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1% ============================================================================2% Section 6: Results3% ============================================================================4\section{Results}\label{sec:results}56\subsection{The price of the complete set}\label{subsec:completeness_results}78Table~\ref{tab:completeness} reports the tests of the pricing restriction for the Fed group (first column) and, for comparison, the election group discussed in Section~\ref{sec:robustness}. Over the 146 days of the main sample the sum of the five state prices averages 1.0167 with a standard deviation of 0.0261; the median is 1.0093, the minimum 0.9605 and the maximum 1.108. The mean overround of 1.67 percentage points has a HAC standard error of 0.44 and is therefore significantly different from zero ($t=3.80$). It is, however, small relative to the quoted spreads of the first weeks and it is concentrated early in the sample: the mean of $\Sigma_t$ is 1.051 (standard deviation 0.029) over the first thirty days and 1.0040 (standard deviation 0.0036) over the last thirty. On 84 percent of days $\Sigma_t$ exceeds one; on 44 percent of days it is within one percentage point of one, on 65 percent within two and on 87 percent within five.910\input{../results/tables/table3_completeness.tex}1112Panel~C evaluates the executable bounds of equation~\eqref{eq:bounds}. The lower bound $e^{-r_t\tau_t}$ averages 0.989 over the sample and reaches 0.981 at the start, so that $\Sigma_t$ exceeds the buy-and-hold bound by 2.8 percentage points on average and is never below it. Because the overround is positive, the binding bound is the upper one: $\Sigma_t$ lies within $[\delta_t-0.005,\,1.005]$ on 29 percent of days and above 1.005 on 65 percent. Taken at face value this would mean that selling the complete set and converting the NO legs was profitable on most days. The order-book snapshot in Table~\ref{tab:orderbook} shows why the face-value reading is wrong. On 22 May 2026 all five contracts were quoted with one-tick spreads of 0.001, the sum of the best bids was 0.999 and the sum of the best asks 1.004; the daily close recorded for that day, 1.0015, is exactly the sum of the midpoints. Neither direction of the round trip was profitable at the quotes, and the difference between the recorded $\Sigma_t$ and the executable bid-side sum is the half-tick on each of five contracts. Earlier in the sample, when the Roll estimate of the effective spread on the ``No change'' claim alone was between one and three cents (Figure~\ref{fig:liquidity}, Panel~A), a recorded overround of two to five percentage points is of the same order as the spread costs of the five legs. We conclude that the law of one price holds for the complete set within executable bounds, and that the recorded overround is a measurement artefact of last-trade and midpoint prices in thin low-priced claims rather than an unexploited arbitrage.1314\input{../results/tables/table9_orderbook.tex}1516Panel~D confirms that deviations are transitory. The first-order autocorrelation of $\Sigma_t-1$ is 0.72, implying a half-life of 2.1 days, and the Dickey--Fuller statistic of $-2.82$ rejects a unit root at the 10 percent level ($p=0.055$) in a sample of 146 observations. The regression of the deviation on time to resolution has a slope of 0.113 per year (HAC $t=4.87$): the overround falls by about one percentage point per month as the meeting approaches, which is the pattern expected if spreads narrow and arbitrage capital concentrates as the resolution date nears. Figure~\ref{fig:completeness} plots $\Sigma_t$ against the bounds and Figure~\ref{fig:arbitrage} the distribution of the deviations.1718\begin{figure}[htbp]19 \centering20 \includegraphics[width=\textwidth]{../figures/fig2_completeness_test.pdf}21 \caption{Price of the complete set of five state claims, $\Sigma_t$, with the executable bounds of equation~\eqref{eq:bounds}: the discounted value of one dollar at resolution (dashed, computed from the effective federal funds rate) and one (solid). The shaded band widens each bound by 0.5 percentage points for spread costs.}22 \label{fig:completeness}23\end{figure}2425\begin{figure}[htbp]26 \centering27 \includegraphics[width=\textwidth]{../figures/fig12_arbitrage_violations.pdf}28 \caption{Distribution of the deviation $\Sigma_t-1$ (percentage points) for the Fed group (Panel~A, $T=146$) and the election group (Panel~B, $T=304$). The dashed line in Panel~A marks 0.5 percentage points, the sum of five half-ticks. The single election-group observation above 50 points is an isolated print of 0.50 on an otherwise inactive contract on 9 April 2024.