\section{Methodology}
\label{sec:methodology}

This section describes our NLP framework for measuring tone and novelty (Section~\ref{sec:nlp_framework}), presents the econometric specifications we use to estimate the effects of these measures on asset returns and volatility (Section~\ref{sec:econometric}), and states the testable hypotheses together with the specific coefficient restrictions that each one implies (Section~\ref{sec:hypotheses}).

\subsection{Theoretical Framework}
\label{sec:nlp_framework}

\subsubsection{Conceptual Foundation}

We decompose each FOMC statement into two dimensions: policy tone (hawkish vs.\ dovish) and informational novelty (distance from the previous statement). To illustrate why context-aware methods are needed for this decomposition, consider the difference between dictionary-based and machine learning approaches. A dictionary approach aggregates sentiment as:
\begin{equation}
S_{\text{dict}} = \frac{1}{N} \sum_{i=1}^{N} \mathbb{1}_{[\text{word}_i \in D_{\text{pos}}]} - \mathbb{1}_{[\text{word}_i \in D_{\text{neg}}]}
\end{equation}
where $D_{\text{pos}}$ and $D_{\text{neg}}$ are predetermined word lists. The dictionary approach counts words one at a time, without regard to their position or neighbors. A transformer-based model instead produces $S_{\text{ML}} = f_{\theta}(x_1, x_2, \ldots, x_N; C)$, where $f_{\theta}$ is a neural network with learned parameters $\theta$, and $x_i$ are token representations that depend on surrounding words and document-level context $C$. To see why context matters, note that ``the policy remains accommodative'' and ``the policy is no longer accommodative'' contain the same keyword (``accommodative'') but have opposite meanings; a dictionary assigns similar scores to both, while a transformer assigns opposite scores.

Our approach builds on three literatures. The information processing literature predicts that markets react more strongly to genuinely new information than to repetition \citep{grossman1980,fama1970}. The central bank communication literature shows that policy statements move asset prices independently of rate decisions \citep{gurkaynak2005,campbell2012}. The market microstructure literature demonstrates that high-frequency data can isolate the real-time incorporation of new information into prices \citep{andersen2003,rosa2013}.

\begin{definition}[FOMC Statement Representation]
Each FOMC statement $F_t$ released at time $t$ is mapped to a vector $\mathbf{f}_t \in \mathbb{R}^d$ in $d$-dimensional semantic space through domain-specific transformer embeddings. We employ two architecturally distinct models: MiniLM ($d = 384$) and BERT ($d = 768$), combined into a separation-weighted ensemble.
\end{definition}

Each model maps a statement to a real-valued vector in which semantically similar texts (e.g., ``accommodative'' and ``easy policy'') are represented by nearby points, enabling cosine similarity comparisons and PCA. Unlike bag-of-words representations, where each coordinate is a word count and most are zero, these vectors are dense: every coordinate carries information. The output dimensionality ($d = 384$ for MiniLM, $d = 768$ for BERT) is fixed by the model architecture---MiniLM-L6-v2 has 6 layers and 384 hidden units per layer; BERT-base has 12 layers and 768 hidden units \citep{kenton2019}---and is not a hyperparameter we select. We adapt both architectures to the FOMC domain using a two-stage training procedure.

\paragraph{Stage 1: TSDAE Unsupervised Pre-Training.} We first apply Transformer-based Sequential Denoising Auto-Encoder (TSDAE) \citep{wang2021tsdae} to adapt each model to the FOMC domain vocabulary:
\begin{equation}
\mathcal{L}_{\text{TSDAE}} = - \sum_{i=1}^{N} \log P\bigl(x_i \mid \tilde{x}_i; \theta_{\text{enc}}, \theta_{\text{dec}}\bigr),
\end{equation}
where $\tilde{x}_i$ is a corrupted version of sentence $x_i$ (token deletion with probability 0.6). This unsupervised stage teaches domain-specific vocabulary and structure without requiring labeled pairs, training for 3 epochs (MiniLM) or 2 epochs (BERT) with learning rate $3 \times 10^{-5}$.

\paragraph{Stage 2: Supervised Contrastive Learning with MNRL.} We then fine-tune using Multiple Negatives Ranking Loss (MNRL):
\begin{equation}
\mathcal{L}_{\text{MNRL}} = - \frac{1}{B} \sum_{i=1}^{B} \log \frac{\exp\bigl(\text{sim}(a_i, p_i) / \tau\bigr)}{\sum_{j=1}^{B} \exp\bigl(\text{sim}(a_i, p_j) / \tau\bigr)},
\end{equation}
where $(a_i, p_i)$ are positive pairs, $\text{sim}(\cdot, \cdot)$ denotes cosine similarity, $\tau$ is the temperature parameter, and $B$ is the batch size. In-batch negatives provide contrastive signal without explicit negative sampling. We construct approximately 4,000 training pairs per model using the six pairing strategies described in Section~\ref{sec:data}.

