spb/phd_thesis Public

1\section{Mathematical Proofs}2\label{sec:proofs}34This appendix provides formal mathematical derivations for the key theoretical relationships presented in Section~\ref{sec:methodology}.56\subsection{Derivation of Pure Stance Definitions}78\begin{theorem}[Pure Stance Characterization]9The pure dovish and hawkish stance measures satisfy the relationships given in Definition~\ref{def:pure_stances}.10\end{theorem}1112\begin{proof}13\textbf{Case 1: Pure Dovish Statement.} Assume the Federal Reserve releases a statement that exactly matches the dovish counterfactual: $F_t = F_t^D$.1415The tone measure becomes:16\begin{align}17\text{Tone}_t &= \frac{\text{sim}(F_t, F_t^H) - \text{sim}(F_t, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\18&= \frac{\text{sim}(F_t^D, F_t^H) - \text{sim}(F_t^D, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\19&= \frac{\text{sim}(F_t^D, F_t^H) - 1}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\20&= -121\end{align}2223The novelty measure is:24\begin{equation}25\text{Novelty}_t = 1 - \text{sim}(F_t, F_{t-1}) = 1 - \text{sim}(F_t^D, F_{t-1})26\end{equation}2728Therefore, the stance measure is:29\begin{align}30\text{Stance}_t &= \text{Novelty}_t \times \text{Tone}_t \notag \\31&= \left(1 - \text{sim}(F_t^D, F_{t-1})\right) \times (-1) \notag \\32&= -\left(1 - \text{sim}(F_t^D, F_{t-1})\right) \notag \\33&= \text{Stance}_t^{dove}34\end{align}3536\textbf{Case 2: Pure Hawkish Statement.} Assume $F_t = F_t^H$:3738The tone measure becomes:39\begin{align}40\text{Tone}_t &= \frac{\text{sim}(F_t^H, F_t^H) - \text{sim}(F_t^H, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\41&= \frac{1 - \text{sim}(F_t^H, F_t^D)}{1 - \text{sim}(F_t^D, F_t^H)} \notag \\42&= 143\end{align}4445Therefore:46\begin{align}47\text{Stance}_t &= \left(1 - \text{sim}(F_t^H, F_{t-1})\right) \times 1 = \text{Stance}_t^{hawk}48\end{align}4950This completes the proof. \hfill $\square$51\end{proof}5253\subsection{Derivation of Dovish Weight Parameter}5455\begin{theorem}[Dovish Weight Parameter Formula]56The weight parameter $w_t$ in the weighted stance representation has the form given in Definition~\ref{def:weighted_stance}.57\end{theorem}5859\begin{proof}60From the weighted stance equation:61\begin{equation}62\text{Stance}_t = w_t \cdot \text{Stance}_t^{dove} + (1 - w_t) \cdot \text{Stance}_t^{hawk}63\end{equation}6465Substituting the expressions for pure stances from Theorem 1:66\begin{align}67\text{Stance}_t &= -w_t\left(1 - \text{sim}(F_t^D, F_{t-1})\right) + (1 - w_t)\left(1 - \text{sim}(F_t^H, F_{t-1})\right) \notag \\68&= 1 - 2w_t + w_t\left(\text{sim}(F_t^D, F_{t-1}) + \text{sim}(F_t^H, F_{t-1})\right) - \text{sim}(F_t^H, F_{t-1})69\end{align}7071Collecting the terms in $w_t$, this reads72\begin{equation}73\text{Stance}_t = 1 - \text{sim}(F_t^H, F_{t-1}) - w_t\left(2 - \text{sim}(F_t^D, F_{t-1}) - \text{sim}(F_t^H, F_{t-1})\right).74\end{equation}75Equating with $\text{Stance}_t = \left(1 - \text{sim}(F_t, F_{t-1})\right) \times \text{Tone}_t$ and solving for $w_t$:76\begin{equation}77w_t = \frac{1 - \text{sim}(F_t^H, F_{t-1}) - \left(1 - \text{sim}(F_t, F_{t-1})\right) \times \text{Tone}_t}{2 - \text{sim}(F_t^D, F_{t-1}) - \text{sim}(F_t^H, F_{t-1})}78\end{equation}7980This establishes the formula. \hfill $\square$81\end{proof}8283\subsection{Proof of MPS Decomposition}8485\begin{theorem}[Policy Stance Surprise Decomposition]86The policy stance surprise admits the decomposition given in Proposition~\ref{prop:mps}.87\end{theorem}8889\begin{proof}90From the definitions:91\begin{align}92\text{MPS}_t &= \text{Stance}_t - \mathbb{E}_{t-\Delta}[\text{Stance}_t] \notag \\93&= \text{Novelty}_t \times \text{Tone}_t - (1 - 2p_{t-\Delta}) \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} \notag \\94&= \left(\overline{\text{Novelty}}_{t \mid t-\Delta} + \varepsilon_t\right) \times \text{Tone}_t - (1 - 2p_{t-\Delta}) \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} \notag \\95&= \overline{\text{Novelty}}_{t \mid t-\Delta} \times \text{Tone}_t - \overline{\text{Novelty}}_{t \mid t-\Delta} + 2p_{t-\Delta} \cdot \overline{\text{Novelty}}_{t \mid t-\Delta} + \varepsilon_t \times \text{Tone}_t \notag \\96&= \overline{\text{Novelty}}_{t \mid t-\Delta}(\text{Tone}_t + 2p_{t-\Delta} - 1) + \text{Tone}_t \cdot \varepsilon_t97\end{align}9899This establishes the decomposition. \hfill $\square$100\end{proof}101102\subsection{Economic Interpretation}103104The mathematical results provide several economic insights. First, the pure stance characterization shows that our measures correctly identify extreme policy communications, with dovish statements receiving negative stance values and hawkish statements receiving positive values. Second, the weight parameter derivation reveals how actual policy communications can be understood as weighted averages of extreme alternatives. Third, the MPS decomposition shows that policy surprises have two distinct sources: unexpected tone conditional on expected information content, and unexpected information content weighted by actual tone.105106