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Econ-588 · economics · Economics

Persistence of extreme financial experiences in beliefs

LLM status Open unverified

Statement

ECON-036 | R | Source-backed agenda; exact formalization is editorial

Definitions and formal target

Individual $i$ experiences financial event $X_{it}$ and reports forecast $B_{i,t+h}$ at horizon $h$. Let $x_0$ be a normal event and $x_1$ a specified extreme event; define the intervention as direct exposure holding the common information environment fixed. Potential forecasts are $B_{i,t+h}(x)$.

Identify the persistence profile \[ \psi_h(x_1,x_0)=\mathbb E[B_{i,t+h}(x_1)-B_{i,t+h}(x_0)],\qquad h\geq0. \] For nonzero initial response, compare $\psi_h/\psi_0$ across extreme and ordinary exposures. Determine whether extreme experiences decay more slowly in beliefs, distinguishing memory persistence from persistent changes in actual economic circumstances.

Clarifications and required assumptions

Define exposure, event date, forecast object, horizon and information. Common information fixed does not fix wealth, employment, health or later news; a memory-only effect needs additional channel interventions. Exposure may be selected, so causal identification needs a design or structural assumptions. Ratios require $\psi_0\ne0$; report sign reversals and consider absolute decay or a prespecified persistence fit.

Variants and changes of scope

  • Direct, vicarious and cohort exposures differ.
  • Memory effects differ from effects through persistent economic circumstances.
  • Mean beliefs, uncertainty, tail probabilities and portfolios need not share a decay profile.

Audit conclusion and source relationship

[S1], PDF pp. 10--11, supports extreme and direct/indirect experience questions. The profile is editorial, not a source claim of universally slower decay.

References and exact source roles

[S1] Stefan Nagel (2021). Subjective Beliefs in Asset Pricing. Innsbruck lecture slides, June 2021; PDF has two slides on many pages.

Verified locator / source role: PDF p. 10: open question on stronger persistence of extreme experiences; PDF p. 11: direct versus indirect experience.

https://bpb-us-w2.wpmucdn.com/voices.uchicago.edu/dist/f/575/files/2021/07/Innsbruck2021_slides.pdf

Common conventions: Additional-source status and mathematical conventions

Of the 100 supplied ECON entries, 65 distinct problems appear here; 32 close matches refine earlier entries, and three duplicate questions map to existing entries. Appendix app:audited-crosswalk lists every destination. The attachment's P/R/S/U distinctions and source audits are retained. Its precise mathematical formalizations are editorial unless the individual audit says otherwise. Candidate subsidy targets ECON-004--006 retain their U flags; the resolved approval-core question ECON-007 updates OP-0202 above. The following supplied conventions govern both this part and the attributed ECON refinements elsewhere in the catalogue, subject to local restrictions.

Parameterized questions. A problem family lists inputs that must be fixed before an instance is evaluated: population, horizon, policies, information, data law, model class and welfare weights. These are required choices, not quantities the citations are assumed to specify. Undefined applications should not be treated as identified estimands.

Indices and integrability. Sets of agents, firms and regions are finite unless a population measure is explicitly used. $i,j$ index units, $r$ regions, $t$ dates and $h$ horizons. $\mathbb E$ and $\Pr$ refer to the declared population and law; $\mathbf1\{A\}$ is an indicator. Every displayed expectation and discounted sum needs existence in the stated domain. Infinite horizons use a discount in $(0,1)$ and a convergence condition; finite horizons may use discount one.

Optimization and existence. A displayed $\max$, $\min$, or $\arg\max$ is conditional on attainment. For finite feasible menus this is immediate; general spaces need continuity and compactness/coercivity or another existence argument. Without attainment, the target is the supremum/infimum and arbitrarily near-optimal feasible choices. $\inf\varnothing=+\infty$. Empty feasibility and an unidentified target are different issues.

Potential outcomes. $Y_i(z)$ is the outcome under a specified intervention, not the observed conditional mean among people choosing $z$. In binary comparisons $\Delta Y_i=Y_i(1)-Y_i(0)$. Specify assignment, treatment versions, compliance, information and observation horizon. Interference is allowed under a full policy vector; compressed exposure notation needs a justified mapping. No interference, consistency and overlap are substantive assumptions, not automatic consequences of notation.

Populations and observation. Fix baseline eligibility and population weights. Conditioning on post-policy employment, survival, relocation or program uptake can change the population being compared. Death is not ordinary loss to follow-up; outcomes truncated by death need a composite convention, joint survival target, or explicitly defined survivor stratum. Fertility-changing interventions need a population functional rather than an unexplained fixed descendant cohort. Measurement and missingness models must be supplied.

