Empirical test of an Incomplete Information Model

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Empirical test of an Incomplete Information Model

Is it possible to represent an incomplete information economy as a hybrid economy?

Sigmund Ellingsrud

Master’s Thesis in Economics at the University of Oslo

Department of Economics University of Oslo

Norway

May 2019

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Empirical test of an Incomplete Information Model

Is it possible to represent an incomplete information economy as a hybrid economy?

Sigmund Ellingsrud

¹

Abstract

As shown by Angeletos and Huo [2018], an incomplete information economy is observationally equivalent to an economy with a myopic and anchoring representative-agent under certain restrictions. Using the seminal model from Smets and Wouters [2007], I test whether an estimated New Keynesian Phillips Curve on Norwegian and US data is consistent with the restrictions under which this equivalence holds. The evidence suggest there is empirical support for the observational equivalence result.

1Contact at [email protected]. I’d like to thank my supervisor Ragnar Nymoen for many wise and encouraging comments. In addition, I’d like to thank Steinar Holden for valuable comments and initiating my interest in the topic. They have both have greatly aided the quality of this work. Remaining errors are my own.

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List of Tables

1 Excerpts from Table 3 . . . 34

2 Prior information (parameters) . . . 41

2 (continued) . . . 42

3 Posterior Parameters Estimation Original Sample USA . . . 43

4 Posterior Parameters Estimation Reduced Sample USA . . . 44

5 Posterior Parameters Estimation Full Sample USA . . . 45

6 Posterior Parameters Estimation Norwegian Data Q1 1995 - Q4 2018 . . 46

7 Posterior Parameters Estimation Norwegian Data Q1 1981 - Q4 2018 . . 47

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List of Figures

1 Robustness check on Norwegian data . . . 39

2 Robustness check on US data . . . 40

3 Scatterplot ofωf and ωb drawn from the original posterior distribution . 48 4 Scatterplot ofω_f and ω_b drawn from my US data in the original period. . 49

5 Scatterplot ofω_f and ω_b drawn from my US data in the full period. . . . 50

6 Scatterplot ofωf and ωb drawn from the Norwegian data Q1 1995 - Q4 2018 51 7 Scatterplot ofω_f and ω_b drawn from the Norwegian data Q1 1981 - Q4 2018 52 8 Optimal π₁^∗ plotted together with scatter for full US sample. . . 53

9 Joint scatterplot US data . . . 53

10 Estimated KDE with confidence regions and level curve from Angeletos & Huo. The blue regions are 0.5, 0.90, 0.95 and 0.99 respectively. The red line isωb = 1− _ρ¹2δωf, the AH level curve. Based on original posterior distribution. . . 54

11 Estimated KDE with confidence regions and level curve from Angeletos & Huo. The blue regions are 0.5, 0.90, 0.95 and 0.99 respectively. The red line is ω_b = 1− _ρ¹2δω_f, the AH level curve. Based on US data in original period Q1 1948 - Q4 2004. . . 55

12 Estimated KDE with confidence regions and level curve from Angeletos & Huo. The blue regions are 0.5, 0.90, 0.95 and 0.99 respectively. The red line is ω_b = 1− _ρ¹2δω_f, the AH level curve. Based on US data in full period Q1 1948 - Q4 2018. . . 56

13 Estimated KDE with confidence regions and level curve from Angeletos & Huo. The blue regions are 0.5, 0.90, 0.95 and 0.99 respectively. The red line is ωb = 1− _ρ¹2δωf, the AH level curve. Based on Norwegian data Q1 1995 - Q4 2018. . . 57

14 Estimated KDE with confidence regions and level curve from Angeletos & Huo. The blue regions are 0.5, 0.90, 0.95 and 0.99 respectively. The red line isω_b = 1− _ρ¹2δω_f, the AH level curve. Based on full Norwegian data Q1 1981 - Q4 2018. . . 58

15 The raw variables of the US time series used in SW . . . 67

16 The transformed variables of the US time series used in SW . . . 68

17 The growth variables of the US time series used in SW . . . 69

18 The raw variables of the Norwegian time series used in SW . . . 71

19 The transformed variables of the Norwegian time series used in SW . . . 72

20 The growth variables of the Norwegian time series used in SW . . . 73

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1 Introduction

An economic model with a representative agent makes an at times a subtle, but significant assumption: Regardless of the representative agents’ information set, it will always include knowing what others know. Subtle because information about others is irrelevant when you’re the sole agent. Significant because it implicitly assumes a remarkable level of coordination: It models an economy where everyone knows everything about all the other agents in the economy, both their preferences and their information sets. In a model with heterogeneous agents and heterogeneous information, this coordination breaks down. Not only are agents required to form expectations about the dynamics of the fundamentals in the model, they must form higher order expectations as well. I.e., expectations of other agents’ expectations. And expectations of other’s expectations of their expectations and so on. In such a model, it’s possible that everyone produce more despite believing output will fall because they believe that everyone else believes output will rise and therefore raise their production, as the sentiment-shock inAngeletos and La’O [2013], a version of the island-model first proposed by Lucas [1972]. Such behaviour resemble notions like animal spirits inKeynes [1936] or Akerlof and Shiller [2010] and irrational exuberance inShiller [2016] andGreenspan[1996]. Both these concepts suggest some kind of deviation from the default assumptions of rationality found in standard microeconomic textbooks like Mas- Colell et al.[1995]. However, a recent strain of macroeconomic literature proposes that the dynamics of demand driven business cycles can be explained merely with heterogeneous agents possessing heterogeneous information while maintaining rational expectations. One such model, Angeletos and Huo[2018], proposes an incomplete information economy and shows that under certain restrictions it produces macroeconomic dynamics identical to that from an economy with a representative agent that is myopic and anchoring. Furthermore, they apply the theory to numerous examples, amongst them the hybrid New Keynesian Phillips Curve (NKPC) as found in e.g. Smets and Wouters [2007]. This thesis explores that particular application. Specifically, I test whether the restrictions necessary for the AH-model’s observational equivalence with the hybrid economy model hold with US and Norwegian data.

