to BLE, we consider rational expectations equilibrium first., 4.. Theoretical results Comparing the NK model (4.5) with the general framework (??) we note that a = 0 and b 0 = 0. The rational expectation equilibrium fixed point equations ( ) then simplify to ξ = 0 (4.6) η = Bηρ+C. (4.7) In the case φ π >, then the rational expectations equilibrium is the stable stationary process with the mean x = 0. (4.8) In the special case of ρ with ρ i = ρ j = ρ for i,j =,2,,n and φ π >, the rational expectation equilibrium x t satisfies x t = (I ρb) Cu t. (4.9) Thus its covariance is Σ x = E(x t x )(x t x ) = ( ρ 2 ) (I ρb) CΣ Σ ε [(I ρb) C]. (4.0) Furthermore, the firstorder autocorrelation of the ielement x i of x is equal to ρ. That is, in this case the persistence of the REE coincides exactly with the persistence of the 5
16 exogenous driving force u t and the firstorder autocorrelations of output gap and inflation are the same, i.e. symmetric, equal to the autocorrelation in the driving force. Agents are assumed to be boundedly rational and, as in the general setup in Section 3, agents use a simple univariate linear rule to forecast the output gap y t and inflation π t of the economy. The use of a simple AR() process (2.3) is supported by evidence from the learningtoforecast laboratory experiments in the NK framework in Assenza et al. (20) and Pfajfar and Zakelj (202). The actual law of motion (4.5) becomes x t = B[α+β 2 (x t α)]+cu t, (4.) u t = ρu t +ε t. For the actual law of motion (4.), the means and firstorder autocorrelations are α = x = 0, G(α,β) = (G (β,β 2 ),G 2 (β,β 2 )) = (corr(y t,y t ),corr(π t,π t )). In order to obtain analytic expressions for G (β,β 2 ) and G 2 (β,β 2 ), in the following we focus on the symmetric case with ρ = ρ 2 = ρ. The firstorder autocorrelations of output gap and inflation are obtained through complicated calculations (see Appendix C) 2 : G (α,β,β 2 ) = f g (4.2) G 2 (α,β,β 2 ) = f 2 g 2 (4.3) 2 Numerical computations of the firstorder autocorrelation coefficients of output gap and inflation based on simulated time series generated by the model are completely consistent with those calculated basedonthefollowingequations. HencealthoughtheexpressionsofG andg 2 areextremelycomplicated, they are easily obtained and confirmed in numerical simulations. 6
17 where f = σ 2 (ρ+λ +λ 2 λβ2 2 )[ λβ2 2 (ρ+λ +λ 2 )]+[λβ2 2 (ρλ +ρλ 2 +λ λ 2 ) ρλ λ 2 ][(ρλ +ρλ 2 +λ λ 2 ) λβ2ρλ 2 λ 2 ] (ϕφ π (ρ+λ +λ 2 ) ϕβ2)) 2 [ϕφ π ϕβ2 2 (ρ+λ +λ 2 )]+[ϕβ2 2 (ρλ +ρλ 2 +λ λ 2 ) ϕφ π ρλ λ 2 ] [ϕφ π (ρλ +ρλ 2 +λ λ 2 ) ϕβ2 2 ρλ λ 2 ], g = σ 2 [(+λ 2 β2 4 ) 2λβ2 2 (ρ+λ +λ 2 )+(+λ 2 β2 4 )(ρλ +ρλ 2 +λ λ 2 )] ρλ λ 2 [(+λ 2 β2 4 )(ρ+λ +λ 2 ) 2λβ2 2 (ρλ +ρλ 2 +λ λ 2 )+(+λ 2 β2 4 )ρλ λ 2 ] [((ϕφ π ) 2 +ϕ 2 β2 4 ) 2ϕφ πϕβ2 2 (ρ+λ +λ 2 )+((ϕφ π ) 2 +ϕ 2 β2 4 )(ρλ +ρλ 2 +λ λ 2 )] +σ 2 2 +σ 2 2 ρλ λ 2 [((ϕφ π ) 2 +ϕ 2 β2 4 )(ρ+λ +λ 2 ) 2ϕφ π ϕβ2 2 (ρλ +ρλ 2 +λ λ 2 ) +((ϕφ π ) 2 +ϕ 2 β2 4 )ρλ λ 2 ], f 2 = σ 2 γ 2 [(ρ+λ +λ 2 ) ρλ λ 2 (ρλ +ρλ 2 +λ λ 2 )] +σ2 2 [(+ϕφ y )(ρ+λ +λ 2 ) β] 2 [(+ϕφ y ) β 2 (ρ+λ +λ 2 )]+[β 2 (ρλ +ρλ 2 +λ λ 2 ) (+ϕφ y )ρλ λ 2 ] [(+ϕφ y )(ρλ +ρλ 2 +λ λ 2 ) βρλ 2 λ 2 ], g 2 = σ 2 γ 2 [+ρλ +ρλ 2 +λ λ 2 ρλ λ 2 (ρ+λ +λ 2 ) (ρλ λ 2 ) 2 ] [((+ϕφ y ) 2 +β) 2(+ϕφ 4 y )β(ρ+λ 2 +λ 2 )+((+ϕφ y ) 2 +β) 4 +σ 2 2 (ρλ +ρλ 2 +λ λ 2 )] ρλ λ 2 [((+ϕφ y ) 2 +β 4 )(ρ+λ +λ 2 ) 2(+ϕφ y )β 2 (ρλ +ρλ 2 +λ λ 2 )+((+ϕφ y ) 2 +β)ρλ 4 λ 2 ], λ +λ 2 = β2 +(γϕ+λ+λϕφ y )β 2 2 +γϕφ π +ϕφ y, λ λ 2 = λβ 2 β2 2 +γϕφ π +ϕφ y. From these expressions, it is easy to see G (α,β,β 2 ) and G 2 (α,β,β 2 ) are analytic functions with respect to β and β 2, independent of α. The actual law of motion (4.5) depends on ten parameters ϕ, λ, γ, φ y, φ π, ρ, σ 2 and σ2 2. Only the ratio σ2 /σ2 2 of noise terms matters for the persistence G i(β,β 2 ) in (4.2) and (4.3). Hence, the existence of BLE (α,β ) depends on seven parameters ϕ, λ, γ, ρ, φ y, φ π and σ 2/σ2 2. Using Proposition we have the following property for the New Keynesian model: 7
18 Corollary If γ(φ π )+( λ)φ y > 0, there exists at least one BLE (α,β ), where α = 0 = x. Proof: See Appendix D. It is useful to discuss the special case with an exogenous i.i.d. shock u t, that is, ρ = 0 (no autocorrelation in the driving variable). It is easy to see that G (α,0,0) ρ=0 = 0, G 2 (α,0,0) ρ=0 = 0. That is to say (α,0,0) is a BLE for ρ = 0, where α = 0. Hence in this case the BLE coincides with the rational expectation equilibrium Numerical analysis We illustrate these results by numerical simulations for empirically plausible parameter values. We calibrate the parameters 3 : γ = 0.04,ϕ =,λ = 0.99,ρ = ρ 2 = ρ = 0.5, σ 2 σ = 0.5. As shown theoretically above, the numerical results are independent of the selection of the parameter values within plausible ranges, sample paths, initial values and distribution of shocks. Figureillustratestheexistence ofauniqueble(β,β 2 ) = (0.9,0.9592)4. Inorderto obtain (β,β 2 ), we calculate the corresponding fixed point β 2 (β ) satisfying G 2 (β,β 2 ) = β 2 for each β and the corresponding fixed point β (β 2 ) satisfying G (β,β 2 ) = β for each β 2 as shown in Figure 5. Hence their intersection point (β,β 2 ) satisfies G (β,β 2 ) = β andg 2 (β,β 2) = β 2. AttheBLE,thefirstorderautocorrelationcoefficients ofoutputgap and inflation (β,β 2 ) = (0.9,0.9592) are substantially higher than those at the REE, i.e. ρ(0.5), which means in the BLE the persistence of output gap and inflation are amplified. In order to indicate the persistence amplification more clearly, Figure 2 indicates the autocorrelation functions of output gap and inflation at the BLE and the REE. Clearly the autocorrelation function (ACF) at the BLE is much more