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% Section 5: Methodology
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\section{Methodology}\label{sec:methodology}

This section presents the econometric methods employed to test the hypotheses developed in Section~\ref{sec:theoretical}. We describe five methodological components: (i) Arrow--Debreu completeness tests, (ii) event study methodology, (iii) Granger causality and vector autoregression, (iv) GARCH-family volatility models, and (v) the HAR-RV model for realized volatility. For each method, we specify the estimating equations, identification strategy, and inference procedures.

\subsection{Arrow--Debreu Completeness Tests}\label{subsec:completeness_method}

To test Hypothesis~\ref{hyp:completeness}, we construct the daily state price sum $\Sigma(t) = \sum_{j=1}^{5} p_j(t)$ for the Federal Reserve rate decision market and evaluate its properties against the theoretical predictions of the Arrow--Debreu framework.

\subsubsection{Mean Deviation Test}

The primary test statistic is the time-series average of $\Sigma(t)$:
\begin{equation}\label{eq:mean_test}
  \bar{\Sigma} = \frac{1}{T} \sum_{t=1}^{T} \Sigma(t),
\end{equation}
where $T = 146$ is the number of trading days. Under the null hypothesis that the market is Arrow--Debreu complete with a risk-free rate close to zero, $\bar{\Sigma} = 1$. We test $H_0: \mathbb{E}[\Sigma(t)] = 1$ against $H_1: \mathbb{E}[\Sigma(t)] \neq 1$ using a $t$-test with Newey--West standard errors \citep{newey1987simple} to account for potential serial correlation in $\Sigma(t)$.

\subsubsection{Positivity Test}

Hypothesis~\ref{hyp:completeness}(b) requires that all state prices remain strictly positive. We verify this condition by checking whether $\min_{j,t} p_j(t) > 0$ across all five contracts and all trading days.

\subsubsection{Stationarity of Deviations}

To assess whether deviations of $\Sigma(t)$ from unity are temporary (consistent with transaction-cost frictions) or persistent (suggesting structural mispricing), we apply the Augmented Dickey--Fuller test to the series $\delta(t) = \Sigma(t) - 1$, testing the null of a unit root against the alternative of stationarity. Rejection of the unit root null would indicate that arbitrage forces push the state price sum back toward unity, consistent with Hypothesis~\ref{hyp:no_arb}(b).

\subsection{Event Study Methodology}\label{subsec:event_method}

We employ the standard event study methodology of \citet{mackinlay1997event} to assess the impact of macroeconomic announcements on prediction market prices. The procedure comprises four steps.

\subsubsection{Step 1: Estimation Window}

For each event $k$ with announcement date $\tau_k$, we define the estimation window as the 60 trading days ending 4 days before the event: $[\tau_k - 63, \tau_k - 4]$. The constant mean return model is estimated over this window:
\begin{equation}\label{eq:est_window}
  \hat{\mu}_j^{(k)} = \frac{1}{60} \sum_{t=\tau_k - 63}^{\tau_k - 4} r_j(t).
\end{equation}

We use a 60-day estimation window as a compromise between statistical precision (which favors longer windows) and parameter stability (which favors shorter windows). The 3-day gap between the estimation window and the event window prevents contamination of the benchmark model by pre-announcement information leakage.

\subsubsection{Step 2: Abnormal Returns}

Abnormal returns for contract $j$ around event $k$ are computed as
\begin{equation}\label{eq:ar_method}
  AR_j^{(k)}(t) = r_j(t) - \hat{\mu}_j^{(k)}, \quad t \in [\tau_k - w, \tau_k + w],
\end{equation}
where $w$ is the half-width of the event window (3 days for FOMC and CPI events, 5 days for election events).