}29 \label{fig:arbitrage}30\end{figure}3132Two further remarks are in order. The positivity condition is satisfied throughout---the lowest recorded price is 0.002 for the ``50+ bps increase'' claim---but, as noted in Section~\ref{sec:theoretical}, this is guaranteed by the price grid and carries no information. And the earlier version of this paper reported a mean of 1.0205, a standard deviation of 0.0527 and a maximum absolute deviation of 0.57; those figures include the listing day, on which the five seeded quotes summed to 1.57 before any trade had occurred, and the maximum deviation they reported was that artefact.3334\subsection{The risk-neutral distribution of the June decision}\label{subsec:rnd_results}3536Figure~\ref{fig:rn_density} shows the normalised distribution $q_{j,t}$ at five dates and Figure~\ref{fig:eq_path} the implied expected policy change of equation~\eqref{eq:eq} and its daily changes. At the start of the sample the distribution is bimodal, with 36 percent of risk-neutral mass on ``No change'' and 53 percent on a 25-basis-point cut; the expected policy change is $-15.5$ basis points and its risk-neutral standard deviation 21.7 basis points. By 22 May 2026 the distribution is concentrated on ``No change'' (0.978), the expected change is $-0.1$ basis points and the standard deviation 5.3 basis points. The expected change never exceeded $-0.05$ or fell below $-17.4$ basis points. Its path is dominated by a small number of large moves: $+6.0$ basis points on 18 March 2026 (FOMC), $-1.7$ basis points on 9 January 2026 (employment report), and $+1.3$ and $+1.2$ basis points on the CPI releases of 11 March and 11 February.3738\begin{figure}[htbp]39 \centering40 \includegraphics[width=\textwidth]{../figures/fig3_risk_neutral_density.pdf}41 \caption{Risk-neutral distribution of the June 2026 decision, $q_{j,t}=p_{j,t}/\Sigma_t$, at five dates spanning the sample.}42 \label{fig:rn_density}43\end{figure}4445\begin{figure}[htbp]46 \centering47 \includegraphics[width=\textwidth]{../figures/fig4_expected_policy_change.pdf}48 \caption{Upper panel: risk-neutral expected change in the upper bound of the target range at the June 2026 meeting, $\mathbb{E}^{\mathbb{Q}}_t[\Delta r]$ (basis points), with FOMC statements (triangles), CPI releases (circles) and employment reports (squares). Lower panel: daily changes; release days in colour.}49 \label{fig:eq_path}50\end{figure}5152\subsection{Scheduled releases}\label{subsec:event_results}5354Table~\ref{tab:event_study} examines Hypothesis~\ref{hyp:info}. Panel~A lists every release in the sample with the day-0 change and the cumulative changes over three- and seven-day windows, for both the expected policy change and the ``No change'' state price. The four FOMC statements produced day-0 changes in the modal state price of $+1.5$, $-11.5$, $+23.0$ and $0.0$ percentage points. The January statement was read as making a June cut more likely---the ``25 bps decrease'' price rose by six points and the two tightening states, which had been nearly worthless, were briefly bid up to 0.054 and 0.037, so that the expected policy change actually rose slightly while its dispersion increased; the March statement removed most of the probability of a cut and moved the expected change by six basis points. The May statement, one week before the end of the sample, left the distribution unchanged. Among the data releases, the employment report of 6 March 2026 lowered the ``No change'' price by 12 points on the day and the CPI release of 11 February raised it by 7 points; the remaining releases moved it by less than five points.5556\input{../results/tables/table4_event_study.tex}5758Panel~B pools the releases. On the thirteen release days the absolute change in the expected policy change averages 1.15 basis points against 0.51 on the other 132 days (ratio 2.3); the variance ratio is 4.9, the Brown--Forsythe test rejects equal variances ($p=0.008$) and the Mann--Whitney test rejects equal distributions ($p=0.012$). For the ``No change'' state price the absolute change averages 5.5 against 1.4 percentage points, the variance ratio is 14.2 and both tests reject at the 1 percent level. By release type, the excess variance is concentrated on FOMC days (variance ratios of 10.6 and 30.4) and, for the state price, on employment-report days (variance ratio 4.5, Brown--Forsythe $p=0.023$); the four CPI releases in the sample show no excess variance in either measure. The absence of a CPI effect should not be over-interpreted: four observations cannot distinguish a small effect from none, and the CPI prints of the period may simply have been close to expectations. Over the same dates the absolute daily change in the ten-year Treasury yield was 3.3 basis points on release days and 3.3 on other days, so that the Treasury market shows no release-day excess variance at the daily frequency in this particular sample either; the difference between the two markets is that the Polymarket contract prices a single meeting and therefore reacts almost exclusively to information about that meeting.5960The last row of Panel~B compares weekend and weekday observations among non-release days. Weekend changes in the expected policy change have a standard deviation about 0.6 times that of weekday changes (variance ratio 0.33, Brown--Forsythe $p=0.11$) and weekend changes in the modal state price a variance ratio of 0.34 ($p=0.053$). Weekend trading therefore carries real but reduced information flow, roughly one third of weekday variance, which is close to what \citet{french1986stock} found for equity returns over non-trading periods and consistent with the view that most of the news relevant to the June decision arrives during business hours. Figure~\ref{fig:event_study_cars} displays the distributions.6162\begin{figure}[htbp]63 \centering64 \includegraphics[width=\textwidth]{../figures/fig5_event_study_cars.pdf}65 \caption{Absolute daily changes in the risk-neutral expected policy change (Panel~A, basis points) and in the ``No change'' state price (Panel~B, percentage points) on non-release days and on employment-report, CPI and FOMC days. Horizontal bars are group means.