persistent than at the REE. 3 We use this calibration in order to match the stylized facts of autocorrelation functions of output gap and inflations in the different cases of the Taylor rule. We follow Clarida et al. s calibration ( where γ = 0.3,ϕ =,λ = 0.99) to set ϕ =,λ = 0.99 and choose γ between γ = in Woodford (2003) s calibration and γ = 0.3 in Clarida et al. s calibration. For the exogenous shocks, we set the ratio of shocks fracσ 2 σ = 0.5, which is a possible case as shown in Fuhrer (2006). 4 It is easy to see (α,α 2 ) = (0,0). 5 Our simulations indicate G 2(β 2) < for each β and G (β ) < for each β 2, which suggests the unique BLE is stable under learning. 8
19 β 2 * (β ) (β *, β2 * )=(0.9, ) β β * (β 2 ) β Figure : BLE (β,β 2) = (0.9,0.9592) is the intersection point of the line β 2(β ) and the line β (β 2), where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,φ π =.5,φ y = 0.5, σ 2 σ = 0.5. Also the autocorrelation functions of output gap and inflation are similar to empirical work. In particular, the firstorder autocorrelation coefficients of output gap is about 0.9 and after about 5 periods its values decays to about 0.5, which is consistent with the empirical work, see for example Fuhrer (2006). Furthermore, the autocorrelation function of inflation with high persistence and lag of about 30 is also close to empirical work calculated with U.S. inflation data. Figure 3 shows the firstorder autocorrelations β of output gap and β 2 of inflation, i.e. the BLE, as a function of the autoregressive coefficient ρ of exogenous shocks. This figure clearly shows the persistence amplification along BLE, with much higher ACF than under RE, especially for ρ > 0.4. For ρ 0.5 we have β 0.9 and β 2 that output gap and inflation has significantly high persistence. > 0.95, implying Similar to Hommes and Zhu (204), if the BLE is unique, it is stable under SAClearning. Figure 4 illustrates that β,t converges to β and β 2,t converges to β 2 as t tends to infinity, that is, the BLE (β,β 2 ) = (0.9,0.9592) is stable under SAClearning. 4.2 Alternative specifications for setting interest rates The monetary policy Taylor rule (4.4) sets the interest rate in response to contemporaneous output gap y t and inflation π t. This contemporaneous rule assumes that the 9
20 ACF of output gap y at the BLE ACF of output gap y at the REE ACF of inflation π at the BLE ACF of inflation π at the REE AutoCorr. Func. of y AutoCorr. Func. of π Lag k Lag k (a) (b) Figure 2: Autocorrelation functions of output gap y and inflation π for contemporaneous Taylor rule at theble (β,β 2 ) = (0.9,0.9592)and atthe REE (β,β 2 ) = (0.5,0.5), where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,φ π =.5,φ y = 0.5, σ 2 σ = * β i β 2 * β * σ /σx * x σ /σπ * π σ /σy * y ρ ρ (a) (b) Figure 3: Effects of ρ with contemporaneous Taylor rule, a) β i(i =,2) with respect to ρ; b) the ratio of variances (σy 2/σ2 y, σ2 π /σ2 π ) with respect to ρ at the corresponding BLE (β,β 2), where λ = 0.99,ϕ =,γ = 0.04,φ π =.5,φ y = 0.5, σ 2 σ =
