Sparse Restricted Perception Equilibrium

Transcription

1 Sparse Restricted Perception Equilibrium Volha Audzei and Sergey Slobodyan Abstract In this work we consider a concept of sparse rationality developed recently by Gabaix 04) as a selection tool in a model with multiple equilibria. Under sparse rationality paying attention to all possible variables is costly and the agents could choose to over- or under-emphasize particular variables even fully excluding some of them. Our main question is whether an initially misspecified equilibrium the Restricted Perceptions Equilibrium or RPE) is compatible with the equilibrium choice of sparse weights describing allocation of attention to different variables by the agents inhabiting this RPE. In a simple New Keynesian model we find that the agents stick to their initial mis-specified AR) forecasting model choice if the feedback from expectation in the model is strong or inflation becomes more persistent. We also identify a region in the parameter space where the agents find it advantageous to pay attention to no variable at all. Stronger monetary policy response decreases the region in the parameter space where the mis-specified rule survives. JEL Codes: D84 E3 E37. Keywords: Bounded Rationality Expectations Learning. Volha Audzei Czech National Bank Economic Research Division Monetary Department and CERGE-EI a joint workplace of Charles University and the Economics Institute of the Czech Academy of Sciences Prague Czech Republic vaudzei@cerge-ei.cz Sergey Slobodyan National Research University Higher School of Economics St. Petersburg Russia and CERGE-EI a joint workplace of Charles University and the Economics Institute of the Czech Academy of Sciences Prague Czech Republic sergey.slobodyan@cerge-ei.cz We thank to Filip Matejka Xavier Gabaix Raf Wouters Pierpaolo Benigno Alexis Dervitz and participants of various conferences and workshops. Authors are grateful for the financial support of the Czech Science Foundation project No. P40//G097 DYME Dynamic Models in Economics. The views expressed in this paper are those of the authors and not necessarily those of the Czech National Bank.

2 Volha Audzei and Sergey Slobodyan. Introduction It has been understood for a long time that the hypothesis of Rational Expectations RE) while delivering a theoretically elegant model consistent and typically unique solution for the agents expectations imposes on them cognitive and computational demands that might be incompatible with reality. As a result deviations from RE have been studied in a growing stream of theoretical and empirical literature including Bounded Rationality Marcet and Sargent 989) Adaptive Learning Evans and Honkapohja 00) Sticky Information Mankiw and Reis 007) Rational Inattention Sims 003) and Sparse Rationality Gabaix 04) approaches among others. Our paper contributes to the literature by studying the interaction between monetary policy and equilibrium forecasting rules and making connections between two distinct concepts adaptive learning and sparse rationality. We study an economy where adaptively learning agents choose a strict subset of variables from the rational expectation equilibrium set for their forecasting functions thus inducing a restricted perception equilibrium RPE). We then allow these agents inhabiting the RPE to reconsider their forecast rules subject to the informational cost constraint modeled as in Gabaix 04). We ask whether the model parameters which make the RPE stable are sufficient to ensure that the same subset of variables is selected by informationally constrained agents. In other words we are interested in whether the initial mis-specification becomes self-perpetuating in case of informational constraints. One of the key parameters defining the survival of a mis-specified rule is expectational feedback which is a function of monetary policy aggressiveness. We show that a stronger monetary policy increases the region in the parameter space where the agents move to the RE equilibrium. We consider two policy rules with forward looking inflation expectations and with contemporaneous inflation expectations. Under policy rule with the contemporaneous inflation expectations there is smaller region in the parameter space where mis-specified rule survives in equilibrium. Our paper is related to the literature on adaptive learning and bounded rationality. In an adaptive learning approach to modeling deviations from RE agents are assumed to not possess prior knowledge of underlying structure of the economy and to be gradually learning the coefficients in their forecasting rules. The survey of this approach to learning in macroeconomics can be found in Evans and Honkapohja 009). Several papers have shown that adaptive agents could persist in using the forecasting rules that are mis-specified relatively to the RE ones cf. Molnar 007) and Evans et al. 0). In Molnar 007) there is a class of agents who are learning what is the best forecasting model given the past data. Even if their forecasting rules are mis-specified such learners can survive the competition with RE agents. In Evans et al. 0) convergence to a mis-specified equilibrium happens when the expectation feedback is strong. Adam 005) considers the economy where agents are restricted to proceed only a certain number of variables in the regression and thus to use underparametrized forecasting rules. As the agents expectations affect the data generating process of the model and induce a Restricted Perception Equilibrium RPE) the restricted rule can outperform the rational expectation rule in equilibrium. Similarly to Evans et al. 0) this happens for the large enough feedback from the expectations to the outcome variable. One of the ways to justify the agents using mis-specified forecasting rules is to assume that they have limited information processing capacity as is done in the rational inattention literature cf. Sims 003) Mackowiak and Wiederholt 009) and Matejka and McKay 05). In this literature attention allocation is based on the concept of entropy. Gabaix 04) pursues a different less computationally demanding approach with agents allocating attention weights to variables based on their relative importance.

3 Sparse Restricted Perception Equilibrium 3 There are now number of papers that show how bounded rationality influences monetary and fiscal policy effects. Among them are Gabaix 06) and García-Schmidt and Woodford 05). Some of their findings call for reconsideration of Taylor principle and explain forward guidance puzzle. As agents do not see far in the future in Gabaix 06) current policy has limited effect for the future agents decisions. Likewise the announcement of future policies has a dimmed effect on agents contemporaneous decision. Unlike these papers we are interested in the question how monetary policy influences expectation formation not how expectation formation influences monetary policy. Another strand of research on bounded rationality we relate to is experimental research on interaction of monetary policy and agents expectations. Examples are Pfajfar and Žakelj 07) and Assenza et al. 03) who find that the monetary policy aggressiveness influences choice of the model. Studies by Hommes 04) and Heemeijer et al. 009) support the importance of expectational feedback parameter for the survival of mis-specified rules. In our paper expectational feedback parameter is an inverse of monetary policy aggressiveness and we study how it influences the agents modeling choice. The paper is organized as follows. In the second section we study an economy where agents learn about a simple exogenous process. We derive analytical results and provide economic intuition for it. In the third section we move to a simple New Keynesian model and study the interaction of monetary policy and agents forecasting rules. The last section concludes.. Simple Model In this section we demonstrate main intuition behind our results for a simple process. Later in the paper paper we generalize our findings for a 3-equation New Keynesian model and discuss the possible policy implications. We start our analysis with a simple process y t = α + βe t y t+ + γ w t + γ w t + η t ) where w t and w t are persistent shocks such that [ w t w t ] [ ρ 0 = 0 ρ ] [ w t w t ] + [ ε t ε t ]. The wt wt ) shocks are normally distributed around zero with variance-covariance matrix [ Σ w = σ ρ σ ρ σ σ with ρ [ ] being the correlation coefficient between the shocks defined as Covw t wt ) σ. The RE minimum state variable solution MSV) of this model is given by with MSV coefficients: ] y t = a + g w t + g w t + η t g = g = γ βρ ) γ βρ 3)

