Python 56.2%
TeX 43.7%
1% ============================================================================2% Appendix3% ============================================================================4\newpage5\begin{appendices}67\section{Conditional Volatility of the Modal State Price}\label{app:volatility}89This appendix reports the volatility models that an earlier version of the paper presented as a principal result. The series is the daily log return on the ``No change'' claim over the main sample (145 calendar-day returns, listing day excluded), in percent; it has a mean of 0.69, a standard deviation of 6.67, a minimum of $-31.6$ (28 January 2026) and a maximum of $+30.9$ (18 March 2026), both FOMC days.1011Before estimating conditional-variance models we test for their premise. Engle's ARCH-LM statistic with five lags is 4.92 ($p=0.43$) and the Ljung--Box statistic on squared returns at ten lags is 10.7 ($p=0.38$): neither rejects conditional homoskedasticity. The Ljung--Box statistic on the returns themselves is 20.7 ($p=0.023$), so there is some evidence of serial correlation in the mean, consistent with the underreaction predicted by \citet{ottaviani2015price} or with stale prices on inactive days, but not of volatility clustering of the kind that GARCH models are designed to capture.1213Table~\ref{tab:garch} reports GARCH(1,1) and GJR-GARCH(1,1) estimates with Gaussian and Student-$t$ innovations and Bollerslev--Wooldridge robust standard errors. Three features stand out. The persistence $\alpha_1+\beta_1$ (plus $\gamma_1/2$ for the asymmetric models) equals one in all four specifications, and the constant $\omega$ is zero to three decimal places in three of them: the estimates lie on the boundary of the parameter space, where the usual asymptotic inference does not apply. The asymmetry parameter $\gamma_1$ of the GJR model is about 0.30 in both distributions, but its robust standard error is 0.72 under Gaussian innovations and the symmetric ARCH coefficient is estimated at exactly zero; the Student-$t$ degrees of freedom are below three, so that the fitted innovation distribution has infinite kurtosis and barely finite variance. The large likelihood gains of the asymmetric and Student-$t$ specifications therefore reflect the fitting of two or three announcement-day observations rather than a pervasive leverage effect, and we do not interpret $\gamma_1$ as evidence of one. The earlier version reported a GJR coefficient of 0.30 as a ``novel leverage effect'' with an AIC improvement of 46 over the symmetric model; the coefficient is reproduced here, but the inference is not.1415\input{../results/tables/table7_garch.tex}1617Table~\ref{tab:har_rv} reports the heterogeneous autoregressive regression of \citet{corsi2009simple} with the daily absolute return as the volatility proxy and, alternatively, the squared return, using the standard definitions of the weekly and monthly components as averages of the daily proxy over five and twenty-two days. With absolute returns the weekly component is positive and significant ($\beta_w=0.63$, $t=2.24$) and the regression explains 15 percent of the variation in next-day absolute returns; with squared returns, which are dominated by the two FOMC days, the fit collapses ($R^2=0.02$, joint $p=0.49$). Persistence in the absolute size of daily moves at the weekly horizon is thus detectable, and is plausibly linked to the clustering of macroeconomic releases within the month, but the evidence is modest. Figure~\ref{fig:volatility} plots the returns and the fitted conditional standard deviations.1819\input{../results/tables/table8_har_rv.tex}2021\begin{figure}[htbp]22 \centering23 \includegraphics[width=\textwidth]{../figures/fig9_volatility.pdf}24 \caption{Panel~A: daily log returns (percent) on the ``No change'' claim, 12 December 2025 to 22 May 2026. Panel~B: conditional standard deviation from the Gaussian GARCH(1,1) and the Student-$t$ GJR-GARCH(1,1) of Table~\ref{tab:garch}.}25 \label{fig:volatility}26\end{figure}2728\section{Variable Definitions}\label{app:variables}2930\begin{table}[htbp]31\centering32\small33\begin{tabular}{l p{10.5cm}}34\toprule35\textbf{Symbol} & \textbf{Definition} \\36\midrule37$p_{j,t}$ & Daily closing price of the YES token of contract $j$, in USDC per \$1 payoff, labelled by the UTC day it closes (the CLOB daily point stamped 00:00 UTC of day $t+1$). \\[0.3em]38$\Sigma_t$ & Price of the complete set, $\sum_{j} p_{j,t}$, computed on days when all $N$ contracts have a close. \\[0.3em]39$\delta_t$ & Discounted value of \$1 at resolution, $\exp(-r_t\tau_t)$, with $r_t$ the effective federal funds rate (FRED DFF) and $\tau_t$ the time to resolution in years. \\[0.3em]40$q_{j,t}$ & Risk-neutral probability of state $j$, $p_{j,t}/\Sigma_t$. \\[0.3em]41$\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ & Risk-neutral expected change in the upper bound of the target range at the June 2026 meeting, $\sum_j q_{j,t}x_j$ with $x=(-50,-25,0,25,50)$ basis points. \\[0.3em]42$\Delta p_{\text{No change},t}$ & Daily change in the ``No change'' state price, in percentage points. \\[0.3em]43$r_t$ (returns) & Daily log return on the ``No change'' claim, in percent, used only in Appendix~\ref{app:volatility}. \\[0.3em]44TNX & Daily change in the CBOE ten-year Treasury yield index, in basis points, on the U.S.