21 β,t β 2,t t t (a) First autocorrelation of y t (b) First autocorrelation of π t Figure4: Timeseries ofβ,t andβ 2,t under SAClearning converging totheble(β,β 2 ) = (0.9,0.9592), where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,φ π =.5,φ y = 0.5, σ 2 σ = 0.5. monetary authority observes current output gap and inflation. Here we consider two alternative specifications for setting interest rates, widely used in the literature, and perhaps under more realistic informational assumptions of either forward looking expectations (e.g. through survey data) or observing lagged variables A forwardlooking monetary policy rule As shown in Bullard and Mitra (2002) and Bullard et al. (2008), another Taylortype interest rate rule is to assume that the monetary authorities set their interest rate instrument in response to their forecasts of output and inflation deviations. This leads to the forward expectations specification for the interest rate equation, where (4.4) is replaced with Thus the system (4.5) becomes i t = φ π Ê t π t+ +φ y Ê t y t+. (4.4) x t = BÊtx t+ +Cu t, u t = ρu t +ε t, (4.5) where B = ϕφ y ϕ( φ π ), C = 0. γ( ϕφ y ) γϕ( φ π )+λ γ In this case we canhave similar results as the baseline model as shown in Figures5and 6. The difference from the baseline model mainly lies in the expression of the firstorder 2
22 autocorrelations. Similar to the above two rules, in the case of the forward expectations rule, the firstorder autocorrelations (4.2) and (4.3) become 6 where G (β,β 2 ) = f g (4.6) G 2 (β,β 2 ) = f 2 g 2 (4.7) f = σ 2 (ρ+λ +λ 2 λβ2 2 )[ λβ2 2 (ρ+λ +λ 2 )]+[λβ2 2 (ρλ +ρλ 2 +λ λ 2 ) ρλ λ 2 ][(ρλ +ρλ 2 +λ λ 2 ) λβ2 2 ρλ λ 2 ] +σ2 2 (ϕ( φ π )β2 2 )2 [(ρ+λ +λ 2 ) ρλ λ 2 (ρλ +ρλ 2 +λ λ 2 )], g = σ 2 [(+λ 2 β2 4 ) 2λβ2 2 (ρ+λ +λ 2 )+(+λ 2 β2 4 )(ρλ +ρλ 2 +λ λ 2 )] ρλ λ 2 [(+λ 2 β2 4 )(ρ+λ +λ 2 ) 2λβ2 2 (ρλ +ρλ 2 +λ λ 2 )+(+λ 2 β2 4 )ρλ λ 2 ] (ϕ( φ π )β2) 2 2 [+ρλ +ρλ 2 +λ λ 2 ρλ λ 2 (ρ+λ +λ 2 ) (ρλ λ 2 ) 2 ], +σ 2 2 f 2 = σ 2 γ 2 [(ρ+λ +λ 2 ) ρλ λ 2 (ρλ +ρλ 2 +λ λ 2 )] +σ2 2 [(ρ+λ +λ 2 ) ( ϕφ y )β] 2 [ ( ϕφ y )β 2 (ρ+λ +λ 2 )]+[( ϕφ y )β 2 (ρλ +ρλ 2 +λ λ 2 ) ρλ λ 2 ] [(ρλ +ρλ 2 +λ λ 2 ) ( ϕφ y )βρλ 2 λ 2 ], g 2 = σ 2 γ 2 [+ρλ +ρλ 2 +λ λ 2 ρλ λ 2 (ρ+λ +λ 2 ) (ρλ λ 2 ) 2 ] [+(( ϕφ y )β) 2 2 2( ϕφ y )β(ρ+λ 2 +λ 2 )+(+(( ϕφ y )β) 2 2 ) +σ 2 2 (ρλ +ρλ 2 +λ λ 2 )] ρλ λ 2 [(+(( ϕφ y )β 2 )2 )(ρ+λ +λ 2 ) 2( ϕφ y )β 2 (ρλ +ρλ 2 +λ λ 2 )+(+(( ϕφ y )β) 2 2 )ρλ λ 2 ], λ +λ 2 = ( ϕφ y )β 2 +(γϕ( φ π)+λ)β 2 2, λ λ 2 = λ( ϕφ y )β 2 β As in the baseline model above, the firstorderautocorrelationsof output gap and inflation calculated based on the time series generated by the model using the statistical definition of autocorrelation are consistent with those calculated based on the following equations. Hence although the expressions of G and G 2 are extremely complicated, it can be sure that they are correct. 22
