Signals and learning in a new Keynesian economy

Journal of Macroeconomics 40 (2014) 114–131

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Journal of Macroeconomics journal homepage: www.elsevier.com/locate/jmacro

Signals and learning in a new Keynesian economy Stefano Marzioni ⇑ LUISS Guido Carli and CASMEF (Arcelli Center for Monetary and Financial Studies), Viale Romania 32, Rome 00197, Italy

a r t i c l e

i n f o

Article history: Received 21 November 2012 Accepted 3 March 2014 Available online 25 March 2014 JEL Classification: E37 E43 E52 E58 Keywords: Gaussian signals Adaptive learning DSGE models Monetary policy

a b s t r a c t This paper aims at assessing whether, and how, communication of central bank’s forecast might affect economic dynamics. In a simple new Keynesian environment it is assumed that private sector conditions its own expectations to central bank’s forecasts. Private sector’s prior expectations are estimated in each period in accordance with the adaptive learning scheme, and successively updated with a signal based on central bank’s forecasts. Using both analytical and numerical calculations it is shown that the economy’s dynamics is affected by central bank’s ability to correctly assess the effect of the signal. In particular, if the central bank takes into account the impact of signals on private agents’ expectations the economic dynamics is less volatile. Moreover, if a fundamentals based signal includes a stochastic component unrelated to the economy, the strategy of communicating expectations to the private sector may perform worst than in the case of a totally uninformative signal. Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction In the recent decades monetary policy strategies have increasingly hinged on expectations management. In order to persuade rational (or near rational) agents about economic perspectives, major central banks often communicate policy intentions (including targets), increase operational transparency or try to convey information from a supposedly more advanced economic model. For instance, communication of forecasts about output and inflation may be interpreted as a channel to improve agents’ own forecasts and the overall economic awareness. The purpose of this paper is to assess the role of macroeconomic forecasts, as communicated by central bank to the private sector, in a general equilibrium model. In the model proposed below the private sector extracts information from signals issued by the policy maker. Such signals are then weighted in accordance with the signal’s past performance in terms of prediction, and become part of their expectations about the economy’s state variables. The mechanism used by agents to generate expectations consists of two steps. In the first step agents determine their prior about expectations. In the second step agents update their prior with a new piece of information (i.e. the signal by central bank). The final expected value of the vector of states variable is hence a combination of new and old information. The weight assigned to new information changes in each period and is determined by the degree of comovement between the signal and the variables and the signals variance. Agents are adaptive learners, in accordance with the framework analyzed in Evans and Honkapohja (2001). Agents learn adaptively the parameters of the rational expectation equilibrium by estimating it period by period in order to compute their prior expectation. Next, agents update the prior accordingly with the signal from central bank. Agents’ expectations are computed on the basis of a guessed solution, of observable data and of signals from the central bank, which together ⇑ Tel.: +39 0685225980. E-mail address: [email protected] http://dx.doi.org/10.1016/j.jmacro.2014.03.002 0164-0704/Ó 2014 Elsevier Inc. All rights reserved.

S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

115

constitute the private sector’s information set. By assumption, the policy maker is endowed with a deeper knowledge of the economy than the private sector. Indeed, it is assumed that the central bank knows the structural equations of the economy, and therefore computes expectations according to the rational expectations (RE) solution of the model. The framework defined by the structural assumptions implies the existence of asymmetric information between agents and policy maker. This is a situation that can arise in a number of context. Though it may seem unreasonable to assume that the policy maker knows exactly the structure of the economy, this assumption has the advantages of analyzing the effect of information spreading throughout the economy from the largest possible information set. Thus, the signal sender is artificially put in a privileged position with respect to agents. The adaptive learning (AL) approach in macroeconomics has been extensively explored in the literature. In this paper it constitutes the mechanism through which agents determine their prior expectation in each period. Behind the mathematical framework which allows to determine the convergence conditions of a given system of differential equations, lies the simple economic intuition of relaxing the strongest axiom of the rational expectations (RE) theory, i.e. the full knowledge of the model underlying the economy. As Orphanides and Williams (2005) argue, one of the most relevant criticism to the RE approach is that it assumes that agents have far more information about the structure of the economy than could reasonably be expected by any real economic agent in the real world. Moreover, in the RE approach it is assumed that agents believe (or know) that the model economy is an accurate representation of the economy they operate in, with no endogenous verification within the model. According to the adaptive learning approach, instead, private agents may or may not be aware of the structure of the economy they operate in, but they act at best of their possibilities. Economic decisions are therefore supported by econometric estimations that are based on all the available information that is perceived as relevant. Expectations are ‘‘adaptive’’ because they are adjusted in each period in the measure of the forecast error. Nevertheless, the expectations’ adaptive form arises from the recursive form that the econometric estimations may assume in certain cases. In particular, if estimates in each period are computed by ordinary least squares, then this method can be expressed in a recursive form. The recursive least squares (RLS) method allows to adjust the estimated coefficients proportionally to the error that arises by comparing the actual new data with the projection of the coefficients on the old information set. This paper departs from the existing literature on learning by assuming that agents’ final expectations (i.e. expectations that actually determine the economy’s dynamics) result from the combination of standard adaptive learning expectations and information from external sources. In the vast existing literature focusing on new Keynesian models with adaptive learning agents there are examples of different information sets co-existing in the same economy. Berardi (2009) focuses on the econometric evidence of expectation shocks affecting real variables. Ferrero and Secchi (2010) analyze the impact of interest rate path communication as well as expected inflation and output gap under a Taylor type rule monetary policy regime. In this paper it is assumed that monetary policy is optimal under RE and that the weight that private sector gives to the signal from central bank is endogenous. This approach is different because in Ferrero and Secchi (2010) the weights assigned to the signals from central bank are fixed exogenously. Moreover, in the present paper it is also considered the case of disturbed signals. Muto (2011) introduces a framework, where private agents refer to the central bank forecasts, by assuming that the central bank has only imperfect knowledge of the economy’ structure and responds to the bank’s own expectations. In the current paper the central bank has perfect structural knowledge of the economy and responds to private agents’ expectations (rather than the bank’s own expectations) in the monetary policy rule. The main finding in Muto’s paper is that the central bank, by following a Taylor type rule, should respond more aggressively to expected inflation than the extent suggested by the Taylor principle. This result arises because of the interaction between central bank’s and private sector’s expectations and is different from the findings in the present paper. Most of that difference is accounted for by the different information contained in central bank’s signal. Since in Muto’s framework central bank has no privileged information with respect to the private sector, it introduces a further shock propagation mechanism. Since the weight agents assign to central bank’s signal is always equal to one monetary policy should offset this kind of shocks by reacting more strongly to private sector’s expectations. This may alter E-stability, as well as determinacy, conditions. The main difference between the current paper and Muto (2011) lies in the interaction mechanism between central bank’s and private agents’ expectations, as well as in the assumption of central bank having perfect structural knowledge of the economy. Actually, in this paper the interaction mechanism between expectations from different sources is based on a two-step approach, which combines adaptive learning and updating with signals. Instead, in Muto (2011) a one-step approach of adaptive learning is adopted, which is more in line with traditional adaptive learning framework. Just like in Muto (2011), in this paper private agents utilize central bank’s forecasts as sources of information. Nevertheless, in this paper central bank has perfect structural knowledge of the economy. The two-steps approach is motivated by the observation of current major central bank’s practices, which often publish forecasts on a regular basis in order to increase the degree of transparency and accountability of policy making. However, since agents are bounded in their information set, they don’t have the theoretical instruments to rationally believe communications from the policy maker, even though central bank perfectly knows the true model of the economy. Hence, agents are not able to recognize whether or not using signals would improve their ability to make decisions. Thus, they have to learn the reliability of all the signals from an external source. This is exactly done by adjusting the weight which is assigned to the signal. It follows that information flowing from one side of the economy (central bank) to another (private sector) has to be treated as a signal and, as a such, it has to be processed before being included into agents’ information set.