\paragraph{Dual-Model Ensemble.} The two models are combined using separation-weighted averaging:
\begin{equation}
\widehat{\text{Tone}}_{t,k}^{\text{ens}} = w_k^{\text{MiniLM}} \cdot \text{Tone}_{t,k}^{\text{MiniLM}} + w_k^{\text{BERT}} \cdot \text{Tone}_{t,k}^{\text{BERT}},
\end{equation}
where the weights are proportional to axis separation quality:
\begin{equation}
w_k^{m} = \frac{\text{Sep}_k^{m}}{\text{Sep}_k^{\text{MiniLM}} + \text{Sep}_k^{\text{BERT}}}, \quad m \in \{\text{MiniLM}, \text{BERT}\}.
\end{equation}
For each statement, we measure inter-model confidence as:
\begin{equation}
\text{Conf}_{t,k} = \frac{1}{2}\bigl[\mathbb{1}[\text{sign}(\text{Tone}_{t,k}^{\text{M}}) = \text{sign}(\text{Tone}_{t,k}^{\text{B}})] + (1 - |\text{Tone}_{t,k}^{\text{M}} - \text{Tone}_{t,k}^{\text{B}}| / \max_j |\text{Tone}_{j,k}^{\text{M}} - \text{Tone}_{j,k}^{\text{B}}|)\bigr].
\end{equation}
Mean ensemble confidence is 0.837 for novelty and ranges from 0.562 to 0.806 for tone across axes. This dual-model approach reduces model-specific biases: MiniLM (33M parameters, 384 dimensions) provides computational efficiency and strong separation on policy stance, while BERT (110M parameters, 768 dimensions) captures richer contextual relationships. The ensemble achieves balanced performance across all semantic dimensions.

\subsubsection{Semantic Dimensions}

\begin{definition}[Semantic Similarity Measure]
The semantic similarity between text vectors $\mathbf{u}$ and $\mathbf{v}$ is quantified using cosine similarity:
\begin{equation}
\text{sim}(\mathbf{u}, \mathbf{v}) = \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}||_2 \cdot ||\mathbf{v}||_2} = \frac{\sum_{i=1}^{d} u_i v_i}{\sqrt{\sum_{i=1}^{d} u_i^2} \cdot \sqrt{\sum_{i=1}^{d} v_i^2}}
\end{equation}
\end{definition}

The cosine similarity measure ranges from $-1$ to $1$, where values approaching $1$ indicate high semantic similarity, values near $0$ indicate orthogonal or unrelated content, and values approaching $-1$ indicate semantic opposition. This metric is particularly well-suited for high-dimensional text embeddings because it captures angular distance between vectors while being invariant to vector magnitude differences that might arise from statement length variations.

\begin{definition}[Statement Novelty]
The informational novelty of statement $F_t$ relative to the immediately preceding statement is defined as:
\begin{equation}
\text{Novelty}_t = 1 - \text{sim}(\mathbf{f}_t, \mathbf{f}_{t-1})
\end{equation}
\end{definition}

This novelty measure captures the degree to which current policy communications represent genuine departures from recent messaging patterns. Higher novelty values indicate statements containing substantively new informational content relative to recent communications, while lower values suggest continuity with established messaging. The measure is bounded between $0$ and $2$, where $0$ indicates identical statements and $2$ indicates maximally opposed statements.

To construct our policy tone measure, we develop a data-driven approach based on PCA-based semantic axis construction that replaces manually selected reference dates with an algorithmic procedure, eliminating researcher degrees of freedom.

