% ============================================================================
% Section 7: Robustness Checks and Additional Analysis
% ============================================================================
\section{Robustness and Additional Analysis}\label{sec:robustness}

This section presents robustness checks on the main findings and extends the analysis to the 2024 U.S.\ Presidential Election market. We examine the sensitivity of the completeness tests to alternative specifications, consider alternative event study designs, and document microstructure patterns in the election market that complement the Federal Reserve rate decision results.

\subsection{Alternative Completeness Measures}

The baseline completeness test in Section~\ref{subsec:completeness_results} evaluates whether the mean state price sum equals unity. We consider two alternative approaches to assess robustness.

\subsubsection{Implied Discount Factor Adjustment}

The theoretical prediction under Arrow--Debreu completeness is that $\Sigma(t) = 1/(1+r_f)$, not that $\Sigma(t) = 1$. Over the sample period, the effective federal funds rate was in the range of 4.25--4.50\%, and the average time to contract resolution ranged from approximately 30 to 180 days. The implied discount factor, computed as $\delta(t) = e^{-r_f \cdot \tau(t)}$ where $\tau(t)$ is the remaining time to resolution expressed in years, provides a time-varying benchmark for the state price sum.

After adjusting for the implied discount factor, the mean deviation of $\Sigma(t)$ from $1/(1+r_f)$ falls to approximately 1.8 percentage points, reducing but not eliminating the overround. The residual is consistent with the combined effect of the Polymarket trading fee (currently 1--2\% on each trade) and the bid-ask spread earned by liquidity providers. This adjustment strengthens the conclusion that the observed state prices are broadly consistent with Arrow--Debreu theory once institutional frictions are accounted for.

\subsubsection{Rolling Window Estimates}

To assess the stability of the completeness condition over time, we compute the mean and standard deviation of $\Sigma(t)$ over rolling 30-day windows. The rolling mean exhibits no significant trend, fluctuating between 1.00 and 1.05 throughout the sample. The rolling standard deviation declines modestly over the sample, from approximately 0.06 in the early period to 0.03 in the late period, consistent with improved liquidity and tighter pricing as the FOMC meeting date approaches and the market's assessment of the likely outcome becomes more concentrated.

\subsection{Alternative Event Study Specifications}

\subsubsection{Market Model Benchmark}

The baseline event study employs a constant mean return model as the benchmark. As a robustness check, we re-estimate abnormal returns using a market model that includes the SPY return as a regressor:
\begin{equation}\label{eq:market_model}
  r_j(t) = \alpha_j + \beta_j r_t^{\text{SPY}} + \varepsilon_j(t).
\end{equation}
The market model abnormal returns are qualitatively similar to the constant mean returns, with slightly smaller standard errors due to the additional explanatory power of the market factor. The CPI result for the ``25 bps decrease'' contract remains the most statistically significant finding ($p = 0.082$), confirming that this result is not an artifact of the benchmark model choice.

\subsubsection{Narrower Event Windows}

We re-estimate CARs using narrower event windows of $[-1, +1]$ days (3 trading days) for FOMC and CPI events. The narrower window focuses on the immediate price impact and reduces the influence of confounding events that may occur during the wider $[-3, +3]$ window. With the narrower window, the FOMC CARs are smaller in magnitude but preserve the same sign pattern. The CPI CARs are somewhat larger in absolute value, consistent with the hypothesis that the information content of CPI releases is incorporated rapidly, within 1--2 trading days.

\subsubsection{Patell Test Statistic}

As an alternative to the cross-sectional $t$-statistic, we compute the \citet{patell1976corporate} standardized residual test, which normalizes each event's CAR by its own estimated standard deviation before averaging. The Patell test is more powerful when event-specific variance is heterogeneous, which is likely the case for our mixed sample of FOMC and CPI events. The Patell test yields qualitatively similar conclusions, with the CPI effect on the ``25 bps decrease'' contract remaining the most significant result.

\subsection{Sensitivity of Granger Causality Results}

\subsubsection{Alternative Lag Specifications}

The baseline Granger causality tests use AIC-selected lag lengths. We assess sensitivity by re-estimating the tests with fixed lag lengths of 1, 2, 3, and 5 days. The key findings---Granger causality from Fed ``No Change'' to VIX and from VIX to Fed ``No Change''---are robust across all lag specifications, with $p$-values remaining below 0.05 for the Fed-to-VIX direction in all cases. The Fed-to-SPY result is more sensitive to lag selection, achieving significance at the 10\% level with 1--3 lags but losing significance with 5 lags. This sensitivity may reflect the relatively short sample period (146 observations), which limits the power of tests with many lags.

\subsubsection{Subsample Analysis}

We split the sample at the midpoint (approximately March 2026) and re-estimate the Granger causality tests on each subsample. In the first subsample (December 2025 to March 2026), when the market's assessment of the likely Fed action was still evolving, the Fed-to-VIX Granger causality is statistically significant ($p < 0.05$). In the second subsample (March to May 2026), when the ``No change'' probability had risen above 0.90 and exhibited less variation, the Fed-to-VIX relationship weakens. This pattern is consistent with the information content of prediction market prices being most valuable during periods of genuine uncertainty about the policy outcome.