An important part of recent development of macroeconomic models is the attempt at reconciling traditional, Keynesian ideas about short-run macroeconomic dynamics with the first principles of the Real Business Cycle (RBC) framework. This is because without a micro-founded model, the lack of structural parameters subjects the theory to parameters that might not be policy invariant, thus being subject to the Lucas critique Lucas [1976].

However, a conventional RBC-model following Kydland and Prescott [1982] leaves no room for fiscal or monetary policy, explaining business cycles as efficient reactions to technological innovations. A negative demand shock, for instance, is paradoxical in such a model. If consumption suddenly drops, the representative agent becomes poorer.

Following standard microeconomic reasoning, the income effect cause a drop in demand for both consumption and leisure. Implying that hours worked, and thus output, increase.

This is one of several oddities about the RBC-framework, that has led the field towards an attempt to evolve the class of models to account for the conventional business cycle intuition followingKeynes [1936], while keeping its advantage; rigorous microfoundations suitable for sharp, testable predictions.

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The New Keynesian model as seen in e.g.Gali[2008] is such an extended RBC-model. The behavior of agents is still derived from optimizing behavior, equating marginal utilities with relative prices. Two mechanisms are introduced which in tandem ensures the non- neutrality of monetary policy: Monopolistic competition and Calvo pricing Calvo [1983].

Monopolistic competition implies that each firm no longer takes the market price as given, deciding on its own optimal price/quantity pair for maximum expected profits. Calvo pricing introduces an inability for these firms to adjust prices at will. Every period a fraction is allowed to adjust their price, while the remaining fraction must retain their old price. The simple form of this New Keynesian framework produce three key equations, one of which is the New Keynesian Phillips Curve. The NKPC derived from a simple New Keynesian model like in Gali [2008], takes the form

π_t=κy_t+δEtπ_t+1 (1)

Where πt represents inflation from period t−1 to period t, yt the output gap in period t, κ andδ are parameters depending on the underlying parameters of the model and Et

the (rational) expectations operator conditional on information from the first period until t−1.

However, it seems to be an empirical feature that the New Keynesian Phillips Curve (NKPC) fits the data better if augmented with myopia and anchoring, yielding the so-

called hybrid NKPC found in Smets and Wouters [2007] and Smets and Wouters [2003].

With myopia it’s meant an excessive discounting of the future relative to the baseline model. With anchoring it’s meant the parameter on lagged inflation is positive (zero in the baseline model).

The hybrid NKPC takes the form

π_t=κy_t+δω_fEπ_t+1+ω_bπt−1. (2) It’s identical to the NKPC except it has extra discounting of the future, i.e. myopia, ω_f and an added anchoring term with parameter ω_b. Seminal estimations of the New Keynesian model, like Smets and Wouters [2007], employ this variant as it seems to have a superior empirical performance.

Conventional microeconomic foundations of New Keynesian Economics, as in the model implying (1), permit neither myopia nor anchoring. As a result, several strains of the macroeconomic literature aim to close the gap between the theoretical framework and the empirical observation. There’s numerous such attempts, but this thesis examines the one made byAngeletos and Huo [2018]. They present an incomplete information model where agents get idiosyncratic private signals about the underlying fundamental(s).

Angeletos and Huo[2018] base their analysis on a conceptually intuitive, but computation- ally difficult, concept: When information is incomplete, every agent also has incomplete information about what others know and thus expects. The framework with a representative agent does not merely assume full information about the probability distributions of the fundamental variables, it also assumes agents have full information about other agents expectations. Every agent knows what every other agents knows. This is trivially true

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in any representative agent model, and eliminates the possibility of asking the question of what happens when agents are uncertain of not only the state variables, but others expectations of those, others expectations of your expectations of those and so forth.² As it turns out, under a set of restrictions, extending a standard model with idiosyncratic information alone provides observational equivalence with (2). Every version of Angeletos

& Huo’s incomplete information model can be represented by an observationally equivalent hybrid NKPC, but every hybrid NKPC cannot be derived from a version of the incomplete information model. Instead of having to estimate a model containing (infinite) layers of higher-order expectations, it’s possible to test whether estimated hybrid NKPC’s falls within the range consistent with the AH-model. Angeletos and Huo[2018] does this using US data from Coibion and Gorodnichenko [2015] and Gali et al. [2005]. This thesis replicate their application on the hybrid NKPC and test it against estimations of the Smets & Wouters model.