\begin{definition}[Data-Driven Semantic Axis Construction]
\label{def:pca_axes}
Let $\mathbf{E} \in \mathbb{R}^{N \times d}$ be the L2-normalized embedding matrix. We compute the first $K = 8$ principal components:
\begin{equation}
\mathbf{E} = \mathbf{U} \boldsymbol{\Sigma} \mathbf{V}^\top + \boldsymbol{\epsilon}.
\end{equation}
For each semantic dimension $k \in \{1, \ldots, 4\}$ (policy stance, risk assessment, policy communication, economic priority), we define keyword banks $\mathcal{K}_k^+$ (positive pole) and $\mathcal{K}_k^-$ (negative pole), and compute a keyword differential score:
\begin{equation}
\Delta_i^{(k)} = \sum_{w \in \mathcal{K}_k^+} \mathbb{1}[w \in \text{text}_i] - \sum_{w \in \mathcal{K}_k^-} \mathbb{1}[w \in \text{text}_i].
\end{equation}
We match each semantic axis to the PC maximizing the absolute Pearson correlation:
\begin{equation}
j^*(k) = \arg\max_{j \in \{1,\ldots,K\}} \bigl|\text{Corr}(\mathbf{v}_j, \boldsymbol{\Delta}^{(k)})\bigr|.
\end{equation}
\end{definition}

The reference selection uses a hybrid seed-and-percentile approach: (1) use 3 expert-selected dates per pole to define an initial axis direction $\hat{\mathbf{d}}_k = \bar{\mathbf{e}}_{\text{pos}} - \bar{\mathbf{e}}_{\text{neg}}$; (2) project all embeddings onto this direction: $s_i = \mathbf{e}_i^\top \hat{\mathbf{d}}_k / \|\hat{\mathbf{d}}_k\|$; (3) select $M = 5$ statements with highest and lowest projection scores; (4) iteratively refine by recomputing the axis direction from selected references and re-selecting. This yields 5 reference statements per pole (compared to 2 in manual selection), producing more stable centroids.

For each axis $k$, we measure discrimination quality using the separation metric:
\begin{equation}
\text{Sep}_k = 1 - \cos(\bar{\mathbf{c}}_k^+, \bar{\mathbf{c}}_k^-),
\end{equation}
where $\bar{\mathbf{c}}_k^+, \bar{\mathbf{c}}_k^-$ are the L2-normalized positive and negative centroids. All axes achieve separations well above the 0.04 minimum threshold, with ensemble separations ranging from 0.248 to 0.302.

\begin{definition}[Policy Tone]
The directional tone of statement $F_t$ on semantic axis $k$ is measured as:
\begin{equation}
\text{Tone}_{t,k} = \frac{\cos(\mathbf{f}_t, \bar{\mathbf{c}}_k^+) - \cos(\mathbf{f}_t, \bar{\mathbf{c}}_k^-)}{\text{Sep}_k}, \quad \text{clipped to } [-1, +1].
\end{equation}
\end{definition}

This tone measure captures the relative proximity of actual statements to the positive (hawkish) and negative (dovish) reference centroids for each semantic axis, normalized by the axis separation. The primary variable used in regressions is the \emph{total policy stance tone}. In the regression tables, the z-scored stance variable is oriented so that positive values indicate more accommodative (dovish) communication; positive coefficients therefore measure the response to a one-standard-deviation dovish shift. The composite MPS measure defined below retains the hawkish-positive orientation. Relative to counterfactual-based methods, this approach is reproducible, does not depend on particular date choices, and extends to semantic dimensions other than the hawkish-dovish spectrum.

\subsubsection{Integrated Policy Stance Measurement}

\begin{definition}[Monetary Policy Stance]
The overall monetary policy stance is defined as:
\begin{equation}
\text{Stance}_t = \text{Novelty}_t \times \text{Tone}_t
\end{equation}
\end{definition}

\textbf{Note on originality:} Definitions 1--11 and Proposition~\ref{prop:mps} are our own contributions. They apply the standard sentence-embedding framework from the NLP literature to monetary policy communications with a new decomposition structure.

The multiplicative form means that stance is large only when a statement is both novel \textit{and} directionally clear. We adopt a multiplicative rather than additive form for two reasons. First, it ensures that Stance is zero whenever either component is zero: repetitive content has no impact regardless of its tone, and novel but directionally neutral content has none either. Second, it captures the reinforcing interaction between the two dimensions: novel information amplifies the impact of tone, and clear directional tone amplifies the impact of novelty.

