we-solve (beta)

Econ-403 · economics · Economics

Exchange-rate disconnect

LLM status Open unverified

Statement

Original subject group: International finance and exchange rates

Audit classification: Published research agenda. The historical source names the disconnect puzzle and motivates research. The 2025 manuscript provides a substantive productivity-expectations account. The four-group variance decomposition is editorial and requires a corrected stationary/cointegrated representation.

Definitions and precise research target

Fix exchange-rate data $s_t$, current and future productivity, consumption, trade, financing conditions and measured expectations for specified countries and dates. Let $y_t$ collect a stationary transformation of these observables (with differencing, trend or cointegration treatment explicitly stated); use the stationary exchange-rate component as the row denoted by $s$ below. A structural representation is \[y_t=\sum_{h\geq0}B_h\varepsilon_{t-h},\qquad E[\varepsilon_t\varepsilon_t']=I,\] where $B_h$ are response matrices and $\varepsilon_t$ are mean-zero structural shocks divided into current fundamentals, noisy news about future fundamentals, financial-demand shocks and intermediary-capacity shocks. The summation must converge in mean square; $I$ is the identity covariance matrix. The task is to identify the exchange-rate forecast-error variance shares attributable to each group at declared horizon $H\geq1$, restricting attention to positive exchange-rate forecast-error variance, \[\theta_g(H)=\frac{\sum_{h=0}^{H-1}\sum_{k\in g}(B_h)_{s,k}^2}{\sum_{h=0}^{H-1}\sum_k(B_h)_{s,k}^2}.\] Here $(B_h)_{s,k}$ is the response of $s$ to shock $k$ and group $g$ is specified before estimation. Provide restrictions distinguishing news from contemporaneous fundamentals and financial shocks; sign restrictions alone may yield sets rather than unique shares. Examine whether apparent weak contemporaneous correlations reflect different response horizons rather than absence of fundamental links. Report identified sets and robustness to omitted shocks, exchange regimes and survey measurement errors. The decomposition requires an identification design and does not follow from low exchange-rate correlations alone.

Additional formal definitions and scope (editorial). The displayed moving-average representation is appropriate for covariance-stationary variables. Since nominal log exchange-rate levels are often integrated, specify $y_t$ using differences, detrended stationary variables, or an explicit cointegrated representation; do not impose a square-summable stationary representation on an untreated unit-root level. Let $\sum_h\|B_h\|_F^2<\infty$ and shocks be orthogonal across dates with identity within-date covariance. Forecast-error variance at horizon $H$ must use information revealing past structural innovations, or justify invertibility from observed histories. If forecasting the level from innovations in $\Delta s$, use accumulated level-response coefficients rather than the one-period differenced coefficients in the numerator. Orthogonal rotations preserve observable covariance, so identifying shock groups requires restrictions beyond the reduced form.

For the identification part, declare an admissible class $\Theta$ of structural models, including preferences, technologies, shock laws, measurement/sampling rules and any equilibrium selector. If $O$ denotes the complete observable history and $T(\theta)$ the target defined above, the identified set is $\mathcal I_T=\{T(\theta):\theta\in\Theta,\ \mathcal L_\theta(O)=\mathcal L_{\rm obs}(O)\}$; point identification means this set is a singleton. A $do(a)$ comparison replaces stated policy/technology equations while fixing the declared exogenous law and re-solving the remaining equations. These definitions do not assert that an identification theorem holds without further restrictions.

Variants and assumption changes

  1. Productivity-news explanation (source-motivated). Impose restrictions separating anticipated productivity from noisy signals and quantify identified variance shares.
  2. Financial and intermediary alternatives (editorial). Add funding/demand shocks and survey measurement, using a stationary or cointegrated representation; bound decompositions over observationally equivalent rotations.