The only difference between the AH-model and the New Keynesian theory in Gali[2008]

is the introduction of idiosyncratic information. Each firm over an indexi∈(0,1) solves the problem

P_it^∗ = arg max

Pit

∞

X

k=0

(δθ)^kE^it

Qt|t+k

PitYit+k|t−Pt+kΨ_t+kYi,t+k|t

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subject to the set of constraints Y_it+k =_P^P^it

t+k

−

Y_t+k. Q_t|t+k is the discount factor from t to t+k, Y_t+k and P_t+k are output and price level in period t+k, P_it the price of firm i set in period t, Yi,t+k|t is the output of firm i in period t+k that last sat a new price in period t and Ψ_t+k is the real marginal cost in period t+k. Finally, is the elasticity of substitution.

The solution to this problem is

π_t= (1−δθ)(1−θ) θ

∞

X

k=0

(δθ)^kEt[ψ_t+k] +δ(1−θ)

∞

X

k=0

(δθ)^kEt[π_t+k+1] (4)

Where π_t is inflation and E^t is the average expectations at time t. If assuming the information is common and complete, this solution reduces to

π_t=κψ_t+δE^t[π_t+1]. (5) Using the law of iterated expectations. This is the NKPC found in e.g. Gali [2008]. An- geletos and Huo [2018] assumes that ψ_t follows an AR (1) process

ψ_t=ρψt−1+η_t (6)

2This concept is known as strategic uncertainty, and is treated in numerous papers. SeeAngeletos and Lian[2016]

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With η_t∼ N(0,1). Furthermore, it’s assumed each agent receives the following private signal each period t

x_it=ψ_t+u_it, u_it ∼ N(0, σ²) (7) With this information structure, (4) can no longer be reduced to (5). The incomplete information Phillips Curve cannot readily be tested. But the main result of Angeletos and Huo [2018] proves an observationally equivalent formulation that is testable. They derive that for some σ >0,ω_f <1 and ω_b >0, the equation

π_t =κψ_t+δω_bEt[π_t+1] +ω_fπt−1 (8) replicates the inflation dynamics of the incomplete information economy as long as the restriction

ω_b = 1− 1

ρ²δω_f (9)

holds. Equation (8) is exactly the hybrid NKPC applied in e.g.Smets and Wouters [2007]

and Gali et al. [2005], and is therefore possible to estimate. By checking whether existing estimates of the hybrid NKPC satisfies the restriction posited in (9), one can test whether AH’s proposed incomplete information economy can be represented by the hybrid NKPC.

If it can, it establishes empirical support for a theoretical link between microeconomic coordination problems and macroeconomic inflation dynamics. It’s worth noting that there are multiple reasons for why discussing models with incomplete information have merit, of which this thesis touch merely on a small fraction. For comprehensive discussions, seeAngeletos and Lian [2016], Angeletos [2018] or Angeletos and Huo [2018].

The main result is that across five different data-sets, three on US data and two on Norwegian data, the restrictions of the model proposed by Angeletos & Huo is consistent with the estimated Smets & Wouters model. This result holds across a broad range of parameters in the AH-model. I argue this is a natural starting point for an empirical examination. If I altered the models for some reason, it would potentially obfuscate the reasoning for why the results ended up the way they did. In particular, both the Smets & Wouters model and it’s criticisms are well known. Thus, I’ve tried to keep my implementation as close to the original as possible not only for simplicity but also for clarity and comparability to previous work. My thesis makes no real theoretical contribution, as both models are taken as given.

There are numerous choices made in the testing regime that might influence the conclusion.

For instance, it might be more pertinent to estimate an open-economy DSGE model for Norwegian data. Although that would firstly require theoretical advances since the hybrid NKPC no longer takes the same form. In addition, there are free parameters chosen in the estimation procedure. E.g. both when deciding the prior distribution in the Smets

& Wouters model, and when calibrating the Angeletos & Huo model. Thus, the results should be interpreted cautiously. The correct description is perhaps that this offering gives an initial empirical examination of the AH-model, rather than a comprehensive test.

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To partially combat this weakness, I implement a non-exhaustive list of robustness checks in Section 5.

A motivating example At first glance, it’s not obvious how powerful the coordination effect of common information really is. Originally designed as a logic puzzle, the blue eyes island puzzle functions as a thought experiment showing precisely the potential power of coordination from common information.³. Consider an island consisting of 100 people with brown eyes and 100 people with blue eyes. They are all perfectly rational, although some wedge makes it impossible for them to communicate. No one knows the color of their own eyes, but can perfectly observe everyone else’s eye color at any given moment.

Every night a ferry comes to the island and takes with it every person who’s been able to deduce his own eye color. At the beginning of the first day, a social planner appears and tells the island he sees at least one person with blue eyes. Following this information, who leaves the island and when?

Every person on the island already sees at least one person with blue eyes, so at first glance the social planners interference seems pointless. However, the answer is that all the blue eyed people leave on the 100th day while the brown eyed people never leave. The reason that the fact makes a difference is because of the ability to infer what others know.