In the following two definitions, $\mathbf{f}_t^D$ and $\mathbf{f}_t^H$ denote the embeddings of counterfactual statements located at the dovish and hawkish reference centroids of the policy stance axis, $\bar{\mathbf{c}}^-$ and $\bar{\mathbf{c}}^+$; they represent what the Fed would have released had it issued a maximally dovish or maximally hawkish statement at meeting $t$.

\begin{definition}[Pure Dovish and Hawkish Stances]
\label{def:pure_stances}
The extreme stance scenarios are defined as:
\begin{align}
\text{Stance}_t^{dove} &= -\left(1 - \text{sim}(\mathbf{f}_t^D, \mathbf{f}_{t-1})\right) \\
\text{Stance}_t^{hawk} &= 1 - \text{sim}(\mathbf{f}_t^H, \mathbf{f}_{t-1})
\end{align}
\end{definition}

\begin{definition}[Weighted Policy Stance Representation]
\label{def:weighted_stance}
The overall stance can be expressed as a convex combination of extreme stances:
\begin{equation}
\text{Stance}_t = w_t \cdot \text{Stance}_t^{dove} + (1 - w_t) \cdot \text{Stance}_t^{hawk}
\end{equation}
where the dovish weight parameter $w_t$ is computed (not estimated) as a deterministic function of the observed similarities:
\begin{equation}
w_t = \frac{1 - \text{sim}(\mathbf{f}_t^H, \mathbf{f}_{t-1}) - \left(1 - \text{sim}(\mathbf{f}_t, \mathbf{f}_{t-1})\right) \cdot \text{Tone}_t}{2 - \text{sim}(\mathbf{f}_t^D, \mathbf{f}_{t-1}) - \text{sim}(\mathbf{f}_t^H, \mathbf{f}_{t-1})}
\end{equation}
\end{definition}

This weighted representation provides an alternative interpretation of policy stance as a position along the spectrum between extreme dovish and hawkish alternatives. The weight parameter $w_t$ is not a free parameter to be estimated; rather, it is algebraically derived from the observed textual similarities.

\subsubsection{Market Surprise Identification}

\begin{definition}[Novelty Decomposition]
The total novelty can be decomposed into predictable and unpredictable components:
\begin{equation}
\text{Novelty}_t = \overline{\text{Novelty}}_{t \mid t-\Delta} + \varepsilon_t
\end{equation}
where $\overline{\text{Novelty}}_{t \mid t-\Delta} \equiv \mathbb{E}_{t-\Delta}[\text{Novelty}_t]$ represents the expected novelty based on information available $\Delta$ periods before the announcement, and $\varepsilon_t$ represents the novelty surprise component with $\mathbb{E}_{t-\Delta}[\varepsilon_t] = 0$ and $\text{Var}(\varepsilon_t) = \sigma_{\varepsilon}^2 < \infty$. We assume $\varepsilon_t$ is covariance-stationary with $\text{Cov}(\varepsilon_t, \varepsilon_{t-j}) \to 0$ as $j \to \infty$.
\end{definition}

\begin{definition}[Expected Policy Stance]
Under rational expectations, the expected policy stance is:
\begin{equation}
\mathbb{E}_{t-\Delta}[\text{Stance}_t] = \mathbb{E}_{t-\Delta}\left[\text{Novelty}_t \times \text{Tone}_t\right]
\end{equation}
\end{definition}

\begin{assumption}[Conditional Independence or Zero Covariance]
\label{ass:indep}
We assume either (i) conditional independence: $\text{Novelty}_t \perp \text{Tone}_t | \mathcal{I}_{t-\Delta}$, where $\mathcal{I}_{t-\Delta}$ is the information set at time $t - \Delta$, or (ii) the weaker condition of zero conditional covariance:
\begin{equation}
\text{Cov}_{t-\Delta}(\text{Novelty}_t, \text{Tone}_t) = 0
\end{equation}
\end{assumption}

This assumption is plausible because novelty captures whether the Fed changes its messaging structure (how information is presented), while tone captures the directional policy stance (hawkish/dovish). Empirically, the correlation between novelty and tone in our sample is 0.12 (Figure~\ref{fig:D6}), supporting the assumption that these are largely independent dimensions. Under this assumption:
\begin{equation}
\mathbb{E}_{t-\Delta}[\text{Stance}_t] = \mathbb{E}_{t-\Delta}[\text{Novelty}_t] \times \mathbb{E}_{t-\Delta}[\text{Tone}_t]
\end{equation}

Further assuming, for tractability, that tone is binary ($\text{Tone}_t \in \{-1, +1\}$, taking the dovish value $-1$ with probability $p_{t-\Delta}$), so that $\mathbb{E}_{t-\Delta}[\text{Tone}_t] = 1 - 2p_{t-\Delta}$:
\begin{equation}
\mathbb{E}_{t-\Delta}[\text{Stance}_t] = (1 - 2p_{t-\Delta}) \cdot \overline{\text{Novelty}}_{t \mid t-\Delta}
\end{equation}

\begin{definition}[Policy Stance Surprise]
The communication-based policy stance surprise is defined as:
\begin{equation}
\text{MPS}_t = \text{Stance}_t - \mathbb{E}_{t-\Delta}[\text{Stance}_t]
\end{equation}
\end{definition}