\subsubsection{Step 3: Cumulative Abnormal Returns}

Cumulative abnormal returns over the event window are
\begin{equation}\label{eq:car_method}
  CAR_j^{(k)} = \sum_{t=\tau_k - w}^{\tau_k + w} AR_j^{(k)}(t).
\end{equation}

\subsubsection{Step 4: Cross-Sectional Testing}

The null hypothesis is that the mean CAR across events is zero: $H_0: \mathbb{E}[\overline{CAR}_j] = 0$. The cross-sectional test statistic is
\begin{equation}\label{eq:cross_sec}
  t_j = \frac{\overline{CAR}_j}{s_j / \sqrt{K}},
\end{equation}
where $\overline{CAR}_j = K^{-1} \sum_{k=1}^{K} CAR_j^{(k)}$, $s_j$ is the cross-event standard deviation of CARs, and $K$ is the number of events. Under the null hypothesis and standard regularity conditions, $t_j$ follows a $t$-distribution with $K - 1$ degrees of freedom \citep{boehmer1991event}.

The event study methodology relies on two identifying assumptions. First, the constant mean model must be a reasonable benchmark for expected returns in the absence of the event. For prediction market contracts, which have no systematic risk exposure in the CAPM sense, the constant mean model is more appropriate than market model or factor model alternatives. Second, events must be exogenous to the prediction market---that is, the FOMC and CPI announcement dates must not be determined by prediction market prices. This assumption is clearly satisfied, as these dates are set on predetermined schedules.

\subsection{Granger Causality and Vector Autoregression}\label{subsec:granger_method}

To assess information linkages between prediction markets and traditional financial markets (Hypothesis~\ref{hyp:info}(c)), we employ pairwise Granger causality tests \citep{granger1969investigating} and estimate a vector autoregressive (VAR) model \citep{sims1980macroeconomics}.

\subsubsection{Pairwise Granger Causality}

Consider two stationary time series $x_t$ and $y_t$. We say that $x$ Granger-causes $y$ if lagged values of $x$ contain information useful for predicting $y$ beyond the information contained in lagged values of $y$ alone. Formally, we estimate the unrestricted model
\begin{equation}\label{eq:granger_unrestricted}
  y_t = \alpha + \sum_{\ell=1}^{L} \beta_\ell y_{t-\ell} + \sum_{\ell=1}^{L} \gamma_\ell x_{t-\ell} + \varepsilon_t,
\end{equation}
and the restricted model
\begin{equation}\label{eq:granger_restricted}
  y_t = \alpha + \sum_{\ell=1}^{L} \beta_\ell y_{t-\ell} + u_t,
\end{equation}
where $L$ is the lag length selected by the Akaike Information Criterion \citep{akaike1974new}. The null hypothesis $H_0: \gamma_1 = \gamma_2 = \cdots = \gamma_L = 0$ is tested using an $F$-test. Rejection of $H_0$ implies that $x$ Granger-causes $y$.

We conduct pairwise Granger causality tests among four return series: the Fed ``No Change'' contract, SPY, BTC, and VIX. This yields 12 directional tests (four variables, each pair in both directions), reported in Table~\ref{tab:granger}.

\subsubsection{Vector Autoregression}

To capture the joint dynamics of prediction market returns and traditional asset returns, we estimate a VAR model of order $L$:
\begin{equation}\label{eq:var}
  \mathbf{y}_t = \boldsymbol{\mu} + \sum_{\ell=1}^{L} \mathbf{A}_\ell \mathbf{y}_{t-\ell} + \boldsymbol{\varepsilon}_t,
\end{equation}
where $\mathbf{y}_t = (r_t^{\text{Fed}}, r_t^{\text{SPY}}, r_t^{\text{BTC}}, r_t^{\text{VIX}})'$ is the $4 \times 1$ vector of returns, $\boldsymbol{\mu}$ is a $4 \times 1$ vector of intercepts, $\mathbf{A}_\ell$ are $4 \times 4$ coefficient matrices, and $\boldsymbol{\varepsilon}_t$ is a $4 \times 1$ vector of innovations with $\mathbb{E}[\boldsymbol{\varepsilon}_t] = \mathbf{0}$ and $\mathbb{E}[\boldsymbol{\varepsilon}_t \boldsymbol{\varepsilon}_t'] = \boldsymbol{\Omega}$. The lag length $L$ is selected by minimizing the AIC over candidate lag orders $L \in \{1, 2, \ldots, 8\}$.