To understand the solution, it’s helpful to start assuming there’s only one inhabitant and iterate. Imagine an island with only one blue-eyed person. Trivially, he’d leave on the first day. Then imagine an island with two blue-eyed people. On the first day, both would see one blue-eyed person and thus stay. On the second day, they’d both realize that if they had brown eyes the other person would’ve left the day before. Since he would then see zero people with blue eyes and infer his own color. Thus, both leave on day two. With three blue-eyed people, they’d all stay the first two days. But on day three, they’d induce that if any of the other two saw only one blue-eyed person, they’d leave on day two. Thus, all three leave on day three. Extending the argument by induction, the answer becomes all blue-eyed people leave on day 100. Had there been 150 blue-eyed people on the island, it would’ve been on day 150. The salient feature of this thought experiment to take away is the following: Everyone already knew that there was at least one blue-eyed person on the island. In fact, everyone already knew that there was at least 99. The piece of information provided by the social planner was therefore irrelevant to the information set about the fundamentals, but relevant to the information about others’ information about the fundamentals. It shows the remarkable potential for coordination that stems from common information, revealing why one must have good reasons to assume it away when modeling systems of people such as an economy.

The rest of the thesis is organised as follows. Section 2 describes the relation to existing literature. Section 3 outlines the two models used. Section 4 describes the data-sets used for estimation. Section5 describes the estimation methods and results. Section 6 provides a non-exhaustive discussion on broader topics connected to the thesis. Section 7 concludes.

The analysis is performed using a combination of Dynare Adjemian et al. [2017], Ju-

3This particular adoption of the puzzle is fromhttps://www.xkcd.com/blue_eyes.html

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lia Bezanson et al. [2017], Python van Rossum [1995] and R R Core [2018], Hyndman [2018] and Sax and Edellbuette [2018]. All data-sets, programs and results are available on request.

2 Literature

There is a growing number of articles in similar vein to Angeletos and Huo[2018] authored or co-authored by Angeletos. For a good overview, I’d recommend the chapter in the Handbook of Macroeconomics on the topic of incomplete information Angeletos and Lian [2016]. An even more recent overview is found in Angeletos [2018], which aptly describes the nuances of models with incomplete information versus conventional New Keynesian models and even Real Business-cycle models. In addition to the model discussed in this thesis, perhaps the two most noteworthy other works are the model with sentiment shocks Angeletos and La’O [2013] and the work using an incomplete information model to analyze the forward guidance puzzle Angeletos and Lian [2018] The former model is an island model following Lucas [1972]. Each agent lives on an island and is randomly matched with another agent on another island for trade each period. However, the production decision is made prior to knowing the match, introducing strategic uncertainty about the terms of trade. Angeletos and La’O[2013] show that by varying the sentiment, i.e. higher order beliefs, business cycles can arise.

Broadly, this class of models has grown from early attempts at modeling regime change.

Perhaps the most famous contribution being the work on bank runs by Diamond and Dybvig[1983]. Models of regime change is designed to analyze whether the players decide to attack or not attack some object. In the aforementioned paper, this constitutes whether to initiate a bank run or not. The problem with the early models of regime change was the existence of multiple equilibria. However, important contributions by Morris and Shin [2002], Morris and Shin[1998] and van Damme and Carlsson [1993] spurred a version of these models where small perturbations of the agents information sets provided regime change models with unique equilibria. This literature, known as the global games literature, can be viewed as either the predecessor or the parent class of the incomplete information model studied in this thesis.

The literature on uncertainty shocks following Bloom [2009] is similar but different. It seeks to explain similar macroeconomic phenomena, but the mechanism proposed leaves no room for coordination failure. That paper uses shocks to higher moments, shocks that can at times produce macroeconomic dynamics with similar sign as the AH-model. Since uncertainty in Bloom’s paper is postulated as something that shifts from shocks. For instance, a negative oil price shock can increase uncertainty and this decrease investment in oil, implying the oil price may decrease even more. Thus, it is related by interest but not so much by method.

The same can be said about the research project of Xavier Gabaix on rational inattention, for example Gabaix [2017], Gabaix [2016] and Gabaix [2015]. Gabaix derives a comprehensive microeconomic theory with rational inattention, and applies it amongst others to a New Keynesian model. In this class of models, only myopia is permitted by

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the theory, not anchoring. This is empirically the most relevant difference between the AH-model and this set of papers. However, Gabaix considers a deeper level of endogeneity, attempting to explain why information is sparse. That incomplete information stems from rational inattention is a plausible explanation, and one that the AH-model stays silent on.

Angeletos & Huo has an exogenous information process, leaving the interpretation open.

Another seminal contribution on modelling rational inattention is Sims[2003], while a vastly different approach is one with sticky information akin to sticky prices following Mankiw and Reis [2002] and Mankiw and Reis [2007].

For estimation of the medium-scale DSGE model from Smets and Wouters[2007] I use conventional methods described in e.g. Fern´andez-Villaverde et al. [2015], Herbst and Schorfheide [2014] or the Dynare manual Adjemian et al.[2017]. There is an important literature discussing Bayesian estimation of DSGE models. For example regarding different algorithms for sampling the posterior distribution, like sequential Monte Carlo sampling Herbst and Schorfheide [2013]. However, it’s been beyond the scope of this thesis to thoroughly explore the varying methods, tools and techniques and is only briefly discussed.

Note that this reduction weakens the robustness of the obtained results. For example, exploring alternative algorithms or priors would be a natural extension should one desire a more complete empirical examination. Discussion of other possible iterations of estimating DSGE models, such as nonlinear or continuous time versions is also omitted to keep the scope reasonable.