We use the abbreviation MPS (monetary policy surprise) for brevity, but emphasize that this measure is conceptually distinct from the traditional rate-based monetary policy surprise of \citet{kuttner2001}, which measures the unexpected component of the federal funds rate decision using futures prices. Our MPS is a \textit{communication} surprise---it captures unexpected variation in the textual content of FOMC statements, not in the quantitative rate decision. The two measures are complementary and potentially orthogonal: a meeting can produce a large rate surprise with a routine statement, or a novel and directional statement with a fully anticipated rate decision. Our measure captures the latter dimension, which has become increasingly important as forward guidance and qualitative communication have gained prominence in the monetary policy toolkit.

\begin{proposition}[MPS Decomposition Structure]
\label{prop:mps}
The monetary policy surprise admits the following decomposition:
\begin{equation}
\text{MPS}_t = \overline{\text{Novelty}}_{t \mid t-\Delta}(\text{Tone}_t + 2p_{t-\Delta} - 1) + \text{Tone}_t \cdot \varepsilon_t
\end{equation}
\end{proposition}

This decomposition reveals that policy surprises consist of two distinct components: (1) surprises arising from unexpected tone conditional on expected novelty, and (2) surprises arising from unexpected novelty, weighted by the actual tone of the communication. The proof of Proposition~\ref{prop:mps}, together with the derivations underlying Definitions~\ref{def:pure_stances} and~\ref{def:weighted_stance}, is in Appendix~\ref{sec:proofs}.

\subsection{Econometric Framework}
\label{sec:econometric}

\subsubsection{Panel Data Structure}

The data are organized as a three-dimensional panel structure with observations indexed along three dimensions:
\begin{itemize}
\item Asset identifier $i \in \{1, 2, \ldots, I\}$ where $I$ represents the total number of financial instruments
\item Event identifier $j \in \{1, 2, \ldots, J\}$ corresponding to FOMC announcement dates where $J = 148$
\item Relative minute $m \in \{-120, \ldots, +120\}$ measured from announcement time, with the primary post-announcement window covering $m \in \{0, \ldots, +45\}$
\end{itemize}

\subsubsection{Identification Strategy}

Our identification relies on the fact that FOMC statements are drafted and finalized before the announcement and released at a scheduled time (typically 2:00 PM ET). The text at $t=0$ therefore cannot be influenced by market reactions at $t>0$, satisfying the timing restriction for causal interpretation.

Our identification relies on three core conditions:

\textbf{Identification Condition 1 (Exogenous Timing).} The precise minute-by-minute timing of FOMC announcements is predetermined and exogenous to short-term market movements within our event windows.

\textbf{Identification Condition 2 (Information Concentration).} Within our 45-minute post-announcement event windows, FOMC statements represent the dominant source of monetary policy-relevant information, with minimal contamination from other systematic news sources.

\textbf{Identification Condition 3 (Market Efficiency).} Financial markets rapidly incorporate new information from FOMC statements into prices, enabling causal interpretation of immediate price movements following announcements.

We acknowledge two potential threats to this identification strategy. First, the Federal Reserve may adjust its communication in response to broader financial conditions observed before the meeting. We mitigate this concern by including event fixed effects $\delta_j$ that absorb all meeting-level heterogeneity. Second, FOMC announcements may coincide with other information releases (e.g., Summary of Economic Projections, press conference expectations). Our focus on the immediate post-announcement window limits contamination from subsequent press conference content, which typically begins 30 minutes after the statement release.

\subsubsection{Dynamic Response Function Estimation}

Our primary empirical specification estimates dynamic impulse response functions:
\begin{equation}
r_{i,m}^{(j)} = \alpha_i + \delta_j + \sum_{k=0}^{+45} \beta_k \cdot \mathbb{1}_{[m=k]} \cdot \text{MPS}_j + \varepsilon_{i,m,j}
\label{eq:dynamic}
\end{equation}
where $r_{i,m}^{(j)}$ denotes the log return of asset $i$ at minute $m$ relative to monetary policy announcement $j$, $\alpha_i$ represents asset fixed effects, $\delta_j$ captures event fixed effects, and $\beta_k$ measures the marginal impact of monetary policy surprises occurring exactly $k$ minutes relative to the announcement.

