Expectation Formation and Rationally Inattentive Forecasters

Transcription

1 Expectation Formation and Rationally Inattentive Forecasters Javier Turén UCL October 8, 016 Abstract How agents form their expectations is still a highly debatable question. While the existing evidence stresses the importance of noisy information models to characterize the process of expectation formation, still there is no consensus on the causes of these frictions and their relevance, from an empirical point of view. This paper contributes to this literature by answering these two questions. Based on a model with rational inattentive forecasters, I characterize the source of these noisy signals as a consequence of the agent s limited attention. The evolution of the agent s attention along with its main components are characterized through the model. Preliminary simulations, suggest that the attention level changes significantly as a function of the forecast horizon. These results are important to understand the deviations from rationality, typically observed in the data. Keywords: Rational Inattention, Professional Forecasters, Expectation Formation, Endogenous Attention. JEL Classification: E7, E37, D80, D84. to be added javier.roman.1@ucl.ac.uk 1

2 1 Extended Abstract How do economic agents form their expectations has been a long debated question from a theoretical and empirical point of view. While the assumption regarding agent s rationality is commonly undisputed, the empirical evidence is at odds with this feature, Mankiw, Reis and Wolfers (004) and Pesaran and Weale (006). Moreover, the presence of economic policies aiming to manipulate people s expectation (e.g. Inflation Targeting and Forward Guidance) not only asks for a deeper understanding of the main characteristics behind this process, but also on the main causes driving the deviations from rationality. The existing evidence argues that the presence of information frictions are crucial to understand the main features of this process. However, still there is no consensus on the causes of these frictions and how relevant they are from an empirical point of view. This paper contributes to this literature by answering these two questions. Relying on the literature on Rational Inattention, this paper model and introduce a novel way to determine and estimate the actual level of attention that economic agents give to different macroeconomic outcomes. Based on evidence from surveys of expectations, noisy information models are presented as one of the main causes behind these rigidities, Coibion and Gorodnichenko (01) and Coibion and Gorodnichenko (015). It is been argued that while agents are constantly tracking and observing current information, the true realization is only observed with some exogenous noise. The importance of these models goes beyond the information rigidity and they have been used also to model why do agents disagree, Patton and Timmermann (010), to determine the importance of Public Information Manzan (011), and even to understand the strategic behavior among professional forecasters, Ottaviani and Sørensen (006). Moreover, while all these evidence focused on professional forecasters, recently the presence of noisy information have been documented even across firms, Coibion, Gorodnichenko and Kumar (015). While important, the straightforward questions are then: why do agents observe such noisy signals? And more importanly, what are the underlying reasons that prevent agents to fully observe the actual realizations? From a theoretical point of view, these questions have been answered based on Rational Inattention Models, Sims (003), Maćkowiak and Wiederholt (009) and Matějka (015). Under rational inattention, agents faced a limited capacity to observe and collect all the existing information. Hence, the problem is not linked with the lack of current information (in contrast to other theories, such as the well known sticky information model, Mankiw and Reis (001)), it is driven by the fact that people does not have the mental capacity to process and keep track of all the available information.

3 While the implications and importance of Rational inattention models are unquestionable, the empirical evidence linking this strand of the literature with the expectation formation process is still very scarce. At first, this could be surprising based on the obvious links between Rational Inattention and the aforementioned noisy information models. However, models with rationally inattentive agents are not trivial to solve and there is no obvious data to estimate these models. The main problem is the difficulty to determine what are the actual noisy signals that agents used to update their expectations. Just recently, Mackowiak, Matejka, Wiederholt et al. (016) proposed a procedure to deal with this, but it only focuses on its steady state implications. This paper introduces a new multi-horizon endogenous attention model with heterogeneous agents that characterize the evolution of attention based on individual characteristics and the forecast horizon. Through the model, I characterize the evolution of both agent s accuracy and disagreement, as a function of the underlying attention. Finally, using surveys of professional forecasters as a proxy for agent s expectations, it is possible to pin down the model s main parameters to estimate the actual evolution of attention. In terms of data, this paper relies on Bloomberg s ECFC Survey of Professional Forecasters. While this is just one of the different surveys on expectations, it has some distinct features that makes it relevant for this analysis. Particularly (and in line with one of the main assumptions of Rational Inattention models) agents can update their forecasts at any time they want. Thus, in principle, the observed forecasts should represent the latest piece of available information. Preliminary simulations of the model show that the level of attention changes as a function of the horizon. Particularly, forecasters increased their attention between 14 to 8 months before the variable is finally known. Afterwards, there is a monotonic decrease in the level of attention as the release date approaches. These results can be rationalized as agents providing higher effort to collect information at longer horizons when the uncertainty (driven by the lack of information), about the final value of the forecasted variable, is higher. The results are prominent, in order to explain the mixed evidence regarding deviations from rationality at different horizons reported in the literature, e.g. Patton and Timmermann (01). Model The model is embedded in a fixed event forecast scheme. There are N different agents that for H consecutive months will provide a forecast for a macroeconomic release that is known by the 3