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The simple framework in this paper allows agents to include more information in the expectations formation process than in standard adaptive learning economies. The additional sources can be added even though they do not directly enter the agents’ information set. This setup also allows the comparison of the economy’s dynamics under standard assumptions as well. Final expectations are based on agents’ prior expectations and are conditioned to central bank’s forecasts. The former are computed by standard AL algorithms, the latter are treated like gaussian signals. The numerical analysis is conducted with both disturbed signals (i.e. a i:i:d. shock hits the signal) and clean signals (i.e. not hit by shocks, and thus reflecting exactly the central bank’s forecast). This approach allows to capture distortions that may occur in the communication process. Indeed, often forecasters in the real world take into account many different information sources. As a result, noise would reflect the dispersion of the consensus about a forecast, and thus pessimistic or optimistic moods. Nevertheless, in this paper the information setup allows for the presence of two players only, namely the central bank and the group of homogeneous private agents, and thus of two information sources only. Hence, the noise captures all the factors that in the real world may determine a departure from the ‘‘true’’ value according to the central bank’s model, i.e. procedural errors, as well as communication difficulties due to agents’ lack of the necessary knowledge to appropriately interpret and utilize the forecast in their own model. The noise may also reflect imperfections in the information transmission procedures. Indeed, central banks’ communication is often made of statements intended to make its action more predictable, as well as of numerical forecasts. Hence, since there is no clearly superior rule of conduct in is field, it may happen that the whole central bank’s communication may suffer, in practice, from poor design or poor implementation, which in turn can lead to an imperfect signal. Since in all cases the noise summarizes non-numerical imperfections affecting information, the analysis conducted below can be useful in assessing the importance of communicating numerical forecasts in a framework otherwise characterized by unreliable information. The results in this paper show that communicating numerical forecasts improves the economy’s welfare if noise is relatively low. As long as noise increases because agents embed it into their forecasts, these tend to have less predictive power. The numerical analysis is conducted by Monte Carlo simulations. Therefore, the asymptotic economic performance is depicted for different ‘‘expectational regimes’’. Expectational regimes differ essentially by the nature of the signal and by whether the central bank takes into account the effect of the signal on expectations or not. In the former case the results rely on simulated convergence only. In the latter case the analysis of convergence under learning and E-stability conditions are provided. The results of simulation show that cleanness of the signal (i.e. the signal-to-noise ratio) is what really matters in determining a better economic outcome. Higher noise’s variance has a negative effect on average output gap and a positive effect on average inflation, hence the expected loss is higher. Moreover, when the average noise-to-signal ratio is too high, i.e. a sufficiently large share of the signal is orthogonal to the RE expected value, convergence is not always reached if the central bank does not take into account the effect of the noise on expectations. As for the short run a noisy signal has different outcomes with respect to the standard adaptive learning model. Actually, the impulse response function analysis shows that the reversion to the steady state value might occur by a smoother path for lower noise-to-signal ratios. However, the effect of the noise can be offset by the policy maker by the adoption of a proper informative regime. Indeed, when a cost shock occurs at the steady state, the economy’s reaction is smoother if the policy maker takes into account the effect of communication on agents’ expectations. The remainder of the paper is structured as follows. In Section 2 it is introduced a model based on the standard AL literature. Then the signal is defined and its effect on economy’s stability under learning is analized. In Section 3 the model is simulated by taking into account different kinds of signal and the results are compared to the standard model without signal. The effects of a disturbed signal are analyzed as well. Section 4 summarizes and concludes. 2. The model 2.1. The new Keynesian framework The economy can be described by a standard new Keynesian log-linear framework as developed in Clarida et al. (1999).1 The model consists of the two following structural equations:

  ps xt ¼ u it  Eps t ptþ1 þ Et xtþ1 þ g t ;

ð1Þ

pt ¼ kxt þ bEpst ptþ1 þ ut ;

ð2Þ

where xt is the output gap, pt is the inflation rate and it is the nominal interest rate. All the parameters in (1) and (2) are ps positive. 0 < b < 1 is the discount factor of the representative firm. Eps t ptþ1 and Et xtþ1 denote private sector expectations of inflation and output gap next period. In this subsection it is assumed that expectations are rational. From the subsequent subsection on equilibria outside the REE will be discussed. g t and ut denote observable first-order autoregressive shocks, ~ t and g t ¼ qg t1 þ g ~t , where q; l 2 ½0; 1Þ, and u ~ t  Nð0; r2u Þ; g~t  Nð0; r2g Þ. g t determined respectively by ut ¼ qut1 þ u represents shocks to government purchases as well as shocks to potential GDP. ut represents any cost push shocks to marginal shocks other than those entering through xt . The policy maker’s objective function at time t has the usual form 1

The full model description is available in Galí (2008) or in Woodford (2003).

S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

min Et

1   X bj p2tþj þ ax2tþj ;

117

ð3Þ

j¼0

where a > 0. As shown in McCallum and Nelson (2004), the optimality condition under commitment, in a timeless perspective is

kpt ¼ aðxt  xt1 Þ;

ð4Þ

which rules out dynamic inconsistency issues. The policy implementation of the optimality condition (4) requires an interest rate rule. Evans and Honkapohja (2006) show that an interest rate rule that implements (4) and under which the economy is E-stable, is the following expectation-based optimal rule

it ¼ dL xt1 þ dp Eps ptþ1 þ dx Eps xtþ1 þ du ut þ dg g t ;

ð5Þ

where

dL ¼



"

a

u a þ k2

dp ¼ 1 þ

;

dx ¼ dg ¼ u1 ;

du ¼



u a þ k2

k



#

bk

u a þ k2

 ;

:

The optimal rule in Eq. (5) is obtained by combining the optimality condition (4) with the price-setting Eq. (2) and the dynamic IS (1), solving for it and treating expectations as given. As Evans and Honkapohja (2006) argue, by setting it according to Eq. (5), the policy maker recognizes the possibility that private agents may have non-rational expectations. Hence, central bank reacts to given expectations as well as to fundamentals. They show in their Proposition 3 that under RE this rule let the equilibrium to be uniquely determined and stable under learning. The model can be represented compactly as follows:

yt ¼ MEps ytþ1 þ Nyt1 þ Pv t ;

ð6Þ

and

v t ¼ F v t1 þ v~ t ;

M¼4

0

0

pt  ; v t ¼ ½g t ut  ; v~ t ¼ ½g~t u~t  and

where yt ¼ ½xt

2

ð7Þ

0

0

bk ðaþk2 Þ

0

ba ðaþk2 Þ

3

5;

" N¼

a

ðaþk2 Þ

ak ðaþk2 Þ

0 0

# ;

" P¼

0

k ðaþk2 Þ

0

a

# ;

 F¼

ðaþk2 Þ



l 0 : 0 q

The RE solution of the model, also called minimum state variable (MSV), is yt ¼ Byt1 þ Cv t , whose coefficients, found by the undetermined coefficients method are the following:

B¼



where bx ¼

bx

0

bp

0

;

pffiffiffiffiffiffiffiffiffiffi

c

c2 4b

2b

C¼



0 cx ; 0 cp

; bp ¼ ak ð1  bx Þ; cx ¼ ½k þ bbp þ að1  bqÞ=k

1

and cp ¼ ak cx .

2.2. Signals Since the analysis is concerned with the consequences of the coexistence of different information sets, the information structure is crucial in this model. In particular, different information sets are available to the private sector and to the central bank. The private sector’s information set at time t is composed of yt1 ; v t ; F; st . Some theoretical background for the framework utilized in this Section is provided in Appendix A by means of a one-dimensional example. In this Section the central bank’s information set instead includes yt1 ; v t ; M; N; P and F. In each period the activity of private agents and policy maker takes place in a predetermined order. In particular, at the beginning of period t the states vector yt1 is observed. Then the vector v t , containing innovations, realizes. The private sector computes its prior according to the adaptive learning algorithm only. As v t realizes, also the central bank has enough information to compute its own expectations, which in turn are disclosed. The private sector treats central bank’s forecasts as a signal about the real state of nature and, before making economic decisions in t, conditions its final expectations to the signal. The prior that has been previously computed is updated such that the private sector the expectations that are relevant to its economic activity. Finally, the central bank sets the appropriate level of the interest rate and yt realizes. The private sector learns adaptively the RE coefficients of the MSV solution yt ¼ Byt1 þ Cv t , as stated above. According to the data collected in the vector zt ¼ ð1; yt1 ; v t Þ, the prior about the next period’s value of vector yt is 2 E fytþ1 jzt g ¼ ðI þ bÞa þ b yt1 þ ðbc þ cF Þv t , where a; b and c are estimated parameters, whose time subscript is omitted for ease of notation.