23 ACF of output gap y at the BLE ACF of output gap y at the REE ACF of inflation π at the BLE ACF of inflation π at the REE AutoCorr. Func. of y AutoCorr. Func. of π Lag k Lag k (a) (b) Figure 5: Autocorrelation functions of output gap y and inflation π with forward looking Taylor rule at the BLE (β,β 2 ) = (0.8326,0.9605), where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,φ π =.5,φ y = 0.5,σ 2 /σ = β 2 * σ /σπ * π 0.6 * β i β * 2 2 σ /σx * x σ /σy * y ρ ρ (a) (b) Figure 6: Effects of ρ with forward looking Taylor rule a) β i(i =,2) with respect to ρ; b) the ratio of variances (σy 2/σ2 y, σ2 π /σ2 π ) with respect to ρ at the corresponding BLE (β,β 2), where λ = 0.99,ϕ =,γ = 0.04,φ π =.5,φ y = 0.5, σ 2 σ =
24 4.2.2 A lagged monetary policy rule As argued e.g. in Bullard and Mitra (2002), it may be viewed as realistic for central bank practice to posit that the monetary authorities must react to last quarter s observations on inflation and the output gap as contemporaneous values are not known yet. This leads to the lagged data specification for the interest rate equation, where (4.4) is replaced by Thus the system (4.5) becomes x t = Ax t +Bx e t+ +Cu t, i t = φ π π t +φ y y t. (4.8) u t = ρu t +ε t, (4.9) where A = ϕφ y ϕφ π, B = ϕ, C = 0. γϕφ y γϕφ π γ γϕ+λ γ In this case the expression of the rational expectations equilibrium is different from the above two cases. However, we can have similar results as the above two cases for the existence and stability of BLE as shown in Figures 7 and 8. The difference from the baseline model just lies in the expression of the firstorder autocorrelations. In the case of the lagged rule, following similar calculations as in the baseline model, the firstorder autocorrelations (4.2) and (4.3) become where G (β,β 2 ) = f g (4.20) G 2 (β,β 2 ) = f 2 g 2 (4.2) f = σ 2 (ρ+λ +λ 2 λβ2 2 )[ λβ2 2 (ρ+λ +λ 2 )]+[λβ2 2 (ρλ +ρλ 2 +λ λ 2 ) ρλ λ 2 ][(ρλ +ρλ 2 +λ λ 2 ) λβ2 2 ρλ λ 2 ] +σ2 2 (ϕφ π ϕβ2 2 )2 [(ρ+λ +λ 2 ) ρλ λ 2 (ρλ +ρλ 2 +λ λ 2 )], g = σ 2 [(+λ 2 β2) 2λβ 4 2(ρ+λ 2 +λ 2 )+(+λ 2 β2)(ρλ 4 +ρλ 2 +λ λ 2 )] ρλ λ 2 [(+λ 2 β2 4 )(ρ+λ +λ 2 ) 2λβ2 2 (ρλ +ρλ 2 +λ λ 2 )+(+λ 2 β2 4 )ρλ λ 2 ] (ϕφ π ϕβ2) 2 2 [+ρλ +ρλ 2 +λ λ 2 ρλ λ 2 (ρ+λ +λ 2 ) (ρλ λ 2 ) 2 ], +σ