4 end of year T. Agents are labeled i = 1,..., N while the forecast horizon h = 1,..., H within year t = 1,..., T is given by t h, i.e. h months before the variable is released at year t. The agents are interested in tracking x t h, which follows a process: x t h = φx t h 1 + ɛ t h ɛ t h iid N(0, σ ɛ ) (1) The variable x t h, represents some monthly (annualized) macroeconomic variable such as: inflation or unemployment rate at time t h. Moreover, φ < 1, i.e. the process is stationary. It is assumed that at each time t h, the agents observes: s i,t h = x t h + η t h + ψ i,t h () x t h j = x t h j + η t j j = 1,..., H (3) iid Where η t j N(0, ση) iid ψ i,t h N(0, σi,h,ψ ). At each time, agents observe a noisy signal s i,t h, about the current value of x t h which is affected by measurement error η t and also by idiosyncratic noise, ψ i,t h. Importantly, the precision of the current signal is determined by the agent at each horizon. Additionally, agent s observe all past realizations of x t j although contaminated by the exogenous measurement error. Agent i information set at time t h: I i,t h = {s i,t h, x t h 1, x t h,...} (4) The idea behind the information structure is trying to replicate the timing of the release of public macroeconomic figures. Typically, current macroeconomic variables are always observe with delay and also even the published official statistics are always subject to revisions. Each forecaster wants to minimize: min α i E[(x y t E(x y t I i,t h )) ] + λκ i,t h (5) κ i,t h Where: x y t = x t h (6) 4

5 The objective of each forecaster is to minimize the mean square forecast error of x y t, which is the yearly realization of the variable x t, i.e. the (annualized) sum of monthly realizations within the target year t. In the loss function (5), α i represents the agent s preference for accuracy and λ stands for a fixed cost parameter. Following Sims (003), each agent faces a constraint related to the amount of information that they can process at each period of time. The constraint is defined as entropy reduction: H(x t h I i,t h 1 ) H(x t h I i,t h )] κ i,t h (7) Entropy is defined by H(.) in equation (7). The left hand side of equation (7) stands for the difference between prior and posterior uncertainty about x t h, which is bounded by κ i,t h. Thus, while any agent is able to increase κ i,t h at any time, they must be willing to pay the cost λ. The model can be rationalized as follows. Any forecaster at any point in time, can choose to put more attention to the current release of x t h. More attention leads to a higher precision on the observed signal. However, since the attention is limited, more effort comes at a cost λ. To start dealing with the model, it is useful to state the problem in a State Space form. This is presented in Appendix A. The model is solved in two steps. Firstly, the optimal forecasts for each t h are determined given some exogenous precision on the signal. Given the precision and based on the objective function, the agents rely on the Kalman Filter to obtain the optimal forecast for x t h. Secondly, the optimal choice for the distribution of the signal is determined, given the constraint. 1 Lemma 1 Optimal Forecast. Based on the AR(1) process for x t h, the definition of x y t and given a specific signal precision for each agent i, the optimal forecast at time t h: E(x y t I i,t h ) = φh 11 (1 φ 1 ) x i,t h t h 1 φ if h > 11 (8) = (1 φh+1 ) 1 φ x i,t h t h if h = 11 (9) = (1 φh+1 ) 1 φ x i,t h t h + j=h+1 x t j if h < 11 (10) 1 Given the normality assumption for the AR(1) and the quadratic objective function, it can be proved that the distribution of the noise would be normal as well. 5

6 Proof: Appendix B Where x i,t h t h = E(x t h I i,t h ), i.e. the expected value of x t h after observing the signal s i,t h at time t h, which is defined as: x i,t h t h = θ i,h x i,t h t h 1 + (1 θ i,h )s i,t h (11) θ i,h = σ i,h,ψ σ x i,t h t h 1 + σ i,h,ψ σ i,h,ψ = ση + σi,h,ψ (13) In words, the optimal forecast of x t h given a specific precision of the signal σi,h,ψ for each agent, is given by the lineal combination between the prior, i.e. x i,t h t h 1 and the signal, s i,t h. The more precise the signal, more weight is attached to the additional piece of information. Following Lemma 1, we can no rewrite the objective function (5) as follows. (1) Lemma Objective Function Proof: Appendix C. min α i [Φ h V ar(x t h I i,t h ) + C h ] + λκ i,t h (14) κ i,t h Now, Φ h is a function of the persistence at different horizons, V ar(x t h I i,t h ) is the conditional variance of x t h given the agent i information set at each horizon and C h stands for a constant term that varies with h, but is unaffected by the forecaster effort. With the solution for the problem to any exogenous signal, we can now derive the optimal level of attention. Firstly, we need to work out an expression for the constraint on attention. This is emphasized in lemma 3. Lemma 3 Restriction on attention Given the normality assumptions, it is possible to rewrite (7) as a function of the conditional variances: Where σ x i,t h t h 1 corresponds to the conditional variance of x t h given the previous information, and σ i,h,ψ = σ η + σ i,ψ ( 1 σ log xi,t h t h 1 σ i,h,ψ + 1 ) κ i,t h (15) is the chosen precision on attention. 6