\subsubsection{Semantic Decomposition Specification}

To examine the differential effects of policy tone versus informational novelty, we estimate:
\begin{equation}
r_{i,m}^{(j)} = \alpha_i + \delta_j + \sum_{k=0}^{+45} \left[\beta_k^{(T)} \cdot \mathbb{1}_{[m=k]} \cdot \text{Tone}_j + \beta_k^{(N)} \cdot \mathbb{1}_{[m=k]} \cdot \text{Novelty}_j\right] + \varepsilon_{i,m,j}
\label{eq:decomp}
\end{equation}

The tone coefficients $\{\beta_k^{(T)}\}$ measure the dynamic response to hawkish versus dovish content, while the novelty coefficients $\{\beta_k^{(N)}\}$ capture the response to informational content, independent of directional bias.

\subsubsection{Rolling Realized Measures}

In addition to return effects, we examine how policy communications affect market volatility using rolling realized measures computed from 1-minute log returns.

\paragraph{Realized Variance.} We construct NA-tolerant rolling realized variance in $K$-minute windows:
\begin{equation}
\mathrm{RV}_K(t) = \frac{K}{n_{\text{valid}}} \sum_{s = t-K+1}^{t} r_s^2 \cdot \mathbb{1}[r_s \text{ valid}],
\end{equation}
where $n_{\text{valid}}$ is the number of non-missing observations in the window. We require at least 80\% valid observations.

\paragraph{Realized Beta.} We compute rolling beta relative to the ES (S\&P 500) benchmark:
\begin{equation}
\hat{\beta}_K(t) = \frac{\sum_{s=t-K+1}^{t} r_{i,s} \cdot r_{m,s}}{\sum_{s=t-K+1}^{t} r_{m,s}^2},
\end{equation}
where $r_{m,s}$ denotes the ES benchmark return. We use $K = 5$ for panel minute-level regressions and $K = 30$ for event-level analysis.

\subsubsection{Panel Minute-Level Regressions}

For each ticker, we estimate:
\begin{equation}
\log(\mathrm{RV}_{i,t}) = \alpha + \beta \cdot (\text{Post}_t \times \text{Semantic}_i) + \varepsilon_{i,t},
\label{eq:panel}
\end{equation}
where $\text{Post}_t = \mathbb{1}[t > 0]$ indicates post-announcement minutes, and $\text{Semantic}_i$ is either \emph{Stance} or \emph{Novelty} (z-scored). Standard errors are clustered by event date.

\subsubsection{Event-Level Regressions}

For each event, we compute pre/post changes:
\begin{equation}
\Delta \log(\mathrm{RV})_i = \overline{\log(\mathrm{RV})}_{\text{post}} - \overline{\log(\mathrm{RV})}_{\text{pre}},
\end{equation}
and regress on semantic measures:
\begin{equation}
\Delta Y_i = \alpha + \beta_1 \text{Stance}_i + \beta_2 \text{Novelty}_i + \beta_3 (\text{Stance}_i \times \text{Novelty}_i) + \varepsilon_i,
\label{eq:event}
\end{equation}
with Newey--West HAC standard errors. We estimate this specification for five dependent variables: $\Delta\mathrm{RV}$, $\Delta\log(\mathrm{RV})$, RV ratio, $\log$(RV ratio), and $\Delta\beta$.

\subsubsection{Local Projection Impulse Response Functions}

Following \citet{jorda2005}, for each horizon $h \in \{1, 2, 3, 5, 10, 15, 20, 30, 45, 60, 90, 120\}$ minutes post-announcement:
\begin{equation}
\text{CumRet}_i(0 \to h) = \alpha + \beta_1^{(h)} \text{Stance}_i + \beta_2^{(h)} \text{Novelty}_i + \beta_3^{(h)} (\text{Stance}_i \times \text{Novelty}_i) + \varepsilon_i^{(h)}.
\label{eq:irf}
\end{equation}
Abnormal cumulative returns are computed as $\text{CAR}_i(h) = \text{CumRet}_i(h) - \text{CumRet}_i^{\text{ES}}(h)$. This local projection approach is robust to misspecification of the data-generating process and allows horizon-specific inference without imposing a parametric impulse response shape. Because the post-meeting press conference typically begins 30 minutes after the statement release, we interpret responses at horizons beyond $h = 30$ as the joint effect of the statement and early press-conference communication.

