spb/wp8_uqo

1% ============================================================================2% Section 5: Methodology3% ============================================================================4\section{Methodology}\label{sec:methodology}56\subsection{Tests of the pricing restriction}\label{subsec:completeness_method}78For each day of the main sample we compute $\Sigma_t$ from equation~\eqref{eq:sp_sum} and the deviation $\Sigma_t-1$. The mean deviation is tested against zero with a heteroskedasticity- and autocorrelation-consistent standard error \citep{newey1987simple} with five lags, since $\Sigma_t-1$ is serially correlated. We report the share of days on which $|\Sigma_t-1|$ is within 0.5, 1, 2 and 5 percentage points, and the share of days on which $\Sigma_t$ lies within the executable bounds \eqref{eq:bounds}, with the lower bound computed as9\begin{equation}\label{eq:lower}10  \delta_t = \exp\!\big(-r_t\,\tau_t\big),11\end{equation}12where $r_t$ is the effective federal funds rate on day $t$ and $\tau_t$ the number of days to the resolution date divided by 365, and with $c_t=0.005$. The dynamics of the deviation are characterised by the first-order autoregressive coefficient $\phi$ of $\Sigma_t-1$, the implied half-life $-\ln 2/\ln\phi$, and the augmented Dickey--Fuller statistic with a constant and AIC-selected lag length. Because the lower bound rises towards one as $\tau_t\to 0$, we also regress $\Sigma_t-1$ on $\tau_t$ with HAC standard errors; a positive slope indicates that the overround shrinks as resolution approaches. The same statistics are computed for the election group with $T$ equal to 5 November 2024.1314The order-book snapshot provides a direct check of the bounds at one point in time. For each contract we record the best bid and ask, the quoted and relative spread, the depth at the best quotes, and the notional resting within one cent of the best quote; the sum of best bids is the proceeds from selling one complete set at market and the sum of best asks the cost of buying one.1516\subsection{Announcement-day analysis}\label{subsec:event_method}1718Let $y_t$ denote either the daily change in the risk-neutral expected policy change, $\Delta\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ in basis points, or the daily change in the ``No change'' state price in percentage points. Under the risk-neutral martingale property both have conditional mean close to zero, so that no estimation window is needed to define an abnormal change; the information content of a release is identified by the variance of $y_t$ rather than by its mean, as in \citet{ederington1993markets}. For each release type we report the change on day 0 and the cumulative changes over $[-1,+1]$ and $[-3,+3]$ calendar days for every event in the sample, and we compare the distribution of $|y_t|$ on the $K$ release days with its distribution on the remaining days using three statistics: the ratio of means, the ratio of variances with the Brown--Forsythe test of equal variances (the Levene test centred at the median, robust to the heavy tails documented in Table~\ref{tab:summary_stats}), and the one-sided Mann--Whitney test that $|y_t|$ is stochastically larger on release days. With $K$ between 4 and 13 these tests have limited power, which is why we report them for all releases pooled as well as by type. The same comparison between weekend and weekday observations among non-release days tests whether the market incorporates information when conventional markets are closed \citep{french1986stock}.1920The earlier version of this paper reported cumulative abnormal returns for each contract with a constant-mean benchmark \citep{mackinlay1997event} and a cross-sectional $t$-test over two or three events. We abandon that design for three reasons: simple returns on claims priced at a fraction of a cent are dominated by tick effects, the cross-sectional $t$-statistic with two events has one degree of freedom, and the sixty-day estimation window excluded two of the four FOMC meetings and two of the five CPI releases from the sample.2122\subsection{Lead--lag tests}\label{subsec:granger_method}2324Lead--lag relations are tested on the U.S.\ trading-day grid with five variables: the daily change in $\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ (basis points), the change in the ten-year Treasury yield (basis points) and the log changes in SPY, VIX and BTC (percent). For a pair $(x_t,y_t)$ the bivariate vector autoregression of order $p$,25\begin{equation}\label{eq:granger_unrestricted}26  y_t = \alpha + \sum_{\ell=1}^{p} \beta_\ell\, y_{t-\ell} + \sum_{\ell=1}^{p} \gamma_\ell\, x_{t-\ell} + \varepsilon_t,27\end{equation}28is estimated by least squares and the hypothesis $\gamma_1=\cdots=\gamma_p=0$ is tested with the conventional $F$-statistic and with a Wald statistic using the Newey--West covariance with five lags. The lag order is chosen once, by the Akaike information criterion \citep{akaike1974new} applied to the five-variable system over $p\in\{0,\ldots,5\}$; because the criterion selects $p=0$, as do the Schwarz \citep{schwarz1978estimating} and Hannan--Quinn criteria, the main tests use $p=1$ and we report $p$-values for $p\in\{1,2,5\}$ as a sensitivity check. We deliberately do not select, for each pair, the lag that minimises the $p$-value: that procedure, used in the earlier version, is a search over five correlated tests and its nominal size is far below its actual size. The full five-variable VAR(1) is also reported equation by equation with HAC $t$-statistics \citep{lutkepohl2005new}.2930Two limitations of the design are inherent in the data. Daily frequency is coarse relative to the speed at which both markets react to scheduled releases, so that a lead of minutes or hours, if it existed, would not be detectable; and the sample contains 110 trading days, so that only large predictive relations could be found. The tests are therefore informative about the absence of daily-horizon predictability, not about intraday price discovery.3132\subsection{Spread estimation}\label{subsec:liquidity_method}3334The implicit effective spread of \citet{roll1984simple} is estimated from the first-order serial covariance of daily price changes over a rolling twenty-day window, $\widehat{S}_t=2\sqrt{-\widehat{\mathrm{Cov}}(\Delta p_t,\Delta p_{t-1})}$ when the covariance is negative and zero otherwise. With one price per day the estimator captures the bid--ask bounce only partially and is reported as a descriptive complement to the quoted spreads of the order-book snapshot. The Amihud illiquidity ratio \citep{amihud2002illiquidity} cannot be computed because the price history does not include daily volume; the earlier version of this paper referred to it in error.3536\subsection{Volatility models}\label{subsec:vol_method}3738The conditional volatility of daily log returns on the ``No change'' claim is examined in Appendix~\ref{app:volatility} with GARCH and GJR-GARCH models \citep{bollerslev1986generalized,glosten1993relation} under Gaussian and Student-$t$ innovations, estimated by quasi-maximum likelihood with robust standard errors, and with the heterogeneous autoregressive regression of \citet{corsi2009simple} using absolute and squared daily returns as volatility proxies. Engle's ARCH-LM test \citep{engle1982autoregressive} and Ljung--Box statistics on squared returns are reported alongside, because the presence of conditional heteroskedasticity is itself in question in a sample of 145 observations dominated by a handful of announcement days.39