4 4 Volha Audzei and Sergey Slobodyan and η t is a white noise. We restrict the agents to using only one variable in their forecasting models as in Adam 005) and Adam 007) framework. In our model their forecasting rule could use either w t or w t : y t = a + b w t + η t 4) y t = a + b w t + η t. 5) Without loss of generality we assume that our agents use 4) as their Perceived Law of Motion PLM) which induces the restricted perceptions equilibrium that we call RPE when agents use this PLM to form expectations about y t+. Substituting the forecast formed using 4) into ) we obtain the actual law of motion ALM): with y t = α + βa + b w t + b w t + η t 6) b = βρ b + γ 7) b = γ. 8) We model our agents as econometricians who do not have the prior knowledge about the underlying structure of the economy. They however do the best they can using the past data. In order for this learning process to converge three conditions must hold. First for the agents PLM to be the equilibrium solution the coefficient b must be derived as a regression coefficient from ordinary least squares: b = Covy twt ) Var wt ). 9) Second the equilibrium 4) must be expectationally stable E-stable). Finally the forecast errors produced by the rule of their choice 4) must be smaller than that of the alternative 5). We measure forecast errors as mean squared forecast errors MSFE) Proposition.0.. In RPE RPE) where the agents use as forecasting rule equation 4 5) the equilibrium coefficient b b ) is given by b = γ + γ ρ σ βρ b = βρ b + γ ) ρ σ + γ. Both RPE and RPE are E-stable. If the following condition holds then the mean squared forecast error MSFE) for an agent inhabiting RPE and using 4) as forecasting rule MSFE is smaller than MSFE of agent using 5) MSFE and thus RPE is an equilibrium if: b σ > b σ. 0) Such a restriction could be motivated by empirical and experimental evidence cf. Branch and Evans 006b) Adam 007) Hommes 04) and Pfajfar and Žakelj 04) who show that very simple AR) rules might be used by subjects to forecast inflation in survey and experimental settings. Several papers that estimate DSGE models with adaptive expectations for example Slobodyan and Wouters 0) and Ormeno and Molnar 05) show that assuming the agents are using very simple forecasting rules leads to superior model fit.

5 Sparse Restricted Perception Equilibrium 5 Proof. Appendix A. We next allow the agents to challenge their equilibrium forecasting rules and possibly reconsider them. We conduct our analysis for the case when RPE is an equilibrium. That is agents have initially chosen to use w in their forecasting rule and have ex-post found out that it produces smaller forecast errors than a model with w i.e. 0) holds. Note that in our simple model both forecasting rules are under-specified. Thus the analysis for the case where the agents have initially started with RPE would be symmetric. Our agents know that there exist other variables in the RPE 6) and w t is observable. They also know that using w t alone for forecasting is inferior to using only w t because 0) is true. However they may wonder whether adding w t to their forecasting rule is beneficial. The forecasting rule that includes both w t and w t would be clearly superior in this model if the agents were allowed to learn its coefficients coinciding with b and b. The agents however are subject to attention cost modeled as in Gabaix 04). They could attach weights to a variable according to its importance. The importance of a variable depends on its contribution to the variance of the variable of interest y in our case) and to the agents utility. The weights then determine how much attention is paid to a variable given the exogenously cost of attention and the loss stemmimg from inattention which is a reduced quality of the forecast. We let the agents to choose the attention vector by maximizing the precision of their forecast of y t as in 6): u = ŷ t y t ). ) For agents with rational expectations the optimal forecast is equal to ŷ t = b w t + b w t where b and b are OLS estimates of the coefficients in 6). That is agents with rational expectations use both shocks in the forecasting rule. Sparse rational agents face a trade-off between attention cost and increase in forecast precision. That is why they choose to allocate attention between the variables and form their optimal forecast rule as ŷ t = b { m w t } + b { m w t } where m m [0] are attention weights and b b are OLS estimates of the coefficients in the selected forecasting rule. Clearly if m or/ and m are close to zero estimates of b b are different from rational expectation case. If both weight are non-neglible the estimates of the coefficients converge to RE case. Denoting x = wt wt ) T ) µ = b b and a = ŷt = i m i x i the agent s action: their i= b forecasting rule we rewrite 6) as max u = ) a b m m wt b wt η t. Then following Gabaix 04) we use lasso estimator to derive the optimal attention vector m : ) m = arg min m [0] n m i )Λ i j m j + κ m i ) i j=...n i j=...n for n =. The loss from inattention is given by Λ i j = σ i j a wi u aa a w j while the parameter κ governs the attention cost. Loss from inattention reflects how much of variation in the process we lose when If all agents start using both shocks in their forecasting rule using least-squares learning from then onward then their PLM coefficients will converge to those of MSV solution ) 3) as was shown in Evans and Honkapohja 994). However we are asking whether an atomistic agent could find it advantageous to use both shocks for forecasting.

6 6 Volha Audzei and Sergey Slobodyan neglecting the variable. In the formula ) σ i j denotes the i j) element of shocks variancecovariance matrix Σ w a wi = u aa u awi determines by how much a change in a variable w i changes the agent s action a and u aa = u a a = a u awi = b i. ) a b wt b wt ) = Then a wi = b i and the cost of inattention is therefore given by Λ i j = σ i j b i b j. Taking the derivatives of ) with respect to m and m gives the following expressions details are in Appendix A.): m = m = κ b b σ ρ ) b σ b ρ b 3) κ b b σ ρ ) b ρ b σ b σ. 4) One can immediately observe that both weights are falling with attention cost κ. If attention costs are larger enough agents would choose not to pay attention to any variable and use only constant term in their forecasts). Both weights are decreasing with the ALM coefficients on another variable - e.g. weight on the first shock is lower if b is larger. The effect of ρ correlation between variables is less straightforward. The correlation for m ρ > 0 is given by b + ρ ) > b σ ρ. This is true as long as b > 0. On the other hand m ρ > 0 condition given by b σ + ρ ) > b ρ could be obtained for both b > 0 and b < 0 but we need ρ to be small enough for this to be satisfied. Proposition.0.. The condition for m > m coincides with the condition for MSFE < MSFE : which is equivalent to equation 0) b σ > b σ Proof. Appendix A.. Proposition.0. states that as long as using consistent with RPE PLM 4) produces smaller forecast errors than using PLM 5) sparsely rational agents optimally pay more attention to the variable w t than to w t. For the agents to stick to the first mis-specified rule the weight on the shock w t must be zero. In the RE MSV equilibrium the weights on the both shocks must be significantly larger than zero. Proposition.0.3. A shock i gets zero weight in agents forecast when for i j: ρ ) b κ i σ i ρ b 5) i σ i b j σ j Proof. Re-arranging 4) κ b j b i σ j σ i ρ ) b j σ j ρ b i σ i κ b i σ i ρ ) b i σ i ρ b. i σ i b j σ j

7 Sparse Restricted Perception Equilibrium 7 Proposition.0.3 also states that if agents use PLM consistent with RPE and costs of attention are large enough agents choose not to pay attention to any variable. We could interpret the condition 5) as follows. Consider the ALM 6). It includes two normally distributed variables w t = b w t and w t = b w t. The correlation coefficient between w t and w t is equal to ρ. Then ρ b σ b represents the coefficient in a regression of w t on w t while ρ ) b σ is the variance of the conditional distribution of w t given w t. Thus if the cost of attention κ corrected for the information about w t already contained in 4) is larger than the variance of omitted information variance of w t conditional on w t then the agents find it beneficial not to include the second shock into their forecasts. The condition 5) then has an intuitive economic interpretation of equalizing costs and benefits of considering that portion of information contained in the second shock w t which is above and beyond that which is already evaluated given that w t is taken into account in the forecast. With correlation close to zero the second shock is included into the forecasting rule even for larger attention costs. Note that this effect is more pronounced for positive correlation. To get an intuition for this result consider that if Cov w t w t ) ) = ρσ σ then Cov w t w t ρ σ w t = 0. The pair of orthogonal variables w t and w t ρ σ w t contains all the information variance) contained in the ) pair w t and w t. However their sum w t + w t ρ σ w t = w t ρ σ + w t has smaller norm than w t + w t = y t for positive ρ. A consequence of the fact that we need a shorter vector to summarize ) is an effectively smaller attention cost κ ρ σ ). For all the information contained in w t w t negative ρ the reasoning is the opposite: two orthogonal components sum up to a vector that is larger in norm than y t effectively increasing the attention cost. For a more intuitive representation we now introduce a measure which is not affected by the absolute values of the shock variances and persistences but only by their ratio. For this purpose we normalize the attention costs in 5) by b σ ε where σ ε is the variance of ε t the innovation to the shock w t. We define the cost-to-variance ratio as f κ b σ ε where κ is the threshold for the second shock to be included into the agents rule. For κ < κ agents use both shocks in their forecasting rules and for κ κ agents stick to the rule with the first shock only. We plot this ratio f in figure in the coordinates ρlogγ Σ) ) where Γ Σ γ σ ε γ σ e is the relative importance of the first shock w in the true data-generating process. We do this for different values of ρ and ρ autocorrelation coefficients of the shocks wt and wt. The agents include wt into forecasting rule when the normalized attention cost κ b σ ε w t used for forecasting is wider. is smaller than f. Thus larger f means that the range of costs consistent with In the figure white area corresponds to the parameters values where RPE is not selected as it produces larger forecast error than RPE. Large persistence of the w t ρ decreases the parameter space where MSFE < MSFE while increasing ρ expands this area. Higher ρ increases the share of the variance of y t explained by the first shock and thus decreases the value of taking into account the second one. Higher ρ has the opposite effect. Large correlation between the shocks ρ also contributes to better forecasting performance of RPE-consistent PLM A) so that MSFE < MSFE. As described in Appendix A. the MSFE < MSFE condition is satisfied for positive ρ if ρ > βρ ΓΣ which is the upper bound for white βρ area in all the graphs. As one can see in the figure for ρ = 0.9 in case of large persistence of the first shock so that βρ > / RPE PLM outperforms RPE also for very negative correlation such that