25 ACF of output gap y at the BLE ρ k ACF of inflation π at the BLE ρ k AutoCorr. Func. of y AutoCorr. Func. of π Lag k Lag k (a) (b) Figure 7: Autocorrelation functions of output gap y and inflation π with lagged Taylor rule at the BLE (β,β 2 ) = (0.7746,0.9628),where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,φ π =.5,φ y = 0.5,σ 2 /σ = 0.5. f 2 = σ 2 γ 2 [(ρ+λ +λ 2 ) ρλ λ 2 (ρλ +ρλ 2 +λ λ 2 )] +σ2 2 [(ρ+λ +λ 2 ) (β 2 ϕφ y)] [ (β 2 ϕφ y)(ρ+λ +λ 2 )]+[(β 2 ϕφ y)(ρλ +ρλ 2 +λ λ 2 ) ρλ λ 2 ] [(ρλ +ρλ 2 +λ λ 2 ) (β 2 ϕφ y )ρλ λ 2 ], g 2 = σ 2 γ 2 [+ρλ +ρλ 2 +λ λ 2 ρλ λ 2 (ρ+λ +λ 2 ) (ρλ λ 2 ) 2 ] [(+(β 2 ϕφ y) 2 ) 2(β 2 ϕφ y)(ρ+λ +λ 2 )+(+(β 2 ϕφ y) 2 ) +σ 2 2 (ρλ +ρλ 2 +λ λ 2 )] ρλ λ 2 [(+(β 2 ϕφ y) 2 )(ρ+λ +λ 2 ) 2(β 2 ϕφ y) (ρλ +ρλ 2 +λ λ 2 )+(+(β 2 ϕφ y ) 2 )ρλ λ 2 ], λ +λ 2 = β 2 ϕφ y γϕφ π +(γϕ+λ)β 2 2, λ λ 2 = λ(β 2 ϕφ y)β Effects of monetary policies Figure 9(a) suggests that the first order autocorrelation of output gap β becomes smaller as policymakers more aggressively respond to output deviations (i.e. large φ y ), while Figure 9(b) suggests that the first order autocorrelation of inflation β2 becomes smaller as policymakers more aggressively respond to inflation deviations (i.e. large φ π ). That is, large response of policymaking tends to cause relatively low persistence. 25
26 β 2 * β i * β * ρ Figure 8: Effects of ρ with lagged Taylor rule, i.e. β i(i =,2) with respect to ρ, where λ = 0.99,ϕ =,γ = 0.04,φ π =.5,φ y = 0.5, σ 2 σ = 0.5. As shown in Fuhrer (2009), intuitively, when policy maker is more aggressive to adjust inflation or output to targets, the inflation or output tends to fluctuate more frequently and thus lead to low persistence. Figures 0 and suggests the similar results for the two alternative Taylortype interest rules. Furthermore, we find that if φ y = 0, that is, policy makers only care about inflation, not output, the effects of adjustment are relatively larger (the changing ranges of β2 are relatively larger) no matter which Taylortype interest rule is adopted. If φ y becomes large ( such as φ y = 0.5), the effects of adjusting φ π on inflation becomes relatively small, which leads to larger β2 for φ y = 0.5 than for φ y = 0 as φ π becomes large enough as shown in Figures 9(b), 0(b) and (b). 5 Concluding remarks We have further applied our behavioral learning equilibrium concept to the higher dimensional New Keynesian model. Boundedly rational agents use a univariate linear forecasting rule to output gap and inflation and in equilibrium correctly forecast the unconditional mean and firstorder autocorrelation of output gap and inflation. Hence, to a first order approximation the simple linear forecasting rule is consistent with observed market realizations. Sample autocorrelation learning means that agents are gradually updating the two coefficients sample mean and firstorder autocorrelation of their lin 26
27 β * 0.9 φ y =0 φ y = β 2 * 0.9 φ y = φ y = φ π φ π (a) (b) Figure 9: Effects of monetary policy with contemporaneous Taylor rule, where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5, σ 2 σ = 0.5. φ y =0 0.9 φ y = β * * β 2 φ y =0.5 φ y = φ π φ π (a) (b) Figure 0: Effects of monetary policy with forward looking Taylor rule, where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,σ 2 /σ =
