## Deciphering Delphic Guidance: The Bank of England and Brexit (wpiea2024160-print-pdf)

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### Executive Summary
- Examines the Bank of England’s (BoE) monetary policy response to the June 2016 advisory referendum (Brexit) and its impact on UK government bond yields.
- Central questions:
  - The impact of BoE's monetary policies on government bond yields.
  - The role of BoE communication strategy on market perceptions of risk.
  - The overall effectiveness of these measures in managing Brexit-triggered uncertainties.
- Approach: develop a model of sovereign debt supply to understand drivers of bond yield risk premia and complement it with an empirical analysis of BoE actions and communications.
- Key executive findings:
  - BoE adopted a multilayered or "mixed strategy" combining conventional and unconventional tools plus communication.
  - BoE communication, particularly Brexit-related communication, significantly shaped market expectations and reduced risk premia.
  - The mixed strategy contributed to lowering the term premium and to adjusting market expectations toward more expansionary monetary policy.
  - QE communication, Brexit-related communication, and the BoE’s tone interacted dynamically; their relative impacts varied over time.

### Introduction and research questions
- Background and immediate market reactions:
  - UK referendum on EU membership: 23 June 2016.
  - Market moves the day after the referendum:
    - UK stock market contracted by 3%.
    - Pound depreciated by approximately 8% against the US dollar.
    - UK 10-year government bond yields declined by 30 basis points.
  - On 4 August 2016 the BoE cut policy rates from 0.50% to 0.25% and expanded QE (increase in stock of UK government bonds purchased by £60 billion noted).
- Research questions:
  1. How did the BoE’s monetary policy (conventional and unconventional) influence UK government bond yields after Brexit?
  2. What role did the BoE’s communication strategy, including forward guidance, play in shaping markets’ risk perceptions?
  3. Was the BoE’s approach effective in navigating Brexit-related uncertainty?
- Analytical strategy:
  - Theoretical sovereign debt supply model.
  - Empirical decomposition of UK 10-year yields into risk-free rate expectations and term premium using a three-step regression (Adrian et al., 2013).
  - Text mining on BoE monetary policy summaries and MPC minutes (January 1999–December 2019) to build tone, QE, and Brexit indices.
  - Narrative approach to extract policy shocks orthogonal to real-time forecasts.
  - Local projections to estimate dynamic causal effects up to a 12-month horizon.

### Methods and novel contributions
- Theoretical model highlights:
  - Households allocate wealth among money, one-period nominal bonds (T-bills), and long-term consols; liquidity preference parameter 휙.
  - No-arbitrage with trading frictions: q'_t = q_t e^{ψ(b_t − b̄)}; larger debt stock b_t above steady-state b̄ reduces market price of consols relative to T-bills.
  - Interpretation: reductions in net supply of long-term bonds (e.g., QE) are expected to lower term premia; higher demand for liquidity or increases in net supply raise term premia.
- Text-mining and indices:
  - Corpus: BoE monetary policy summaries and MPC minutes (January 1999–December 2019).
  - Tone index (hawkishness vs dovishness) constructed via dictionary with tf-idf weights; index bounded between +1 (hawkish) and −1 (dovish).
  - QE communication index via LDA with K = 44 topics (robustness K ∈ [40,60]); Topic 19 interpreted as QE-related; QE topic proportion spikes correspond to QE announcements (first QE 2009; second QE October 2011; third QE August 2016).
  - Brexit-related communication index via LDA; Topic 16 interpreted as Brexit-related; Brexit index peaks around 14 July 2016.
  - LDA hyperparameters: α = 50/K and η = 0.1.
- Identification of communication shocks:
  - Orthogonalized to contemporaneous macro information using real-time NIESR forecasts for GDP growth, inflation, and unemployment.
  - Policy shocks defined: policy rate shock (ξ_m^{MP}); tone shock (ξ_m^{TONE}); QE shock (ξ_m^{QE}); Brexit shock (ξ_m^{Brexit}).
  - Notable extracted shock magnitudes/events:
    - Policy rate shock index falls by more than 50 basis points during the Great Recession of 2009.
    - QE communications shock rises by more than five standard deviations in 2009 at first QE announcement.
    - Brexit-related communication increased ahead of the June 2016 referendum and spikes after the vote.