The VAR model allows us to trace the dynamic propagation of shocks across markets through impulse response functions and to decompose forecast error variance into contributions from each variable through variance decomposition analysis. These tools complement the pairwise Granger causality tests by providing a system-level perspective on information transmission.

\subsection{GARCH-Family Volatility Models}\label{subsec:garch_method}

To characterize the conditional volatility dynamics of prediction market returns (Hypothesis~\ref{hyp:vol}), we estimate two members of the GARCH family: the standard GARCH(1,1) model and the GJR-GARCH(1,1) model.

\subsubsection{GARCH(1,1)}

The GARCH(1,1) model of \citet{bollerslev1986generalized} specifies the conditional variance of returns as
\begin{align}
  r_t &= \mu + \varepsilon_t, \quad \varepsilon_t = \sigma_t z_t, \quad z_t \sim \mathcal{N}(0,1), \label{eq:garch_mean} \\
  \sigma_t^2 &= \omega + \alpha_1 \varepsilon_{t-1}^2 + \beta_1 \sigma_{t-1}^2, \label{eq:garch_var}
\end{align}
where $\omega > 0$, $\alpha_1 \geq 0$, $\beta_1 \geq 0$, and the stationarity condition $\alpha_1 + \beta_1 < 1$ ensures that the unconditional variance is finite. The parameter $\alpha_1$ measures the impact of yesterday's squared shock on today's conditional variance (the ARCH effect), while $\beta_1$ measures the persistence of past conditional variance (the GARCH effect). The sum $\alpha_1 + \beta_1$ measures the overall persistence of volatility shocks: values close to unity indicate near-integrated volatility.

\subsubsection{GJR-GARCH(1,1)}

The GJR-GARCH(1,1) model of \citet{glosten1993relation} augments the GARCH variance equation with an asymmetric leverage term:
\begin{equation}\label{eq:gjr_var}
  \sigma_t^2 = \omega + \alpha_1 \varepsilon_{t-1}^2 + \gamma_1 \varepsilon_{t-1}^2 \mathbf{1}\{\varepsilon_{t-1} < 0\} + \beta_1 \sigma_{t-1}^2,
\end{equation}
where $\gamma_1 \geq 0$ is the leverage coefficient. When $\gamma_1 > 0$, negative shocks ($\varepsilon_{t-1} < 0$) increase the conditional variance by $\alpha_1 + \gamma_1$, while positive shocks increase it by only $\alpha_1$. The standard GARCH model is nested as the special case $\gamma_1 = 0$.

Model comparison between the GARCH(1,1) and GJR-GARCH(1,1) is based on the Akaike Information Criterion:
\begin{equation}\label{eq:aic}
  \text{AIC} = -2 \ln \hat{\mathcal{L}} + 2k,
\end{equation}
where $\hat{\mathcal{L}}$ is the maximized log-likelihood and $k$ is the number of estimated parameters. The model with the lower AIC is preferred. Since the GJR-GARCH has one additional parameter ($\gamma_1$) relative to the standard GARCH, a reduction in AIC of more than 2 constitutes evidence in favor of the asymmetric specification.

Both models are estimated by maximum likelihood under the assumption of conditionally Gaussian innovations. To ensure robustness, we also compute quasi-maximum likelihood (QML) standard errors \citep{white1980heteroskedasticity}, which remain consistent even if the true conditional distribution departs from Gaussianity.

\subsection{HAR-RV Model}\label{subsec:har_method}

To model the dynamics of realized volatility and test Hypothesis~\ref{hyp:vol}(c), we employ the Heterogeneous Autoregressive model of Realized Volatility (HAR-RV) proposed by \citet{corsi2009simple}. The HAR-RV captures long-memory properties of realized volatility through a cascade of volatility components at daily, weekly, and monthly horizons.