\ trading-day grid. \\[0.3em]45SPY, VIX, BTC & Daily log changes (percent) in the SPDR S\&P~500 ETF, the CBOE Volatility Index and Bitcoin (USD), on the U.S.\ trading-day grid. \\[0.3em]46$\widehat{S}_t$ & Roll (1984) spread estimate, $2\sqrt{-\widehat{\mathrm{Cov}}(\Delta p_t,\Delta p_{t-1})}$ over a rolling twenty-day window, zero when the covariance is positive. \\[0.3em]47$RV^{(d)}_t, RV^{(w)}_t, RV^{(m)}_t$ & Daily volatility proxy ($|r_t|$ or $r_t^2$) and its averages over the past 5 and 22 calendar days. \\48\bottomrule49\end{tabular}50\caption{Variable definitions.}51\label{tab:variables}52\end{table}5354\section{Reconciliation with the Earlier Version}\label{app:reconciliation}5556The first version of this paper (May 2026) reported several results that the present version does not reproduce. This appendix records the differences and their causes so that readers of both versions can reconcile them; the full audit trail is in the repository's revision log.5758The earlier completeness statistics (mean 1.0205, standard deviation 0.0527, maximum absolute deviation 0.570, 146 observations) are reproduced exactly when the listing day is included: the seeded quotes of 10 December 2025 summed to 1.57 before any trade. Excluding that day gives the figures of Table~\ref{tab:completeness}. The earlier text attributed the 2 percent overround to transaction costs ``and the time value of money''; as shown in Section~\ref{sec:theoretical}, the time value of money lowers the price of the complete set and cannot contribute to a positive overround. The earlier text also stated that the effective federal funds rate was 4.25--4.50 percent over the sample; the FRED series shows 3.62--3.64 percent.5960The earlier event study reported cumulative abnormal returns of $-0.615$ and $-0.999$ for individual contracts with $p$-values of 0.107 and 0.095 from two and three events respectively; the contract labels in the text did not match those in the table. The design is replaced by the announcement-day analysis of Section~\ref{sec:results}, which uses all thirteen releases in the sample and a variance-based identification.6162The earlier Granger tests reported $F=4.96$ ($p=0.001$) for the Fed contract to the VIX, $F=2.73$ ($p=0.049$) for the VIX to the Fed contract and $F=2.42$ ($p=0.072$) for the Fed contract to the SPY, described as using AIC-selected lags. The code selected, for each pair, the lag in $\{1,\ldots,5\}$ with the smallest $p$-value, and the Polymarket series was labelled one day late, so that a Polymarket ``return'' on day $t$ was in fact the change to the evening of day $t-1$ and the regression of VIX changes on lagged Polymarket changes effectively used information from the same evening. With the calendar corrected but the minimum-$p$ selection retained, the three $p$-values become 0.064, 0.370 and 0.258; with the lag fixed by information criteria they are 0.533, 0.589 and 0.345 (Table~\ref{tab:granger}). The earlier VAR table contained only the intercepts because the information criterion had selected zero lags, which the text did not report.6364The earlier abstract and summary referred to Hasbrouck information shares, to Treasury futures, to a lead of ``15--45 minutes'' and to Amihud illiquidity ratios. None of these was computed: the data are daily, contain no Treasury futures and no daily volume, and the series are not cointegrated, so that information shares are undefined. The earlier description of the election market as converging to 0.58 after the election and of a settlement delay was incorrect; the sample ends on election evening (Section~\ref{sec:robustness}). The earlier robustness section described market-model, Patell, Student-$t$, EGARCH, subsample, cross-market, Johansen and GMM analyses whose results were not produced by the code; they have been removed, and the analyses that were feasible with the data (Student-$t$ innovations, alternative lag orders, alternative valuation of open-ended states, listing-day treatment) are reported in Sections~\ref{sec:robustness} and Appendix~\ref{app:volatility}.6566\end{appendices}67