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% Section 6: Results
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\section{Results}\label{sec:results}

This section presents the empirical findings organized around the four hypotheses developed in Section~\ref{sec:theoretical}. We begin with the Arrow--Debreu completeness tests, proceed to the event study and information linkage analyses, and conclude with the volatility dynamics results.

\subsection{Arrow--Debreu Completeness}\label{subsec:completeness_results}

Table~\ref{tab:completeness} reports the results of the Arrow--Debreu completeness tests for the Federal Reserve rate decision market. The mean state price sum across the 146 daily observations is 1.0205, with a standard deviation of 0.0527. The mean deviation from unity is 0.0248, and the maximum absolute deviation is 0.5700. All individual state prices remain strictly positive throughout the sample, confirming that the non-negativity condition required by the Arrow--Debreu framework is satisfied.

\input{../results/tables/table3_completeness.tex}

These results are strongly supportive of Hypothesis~\ref{hyp:completeness}. The mean state price sum of 1.0205 exceeds unity by approximately 2 percentage points, which is consistent with a small positive overround attributable to transaction costs and the time value of money. Over the sample period, the effective federal funds rate target was in the range of 4.25--4.50\%, implying a daily risk-free rate of approximately 0.012\%. Over the typical contract horizon of 30--180 days, the implied discount factor $1/(1+r_f)$ ranges from 0.9988 to 0.9926, which alone does not fully account for the 2\% overround. The residual deviation is plausibly attributable to the Polymarket trading fee, gas costs associated with blockchain transactions, and the compensation earned by liquidity providers for bearing adverse selection risk.

Figure~\ref{fig:completeness} plots the state price sum $\Sigma(t)$ over the sample period. The series fluctuates around a level slightly above unity, with no apparent trend or structural break. The absence of persistent departures from unity is consistent with Hypothesis~\ref{hyp:no_arb}(b), which predicts that arbitrage forces push the state price sum back toward its equilibrium value when temporary deviations occur.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig2_completeness_test.pdf}
  \caption{State price sum $\Sigma(t)$ for the Federal Reserve rate decision market, December 2025 to May 2026. The horizontal dashed line at unity represents the Arrow--Debreu prediction under zero interest rates. The series fluctuates around 1.02 with standard deviation 0.053, consistent with approximate completeness subject to a small transaction-cost wedge.}
  \label{fig:completeness}
\end{figure}

The finding that all state prices are strictly positive is a necessary condition for the existence of a unique positive state-price density, which in turn is equivalent to market completeness under the Fundamental Theorem of Asset Pricing (Proposition~\ref{prop:no_arb}). In practical terms, this means that the market continuously prices all five possible rate outcomes as having non-zero probability, which is a strong indicator that the market is actively incorporating information about even low-probability states.

Figure~\ref{fig:arbitrage} provides further evidence on no-arbitrage conditions by plotting the time series of arbitrage violations---instances in which the state price sum deviates from the no-arbitrage bounds by more than a threshold calibrated to transaction costs. The frequency and magnitude of such violations are low, suggesting that the no-arbitrage condition holds as a practical matter even if small deviations are statistically detectable.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig12_arbitrage_violations.pdf}
  \caption{Arbitrage violations in the Federal Reserve rate decision market. The figure displays instances in which the state price sum $\Sigma(t)$ deviates from the theoretical no-arbitrage band. Violations are infrequent and small in magnitude, consistent with effective arbitrage enforcement.}
  \label{fig:arbitrage}
\end{figure}

\subsection{Event Study Results}\label{subsec:event_results}

Table~\ref{tab:event_study} reports cumulative abnormal returns (CARs) for the five Fed rate decision contracts around FOMC announcements, CPI releases, and election key dates. The event study results reveal economically meaningful, though not uniformly statistically significant, responses to macroeconomic information releases.

\input{../results/tables/table4_event_study.tex}

For FOMC announcements, the ``50+ bps decrease'' contract exhibits a CAR of $-0.615$ over the $[-3,+3]$ window, with a $t$-statistic of $-5.89$ and a $p$-value of 0.107. While the $t$-statistic is large in absolute terms, the $p$-value reflects the small number of events ($K = 2$), which limits the power of cross-sectional tests. The ``No change'' contract shows a near-zero CAR of $-0.041$ ($t = -0.08$, $p = 0.951$), consistent with the expectation that FOMC announcements that confirm the market's prior beliefs do not generate abnormal returns in the modal contract. The ``25 bps decrease'' contract exhibits a CAR of $-0.054$ ($t = -0.21$, $p = 0.866$), again close to zero.