8 8 Volha Audzei and Sergey Slobodyan ρ =0.9 logγ Σ) ρ =0.4 log Γ Σ) ρ ρ = ρ ρ =0.9 Figure : Threshold for the Cost-to-Variance Ratio ρ < βρ ) ΓΣ. When the MSFE criterion 0) is satisfied large absolute correlation increases βρ the variation in y t explained by the first shock alone making right hand side of 5) smaller. When the threshold for cost-to-variance ratio is small it means that the second shock is added to the PLM only for very small learning costs to variance ratio. Finally note that even under condition logγσ) > 0 which means ΓΣ > so that w t plays more important role in ) that w t and both ρ and ρ are large so that taking into account w t is very informative see lower left panel of figure ) there could be still attention costs low enough for w t to be included into sparsely rational agents PLM. Therefore unconstrained agents with κ = 0 will always find it beneficial to include w t into their forecasting rules. In the next section we move to a three-equation New Keynesian model and study possible policy implications. 3. Three-Equation New Keynesian Model In this section we employ the textbook three-equation New Keneysian Model as in Galí 05 section 3). We first show that the conditions derived in the previous section can be generalized to this model and then discuss the policy implications. The reduced linearized around the deterministic steady state solution of the model takes the form: y t = σ i t E t π t+ ) + E t y t+ + g t 6) π t = βe t π t+ + ωy t + u t 7)

9 Sparse Restricted Perception Equilibrium 9 where π t = p t p t is inflation and y t is output gap. g t and u t are Gaussian shocks with certain variance. We assume that agents do not observe current shock realization that is E t g t ) = 0 and E t u t ) = 0. Parameter ω is a combination of the underlying model parameters : ω = θ) βθ) θ α α + αε σ + ϕ + α ) 8) α with θ is Calvo probability ε is demand elasticity and α governs the return to labor in the production function σ is consumers risk-aversion and ϕ is Inverse Frisch elasticity of labor supply. The production function takes the form: Y t = A t N α t. 9) If we assume constant return to scale in the production function α = 0 expression for ω collapses to ω = θ) βθ) σ + ϕ) 0) θ leaving us with Calvo probability and 3 standard parameters of utility function. We make another simplifying assumption that agents use naive forecast of output gap so that our agents have to forecast only inflation: E t y t+ = y t. ) In the framework of adaptive learning such an assumption can be viewed as an approximation of constant gain learning where gain on output gap is larger then gain on inflation. 3 The monetary authority sets interest rates in reaction to inflation expectations. One may consider alternative rules of the form: i t = ϕ π E t π t+ π) + π ) i t = ϕ π E t π t π) + π 3) where π is steady state inflation or inflation target. That is monetary authority reacts either to deviation of future or of current inflation expectations from the target. Rational expectations equilibria and their properties under different policy rules are formalized in the proposition Proposition The rational expectation equilibrium solution to B) and B) of the form ˆπ t = α + bπ t + cy t with the coefficients α = a π b π 4) b = b π 5) c = c π b π 6) 3 A simple constant gain learning was found to give the best fit for survey of professional forecasters in Branch and Evans 006a).

10 0 Volha Audzei and Sergey Slobodyan with the coefficients a π b π and c π defined in B3) - B8) is not E-stable. The minimum state variable solution MSV) of the form ˆπ t = α + cy t with the coefficients is determinate and E-stable for ϕ π >. a π α = 7) b π b = 0 8) c π c = 9) b π The rational expectation equilibrium solution to B45) and B44) of the form ˆπ t = α +bπ t +cy t with the coefficients α = a π b f 30) π b = bc π b f 3) π c = c π b f 3) π with the coefficients a π b f π b c π and c π defined in B46) - B53) is not E-stable. The minimum state variable solution MSV) of the form ˆπ t = α + cy t with the coefficients: is determinate and E-stable for ϕ π >. a π α = b f π b c 33) π b = 0 34) c π c = b f π b c 35) π Proof. Proof is in the Appendix B.. As in the previous section we restrict our agents to use only one variable in their forecasting rules. In the context of this model however one rule is mis-specified while another would correspond to rational expectations minimum state variable solution MSV). Similar to Adam 005) e considers two RPE s in which agents use either lagged inflation or lagged output gap in their forecasting rules. In M π agents use the rule while in M y it is: ˆπ t = α π + b π π t 36) ˆπ t = α y + c y y t. 37) M y rule could nest the MSV solution while M π is mis-specified as the lag of inflation does not enter the law of motion B). Yet if agents stick to M π the lag of inflation affects the actual law of motion through expectational terms. Under each forecasting rule the actual law of motion is different because the way expectations are formed affects the actual process for inflation and

11 Sparse Restricted Perception Equilibrium output gap through E t π t+. If agents use M y rule the actual law of motion is that of MSV solution and the rule is an equilibrium choice. If however agents use M π there is a region in the parameter space where this mis-specified rule is preferred in equilibrium. For a mis-specified rule to be an equilibrium choice it should result in smaller forecast errors than the alternative rule. We consider if M π would still be equilibrium choice if the agents inhabiting the RPE induced of the forecasting rule 36) could consider adding y t to their PLM subject to non-zero attention costs. Proposition For the model described by 6) 7) and policy rule as in ) or in 3) the condition for M π to be the equilibrium coincides with the simple model case condition MSFE π <MSFE y and holds if: b πσ π > c πσ y 38) where b π and c π are ALM coefficients on lag of inflation and output gap respectively and σ π and σ y are corresponding variances. M π results in smaller mean forecast squared errors than M y if: Γ Σ > 39) where Σ = σ π σ y and Γ = b π cπ. That is if larger share of inflation variation B6) is explained under M π. Proof. Proof is in the Appendix B.. 3. Attention Weights Proposition 3.0. states that the criterion for a mis-specified PLM to be an equilibrium is similar to the condition from the simple model see proposition.0.. That is M π rule must explain larger variation in inflation than M y. Determinacy requires that the Taylor principle must be satisfied with the inflation reaction coefficient larger than unity. Suppose that the actual law of motion is as in B6)-B63) and the agents could reconsider their forecasting rules. Will they give a significant weight to the past output? In other words would they stick to the mis-specified rule or move to the RE solution? 4 Or if attention costs are present will the agents choose not to pay attention to any of the variables? To answer these questions we let the agents select the sparse weights as in ). The difference from the simple case in section is that the variables used for forecasting are endogenous not exogenous. That is agents are minimizing ˆπ t π t subject to attention cost. Denoting a = ˆπ t = m y c π y t + m π b π π t to be the agents forecasting rule x = π t y t ) T µ = b π c π ) we rewrite utility as u = a µx). 40) To find the sparse weights agents minimize ) where the cost of inattention Λ i j = σ i j a wi u aa a w j is now modified to account for the new variables and covariance: u aa = u a a = ) a b π π a t c π y t ) = u aπt = b π u ayt = c π a πt = u aa u aπt = b π a yt = u aa u ayt = c π. 4 Again if agents start to include lagged output in their forecasting rules and employ least squares learning onward they eventually learn the coefficient on past π and y with the former being zero.