Empirically, this thesis is related to recent work on survey expectations. Coibion and Gorodnichenko [2015] analyze the informational frictions in a number of surveys by regressing ex-post forecast errors on ex-ante forecast revisions. Showing this can serve as an estimation of the degree of information rigidity in noisy-information models like Sims [2003] or sticky-information models like Mankiw and Reis [2002]. The significant regression coefficients suggests that there is microeconomic survey evidence of information rigidity. Albeit more complicated, a similar exercise is possible for the model in Angeletos and Huo [2018]. Basselier et al.[2018] is another example of using survey expectations in macroeconomic models, while Bachmann and Zorn [2018] studies survey data and the relation with aggregate investment in Germany.

It also worth noting that one broader but still highly relevant question is that of identifying shocks which cause business cycles. On this topic, Mian and Sufi [2014] studied the drop in employment during the financial crisis. According to their analysis, a significant part of the negative shock was attributable to reduced housing net worth causing reduced consumer demand. Although this evidence is not directly related to the models discussed in this thesis, it’s worth noting that macroeconomic dynamics seems to be capable of producing demand driven business cycles broadly consistent with these results.

Generally, the work on incorporating incomplete information models is also consistent with less technical discussions on the principles of macroeconomics like Akerlof [2002] and Akerlof and Shiller [2010]. Furthermore, the relationship with results from behavioral economics depends on nuances in how incomplete information is interpreted. For example, myopia can arise from a mechanism such as hyperbolic discounting Laibson [1997], from incomplete information or from rational inattentionGabaix[2016]. The AH-model doesn’t

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really take a stance of what generates the informational rigidity. Therefore, it’s difficult to claim the AH-model is inconsistent with evidence on e.g. hyperbolic discounting.⁴

3 Model

3.1 The Angeletos & Huo Model

The model in Angeletos and Huo [2018] is a type of global game model made popular by Morris and Shin [2002] . The framework has discrete timed t and a continuum of agents i∈(0,1). At each t, every agenti chose an action a_it. The best response is specified as:

a_it =Eit

φξ_t+βa_it+1+γa_t+1

(10) Where ξ_t is an exogenous fundamental and Eit the rational expectations operator conditional on the information set available at time t. The information set at time t is the history of signals until t. a_t is the average action across agents, i.e.^R₀¹ia_itdi. φ is a parameter satisfyingφ > 0 andφ∈(0,1). β and γ are parameters satisfying β, γ ∈(0,1) and β+γ < 1⁵ φ is a parameter satisfying φ > 0 and φ ∈(0,1). Equilibrium requires that every player iplays his best response.

Starting with a_i,t =Ei,t

φξ_t+βa_i,t+1+γa_t+1

and iterating once yields a_i,t+1 =Ei,t+1

φξ_t+1+βa_i,t+2+γa_t+2

Inserting in the best response for t+ 2 a_i,t =Ei,t

"

φξ_t+β

Ei,t+1[φξ_t+1+βa_i,t+2+γa_t+2]

+γa_t+1

#

iterate once more to get a_i,t =Ei,t

"

φξ_t+β

Ei,t+1[φξ_t+1+β[Ei,t+2

φξ_t+2+βa_i,t+3+γa_t+3

] +γa_t+2]

+γa_t+1

#

Rearrange and collect terms to get

4For instanceDellavigna and Malmendier[2006]. Note also that there is an issue with aggregation to my knowledge unsolved.

5Note from (14) why this is necessary. If not, then the system is no longer stable and it’ll permit explosive behavior. Which removes a model’s analytical value.

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a_i,t =E^i,t

"

φξ_t+βE^i,t+1[φξ_t+1] +β²E^i,t+2[φξ_t+2] + γa_t+1+βEi,t+1[γa_t+2] +β²Ei,t+w[γa_t+3] +

β³E^i,t+2[a_i,t+3]

#

Showing that (10) can be iterated forward to yield the following expression, using the fact that at time t, Ei,tEi,t+1x_t=Ei,tx_t:⁶

a_i,t =

∞

X

k=0

β^kEi,t[φξ_t+k] +γ

∞

X

k=0

β^kEi,t[a_t+k+1] (11)

Taking the average of this expression results in the necessary equilibrium condition. Recall that:

a_t=

Z 1 0

ia_i,tdi (12)

Implying a_t =

Z 1 0

i

P∞

k=0β^kEi,t[φξ_t+k] +γ^P^∞_k=0β^kEi,t[a_t+k+1]

di a_t =

Z 1 0

i

P∞

k=0β^kEi,t[φξ_t+k]

di+

Z 1 0

i

γ^P^∞_k=0β^kEi,t[a_t+k+1]

di a_t =φ

∞

X

k=0

β^k

Z 1 0

iEi,t[ξ_t+k]di+γ

∞

X

k=0

β^k

Z 1 0

iEi,t[a_t+k+1]di

And, when denoting the average expectations as Et, averaging using 11becomes

a_t=φ

∞

X

k=0

β^kE^t[ξ_t+k] +γ

∞

X

k=0

β^kE^t[a_t+k+1] (13)

Note that it’s possible to nest the case of complete information, i.e. remove the element of heterogeneous information. In that case, the average expectation in (13) will just be the expectation of the representative agent. The law of iterated expectations thus apply and (13) reduce to

a_t=φξ_t+δEt[a_t+1] (14) Withδ= β+γ ∈(0,1). The difference between agents in the AH-model is their information.

Removal of that heterogeneity collapses the model into the standard framework with a representative agent, reducing 10 to14.