\subsubsection{Pre-Announcement Placebo Test}

We estimate the same specification~\eqref{eq:irf} for $h$ minutes \emph{before} the announcement:
\begin{equation}
\text{CumRet}_i(-h \to 0) = \alpha + \beta_1^{(-h)} \text{Stance}_i + \beta_2^{(-h)} \text{Novelty}_i + \beta_3^{(-h)} (\text{Stance}_i \times \text{Novelty}_i) + \varepsilon_i^{(-h)}.
\label{eq:placebo}
\end{equation}
Under the null that announcement content is not anticipated, all $\beta^{(-h)}$ should be zero. This placebo test provides a direct validation of our event study design.

\subsubsection{Statistical Inference and Robustness}

We subject all event-level results to five alternative inference methods:

\begin{enumerate}
\item \textbf{Newey--West HAC standard errors}: Robust to heteroskedasticity and autocorrelation.
\item \textbf{Wild bootstrap} ($B = 1{,}999$, Rademacher weights): Robust to heteroskedasticity with improved finite-sample properties.
\item \textbf{Clustered standard errors} (HC1, by event date): Accounts for cross-sectional dependence within events.
\item \textbf{Quantile regression} ($\tau = 0.5$): Robust to outliers and heavy tails in the dependent variable.
\item \textbf{Permutation test} ($B = 4{,}999$): Gold standard---shuffles semantic labels across events preserving panel structure, providing exact $p$-values under the null.
\end{enumerate}

All $p$-values in panel and IRF regressions are adjusted for multiple testing using the Benjamini--Hochberg (BH) procedure within each family of tests (per dependent variable and model type) \citep{benjamini1995controlling}. We use only standard significance levels: 1\%, 5\%, and 10\%.

\textbf{Sub-Period Stability.} We examine coefficient stability across six Fed policy regimes: crisis and recovery (2008--2012), post-crisis normalization (2013--2015), pre-pandemic tightening (2016--2019), pandemic response (2020--2021), and inflation tightening (2022--2025).

\textbf{Alternative Semantic Measures.} The dual-model ensemble itself provides a built-in robustness check: by comparing results from MiniLM-only, BERT-only, and ensemble measures, we verify that findings are not driven by model-specific artifacts.

\subsection{Hypotheses and Testable Predictions}
\label{sec:hypotheses}

We now state our hypotheses and, for each one, the coefficient restriction that operationalizes it in the specifications of Section~\ref{sec:econometric}. Recall the sign conventions: the z-scored stance regressor in equations~\eqref{eq:decomp}--\eqref{eq:irf} is dovish-positive, while the composite MPS in equation~\eqref{eq:dynamic} is hawkish-positive. Table~\ref{tab:hyp_map} summarizes the mapping from hypotheses to coefficients, equations, and the tables in which each test is reported.

\textbf{Hypothesis 1 (Tone Effects on Returns).}
\begin{itemize}
\item \textbf{H1a.} Dovish tone increases returns on risk assets (equities, commodities). \textit{Test:} $\beta_k^{(T)} > 0$ in the semantic decomposition~\eqref{eq:decomp} and $\beta_1^{(h)} > 0$ in the local projection~\eqref{eq:irf} for ES and CL.
\item \textbf{H1b.} Dovish tone decreases returns on safe-haven assets, as accommodative policy triggers portfolio rebalancing away from safe havens toward risk assets. \textit{Test:} $\beta_1^{(h)} < 0$ in~\eqref{eq:irf} for GC, ZN, and ZF.
\item \textbf{H1c.} Tone effects strengthen with the horizon as markets progressively process policy implications. \textit{Test:} $|\beta_1^{(h)}|$ increasing in $h$ in~\eqref{eq:irf} for ES.
\end{itemize}

\textbf{Hypothesis 2 (Novelty Effects on Volatility).}
\begin{itemize}
\item \textbf{H2a.} High novelty increases market volatility through information processing complexity. \textit{Test:} $\beta_2 > 0$ in the event-level regression~\eqref{eq:event} with $\Delta\mathrm{RV}$ as the dependent variable, and $\beta > 0$ in the panel regression~\eqref{eq:panel} with $\text{Semantic} = \text{Novelty}$, most directly for VX.
\item \textbf{H2b.} Novelty effects dissipate rapidly as markets resolve uncertainty. \textit{Test:} $|\beta_2^{(h)}|$ decreasing in $h$ in~\eqref{eq:irf} for VX.
\item \textbf{H2c.} Novelty has minimal effects on directional returns. \textit{Test:} $\beta_2^{(h)} \approx 0$ (statistically indistinguishable from zero) in~\eqref{eq:irf} for the return contracts (ES, CL, GC, DX).
\end{itemize}