For CPI releases, the most notable finding is the large negative CAR of $-0.999$ for the ``25 bps decrease'' contract ($t = -3.01$, $p = 0.095$), which is statistically significant at the 10\% level. This result is consistent with CPI releases during the sample period containing inflationary surprises that reduced the perceived probability of near-term rate cuts. Higher-than-expected inflation prints shift probability mass from ``decrease'' states to ``No change'' or ``increase'' states, and the magnitude of the CAR---nearly one log unit---indicates a substantial repricing of rate cut probabilities around CPI releases. The ``No change'' contract shows a CAR of $-0.228$ ($t = -2.83$, $p = 0.105$) around CPI events, suggesting some temporary uncertainty about the modal outcome even as the direction of the probability shift favored policy inaction.

For election key dates, the CAR for the election contracts is $-0.057$ ($t = -0.88$, $p = 0.470$), which is not statistically significant. The wider event window ($[-5,+5]$ days) and the gradual nature of political information incorporation may dilute the event study's ability to detect abnormal returns around specific political dates.

Figure~\ref{fig:event_study_cars} displays the cumulative abnormal returns graphically, showing the temporal pattern of CARs within the event windows. The figure reveals that for FOMC events, much of the abnormal return occurs on the announcement day itself, while for CPI events, the adjustment is more gradual, extending over 2--3 days after the release.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig5_event_study_cars.png}
  \caption{Cumulative abnormal returns (CARs) around macroeconomic events. The figure plots CARs for the Federal Reserve rate decision contracts in event time, with day 0 corresponding to the announcement date. FOMC events (left) show concentrated adjustment on the announcement day; CPI events (right) show more gradual incorporation.}
  \label{fig:event_study_cars}
\end{figure}

\subsection{Granger Causality and Information Linkages}\label{subsec:granger_results}

Table~\ref{tab:granger} reports the results of pairwise Granger causality tests among the Fed ``No Change'' contract returns, SPY returns, BTC returns, and VIX returns. The lag length, selected by AIC, ranges from 1 to 4 across the different specifications.

\input{../results/tables/table5_granger.tex}

The most striking finding is the strong Granger causality from the Fed ``No Change'' contract to VIX returns ($F = 4.96$, $p = 0.001$), which is significant at the 1\% level. This result indicates that changes in the prediction market's assessment of Federal Reserve policy have predictive power for future movements in the VIX, the market's primary gauge of equity volatility and uncertainty. The economic interpretation is intuitive: as the probability of ``No change'' increases, uncertainty about monetary policy decreases, which should be associated with lower realized and implied volatility. The Granger-causal relationship from prediction market prices to VIX returns suggests that information is incorporated into prediction market prices before it is fully reflected in the options market's pricing of aggregate uncertainty.

In the reverse direction, VIX returns Granger-cause Fed ``No Change'' returns ($F = 2.73$, $p = 0.049$), significant at the 5\% level. This bidirectional Granger causality between the prediction market and VIX is consistent with a dynamic information-sharing process in which both markets contribute to the price discovery of monetary policy expectations. Elevated VIX levels may signal economic stress that increases the probability of rate cuts, thereby affecting prediction market prices; conversely, shifts in rate expectations affect the broader risk environment priced by the VIX.

The Fed ``No Change'' contract also Granger-causes SPY returns at the 10\% level ($F = 2.42$, $p = 0.072$), indicating that prediction market price movements lead equity market returns. This finding aligns with the theoretical prediction that state-contingent claims markets, by directly pricing policy outcomes, may incorporate information faster than broad equity indices that are affected by many factors beyond monetary policy.

Bitcoin returns do not exhibit statistically significant Granger causality with the Fed ``No Change'' contract in either direction (Fed $\to$ BTC: $F = 1.51$, $p = 0.196$; BTC $\to$ Fed: $F = 1.95$, $p = 0.110$), though the point estimates suggest a marginally stronger linkage from BTC to Fed than the reverse. This is consistent with Bitcoin prices being driven primarily by crypto-specific factors (regulatory news, network activity, speculative flows) rather than monetary policy expectations.

