Behavioral Implications of Causal Misperceptions Part II
Ran Spiegler (TAU & UCL)
ES Winter School, Delhi
December 2019
β’ DAGs represent subjective causal models.
β’ The factorization formula πππ π represents systematic belief
distortion due to fitting a wrong causal model to long-run data
β’ Personal equilibrium: Subjective EU maximization w.r.t πππ π
β’ Bayesian-network tool (d-separation) helps understanding when to watch out for equilibrium effects
Part 1 Recap
β’ Interacting with/among agents who form beliefs according to the model presented in Part I:
β Can a rational principal systematically fool an agent with causal misperceptions?
β Strategic interaction among players, where the notion of a playerβs type includes his DAG
Interaction
β’ Interacting with/among agents who form beliefs according to the model presented in Part I:
β Can a rational principal systematically fool an agent with causal misperceptions?
β Strategic interaction among players, where the notion of a playerβs type includes his DAG
Interaction
A Leader-Follower Setting
β’ The leader observes a state of Nature and takes an action.
β’ The follower then observes a signal and forms a belief about other variables (including endogenous variables that depend on his belief).
β’ The leaderβs payoff is an indirect linear function of the followerβs belief regarding a particular variable.
A Leader-Follower Setting
β’ The leader is rational; the follower has a subjective DAG.
β’ An ex-ante perspective: Choose leaderβs strategy and
followerβs belief to maximize the leaderβs expected payoff
β Subject to the constraint that the followerβs belief factorizes the objective distribution according to his DAG.
β’ π π β {0,1} indicates whether a firm sponsors a review of its product after privately observing the productβs quality ππ.
β’ Review content π‘π‘ is some stochastic function of ππ and π π .
β’ A consumer observes π‘π‘ and forms a quality estimate ππ.
β’ ππ is an objective long-run joint distribution over ππ, π π , π‘π‘.
An Example: Manipulating Reputation
β’ The firmβs payoff is ππ β πππ π , where ππ > 0.
β’ Rational-expectations quality estimate: ππ = πΈπΈ ππ π‘π‘ for every π‘π‘.
βΉ The ex-ante optimal strategy is to play π π = 0 for all ππ.
β’ The consumer has a subjective causal model. E.g.:
β βNaΓ―veβ model: ππ β π‘π‘ β π π β βCynicalβ model: ππ β π π β π‘π‘
Manipulating Reputation
β’ Consider the βcynicalβ model π π : ππ β π π β π‘π‘.
β’ πππ π ππ π‘π‘ = βπ π ππ π π π‘π‘ ππ(ππ|π π )
β’ The consumerβs quality estimate: πΈπΈπ π ππ π‘π‘ = βππ πππ π (ππ|π‘π‘)ππ
β’ πΈπΈ ππ = βπ‘π‘ ππ(π‘π‘)πΈπΈπ π ππ π‘π‘
β’ The question: πΈπΈ ππ = πΈπΈ ππ ?
Manipulating Reputation
π¬π¬ ππ = οΏ½
ππππ ππ π¬π¬πΉπΉ π½π½ ππ =
= οΏ½
π‘π‘οΏ½
ππ οΏ½
π π ππ(π‘π‘)ππ π π π‘π‘ ππ(ππ|π π )ππ = οΏ½
π‘π‘οΏ½
πποΏ½
π π ππ(π π )ππ π‘π‘ π π ππ(ππ|π π )ππ
= οΏ½
πποΏ½
π π ππ(π π )ππ(ππ|π π )ππ οΏ½
π‘π‘ππ π‘π‘ π π = οΏ½
ππ οΏ½
π π ππ(ππ)ππ(π π |ππ)ππ
= οΏ½
ππππ ππ ππ οΏ½
π π ππ π π ππ = π¬π¬(π½π½)
Algebra Says Yes
β’ The consumerβs quality estimate is correct on average, despite his misperception that π‘π‘ β₯ ππ|π π .
β The firmβs ex-ante optimal strategy coincides with the rational-expectations prediction.
β’ Which DAGs make systematically biased estimates possible?
β’ Would standard parametrizations of ππ eliminate this possibility?
The Questions
General Analysis
β’ I focus now on the followerβs belief, ignoring the leader.
β’ π₯π₯0, π₯π₯1, β¦ , π₯π₯ππ is a collection of real-valued economic variables.
β’ An agent has a subjective causal model represented by a DAG whose set of nodes is ππ β {0, β¦ , ππ}, 0 β ππ.
β’ The agent observes π₯π₯0 and forms an estimate/forecast ππππ of π₯π₯ππ, ππ β ππ\{0}.
General Analysis
β’ ππ : An objective joint distribution over π₯π₯0, π₯π₯1, β¦ ,π₯π₯ππ, (ππππ)ππβππ β Full support over π₯π₯0, π₯π₯1, β¦ , π₯π₯ππ
β Defining ππ over (ππππ)ππβππ is not necessary for the general analysis.