6This is due to the law of iterated expectations, or the law of iterated projections as it’s sometimes referred as. See e.g. Section 4.5 inHamilton[1994]

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Note that (14), with particular choice of parameters, can represent several key equations used in economics. By letting

a_t =π_t ξ_t= ˜y_t φ=κ δ=β

Equation 14becomes

π_t=κy˜_t+βEt[π_t+1]. (15) I.e., the New Keynesian Phillips Curve from e.g.Gali [2008]. Similarly, the model also nests the Dynamic IS equation from New Keynesian models by letting the action be consumption and the fundamental the interest rate.

Iterating 14forward yields:

a_t =φ

∞

X

k=0

δ^kEt[ξ_t+k] (16)

From this, a key difference between the incomplete information model, (11), and the complete information benchmark, (16), emerge: The full information case relies only on the expectations on future value of fundamentals, while in the incomplete case the expectations of future average actions matter. Since the average action again is a function of all agents actions, it follows that this term reflects the effect of higher-order beliefs. An agent i must form an expectation of the other agents expectation, their expectations of his expectations of their expectations, and so on.

The incomplete information model above is intractable. Conceptually, it’s because of the infinite degree of higher-order expectations.Angeletos and Huo[2018] provide an example, even under simplifying assumptions, conveying how quickly the degree of expectations grow. The authors show, that under certain assumptions, it’s possible to reduce this complexity into a tractable problem.

Assumption 1. The fundamental,ξ_t follows an AR (1) process

ξ_t=ρξt−1+η_t= 1

1−pLη_t (17)

η_t∼ N(0,1) (18)

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Where ρ∈(0,1) is the AR (1) parameter, Lis the lag operator, and η_t the period t shock to the time series. Furthermore it’s assumed that:

Assumption 2. Agenti receives a noisy signalx_it about the fundamental.

x_it=ξ_t+u_it (19)

Where uit ∼ N(0, σ²). This decides the structure of the incomplete information. Every period an agent receives a noisy signal about the state of the fundamental. How much of an information wedge there is is parameterized by the variance of the noise term u_it. Since all agents receive different signals, they have different information about the fundamental and no longer full information about what the other agents expect. Note that this means the information structure is exogenous, ruling out endogenous signals within the model. In addition, it implies that what this models calls information may or may not correspond to what we’d characterise as information in the real world. Consider an agent who observes the world perfectly, but some cognitive bias leads him to infer a noisy signal where there shouldn’t be one⁷. It could also represent an agent with rational inattention, accepting a noisy signal believing more precision would be too costly⁸. This theory is silent on what causes the informational friction and its precise interpretation. This isn’t necessarily a flaw, since this certain specification allows the reduction of the state space from the infinite higher-order expectations to a model where it’s as if there’s a representative agent.

Applying the two assumptions above and methods from Huo and Takayama [2018], Angeletos and Huo [2018] prove that the equilibrium exists, is unique and that the aggregate outcome follows:

a_t=

1− υ ρ

1 1−υL

a^∗_t (20)

Where a^∗_t is the equilibrium process under perfect information, that is

a^∗_t =

φ 1−ρδ

1 1−ρL

η_t (21)

And υ is a scalar given by a certain relationship between ρ, σ², β and γ, i.e. the fundamental parameters of the model. The proof is rather complex, thus omitted from this discussion. It can be found in Angeletos and Huo [2018].

Equation (21) follows from the assumptions above, which implies that Et[ξ_t+1] =ρξ_t and that ξ = _1−ρL¹ ηt. In this case:

7There are numerous such phenomena discussed in behavioral economics. Hyperbolic discounting, the gamblers fallacy or anchoring against a reference is a non-exhaustive list.

8For an alternative approach, Gabaix[2016] derives a model with rational inattentive agents and arrives at an Euler equation with myopia, not myopia and anchoring as in this model

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a_t =φξ_t+δE^t[a_t+1]

a_t =φξ_t+δρa_t

a_t(1−δρ) =φξ_t

a_t = _1−δρ^φ _1−ρL¹ η_t

By inserting a^∗_t ina_t I get at = (1−^υ_ρ)(_1−υL¹ )((1−ρδ)(1−ρL)^φ )

Extracting the terms without the lag-operator and lettingψ = (1−^υ_ρ)(_1−ρδ^φ ), the expression becomes

a_t =ψ(1−υL)⁻¹(1−ρL)⁻¹η_t

Φ₁(L)a_t=η_t with

Φ₁(L) = _ψ¹ −^υ+ρ_ψ L+^ρυ_ψL².

The lag polynomial on an AR(2) process.

