# WP47 — Design and pre-registered analysis plan

Title (working): The Yield Curve at One-Minute Resolution: Nelson–Siegel Factor-Mimicking Portfolios from Treasury Futures Author: Guillaume Manseau (Université Laval) · Date frozen: 2026-11-09 (git tag design-frozen, set before any result of §4–§6 is inspected)

# 1. Question and contribution

The Nelson–Siegel (NS) curve and its dynamic version (Diebold–Li 2006) summarise the Treasury yield curve by three factors — level, slope and curvature — observed at the daily or monthly frequency because the inputs are end-of-day yields. Nothing in the model requires that frequency. This paper asks what the three NS factors look like minute by minute, over the full 23-hour CME session, from 2008 to 2026.

Contribution. (i) A construction: factor-mimicking portfolios of Treasury futures that reproduce daily NS factor changes and can be applied to one-minute futures returns, yielding intraday level, slope and curvature changes for ~4 700 sessions and ~6 million minutes. The construction is validated out of sample against daily factors that are never used to form the weights. (ii) New facts: the intraday anatomy of the three factors around FOMC statements, CPI and Employment releases (which factor moves, how much of the daily factor variance is realised in a 30-minute window, how the mix has changed from the zero-lower-bound era to the 2022–2026 tightening/easing cycle); the intraday periodicity of factor volatility; the realized covariance structure of the factors (the sign and stability of level–slope correlation at high frequency, Epps-type attenuation). (iii) A forecasting application: HAR models on realized factor variances versus daily-data benchmarks.

Closest literature: Andersen, Bollerslev, Diebold & Vega (2007) on announcement effects in bond futures (single contracts, not factors); Gürkaynak, Sack & Swanson (2005) and Swanson (2021) on target/path/LSAP factors from FOMC-window asset moves (factor structure of the surprise, not of the curve); Fleming & Remolona (1999) on price formation around announcements; Hansen (2023, JFEC) on a joint model of yields and realized yield-curve covariation from high-frequency data (daily realized measures, not NS factors nor a futures-based intraday construction); Cieslak & Povala (2016) on information in the Treasury market; Diebold & Rudebusch (2013) book. We did not find a paper that constructs minute-frequency NS factors from futures and validates them against daily factors (searches of 2026-11-09 documented in docs/literature_search.md).

# 2. Data

Session convention. A "yield day" t runs from 15:30 ET on business day t−1 to 15:30 ET on day t (H.15 CMT yields are read at approximately 15:30 ET). Intraday returns are one-minute log price changes computed within the CME session only (no return across the 17:00–18:00 ET maintenance break; no return across a roll splice; first bar after a gap > 5 minutes starts a new segment). Minutes without a bar carry a zero return for the grid but are flagged; a minute enters the cross-section only if all six (or five, pre-2016) roots trade.

Known limitations to state: futures returns reflect the cheapest-to-deliver, carry and the futures–cash basis; the construction identifies changes in factors, not levels; the 1-minute bars are open-to-open prices built from trades.

# 3. Construction

3.1 Daily factors. For each day t, fit $y_t(\tau)=\beta_{1t}+\beta_{2t}\frac{1-e^{-\lambda\tau}}{\lambda\tau} +\beta_{3t}\left(\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}\right)$ by OLS on the 11 CMT maturities with $\lambda=0.7308$ per year (curvature loading maximised at 30 months, Diebold–Li), giving $\beta_t=(L_t,S_t,C_t)$ in percentage points. Robustness: $\lambda$ estimated by NLS on the full sample; GSW zero yields at 1–30 years.

3.2 Mimicking portfolios. Let $r_t\in\mathbb{R}^K$ be the vector of daily futures log returns (percent) over the yield day, $K=5$ (2008–2015) or $6$ (2016–). For each factor $k$ estimate by OLS on a trailing window $\Delta\beta_{kt}=a_k+w_k' r_t+e_{kt}$. Weights $w_k$ are re-estimated every calendar quarter using the previous 3 years (≥ 500 days) and applied to the following quarter only (strictly out of sample). Pre-specified alternatives: expanding window; full-sample weights (in-sample upper bound); structural inversion using cheapest-to-deliver durations.

3.3 Intraday factors. For minute m of day t, $\Delta\hat\beta_{t,m}=\hat W_{q(t)}, r_{t,m}$ (no intercept). Realized factor variance $RV^{(k)}t=\sum_m \Delta\hat\beta^2{k,t,m}$, realized covariance analogously, at 1, 5 and 30 minutes.

# 4. Hypotheses (frozen)

• H1 (fidelity). Out of sample, $\sum_m\Delta\hat\beta_{k,t,m}$ explains daily $\Delta\beta_{kt}$ with $R^2\ge0.90$ for level and slope and $\ge0.60$ for curvature; the slope coefficient of daily on cumulated intraday is within $[0.9,1.1]$. Fidelity is lower before 2016 (five roots) and in 2008–2009.
• H2 (announcement anatomy). In the [−5, +25]-minute window around FOMC statements the share of the yield day's realized variance is largest for slope, then level, then curvature; CPI and Employment releases load mainly on level and slope with a larger slope share after 2021 than during 2009–2015 (ZLB), when short-rate expectations were pinned. Variance shares in announcement windows are at least 10× the unconditional share of a 30-minute window (30/1380 ≈ 2.2%).
• H3 (periodicity). Factor volatility has a pronounced intraday pattern: a 08:30 ET spike, a maximum between 08:30 and 11:00 ET, a minimum during the Asian session; curvature volatility is more concentrated in US hours than level volatility.
• H4 (covariation). The 1-minute level–slope realized correlation is negative on average, less negative at 5 and 30 minutes (Epps effect), and its sign/magnitude differs between the ZLB period and 2022–2026.
• H5 (forecasting). HAR models on log realized factor variances beat (lower QLIKE and MSE, Diebold–Mariano at 5%) a GARCH(1,1) fitted to daily factor changes and a rolling 22-day variance, at horizons of 1, 5 and 22 days, out of sample 2016–2026 with weights re-estimated monthly.

Secondary/descriptive (not hypotheses): factor drift in the hours before announcements (pre-FOMC drift, Lucca–Moench 2015), cumulative response paths, decomposition by meeting type (press-conference vs not).

# 5. Inference and evaluation

Newey–West (bandwidth 1.3 T^{1/2}) for daily regressions; block bootstrap (daily blocks) for event-window variance shares; Diebold–Mariano with HAC for forecast comparisons; all tests two-sided, 5%. Samples: full (2008–2026), sub-periods 2008–2009 (crisis), 2010–2015 (ZLB), 2016–2019 (normalisation), 2020–2021 (pandemic ZLB), 2022–2026 (tightening/easing).

# 6. Outputs

Tables: data coverage; daily factor summary; mimicking weights and OOS fidelity (H1); announcement variance shares (H2); intraday periodicity (H3, figure); realized correlations (H4); HAR forecast evaluation (H5). Figures: intraday factor paths on selected FOMC days; average cumulative response; intraday volatility signature; time series of realized factor volatility; rolling fidelity $R^2$.

# 7. What would falsify the construction

If H1 fails (R² < 0.6 for level or slope), the paper is re-scoped to a methods note on why futures do not span the NS factors (basis/CTD/term-premium reasons) and the event-study results are not reported as factor results.