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The private sector’s posterior expectations of the state vector, conditioned to a gaussian signal st are computed as follows:

Eps fytþ1 jzt ; st g ¼ E fytþ1 g þ Ut ðst  Efst gÞ; 0

ð8Þ

1

where Ut ¼ Cov ðy; sÞ VarðsÞ . Henceforth, the conditioning will be omitted for ease of notation. The signal provided by the central bank is a bivariate vector containing its own forecast of ytþ1 :

~t þ CF v t ; st ¼ Ecb fytþ1 g ¼ By

ð9Þ

As the value of yt is not directly observable in t nor from private agents neither from the central bank, the latter has to com~t . The policy maker knows the structure of the economy, so it uses this piece of information to compute pute a nowcast y ~t ¼ ME fytþ1 g þ Nyt1 þ Pv t , which represents the nowcast over the current period. It is here assumed that the central bank y computes a theoretical value of yt based on private sector’s prior instead of the actual updated expectations, i.e. assuming that Eps fg ¼ E fg. This implies that the central bank computes the actual value of ytþ1 under the assumption that the signal ~t ) and to the agents’ information set is then based on is ignored. The expected value of st conditional to the signal (including y ~t and is Efst g ¼ a þ by ~t þ cF v t . y 2.3. Adaptive learning Real time learning is implemented according to the guidelines provided in chapters 6 and 10 in Evans and Honkapohja (2001). The algorithm includes a time-decreasing gain and, in addition to the algorithm in the original book, it also includes the equations describing the dynamics of the parameters in Ut . The following equation represent the moment matrix updating rule:

  Rt ¼ Rt1 þ t 1 zt z0t  Rt1 ;

ð10Þ

where zt ¼ ð1; y0t1 ; v 0t Þ. In order to convert the system in a convenient standard form, a timing change is made on Rt . Therefore, it is defined the artificial variable St , such that St1 ¼ Rt , in order to get the following relationship:

  St ¼ St1 þ t1 zt z0t  St1 þ t 2



 t  0 zt zt  St1 : tþ1

ð11Þ

The parameters updating equation is

 0 0 nt ¼ nt1 þ t1 R1 t zt1 yt1  nt1 zt1 :

ð12Þ

The algorithm is completed by adding a recursive formulation of the two functions determining the current value of

Ut ¼ C 0t V 1 t , where C t ¼ cov ðy; sÞ and V t ¼ v arðsÞ:   C t ¼ C t1 þ t 1 yt s0t  C t1 ;

ð13Þ

  V t ¼ V t1 þ t1 st s0t  V t1 :

ð14Þ

n0t

where ¼ ða; b; cÞ. The real-time learning algorithm above is in the form ht ¼ ht1 þ t 1 Hðht1 ; X t Þ þ t 2 qt ðht1 ; X t Þ, where ht ¼ ðnt ; St ; V t ; C t Þ and X t ¼ ð1; yt1 ; v t ; yt2 ; v t1 Þ, whose convergence has been demonstrated to be driven by E-stability under conditions A and B in Section 6.2.1 in Evans and Honkapohja (2001). In order to prove that also the algorithm (11)–(14) converges, and that the convergence is driven by the E-stability conditions provided in Section 2.4, it is needed to analyze the equations governing vector X t which is expressed as X t ¼ Aðnt1 ; C t1 ; V t1 ÞX t1 þ BW t , where

 W 0t ¼ 1

0t



;

0

0 0 0 B B T a ðnt1 ; C t1 ; V t1 Þ T b ðnt1 ; C t1 ; V t1 Þ T c ðnt1 ; C t1 ; V t1 Þ B 0 0 F A¼B B B 0 I 0 @ 0 0 I 0

1 0

1

C B B0 0C C B C B¼B B 0 I C: C B 0 0 A @ 0 0

0 0

1

C 0 0C C 0 0C C; C 0 0A 0 0

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S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

Zeros and Is in previous matrices represent respectively matrices of zeros and identity matrices of appropriate dimensions. By substituting yt ¼ T ðnt ; C t ; V t Þzt into Eq. (12) the following expression is obtained: 0 nt ¼ nt1 þ t 1 S1 t1 zt1 zt1 ðT ðnt1 ; C t1 ; V t1 Þ  nt1 Þ;

which is an expression for parameters dynamics. The stability of the above system of equations has to be intended as local around a fixed point of T ðn; C; V Þ. Following Evans and Honkapohja (1998), a definition of the algorithm’s domain D is re   C; V be a fixed point of T ðn; C; V Þ and the eigenvalues of b be strictly inside the unit circle. Given that quired. Let n;   z0t ðn; C; V Þ ¼ 1; y0t1 ðn; C t1 ; V t1 Þ; v 0t , and recalling that yt1 ¼ T ðnt1 ; C t1 ; V t1 Þzt1 , then zt ðn; C; V Þ is a stationary process     b be an open set around n; C; V . Let Mz ðn; C; V Þ ¼ E zt ðn; C; V Þz0t ðn; C; V Þ , and let D for all ðn; C; V Þ sufficiently near     0 b   n is the only fixed point of T; S and V are invertible, the roots of n; S; C; V , where S ¼ E zt zt so that for all ðn; S; C; V Þ 2 D; b are strictly inside the unit circle. Given that the decreasing gain function is t1 , with this construction the conditions A b is sufficiently small, conditions B in the same section are met since the moments in Section 6.2.1 are met. Provided that D of W t are bounded. The associate ODE of the algorithm (11), (12), (13) and (14) are computed taking expectations and limits, as follows:

dn ¼ S1 Mz ðT ðn; C; V Þ  nÞ; ds dS ¼ Mz  S; ds dC ¼ T ðn; C; V ÞMz K ðnÞ0  C; ds dV ¼ K ðnÞM z K ðnÞ0  V; ds where K ðnÞ is such that st ¼ K ðnÞzt . The first two equations are standard, and as long as S ! M z , convergence is driven by conditions T ðn; C; V Þ  n, which turns out to be the E-stability condition. The two remaining equations converge as long as the convergence of n is granted. The third equation shows that the recursively estimated covariance converges to its theoretical value, as Cov ðT n ðnÞz; KzÞ ¼ T n ðnÞCov ðz; zÞK 0 . As long as VarðKzÞ ¼ KVar ðzÞK 0 , the fourth equation as well converges to its asymptotic value. The algorithm in Eqs. (10)–(12) is used to simulate the economy under learning in the following section. Further details about the link between e-stability and real-time learning can be found in the original book from Evans and Honkapohja. 2.4. E-Stability As shown in Section 2.3, recursive least squares learning is locally convergent to the REE, and the stability of the main associated ODE is driven by the equation dn=ds ¼ Tðn; C; VÞ  n, where n ¼ ða; b; cÞ. In this Section it will be shown that such a condition corresponds to the E-stability condition derived for model (6)–(9). Then such conditions are derived in closed form. The analysis of the stability under learning is conducted according to the methodology proposed in Evans and Honkapohja (2001). According to Evans and Honkapohja (2006) the model without signal is determinate under RE and is e-stable for all parameter values. According to the structural equations of the economy, described by Eqs. (6)–(9), the actual value of yt is given by the following actual law of motion (ALM):

yt ¼ MEps fytþ1 g þ Nyt1 þ Pv t 2

2

¼ fM½bc þ cF þ UðC  cÞ þ UðB  bÞM ðbc þ cF Þ þ UðB  bÞP þ Pgv t þ fM½b þ UðB  bÞMb þ UðB  bÞN þ Ngyt1 þ M ðI þ bÞa þ M UðB  bÞM ðI þ bÞa; where U is a combination of estimated variances and covariances of the signal and the state variables, such that U ¼ Cov ðy; sÞ0 VarðsÞ1 . The evolution of the estimated parameters is described by the following mapping:

T ða; b; cÞ ¼ ½M ðI þ bÞ þ UðB  bÞMðI þ bÞ; 2

2

2

Mb þ MUBMb  MUbMb þ M UBN  MUbN þ N; Mðbc þ cF þ UðC  cÞF þ UðB  bÞM ðbc  cF Þ þ UðB  bÞPÞ þ P: By computing the Jacobian of vecT j , for j ¼ fa; b; cg, the stability conditions of the system d=dsða; b; cÞ ¼ T ða; b; cÞ  ða; b; cÞ can be computed. Thus, the economy described by Eqs. (6)–(9) is asymptotically stable under learning if the eigenvalues of the following matrices have real part less than unity :

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S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

  DT a b ¼ M ðI þ bÞ þ UðB  bÞMðI þ bÞ;   0 20 DT b b ¼ ½b  ðM þ M UBM  M UbMÞ þ ½I  ðMb þ M UBMb  M UbMbÞ  N0  MU  b M  U;   c ¼ I  ½Mb þ UðB  bÞMb þ F 0  ½M ðI  UÞ þ UððB  bÞM: DT c b;  ¼ B, are The eigenvalues of DT a evaluated at b