}66 \label{fig:event_study_cars}67\end{figure}6869\subsection{Lead--lag relations with conventional markets}\label{subsec:granger_results}7071Table~\ref{tab:granger} reports the Granger tests of Hypothesis~\ref{hyp:leadlag} on 110 trading days from 12 December 2025 to 21 May 2026. With the lag order chosen by information criteria, none of the eight directional tests rejects the null at the 10 percent level: the $p$-values range from 0.19 to 0.90 for the conventional $F$-test and from 0.20 to 0.88 for the HAC Wald test. The results are stable at two lags. At five lags two cells fall below 10 percent---the expected policy change to the VIX ($p=0.059$, HAC $p=0.033$) and to Bitcoin ($p=0.006$, HAC $p=0.317$)---but these appear in a block of 24 additional tests, are not supported by the information criteria, and in the Bitcoin case do not survive HAC inference. We read Table~\ref{tab:granger} as consistent with the null of no daily-horizon predictability in either direction.7273\input{../results/tables/table5_granger.tex}7475Table~\ref{tab:var} reports the five-variable VAR(1). No lagged variable is significant in the equation for the expected policy change, whose $R^2$ is 0.018, and the lagged expected policy change is insignificant in the four other equations. The contemporaneous correlations, not shown in the table, are informative about the direction of the common shocks: the daily change in the expected policy change has a correlation of 0.26 with the change in the ten-year yield and of $-0.15$ with the SPY return, with near-zero correlations with the VIX and Bitcoin. The Polymarket contract on the June meeting and the Treasury market thus respond to the same macroeconomic news within the day, in the direction one would expect---news that raises the expected policy rate raises long yields---but neither market's daily change predicts the other's. Figure~\ref{fig:granger_heatmap} shows the complete matrix of $p$-values.7677\input{../results/tables/table6_var.tex}7879\begin{figure}[htbp]80 \centering81 \includegraphics[width=0.75\textwidth]{../figures/fig7_granger_heatmap.pdf}82 \caption{Granger-causality $p$-values (bivariate VAR(1), 110 trading days) from the column variable to the row variable. ``Fed'' denotes the daily change in the Polymarket-implied expected policy change.}83 \label{fig:granger_heatmap}84\end{figure}8586The earlier version of this paper reported that the Fed contract Granger-caused the VIX at the 1 percent level and the SPY at the 10 percent level, and that the VIX Granger-caused the Fed contract at the 5 percent level. Appendix~\ref{app:reconciliation} shows that these results were produced jointly by the one-day misalignment of the Polymarket calendar and by the selection, for each pair, of the lag with the smallest $p$-value among five; with the alignment corrected, the same selection procedure yields $p$-values of 0.06, 0.37 and 0.26 for the three relations, and with the lag fixed by information criteria the $p$-values are 0.53, 0.59 and 0.35.8788\subsection{Spreads and depth}\label{subsec:liquidity_results}8990Figure~\ref{fig:liquidity} completes the microstructure evidence. Panel~A plots the Roll estimate of the effective spread for the two main claims over rolling twenty-day windows. For the ``No change'' claim the estimate averages 1.3 cents, falling from 1.6 cents over the first month to 1.1 cents over the last; in 32 percent of windows the serial covariance of price changes is positive and the estimator is set to zero, a familiar limitation of the Roll measure at daily frequency when price changes are dominated by information rather than by bounce. Panel~B shows the cumulative depth of the order book on 22 May 2026 as a function of the distance from the midpoint. The modal claim is deep on both sides---about \$0.75 million bid and \$0.60 million offered within one cent of the best quotes---whereas the four low-priced claims are quoted at one-tick spreads with substantial depth on the bid side but only a few thousand dollars offered within one cent; a buyer of a complete set at market would therefore have moved the low-priced legs by several ticks even for a modest size. The relative spread ranges from 0.1 percent for the modal claim to 40 percent for the ``50+ bps increase'' claim quoted at 0.002--0.003: in relative terms the tails of the state space are expensive to trade, which is one more reason to treat the recorded prices of those claims, and the overround they generate, with caution.9192\begin{figure}[htbp]93 \centering94 \includegraphics[width=\textwidth]{../figures/fig10_liquidity.pdf}95 \caption{Panel~A: Roll (1984) estimate of the effective spread (cents) for the ``No change'' and ``25 bps decrease'' claims, rolling twenty-day windows. Panel~B: cumulative resting depth of the five order books on 22 May 2026 (USD thousand, log scale) as a function of the distance from the midpoint (cents, log scale); ask side solid, bid side dashed.}96 \label{fig:liquidity}97\end{figure}98