Among the traditional assets, BTC Granger-causes VIX ($F = 3.57$, $p = 0.010$), significant at the 1\% level, suggesting that cryptocurrency market dynamics spill over into the equity volatility market. SPY does not Granger-cause BTC ($F = 0.30$, $p = 0.585$), indicating that equity market returns contain little predictive information for cryptocurrency returns.

Figure~\ref{fig:granger_heatmap} presents the Granger causality results in heatmap form, providing a visual summary of the directional information linkages across the four markets.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig7_granger_heatmap.pdf}
  \caption{Granger causality heatmap. Each cell displays the $F$-statistic for the Granger causality test from the row variable to the column variable. Darker shading indicates stronger Granger-causal relationships. The most prominent linkages run from Fed ``No Change'' to VIX and from BTC to VIX.}
  \label{fig:granger_heatmap}
\end{figure}

\subsection{VAR Estimates}\label{subsec:var_results}

Table~\ref{tab:var} reports the coefficient estimates from the four-variable VAR model. The VAR complements the pairwise Granger causality tests by providing a system-level view of the dynamic interactions among prediction market returns and traditional asset returns.

\input{../results/tables/table6_var.tex}

The VAR estimates confirm the patterns identified in the Granger causality analysis. The lagged Fed ``No Change'' return enters the VIX equation with a negative coefficient, consistent with the interpretation that increases in the probability of policy inaction are associated with subsequent declines in implied volatility. The lagged VIX return enters the Fed ``No Change'' equation with a positive coefficient, consistent with the interpretation that increases in market uncertainty are associated with subsequent increases in the probability that the Fed holds rates steady (a ``flight to safety'' in monetary policy expectations).

The intercept terms across all four equations are close to zero, indicating no significant drift in any of the return series over the sample period---a finding consistent with the approximate martingale property expected of efficient market prices.

\subsection{Volatility Dynamics}\label{subsec:vol_results}

\subsubsection{GARCH and GJR-GARCH Estimates}

Table~\ref{tab:garch} reports the parameter estimates for the GARCH(1,1) and GJR-GARCH(1,1) models estimated on the Fed ``No Change'' contract returns.

\input{../results/tables/table7_garch.tex}

The GARCH(1,1) estimates reveal substantial volatility clustering. The ARCH coefficient $\alpha_1 = 0.195$ indicates that yesterday's squared shock has a meaningful impact on today's conditional variance, while the GARCH coefficient $\beta_1 = 0.805$ indicates high persistence. The sum $\alpha_1 + \beta_1 = 1.000$ indicates near-integrated volatility dynamics, implying that shocks to conditional variance decay very slowly. The conditional mean $\mu = 0.150$ is positive but small. The GARCH(1,1) model achieves an AIC of 940.68.

The GJR-GARCH(1,1) model provides a substantially better fit, with an AIC of 894.19---a reduction of 46.49 relative to the standard GARCH. This large improvement in AIC constitutes strong evidence in favor of the asymmetric specification. The leverage coefficient $\gamma_1 = 0.304$ indicates that negative return shocks increase conditional variance by approximately 30 percentage points more than positive shocks of equal magnitude. The symmetric ARCH coefficient falls to $\alpha_1 \approx 0.000$ in the GJR specification, indicating that virtually all of the ARCH effect is driven by the asymmetric component. The GARCH coefficient rises to $\beta_1 = 0.848$, maintaining high persistence.

The presence of leverage effects in prediction market returns is a novel empirical finding. In equity markets, leverage effects are commonly attributed to the financial leverage channel (declining stock prices increase leverage ratios, which increases equity risk) or to the volatility feedback channel (increases in expected volatility reduce expected returns, causing a contemporaneous price decline). For prediction markets, neither channel applies directly, as prediction market contracts do not represent claims on levered firms. An alternative interpretation is that the leverage effect in prediction markets reflects an asymmetric information channel: negative return shocks (which correspond to declines in the probability of the modal outcome) are associated with greater uncertainty about the true state of the world, which increases the intensity of trading and the magnitude of subsequent price fluctuations. This interpretation is consistent with the theoretical models of \citet{kyle1985continuous} and \citet{glosten1985bid}, in which increases in information asymmetry lead to wider spreads and greater price impact.