Now consider instead a common economy with a representative agent described with the following process

at=φξt+δωfE^t[a_t+1] +ωbat−1 (22) With ω_f <1 and ω_b >0. Letting those two parameters equal 1 and 0 respectively will reduce this to the perfect information benchmark in (14). Following the same assumption on the process of the fundamental,Eta_t+1 =ρa_t. Thus

a_t =φξ_t+δω_fEt[a_t+1] +ω_bat−1

a_t =φξ_t+δω_fρa_t+ω_bLa_t

a_t(1−δω_fρ−ω_bL) =φξt

a_t(1−δω_fρ−ω_bL) =φ_1−ρL¹ η_t

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a_t(1−δω_fρ−ω_bL)(1−ρL) =φη_t

Φ₂(L)a_t=η_t With

Φ₂(L) = _φ¹(1−δω_fρ−(ω_b+ρ−δω_fρ²)L+ω_bρL²)

Since the solution of both models follows an AR (2) process, it’s possible to state that these two processes are identical if they have identical lag polynomials. I.e., when

Φ₁(L) = Φ₂(L) (23)

It’s possible to re-write the hybrid economy AR(2) process as

a_t= ζ₀

1−ζ₁Lξ_t (24)

With

ζ₀ = φζ₁

ω_b −ρδω_fζ₁ (25)

and

ζ₁ = 1

2ωfδ(1−^q1−4δω_fω_b) (26) since the incomplete information model’s process can be written as:

a_t=

1−υ ρ

1 1−υL

φ 1−ρδ

ξ_t (27)

setting

ζ₀ =

1− υ ρ

φ 1−ρδ

(28)

ζ₁ =υ (29)

equates the two processes. That is, when the above two equations hold, the incomplete information economy and the hybrid economy displays identical dynamics. In other words, they’re observationally equivalent. It’s now possible to combine (26), (25), (29) and (28)

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to obtain the set of pairs (ω_f, ω_b) required for the observational equivalence. Combine (26) and (29) to get

υ = _2ω¹

fδ(1−^q1−4δω_fωb) 2ω_fδυ = 1−^q1−4δω_fω_b

q1−4δω_fω_b = 1−2ω_fδυ

using the formula for expanding quadratic expressions 1−4δω_fω_b = 1−4ω_fδυ+ 4ω_f²δ²υ²

−δωfωb =−ωfδυ+ω_f²δ²υ²

−ω_b =−υ+δυ²ω_f

ω_b =υ(1−δυω_f).

Now combine (25) with (28) and insert the above expression for ω_b

1−^υ_ρ

φ 1−ρδ

= _ω ^φζ¹

b−ρδω_fζ1

1

1−ρδ − _ρ(1−ρδ)^υ = _ω ^υ

b−ρδω_fυ

ρ−υ

ρ(1−ρδ) = _ω ^υ

b−ρδω_fυ

ω_b−ρδω_fυ = ^{υρ(1−ρδ)}_ρ−υ

υ(1−δυω_f)−ρδυω_f = ^{υρ(1−ρδ)}_ρ−υ

1−δυω_f −ρδω_f = ^ρ(1−ρδ)_ρ−υ

−δυωf −ρδωf = ^ρ(1−ρδ)_ρ−υ −^ρ−υ_ρ−υ

−δ(υ+ρ)ω_f = ρ(1−ρδ)−ρ+υ ρ−υ

δ(ρ+υ)ω_f = ^{ρ−υ−ρ+ρ}_ρ−υ ²^δ

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Rearrange, cancel and apply the formula for the conjugate product to the denominator to get:

ωf = ρ²δ−υ

δ(ρ² −υ²) (30)

Insert into the expression for ω_b: ω_b =υ(1−δυω_f)

ω_b =υ−δυ²_δ(ρ^ρ²2^δ−υ−υ²)

ω_b = ^υδ(ρ²^−υ_δ(ρ²^)−δυ2−υ²²^(ρ) ²^δ−υ)

ωb = ^υ(ρ²^−υ²_ρ^)−υ₂_−υ²₂^(ρ²^δ−υ)

ω_b = ^υ[^(ρ²^−υ²^)−υ(ρ²^δ−υ)]

ρ²−υ²

ω_b = ^υ(ρ_ρ²2^−υρ−υ²²^δ)

Meaning the expression for ωb is

ω_b = υρ²(1−δυ)

ρ²−υ² (31)

It’s now possible to combine (30) and (31) to obtain the restriction imposed by the AH-model on the parameters of the hybrid economy. Staring with

ωf = _δ(ρ^ρ²2^δ−υ−υ²)

Now choose a convenient way to add 0 ωf = ^ρ²^δ−υ+δυ_δ(ρ₂_−υ²₂^−δυ₎ ²

And a convenient way to multiply with 1

ω_f = ρ²(ρ²δ−υ+δυ²−δυ²)

ρ²(δ(ρ²−υ²)) (32)

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Multiply out the numerator to get ω_f = ^δρ⁴^−δυ_ρ2²δ(ρ^ρ²^−ρ²−υ²^υ+δυ²) ²^ρ²

Splitting the expression yields ω_f = _ρ^δρ2δ(ρ⁴^−δυ²−υ²^ρ²²) − _ρ¹2δ

ρ²υ+ρ²υ²δ ρ²−υ²

Now recognize that

ρ²υ+ρ²υ²δ ρ²−υ² =ω_b And that

δρ⁴−ρ²υ²δ ρ²δ(ρ²−υ²) = 1

We obtain the following relationship between ω_f and ω_b

ωf = 1− 1

ρ²δωb (33)

Now what does this mean in words? It means that the incomplete information economy can be represented as a hybrid economy if and only if condition (33) holds. Because whenever that condition holds the coefficients on the two AR (2) processes are identical.

If you consider the set of all possible hybrid economies, only this subset will be consistent with the idea that an incomplete information economy can be represented by a hybrid economy. In other words, it’s true that for a givenσ, the incomplete information economy can be represented by a hybrid economy. But it’s not true that a hybrid economy can always be represented as an incomplete information economy. Thus, with an unrestricted estimation of ω_f andω_b it’s possible to test whether condition (33) holds in some data.

