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1% ============================================================================2% Appendix3% ============================================================================4\newpage5\begin{appendices}67\section{Market Descriptions}\label{app:markets}89This appendix provides detailed descriptions of the Polymarket contract markets analyzed in the paper.1011\subsection{Federal Reserve Rate Decision Market (June 2026 FOMC)}1213The Federal Reserve rate decision market on Polymarket consists of five binary outcome contracts corresponding to the possible actions taken by the Federal Open Market Committee (FOMC) at its scheduled June 2026 meeting regarding the target federal funds rate. The contracts are structured within Polymarket's neg-risk framework, which enforces mutual exclusivity and collective exhaustiveness: exactly one contract resolves to \$1 and all others resolve to \$0.1415\begin{itemize}16 \item \textbf{Contract 1: 50+ bps decrease.} This contract resolves to \$1 if the FOMC lowers the target federal funds rate by 50 basis points or more at the June 2026 meeting. The contract covers all scenarios in which the rate is reduced by 50 bps, 75 bps, 100 bps, or any larger decrement.1718 \item \textbf{Contract 2: 25 bps decrease.} This contract resolves to \$1 if the FOMC lowers the target federal funds rate by exactly 25 basis points. It does not include rate cuts of other magnitudes.1920 \item \textbf{Contract 3: No change.} This contract resolves to \$1 if the FOMC leaves the target federal funds rate unchanged at its current level (4.25--4.50\% at the start of the sample period).2122 \item \textbf{Contract 4: 25 bps increase.} This contract resolves to \$1 if the FOMC raises the target federal funds rate by exactly 25 basis points.2324 \item \textbf{Contract 5: 50+ bps increase.} This contract resolves to \$1 if the FOMC raises the target federal funds rate by 50 basis points or more.25\end{itemize}2627The resolution source for all five contracts is the official FOMC statement released at the conclusion of the meeting. The sample period extends from December 2025 to May 2026, providing 146 daily observations.2829\subsection{2024 U.S.\ Presidential Election Market}3031The 2024 U.S.\ Presidential Election market on Polymarket consists of 17 binary outcome contracts, each corresponding to a potential winner of the 2024 presidential election. The contracts are also structured within the neg-risk framework: exactly one candidate contract resolves to \$1 (the winner) and all others resolve to \$0.3233The 17 candidates include the major-party nominees (Donald Trump, Kamala Harris) and 15 additional candidates or potential candidates whose contracts were listed on the platform at various points during the sample period. The sample spans 307 daily observations, covering the period from the establishment of the market through post-election settlement.3435The resolution source is the official certification of the presidential election results. Donald Trump won the election, and his contract resolved to \$1 upon certification. The final trading price for Trump's contract was 0.58, reflecting the fact that some secondary market trading continued between the election date and the formal settlement of contracts on the blockchain.3637\section{Variable Definitions}\label{app:variables}3839This appendix provides precise definitions of all variables used in the empirical analysis.4041\begin{table}[htbp]42\centering43\small44\begin{tabular}{l p{10cm}}45\toprule46\textbf{Variable} & \textbf{Definition} \\47\midrule48$p_j(t)$ & Closing price of Polymarket contract $j$ on trading day $t$. Prices are denominated in USDC and range from 0 to 1. \\[0.3em]49$r_j(t)$ & Log return on contract $j$: $r_j(t) = \ln(p_j(t) / p_j(t-1))$. Undefined when $p_j(t-1) = 0$. \\[0.3em]50$\Sigma(t)$ & State price sum: $\Sigma(t) = \sum_{j=1}^{N} p_j(t)$. Equals $1/(1+r_f)$ under Arrow--Debreu completeness. \\[0.3em]51$q_j(t)$ & Risk-neutral probability: $q_j(t) = p_j(t) / \Sigma(t)$. Sums to unity by construction. \\[0.3em]52$AR_j(t)$ & Abnormal return: $AR_j(t) = r_j(t) - \hat{\mu}_j$, where $\hat{\mu}_j$ is the mean return over the 60-day estimation window. \\[0.3em]53$CAR_j(t_1, t_2)$ & Cumulative abnormal return: $CAR_j(t_1, t_2) = \sum_{t=t_1}^{t_2} AR_j(t)$. \\[0.3em]54$RV_t^{(d)}$ & Daily realized volatility proxy: $RV_t^{(d)} = |r_t|$ (absolute daily return). \\[0.3em]55$RV_t^{(w)}$ & Weekly realized volatility: $RV_t^{(w)} = \frac{1}{5} \sum_{i=0}^{4} RV_{t-i}^{(d)}$. \\[0.3em]56$RV_t^{(m)}$ & Monthly realized volatility: $RV_t^{(m)} = \frac{1}{22} \sum_{i=0}^{21} RV_{t-i}^{(d)}$. \\[0.3em]57$\sigma_t^2$ & Conditional variance from GARCH/GJR-GARCH models. \\[0.3em]58Roll spread & Implicit effective spread: $\text{Roll} = 2\sqrt{-\text{Cov}(r_t, r_{t-1})}$ when autocovariance is negative; 0 otherwise. \\[0.3em]59ILLIQ$_t$ & Amihud illiquidity ratio: $\text{ILLIQ}_t = |r_t| / V_t$, where $V_t$ is dollar trading volume. \\[0.3em]60SPY$_t$ & Daily log return on the SPDR S\&P 500 ETF Trust. \\[0.3em]61BTC$_t$ & Daily log return on Bitcoin (BTC/USD). \\[0.3em]62VIX$_t$ & Daily log change in the CBOE Volatility Index. \\[0.3em]63\bottomrule64\end{tabular}65\caption{Variable definitions.}66\label{tab:variables}67\end{table}6869\section{Additional Robustness}\label{app:robustness}7071This appendix reports supplementary robustness checks referenced in the main text.7273\subsection{Unit Root Tests on Return Series}7475Augmented Dickey--Fuller (ADF) tests are conducted on all return series to confirm stationarity. The null hypothesis of a unit root is rejected at the 1\% significance level for all five Fed rate decision contract return series (ADF statistics ranging from $-7.2$ to $-11.4$, all below the 1\% critical value of $-3.48$), as well as for SPY, BTC, and VIX return series. These results confirm that the return series are stationary and suitable for the time-series analyses (Granger causality, VAR, GARCH) employed in the paper.7677\subsection{Multicollinearity Diagnostics for HAR-RV}7879The three regressors in the HAR-RV model---daily, weekly, and monthly realized volatility---are constructed from overlapping observations and are therefore correlated by construction. We compute variance inflation factors (VIFs) for the HAR-RV regression: VIF$(RV^{(d)})$ = 1.34, VIF$(RV^{(w)})$ = 2.87, VIF$(RV^{(m)})$ = 2.41. All VIFs are below the conventional threshold of 10, indicating that multicollinearity is not a serious concern for coefficient estimation, though it may contribute to the imprecision of individual coefficient estimates.8081\subsection{Residual Diagnostics for GARCH Models}8283We examine the standardized residuals from the GJR-GARCH(1,1) model to assess model adequacy. The Ljung--Box test for serial correlation in the squared standardized residuals fails to reject the null of no autocorrelation at the 5\% level for lag orders up to 20, indicating that the GJR-GARCH specification adequately captures the conditional variance dynamics. The standardized residuals exhibit excess kurtosis of approximately 4.2, exceeding the Gaussian value of 3, which motivates the Student-$t$ innovation distribution considered as a robustness check in Section~\ref{sec:robustness}.8485\subsection{Johansen Cointegration Test}8687Given the potential for long-run equilibrium relationships between prediction market prices and traditional financial market variables, we conduct the \citet{johansen1991estimation} trace test for cointegration among the level series of the Fed ``No Change'' contract price, SPY, BTC (in logs), and VIX. The trace statistic fails to reject the null of zero cointegrating vectors at the 5\% significance level, indicating no stable long-run equilibrium relationship among the level series. This result is consistent with the use of return-based (rather than level-based) specifications in our main analysis and confirms that the Granger causality and VAR analyses do not suffer from omitted error-correction terms.8889\subsection{Hansen's Generalized Method of Moments}9091As a robustness check on the Euler equation implied by the Arrow--Debreu framework, we estimate the stochastic discount factor using \citet{hansen1982generalized} GMM. The moment conditions are derived from the asset pricing restriction that $\mathbb{E}[m_{t+1} R_{t+1}^j] = 1$ for each contract $j$, where $m_{t+1}$ is the stochastic discount factor and $R_{t+1}^j$ is the gross return. Using a linear specification for the stochastic discount factor, $m_{t+1} = a + b \cdot f_{t+1}$, where $f_{t+1}$ is a factor (we use the return on an equally weighted portfolio of the five contracts), the GMM $J$-test fails to reject the overidentifying restrictions at the 5\% significance level. This result provides additional support for the no-arbitrage interpretation of prediction market prices, as the data are consistent with the existence of a well-behaved stochastic discount factor that prices all five contracts.9293\end{appendices}94