Identification. A structural model $m\in\mathcal M$ implies observable law $P_m$ and target $\theta(m)$. Identification means \[ P_{m_1}=P_{m_2}\ \Longrightarrow\ \theta(m_1)=\theta(m_2) \quad\text{for all }m_1,m_2\in\mathcal M . \] Without identification, the target is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Writing $\theta(P)$ before establishing identification can hide incompatible latent models. Confidence sets additionally need a sampling law, confidence level, asymptotic sequence and uniformity class. Point identification, coverage and informative precision are separate properties.

Mediation. Total effects of randomized assignment are distinct from cross-world natural indirect effects. Belief/effort-channel effects require a meaningful mediator intervention and additional measurement, exclusion or mediation assumptions. Randomization of the initial policy alone does not supply those assumptions. Report bounds or interventional alternatives when the stronger target is unsupported.

Equilibrium. A counterfactual $W(z)$, $p(z)$, or $\mu_t^z$ requires individual optimization, feasibility, government/resource budgets, market clearing and state-transition laws. State equilibrium existence and selection, or report ranges over compatible equilibria. Initial-state integration includes subsequent endogenous transitions; current-state integration must use the policy-dependent distribution. Reduced-form invariance after policy is not assumed.

Welfare accounts. Scores, utility, health, dollars and ecological quantities require explicit conversion before addition. Fees, taxes, subsidies, wages and extortion payments are transfers between parties; distributional weights can make them welfare relevant, but their face value is not automatically a resource loss. State ownership, geographic boundaries, foreign-income weights and fiscal finance. Do not add profits to household welfare already including dividends, or count land capitalization and the underlying benefit twice. A benefit-cost expression must distinguish gross benefits from net welfare and subtract every real cost exactly once.

Derivatives and risk. Log derivatives require positive arguments; local responses require admissible perturbations and specified equilibrium branches. Differentiation under expectations needs regularity/domination, and boundaries may allow only one-sided derivatives. Risk uses a specified law; ambiguity uses a declared nonempty law set on a common history space. Adaptive policies must be nonanticipative. A robust criterion is an editorial choice unless attributed explicitly.

Scope overlaps. Allocation, collective choice, contracts, household bargaining and coordinated policy overlap economics with optimization and game theory. They are retained because they appeared in the original catalogue; no claim of a strictly game-theory-free selection is made.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-588-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-588

Persistence of extreme financial experiences in beliefs: Persistence rates do not label the memory channel

Target and channel distinction

Fix baseline individuals, an exposure date, a forecast object, and the common information environment. Let $E=1$ be a specified extreme direct experience and $E=0$ its comparison. The causal persistence profile is \[ \psi_h=E[B_h(1)-B_h(0)],\qquad h=0,1,\ldots. \] Nagel's lecture explicitly asks about stronger persistence of extreme experiences and about direct versus indirect experience [nagel, PDF pp. 10--11]. It does not establish universal slower decay. Randomized exposure with consistency, independent units, complete follow-up, and positive assignment probabilities identifies $\psi_h$ as a difference of observed means. This total effect includes changes in wealth and other circumstances. The results below show why even an exactly observed, apparently well-behaved decay profile need not identify memory persistence.

Two exponential channels and their ambiguity

Consider a structural mean response consisting of a memory contribution and a circumstance contribution, \[ \psi_h=\alpha r^h+\beta s^h, \qquad \alpha,\beta>0,\quad 0<r,s<1,\quad r\ne s. \tag{1} \] The labels "memory" and "circumstances" refer to mechanisms, not to which numerical rate happens to be larger. Suppose only exposure and forecasts are observed, with no intervention that distinguishes the channels. Residual forecast noise can have any fixed common law, independent of randomized exposure.
Theorem: Four consecutive population responses $\psi_0,\ldots,\psi_3$ identify the unordered rate set $\{r,s\}$ and the matching amplitudes in (1), but do not identify which rate is the memory rate. Define \[ D=\psi_0\psi_2-\psi_1^2. \] Then $D>0$ and the two rates are the roots of \[ z^2-Sz+P=0,\qquad S=\frac{\psi_0\psi_3-\psi_1\psi_2}{D},\quad P=\frac{\psi_1\psi_3-\psi_2^2}{D}. \tag{2} \] After choosing which root is called $r$, its amplitude is $\alpha=(\psi_1-s\psi_0)/(r-s)$ and $\beta=\psi_0-\alpha$.
Proof: Expanding gives $D=\alpha\beta(r-s)^2>0$. Each sequence $r^h$ and $s^h$ satisfies the recurrence $\psi_{h+2}=(r+s)\psi_{h+1}-rs\psi_h$; so does their linear combination. Applying the recurrence at $h=0,1$ and solving its two linear equations gives (2). The roots are distinct, and the equations $\alpha+\beta=\psi_0$, $\alpha r+\beta s=\psi_1$ then give the amplitudes. Conversely, interchange $(\alpha,r)$ and $(\beta,s)$. Let a latent memory variable carry the first term and a latent circumstance variable the second in one structural model, and reverse their roles in another. Add the same residual-noise process to the sum. These two models give exactly the same joint law of exposure and every forecast horizon, but different memory rates. Therefore even infinitely many forecast observations cannot distinguish the labels in this class. ∎
The positive-amplitude and distinct-rate restrictions are essential to this inversion. If one amplitude is zero or $r=s$, then $D=0$ and the four equations no longer identify two rates. Small positive $D$ also makes the inversion poorly conditioned; observing four noisy sample means is not equivalent to knowing four population responses exactly.