28 φ y = φ y = β * φ y =0.5 β 2 * 0.9 φ y = φ π φ π (a) (b) Figure : Effects of monetary policy with lagged Taylor rule, where λ = 0.99,ϕ =,γ = 0.04,ρ = 0.5,σ 2 /σ = 0.5. ear rule. In the long run, agents thus learn the best univariate linear forecasting rule, without fully recognizing the more complex structure of the economy. Even in the higherdimensional system, BLE still exists and learnable under some conditions. We further investigate the effects of monetary policies under this misspecification BLE equilibrium. Acknowledgements We are grateful to the EU 7th framework collaborative project MacroRisk Assessment and Stabilization Policies with New Early Warning Signals (Rastanews), grant no and the. Cars Hommes acknowledges financial support from the INETCIGI Research Grant Heterogeneous Expectations and Financial Crises (HExFiCs). Mei Zhu also acknowledges financial support from NSFC(40365). 28
29 Appendix A Mean of the rational expectations equilibrium x = ξ +ηū = (I b ) b 0 +(I b ) b ηa+η(i η(i ρ) a = (I b ) b 0 +(I b ) [b η(i ρ)+(i b )η](i ρ) a = (I b ) [b 0 +b 2 (I ρ) a]. B Proof of Proposition 2 (stability SAClearning) Set γ t = ( + t). For the state dynamics equations in (3.8) and (2.8) 7, since all functions are smooth, the SAClearning rule satisfies the conditions (A.A.3) of Section 6.2. in Evans and Honkapohja (200, p.24). In order to check the conditions (B.B.2) of Section 6.2. in Evans and Honkapohja (200, p.25), we rewrite the system in matrix form by X t = A(θ t )X t +B(θ t )W t, where θ t = (α t,β t,r t ),X t = (,x t,x t,u t ) and W t = (,v t,ε t ), b α( β 2 )+b 2 a b β 2 0 b 2 ρ A(θ) =, 0 I 0 0 a 0 0 ρ B(θ) = I I b I. As shown in Evans and Honkapohja (200, p.86), A(θ) and B(θ) clearly satisfy the Lipschitz conditions and B is bounded. Since u t and ε t are assumed to have bounded 7 For convenience of theoretical analysis, one can set S t = R t. 29
30 moments, condition (B.) is satisfied. Furthermore, the eigenvalues of matrix A(θ) are 0 (double), ρ and the eigenvalues of b β 2. According to the assumption, all eigenvalues of A(θ) are less than in absolute value. Then it follows that there is a compact neighborhood including the BLE solution (α,β ) on which the condition that A(θ) is bounded strictly below is satisfied. Thus the technical conditions for Section 6.2. of Chapter 6 in Evans and Honkapohja (200) are satisfied. Moreover, since x t is stationary, then the limits σ 2 i := lim t E(x i,t α i ) 2, σ 2 x i x i, := lim t E(x i,t α i )(x i,t α i ) exist and are finite. Hence according to Section 6.2. of Chapter 6 in Evans and Honkapohja (200, p.26), the associated ODE is Furthermore, J G(α,β ) = dα dτ = x(α,β) α, dβ dτ = R [σ 2 xx βσ 2 ], dr dτ = σ2 R. [I b (β ) 2 ] ( b ) G (β ) G n (β ) Hence a BLE corresponds to a fixed point of the ODE(B). Furthermore, the SAClearning (α t,β t ) converges to the stable SCEE (α,β ) as time t tends to.. C Firstorder autocorrelation coefficients of output gap and inflation Now we try to calculate G(α,β). Define z t = x t Ex t. Then in order to obtain G(α,β), we first calculate E(z t z t ) and E(z t z t). Rewrite model (4.) into z t =Bβ 2 z t +Cε t +CρIε t +. (C.) 30