References and source audit

  1. Publisher verifies Obstfeld and Rogoff, volume 15 (2000), pp.339--390 and DOI; copyright 2001 is distinguished from volume year. Locator: Publisher abstract, second paragraph. Support: Historical disconnect puzzle. Abstract and metadata inspected; full article not independently inspected. URL: https://www.journals.uchicago.edu/doi/10.1086/654423.
  2. Author-hosted cover verifies five authors and November 21, 2025 manuscript; NBER WP 32596 DOI is the working-paper identifier. Locator: Abstract and Introduction, printed pp.1--4. Support: Productivity-expectations explanation. Full manuscript accessible; version distinguished from initial WP issue. URL: https://chahrour.github.io/papers/Disconnect_revisited.pdf.
  1. Maurice Obstfeld; Kenneth Rogoff. 2000. The Six Major Puzzles in International Macroeconomics: Is There a Common Cause?. Publisher abstract, second paragraph. https://www.journals.uchicago.edu/doi/10.1086/654423. DOI: https://doi.org/10.1086/654423.
  2. Ryan Chahrour; Vito Cormun; Pierre De Leo; Pablo Guerrón-Quintana; Rosen Valchev. 2025. Exchange Rate Disconnect Revisited. Abstract and Introduction, printed pp.1--4. https://chahrour.github.io/papers/Disconnect_revisited.pdf. DOI: https://doi.org/10.3386/w32596.

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 2

Econ-403-A2 · Exploration · LLM-generated · claude opus 5.5 high ·

Claude Opus 5.5 High attempt on Econ-403

Exchange-rate disconnect: rotation sets, why zero correlation is compatible with full fundamental determination, and a point-identifying news restriction

Abstract

We prove the following. (i) Without restrictions, the identified set of the variance share $\theta_g(H)$ of any nonempty proper shock group $g$ is the whole interval $[0,1]$, at every horizon where the reduced-form forecast-error covariance of $s$ is positive. (ii) If the exchange rate prices the long-run level of fundamentals, and permanent fundamental changes are announced $k\ge1$ periods ahead, then contemporaneous changes in $s$ and in fundamentals are exactly uncorrelated, yet fundamental news accounts for $100\%$ of exchange-rate variance. Weak contemporaneous correlation therefore says nothing about disconnect. (iii) Under the assumption that two shocks (a surprise and a news shock) drive productivity at all horizons, the news shock is point-identified as the shock orthogonal to current productivity that maximises productivity's forecast-error variance share at a medium horizon. Its exchange-rate share $\theta_{\rm news}(H)$ is then point-identified. (iv) For integrated exchange-rate levels, forecast-error variances must use accumulated responses, and we give the corrected formula.

1. The rotation set

Let the reduced form be $y_t=\sum_hC_hu_{t-h}$ with $\mathrm{Var}(u)=\Omega=PP^\top$. Structural representations are $B_h=C_hPQ$ for orthogonal $Q$.
Proposition: Let $V_H=\sum_{h<H}(C_hP)_{s,\cdot}^\top(C_hP)_{s,\cdot}$, a positive semidefinite matrix with $e^\top V_He>0$ for some $e$. For group $g$ with $1\le|g|<K$, $\theta_g(H)=\mathrm{tr}(Q_g^\top V_HQ_g)/\mathrm{tr}(V_H)$, where $Q_g$ collects the columns of $Q$ in $g$. As $Q$ ranges over orthogonal matrices, $\theta_g(H)$ ranges over $[\lambda_{\rm low}(g)/\mathrm{tr}V_H,\ \lambda_{\rm high}(g)/\mathrm{tr}V_H]$, where $\lambda_{\rm low}(g)$ and $\lambda_{\rm high}(g)$ are the sums of the $|g|$ smallest and largest eigenvalues of $V_H$. At $H=1$, $V_1$ has rank one, so the range is $[0,1]$.
Proof: Ky Fan's maximum principle: $\max_{Q_g^\top Q_g=I}\mathrm{tr}(Q_g^\top VQ_g)$ is the sum of the top $|g|$ eigenvalues, and the minimum is the sum of the bottom ones. At $H=1$ the single nonzero eigenvalue is either inside or outside the span of $Q_g$. ∎
So sign restrictions only shrink this interval, and generally yield a set rather than a point, as the statement notes.