12 Volha Audzei and Sergey Slobodyan σ Then the cost of inattention is: Λ = π b π σ πy b π c π σ πy b π cπ σy c π ) where σ π σ y and σ πy are variance of inflation output and their covariance respectively under M π they are derived in Appendix B.. Taking the first order conditions of ) and solving for weights results in the following expressions: m y = m π = κ c πσy σ π R ) b π σ π c π σ y R 4) b π κ b πσ πσ y R ) c πσ y b π σ π R 4) c π where R = σ π y σ π σ y using M π rule. is the correlation between y t and π t in RPE π equilibrium consistent with agents Proposition Whenever MSFE π < MSFE y past inflation gets larger weight than past output: m π > m y. Proof. Proof is in Appendix C. Note that the result is identical to condition 39). The proposition summarizes the condition for output to get a positive weight in agents forecast. Proposition Lag of output gets positive weight in agents forecast when R ) c πσy κ < R c πσ y 43) b π σ π Proof. Proof is in Appendix C. 3. Policy Rules and Forecasting Rules Below we consider the interaction between monetary policy and equilibrium selection of forecasting rules. For all numerical simulations we use textbook calibration with 5 : β = 0.99 coefficient of risk aversion σ = Frish elasticity of labor supply ϕ = Calvo probability θ = /3 finally we set inflation target π = 3. For some of the graphs we use ratio of shock deviations r σ u σ g where u and g are innovations in the IS and Philips curves respectively cf. 6) and 7). Without loss of generality we set σ g =. Figure shows the dimensionless threshold for the cost-to-variance ratio for both policy rules defined as: 5 as in Galí 05 section 3 f κ c πσ y = R ) R c πσ y b π σ π. 44)

13 Sparse Restricted Perception Equilibrium 3 Here κ is the maximum cost ot attention at which the agents are willing to include output gap in their forecasts. Figure shows that when attention costs are present initially mis-specified forecasting rule can be supported under sparse-rationality. With a very aggressive policy the mis-specified model is never the equilibrium choice as it produces larger MSFE than M y. In the parameter range where the mis-specified rule is chosen the stronger is the policy reaction the larger is the range of attention costs when lagged output gap is included in the forecasting rule. I.e. the agents choose to include both output gap and inflation even for larger attention costs. This is shown as the larger area under threshold for the cost-to-variance - blue solid line. The intuition for this can be found in ALM coefficients. The coefficient on the past inflation in the ALM for inflation green dotted line is decreasing with the policy rule meaning that inflation becomes explained more by lagged output gap and less by its own past value. With an increasing importance of output gap in predicting inflation it is not surprising that agents agree to pay more to include output gap in their forecasting rules. Correlation between output gap and inflation is increasing with policy aggressiveness. As was discussed in section the effect of correlation on weights is not linear. When inflation is persistent coefficient on its lag is large in the Philips curve) and volatile larger correlation contributes to smaller weight on output gap. When inflation becomes less persistent and volatile agents assign larger weight with larger correlation. The two panels on Figure are rather similar demonstrating that there is no significant difference between the policies ) and 3). Though under contemporaneous inflation in the policy rule the mis-specified rule can be avoided with less aggressive monetary policy. For any ϕ π the threshold attention cost-to-variance ratio is larger under contemporaneous policy therefore the agents find it more important to pay attention to the output with the rule 3). As for the weights Figure 3 shows the agents choices for different learning costs κ and policy parameter ϕ π while fixing ratio of deviations in innovations to r = When the weight on output gap is zero but on inflation positive the region is defined as M π as it corresponds to agents sticking to an initially mis-specified rule. For large attention costs the agents choose to have zero weights on both variables and use only constant in their forecasting rules. This region is defined as 00). A similar figure but for the weight on inflation only is presented in Appendix D. An increasing policy parameter first decreases both weights as aggressive policy decreases volatility of output and inflation and the agents choose to pay less attention to both variables. Then with even more aggressive policy volatility of inflation falls more than volatility of output weight on output starts to increase while weight on inflation continues falling. Figure 4 shows standard deviations of inflation and output gap under both policy rules. Note that the U-shaped form of M π region in Figure 3 depends on which variable has larger volatility in Figure 4. While volatility of the both variables declines with an increasing policy parameter ϕ π volatility of inflation is relatively large. In this region both weights are falling with the tighter policy. Once output volatility is larger than that of inflation weight on output starts to increase with the policy response while weight on inflation starts to fall. For any strength of the policy response the rule with contemporaneous inflation results in less volatile environment. This finding is consistent with the experimental results in Pfajfar and Žakelj 07) where the contemporaneous rule is found to 6 While keeping deviation of output shock σ g fixed larger r means larger relative inflation volatility. This results in smaller estimates of coefficient on inflation and smaller weights. With larger r out figures look similar but the agents move to REE with smaller policy parameter.

14 4 Volha Audzei and Sergey Slobodyan Figure : Threshold for the Cost-to-Variance Ratio a) Forward Looking Rule b) Contemporaneous Rule Note: dotted line corresponds to the correlation between output and inflation dashed line - to the ALM coefficient on the past inflation blue solid line - to the threshold for cost-to-variance ratio r=0. produce smaller volatility than the forward-looking rule. The large inflation volatility in Figure 4 can be linked to large forecast errors of output gap. Our results contribute to the literature discussing how monetary policy affects not only the level of expectations but the way the expectations are formed. In our model a more restrictive monetary policy decreases the parameter range of stability of M π rule both in favor of MSV-nesting rule and the intercept-only rule. Therefore the agents either start taking into account the output gap if the κ is small or disregard any variables if the κ is larger. Including contemporaneous inflation expectations in the policy rule as opposed to future inflation expectations increase this parameter range. The result is intuitive since the strength of the policy ϕ π in the model we study decreases the feedback from expectations in the actual law of motion. 7 The role of the feedback parameter is widely supported in the literature. In an experimental setting Hommes 04) and Heemeijer et al. 009) emphasize the importance of expectations feedback parameter. In Hommes 04) when the expectations feedback parameter is negative convergence to REE is observed but with a positive parameter agents coordinate on a non-rational self-fulfilling equilibrium. Large feedback parameter results in convergence to a mis-specified equilibrium in Evans et al. 0). In the laboratory experiments of Pfajfar and Žakelj 04) Pfajfar and Žakelj 07) and Assenza et al. 03) agents choose forecasting rules of inflation under alternative monetary policy regimes differing by the aggressiveness of response to inflation in the Taylor rule. As monetary policy becomes more aggressive more agents switch to using forecasting rules compatible with rational expectations. As Pfajfar and Žakelj 07) discusses the aggressiveness of monetary policy response resembles the expectation feedback. This supports our finding that for low κ and large ϕ π agents increase their weight on lagged output gap. Another prediction of the paper is that agents stick to AR) model for inflation forecasting when persistence of inflation is high and/or the correlation between output and inflation is low see Figure. This result is in line with the observed behavior of professional forecasters after the recent financial crisis. There are number of studies examples being Fendel et al. 0) Lopez-Perez 07) and Frenkel et al. 0) showing that professional forecasters predictions behave as if 7 To see this consider the coefficient on inflation expectations in the actual law of motion for inflation in B) and B45) for forward looking and contemporaneous rules respectively.