### Data and estimation
- Sample: monthly frequency January 1999–December 2019 (irregular from late 2016 to accommodate MPC meeting frequency change); excludes post-December 2019 to avoid COVID-19 confounding.
- Yield decomposition: UK 10-year bond yields from NIESR Sovereign Bond Term Premia Series, decomposed into risk-free rate expectations and term premium (Adrian et al., 2013).
- Macroeconomic and financial series: monthly GDP growth (NIESR), exchange rate, FTSE100, VIX (Datastream), CPI inflation (ONS).
- Estimation details:
  - Three-step estimation method (Adrian et al., 2013) applied to zero-coupon gilt yields with five pricing factors extracted via principal components.
  - Equations referenced: (16), (17), (18), (19), (20).
  - Monthly bond yield components then used with daily data and estimated parameters to construct daily-frequency time series.

### Main empirical findings (effects on yield curve components)
- Validation: GDP growth shocks
  - GDP growth increase → rise in risk-free rate expectations.
  - GDP growth increase → decrease in term premium.
- Policy rate shock
  - Increase in policy rate → increase in risk-free rate expectations.
  - Policy rate shock → reduction in risk premium, but not statistically significant.
- QE communication shock
  - QE-related communication had a noticeable and statistically significant effect in reducing the term premium, particularly over a 1 to 5 month horizon.
  - QE communication influenced expectations of risk-free rates only marginally (near-zero nominal rates caveat).
- Tone (hawkish vs dovish) shock
  - Tone shocks contributed to reduction in term premium, with effects becoming statistically significant beyond month 11.
  - Hawkish tone shocks raised future short-term interest rate expectations.
- Brexit communication shock
  - Brexit-related communication significantly reduced the term premium, most strongly over months 4–8 after shocks.
  - Brexit communication did not have a strong relationship with short-term rate expectations.
- Complementarity and timing
  - QE communication: most effective over 1–5 months.
  - Brexit-related communication: most effective over 4–8 months.
  - Tone: most effective at horizons greater than 11 months.
  - Cumulative quantitative effect: combined influence of QE, Brexit communication, and tone decreased the risk premium by about 30 basis points (within a 95% confidence interval), with each channel contributing broadly similar magnitudes but at different horizons.

### Robustness and identification notes
- QE shock variable constructed from March 2009 when minutes began including substantial purchases-related content.
- Shocks orthogonalized to contemporaneous macro forecasts to isolate exogenous communication.
- Exogenous dummies included for:
  - Global Financial Crisis (December 2007–June 2009).
  - Euro area sovereign crisis (May 2010–July 2012).
  - Brexit (starting July 2016).
- Robustness: cutting sample at end-2016 (pre-Brexit referendum) removes significance of Brexit-related communications as expected; QE and forward guidance effects persist.

### Policy implications and interpretation
- Immediate post-referendum bond market movements were driven largely by revised expectations about accommodative monetary policy that offset Brexit-related risk premia.
- The significant fall in the term premium following the referendum appears primarily due to market expectations of a new asset purchase programme (QE) that reduced net supply of long-term debt.
- The BoE’s mixed strategy—combining conventional measures, unconventional measures (QE), and proactive communication (forward guidance, tone, Brexit-specific messaging)—acted complementarily to reduce the term premium and stabilize markets.
- Implications for central banks:
  - Strategic, well-timed communication can materially complement conventional and unconventional monetary tools in reducing term premia and managing market expectations during episodes of elevated uncertainty.
  - A dynamic and adaptive policy framework for communication enhances the ability to influence long-term interest rates and manage market risk perceptions during large shocks.

### Annex highlights
- Annex I — Long-term bond yield decomposition:
  - Uses the three-step method of Adrian et al. (2013) with five principal components and an affine pricing kernel to recover expectations about short rates and the term premium.
  - Focuses on period since the Zero Lower Bound (ZLB), from 2009 onward.
  - Historical observations: expectations about risk-free rates declined since 2009; term premium increased sharply during the financial crisis then trended downward; since 2016 expectations decoupled from term premium and increased again.
- Annex II — LDA algorithm implementation:
  - Corpus: BoE meeting minutes January 1999 to August 2018.
  - Gibbs sampling algorithm (Griffiths and Steyvers (2004)) used to approximate posterior.
  - Hyperparameters set as α = 50/K and η = 0.1.
- Annex III — Uncertainty indices:
  - Correlation facts preserved exactly:
    - Correlation between text-based Brexit-related and general uncertainty in the post-Brexit sample: 0.65.
    - Correlation of Brexit-specific uncertainty with Economic Policy Uncertainty (EPU): 0.4 (post-Brexit sample or full sample).
    - Correlation of combined general plus Brexit-related measures with EPU across entire sample: 0.56.

*Source: IMF Working Paper — Deciphering Delphic Guidance: The Bank of England and Brexit (wpiea2024160-print-pdf).*

### Executive Summary ......................................................................................................