Unrestricted here means unrestricted relative to the condition imposed by the theory of the incomplete information model.⁹

Application on the New Keynesian Phillips Curve Consider the production side of a New Keynesian economy like in Gali [2008], with the only difference that agents now have heterogeneous information. There’s a continuumi∈(0,1) of firms producing a differentiated good in monopolistic competition. Meaning firms are able to set their own price to maximize profits, but Calvo pricing is introduced to only allow infrequent adjustment. I.e., every period only a fraction 1−θ of the firms are allowed to set a new price. The remainingθ must keep their old price. The optimal price is, as usual, obtained by solving

9Note that it’s impossible to undertake any estimation that are truly unrestricted, as any designed test must be a joint test of the theory but also the test design. However, such a debate quickly enters a philosophical realm outside the scope of this thesis.

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P_i,t^∗ = arg max

Pi,t

∞

X

k=0

θ^kEi,t

"

Qt|t+k

P_i,tYi,t+k|t−Ψ_t+k(Yi,t+k|t)

#

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I.e. the sum of discounted expected profits over all future periods weighted by the probability of changing the price. The firms are facing the constraints of the demand functions for the good in each period.

Yi,t+k =

Pi,t

P_t+k

−

Yt+k (35)

and use the technology

Y_i,t+k =A_i,t+kN_i,t+k^1−α (36)

In this setup, Qt|t+k is the stochastic discount factor between t and t+k, P_i,t the price of good iat t, P_t andY_t the aggregate price and goods indices att,Yi,t+k|t is the quantity produced of good iconditional on that the firm hasn’t changed its price since periodt. Ψ_t is the cost function at t andθ is the probability of a firm keeping its price. Since there’s a continuum of firms, 1−θ also corresponds exactly to the fraction of firms resetting their price¹⁰

Taking the relevant first-order condition, solution log-linearised around the steady state is

p^∗_it = (1−δθ)

∞

X

k=0

(δθ)^kEit[ψ_t+k+p_t+k] (37)

Where lower-case letters denote the natural logarithms of the upper case variables, as is conventional. Assuming firms don’t extract information from the current price level, 37 can be restated in terms of inflation:

p^∗_it−pt−1 = (1−δθ)

∞

X

k=0

(δθ)^kEit[ψ_t+k+π_t+k] (38)

Every period a fraction θ adjusts their price, leaving the price level each period as

p_t= (1−θ)

Z

p^∗_i,tdi+θpt−1 (39) And inflation each period becomes

10This follows from the law of large numbers, as the empirical mean equals the true mean in the limit.

Technically, it’s only almost surely.

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π_t=p_t−p_t−1

π_t= (1−θ)^R p^∗_i,tdi+θpt−1−pt−1

Collecting terms yields

π_t=p_t−pt−1 = (1−θ)(

Z

p^∗_i,tdi−pt−1) (40) Combining (38) and (40) yields:

π_t= (1−δθ)(1−θ) θ

∞

X

k=0

(δθ)^kEt[ψ_t+k] +δ(1−θ)

∞

X

k=0

(δθ)^kEt[π_t+k+1] (41)

This incomplete information version of the New Keynesian Phillips Curve may have conceptually beneficial properties, but cannot be readily estimated. In fact, to test such a relationship one needs a set of subjective expectations which is not available in any dataset. It’s even possible to question whether it’ll ever be available, given the difficulty of collecting expectations that are both unbiased and frequent enough to reflect the true subjective expectations. However, it’s now possible to apply the result of observational equivalence derived earlier. By letting

• ξ_t =ψ_t

• a_t = π_t

• φ= (1−θδ)(1−θ) θ

• β =δθ

• γ =δ(1−θ) We get

a_t=φ

∞

X

k=0

β^kEt[ξ_t+k] +γ

∞

X

k=0

β^kEt[a_t+k+1] (42)

Which is exactly 13, the equilibrium solution of the incomplete-information model (as long as the previous assumptions are maintained). Thus, following earlier analysis, it’s possible to re-state the incomplete-information New Keynesian Phillips Curve as

π_t =κψ_t+δω_fE^t[π_t+1] +ω_bπ_t−1 (43) As long as

ω_f = 1− 1

ρ²δω_b (44)

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This concludes the model section of this thesis. In conventional estimations of the hybrid NKPC, which is identical in form to (43), the restriction provided by this theory is not present. Therefore, there is no reason to expect any theoretical directions as to whether the estimates of the hybrid NKPC is inside or outside the set of ω_f andω_b consistent with the theory from Angeletos and Huo [2018]. Thus, those estimations provide a reasonable test of this theory. Which is what the rest of this thesis will concern itself about.

Empirical test of an Incomplete Information Model

Empirical test of an Incomplete Information Model

Is it possible to represent an incomplete information economy as a hybrid economy?

Sigmund Ellingsrud

Master’s Thesis in Economics at the University of Oslo

Department of Economics University of Oslo

Norway

May 2019

Empirical test of an Incomplete Information Model

Is it possible to represent an incomplete information economy as a hybrid economy?

Sigmund Ellingsrud

Abstract

Contents

List of Tables

List of Figures

1 Introduction

2 Literature

3 Model

3.1 The Angeletos & Huo Model