7 Proof: Appendix D The restriction on attention can now be directly interpreted as a signal-to-noise ratio, where a higher information capacity chosen for time t h, would lead to a more precise signal. Given equation (15), now we solve the forecasters problem for any horizon h. Proposition 1 Multi-horizon optimal attention Each agent i = 1,..., N at each horizon h = 1,..., H minimize equation (14) subject to (15). Then if: σ i,h,ψ = σ x i,t h t h 1 κ i,t h 1 σ η The optimal level of attention at any horizon h: ( ) κ i,t h = 1 αi log ln()φ h σx i,t h t h 1 if α i ln()φ h σx λ i,t h t h 1 > λ (16) = 0 Otherwise (17) Where the prior variance σx i,t h t h 1 is equal to the projected posterior variance at t h 1 plus the variance of the process, i.e. σx i,t h t h 1 = φ σx i,t h 1 t h 1 + σɛ. Proof: Appendix E The equation for κ i,t h is now the dynamic provision for attention. As expected, the agent would exert higher effort depending on their preferences for accuracy and the volatility of the forecasted process. Likewise, the cost of attention decreases the accuracy. However, there are some interesting dynamic features. As mentioned in the introduction, the optimal provision of attention depends directly on the forecast horizon. Moreover, it is clear how the evolution of attention is tied to the persistence of the forecast process. 7

8 3 Appendix Appendix A: State Space Form The state equation can be written: ξ t h = x t h x t h 1. x t h 11 = φ x t h 1 x t h. x t h 1 + ɛ t h 0. 0 (18) ξ t h = Gξ t h 1 + ε t h ε t h iid N(0, Ω ε ) While the observation equation: y i,t h = s i,t h x t h 1. x t h 11 = x t h x t h 1. x t h 11 + η t h + ψ i,t h η t h 1. η t h 11 (19) y i,t h = F ξ t h + ν i,t h ν i,t h iid N(0, Ω i,h,ν ) Where: Ω ε = σɛ and Ω i,h,ν = ση + σi,h,ψ ση ση ση From the two expression we can compute ξ t h = E(ξ t h I i,t h ) for each individual using the Kalman Filter. Since the precision of the signal varies with the horizon h, we cannot rely on the Steady State solution for the Filter. Appendix B: Lemma 1 Proof. Let s start by showing how to derive the unconditional mean of x y t at different horizons t h. As stated by equation (6), the forecasted variable is equal to the last twelve 8

9 realizations of x t : x y t = x t h = x t + x t x t 11 h=0 Given the process for x t : x t = φx t 1 + ɛ t = φ x t + ɛ t + φɛ t 1 =... = φ 1 x t 1 + φ j ɛ t j Hence, its is possible to find expressions for all x t h with h = 0..., 11: 10 x t 1 = φ 11 x t 1 + φ j ɛ t j 1 for h = 1 x t = φ 10 x t 1 + =... 9 φ j ɛ t j for h = x t 11 = φx t 1 + ɛ t 11 for h = 11 All the expression are a function of the first monthly realization that does not correspond to the target year, x t 1. Then, the annual realization of x y t at h = 1 is: x y t = φ(1 φ1 ) 1 φ x t φ j+1 1 φ ɛ t j By continuing with the recursion, at horizon t h with h > 11: x y t = φh 11 (1 φ 1 ) x t h + 1 φ 1 φ j+1 1 φ ɛ h 1 t j + j=1 Likewise, the annual realization within the target year, i.e, h 11: x y t = φ(1 φh ) 1 φ x t h + j=h x t j + φ j 11 (1 φ 1 ) ɛ t j (0) 1 φ 1 φ j+1 1 φ ɛ t j (1) Hence, under full information, the optimal forecast at time t h is given by the expected value of equations (0) and (1), depending if we are outside or inside the target year, i.e. 9

