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1% ============================================================================2% Section 4: Data3% ============================================================================4\section{Data}\label{sec:data}56This section describes the data sources, sample construction, variable definitions, and preliminary statistical properties of the prediction market and financial market data used in our analysis.78\subsection{Polymarket Data}910Polymarket is a blockchain-based prediction market platform that enables users to trade binary outcome contracts settled on the Polygon network. Each contract pays \$1 if the specified outcome occurs and \$0 otherwise, with prices reflecting the market's implied probability of the event. We collect daily closing prices, trading volumes, and liquidity measures for two sets of Polymarket contracts.1112\subsubsection{Federal Reserve Rate Decision Market}1314Our primary dataset consists of daily observations on five contracts corresponding to the possible outcomes of the Federal Reserve's June 2026 FOMC rate decision. The five mutually exclusive and collectively exhaustive states are:1516\begin{enumerate}17 \item \textbf{50+ bps decrease}: The FOMC lowers the target federal funds rate by 50 basis points or more.18 \item \textbf{25 bps decrease}: The FOMC lowers the target federal funds rate by exactly 25 basis points.19 \item \textbf{No change}: The FOMC leaves the target federal funds rate unchanged.20 \item \textbf{25 bps increase}: The FOMC raises the target federal funds rate by exactly 25 basis points.21 \item \textbf{50+ bps increase}: The FOMC raises the target federal funds rate by 50 basis points or more.22\end{enumerate}2324The sample spans 146 daily observations from December 2025 through May 2026. These contracts are traded within Polymarket's neg-risk framework, which enforces mutual exclusivity and collective exhaustiveness: exactly one contract in the group resolves to \$1, and the remaining four resolve to \$0. This structural feature makes the Fed rate decision market an ideal testing ground for Arrow--Debreu completeness conditions.2526Table~\ref{tab:summary_stats} reports summary statistics for the five contracts. The ``No change'' contract has the highest mean price at 0.608, reflecting the market's prevailing expectation of policy inaction. The ``25 bps decrease'' contract has a mean price of 0.304, indicating meaningful probability mass on a single rate cut. The two extreme states (50+ bps decrease and 50+ bps increase) have mean prices of 0.058 and 0.016 respectively, consistent with the market assigning low probability to aggressive rate moves. All five price series exhibit non-normal distributions, as evidenced by significant Jarque--Bera statistics: the ``25 bps increase'' contract displays extreme positive skewness (3.98) and leptokurtosis (30.46), reflecting its status as a low-probability, high-surprise event.2728\input{../results/tables/table2_summary_stats.tex}2930\subsubsection{2024 U.S.\ Presidential Election Market}3132Our secondary dataset comprises daily observations on 17 candidate contracts from the 2024 U.S.\ Presidential Election market, observed over 307 trading days. This market provides a richer cross-sectional dimension for analyzing microstructure properties, liquidity dynamics, and multi-contract pricing relationships. The election market resolved on November 5, 2024, but Polymarket continued trading after the election date, providing observations on post-resolution price convergence.3334Donald Trump won the election, and his contract converged to a final price of 0.58---notably below the \$1 resolution value, indicating that some trading continued in the secondary market before final settlement. The election market's larger cross-section and longer time series complement the Fed rate decision market's cleaner state-space structure.3536\subsection{Financial Market Data}3738To assess the information linkages between prediction markets and traditional financial markets, we collect daily returns data for three benchmark assets:3940\begin{itemize}41 \item \textbf{SPY}: The SPDR S\&P 500 ETF Trust, representing the U.S.\ equity market. Daily closing prices are obtained from standard financial data providers.42 \item \textbf{BTC}: Bitcoin, the largest cryptocurrency by market capitalization. Bitcoin is included both as a benchmark for blockchain-native assets and as a proxy for speculative risk appetite in the digital asset ecosystem.43 \item \textbf{VIX}: The CBOE Volatility Index, which measures the market's expectation of 30-day forward-looking volatility implied by S\&P 500 index options. The VIX is included as a proxy for aggregate uncertainty and risk aversion.44\end{itemize}4546Daily returns are computed as log returns: $r_t = \ln(P_t / P_{t-1})$, where $P_t$ is the closing price on day $t$. For the prediction market contracts, the same return definition is applied, with the convention that returns are undefined (and excluded) on days when the previous day's price is zero.4748\subsection{Event Data}4950We identify macroeconomic announcement events within the sample period for the event study analysis. The events fall into three categories:5152\begin{itemize}53 \item \textbf{FOMC announcements}: We identify 2 scheduled FOMC meeting dates that fall within the sample period. FOMC announcements are released at 2:00 PM Eastern Time and are accompanied by the statement, Summary of Economic Projections (when available), and the press conference.54 \item \textbf{CPI releases}: We identify 3 Consumer Price Index releases by the Bureau of Labor Statistics within the sample period. CPI releases occur at 8:30 AM Eastern Time and provide the most closely watched measure of consumer inflation.55 \item \textbf{Election key dates}: We identify 3 politically significant dates in the election sample, including debate dates, major endorsement events, and the election day itself.56\end{itemize}5758Event windows are defined symmetrically around each announcement date. For FOMC and CPI events, we use a $[-3, +3]$ day window (7 trading days), with the estimation window for the constant mean model spanning 60 trading days prior to the event window. For election events, we use a wider $[-5, +5]$ day window to accommodate the possibility of slower information incorporation in political markets.5960\subsection{Variable Construction}6162We construct the following variables for our empirical analysis:6364\textbf{State price sum.