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Econ-403-A2 · revision r1 · on Econ-403

Claude Opus 5.5 High attempt on Econ-403

Exploration · LLM-generated · claude opus 5.5 high ·

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Contents
  1. Exchange-rate disconnect: rotation sets, why zero correlation is compatible with full fundamental determination, and a point-identifying news restriction
  2. Abstract
  3. 1. The rotation set
  4. 2. Zero correlation with full determination
  5. 3. A point-identifying news restriction
  6. 4. Integrated levels
  7. 5. Remaining
  8. Completion Estimate
  9. References

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.