we-solve (beta)

Econ-506 · economics · Economics

Origins of business cycles

LLM status Partially solved unverified

Statement

Original subject group: Business cycles and macro dynamics

Audit classification: Published research agenda. The source leaves belief-versus-short-run-news interpretation unresolved; the four-way 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 country, a quarterly sample, and a finite vector $z_t$ containing log output, consumption, investment, hours, prices, and observed financial conditions. Assume a stable reduced-form vector autoregression with innovation covariance $\Sigma$, whose moving-average coefficients are $\Psi_h$. A structural representation has $u_t=B\varepsilon_t$, $BB'=\Sigma$, and orthogonal unit-variance innovations $\varepsilon_t$. Require a positive output forecast-error variance at every evaluated horizon. Partition their labels into technology, demand, finance, and beliefs. Let $\mathcal B$ be the matrices satisfying explicitly reported timing, sign, and external-instrument restrictions; belief innovations must exclude innovations in the specified observed fundamental-information set. For output-selector $e_Y$, define the horizon-$H$ variance share of class $k$ by \[ \theta_k(B,H)=\frac{\sum_{h=0}^{H-1}\sum_{j\in k}(e_Y'\Psi_hB e_j)^2} {\sum_{h=0}^{H-1}e_Y'\Psi_h\Sigma\Psi_h'e_Y}. \] The target is the joint identified set of these shares over $B\in\mathcal B$, with sampling uncertainty and sensitivity to omitted information. Determine which additional independently justified measurements shrink that set enough to distinguish competing explanations of the main business-cycle component. A decomposition obtained from one selected rotation is insufficient: observationally equivalent rotations and alternative reasonable shock classifications must be reported. The exercise concerns a stated sample and model class, not universal fractions valid across all economies.

Take $z_t\in\mathbb R^m$, $\Sigma\succ0$, $\Psi_0=I_m$, and a partition $(J_k)_{k=1}^4$ of $\{1,\ldots,m\}$. The proposed shock classes are mutually exclusive labels, not automatically distinct economic mechanisms. Define $\mathcal B(P)$ using the population observable law $P$ and the specified restrictions; the target is $\mathcal I_H(P)=\{(\theta_k(B,H))_{k=1}^4:B\in\mathcal B(P)\}$. Empty $\mathcal B(P)$ means the restrictions are falsified. News is an innovation that revises a conditional forecast of a specified future fundamental; a belief shock is a separately defined subjective-forecast revision satisfying the declared exclusion. Orthogonality alone does not prove that a shock is nonfundamental.

Variants and assumption changes

The following are editorial variants, not additional published open conjectures.
  1. Frequency-domain attribution. For a fixed band $D\subset[-\pi,\pi]$, replace forecast-horizon sums by integrated squared transfer functions over $D$. Require an integrable spectral density and positive band variance; identify band shares separately from horizon shares.
  2. External-instrument restriction. Add observed $Z_t$ with $E[Z_t\varepsilon_{jt}]=0$ for every nontarget shock and nonzero covariance with the target. Compare the resulting identified set with sign-only restrictions, allowing bounded exclusion violations.

References and source audit

The source leaves belief-versus-short-run-news interpretation unresolved; the four-way attribution is editorial.
  1. George-Marios Angeletos; Fabrice Collard; Harris Dellas. 2020. Business-Cycle Anatomy. Section III.C, final paragraph, printed p.3046; metadata on p.3030. https://economics.mit.edu/sites/default/files/publications/MBC_AER_published.pdf. DOI: https://doi.org/10.1257/aer.20181174.

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-506-A1 · Partial result · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-506

Business-cycle attribution as a constrained rotation problem

Scope

The task is a joint identified set for four labeled variance shares. A particular orthogonal rotation is an admissible explanation only after its economic restrictions have been stated. This attempt derives exact unrestricted coordinate bounds, an exact joint set at horizon one, quantitative bounds from an external proxy, and a finite approximation certificate. It does not identify the actual contributions of technology, demand, finance and beliefs in a country without its data and maintained economic restrictions.

Let $\Sigma=SS'$ with nonsingular $S$, and write every impact matrix as $B=SQ$, $Q'Q=I_m$. For a positive-variance horizon define \[ A_H=\sum_{h=0}^{H-1}S'\Psi_h'e_Ye_Y'\Psi_hS,\qquad t_H=\operatorname{tr}(A_H)>0. \] For the diagonal projector $P_k$ selecting the $d_k$ labels of class $k$, \[ \theta_k(Q,H)=\operatorname{tr}(P_kQ'A_HQ)/t_H. \tag{1} \] The matrix $A_H$ is positive semidefinite. Formula (1) proves that every share is nonnegative and the four shares sum to one. A class with no assigned labels has share zero. No interpretation of a column as a belief shock follows from this identity.

Exact coordinate bounds without economic exclusions

Write the eigenvalues of $A_H$ as $\lambda_1\ge\cdots\ge\lambda_m\ge0$.
Proposition: If the only restriction on $Q$ is orthogonality, the sharp range for a class containing $d$ labels is \[ \left[\frac{\sum_{j=m-d+1}^m\lambda_j}{t_H},\quad \frac{\sum_{j=1}^d\lambda_j}{t_H}\right]. \tag{2} \] For $d=0$ or $m$ the usual empty/full sums apply.
Proof: The selected columns define a rank-$d$ orthogonal projector $\Pi$. In an eigenbasis of $A_H$, its diagonal entries $w_j$ belong to $[0,1]$ and sum to $d$. Consequently $\operatorname{tr}(A_H\Pi)=\sum_j\lambda_jw_j$ lies between the sums of the smallest and largest $d$ eigenvalues. Choosing those eigenspaces attains the bounds. For $0<d<m$, continuously rotating one $d$-dimensional subspace into another connects the minimizing and maximizing subspaces. The trace is continuous on that path, so every intermediate scalar value is attained. An orthonormal basis of the subspace and its complement completes the required $Q$. ∎
This is a statement about a coordinate projection, not a rectangular joint set. For example, two nonempty classes cannot both have share one, even if both coordinate intervals reach one. Likewise, independently selecting an extremizing rotation at each horizon does not produce one jointly feasible impulse-response system.