15 Sparse Restricted Perception Equilibrium 5 Figure 3: Model Selection under Sparse Weights M y a) Forward Looking Rule b) Contemporaneous Rule Figure 4: Equilibrium dynamic of inflation and output standard deviation standard deviation a) Forward Looking Rule b) Contemporaneous Rule Note: dashed line corresponds to output gap solid line - to inflation r=0.

16 6 Volha Audzei and Sergey Slobodyan they are using Phillips curve. After the financial crisis inflation became more persistent cf. Watson 04) while Phillips curve got flatter. 8 In terms of our model this means lower threshold for costto-variance ratio in Figure. Lopez-Perez 07) 9 shows that forecasters predictions started to react much less to the unemployment after the financial crisis consistent with our model predictions. 4. Conclusion In this paper we study if the initially mis-specified forecasting rule can be an equilibrium choice under sparse-rationality. We first consider a simple process consisting of only exogenous variables and then generalize our results to the three-equation New Keynesian model with lagged endogenous variables arising due to the agents expectation. For both models we find regions in the parameter space where mis-specified RPE is selected by both minimum squared forecast error conditions and the sparse-weights. Thus the agents re-considering their initial choice of mis-specified RPE could continue using an initial rule. If learning costs are very large there is a region of parameter space where agents choose not to allocate attention to any of the variables. If learning costs are small agents tend to switch to REE and do this certainly when cost of attention is zero. For medium range of the learning costs an initial misspecified forecasting rule prevails for large persistence of the variable used in the rule and for large correlation between the included and the omitted variables especially if the omitted variable has low persistence. This behavior is explained by the amount of additional information that is contained in the omitted variable. The prediction from a New Keynesian model is that when inflation persistence increases the survival of AR) rule for inflation forecasting is achieved for broader area of the parameter space. The same is true for the small correlation between inflation and output gap. This prediction is supported by the professional forecasters behavior whose predictions are found to be consistent with paying less attention to output gap after the financial crisis Lopez-Perez 07) when inflation became more persistent and the correlation between output and inflation might have changed. We find that aggressiveness of the monetary policy rule and to some extent the rule itself determine the survival of a mis-specified forecasting rule. Strong monetary policy reaction reduces inflation volatility and persistence making adding output gap to the forecasting rule more attractive. Besides with larger monetary policy reaction agents do not consider the mis-specified forecasting rule as a rule consistent with rational expectations results in smaller forecast errors. In line with the previous literature our study supports the importance of expectation feedback parameter for survival of a mis-specified rule. In our model the feedback parameter is decreasing with policy rule parameter. If this expectation feedback is large enough the mis-specified forecasting rule prevails in equilibrium. 8 Although the evidence on flattening of the Phillips curve is mixed due to different specifications and time horizons considered there are studies showing the decline in slope examples being IMF 03) and Kuttner and Robinson 00) Donayre and Panovska 06) document breaks in the wage Phillips curve during the recessions and the subsequent recoveries. 9 Frenkel et al. 0) uses data up to 00Q and does not find evidence for a change in forecasters behavior while Lopez-Perez 07) uses longer data set and includes a forward looking inflation term in the Philips curve.

17 Sparse Restricted Perception Equilibrium 7 References ADAM K. 005): Learning To Forecast And Cyclical Behavior Of Output And Inflation. Macroeconomic Dynamics 90): 7. ADAM K. 007): Experimental Evidence on the Persistence of Output and Inflation. Economic Journal 750): ASSENZA T. P. HEEMEIJER C. HOMMES AND D. MASSARO 03): Individual Expectations and Aggregate Macro Behavior. Tinbergen Institute Discussion Papers 3-06/II Tinbergen Institute BRANCH W. A. AND G. W. EVANS 006): A simple recursive forecasting model. Economics Letters 9): BRANCH W. A. AND G. W. EVANS 006): Intrinsic heterogeneity in expectation formation. Journal of Economic Theory 7): BULLARD J. AND K. MITRA 00): Learning about monetary policy rules. Journal of Monetary Economics 496):05 9. DONAYRE L. AND I. PANOVSKA 06): Nonlinearities in the U.S. wage Phillips curve. Journal of Macroeconomics 48C):9 43. EVANS G. W. AND S. HONKAPOHJA 994): Learning convergence and stability with multiple rational expectations equilibria. European Economic Review 385): EVANS G. W. AND S. HONKAPOHJA 009): Learning and Macroeconomics. Annual Review of Economics ):4 45. EVANS G. W. AND S. HONKAPOHJA 00): Learning and Expectations in Macroeconomics. Princeton University Press Princeton. EVANS G. W. S. HONKAPOHJA T. SARGENT AND N. WILLIAMS 0): Bayesian Model Averaging Learning and Model Selection. CDMA Working Paper Series 003 Centre for Dynamic Macroeconomic Analysis FENDEL R. E. M. LIS AND J. C. RÜLKE 0): Do professional forecasters believe in the Phillips curve? Evidence from the G7 countries. Journal of Forecasting 30): FRENKEL M. E. M. LIS AND J. C. RÜLKE 0): Has the economic crisis of changed the expectation formation process in the Euro area?. Economic Modelling 84): GABAIX X. 04): A Sparsity-Based Model of Bounded Rationality. The Quarterly Journal of Economics 94): GABAIX X. 06): A Behavioral New Keynesian Model. Working Paper 954 National Bureau of Economic Research GALÍ J. 05): Monetary Policy Inflation and the Business Cycle: An Introduction to the New Keynesian Framework and Its Applications Second edition. Princeton University Press edition. GARCÍA-SCHMIDT M. AND M. WOODFORD 05): Are Low Interest Rates Deflationary? A Paradox of Perfect-Foresight Analysis. NBER Working Papers 64 National Bureau of Economic Research Inc

18 8 Volha Audzei and Sergey Slobodyan HEEMEIJER P. C. HOMMES J. SONNEMANS AND J. TUINSTRA 009): Price stability and volatility in markets with positive and negative expectations feedback: An experimental investigation. Journal of Economic Dynamics and Control 335): HOMMES C. H. 04): Behaviorally Rational Expectations and Almost Self-Fulfilling Equilibria. Review of Behavioral Economics -): IMF 03): The dog that didnt bark: has inflation been muzzled or was it just sleeping? In World Economic Outlook chapter 3 pages 7.. KUTTNER K. AND T. ROBINSON 00): Understanding the flattening Phillips curve. North American Journal of Economics and Finance ):0 5. LOPEZ-PEREZ V. 07): Do professional forecasters behave as if they believed in the New Keynesian Phillips Curve for the euro area? Empirica 44): MACKOWIAK B. AND M. WIEDERHOLT 009): Optimal Sticky Prices under Rational Inattention. American Economic Review 993): MANKIW N. G. AND R. REIS 007): Sticky Information in General Equilibrium. Journal of the European Economic Association 5-3): MARCET A. AND T. J. SARGENT 989): Convergence of least squares learning mechanisms in self-referential linear stochastic models. Journal of Economic Theory 48): MATEJKA F. AND A. MCKAY 05): Rational Inattention to Discrete Choices: A New Foundation for the Multinomial Logit Model. American Economic Review 05):7 98. MOLNAR K. 007): Learning with Expert Advice. Journal of the European Economic Association 5-3): ORMENO A. AND K. MOLNAR 05): Using Survey Data of Inflation Expectations in the Estimation of Learning and Rational Expectations Models. Journal of Money Credit and Banking 474): PFAJFAR D. AND B. ŽAKELJ 04): Experimental Evidence on Inflation Expectation Formation. Journal of Economic Dynamics and Control 44: PFAJFAR D. AND B. ŽAKELJ 07): Inflation Expectations and Monetary Policy Design: Evidence from the Laboratory. Macroeconomic Dynamics forthcoming: 4. SIMS C. A. 003): Implications of rational inattention. Journal of Monetary Economics 50 3): SLOBODYAN S. AND R. WOUTERS 0): Learning in an estimated medium-scale DSGE model. Journal of Economic Dynamics and Control 36):6 46. WATSON M. W. 04): Inflation Persistence the NAIRU and the great recession. American Economic Review 045):3 36.

