spb/wp47_uqo

1% ============================================================================2\section{Methodology}3\label{sec:methodology}4% ============================================================================56\subsection{Daily Nelson--Siegel factors}7\label{sec:meth-ns}89For each yield day $t$ the eleven CMT yields $y_t(\tau)$, in percent, are fitted by ordinary least squares to10\begin{equation}11  y_t(\tau) = L_t + S_t\,\frac{1-e^{-\lambda\tau}}{\lambda\tau} + C_t\left(\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}\right) + \varepsilon_t(\tau),12  \label{eq:ns}13\end{equation}14with $\lambda = 0.7308$ per year, the value at which the curvature loading peaks at 30 months \citep{diebold2006forecasting}.15$L_t$ is the level (the yield at infinite maturity), $S_t$ the slope (the short rate minus the level, so that $L_t+S_t$ is the16instantaneous short rate $y_t(0)$) and $C_t$ the curvature. The fit has a root mean squared error of \fitRmseBp{} basis points over172008--2026, concentrated at the shortest and longest maturities. Estimating $\lambda$ by non-linear least squares over the same period18gives \lambdaHat{} and reduces the error to \fitRmseLamHatBp{} basis points; we keep the fixed value, which makes the loadings constant19over time and the factors comparable with the literature. Table~\ref{tab:dailyfactors} reports the summary statistics. Daily changes20in level and slope have a correlation of \corrDailyLs{}, a feature that matters below.2122\input{tables/tab_daily_factors}2324\subsection{Factor-mimicking portfolios}25\label{sec:meth-mimicking}2627Let $r_t$ be the $K$-vector of daily futures log returns, in percent, over yield day $t$, with $K$ equal to the number of contracts28available with three years of history: four (2-, 5-, 10-year and bond) at first, five from \firstQuarterFiveRoots{} (ultra bond) and29six from \firstQuarterSixRoots{} (ultra 10-year). For each factor $k\in\{L,S,C\}$ we estimate30\begin{equation}31  \Delta\beta_{kt} = a_k + w_k' r_t + e_{kt}32  \label{eq:mimic}33\end{equation}34by OLS on a trailing window of three years, requiring at least 500 days on which all $K$ contracts have at least 100 minute returns and35including a contract only if it is present on 85 percent of the window's days. The estimation is repeated at the start of every calendar36quarter and the weights $\hat w_k$ are applied to the following quarter only. The weights are therefore out of sample for every day on37which they are used, and the first out-of-sample quarter is 2011Q1. Equation~\eqref{eq:mimic} is the tracking-portfolio regression of38\citet{lamont2001economic}; the fitted value $\hat w_k' r_t$ is the return on the portfolio of futures that best tracks the factor change39in mean-squared-error terms. Because the regressors are returns and the dependent variable is a yield change, the weights have the40dimension of an inverse duration and are negative for the contracts whose maturities drive the factor; Table~\ref{tab:weights} reports41them. An alternative is to invert the duration mapping contract by contract; the regression is preferred because it absorbs the42cheapest-to-deliver, basis and duration-drift effects that the structural mapping would have to model explicitly.4344\subsection{Intraday factors and realized measures}45\label{sec:meth-intraday}4647For minute $m$ of yield day $t$ in quarter $q$,48\begin{equation}49  \Delta\hat\beta_{k,t,m} = \hat w_{k,q}' r_{t,m},50  \label{eq:intraday}51\end{equation}52with no intercept, computed for every minute at which all contracts of the quarter's set are active. Summed over the minutes of a53yield day, $\sum_m \Delta\hat\beta_{k,t,m} = \hat w_{k,q}'\sum_m r_{t,m}$, which equals the out-of-sample fitted value of~\eqref{eq:mimic}54without its intercept when every minute of the day enters the cross-section, and differs from it by the returns of the minutes dropped55for inactivity otherwise. We therefore report two fidelity measures: the out-of-sample $R^2$ of the daily regression itself, and the56$R^2$ of the daily factor change on the \emph{cumulated minute series} over all yield days on which it exists, which is the test of57the object the paper uses. Because the factor changes are linear in the futures returns, the fidelity of any implied yield change58$\Delta\hat y(\tau)$ can be assessed the same way against the corresponding CMT yield. Sections~\ref{sec:results-etf}59and~\ref{sec:robustness} add two checks that do not use daily data at all: an independent ETF construction and a60leave-one-contract-out spanning test.6162Realized factor variances and covariances are computed per yield day from the minute changes aggregated to $h$-minute intervals,63\begin{equation}64  RV^{(k)}_{t,h} = \sum_{j} \Big(\sum_{m\in j}\Delta\hat\beta_{k,t,m}\Big)^2, \qquad65  RC^{(kl)}_{t,h} = \sum_{j} \Big(\sum_{m\in j}\Delta\hat\beta_{k,t,m}\Big)\Big(\sum_{m\in j}\Delta\hat\beta_{l,t,m}\Big),66\end{equation}67for $h\in\{1,5,30\}$ minutes. The pre-registered design named one-minute sampling; the design review (Appendix~\ref{app:design})68argued, and Section~\ref{sec:results-cov} confirms, that one-minute covariances are dominated by tick-size noise, so five-minute69measures are the main ones and the one-minute measures are reported as a signature plot. Realized correlations for a period are ratios70of summed covariances to the geometric mean of summed variances.7172The short rate and implied yields are linear in the factors: $\Delta y(0) = \Delta L + \Delta S$ and73$\Delta y(\tau) = \Delta L + \Delta S\, f_2(\tau) + \Delta C\, f_3(\tau)$ with the loadings of~\eqref{eq:ns}. We report event74results both in factor units and for the implied three-month, two-, five-, ten- and thirty-year yields, because the three factors have75very different scales---the curvature loading is at most 0.30, so a curvature change of ten basis points moves no yield by more than three.7677\subsection{Event windows and placebos}78\label{sec:meth-events}7980For each scheduled release we sum the minute changes over the window from five minutes before to twenty-five minutes after the release81time, and, for FOMC statements, also over $[-5,+15]$ and $[-5,+90]$ minutes, the latter covering the press conference. We report the82standard deviation of the window change across events, its ratio to the standard deviation of placebo windows taken at the same clock83time on three non-announcement days per event, matched on weekday and within 200 days, and the share of the yield day's realized84variance that falls in the window. Confidence intervals resample events (and placebo windows) with replacement, 2{,}000 draws.85Pre-announcement drift is the cumulated change from 24 hours before the release to five minutes before it. The cumulative response path86is the root mean square across events of the cumulated change from 60 minutes before to 120 minutes after the release.8788\subsection{Impulse responses and connectedness}89\label{sec:meth-irf}9091Three dynamic specifications describe how factor moves propagate. First, local projections around releases: for each scheduled92event $e$ of a given type, the cumulated change of a variable $v$ from five minutes before the release to horizon $h$,93$\Delta v_e(h)$, is regressed across events on the window change in the implied 2-year yield, $s_e$,94\begin{equation}95  \Delta v_e(h) = \alpha_h + \beta_h\, s_e + u_{e,h}, \qquad h \in \{25, 60, 120, 240 \text{ minutes};\ 0, 1, 2, 5, 10 \text{ yield days}\},96  \label{eq:lp}97\end{equation}98so that $\beta_h$ is the response at horizon $h$ per basis point of the initial 2-year move; for the 2-year yield itself99$\beta_{25}=1$ by construction and values below (above) one at longer horizons indicate reversal (continuation). Intraday horizons100use the minute series; daily horizons cumulate the futures-based daily factor changes to the end of the release's yield day and of101the following one to ten yield days. Standard errors are heteroskedasticity-robust and 95-percent intervals resample events.102Second, a vector autoregression on five-minute changes in the short rate, the level and the curvature during US hours103(08:00--16:00~ET), estimated on within-day observations only, with lag order chosen by BIC; generalized impulse responses and104forecast-error variance decompositions \citep{pesaran1998generalized} at a one-hour horizon give the Diebold--Yilmaz105connectedness measures \citep{diebold2012better}, by regime. Third, a VAR on the daily log realized variances of the three factors106(five-minute sampling) with the same decomposition at ten days, over the full sample and in 250-day rolling windows, measures how107volatility shocks spill across factors.108109\subsection{Forecasting}110\label{sec:meth-forecast}111112For each factor the target is the realized variance from five-minute sampling cumulated over the next $h\in\{1,5,22\}$ yield days. The113HAR model regresses the log of the target on the log realized variance of the day, its five-day mean and its 22-day mean114\citep{corsi2009simple}; forecasts are exponentiated with the usual half-variance correction. The benchmarks are a GARCH(1,1) with115constant mean fitted to the daily factor changes measured from the CMT yields, iterated $h$ steps ahead and summed, and the 22-day rolling116variance of those changes scaled by $h$. All models are re-estimated at the start of every month on an expanding window, and the117out-of-sample period is January 2016 to November 2026. Losses are QLIKE and mean squared error against the realized target118\citep{patton2011volatility}; differences are tested with \citet{diebold1995comparing} statistics and a Newey--West variance with119bandwidth $\lfloor 1.3\,T^{1/2}\rfloor$ or $h-1$, whichever is larger. The nine HAR-versus-GARCH comparisons under each loss are treated as a family and120reported with Holm-adjusted significance.121122\subsection{Hypotheses}123\label{sec:meth-hypotheses}124125The design and the following hypotheses were frozen before any intraday result was computed (git tag \texttt{design-frozen}). H1:126cumulated intraday changes explain daily factor changes out of sample with $R^2\geq 0.90$ for level and slope and $\geq 0.60$ for curvature,127with a slope coefficient in $[0.9,1.1]$. H2: the realized-variance share of the thirty-minute FOMC window is largest for slope, then level,128then curvature, CPI and Employment releases load mainly on level and slope with a larger slope share after 2021 than during the zero lower129bound, and all shares are at least ten times the unconditional share of 30/1380. H3: factor volatility spikes at 08:30~ET, peaks between13008:30 and 11:00~ET and troughs during Asian hours, more sharply for curvature. H4: the one-minute level--slope realized correlation is131negative and less negative at five and thirty minutes, and differs between the zero-lower-bound years and 2022--2026. H5: HAR beats GARCH(1,1)132and the rolling variance at all three horizons. Section~\ref{sec:results} states for each hypothesis whether it is supported, and133Appendix~\ref{app:design} lists the changes made to the design after the review and after H1 was inspected.134