" 0; 

ab a þ k2

#  ;

which are both less than 1, as b < 1 by assumption. Hence, provided that b ! B, this condition is sufficient for learnability of the parameters in vector a.  DT b b ’s eigenvalue depend on the elements of U. Therefore, conditioning convergence upon structural parameters a; b and k only, could be not enough in order to prevent E-unstable solutions. In Appendix C it is shown that the eigenvalues  of DT b b can be expressed as follows:

 pffiffiffi pffiffiffi bp bk 0;  ; A þ B; A  B ; d where d ¼ a þ k2 and A and B are combinations of structural parameters and elements of U.2 The relationship between the elements of U and the eigenvalue is not straightforward, although some conclusion can be drawn relying on parametric analysis and numerical simulations. Ceteris paribus, an increase of standard deviation of inflation’s signal sp , determines an increase in elements u11 and u21 and a decrease of element u22 . The effect on u12 is unclear. An increase of standard deviation of output gap’s signal determines an increase of elements u12 and u22 , and a decrease of element u11 , while the effect on u21 is unclear. An increase of the estimated correlation between inflation and the related signal, qp;sp let u22 increase and u21 decrease. An increase of the estimated correlation between output gap and the related signal, qx;sx let u11 increase and u12 decrease. The elements that are not mentioned remain unchanged after a variation of any of the correlation between a variable and the related signal. According to the numerical results provided in Fig. 1, in Appendix F, the asymptotic average behavior of U suggests that the e-stability conditions depicted above are sufficient for convergence. Indeed, the elements of Ut play a crucial role in  determining the eigenvalues of DT b b and such elements vary through time. In general, in issuing a signal, one should pay attention to the way the signal will be predicted from the private sector, because a non predictable signal which is also non totally rational, may potentially lead to unexpected and non-learnable dynamics.   c Provided that the parameter b converges to B, the economy is e-stable if also the real part of the eigenvalues of DT c b;   are less than one. Eigenvalues of DT c b; c are

  bðkbbp þ lða þ ku21  au22 ÞÞ bðkbbp þ qða þ ku21  au22 ÞÞ ; 0; 0; ; a þ k2 a þ k2 The non-zero eigenvalues are equal for q ¼ l, and are less than unity under the following conditions:

1 > bqð1  u22 Þ;

k þ bbp > u21 : bq

The first condition is always met provided that u22 > 1  1=bq, i.e. u22 has to be larger than a negative number. According to the calibration used in Section 3, u22 > 0 8t, such that the condition is always satisfied. The second condition is satisfied for

k þ bbp > u21 ; qb which usually hold, given structural parameter values and resulting simulated estimates of u21 . In general u21 is required to , be relatively small, and u22 to be relatively large. This finding is consistent with the E-stability condition related to DT b b which in general moves toward the E-unstable region for relatively small off-diagonal elements of U. 3. Numerical analysis In this Section the model presented above, which will be referred to as information regime R2 , is simulated and compared to several benchmarks and specifications. The two simulated benchmarks are the rational expectations economy (RE) and the pure adaptive learning economy (R0 ). Different specifications of the economy with signal are simulated as well. In particular, the economy is simulated under the assumption that the signal is totally uninformative (information 2

See Appendix C for further details.

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121

regime R1 ), and under the assumption that the central bank includes the effect of its signal on the current level of the states (R3 ). The purpose of this Section is to capture the deviations from the standard new Keynesian model without signal in terms of welfare and in terms of response to shocks. Therefore, both long run and short run simulations are performed. 3.1. Long run simulations The horizon chosen for the simulation (290 periods) is sufficient for most of the estimated parameters in nt to converge to their RE value. The first step consists in generating the pseudo-random structural shocks, i.e. the innovations ut and g t . Then, the timing of actions in each period reflects the assumptions made in Section 2.2 . Therefore, according to the realized shocks (which are assumed to be directly observable) both the signal from central bank and the private sector prior are computed. The former is related to central bank perfect knowledge of the economy’s structural equations. The latter is computed in accordance with real time learning algorithms. Finally, the actual response of the economy is computed and the current realizations of the state variables is generated. It follows that the timing of the simulation is such that the output gap is determined only after expectations are formulated. As remarked above, this assumption is necessary because monetary policy reacts to private sector’s expectations by setting the interest rate consistently with the optimal policy under commitment. Expectations, in turn, are formulated only after the signal is received. In the model analyzed in Section 2 (i.e. R2 ), the central bank is unable to evaluate ex ante the impact of the signal on private sector’s expectations, even though it has the necessary knowledge to do so. Indeed, it is there assumed that the central bank issues a signal under the assumption that it is ignored by the private sector. This assumption simplifies of calculation because it allows a closed form solution. This assumption is relaxed below where it will be shown numerically that the difference between the two regimes is negligible only if the signal is clean, i.e. if it is not affected by any disturbance. As a consequence of the information structure of the economy, yt , the current value of the state variables realizes only at the end of period t. Hence, it is not directly observable until agents and policy maker have made their decisions in the same period.3 Therefore, the value of the state variables becomes observable only with a one-period lag. The difference between private sector and central bank, in terms of information set, is that the central bank knows exactly the structural form of the economy, and utilizes this piece of information in order to construct the true MSV solution (and form expectations accordingly). The private sector instead must generate expectations by estimating in each period a guessed REE. Once the shocks are observed, the private sector is able to compute its prior about expectations by the standard real-time learning mechanism. In the meanwhile, the central bank computes its expectations too, which are different although not necessarily correct. Then, the central bank sends the signal i.e. communicates its expectations to the private sector, which in turn updates its prior by conditioning the ‘‘old’’ information to ‘‘new’’ gaussian information. As a result, the relevant private sector’s expectations are computed. Only the updated expectations are relevant, because these are the only expectations that are used in order to make allocation choices, and therefore are the only entering the structural equations. Once the private sector’s expectations are computed, the central bank is able to fix the interest rate that optimizes its loss function, according to the optimal monetary policy regime. Hence, yt realizes and the period ends. Before starting the update of their prior it is assumed that agents collect and estimate ten years of data. In these ten years the central bank sends the same kind of signal that is supposed to send in the following periods. This assumption is useful to get rid of small sample issues related to estimation of the variances and covariances of states and signals. A shorter data collection period of 20 quarters does not affect the model with clean signals. Nevertheless, it affects the model with disturbed signal in such a way that the matrix U diverges. For this reason it is here assumed a longer time to collect data. Such a period is also comparable with the pre-estimation sample used in the AL literature, for instance, in Milani (2011). In the data-collection period, agents learn adaptively, starting from an arbitrary initial condition or from a simulated steady state, the priors about the MSV coefficients are actually equal to the RE solution and the initial moment matrix is a symmetric matrix of random realization of zero-mean variables with arbitrary variance smaller that the variance of the economy’s shocks. The learning algorithm has a decreasing gain ðt þ 5Þ1 . This implies that the learning improvements affect the prior less as a consequence of the simple passage of time. s1 In each period the private sector estimates U by using sample covariances, such that Ut ¼ Ry;s t Rt . In order to avoid unreasonable results due to singularity issues, a projection facility is added to the algorithm that computes U. It is there assumes that, if the covariance matrix of the signal vector Rst is nearly singular, it is substituted by Rst1 . This can be considered a reasonable assumption because of the nature of the estimation. Throughout the model PS acts as an econometrician, and as a such, it may be reasonable to subordinate the estimation tools to the economic meaning and usefulness of the estimated object. In this case, a singular Rst would not be of any help in adding economic meaning to the signal, that is the purpose of its estimation. Nevertheless, the consequences of a singular Rst would be devastating for the dynamics of the model, as it would let expectations diverge, reflecting this tendency to the whole economy. If this facility were used consistently, such that the validity of the model would be heavily questioned, it would make sense to get rid of it and to re-think the model. But, according to the simulations’ results, its influence is very small in all the configurations of the model that have been tested. 3 This assumption still holds even though the central bank would have enough information to correctly compute it if it could compute the effect of the signal on Eps ytþ1 .