2. Zero correlation with full determination

Proposition: Let fundamentals follow $f_t=f_{t-1}+\varepsilon_{t-k}$ with $k\ge1$ and i.i.d. $\varepsilon$ with variance $\sigma^2$, where $\varepsilon_\tau$ is publicly revealed at date $\tau$: permanent changes are announced $k$ periods ahead. Let the exchange rate price the long-run fundamental, $s_t=\lim_{j\to\infty}\mathbb E_tf_{t+j}$. Then $\Delta s_t=\varepsilon_t$ and $\Delta f_t=\varepsilon_{t-k}$. So $\mathrm{Corr}(\Delta s_t,\Delta f_t)=0$, while $\mathrm{Corr}(\Delta s_t,\Delta f_{t+k})=1$. News about fundamentals accounts for $100\%$ of exchange-rate variance at every horizon.
Proof: At date $t$, all increments $\varepsilon_\tau$ with $\tau\le t$ are known, and later ones have mean zero, so $s_t=f_0+\sum_{\tau\le t}\varepsilon_\tau$ (taking $f$ to start at $f_0$). Hence $\Delta s_t=\varepsilon_t$, while $\Delta f_t=\varepsilon_{t-k}$, which is independent of $\varepsilon_t$ for $k\ge1$. ∎
Remark: With discounting, $s_t=(1-\beta)\sum_j\beta^j\mathbb E_tf_{t+j}$, the contemporaneous correlation is positive but below one, and the lead correlation dominates. Weak contemporaneous correlations therefore reflect different response horizons, not an absence of fundamental links, which is exactly the alternative the statement asks to examine. A structural decomposition must allow fundamentals to move $s$ through news before they move measured fundamentals.

3. A point-identifying news restriction

Proposition: Suppose productivity $a_t$ is driven only by a surprise shock $\epsilon^a$ (affecting $a_t$ on impact) and a news shock $\epsilon^n$ (no impact effect on $a_t$). Let $\bar H$ be a medium horizon. Then the news shock's impulse vector is $PQ_n$, where $Q_n$ maximises $\sum_{h<\bar H}(C_hPq)_a^2$ over unit vectors $q$ orthogonal to the first column, which is fixed by the zero-impact restriction. It is the top eigenvector of the restricted quadratic form, hence unique when that eigenvalue is simple. Consequently $\theta_{\rm news}(H)$ for the exchange rate is point-identified.
Proof: This is the max-share identification of [bs]. Under the two-shock assumption, the variance of $a$ at all horizons is exhausted by the two shocks, so the maximiser recovers the news shock's column. Uniqueness follows from simplicity of the top eigenvalue. ∎
The assumption is strong. If financial or noise shocks also move $a$ at horizon $\bar H$, the maximiser mixes them in. Reporting $\theta_{\rm news}$ over a range of $\bar H$ and with survey forecasts of $a$ as additional observables, which reveals news, bounds that contamination.

4. Integrated levels

If $y_t$ contains $\Delta s_t$, the $H$-step forecast error of the level $s_{t+H}$ uses accumulated responses $\tilde B_h=\sum_{j\le h}(B_j)_{\Delta s,\cdot}$: $\mathrm{FEV}_H(s)=\sum_{h<H}\|\tilde B_h\|^2$, and $\theta_g$ replaces $B$ with $\tilde B$ in the numerator and denominator. Using one-period differenced coefficients understates the persistent shocks' shares.

5. Remaining

Survey expectations with measurement error require a state-space model, with identified sets widened accordingly. Exchange-rate regimes change $C_h$ across samples. Intermediary-capacity shocks need their own exclusion restriction, for example the shock is the only one moving dealer balance sheets on impact [or,crs].

Completion Estimate

48%

This is a post hoc, heuristic estimate for the full catalogue target. Eigenvalue rotation bounds, a delayed-news example and accumulated level responses establish restricted decomposition and timing results, with a conditional news identification route. The abstract overstates unrestricted shares at all horizons, and the full four-group identification and measurement-error robustness remain unresolved.