} For each trading day $t$, the state price sum is computed as $\Sigma(t) = \sum_{j=1}^{5} p_j(t)$, where $p_j(t)$ is the closing price of the $j$-th contract. Under Arrow--Debreu completeness, $\Sigma(t) \approx 1$.6566\textbf{Risk-neutral probabilities.} The risk-neutral probability of state $j$ at time $t$ is computed as $q_j(t) = p_j(t) / \Sigma(t)$. This normalization ensures that risk-neutral probabilities sum to unity.6768\textbf{Abnormal returns.} For the event study analysis, abnormal returns are computed as $AR_j(t) = r_j(t) - \hat{\mu}_j$, where $\hat{\mu}_j$ is the mean return over the 60-day estimation window. Cumulative abnormal returns are computed by summing abnormal returns over the event window.6970\textbf{Realized volatility.} Daily realized volatility for the ``No change'' contract is computed as the absolute value of the daily return, $|r_t|$. Weekly and monthly realized volatility components are computed as the averages of daily realized volatility over the preceding 5 and 22 trading days, respectively, following the HAR-RV convention of \citet{corsi2009simple}.7172\textbf{Liquidity measures.} We compute two liquidity measures. The Roll spread \citep{roll1984simple} is estimated from the serial autocovariance of returns: $\text{Roll}_t = 2\sqrt{-\text{Cov}(r_t, r_{t-1})}$ when the autocovariance is negative, and zero otherwise. The Amihud illiquidity ratio \citep{amihud2002illiquidity} is computed as $\text{ILLIQ}_t = |r_t| / V_t$, where $V_t$ is the dollar trading volume on day $t$.7374\subsection{Descriptive Statistics}7576Table~\ref{tab:descriptive_stats} reports descriptive statistics for the broader set of Polymarket contracts examined, including trading volume and liquidity measures in U.S.\ dollars. The table documents the substantial heterogeneity in market activity across contracts, with trading volumes ranging from under one million to nearly ten million dollars and liquidity (measured as the depth of the order book) varying by more than an order of magnitude.7778\input{../results/tables/table1_descriptive_stats.tex}7980Figure~\ref{fig:state_prices} plots the evolution of state prices for the five Fed rate decision contracts over the sample period. The dominant feature is the secular increase in the ``No change'' contract price from approximately 0.575 at the beginning of the sample to 0.978 by May 2026, accompanied by a corresponding decline in the ``25 bps decrease'' contract from approximately 0.40 to near zero. This pattern is consistent with the market progressively assigning higher probability to policy inaction as incoming macroeconomic data (strong labor markets, persistent inflation) reduced the case for near-term rate cuts.8182\begin{figure}[htbp]83 \centering84 \includegraphics[width=\textwidth]{../figures/fig1_state_prices.pdf}85 \caption{Arrow--Debreu state prices for Federal Reserve rate decision contracts, December 2025 to May 2026. Each line represents the daily closing price of one of the five mutually exclusive outcome contracts. The ``No change'' contract (upper line) rises from approximately 0.575 to 0.978, while the ``25 bps decrease'' contract (declining line) falls from approximately 0.40 to near zero.}86 \label{fig:state_prices}87\end{figure}8889Figure~\ref{fig:rn_density} displays snapshots of the risk-neutral density at selected dates, illustrating the progressive concentration of probability mass on the ``No change'' outcome. Early in the sample, the distribution is bimodal, with substantial mass on both ``No change'' and ``25 bps decrease''; by the end of the sample, the distribution is sharply peaked at ``No change,'' with all other states assigned negligible probability.9091\begin{figure}[htbp]92 \centering93 \includegraphics[width=\textwidth]{../figures/fig3_risk_neutral_density.pdf}94 \caption{Risk-neutral density snapshots for the Federal Reserve rate decision market. Each panel shows the distribution of risk-neutral probabilities across the five outcome states at a selected date. The progressive concentration of mass on ``No change'' reflects the market's increasing confidence in policy inaction.}95 \label{fig:rn_density}96\end{figure}9798\subsection{Stationarity and Distributional Properties}99100Since prediction market prices are bounded between 0 and 1 and converge to either 0 or 1 at resolution, the level series are inherently non-stationary. All time-series analyses in this paper therefore use log returns rather than price levels. Augmented Dickey--Fuller tests reject the unit root null for all return series at the 1\% significance level, confirming stationarity.101102The return distributions exhibit significant departures from normality, as indicated by the Jarque--Bera statistics reported in Table~\ref{tab:summary_stats}. The ``25 bps increase'' contract displays particularly extreme non-normality, with a kurtosis of 30.46 reflecting occasional large price jumps associated with unexpected news. These distributional properties motivate the use of heteroskedasticity-robust inference methods throughout our analysis, including Newey--West standard errors \citep{newey1987simple} and White standard errors \citep{white1980heteroskedasticity}.103