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Each experiment controls the stream of pseudo-randomly generated numbers, such that these may vary across simulations in each experiment, but not across experiments with different regimes. Hence, different regimes are simulated with the same disturbance. 3.1.1. Calibration The values of the structural parameters are taken from the existing literature. The optimal monetary policy regime implies that the only parameters involved in the simplified structural form (and also in the RE solution) are the discount factor b; k and policy maker’s weight on output gap in the loss function a. For these parameters the chosen calibration is the one proposed by McCallum and Nelson (1999). The main results are also compared across different calibrations, namely a quarterly adaptation of the Clarida, Galí and Gertler (CGG) (2000) calibration, and the Woodford (1999) as in Table 1. The preference parameter a is considered a policy parameter and its value varies across the literature. It is here assumed that a ¼ 0:5. The AR coefficients of the structural shocks are l ¼ q ¼ 0:8, and the variance of their i.i.d. innovations are both r2e ¼ 0:0712. Each experiment consists of 500 simulations. Each simulation is 290 periods long, where each period represents a quarter. The total number of periods is such that after the ten years of data-collection, we are able to observe the behavior of the economy along about 60 years. According to experiments with all the mentioned calibrations, this lapse of time is sufficient for estimated parameters a; b, and c to show a pattern of reversion toward the RE values, i.e. A; B and C. 3.1.2. The benchmark models The first benchmark is the pure learning model (R0 ). In this benchmark model the expectations used by agents coincide with their prior, i.e. Eps fytþ1 g ¼ E fytþ1 g. In this framework the signal has no influence on the economy by construction, as the private sector ignores it. This specification of the model is in fact a standard adaptive learning model á la Evans and Honkapohja, and shows results that are entirely predictable by the existing literature. A second benchmark is constituted by the REE values, i.e. the equilibrium that would realize if the private sector shares the same information set of the central bank. In both cases 500 simulations are generated, each 290 periods, under three calibrations and under the assumption that a ¼ 0:5; q ¼ l ¼ 0:8 and r2x ¼ r2p ¼ 0:0712. The results of these two models are summarized in Tables 2–4. The most evident result is that the two model are practically identical. Indeed, some difference exist but are in an extremely low order of magnitude (1010 ). 3.1.3. Results under different signaling regimes In this Subsection some results are shown from the model depicted in Section 2, i.e. R2 . It is also provided results for a specification in which the central bank’s expectations are formed by taking into account the effect that the signal has on the PS’ expectations and on current and future states. This regime is named R3 . Under R3 expectations are formed by solving the model in Eqs. (15)–(18). A comparison between R2 and R3 shows that if the signal is clean, differences are negligible. The result from the model where the signal is clean are compared with those from a model in which the signal is disturbed by a relatively low variance shock. This is obtained by adding a noise disturbance to the signal itself. These results are also compared with a model in which the only signal that is an i.i.d. bivariate zero-mean stochastic process (R1 Þ. The main signaling regime, for which some analytical results are available, henceforth named R2 , is the one described in Section 2. This regime allows an analytical solution in terms of the weight matrix U, and then the discussion of the e-stability conditions. E-stability is granted for elements of U lying in a region for which some examples are given in Fig. 2. One of the main drawbacks of this model is that the central bank while issuing the signal, does not take into account the effect that the signal has on the economy. In practice, the central bank issues a signal while assuming that it will be ignored. In order to check the relevance of this assumption, R2 is compared with R3 , where this assumption is relaxed. As it is evident from Tables 5 and 6 the lower moments are not heavily affected. There exist negligible differences only at a very low order of magnitude. Nevertheless, this is true only if the signal is clean, which is the case when the signal is not hit by disturbance. Table 1 Calibrations.

b k

MN

CGG

W

0.99 0.3

0.99 0.075

0.99 0.024

Table 2 Benchmarks mean values. AL

 x p

RE

MN

CGG

W

MN

CGG

W

0.00043 2:5  105

0.0023 0.001

0.0054 0.0026

0.00043 2:5  105

0.0023 0.001

0.0054 0.0026

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S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131 Table 3 Benchmarks variances. AL

r2x r2p

RE

MN

CGG

W

MN

CGG

W

0.7178 0.1889

1.3320 1.433

0.7978 3.104

0.7178 0.1889

1.3320 1.433

0.7978 3.104

Table 4 Correlations under pure adaptive learning (R0 Þ.

x

x

p

u

g

1

0.1419 1

0.0008 0.0028 1

0.8791 0.6109 0.0005 1

p u g

Table 5 Average values across signaling regimes. R0

R1

R2

R3

Clean

 x p

0.00043 2:5  105

– –

0.00043 2:5  105

0.00043 2:5  105

1%

 x p

– –

0.0049 1.8866e05

4.2825e04 2.3950e05

4.2605e04 2.4601e05

5%

 x p

– –

0.001 7.5033e07

4.3144e04 2.0229e05

4.0184e04 2.8179e05

50%

 x p

– –

0.0034 1.0377e04

– –

1.3685e04 2.1842e05

100%

 x p

– –

0.0073 2.3158e04

– –

1.4191e04 2.8170e05

Table 6 Average variances across signaling regimes. Clean signal.

2 x 2

R0

R1

R2

R3

Clean

r rp

0.7178 0.1889

– –

0.7178 0.1889

0.7178 0.1889

1%

r2x r2p

– –

0.7463 0.2307

0.7179 0.1890

0.7179 0.1889

5%

r2x r2p

– –

0.7181 0.1912

0.7188 0.1909

0.7180 0.1890

50%

r2x r2p

– –

0.7192 0.1916

– –

0.7233 0.1955

100%

r2x r2p

– –

0.7212 0.1953

– –

0.7342 0.2117

Under the regime R3 the central bank uses all the available information, i.e. the knowledge of the structural equations of the economy and the expectations formation mechanism, as well as observation of the states with one lag and of the current shocks. The difference with respect to R2 is that under R3 the central bank computes the effect of the signal on expectations. By computing expectations the same way the private sector does, due to the knowledge of the structural equations, it is able to compute the ALM of the economy, such that it can compute and release a nowcast of yt . Such an information is used to solve a linear system constituted of the following equations:

^ t þ Cv t ; st ¼ By

ð15Þ

^t ¼ M b E ps fytþ1 jst g þ Nytþ1 þ P v t ; y

ð16Þ

  b E ps fytþ1 jst g ¼ E fttþ1 g þ Ut1 st  sex t ;

ð17Þ

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^ sex t ¼ bt yt þ c t F v t þ at :

ð18Þ

^t based Eq. (15) represents how the central bank computes expectations of xtþ1 and ptþ1 at time t, including the nowcast y on a complete information set. The peculiarity of this model with respect to R2 is the presence of Eq. (16), which provides the correct value of the current period states even though these are not observable yet. In fact, this value is computed accordingly to the private sector’s expectations, conditioned on the signal, as in Eq. (17). Under R2 this was not the case, as the current value of yt was computed assuming that expectations coincided with the prior E fytþ1 g instead. In Eq. (17) it is used Ut1 instead of Ut for ease of computation, otherwise the system would be non-linear, as Ut depends on both yt and st . This is not a strong assumption since after the very early computation of U during the data-collection period elements of U tend to be quite constant. Finally, Eq. (18) gives the expected signal by the private sector. This equation is utilized in the expectations updating Eq. (17). It is assumed that the private sector uses central bank’s nowcast. This assumption seems reasonable as long as the central bank makes efforts in order to signal forecasts. Moreover, this assumption is consistent with the spirit of the paper, which consider the private sector unable to know immediately the reliability of central bank’s information. Therefore, the nowcast enters into the expected signal but does not enter into the prior, because the expected signal, as well ^t is considered as a part of the signal, and as such, as the signal itself, is weighted according to its history. In R3 the nowcast y the private sectors does not necessarily think it is correct. 0 0 ^0t b E ps fytþ1 jst g0 sex The vector of the system’s unknowns is ½s0t y t  . At each iteration this system is solved and forecasts are accordingly provided to private agents. The stability under learning of this model is verified numerically only, as the parameters always converged for the given calibrations. In R1 the signal is an i:i:d. exogenous random variable, normally distributed with mean equal to zero and variance equal to 0.00356. In the long run, the simulated economy under R1 with a completely unreliable signal, determines elements of U which are all very close to zero. Therefore, it is possible to conjecture that in the limit the same elements tend to zero, so that the model with a completely random signal, tends to the pure adaptive learning economy. During the data-collection period, the signal is issued exactly the same way than in the following periods. In the data-collection period also the private sector acts the same way than in following periods, by assessing the relevance of the signal as a predictor by computing the matrix U. Thus the only difference between the two periods is that the signal is used only once the data collection period expired. According to Tables 5 and 6 the signal-processing structure is neutral with respect to the benchmarks. The estimated parameters are very close to the REE values, such that also the signal is highly predictable if the signal is clean. Therefore, by construction, as the signal does not carry a relatively large amount of information if the learning transition already occurred, the expectation error (which in turn affects Eps fytþ1 jst g) tends to zero. Things change if the signal is hit by disturbance. If a random shock hits the signal, the choice of the signaling regime is not neutral with respect to the benchmarks. In this case simulations are performed by adding a bivariate random disturbance gt  Nð0; Rg Þ to the signal, where Rg ¼ 0:00712I2 . The variance of the disturbance hitting the signal is proportional to the variance of the original shock, and is respectively 1%; 5%; 50% and 100% of the original shock. According to the simulation’s result, a disturbed noise is harmful to the economy in terms of central bank’s loss function. Both the signaling regimes R2 and R3 , where the signal is driven by a fundamentals-based forecast, perform worse. Regime R2 in some simulations comes out to lead economy to an unstable path if the variance of the disturbance is too high. Regime R3 performs better than R2 because of lower expected losses and also because economy never throughout the simulations. The two regimes differ essentially in the nowcast they base their forecasts on. Under R2 , the central bank’s nowcast does not consider the effect of the noise, while under R3 it does. In the simulations described above, the signal’s disturbance is not observable, such that the information contained in the signal is similar in R2 and R3 . Nevertheless, the economy’s dynamics is affected very differently in the two cases. The performance of the economy under R1 is quite ambiguous (Table 7). Indeed, expected losses are lower than in other regimes in some cases, but their behavior is not straightforward. In fact, expected losses decrease for noise’s variances from 1% to 50% of the generated shock, and then increases for a full shock, with respect to a 50% shock. In this case, as the signal is not useful in order to predict yt ; U quickly reaches values close to zero, therefore making the economy similar to the adaptive learning case, which actually has a lower expected loss. A comparison between R1 and R2 or R3 allows to figure out that by adding a high quality forecast to otherwise noisy information is beneficial to the economy if the noise level is low. The intuition behind this result is that if the noise level is low, a signal composed by central bank expectation plus the noise is essentially driven by central bank expectations. On the other hand, if the noise level is high, the signal is driven by the noise. A purely random signal tends to determine a weight matrix close to zero. Nevertheless, a noise driven signal that retains some Table 7 Expected losses.

