we-solve (beta)

Econ-235 · economics · Economics

Self-fulfilling fluctuations

LLM status Open unverified

Statement

Original subject group: Business cycles and macro dynamics

Audit classification: Explicit published open question. A quantitative bubbles-and-crashes challenge is explicit; the cross-world recession attribution is editorial. Verification concerns the cited version, not first priority or proof that this exact editorial specification remains unsolved on October 8, 2026.

Definitions and precise research target

Editorial mathematical specification.
Fix a nonlinear incomplete-markets model with observed fundamental state $f_t$, endogenous distribution $\mu_t$, and a declared equilibrium-selection variable $b_t$ representing beliefs or coordination. A belief innovation $\eta_t$ is required to be orthogonal to innovations in the specified fundamental-information set, not merely to lagged output. Let equilibrium log output be $y_t=g(f_t,\mu_t,b_t)$. Define a recession as $y_{t+H}-y_t<-q$, for preselected horizon $H$ and threshold $q>0$. For each observed recession episode, construct the counterfactual path $y^{0}_t$ obtained by suppressing $\eta_t$ while retaining the same fundamental innovations, initial distribution, and policy rule. The target is \[ \Theta=\Pr\{y_{t+H}-y_t<-q,\; y^{0}_{t+H}-y^{0}_t\ge -q\}, \] the probability that a recession would be prevented under this intervention on beliefs. Determine an identified set for $\Theta$ from forecast surveys, balance sheets, asset prices, and instruments with explicit exclusion restrictions. Because equilibrium selection is latent and beliefs can react to fundamentals, provide observationally equivalent explanations and falsifiable restrictions. Establish robustness to plausible selection rules and financial-friction specifications. Existence of sunspot equilibria in a particular model does not establish their quantitative importance in actual recessions.

A cross-world event needs a coupling. Specify measurable structural equations $f_{t+1}=F(f_t,\xi_{t+1})$, $b_{t+1}=B(b_t,f_t,\mu_t,\eta_{t+1})$, and $\mu_{t+1}=U(f_t,\mu_t,b_t)$, together with an initial law and joint law of $(\xi,\eta)$. Both worlds use the same initial draw and fundamental innovations; the intervention replaces the specified future $\eta$ sequence by zero and recomputes $b,\mu,y$, rather than holding the endogenous distribution fixed. A sunspot is extrinsic randomness absent from preferences, technology, endowments, and information about fundamentals; a nonfundamental belief shock need not be a rational-expectations sunspot. Let $D$ and $D^0$ be the two recession indicators. Then $\Theta=P(D=1,D^0=0)$, while the fraction of observed recessions prevented is $\Theta/P(D=1)$ when $P(D=1)>0$. Marginal recession probabilities alone do not identify their joint law.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. No cross-world coupling. Identify only $p=P(D=1)$ and $p_0=P(D^0=1)$. Sharp unrestricted coupling bounds are $\max(0,p-p_0)\le\Theta\le\min(p,1-p_0)$; compare these with bounds under a specified monotonicity assumption.
  2. Rational sunspot equilibrium. Require agents' conditional forecasts to coincide with the selected equilibrium law, and let an independent extrinsic signal select equilibria. Determine whether quantitatively important crashes remain feasible under household portfolio moments; this excludes arbitrary subjective forecast errors.

References and source audit

A quantitative bubbles-and-crashes challenge is explicit; the cross-world recession attribution is editorial.
  1. Benjamin Moll. 2020. Research Agenda: Benjamin Moll. Section4.3, Heterogeneity, Bubbles and Crashes. https://economicdynamics.org/research-agenda-moll2020/.

Common conventions: Source mathematical conventions

These conventions apply to each entry unless explicitly replaced.

Models and identification. A primitive model $m$ specifies all probability laws, feasible choices, information, equilibrium and measurement rules used by its target. The admissible class $\mathcal M$ is fixed independently of outcomes. If $P_O(m)$ is its observable probability law and $\tau(m)$ the target, its identified set at observable law $P$ is \[ \mathcal I_\tau(P)=\{\tau(m):m\in\mathcal M,\ P_O(m)=P\}. \] Point identification means that this set is a singleton. An empty set signals incompatibility of data and assumptions. Coordinate bounds need not describe the joint set. Bounds are sharp only if every retained target is attainable or arbitrarily approachable by compatible models. Finite bounds require explicit restrictions on unobserved quantities; unrestricted confounding, hidden wealth, extrapolation or arbitrary equilibrium selection can yield uninformative sets.