At $H=1$, $A_1=vv'$ for $v=S'e_Y\ne0$. In this case the entire joint set is particularly simple: it is the probability simplex on the nonempty classes. Indeed choose any desired shares $s_k\ge0$ summing to one and a vector $w$ of norm $\|v\|$ whose coordinates within class $k$ have squared norm $s_k\|v\|^2$. There exists an orthogonal $Q$ with $Q'v=w$; construct orthonormal bases beginning with the normalized vectors. Substitution in (1) gives every requested share. At horizon one, covariance information alone thus places no relative restriction on nonempty economic labels beyond adding to one.

A proxy direction and bounded exclusion failures

Let $Z$ be a scalar measured instrument and suppose $E[\varepsilon Z]=c$. The observable whitened covariance is \[ v_Z=S^{-1}E[uZ]=Qc. \] For a designated single target column $q_1$, assume $c_1\ge\kappa>0$ and $\|c_{-1}\|_2\le\epsilon$. Orthogonality gives \[ v_Z=c_1q_1+Q_{-1}c_{-1},\qquad \angle(v_Z,q_1)\le\arctan(\epsilon/\kappa). \tag{3} \] The second term is perpendicular to $q_1$, so (3) follows by taking the ratio of its norm to $c_1$. If $\epsilon=0$, the target direction and its sign are identified as $\bar q=v_Z/\|v_Z\|$. Other columns may still rotate freely in the perpendicular space. Thus a single valid proxy identifies one normalized shock, not an entire labeled system.

For $\epsilon>0$, let $\phi=\arctan(\epsilon/\kappa)$. Since $\|q_1-\bar q\|_2\le2\sin(\phi/2)$, \[ \left|\frac{q_1'A_Hq_1-\bar q'A_H\bar q}{t_H}\right| \le \frac{4\|A_H\|_{\rm op}}{t_H}\sin(\phi/2). \tag{4} \] To prove (4), expand the difference as $(q_1-\bar q)'A_Hq_1+\bar q'A_H(q_1-\bar q)$ and apply Cauchy--Schwarz. Intersecting this interval with $[0,1]$ gives a valid, possibly conservative restriction. Weak relevance $\kappa\downarrow0$ destroys its informativeness, as it should. A sample covariance estimate can be included by enlarging the feasible set of $v_Z$ on a simultaneous confidence event; treating it as known would omit sampling error.

These exclusions are economic assumptions about $Z$. A survey revision that covaries with future productivity can represent news rather than an autonomous belief disturbance. Orthogonal residualization against a selected observed information vector only establishes orthogonality to that vector. It cannot exclude unmeasured future-fundamental information.

Computable approximation with a common rotation

Suppose restrictions define a closed subset $\mathcal Q$ of the orthogonal group through finitely many continuous equalities and weak inequalities. The feasible set is compact, possibly empty. When nonempty, its image under all requested shares at finitely many horizons is compact as well. For a single share, \[ |\theta_k(Q,H)-\theta_k(\widetilde Q,H)| \le \frac{2\sqrt{d_k}\|A_H\|_{\rm op}}{t_H}\|Q-\widetilde Q\|_F. \tag{5} \] Writing the selected column matrices as $X,\widetilde X$ gives the bound by two trace Cauchy--Schwarz inequalities, using $\|X\|_F=\sqrt{d_k}$. A finite orthogonal $h$-net exists by compactness. Evaluating each net point and enlarging its share coordinates by (5) therefore covers the unrestricted joint image. For restrictions $f_\ell(Q)\ge0$ with known Lipschitz constants $L_\ell$, retain points satisfying $f_\ell(Q)\ge-L_\ell h$; an equality uses $|f_\ell(Q)|\le L_\ell h$. Every truly feasible matrix has a retained nearby point. This constructs a certified outer approximation, not an exact feasibility test at fixed resolution. It may be exponentially expensive in dimension, and that limitation is explicit.

Let the mesh tend to zero. Any sequence of retained points has a convergent subsequence, and its limit satisfies all restrictions by continuity. Therefore spurious limit points disappear; if the feasible set is empty, sufficiently fine nets have no retained points. For nonempty sets this proves convergence of the outer image in Hausdorff distance. Sampling uncertainty requires an outer confidence set for the reduced-form parameters and proxy moments before performing this projection. Weak identification may leave a wide rotation set despite precise covariance estimates.

What remains unresolved

The calculations identify how restrictions could sharpen attribution, but do not decide which restrictions are justified in the stated country and sample. Nonlinear dynamics, omitted financial measurements, or alternative partitions of labels produce different targets. The statistical problem can be solved conditionally without resolving the economic distinction between short-run news and beliefs. The source motivates that distinction; the benchmark sets above are derived here and do not assign measured universal business-cycle percentages.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt derives sharp unrestricted share bounds, a horizon-one joint simplex, bounded proxy-exclusion sensitivity, and convergent rotation-set approximations. Full business-cycle attribution still requires justified economic shock restrictions, a joint sampling confidence construction, and independently informative measurements separating beliefs from omitted fundamental news.

References

Go to contribution →