Clean 1% 5% 50% 100%

R0

R1

R2

R3

51.7718 – – – –

– 54.6868 52.1075 52.1439 52.4848

51.7718 51.7814 52.1653 – –

51.7718 51.7745 51.7911 52.5753 54.3563

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information due to the presence of the central bank’s forecast, may tend to a weight matrix farther from zero. In this case, noise spreads throughout the economy because the signal is not totally uninformative. 3.2. Impulse response functions The short term economy’s response to a cost shock is simulated by Monte Carlo method. The model is simulated 500 times, then average values are plotted. Private agents collect ten years of data, from t ¼ 1 to t ¼ 40, before starting to use the signal. Since the comparison with R1 is meaningful in the long run only, it is not included in this subsection. The distinction between periods in which data are collected and the following periods, when signal is actually processed, holds for signaling regimes R2 and R3 only, since REE is unaffected by any kind of signal. Once data are collected, the private agents likely have a stable estimation of the MSV solution and of U. The shock occurs at time t ¼ 41. From that period on no disturbance affects costs and aggregate demand. Hence, after the shock occurs, it is possible to observe how the shock is absorbed. In Figs. 3 and 4 the initial point is not the steady state. Instead, in each simulation the response is evaluated from the state at the beginning of signal processing period. This peculiar setup allows to emphasize the role of learning on transition. In fact, learning relies entirely on previous estimates and on previous data, such that the realizations path is also relevant in characterizing agents’ information set. This assumption is relaxed in the next subsection, where simulated steady states are used as starting points. After the shock, agents keep learning. The model is calibrated according to MN with a ¼ 0:5. During the data collection period the autocorrelation coefficient of the AR disturbances are l ¼ q ¼ 0:8, and their i:i:d shocks’ variances are r2g ¼ r2u ¼ 0:0712. The response of the model to a cost push shock is evaluated by comparing the signaling regimes, namely R2 and R3 , with the REE. Simulation results are shown in Fig. 3 and are in line with the long term statistics showed above. In particular, the signaling regimes in which the central bank issues a correct signal (i.e. R2 and R3 in absence of disturbance) differ by the benchmarks negligibly. Indeed, REE, R2 and R3 overlap, even though some difference exist at a very low order of magnitude. This result supports the neutrality of signaling regimes as long as the signal is clean, i.e. in absence of noise. Under the assumption of a noisy signal, the economy’s response under regime R3 is closer to the REE than under R2 , and also shows a smoother path. This result is consistent with long term comparison between R2 and R3 with noisy signal. In Fig. 4 it is emphasized how noise variance affects the deviations from benchmarks. It can be observed that under R2 the economy’s response deviates from RE more than under R3 . For low noise variances (until 5%) deviations are relatively small under R2 as well, and roughly overlap to the clean benchmark. 4. Concluding remarks This paper addresses the question of whether, and how, communication of central bank’s forecasts might affect the dynamics of an economy. The analysis is conducted in a framework where private agents and central bank have heterogeneous information sets. The main finding of this paper are summarized as follows. If the signal is clean (i.e. not hit by any disturbance) the presence of the signal does not alter the conditions of equilibrium stability. When the signal is affected by random disturbance, it is possible that the equilibrium stability is not satisfied or the economic dynamics can be volatile, if the central bank does not internalize the influence of the signal on private agents. If the central bank takes into account the impact of signals on private agents’ expectations, the equilibrium stability is more likely to be satisfied and the economic dynamics to be less volatile. The standard adaptive learning and the RE versions of the economy are taken as benchmarks. Three further information regimes are analyzed. In the first ðR1 Þ the signal is a random disturbance. Under regime R2 the signal is constituted by central bank’s forecasts, which are computed without taking into account the effect of the signal on agents’ expectations. Under regime R3 central bank’s signal takes into account the effect on agents’ forecasts. An analytical closed form solution is provided for the signaling regime R2 . Numerical results are provided for all the signaling regimes. The simulation’s results provide some support in favor of encouraging central banks to signal their own forecasts. In fact, if the signal is clean, then the simulated model differs only negligibly from its rational expectations and pure adaptive learning counterparts. Nevertheless, if the signal is systematically hit by an i.i.d. shock, how the sender uses its information set is crucial in determining the economy’s response. If the central bank does not take into account its role of signal-sender the economy’s fluctuation around the path of convergence after the shock may be heavily altered. In particular, both the asymptotic statistics and the impulse response functions show that central bank’s attitude to communication, as well as the possibility of disturbances on the signal issued, might determine large fluctuations both in inflation and in output gap. It is shown that if the signal is systematically disturbed, expected losses increase with the variance of the noise. Moreover, the economy may become unstable. When the signal is systematically disturbed, a committed central bank which takes into account the effect of its signals to agents’ expectations, is more effective in its policy measures than a central bank neglecting it. The same result may be interpreted in a broader sense if agents tend to rely on noisy information. In that case, if noise variance is relatively low central bank’s signals may reduce the expected loss. For a relatively high noise variance this is not the case because, by providing informative content to the signal, the bayesian updating mechanism would weight more the signal, and the noise would spread through the economy.

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The results from the overall analysis suggest that one way to improve the effectiveness of monetary policy, in presence of noisy signals, is to take into account the effect of the noise variance on expectations. In particular, since in the real world the central bank’s knowledge of the economy’s structural equations may be better than the private sector, but not perfect, in its signal there will always be a random component, unrelated to the real economy, as a consequence of model uncertainty. Thus, trying to include an estimate of the central bank’s prediction error into the structural model, through private sector expectations, as in R3 , might actually reduce the relevance of uncertainty to economic dynamics. Acknowledgements I thank Christian Matthes, Giorgio Di Giorgio, Alessandro Pandimiglio, Marco Spallone, Jordi Galí and an anonymous referee for very helpful comments and suggestions. I also gratefully aknowledge hints and comments from participants to the Macroeconomic Breakfast seminar at Universitat Pompeu Fabra (December 2011). The usual disclaimer applies. Appendix A. A one-dimensional example Expectations in this model are formed by updating a prior as new information is communicated to agents. The signal is driven by the REE-based expectations about the next period’s value of the economy’s states. It is distributed as a gaussian variable, as well as the state of nature, which is, indeed, the vector of the state variables of the economy. Therefore, the probability distribution of the signal depends on the probability distribution of the state of nature. The basic intuition of how the updating procedure works can be explained by making use of the Bayes’ rule. The following example is developed from Chamley (2004).   h; r2h . Then, the Let us assume that the state of nature h is the realization of a scalar normal random variable, i.e. h  N    private signal s has a normal distribution and is defined by s ¼ h þ , where  is a independent of h and   N 0; r2 . Let  noise  us then suppose that the agent’s prior distribution on h is denoted by f ðhÞ  N m; r2 , and that the conditioned distribution of s on h has a density /ðsjhÞ. Therefore, the distribution on h is updated to f ðhjsÞ using Bayes’ rule:

f ðhjsÞ ¼ R

/ðsjhÞf ðhÞ : /ðsjhÞf ðhÞdh

The parameters of the posterior distribution are defined as follows:

r02 ¼

r2 r2 ; m0 ¼ gs þ ð1  gÞm; r2 þ r2

where g ¼ r02 =r2 . If the signal is infinitely precise (i.e. r2 tends to zero) the weight assigned to the signal tends to one. If the signal is totally unreliable (i.e. r2 tends to infinity) the weight assigned to the signal tends to zero. It is worth to remark that in this example the expected values of the signal and of the state of nature coincide, such that the final expected value m0 is just the weighted average between the signal and the prior. The same example can be generalized to a multivariate framework. Let’s consider the case of a bivariate gaussian y. The conditional distribution of a gaussian on another gaussian is gaussian as well. Let also ly be the 2  1 vector of the prior expectations on y, and Ky the 2  2 prior covariance matrix. Then,





lyjs ¼ ly þ KTys K1 s  ls : s If

ls ¼ ly it follows that Kys ¼ cov ðyex ; sÞ ¼ cov ðyex ; yex þ eÞ ¼ Ky :