\textbf{Hypothesis 3 (Policy Surprise Effects).}
\begin{itemize}
\item \textbf{H3a.} Hawkish policy surprises reduce equity returns. \textit{Test:} $\beta_0 < 0$ in the dynamic response function~\eqref{eq:dynamic} for ES.
\item \textbf{H3b.} Policy surprises increase implied volatility. \textit{Test:} $\beta_0 > 0$ in~\eqref{eq:dynamic} for VX.
\item \textbf{H3c.} Surprise effects on volatility are most pronounced immediately after the announcement. \textit{Test:} $|\beta_k|$ in~\eqref{eq:dynamic} largest at $k = 0$ and decaying in $k$ for VX.
\end{itemize}

\textbf{Hypothesis 4 (Cross-Asset Patterns).}
\begin{itemize}
\item \textbf{H4a.} Risk assets respond to tone in the same direction (risk-on/risk-off). \textit{Test:} $\text{sign}\bigl(\beta_1^{(h)}\bigr)$ equal across ES and CL in~\eqref{eq:irf}, opposite for GC, ZN, ZF.
\item \textbf{H4b.} Volatility responses to novelty are asset-specific. \textit{Test:} $\beta_2$ and $\beta_3$ in~\eqref{eq:event} differ in magnitude and significance across the seven contracts.
\end{itemize}

Finally, the validity of the design itself is testable: under no anticipation of statement content, all pre-announcement coefficients in the placebo specification~\eqref{eq:placebo} should be zero, $\beta_1^{(-h)} = \beta_2^{(-h)} = \beta_3^{(-h)} = 0$ for all $h$.

\begin{table}[H]
\centering
\footnotesize
\caption{Mapping of Hypotheses to Coefficients, Equations, and Evidence}
\label{tab:hyp_map}
\begin{tabular}{lp{4.5cm}p{4.3cm}p{3.3cm}}
\toprule
Hyp. & Prediction & Coefficient restriction (equation) & Evidence \\
\midrule
H1a & Dovish tone $\Rightarrow$ risk-asset returns $\uparrow$ & $\beta_1^{(h)} > 0$ in \eqref{eq:irf}, ES/CL & Table~\ref{tab:irf_stance_ret_h} \\
H1b & Dovish tone $\Rightarrow$ safe-haven returns $\downarrow$ & $\beta_1^{(h)} < 0$ in \eqref{eq:irf}, GC/ZN/ZF & Table~\ref{tab:irf_stance_ret_h} \\
H1c & Tone effects build with horizon & $|\beta_1^{(h)}|$ increasing in $h$, ES & Table~\ref{tab:irf_stance_ret_h}; Fig.~\ref{fig:R8} \\
H2a & Novelty $\Rightarrow$ volatility $\uparrow$ & $\beta_2 > 0$ in \eqref{eq:event}, $\Delta$RV & Tables~\ref{tab:panel_logrv_novelty}, \ref{tab:rolling_delta_rv_30min} \\
H2b & Novelty effects decay quickly & $|\beta_2^{(h)}|$ decreasing in $h$, VX & Table~\ref{tab:irf_novelty_ret_h} \\
H2c & Novelty does not move returns & $\beta_2^{(h)} \approx 0$ in \eqref{eq:irf} & Table~\ref{tab:irf_novelty_ret_h}; Fig.~\ref{fig:R9} \\
H3a & Hawkish surprise $\Rightarrow$ equity returns $\downarrow$ & $\beta_0 < 0$ in \eqref{eq:dynamic}, ES & Section~\ref{sec:baseline} \\
H3b & Surprise $\Rightarrow$ implied volatility $\uparrow$ & $\beta_0 > 0$ in \eqref{eq:dynamic}, VX & Section~\ref{sec:baseline} \\
H3c & Surprise effects peak on impact & $|\beta_k|$ max at $k=0$ in \eqref{eq:dynamic}, VX & Section~\ref{sec:baseline} \\
H4a & Homogeneous return response, risk assets & $\text{sign}(\beta_1^{(h)})$ equal, ES/CL in \eqref{eq:irf} & Table~\ref{tab:irf_stance_ret_h} \\
H4b & Heterogeneous volatility responses & $\beta_2, \beta_3$ vary across contracts in \eqref{eq:event} & Tables~\ref{tab:rolling_delta_rv_30min}, \ref{tab:robustness_A} \\
Placebo & No pre-announcement effects & $\beta^{(-h)} = 0$ in \eqref{eq:placebo}, all contracts & Fig.~\ref{fig:placebo_stance}, \ref{fig:placebo_novelty} \\
\bottomrule
\end{tabular}
\end{table}