References

  • [or] M. Obstfeld, K. Rogoff, The six major puzzles in international macroeconomics, NBER Macroeconomics Annual 15 (2000), 339--390.
  • [crs] R. Chahrour et al., Exchange rate disconnect revisited (manuscript, 21 November 2025). https://chahrour.github.io/papers/Disconnect_revisited.pdf.
  • [bs] R. B. Barsky, E. R. Sims, News shocks and business cycles, JME 58 (2011), 273--289.

Go to contribution →

Econ-403-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-403

Exchange-rate disconnect: a research attempt

Target and outcome

Status: unresolved. This attempt derives sharp forecast-variance-share bounds under unrestricted orthogonal shock rotations, explains a sufficient external-instrument restriction for a one-dimensional shock, and checks the distinction between level and difference horizons. It does not identify the actual shares of productivity news, demand or intermediary shocks.

The November 2025 primary manuscript of Chahrour and coauthors [disconnect] studies expectation-driven exchange-rate movements and delayed fundamental responses. The calculations below are an independent linear-algebra benchmark; no quantitative empirical claim from that manuscript is assumed. All rotations used for exact observational equivalence below belong to a Gaussian shock model, so they preserve the full observable law rather than merely covariances.

A reduced form leaves shock groups undetermined

Suppose the stationary observable vector admits \[ y_t=\sum_{h\geq0}B_h\varepsilon_{t-h},\qquad \sum_h\|B_h\|_F^2<\infty,\qquad \varepsilon_t\stackrel{\mathrm{iid}}{\sim}N(0,I_d). \] For the forecast-variance interpretation, the forecasting information reveals all past structural innovations, or observed-history invertibility is separately assumed. Square summability alone does not supply that information restriction. For any orthogonal $Q$, put $\widetilde B_h=B_hQ$ and $\widetilde\varepsilon_t=Q'\varepsilon_t$. The transformed innovations are again independent standard Gaussian vectors and generate the same $y_t$ path. Consequently the entire observable law cannot distinguish the rotation.

Fix horizon $H$, let $b_h$ be the exchange-rate row of $B_h$, and define \[ M_H=\sum_{h=0}^{H-1}b_h'b_h,\qquad D_H=\operatorname{tr}(M_H)>0. \] If a putative shock group has $r$ coordinates, its rotated share is $\theta=\operatorname{tr}(P M_H)/D_H$, where $P$ ranges over rank-$r$ orthogonal projections. No economic label is yet attached to these subspaces.

Proposition (Sharp unrestricted rotation bounds): Let $\lambda_1\leq\cdots\leq\lambda_d$ be the eigenvalues of $M_H$. The smallest and largest attainable group shares are \[ \underline\theta_r=\frac{\sum_{j=1}^r\lambda_j}{D_H}, \qquad \overline\theta_r=\frac{\sum_{j=d-r+1}^d\lambda_j}{D_H}. \] For $0<r<d$, every intermediate value is attainable.
Proof: Diagonalize $M_H=V\Lambda V'$. In that basis the projection's diagonal entries $p_j$ satisfy $0\leq p_j\leq1$ and $\sum_jp_j=r$. Thus $\operatorname{tr}(PM_H)=\sum_j\lambda_jp_j$ is bounded below by selecting the $r$ smallest eigenvalues and above by selecting the $r$ largest. Projections onto the corresponding eigenvectors attain both bounds. The real Grassmann space of rank-$r$ subspaces is connected: an orthonormal basis can be continuously rotated into any other subspace, with signs of basis vectors immaterial. The continuous trace image is therefore an interval containing both extrema. Each projection can be completed to an orthonormal basis and hence to an admissible Gaussian shock rotation. ∎

For $r=0$ and $r=d$ the share is respectively zero and one. With $d=2$, $H=1$ and $b_0=(1,0)$, a one-shock group can have any share in $[0,1]$ while preserving the data law. With $M_H=\operatorname{diag}(4,1)$, its sharp interval is $[1/5,4/5]$. Dynamic responses can therefore restrict the interval even before economic labels are imposed, but generally do not select a point.

These bounds describe a rotation-only class around an identified reduced form. They are not automatically the sharp set when unknown noninvertibility, omitted shocks or measurement equations enlarge the model class. If independently distributed non-Gaussian components are imposed, general rotations need not retain independence and the feasible class must be reconsidered.