Hence

lyjs ¼ Uly þ ðI  UÞs; where U ¼ I  KTy K1 s . The posterior covariance matrix is determined as follows:

Kyjs ¼ Ky  KTys K1 s Ksy ¼ UKy The two matrices of interest are

!

r2x qy ry rp ; Ky ¼ qy ry rp r2p stop

!

r2x þ r2ex qy ry rp : Ks ¼ qy ry rp r2p þ r2ep Also in the multivariate example, in the limit case of an infinitely precise signal, the agents assign no weight to their prior. The posterior value is determined entirely by the signal.

S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131 T

1

lim K K rex ;rep !0 y s

127

¼I ) U¼0

On the contrary, should the signal be completely independent, the posterior value would be entirely determined by the prior. T

1

lim K K rex ;rep !1 y s

¼ 0 ) U ¼ I:

If ls –ly the updating procedure works in the same way, although the posterior mean cannot be interpreted as a weighed average. The weight of the signal is generically counterbalanced by agents’ expectations over the signal itself (i.e. ls ). This signal extraction theory can be used in an adaptive learning framework where both the state of nature and the signal are distributed as gaussians. In this case the prior is determined in accordance with the adaptive learning algorithm, while the signal is determined by the RE expectations over the same economy. The posterior is the result of the bayesian update of the prior, according to the signal and the weight that it is given. Hence the posterior is the actual forecast, which is used in order to sign contracts in this economy. The nature of the updating process is static, since in each period it is reiterated in the same way. Time dependence arises through the estimates of signal covariance matrix, which is history dependent, but does not have any link with future values. Let us think of a univariate state of nature kt which evolves according to the following linear difference equation:

kt ¼ a1 Et ktþ1 þ a2 kt1 þ bzt ; where zt ¼ qzt1 þ t ; a1 ; a2 and b are the true structural parameters and t  Nð0; 1Þ. Let’s assume that agents are adaptive learners and that kt is directly observable. Let’s assume also that a very smart research center somehow discovered the true values of the structural parameters. In order to convince other agents that its model is correct, the research center publishes forecasts according to its supposedly true model of the economy. As a result, a signal based on RE is sent to economic operators. The signal actually consists of the RE forecast, which is computed moving ahead of one period the reduced form of the economy (i.e. the REE). In particular, it can be shown by undetermined coefficients method that the REE is the following:

kt ¼ /k kt1 þ

/z

q

zt ;

where /k and /z are combination of structural known parameters. Therefore, the research center’s forecasts are determined by ERC t ktþ1 ¼ /k kt þ /z zt st . Let us assume that agents have an idea of what the arguments of the REE are, but do not know exactly the parameters /k and /z . Thus, agents’ prior is determined by Et ktþ1 ¼ at þ bt þ ct qzt , where at ; bt and ct are the estimated values of the coefficients in the REE. For t ! 1, convergence occurs when a ! 0; b ! /k and c ! /z . Since the signal is issued on a period basis, the prior must be updated in each period. Hence, the posterior expected value of ktþ1 in each period is Et ktþ1 ¼ Et ktþ1 þ ut ðst  Et st Þ. In this case, as yt is directly observable with no lags, it is possible to invoke the anticipated utility principle proposed in Kreps (1998), which states that in adaptive learning models all the available data are already used to produce the most efficient estimations of the variables that st is aimed at forecasting. Thus, the expected value of the signal, conditioned to the private sectors information set, coincides with the prior on the underlying variable, such that Et st ¼ Et ktþ1 . It follows that Et ktþ1 ¼ ð1  ut ÞEt þ ut st , where ut ¼ rs;k ðtÞ=r2s ðt Þ 2 ½0; 1. The dynamics of a similar multivariate system is treated in the following subsection, where the basic idea exposed above is embedded into a log-linearized new Keynesian framework. Hence, in the following analysis the state of nature is a bivariate vector including output gap and inflation. Appendix B. Derivation of the T-map In order to determine the E-stability conditions it is necessary to compute the Actual Law of Motion of the economy, i.e. the structural reaction to expectations. The ALM is a process yt ¼ D þ Htt1 þ Cv t where D; H and C are combinations of the PS arguments that determine the final expectations EPS t ytþ1 , i.e. a; b; c; B; C and U. The ALM is obtained by substituting for Et .

yt ¼ MEPS ytþ1 þ Nyt1 þ Pv t n h io 2 2 ¼ M ½bc þ cf þ Ut ðC  cÞv t þ bt1 þ ðI þ bÞa þ þUt ðB  bÞ M ðI þ bÞa þ Mbt1 þ M ðbc þ cF Þv t þ Nyt1 þ Pv t þ Nyt1 pv t ¼ M ½bc þ cF þ UðC  cÞF þ UðB  bÞM ðbc þ cF Þ þ UðB  bÞP v t þ Pv t h i 2 2 þ M b þ UðB  bÞMb þ UðB  bÞN yt1 þ Nyt1 þ M ðI þ bÞa þ M UðB  bÞMðI þ bÞa: The previous equation implies what follows:

C ¼ M½bc þ cF þ UðC  cÞF þ UðB  bÞMðbc þ cF Þ þ UðB  bÞP þ P; h

2

2

i

H ¼ M b þ UðB  bÞMb þ UðB  bÞN þ N; D ¼ MðI þ bÞa þ M UðB  bÞMðI þ bÞa: Let T ða; b; c;Þ ¼ ðD; H; CÞ, such that yt ¼ T ða; b; c;Þð1; tt1 ; v t Þ0 . The dynamics of T ðÞ determines the convergence of yt . Therefore, by analyzing the stability of T ðÞ, i.e.

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dða; b; cÞ ¼ T ða; b; c;Þ  ða; b; cÞ; ds one obtains the asymptotic stability under learning of the economy. Appendix C. E-stability conditions I

 2  1 bkðdbx ðbp u12  1Þ þ au22 Þ ab k u21  dbp bx u11 abk2 u22  bdbp ðk þ abx u12 Þ B C d2 d2 d2 d2 B  C    B C B b dbp ða þ kbx u21 Þ  a2 u21 b dbx ða þ kbp u22 Þ  a2 u22 a bðak þ dbp bx Þu21 abðdbp  ðak þ dbp bx Þu22 Þ C B C  B C d2 d2 d2 d2 DT b ðÞ ¼ B C; B C bkb p B C 0 0  0 B C d B C @ A abbp 0 0 0 d ðC:1Þ 0

bkðdbp ðbx u11  1Þ þ au21 Þ

where d ¼ a þ k2 . Its eigenvalues are

w1 ¼ 0;

w3 ¼

w2 ¼ 

bp bk d

  p 2 2  b d 4bp dkbx dbx ðu11 ðbp ku22 þ aÞ þ ku21 ð1  bp u12 ÞÞ þ bp dðau12 þ ku22 Þ þ a2 ðu11 u22  u12 u21 Þ

1 3

2d

þ ðaðdbx  au22 þ ku21 Þ þ bp dkððu11 þ u22 Þbx  1ÞÞ þ

w4 ¼

1  2d3

2

! 

abd2 bx þ bp bd2 ku11 bx þ bp bd2 ku22 bx  bp bd2 k þ a2 ðbÞdu22 þ abdku21



ðC:2Þ

  1  p 2 2   b d 4bp dkbx dbx ðu11 ðbp ku22 þ aÞ þ ku21 ð1  bp u12 ÞÞbp dðau12 þ ku22 Þ þ a2 ðu11 u22  u12 u21 Þ

2d3

þðaðdbx  au22 þ ku21 Þ þ bp dkððu11 þ u22 Þbx  1ÞÞ  þa2 ðbÞdu22 þ abdku21 :