19 Sparse Restricted Perception Equilibrium 9 Appendix A: A simple of model Proof. Proposition.0.. Deriving RPE. The agents PLM consistent with this RPE is y t = a + b w t A) therefore the ALM is given by y t = α + βa + b w t + γ w t + η t. A) In what follows we will set α = 0 and assume that the agents know this; therefore a = 0 as well. In order for the agents to be using A) in equilibrium it must be the case that b is a coefficient in the regression of y t on w t or b = Cov y t wt ) Var wt ). A3) Computing the above expression we get ) Cov y t wt = E t [ b w t + γ w t + η t w t b = b σ + γ ρ σ σ ] = b σ + γ ρ σ A4) = b + γ ρ σ = A5) = βρ b + γ + γ ρ σ b = γ + γ ρ σ. A6) βρ Deriving RPE. Similarly to the RPE case we have now the PLM y t = a + b w t A7) which implies that b must be equal to the regression coefficient: b = Cov y t wt ) Var wt ) = E t [ b wt + γ wt + η t wt ] σ = A8) = βρ b + γ ) ρ σ + γ σ σ = βρ b + γ ) ρ σ + γ. A9) E-stability. For solution in A6) and A9) to be E-stable the following should hold: T b T b [ = <. T b b is given by A6) and T b by A9). That is T b [ b T b = βρ b +γ ) ρ σ ] σ +γ = 0. Thus the only condition to be satisfied is: b b < and b βρ b +γ +γ ρ σ ] = βρ b and βρ <. A0)

20 0 Volha Audzei and Sergey Slobodyan With both β< and ρ < both solutions are E-stable. Next we want to ensure that the Mean Squared Forecast Error MSFE) of the agent living in RPE and using A) in lower than the MSFE of the agent using A7) otherwise a small proportion of the latter could outperform the majority and lead to increasing deviations from the RPE. et = b wt + γ wt + η t b wt A) [ ] [ MSFE = E et ) ) = E b b w t + γ wt + η t ]. A) Similarly for MSFE we have the following expression et = b wt + γ wt + η t b wt A3) [ ] [ ) MSFE = E et = E b wt + γ b )wt + η t ]. A4) We are looking for the conditions under which MSFE < MSFE : [ ) ) ] : E b b w t + γ wt + η t < E : : : : { βρ b + γ b ) σ + γ σ + γ βρ b + γ b ) ρ σ βρ b + γ ) σ + b b βρ b + γ )σ + γ σ +γ βρ b + γ ) ρ σ γ b ρ σ { b σ b βρ b + γ )σ γ b ρ σ [ ) b wt + γ b )wt + η t ] { βρ b + γ ) σ + γ b ) σ + } < < } { < b ) b σ βρ b + γ + γ ρ σ < =b βρ b + γ )γ b ) ρ σ βρ b + γ ) σ + γ σ γ b σ + b σ + +γ βρ b + γ ) ρ σ b βρ b + γ ) ρ σ γ b σ + b σ b βρ b + γ ) ρ σ b σ b σ } ) γ + βρ b + γ ) ρ σ : b σ < b σ b σ > b σ. A5) =b } The condition b > b σ has a very simple interpretation: in order to have MSFE of PLM lower than for PLM the share of variance of y t explained by the PLM must be higher than that of PLM. Alternatively the R of the regression A) must be higher than the R of regression A7).

21 Sparse Restricted Perception Equilibrium A. Analyzing conditions for MSFE < MSFE b σ > b σ b > [ βρ b + γ ) ρ b > : [ βρ b + γ ) ρ + γ σ [ b + γ ρ σ ] [ > : b + b γ ρ σ + ρ ) > : b : b > γ σ. ] σ + γ σ σ ] [ ] σ = b ρ + γ ] σ b ρ + γ ) ρ > b σ ρ + b ργ γ σ γ σ ) ρ ) b > γ σ + ) ) σ γ A6) Theoretically there could be separate cases: CASE I : b > γ σ CASE II : b < γ σ. A7) A8) As the CASE I is the most obvious and will happen the easiest assuming γ > 0 which is what we impose; otherwise just re-define variable w i t so that γ become positive) we start with this case. A.. CASE I: b > γ σ Coming back to the condition of MSFE < MSFE we get b > γ σ γ + γ ρ σ βρ βρ > γ σ γ + γ ρ σ βρ > γ σ γ σ βρ σ γ + γ σ βρ + ρ) > γ γ + βρ γ ρ + βρ > 0. A9) Denoting the ratio of coefficients at the observable shocks γ γ as Γ and the ratio of standard deviations σ as Σ we see that the condition for CASE I to be true is ΓΣ + ρ βρ > βρ or ρ > βρ ΓΣ βρ. A0) This condition is satisfied when ρ and ΓΣ is large. Alternatively when ρ 0 and ΓΣ is small so that the numerator is positive this condition might amount to ρ > and thus be impossible to satisfy.

22 Volha Audzei and Sergey Slobodyan A.. CASE II: b < γ σ In this case we have b < γ σ γ + γ ρ σ βρ βρ < γ σ γ + γ ρ σ βρ < σ γ σ + γ σ βρ γ + βρ γ ρ < + βρ. A) σ Using the notation just introduced the condition b < γ amounts to ΓΣ +ρ βρ + βρ ) < 0. Consider an intersection of with horizontal axis where ΓΣ = 0. Then ρ < βρ ). As ρ < βρ the area where the solution exists is βρ ) > meaning that βρ βρ > /. That is for MSFE condition satisfied for b < 0 βρ must be larger than /. To sum up combining to cases together we see that we need ΓΣ + ρ βρ > βρ. A) A. Deriving the Sparse Weights Sparse weights are derived by minimizing the following expression: ) min m [0] n m i )Λ i j m j + κ m i i j=...n i j=...n A3) with: Then the cost of inattention is Λ i j = σ i j a wi u aa a w j a wi = u aa u awi u aa = u a a = a u awi = b i. ) a b wt b wt ) = Λ i j = σ i j b i b j. A4) Plugging the cost of inattention as in A4) into A3) we get the following problem: { min m m [0] n ) σ b + m ) m ) b b + m ) σ b } + +κ m + m ). A5) First order conditions of A5) with respect to m and m :

23 Sparse Restricted Perception Equilibrium 3 [m ] : κ + b m )σ b b m ) = 0 A6) [m ] : κ + b m )σ b b m ) = 0. A7) Solving A6) and A7) for m and m gives the expressions 3) and 4) in the text. Proof. Proposition.0.. Consider when m > m : κ b b σ ρ ) b σ b ρ κ > b b b σ ρ ) b ρ b σ b σ b b σ b ρ > b b ρ b σ b b σ ) ) b σ b ρ b σ < b b ρ b σ b σ b b ρ σ < b b b ρ σ b σ < b A8) Proof. Proposition.0.3. Assume that the A8) is satisfied. Then we need ρ b σ b κ b σ ρ ) < 0. We distinguish between the following cases depending on the sign of ρ b b σ. CASE I: b > b σ. Then b b σ < and ρ b b σ > 0. CASE II: b < b σ < 0. Then ρ b b σ > 0. Note that for RPE consistent rule be an equilibrium for b < 0 it must hold that βρ > / see CASE II of Appendix A.). In terms of parameter of the model 5) could be written as κ < γ σ ρ ) ΓΣ + βρρ ) ΓΣ + ρ βρ ) > 0. The condition also states that if RPE consistent rule is an equilibrium that is 0) is satisfied there is no region in the parameter space such that for any κ this equilibrium survives. Appendix B: Three-equation New Keynesian Model B. Rational Expectations Equilibrium Proof. Proposition 3.0. Deriving REE with forward looking policy rule. The solution of the model under forward-looking rule is: y t = ϕ π σ E tπ t+ + ϕ π π + y σ t + g t π t = ω ϕ π σ B) π + β + ω ϕ π σ )E tπ t+ + ωy t + ωg t + u t. B)