The Problem
β’ The agentβs estimates:
ππππ = πΈπΈπ π π₯π₯ππ π₯π₯0 = οΏ½
π₯π₯πππππ π (π₯π₯ππ|π₯π₯0)π₯π₯ππ Definition: The agentβs estimates are unbiased if
βπ₯π₯0 ππ π₯π₯0 πΈπΈπ π π₯π₯ππ π₯π₯0 β‘ πΈπΈ(π₯π₯ππ) for every ππ and every ππ β ππ\ 0 . Problem: Which DAGs induce unbiased estimates?
β’ Definition: A DAG is perfect when it contains no
βimmoralitiesβ.
β’ Corollary of Verma-Pearl equivalence: Two perfect DAGs are equivalent if and only if they have the same undirected
version.
β Direction of an individual link is observationally meaningless.
Perfect DAGs
Proposition: The agentβs estimates are unbiased if and only if his DAG is perfect.
β’ Identifying neglect of direct correlation between direct causes as the potential source of systematic errors.
β’ A somewhat more elaborate condition for an unbiased estimate of a given variable relies on the tool of d-separation.
A Result
β’ πππ π π₯π₯ππ β‘ ππ π₯π₯ππ if ππ is an ancestral node in π π .
β’ Because π π is perfect, every ππ can be regarded as ancestral.
β’ Therefore, πππ π π₯π₯ππ = ππ π₯π₯ππ .
οΏ½π₯π₯0ππ π₯π₯0 πππ π π₯π₯ππ π₯π₯0 = οΏ½
π₯π₯0πππ π π₯π₯0 πππ π π₯π₯ππ π₯π₯0 = πππ π π₯π₯ππ = ππ π₯π₯ππ
Proof: If
β’ An imperfect π π must contain an βimmoralityβ, which may or may not involve the node 0 (need to check various cases).
β’ E.g., ππ β 0 β ππ (no direct link between ππ and ππ)
β’ Construct ππ such that π₯π₯ππ and π₯π₯ππ are correlated, π₯π₯0 is some function of their joint realizations, and all other variables are independent. Show that βπ₯π₯0 ππ π₯π₯0 πππ π π₯π₯ππ π₯π₯0 β ππ(π₯π₯ππ).
Proof: Only If (Basic Idea)
Implications for Reputation Example
β’ The βnaΓ―veβ DAG ππ β π‘π‘ β π π and βcynicalβ DAG ππ β π π β π‘π‘ are both perfect.
β The firm cannot enhance its average reputation.
β’ The DAG ππ β π‘π‘ β π π is imperfect; the firm may be able to
manipulate its average reputation, for some parameterization.
β’ ππ β {0,1}
β’ ππ ππ = 1 = 0.5
β’ π‘π‘ = ππ + π π with probability one.
β’ πππ π ππ = 0 π‘π‘ = 2 = πππ π ππ = 1 π‘π‘ = 0 = 0 β Consistent with rational expectations
ππ β π‘π‘ β π π : Specific Parameterization of ππ
β’ In contrast, the firm can play a mixed strategy ππ π π ππ such that πππ π ππ = 1 π‘π‘ = 1 > ππ ππ = 1 π‘π‘ = 1
β’ Average reputation βπ‘π‘ ππ π‘π‘ πππ π ππ = 1 π‘π‘ can exceed E ππ = 0.5β¦
β’ β¦but cannot exceed 9/16.
β’ Optimal strategy when cost of sponsorship is c β (0,0.5): ππ π π = 1 ππ = 1 = 0 ππ π π = 1 ππ = 0 = 0.5 β ππ
ππ β π‘π‘ β π π : Specific Parameterization of ππ
β’ A central bank chooses an action ππ that affects inflation ππ.
β’ The private sector observes ππ and forms an inflation forecast ππ (simultaneously with the realization of ππ).
β’ Real output is π¦π¦ = ππ β ππ + ππ, where ππ ~ ππ 0, ππ2 β An βexpectationalβ Phillips curve
An Example: βMonetary Policyβ
β’ This set-up is quite standard (e.g. Sargent 1999).
β’ If the private sector had rational expectations, it would form the optimal conditional inflation forecast ππ = πΈπΈ(ππ|ππ).
β Because πΈπΈ ππ = πΈπΈ(ππ), the central bank cannot use monetary policy to enhance expected output.
βMonetary Policyβ
β’ Suppose the private sectorβs (imperfect) DAG is π π : ππ β ππ β π¦π¦. β βClassicalβ belief in monetary neutrality.
πππ π ππ, ππ, π¦π¦ = ππ ππ ππ(π¦π¦)ππ ππ ππ, π¦π¦
β’ The private sectorβs inflation forecast:
πΈπΈπ π ππ|ππ = οΏ½
πππππ π ππ ππ ππ = οΏ½
ππ οΏ½
π¦π¦ππ(π¦π¦)ππ ππ ππ, π¦π¦ ππ
βMonetary Policyβ
β’ Suppose the private sectorβs (imperfect) DAG is π π : ππ β ππ β π¦π¦. β βClassicalβ belief in monetary neutrality.