2



þ

1  2d3

abd2 bx þ bp bd2 ku11 bx þ bp bd2 ku22 bx  bp bd2 k ðC:3Þ

Eigenvalue w1 is equal to zero, and w2 is negative for all parameters positive values. w3 and w4 can be represented as A

where

A¼ and

1  2d3

pffiffiffi B,



abd2 bx þ bp bd2 ku11 bx þ bp bd2 ku22 bx  bp bd2 k þ a2 ðbÞdu22 þ abdku21 ;

   B ¼ b2 d2 4bp dkbx dbx ðu11 ðbp ku22 þ aÞ þ ku21 ð1  bp u12 ÞÞ þ bp dðau12 þ ku22 Þ þ a2 ðu11 u22  u12 u21 Þ  2 þ ðaðdbx  au22 þ ku21 Þ þ bp dkððu11 þ u22 Þbx  1ÞÞ :

Eigenvalue w3 is less than 1 if

pffiffiffi pffiffiffi A1< B ) 1A> B

then there exist two possibilities depending on whether B  0 or B < 0: Case I. The B component is negative, hence its square root is an imaginary number. As a such, it is not relevant in order to evaluate e-stability. In fact only the real part of the eigenvalue is required to be smaller than 1. Thus, if B < 0 a necessary and sufficient condition for third eigenvalue to be less than 1, is that A < 1.

B<0 )

pffiffiffi pffiffiffiffiffiffiffi pffiffiffi B ¼ i B ) A þ B < 1 if

A<1

Case II. B P 0, is a standard irrational inequality. It can be solved by imposing that both the members of

1A>

pffiffiffi B

are positive, and then solve by squaring both members,

ð1  AÞ2 >

pffiffiffi2 B

S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

129

such that

1 þ A2  2A > B A<1 are both necessary and sufficient conditions w3 to be less than one. As A < 1 also implies that w4 is less than one, then it is sufficient also for this case. The last two inequalities above describe the region in Fig. 2. Appendix D. E-stability conditions II  After having shown that the eigenvalues of DT b b have the following form:

 pffiffiffi pffiffiffi bp bk 0;  ; A þ B; A  B ; d where d ¼ a þ k2 and A and B are combinations of structural parameters and elements of U, in this Subsection it is simulated the asymptotic behavior of the model analyzed in Section 2. The purpose of this exercise is to support the claim that the model is e-stable, and therefore asymptotic convergence is a reliable result even when simulation are too short to clearly show a stable reversion to the REE values. The exercise consists in a long term simulation of the elements of U and of their distribution. Hence, a three-dimensional region is plotted in order to show the E-stability region as a function of the elements of U. The basic results show that the average values of the elements of matrix U are inside the E-stability region, providing a heuristic support to the convergence claim.  The first eigenvalue of DT b b is a constant smaller than one. The second eigenvalue is necessarily less than one because all parameters involved are positive. A sufficient but not necessary condition for the fourth eigenvalue to have real part smaller than one, is that A < 1. Nevertheless, as it will be shown below, the same condition is necessary for the third eigenvalue pffiffiffi to  be smaller than unity. Thus, a crucial issue for the e-stability condition related to DT b b is the magnitude of A þ B. The magnitude of A and B depends crucially on the elements of U which in turn are combinations of estimated variances and covariances between yt and st . The easiest way to verify the magnitude of the third eigenvalue and then one of the conditions for asymptotic stability under learning of this economy, is to check numerically, for given values of structural parameters a; b and k, whether it holds or not in each period and in the limit. One of the reasons that suggest to utilize this procedure is the time-varying nature of the matrix U. In fact, its elements depend on sample estimations of variances and covariances of st and yt available in each period, as a result of the private sector process of evaluation the signal’s reliability. The covariances involved in the calculation of U are estimated by means of the unbiased sample-variance. In what follows it is provided a three-dimensional e-stability region in terms of the elements of U, given some fixed values for the structural parameters a; b and k. The region describes the values of three out of four elements of U that let the pffiffiffi condition A  B < 1 hold, given the values assigned to the structural parameters a; b and k. By this methodology it can be shown that the way agents interpret the signal, i.e. the weight they give to the signal, maypaffect the asymptotic stability ffiffiffi under learning of an economy. All the four elements of U are involved in the definition of A  B, hence a further assumption is required. In order for the region to be defined, the value of one of the elements of U must be pinned down. E-stability assesses the parameters’ asymptotic convergence properties. A plot of the above mentioned e-stability region can be defined for given structural parameters a; b and k and for one given element of matrix U. In order to select the most suitable U element to this purposes, the limit of the U matrix is firstly simulated. It is chosen the elements of U that, in the limit, has the lower standard deviation. Then it is plotted the E-stability region according to the mean value in the limit of the chosen parameter. By choosing values 50% the mean value does not affect the results. The limit is approximated by the 10,000th iteration in a sample of 100 simulations. Fig. 1 plots the sample distributions of the elements of U at t ¼ 10; 000. The elements that exhibit the smallest dispersion around the mean is u21 , with a mean equal to 0:3587 and variance equal to 0:00063.4 Hence the region is plotted for given values of u21 , corresponding to the mean. Fig. 2 shows the region of e-stability for given values of the structural parameters, i.e. under the assumption that a ¼ 0:5; b ¼ 0:99 and k ¼ 0:3, which is the calibration utilized in the numerical analysis too. The graph is plotted for u21 ¼ 0:385 which is its limit mean value. The three axes represent values of u11 ; u22 and u21 from 10 to 10. It can be observed that the point defined by the average asymptotic value of u11 ; u22 and u21 , i.e. u11 ¼ 0:43; u22 ¼ 6:8 and u21 ¼ 0:35 falls into the stability region, and does not lay on its boundary. Appendix E. Derivation of matrix

!

Cov ðy; sÞ ¼

4

qx;sx rsx rx qx;sp rsp rx ; qp;sx rp rsx qp;sp rp rsp

In an experiment of 500 simulation, each of 290 periods, the mean and the variance of u21 are respectively 0:415 and 0:027. The mean and the variance of

u11 are respectively 0:5 and 0:035.

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!

Var ðsÞ ¼

r2sx grsx rsp ; grsx rsp r2sp 0

1

U ¼ Cov ðy; sÞ VarðsÞ

0 gqp;s rp rsx qx;s rsp rx x x rsx rsp 1 @ ¼ 2 gq r r  q p;sp p sx x;sx rsp rx g 1 2 rsx

gqx;sx rsp rx qp;sx rp rsx r2sp gqx;sp rx rsp qp;sp rp rsx rsx rsp

1 A:

Appendix F See Figs. 1–4.

PHI(1,1)

PHI(1,2)

250

250

200

200

150

150

100

100

50

50

0 0.4

0.5

0.6

0.7

0 6.5

7

PHI(2,1)

8

8.5

7.5

8

PHI(2,2)

250

250

200

200

150

150

100

100

50

50

0 0.2

7.5

0 0.3

0.4

0.5

6

6.5

7

Fig. 1. Distribution of U’s elements at t ¼ 5000 in a sample of 1000 simulations under R2 .

 Fig. 2. Combinations of u11 ; u22 and u12 implying that the real part of the third eigenvalue of DT b b is less than one, under the assumption that u21 ¼ 0:385.

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S. Marzioni / Journal of Macroeconomics 40 (2014) 114–131

Clean signal

Disturbed signal 0.06

0.06

0.04

0.04

x

x

0.02 0.02

0

0 -0.02

-0.02 -0.04 0

2

4

6

8

10

12

14

16

18

20

0

0.06

0.06

0.04

0.04

2

4

6

8

10

12

14

16

18

20

0.02

0.02 REE R2 R3

0 -0.02 0

2

4

6

8

10

12

14

16

18

REE R2 R3

0 -0.02 20

0

2

4

6

8

Quarters

10

12

14

16

18

20

Quarters

Fig. 3. Average IRF under all regimes after a 1% cost push shock.

R3

R2 0.06

0.06

0.04

0.04

x

x

0.02 0

0.02

-0.02

0

-0.04

-0.02 0

2

4

6

8

10

12

14

16

18

20

0.06 Clean 1% 5% 50% 100%

0.04 0.02

0

0 -0.02

4

6

8

10

12

Quarters

14

16

18

20

6

8

10

12

14

16

18

20

Clean 1% 5% 50% 100%

0.02

-0.02 2

4

0.04

0 0

2

0.06

0

2

4

6

8

10

12

14

16

18

20

Quarters

Fig. 4. Noise impact on average IRF under R2 and R3 . Percentages refer to the noise’s variance, as a proportion of the variance of the simulated shock.

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