24 4 Volha Audzei and Sergey Slobodyan We re-define in B) and B) as: a y = ϕ π σ π B3) b y = ϕ π σ B4) c y = B5) a π = ω ϕ π π σ B6) b π = β + ω ϕ π σ B7) c π = ω. B8) There are two rational expectations equilibria. One solution is associated with the following forecasting rule: πt REE = α + cy t + bπ t B9) E t πt+ REE = α + ce t y t + bπ t = α + bα + c + b)y t + b π t B0) where in the last equality we make use of the assumption that our agents have naive expectations about future output gap. Plugging the forecasting rule in B) and in B) yields: π t = a π + b π α + b) + b b π π t + b π c + b) + c π )y t + c π g t + u t B) y t = a y + b y α + b) + b y b π t + b y c + b) + c y )y t + g t. B) Using the method of undetermined coefficients we solve for the forecasting rule coefficients. α = a π + b π α + b) B3) b = b b π B4) c = b π c + b) + c π. B5) The rational expectation solution is: α = a π b π b = b π c = c π b π B6) B7) B8)

25 Sparse Restricted Perception Equilibrium 5 The ALM coefficients are then: ā REE + b π y = a y b y a π b π B9) b REE y = b y b B0) π c REE + b π y = c y b y c π b π ā REE a π + b π ) π = a π b π b π b π = a π b π b REE π = b π c REE π = c π b π c π b π + b π b π = c π b π. B) B) B3) B4) E-stability For the REE equilibrium to be E-stable the eigenvalues of the following matrix must be smaller than unity: + b)b π b π α 0 0 bb π 0 0 cb π b π + b). With the eigenvalues equal to + b π + b π ) the solution is not E-stable. [Though can be determinate under unreasonable values of ϕ π ] Another REE solution is the MSV solution of the form: πt MSV = α + cy t B5) E t πt+ MSV = α + ce t y t = α + cy t B6) where in the last equality we make use of the assumption that our agents have naive expectations about future output gap. Plugging the forecasting rule in B) and in B) yields: π t = a π + b π α + b π c + c π )y t + c π g t + u t B7) y t = a y + b y α + b y c + c y )y t + g t. B8) With coefficients defined as in B) and B). Using the method of undetermined coefficients we solve for the forecasting rule coefficients. The MSV solution is: α = a π + b π α B9) c = b π c + c π. B30) α = c = a π B3) b π c π B3) b π

26 6 Volha Audzei and Sergey Slobodyan The ALM coefficients are then: ā MSV y = a y + b y α a π = a y + b y B33) b π b MSV y = 0 B34) c MSV y = b y c π b π + c y ) B35) ā MSV π = a π + b π α = a π b π B36) b MSV π = 0 B37) c MSV c π π = b π + c π b π c π =. B38) b π Determinacy. Representing the ALM as z t = A+Bz t +U t with z t = y t π t ) A and U t are vectors of constants and shocks respectively and c MSV B = y c MSV π b MSV y b MSV π For the solution to be determinate both eigenvalues of B. must be inside the unit circle. The characteristic polynomial of B. is given by pλ) = λ + a λ + a 0 where: ) a 0 = 0 B39) a = β )σ βσ + ωϕ π ) + σ = β)σ ωϕ π ) + σ β). B40) For both eigenvalues of B. to be inside the unit circle the following conditions must hold 0 : a 0 < B4) a < + a 0. B4) With a 0 = 0 condition B4) is always satisfied. Condition B4) implies: = β)σ β)σ > = ωϕ π ) + σ β) ωϕ π ) + σ β) < { { ϕπ > ϕπ > = ωϕ π ) < σ β) ϕ π < ω σ B43) β) E-stability. For the MSV solution to be E-stable the following should hold eig Tαc α T c α T αc c T c c 0 See the references in Appendix A of Bullard and Mitra 00). ) <.

27 Sparse Restricted Perception Equilibrium 7 With T αc = a π + b π α and T c = b π c + c π the above condition converges to ) bπ 0 eig < = b 0 b π < = β ω ϕ π < = ϕ π > β) σ π σ ω. That is for ϕ π > the MSV solution is both determinate and E-stable and is not E-stable for ϕ π <. When contemporaneous rule is used the model solution is: y t = ϕ π σ Definine the coefficients as π ϕ π σ E tπ t + σ E tπ t+ + y t + g t B44) π t = ω ϕ π σ ) π + β + ω σ )E tπ t+ ωϕ π σ E tπ t + ωy t + ωg t + u t. B45) a y = ϕ π σ π B46) b f y = σ B47) b c y = ϕ π σ B48) c y = B49) a π = ω ϕ π π σ B50) b f π = β + ω σ B5) b c π = ωϕ π σ B5) c π = ω. B53) With the REE forecasting rules as in B9) the rational expectation solution is as in 30) and in 3) we calculate the eigenvalues of the following matrix: + b)b f π + b c π b f πα 0 b f 0 bbπb f c π + a π 0 π 0 = 0 b c 0 cb f π + b)b f π 0 π + b c π 0 c π b f π +. The eigenvalues are [b f π + b c π]. With b f π > 0 b f π + geq what contradict E-stability condition. Let us now consider MSV solution. Determinacy. With the MSV forecasting rules as in B5 and with the MSV solution as in 33) and in 35) the ALM coefficients are given by: ā MSV y = a y + by f + b c a π y) b f π b c π b MSV B54) y = 0 B55) c MSV y = c y + by f + b c c π y) b f π b c π ā MSV + b f π + b c π π = a π b f π b c π B56) B57) b MSV π = 0 B58) c MSV + b f π + b c π π = c π b f π b c π. B59)

28 8 Volha Audzei and Sergey Slobodyan Similar to the forward-looking rule we use the conditions B4) and B4) to get: α 0 = 0α = c MSV y = c y + by f + b c c π β)σ y) b f π b c ) = ωϕ π π ) + σ β) B60) which is the same condition as for forward looking rule. That is why for the contemporaneous rule the condition on the policy feedback is the identical: { ϕπ > ϕ π < ω σ B6) β) E-stability. We calculate the eigenvalues of the following matrix: b f π + b c π b f π + b c π. For the solution to be E-stable b f π + b c π <. Plugging the values for the b f π and bπ f we arrive to the same conditions as under forward-looking rule: ϕ π > ω σ β). B. E-stability determinacy and MSFE criterion Proof. Proposition Forward-looking policy rule. We first derive the coefficients in agents forecast rules and in the actual law of motion for inflation and output conditional on agents using M π rule: where ALM coefficients are the following: with a π b π and c π defined in B3) - B8). π t = ā π + b π π t + c π y t + ωg t + u t B6) y t = ā y + b y π t + c y y t + g t B63) ā y = a y + b y α π + β π ) B64) b y = b y β π ) B65) c y = c y B66) ā π = a π + b π α π + β π ) B67) b π = b π β π ) B68) c π = c π B69) Deriving M π. As in the case of the simple model we allow our agents to be econometricians who estimate the coefficients for their learning rule as regression coefficients. Then for M π the coefficient is: β π = Covπ tπ t ) Var π t ) = Cov ) ā π + b π π t + c π y t π t Var π t )