πππ π ππ, ππ, π¦π¦ = ππ ππ ππ(π¦π¦)ππ ππ ππ, π¦π¦
β’ The rational-expectations forecast:
πΈπΈ ππ|ππ = οΏ½
ππ οΏ½
π¦π¦ππ(π¦π¦|ππ)ππ ππ ππ, π¦π¦ ππ
βMonetary Policyβ
β’ The objective distribution ππ is consistent with the DAG ππ β ππ β π¦π¦
ππ
β’ The private sectorβs DAG ππ β ππ β π¦π¦ distorts the true causal model by omitting ππ and reversing causation between ππ and π¦π¦.
βMonetary Policyβ
β’ Expected output is βππ ππ(ππ) πΈπΈ ππ ππ β πΈπΈπ π ππ|ππ .
β’ Can the central bank play a strategy that induces systematic underestimation of inflation by the private sector, such that expected output will be positive?
β’ Because π π is imperfect, the answer may be affirmative!
βMonetary Policyβ
β’ Impose the following structure:
β ππ, ππ β 0,1
β ππ ππ = 1 ππ = π½π½ππ, where π½π½ β (0,1). β Denote ππ ππ = 1 = πΌπΌ.
Proposition: As ππ2 β 0, the maximal expected output converges to π½π½/4 , attained by the strategy πΌπΌ = 0.5.
βMonetary Policyβ
β’ ππ ππ = 1 ππ = 0, π¦π¦ = 0 for every π¦π¦.
β Therefore, πΈπΈπ π ππ|ππ = 0 is unbiased:
πΈπΈπ π ππ|ππ = 0 = οΏ½
π¦π¦ππ π¦π¦ ππ ππ = 1 ππ = 0, π¦π¦ = 0
β’ In contrast, ππ fluctuates conditional on ππ = 1. The private sector accounts for these fluctuations by the variation in π¦π¦.
Sketch of Proof
β’ Denote πΈπΈπ π ππ|ππ = 1 = ππ(1).
β’ ππ(1) β (0,1) (bounded away from both as ππ2 β 0)
β’ The ex-ante distribution over π¦π¦ is normal around three possible values of the inflationary surprise:
Sketch of Proof
βππ(1) 0 1 β ππ(1)
β’ ππ 1 = βπ¦π¦ ππ π¦π¦ ππ ππ = 1 ππ = 1, π¦π¦
β’ In the ππ2 β 0 limit:
β ππ(π¦π¦) assigns positive weight only to βππ 1 , 0 , 1 β ππ 1 . β ππ ππ = 1 ππ = 1, π¦π¦ = 1 β ππ(1) = 1
β ππ ππ = 1 ππ = 1, π¦π¦ = βππ(1) = 0
β ππ ππ = 1 ππ = 1, π¦π¦ = 0 conditions on a vanishing event!
Sketch of Proof
β’ Letβs guess ππ ππ = 1 ππ = 1, π¦π¦ = 0 = 0 because this helps the central bank β we will verify this guess later.
β’ βΉ In the limit, ππ 1 = ππππ π¦π¦ = 1 β ππ(1) = πΌπΌπ½π½ = πΈπΈ(ππ). β As if the private sector didnβt observe ππ
β’ On average, inflation forecasts are biased downward.
β’ πΈπΈπ¦π¦ = 1 β πΌπΌ 0 β ππ 0 + πΌπΌ π½π½ β ππ 1 = πΌπΌ(π½π½ β πΌπΌπ½π½)
Sketch of Proof
β’ Expected output is πΌπΌ(π½π½ β πΌπΌπ½π½).
β’ Optimizing, we get πΌπΌ = 0.5 and expected output is 0.25π½π½.
β’ We need to verify the guess that ππ ππ = 1 ππ = 1, π¦π¦ = 0 = 0.
β ππ 1 = πΌπΌπ½π½ = 0.5π½π½ < 0.5.
β Therefore, βππ(1) is closer to 0 than 1 β ππ(1).
β Conditional on π¦π¦ = 0, ππ = 0 is infinitely more likely than ππ = 1.
Sketch of Proof
β’ Ironically, the central bank exploits the private sectorβs belief in classical neutrality to generate long-run real effects.
β’ Interrelated aspects of the optimal strategy:
β Randomization
β Dynamic inconsistency
Comments
β’ Biased estimates are less likely when we further restrict ππ.
Proposition: Any DAG induces unbiased estimates whenever ππ is a multivariate normal distribution (MND).
β Idea behind the result: Factorizing a MND according to a DAG yields a MND with undistorted means.
β Lesson for 1970s macro debates
Multivariate Normal Distributions
β’ Setting: Leaderβs payoff is linear in the followerβs belief
β’ When the followerβs DAG is perfect, the leaderβs ex-ante optimum coincides with rational-expectations benchmark
β Graphical properties as diagnostic, non-parametric tests
β Multivariate normal distributions extend the coincidence with rational expectations to all DAGs
Summary: Leader-Follower Model
β’ Systematically biased estimates arise when the followerβs DAG neglects correlation between causes.
β’ The leader tries to create statistical patterns (possibly through randomization) that the follower misperceives.
β’ The optimal strategy may be dynamically inconsistent.