\subsubsection{Realized Volatility Construction}

In the absence of intraday data, we proxy daily realized volatility by the absolute daily return on the Fed ``No Change'' contract:
\begin{equation}\label{eq:rv_daily}
  RV_t^{(d)} = |r_t|.
\end{equation}

Weekly and monthly realized volatility components are constructed as arithmetic averages of daily realized volatility:
\begin{align}
  RV_t^{(w)} &= \frac{1}{5} \sum_{i=0}^{4} RV_{t-i}^{(d)}, \label{eq:rv_weekly} \\
  RV_t^{(m)} &= \frac{1}{22} \sum_{i=0}^{21} RV_{t-i}^{(d)}. \label{eq:rv_monthly}
\end{align}

This construction assumes that absolute returns are a reasonable proxy for integrated volatility at the daily frequency. While the theoretical foundations of the HAR-RV model are based on high-frequency realized volatility estimators \citep{andersen2001distribution,barndorff2002econometric}, the model has been shown to perform well with daily-frequency inputs when intraday data are unavailable.

\subsubsection{HAR-RV Specification}

The HAR-RV model regresses future daily realized volatility on the three temporal components:
\begin{equation}\label{eq:har_rv}
  RV_{t+1}^{(d)} = c + \beta_d RV_t^{(d)} + \beta_w RV_t^{(w)} + \beta_m RV_t^{(m)} + \varepsilon_{t+1}.
\end{equation}

The coefficient $\beta_d$ captures the impact of short-run (daily) volatility fluctuations on next-period volatility, $\beta_w$ captures the medium-run (weekly) contribution, and $\beta_m$ captures the long-run (monthly) contribution. In traditional equity markets, all three coefficients are typically positive and statistically significant, with $\beta_w$ and $\beta_m$ providing substantial incremental explanatory power---a pattern attributed to the heterogeneous investment horizons of different market participant types \citep{corsi2009simple}.

The model is estimated by OLS with Newey--West standard errors \citep{newey1987simple} using 5 lags to account for serial correlation in the residuals, which arises from the overlapping nature of the weekly and monthly realized volatility components. Goodness-of-fit is assessed by $R^2$ and adjusted $R^2$.

\subsection{Liquidity Analysis}\label{subsec:liquidity_method}

We complement the time-series analyses with an assessment of market microstructure quality using two established liquidity measures.

\subsubsection{Roll Effective Spread}

The Roll implicit spread \citep{roll1984simple} is derived from the first-order autocovariance of returns under the assumption that observed price changes reflect the sum of an efficient price innovation and bid-ask bounce:
\begin{equation}\label{eq:roll}
  \text{Roll} = 2\sqrt{-\text{Cov}(r_t, r_{t-1})},
\end{equation}
when $\text{Cov}(r_t, r_{t-1}) < 0$. Positive autocovariance, which can arise from momentum effects, yields an undefined Roll spread; in such cases, we set the Roll spread to zero. The Roll spread is interpreted as an estimate of the effective bid-ask spread and serves as a measure of trading costs.

\subsubsection{Amihud Illiquidity Ratio}

The Amihud illiquidity ratio \citep{amihud2002illiquidity} measures the price impact of trading:
\begin{equation}\label{eq:amihud}
  \text{ILLIQ}_t = \frac{|r_t|}{V_t},
\end{equation}
where $V_t$ is the dollar volume on day $t$. Higher values of ILLIQ indicate that a given volume of trading moves prices by a larger amount, implying lower liquidity. The Amihud ratio is computed on a daily basis and then averaged over rolling windows to produce a time series of liquidity conditions.

\subsection{Inference and Standard Errors}

Throughout the analysis, we employ robust inference procedures. For OLS regressions (HAR-RV), we report Newey--West standard errors with lag truncation determined by the sample size \citep{newey1987simple}. For GARCH models, quasi-maximum likelihood standard errors are used. For event studies, cross-sectional standard errors account for cross-event variation. All hypothesis tests are two-sided unless otherwise noted.

For the Granger causality tests, we report both $F$-statistics and $p$-values, with significance assessed at the 1\%, 5\%, and 10\% levels. Lag lengths are selected by the Akaike Information Criterion to balance model fit and parsimony. For the VAR model, we verify the stability condition (all eigenvalues of the companion matrix lie inside the unit circle) to ensure that impulse response functions and variance decompositions are well defined.