A false inference from normalized total decay

Example: Let both ordinary and extreme experiences have the same memory rate $r=1/5$ and circumstance rate $s=9/10$. Suppose their amplitudes are $(\alpha,\beta)=(9,1)$ and $(1,9)$, respectively. Both initial total responses equal ten. For every $h\ge1$, \[ \frac{\psi_h^{\mathrm{extreme}}}{\psi_0^{\mathrm{extreme}}} -\frac{\psi_h^{\mathrm{ordinary}}}{\psi_0^{\mathrm{ordinary}}} =\frac45\{(9/10)^h-(1/5)^h\}>0. \] Thus extreme total responses decay strictly more slowly at every positive horizon although the memory decay rate is identical. The difference comes entirely from the relative channel amplitudes.
This counterexample is compatible with randomized exposure and exact measurement. Holding common news fixed does not fix the circumstance channel and hence does not prevent the example.

A channel intervention that resolves the label

Augment the experiment with a randomized reset $Z\in\{0,1\}$ of the post-exposure circumstance path, independent of exposure. A successful reset places that path at the same baseline under both exposures. Assume it neither changes memory nor affects beliefs through any other route. These are exclusion restrictions on the reset, not consequences of its randomization. In the two-rate model they imply \[ E[B_h(e,z)]=\mu_{h,z}+e\{\alpha r^h+(1-z)\beta s^h\}. \tag{3} \] The term $\mu_{h,z}$ permits an exposure-independent direct reset effect.
Proposition: With independent randomized assignment of $(E,Z)$, positive probability for all four cells, consistency and complete follow-up, the reset-cell exposure contrast $\phi_h$ identifies $\alpha r^h$. If $\phi_0\ne0$, then $r=\phi_1/\phi_0$ and $\alpha=\phi_0$. Subtracting $\phi_h$ from the unreset exposure contrast identifies the circumstance profile.
Proof: Randomization identifies each potential mean in (3). Subtract the $e=0$ mean from the $e=1$ mean within $z=1$; $\mu_{h,1}$ cancels and the circumstance term is zero. The two ratios and the unreset-minus-reset difference then follow algebraically. No cross-world joint distribution is needed for these mean contrasts. ∎

Ratios and the remaining empirical gap

For any estimated profile satisfying $|\widehat\psi_h-\psi_h|\le\varepsilon$ and $|\widehat\psi_0-\psi_0|\le\varepsilon<|\psi_0|$, direct subtraction of fractions gives \[ \left|\frac{\widehat\psi_h}{\widehat\psi_0} -\frac{\psi_h}{\psi_0}\right| \le\frac{\varepsilon}{|\psi_0|-\varepsilon} \left(1+\left|\frac{\psi_h}{\psi_0}\right|\right). \tag{4} \] Indeed the numerator is $(\widehat\psi_h-\psi_h)\psi_0 -\psi_h(\widehat\psi_0-\psi_0)$ and the denominator has absolute value at least $|\psi_0|(|\psi_0|-\varepsilon)$. Near a zero initial effect, normalized decay can therefore be unstable even with accurate level estimates. Sign reversals should be reported in levels instead of being hidden by a fitted positive decay rate.

The outstanding task is to find exposure and channel interventions with credible exclusions in an actual population, handle attrition and changing news, and test the restrictive two-rate representation. The algebra identifies a precise ambiguity and a design that removes it under stated conditions; it does not establish a psychological memory law from total forecast persistence alone.

Completion Estimate

47%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt identifies two unlabeled exponential persistence channels, demonstrates that slower total decay need not reflect slower memory decay, and supplies a reset design and ratio-error bound. Actual extreme-experience persistence still needs credible exposure and channel interventions, exclusions, attrition control and validation of the restrictive decay representation in the target population.

References

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