31 Thus z t = Bβ 2 z t +Cε t +CρIε t + = (Bβ 2 ) 2 z t 2 +Bβ 2 (Cε t +CρIε t 2 + )+Cε t +CρIε t + = Cε t +[CρI +Bβ 2 C]ε t +[Cρ 2 I +Bβ 2 CρI +(Bβ 2 ) 2 C]ε t 2 + +[Cρ n I +Bβ 2 Cρ n I + +(Bβ 2 ) n CρI +(Bβ 2 ) n C]ε t n + = Cε t +C[ρI C Bβ 2 C] [ρ 2 I C (Bβ 2 ) 2 C]ε t +C[ρI C Bβ 2 C] [ρ 3 I C (Bβ 2 ) 3 C]ε t 2 + +C[ρI C Bβ 2 C] [ρ n+ I C (Bβ 2 ) n+ C]ε t n +. Note ρ is a scalar number and I is a 2 2 identity matrix. Based on i.i.d. assumption of ε t, Ez t z t = ECε t + +C[ρI C Bβ 2 C] [ρ n+ I C (Bβ 2 ) n+ C]ε t n + ε t C + +ε t n [ρn+ I (C (Bβ 2 ) n+ C) ][ρi (C Bβ 2 C) ] C + = CΣC + +C[ρI C Bβ 2 C] [ρ n+ I C (Bβ 2 ) n+ C]Σ [ρ n+ I (C (Bβ 2 ) n+ C) ][ρi (C Bβ 2 C) ] C + = C[ρI C Bβ 2 C] [ρ n+ I C (Bβ 2 ) n+ C]Σ[ρ n+ I (C (Bβ 2 ) n+ C) ] n=0 [ρi (C Bβ 2 C) ] C, where Σ = σ σ2 2 In the following we try to obtain the expression of the matrix Ez t z t and hence we first calculate the matrix n=0 [ρn+ I C (Bβ 2 ) n+ C]Σ[ρ n+ I (C (Bβ 2 ) n+ C) ]. Note that Bβ 2 = Bβ 2 has two eigenvalues 8 +γϕφ π +ϕφ y β2 ϕ( λφ π )β 2 2 γβ 2 (γϕ+λ(+ϕφ y ))β 2 2 λ = [β2 +(γϕ+λ+λϕφ y)β2 2]+ [β 2 +(γϕ+λ+λϕφ y)β2 2]2 4λβ 2β2 2 (+γϕφ π +ϕφ y ), 2(+γϕφ π +ϕφ y ) λ 2 = [β2 +(γϕ+λ+λϕφ y)β2 2] [β 2 +(γϕ+λ+λϕφ y )β2] 2 2 4λββ 2 2(+γϕφ 2 π +ϕφ y ). 2(+γϕφ π +ϕφ y ) 8 In the special case λ = λ 2, although Bβ 2 is not diagonalizable, the expressions of firstorder autocorrelations (4.2) and (4.3) still hold based on the Jordan normal form of matrix Bβ 2. Without loss of generality, in the following we assume λ λ 2. 3.
32 Their corresponding eigenvectors are P = P 2 = [ ϕ( λφπ )β2 2, λ +γϕφ π +ϕφ y [ ϕ( λφπ )β2 2, λ 2 +γϕφ π +ϕφ y Let P = [P,P 2 ]. Then C Bβ 2 C = C P λ 0 0 λ 2 where β 2 ], +γϕφ π +ϕφ y β 2 ]. +γϕφ π +ϕφ y (C P), = =: C P (+ϕφ y)ϕ( λφ π)β 2 2 d d 2 d 3 d 4 +γϕφ π+ϕφ y +ϕφ π (λ ( λ γϕ( λφ π)β2 2 +γϕφ π+ϕφ y +. β 2 +γϕφ π+ϕφ y ) (+ϕφy)ϕ( λφ π)β 2 2 ) β 2 +γϕφ π+ϕφ y +γϕφ π+ϕφ y +ϕφ π (λ 2 ( λ 2 γϕ( λφ π)β 2 2 +γϕφ π+ϕφ y + ) β 2 +γϕφ π+ϕφ y ) β 2 +γϕφ π+ϕφ y Correspondingly (C P) = d d 4 d 2 d 3 d 4 d 2 d 3 d, where d d 4 d 2 d 3 = det(c P) = ϕ( λφ π )β 2 2(λ 2 λ ). Hence C (Bβ 2 ) n+ C = C P λn+ 0 0 λ n+ 2 Thus = d d 4 d 2 d 3 ρ n+ I C (Bβ 2 ) n+ C = d d 4 d 2 d 3 (C P) d d 4 λ n+ d 2 d 3 λ n+ 2 d d 2 (λ n+ 2 λ n+ ) d 3 d 4 (λ n+ λ n+ 2 ) d d 4 λ n+ 2 d 2 d 3 λ n+ d d 4 (ρ n+ λ n+ ) d 2 d 3 (ρ n+ λ n+ 2 ) d d 2 (λ n+ 2 λ n+ ) d 3 d 4 (λ n+ λ n+ 2 ) d d 4 (ρ n+ λ n+ 2 ) d 2 d 3 (ρ n+ λ n+ ).. 32