10 h 1 or h 11. However, since x t h is not directly observable we need to replace it with its optimal prediction: x t h t h. Additionally, when we enter into the target year, the lagged values of x t h are now observable but only subject to the measurement error, x t h j. Appendix C: Lemma Proof. Based on Lemma 1, we can show: E[(x y t E(x y t I i,t h )) ] = Φ h V ar(x t h I i,t h ) + 1 h<11 (10 h)σ η + Ω h,ɛ Φ h = = [ ] φ h 1 (1 φ 1 ) if h > 11 1 φ [ ] 1 φ h+1 if h 11 1 φ Where V ar(x t h I i,t h ) = E[(x t h x t h t h ) ] is the variance of x t h conditioning on the agent information set I i,t h, 1 h<11 is an indicator function equal to one when h < 11 and Ω h,ɛ corresponds to the unforcastable part of x y t, which varies with h. Clearly, these last two expressions are independent of the agent s attention. Therefore, the forecaster can only affect, through her chosen effort, the conditional variance. For completeness, we can derive the expressions for Ω h,ɛ, when h > 11: Ω h,ɛ = σ ɛ (1 φ j+1 ) (1 φ) + σ ɛ h 1 j=1 φ j (1 φ 1 ) (1 φ) 10

11 Where the first expression on the right hand side of the previous equation: σ ɛ (1 φ j+1 ) (1 φ) = = = = σ ɛ (1 φ) σ ɛ (1 φ j+1 ) (1 φ) [(1 φ) + (1 φ ) + + (1 φ 1 ) ] σ ɛ (1 φ) [1 (φ + φ + + φ 1 ) + (φ + φ φ 4 )] [ σɛ 1 φ(1 ] φ1 ) + φ (1 φ 4 ) (1 φ) 1 φ 1 φ While the second expression: h 1 σɛ j=1 φ j (1 φ 1 ) (1 φ) = σ ɛ = σ ɛ = σ ɛ = σ ɛ (1 φ 1 ) (1 φ) h 1 j=1 φ j (1 φ 1 ) (1 φ) (φ + φ φ h ) (1 φ 1 ) φ (1 φ h 4 ) (1 φ) 1 φ φ (1 φ 1 ) (1 φ h 4 ) (1 φ) 3 (1 + φ) The unforcastable noise of the series depends only on parameters from the data generating process, i.e. they are independent of the agents effort. A similar expression can be derived for h < 1. Appendix D: Lemma 3 Proof. Starting from the constraint on attention: H(x t h I i,t h 1 ) H(x t h I i,t h )] κ i,t h Following Maćkowiak and Wiederholt (009), the entropy of two normally distributed variables X and Y, is given by: H(X Y ) = 1 log ((πe) T det(ω X Y )) 11

12 Where Ω X Y stands for the conditional covariance matrix. Since both the state variable x t h and the signals s i,t h are normally distributed, the we can write the constraint as: H(x t h I i,t h 1 ) H(x t h I i,t h )] = 1 log ( ) σ xi,t h t h 1 σ x i,t h t h () While clearly given the State Space representation depicted in Appendix A, the dimension of the conditional variance matrix is 1 1, the only endogenous variable that can actually have an impact on uncertainty reduction is the current distribution of today s signal. Thus, we can simply focus our attention on the (1, 1) element of this matrix. From the recursive expression for the conditional variance: σ x i,t h t h 1 σx i,t h t h = σx i,t h t h 1 (1 σx i,t h t h 1 + σ i,h,ψ Plugging this expression in () and ordering terms, we ended up with expression (15). ) Appendix E: Proposition 1 Proof. Starting from the definition of κ i,t h in equation (15), we can prove: κ i,t h 1 = σ x i,t h t h 1 σ i,h,ψ Hence, using the definition of σ i,h,ψ in order for the problem to be well defined and given some posititive level of attention κ i,t h, we need to impose: σ i,h,ψ = σ x i,t h t h 1 κ i,t h 1 σ η 0 To solve the agent s problem at any t h, we can use the same definition of θ i,h = σ i,h,ψ σ x i,t h t h 1 + σ i,h,ψ solving for the optimal σ i,h,ψ. In terms of the optimization solving for the optimal value of κ i,t h and is exactly the same since there s in a one to one mapping from 1

13 one to the other. Moreover, solving the problem for θ i,h instead of σ i,h,ψ is also the same since θ i,h is a monotonic function of the signal s variance. Then, the restriction on attention can be written as: ( ) 1 1 log κ i,t h θ i,h Clearly, there s no point for any forecaster to not use all his mental capacity to acquire information so the previous condition always holds with equality. Moreover, the current conditional variance, can be written, σ x i,t h t h = θ i,h σ x i,t h t h 1. of θ i,h : Finally, we can now solve the objective function relying on Lemma, 3 and the definition Where the FOC: min α i [Φ h θ i,h σx θ i,t h t h 1 + C h ] + λ 1 ( ) 1 i,h log θ i,h α i Φ h σ x i,t h t h 1 λ ( ) 1 = 0 ln() θ i,h Solving for the optimal value of θ i,h and plugging it back in the definition of κ i,t h we get to the optimal level of attention for each horizon h. 13

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