Causal interventions. A potential outcome $Y_i(p)$ is the outcome under a complete policy regime $p$, including timing, financing, information and market spillovers. The target population and units are fixed before treatment. With interference, use the complete assignment vector or a justified exposure mapping; individual no-interference notation alone cannot identify an economy-wide intervention. A difference of mean potential outcomes does not require the cross-world joint distribution. Conditioning on post-treatment migration, survival, enrollment or employment changes the estimand and needs separate justification.

Statistical statements. An estimator, confidence set, forecast or test specifies a sampling law, model class and loss/coverage criterion. Uniform $(1-\alpha)$-coverage means $\inf_{m\in\mathcal M}\Pr_m\{\tau(m)\in C_n\}\geq1-\alpha$, or an explicitly asymptotic version. The whole parameter space trivially has coverage, so informativeness requires a stated width, risk or power objective. A forecasting improvement is relative to a named benchmark, information set and evaluation population.

Equilibrium and optimization. An equilibrium comprises feasible plans, optimality, beliefs where needed, budget constraints and all market/resource clearing. The primitives or a stated benchmark define each correspondence $\mathcal E(p;m)$. Nonemptiness is assumed or proved, never inferred just from compact policy sets. A selected-equilibrium target uses a declared selection rule; a robust target quantifies over all permitted equilibria. Use suprema or infima unless attainment is established. An $\varepsilon$-optimal policy is feasible and within $\varepsilon$ of the finite optimum value. Report an empty feasible set separately.

Welfare and accounting. Normative utility, interpersonal normalization, social weights, numeraire, discounting and the use of public receipts are primitives. Behavioral utility need not coincide with the welfare criterion. Household welfare includes ownership income and rents once; adding profits again to compensating variation generally double-counts them. A monetary equivalent is defined by an explicit transfer and price convention, with monotonicity and range conditions. Average effects, mechanism contributions, transfers and welfare losses are different targets. Infinite sums require convergence and dynamic feasible plans require terminal or no-Ponzi conditions.

Source versions. A working-paper date, revision date and journal publication year are distinguished. A DOI for a journal version is not automatically the DOI of an earlier manuscript. Locators refer to the stated version and printed pages where specified. A metadata-only check does not verify a claim about a full-text conclusion. Every entry's source subsection narrows the provenance claim accordingly.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-235-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-235

Belief-driven recession attribution with bounded cross-world reversals

Target and scope

Let $D$ indicate a recession in the actual economy and $D^0$ a recession when specified belief innovations are suppressed while the same fundamental shocks and initial state are retained. The target is $\Theta=\Pr(D=1,D^0=0)$. The quantitative research motivation concerns bubbles and crashes [moll]; the cross-world attribution model below is an explicitly restricted identification experiment. We refine marginal coupling bounds using observed covariates and a bound on adverse reversals, and identify the assumptions needed to interpret the refinement causally.