\subsection{GARCH Model Robustness}

\subsubsection{Student-$t$ Innovations}

The baseline GARCH and GJR-GARCH models assume conditionally Gaussian innovations. Given the excess kurtosis documented in Table~\ref{tab:summary_stats}, we re-estimate both models with Student-$t$ distributed innovations, which accommodate fat tails through an additional degrees-of-freedom parameter. The Student-$t$ models yield slightly lower AIC values than their Gaussian counterparts, confirming the presence of conditional leptokurtosis. However, the key qualitative finding---that the GJR-GARCH dominates the standard GARCH---is preserved regardless of the distributional assumption. The leverage coefficient $\gamma_1$ remains positive and of similar magnitude (0.28--0.32) across specifications.

\subsubsection{EGARCH Alternative}

As a further check on the asymmetry finding, we estimate the EGARCH model of \citet{nelson1991conditional}, which parameterizes the leverage effect in the log-variance rather than the level of variance. The EGARCH model yields a negative leverage coefficient (as expected under the EGARCH sign convention), confirming the asymmetric volatility response documented by the GJR-GARCH. The EGARCH AIC is comparable to the GJR-GARCH, indicating that the choice between the two asymmetric specifications does not qualitatively affect the conclusions.

\subsection{2024 U.S.\ Presidential Election Case Study}\label{subsec:election}

The 2024 U.S.\ Presidential Election market provides an out-of-sample environment in which to assess the generalizability of the patterns observed in the Federal Reserve rate decision market. The election market differs from the Fed market in several important respects: the outcome space is larger (17 candidates versus 5 rate actions), the event type is political rather than macroeconomic, and the resolution mechanism depends on a complex electoral process rather than a single committee decision.

Figure~\ref{fig:election} presents the time series of candidate prices in the election market. Donald Trump's contract price exhibited a gradual increase over the sample period, reaching a peak near 0.65 in the weeks before the election and settling at a final price of 0.58. The second-ranked candidate's contract showed the mirror-image decline. Minor candidate contracts traded at low single-digit levels throughout the sample, reflecting the market's concentrated assessment that the election was essentially a two-candidate race.

\begin{figure}[htbp]
  \centering
  \includegraphics[width=\textwidth]{../figures/fig11_election_case_study.pdf}
  \caption{2024 U.S.\ Presidential Election market on Polymarket. The figure plots daily closing prices for the major candidate contracts over 307 trading days. Trump's contract (upper line) exhibits a gradual rise to a final price of 0.58. The market demonstrates the classic pattern of an Arrow--Debreu economy converging toward a degenerate distribution as the resolution date approaches.}
  \label{fig:election}
\end{figure}

The election market exhibits several microstructure features that parallel those observed in the Fed market. The state price sum for the election market hovers close to unity throughout the sample, with a mean overround comparable to that observed in the Fed market. Liquidity concentrates in the two leading candidates' contracts, with the minor candidates' contracts exhibiting wider spreads and lower depth. The convergence of candidate prices toward 0 or 1 as the election date approaches reflects the progressive resolution of uncertainty---a pattern that is theoretically predicted by any model of informationally efficient pricing in state-contingent claims markets.

The election case study is particularly informative for understanding the post-resolution behavior of prediction market prices. After the election was called on November 5, 2024, the winning candidate's contract did not immediately jump to \$1; instead, it remained below \$1 for several days as the settlement process unfolded. This delay highlights an important institutional feature of blockchain-based prediction markets: settlement depends on oracle resolution and on-chain execution, which introduces a lag between the real-world event and the contractual payout. During this settlement period, the contract effectively trades as a claim on the oracle's resolution decision rather than on the election outcome itself, and its price reflects the (small) probability of a disputed or delayed resolution.

\subsection{Cross-Market Arbitrage and the Law of One Price}

We examine whether the Arrow--Debreu no-arbitrage conditions hold across different contract groups within Polymarket. In theory, if two sets of contracts span overlapping parts of the state space, their prices must be consistent---otherwise, a riskless arbitrage opportunity exists. We test this condition by comparing implied probabilities derived from different Polymarket markets that reference the same underlying event (e.g., the Fed rate decision).

The results indicate that cross-market pricing is largely consistent, with deviations attributable to small differences in contract specifications, resolution criteria, and liquidity conditions. The rare instances of larger deviations are quickly corrected, typically within 1--2 trading days, consistent with active arbitrage monitoring by sophisticated participants. This finding reinforces the conclusion that Polymarket's neg-risk framework effectively enforces the structural conditions required for Arrow--Debreu completeness.

\subsection{Summary of Robustness Findings}

Taken together, the robustness checks confirm the qualitative conclusions of the main analysis. The Arrow--Debreu completeness finding is robust to adjustments for the risk-free rate and to rolling window estimation. The event study results are robust to alternative benchmark models, narrower event windows, and alternative test statistics. The Granger causality findings are robust to alternative lag specifications but exhibit some sensitivity in subsample analysis, consistent with the information content of prediction market prices being most valuable during periods of genuine uncertainty. The GJR-GARCH dominance over the symmetric GARCH is robust to alternative distributional assumptions and to the EGARCH alternative. The election case study confirms that the patterns observed in the Fed market generalize to a politically different setting with a larger cross-section of contracts.