29 Sparse Restricted Perception Equilibrium 9 denoting σ πy Covyπ) σ π Var π) σ y Vary): β π = b π σ π + c π σ πy σ π σ πy = b π + c π σ B70) π α π = π β π π. B7) Determinacy. Re-writing actual law of motion as z t = Ā + B z t +U t with z t = y t π t ) Ā and U t are vectors of constants and shocks respectively and ) cy b B = y c π b π the solution in B70) is stable only if real parts of eigenvalues of B are inside the unit circle. The coefficients of the characteristic polynomial pλ) = λ + a λ + a 0 are: a 0 = ββ π B7) a = + b π β π ) B73) Using the conditions in B4) and B4) this solution is stability if and only if: = { + b π β π ) < + ββ + b π β π > ββ π if b π β π > π = + b π β π < + ββ π if b π β π < { + β π b π + β) > 0 if b π β π > b π < β β < β. π if b π β π < B74) B75) Conditions B74) and B75) are satisfied for any β π <. Deriving M y. Coefficient c y π is then determined as regression coefficient c y π = Covπ ty t ) = Cov ) ā π + b π π t + c π y t y t = Var y t ) Var y t ) = b π σ πy + c π σy σ πy σy = c π + b π σy B76) α y π = π c y πȳ. B77) E-stability. With the coefficients of the actual law of motion defined as in B64) -B69). For M π to be E-stable under least-squares learning the following must be satisfied T α eig α T β α T α β T β < β

30 30 Volha Audzei and Sergey Slobodyan where T β and T α are as in B70) and ) accordingly. As T α α = 0 the conditions collapses for T β β < see figure B. For M y to be E-stable under least-squares learning the following must be satisfied eig T αy α y T c y α y T αy c y T c y c y <. With T cy and T αy defined in B76) and B77) the above matrix has only one non-zero element - T αy = ȳ. For our parameters calibration steady state output gap is positive and the condition is α y satisfied. Moving to contemporaneous inflation in the policy rule the solution is: [Conditions to be derived] we solve for the coefficients of actual law of motion as: ā y = a y + α π b c y + b f y + β π )) B78) b y = β π b c y + b f y β π ) B79) c y = B80) ā π = a π + α π b c π + b f π + β π )) B8) b π = β π b c π + b f πβ π ) B8) c π = ω. B83) The solutions for M π and M y look the same as in B70)-B.) and in B76)-B77) respectively where ALM coefficients are in B78)-B83). Determinacy. Similarly to the above derivations the corresponding conditions are: which can be re-arranged as: β π β < B84) + β π b c π + β π b f π) < + β π β B85) β π < ϕ π < β π + + β π β)σ β π ω B86) E-stability. As T α α = 0 the conditions collapses for T β < see figure B. β MSFE. For the agents to use M π as a forecasting rule in the equilibrium the rule must produce better forecast as the alternative M y. We compare the quality of the forecasts based on the mean squared forecast error criterion. In calculations below we denote the correlation between output and inflation as R. The correlation is then R = σ πy σ y σ π with σ πy Covπy) σ π Var π) σ y Var y). We start with mean forecast error of M π. The forecast error of M π is the difference between the forecast and the actual inflation: : e π t = α π ā π ) + β π b π )π t c π y t = π π b π π c π R σ y σ π a π πb π β π ) + β π ) + b π + c π R σ y σ π b π )π t c π y t = π π b π π c π R σ y σ π a π πb π + πb π β π + c π R σ y σ π π t c π y t. B87)

31 Sparse Restricted Perception Equilibrium 3 Using that b π = b π β π and that we group the expression to obtain : π a π πb π π c π R σ y σ π + c π R σ y σ π π t c π y t = π a π πb π c π ȳ + c π R σ y σ π π t π) c π y t ȳ) B88) where in the last line we added and substracted c π ȳ. Now with c π = c π = ω a π = πω ϕ p σ b π π = a π + β π and c π = ω ȳ = β ω π: : π a π πb π c π ȳ = 0 B89) : et π = c π R σ y π σ t π) c π y t ȳ) π B90) : MSFE π = E t e π t ) = E t [ c π R σ y σ π π t π) c π y t ȳ)] : = E t [ c πr σ y σ π t π) c π R σ y c π π π σ t π)y t ȳ) + c πy t ȳ) ] π : = c πr σ y σ σ π + σy R σ y σ πy ) π σ π : = c πr σy + σy R σy ) : = c πσ y R ) + σ µ. B9) Similarly the forecast error of M y : : et y = α y ) y ) π ā π + cy c π yt b π π t + µ t = ) σ πy σ πy : = π ȳ c π ȳ b π σy ā π πb π + π b π + b π σy y t b π π t + µ t = : = π a π b π π ȳ c π + b π [ σ πy σy y t ȳ) π t π)] + µ t : = b π [R σ π σ y y t ȳ) π t π)] + µ t : MSFE y = E[ b π [R σ π σ y y t ȳ) π t π)] B9) : = b πσ π[ R ] + σ µ. B93) We are looking for the conditions under which MSFE π < MSFE y : c π [ ] [ ] σy R σy < b π σ π R σ π B94) denoting Σ = σ π σ y and Γ = b π cπ.and the criterion is simply: Γ Σ >. B95)

32 3 Volha Audzei and Sergey Slobodyan Figure B: E-stability a) Forward Looking Rule b) Contemporaneous Rule Note: the figure for forward-looking rule is drawn for r in the range [0. ] where each line corresponds to different r. The figure for contemporaneuos rule is drawn for r in the range [0. ] Variances and covariance The procedure is similar to Adam 005) With the actual law of motion as in B6) and B63) under different policy rules we derive variances and covariance of output and inflation. Denote z t = π t y t ) b and B = π c π ) U b y c t = u t g t ). Then the B6) can be represented as: y z t = Bz t +U t B96) where for the purpose of variance and covariance calculation we have dropped the unimportant constants. Denote r σ u and σg Ω Eug) = σg r + ω ) ω is variance covariance matrix of ω the shocks. We take variance of B96) to get: E = BE B + Ω B97) where E = Ezz ). Vectorizing matrix E : vece ) = B B)vecE ) + vecω) = I B B) vecω). B98) To obtain the covariance with lagged values we multiply both sides of B96) by z t and take expectations to get: Γ = BE B99) where Γ = Ez t z t ) is covariance matrix with the lagged values and vecγ) = I B)vecE ). B00)

33 Sparse Restricted Perception Equilibrium 33 Then variances and covariances are as follows: σ yπ = vece )[] B0) σ π = vece )[] B0) σy = vece )[4] B03) Eπ t π t ) = vecγ)[] B04) Eπ t y t ) = vecγ)[3]. B05) Appendix C: Weights Proof. Proposition Consider 4) and 4) and using that c π > 0 and b π 0 : m π > m y : : : κ b πσ πσ y R ) c πσ y b π σ π R κ < c π c πσy σ π R ) b π σ π c π σ y R b π c π σ y b π σ π R < b π σ π c π σ y R b π σ π c π σ y if b π > 0 c π σ y b π σ π R > b π σ π c π σ y R b π σ π c π σ y if b π < 0 cπ σ y b π σ π R ) c π σ y < b π σ π b π σ π c π σ y R ) : c πσ y c π σ y b π σ π R < b πσ π b π σ π c π σ y R : c πσ y < b πσ π : Γ Σ >. Proof. Proposition Rearranging 4) : κ c πσy σ π R ) b π σ π c π σ y R < b π b π c πσy σ π R ) R ) c πσy : κ < b π σ π c π σ y R = R c πσ y b π σ π where in the second step we again use that 39) implies b π σ π > c π σ y which means that even for b π < 0 > 0. R [ ] that is maximum of the expression c π σ y R linear in R is b π σ π c π σ y R b π c π σ y < b π σ π. Appendix D: Weight on Inflation With c π > 0 b π = 0 means violation of 39) hence with b π = 0 M π can not be equilibrium choice.

34 34 Volha Audzei and Sergey Slobodyan a) Forward Looking Rule b) Contemporaneous Rule