A sharp finite-stratum result

Let $Z$ take finitely many pre-intervention values with probabilities $w_z>0$. Suppose the available experiment identifies \[ p_z=\Pr(D=1\mid Z=z),\qquad q_z=\Pr(D^0=1\mid Z=z). \] For example, random assignment to an implementable belief-suppression regime, consistency, and no interference could identify these marginals. Orthogonality of a survey residual to fundamental innovations alone does not do so. Assume additionally a declared bound on the probability that suppression causes a recession which otherwise would not occur: \[ r_z=\Pr(D=0,D^0=1\mid Z=z)\le\delta_z,\quad\delta_z\ge0. \] There are no other restrictions on the cross-world coupling conditional on $Z$.
Theorem (Sharp attribution interval): The class is nonempty if and only if $\delta_z\ge\max(0,q_z-p_z)$ for every $z$. When nonempty, its sharp identified set for $\Theta$ is the closed interval $[L,U]$, where \[ L=\sum_z w_z\max(0,p_z-q_z),\qquad U=\sum_z w_z\min\{p_z,1-q_z,p_z-q_z+\delta_z\}. \] If $p=\sum_z w_zp_z>0$, the fraction of actual recessions prevented has sharp interval $[L/p,U/p]$.
Proof: Write $t_z=\Pr(D=1,D^0=0\mid Z=z)$. The four cell probabilities, in the order $(1,0),(0,1),(1,1),(0,0)$, are \[ t_z,\quad q_z-p_z+t_z,\quad p_z-t_z,\quad1-q_z-t_z. \] Nonnegativity and the reversal restriction give exactly $\max(0,p_z-q_z)\le t_z\le \min(p_z,1-q_z,p_z-q_z+\delta_z)$. The interval is nonempty precisely under the stated condition. Every point in it defines a valid four-cell law. To realize such laws on a common structural probability space, draw $Z$ and an independent uniform variable, partition its unit interval into the four specified lengths, and define both potential indicators on that partition. Thus the construction uses a fixed latent draw across worlds, rather than independent draws for the two indicators. Mixing endpoint couplings with a common mixing weight realizes every value between the aggregate endpoints. The denominator $p$ is known and fixed, so division preserves sharpness. ∎

This construction proves sharpness in the declared binary structural class. It does not prove that every constructed coupling can be embedded in a particular rational-expectations incomplete-markets economy. Such economic restrictions could shrink the interval and need a separate existence proof.

Why covariates and reversal information matter

Corollary: Without a reversal restriction, the stratum-adjusted interval is contained in the unconditional Fréchet interval $[\max(0,p-q),\min(p,1-q)]$, where $q=\sum_z w_zq_z$. Under monotonicity $D^0\le D$, the target is identified as $p-q$; that restriction is compatible with the marginals only if $q_z\le p_z$ for every stratum.
Proof: Convexity of $x\mapsto\max(0,x)$ gives the lower-end inequality. For each $z$, $\min(p_z,1-q_z)$ is at most each of its two arguments; summing proves the upper-end inequality. Monotonicity sets $r_z=0$, so the cell identity $t_z-r_z=p_z-q_z$ gives the target and the compatibility requirement. Conversely the cell construction realizes it. ∎

For a concrete example, use two equally probable strata with $(p_1,q_1)=(0.8,0.2)$ and $(p_2,q_2)=(0.2,0.8)$. The unconditional interval is $[0,0.5]$, but stratum information gives $[0.3,0.5]$. Bounds $(\delta_1,\delta_2)=(0.1,0.6)$ tighten this to $[0.3,0.35]$. Overall $p=q$ does not imply zero prevention: prevented and induced recessions can offset. A global monotonicity claim would be refuted by the second stratum's marginals.

Finite-sample uncertainty without choosing a coupling

Suppose separate randomized samples in each arm and stratum yield simultaneous confidence intervals $p_z\in P_z$, $q_z\in Q_z$, with joint coverage at least $1-\alpha$. Form the set of all four-cell arrays satisfying these interval constraints, the known stratum weights, nonnegativity, unit sums, and $r_z\le\delta_z$. Minimize and maximize $\sum_z w_zt_z$ over it.
Proposition: If the maintained reversal restrictions are true, the resulting interval covers $\Theta$ with probability at least $1-\alpha$. Both endpoints are finite linear programs. An empty feasible set reports incompatibility of these intervals and restrictions and cannot be interpreted as a point estimate.
Proof: On the simultaneous-coverage event the true cell array is feasible, so its objective lies between the two optimized values. Compactness of a nonempty feasible polytope yields attained extrema. The event has the asserted probability; no coupling is estimated or selected in this argument. ∎

Remaining problem

An implementable intervention with known exclusion is the central missing input in observational macro data. Forecast-survey residuals need not be extrinsic sunspots, and suppressing beliefs must change endogenous wealth and prices according to the specified equilibrium equations. The note does not establish such an intervention or rational-equilibrium embeddability. It solves a transparent attribution subproblem conditional on marginal identification, rather than the full quantitative recession agenda.

Completion Estimate

35%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp covariate-adjusted recession-attribution bounds with limited cross-world reversals and finite-sample linear-program inference. Observational macro data still lack an identified belief-suppression intervention, and the admissible binary couplings are not shown embeddable in the required incomplete-markets equilibrium.

References

Go to contribution →