Figure~\ref{fig:volatility} plots the conditional volatility series implied by both models, illustrating the time-varying nature of prediction market volatility and the differences between the symmetric and asymmetric specifications.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig9_volatility.pdf}
  \caption{Conditional volatility of the Fed ``No Change'' contract returns estimated by the GARCH(1,1) and GJR-GARCH(1,1) models. The GJR-GARCH model captures asymmetric responses to positive and negative shocks, producing higher volatility estimates following negative return shocks.}
  \label{fig:volatility}
\end{figure}

\subsubsection{HAR-RV Estimates}

Table~\ref{tab:har_rv} reports the HAR-RV model estimates for the Fed ``No Change'' contract. The model achieves an $R^2$ of 0.106 and an adjusted $R^2$ of 0.083.

\input{../results/tables/table8_har_rv.tex}

The daily realized volatility component has a coefficient of $\beta_d = 0.008$ ($t = 0.07$), which is statistically insignificant and economically small. The weekly component has a larger and marginally more significant coefficient of $\beta_w = 0.143$ ($t = 1.33$), while the monthly component contributes $\beta_m = 0.030$ ($t = 0.80$). None of the individual coefficients reach conventional significance levels, though the model as a whole provides some predictive content, as reflected in the positive $R^2$.

The relatively modest $R^2$ of 10.6\% contrasts with the typical performance of HAR-RV models in equity and foreign exchange markets, where $R^2$ values in the range of 30--60\% are common \citep{corsi2009simple,andersen2003modeling}. Several factors may account for this difference. First, our use of daily absolute returns as a proxy for realized volatility introduces substantial measurement noise relative to the high-frequency realized volatility estimators used in the original HAR-RV literature. Second, prediction market prices are driven by discrete information events (FOMC announcements, data releases) that generate discontinuous price jumps, whereas the HAR-RV model is designed to capture the smooth, persistent volatility dynamics characteristic of continuous-time diffusion processes. Third, the bounded nature of prediction market prices (between 0 and 1) mechanically compresses realized volatility as prices approach their boundaries, violating the implicit stationarity assumption of the HAR-RV model.

Despite these limitations, the positive $R^2$ indicates that the heterogeneous structure of realized volatility---the cascade from daily to weekly to monthly horizons---provides some useful information for forecasting prediction market volatility. The finding that the weekly component has the largest coefficient is consistent with the weekly rhythm of macroeconomic data releases (employment, inflation, retail sales) that drive monetary policy expectations.

\subsection{Liquidity Dynamics}\label{subsec:liquidity_results}

Figure~\ref{fig:liquidity} presents the time series of liquidity measures---the Roll effective spread and the Amihud illiquidity ratio---for the Fed rate decision contracts.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig10_liquidity.pdf}
  \caption{Liquidity dynamics for the Federal Reserve rate decision market. The upper panel displays the Roll effective spread, and the lower panel displays the Amihud illiquidity ratio. Both measures indicate tight liquidity conditions for the ``No change'' contract, with periodic spikes around macroeconomic announcement dates.}
  \label{fig:liquidity}
\end{figure}

The liquidity measures reveal two patterns. First, the ``No change'' contract, which is the most actively traded of the five contracts, exhibits relatively tight spreads and low price impact, consistent with the theoretical prediction that liquidity concentrates in contracts where information asymmetry is lowest \citep{glosten1985bid}. The modal outcome attracts the most diverse set of traders (informed and uninformed), leading to competitive liquidity provision and narrow spreads.

Second, both liquidity measures exhibit time variation, with periodic spikes around macroeconomic announcement dates. The widening of spreads and the increase in price impact around FOMC and CPI releases are consistent with the \citet{kyle1985continuous} model, in which increases in information asymmetry around news events lead market makers to widen their quotes to compensate for adverse selection risk. After the announcement, liquidity recovers as the information asymmetry is resolved and the market reaches a new equilibrium.

These microstructure findings complement the completeness and information aggregation results. The observation that prediction market liquidity is sufficient to support tight spreads and low price impact for high-profile contracts suggests that the market's information aggregation function is not unduly impaired by microstructure frictions. At the same time, the periodic deterioration of liquidity around information events highlights the ongoing tension between information aggregation and liquidity provision that characterizes all financial markets \citep{grossman1980impossibility}.