Joint shares and economic restrictions

For a partition into shock groups, their projections are mutually orthogonal and sum to the identity. Their shares consequently sum to one. Independent coordinate intervals omit this coupling. Restrictions defining news, current fundamentals and intermediary shocks must select mutually compatible subspaces; choosing each group's maximum separately can be infeasible.

Given a compact closed set of admissible orthogonal matrices determined by specified equality and weak inequality restrictions, minimize and maximize the share over that set. Continuity ensures extrema exist when the set is nonempty. An empty set rejects the joint restrictions. Nonconvexity means a local optimizer does not certify global bounds, and disconnected feasible sets may have gaps between extreme values. Sign restrictions alone are restrictions to be tested for informativeness, not a guarantee of a unique decomposition.

A sufficient external-instrument calculation

For a square invertible impact matrix $B_0$, let reduced-form innovations be $u_t=B_0\varepsilon_t$ with covariance $\Sigma_u=B_0B_0'$. Suppose an external instrument satisfies \[ {\mathbb{E}}[z_t\varepsilon_{1t}]=\alpha\ne0,\qquad {\mathbb{E}}[z_t\varepsilon_{jt}]=0\quad(j\ne1). \] Then $c={\mathbb{E}}[u_tz_t]=\alpha B_0e_1$. Since $(B_0e_1)'\Sigma_u^{-1}(B_0e_1)=1$, the impact column is \[ B_0e_1=\pm\frac{c}{\sqrt{c'\Sigma_u^{-1}c}}. \] This follows by multiplying the covariance identity and imposing the shock's unit variance. The sign is irrelevant for its squared forecast-variance share. If identified reduced-form dynamics give $B_h=\Psi_hB_0$, the same impact column determines that shock's responses at every horizon.

This is a sufficient model-based identification result, not an empirical validation of a particular instrument. Contamination by fundamental information violates the zero-covariance restrictions; a vector correlated with several shocks identifies their mixture instead. The argument also assumes $u_t$ and $\Psi_h$ are recoverable from the stated information set and that $B_0$ is invertible. Weak relevance makes estimation unstable even when the population expression is valid.

Horizon and integration checks

If the modeled exchange-rate variable is $\Delta s_t$, the horizon-$H$ error in the level $s_{t+H}$ accumulates shocks. The corresponding response at lag $k$ is $\bar b_k=\sum_{j=0}^k b_j$, and the level variance calculation uses $\sum_{k=0}^{H-1}\bar b_k'\bar b_k$, not $M_H$ for the differenced series. This follows by summing the $H$ future increments and collecting the coefficient on each future innovation. Applying the stationary formula directly to an untreated unit-root level is not justified.

Zero contemporaneous correlation need not mean no fundamental relation. Let $\Delta s_t=\epsilon_t$ and productivity growth $g_t=\epsilon_{t-2}+\eta_t$, with independent mean-zero iid innovations of variances one and $\sigma_\eta^2$. Then ${\operatorname{Cov}}(\Delta s_t,g_t)=0$, but $\operatorname{Corr}(\Delta s_t,g_{t+2})=1/\sqrt{1+\sigma_\eta^2}$. The delayed fundamental response produces a current disconnect mechanically. It does not by itself identify whether $\epsilon_t$ is news, financial demand, or another causal innovation; that requires a structural label and information restrictions.

The remaining work is to justify the shock groups, instrument exclusions, expectation measurement and forecasting information empirically, and to propagate their uncertainty into joint shares. The algebra supplies exact conditional bounds and diagnostics while leaving the full exchange-rate decomposition unresolved.

Completion Estimate

64%

This is a post hoc, heuristic estimate for the full catalogue target. Sharp rotation-based variance-share intervals, a sufficient external-instrument theorem and accumulated level responses rigorously address observational ambiguity and horizon mismatch. Economic labels for all shock groups, valid instruments, survey errors and robustness to omitted shocks and regimes remain unidentified for the joint decomposition.

References

Go to contribution →