### Executive Summary

### Introduction and research questions
- Examines the Bank of England’s (BoE) monetary policy response to the June 2016 advisory referendum (Brexit) and its impact on UK government bond yields.
- Central questions:
  - The impact of BoE's monetary policies on government bond yields.
  - The role of BoE communication strategy on market perceptions of risk.
  - The overall effectiveness of these measures in managing Brexit-triggered uncertainties.
- Approach: develop a model of sovereign debt supply to understand drivers of bond yield risk premia and complement it with an empirical analysis of BoE actions and communications.

### Methods and novel contributions
- Develops a model of sovereign debt supply to decompose long-term bond yields into components and to identify drivers of bond yield risk premia.
- Uses text mining and machine learning techniques applied to the BoE’s monetary policy summaries and MPC minutes to construct quantitative measures of:
  - Hawkishness/dovishness of BoE communication (tone).
  - Focus on Quantitative Easing (QE) in communications.
  - Communications specifically addressing Brexit-related uncertainty.
- Employs local projections to infer the causal effect of central bank communications on components of long-term yields.

### Main empirical findings
- The BoE adopted a multilayered or "mixed strategy" in response to the unusual sequence of Brexit-related events that affected UK bond yields.
- BoE communication, particularly Brexit-related communication, played a significant role in shaping market expectations and reducing risk premia.
- The mixed strategy:
  - Contributed to lowering the term premium (a proxy for investors’ duration risk).
  - Contributed to adjusting market expectations toward more expansionary monetary policy, counterbalancing potential Brexit-related increases in risk.
- Complementarity and interplay among communication components:
  - QE communication, Brexit-related communication, and the BoE’s tone interact and their relative impacts vary over time.
  - This dynamic interplay implies that the effectiveness of each communication component changes across different periods, underscoring the need for adaptive communication strategies.

### Policy implications and interpretation
- Evidence supports the effectiveness of a coordinated monetary policy strategy that combines:
  - Conventional and unconventional monetary tools (e.g., QE).
  - Proactive communication (forward guidance) tailored to specific uncertainties (e.g., Brexit).
- The Brexit episode illustrates the importance of central bank communication strategies in times of geopolitical and economic fragmentation.
- Implication for central banks: a dynamic and adaptive policy framework for communication enhances the ability to influence long-term interest rates and manage market risk perceptions during large shocks.

*Source: IMF Working Paper — Executive Summary*

### Introduction

### wpiea2024160-print-pdf - Introduction

### Background and context
- The United Kingdom’s referendum on EU membership on 23 June 2016 generated elevated and prolonged economic uncertainty that was quickly reflected in financial markets.
- Market moves the day after the referendum:
  - UK stock market contracted by 3%.
  - Pound depreciated by approximately 8% against the US dollar.
  - UK 10-year government bond yields declined by 30 basis points.
- On 4 August 2016 the Bank of England (BoE) cut policy rates from 0.50% to 0.25% and expanded its Quantitative Easing (QE) programme (an increase in the stock of UK government bonds purchased by £60 billion is noted as part of unconventional measures).
- The paper focuses on quantifying the impact of central bank communications (forward guidance) and BoE actions on UK long-term government bond yields in the Brexit period.

### Market reactions and timeline highlights
- Immediate post-referendum bond and exchange-rate movements (within one day, market close 23 June 2016 to end of 24 June 2016):
  - Term premium dropped by 10 basis points.
  - Expectations about future risk-free rates fell by approximately 16 basis points.
  - Around 60% of the decline in bond yields can be attributed to changed expectations about future short-term rates.
- Subsequent notable events influencing yields include the MPC announcement on 4 August 2016, Theresa May’s 2016 party conference speech, and the publication of the Phase 1 agreement which affected the risk premium and exchange rate.
- From June 2016 to early 2020, the BoE implemented two rate hikes:
  - November 2017: from 0.25% to 0.50%.
  - August 2018: from 0.50% to 0.75%.
- By the end of the Brexit transition period in January 2020 the yield curve had flattened significantly.

### Research questions and approach
- Three primary questions:
  1. How did the BoE’s monetary policy (conventional and unconventional) influence UK government bond yields after Brexit?
  2. What role did the BoE’s communication strategy, including forward guidance, play in shaping markets’ risk perceptions?
  3. Was the BoE’s approach effective in navigating Brexit-related uncertainty?
- Analytical strategy:
  - Theoretical model: a stylized sovereign debt supply model linking liquidity preference, bond supply, and term premium.
  - Empirical decomposition: standard three-step regression (Adrian et al., 2013) to split UK 10-year yields into risk-free rate expectations and term premium.
  - Text mining on BoE monetary policy summaries and MPC minutes (January 1999–December 2019) to construct three indices: a tone (hawkish vs dovish) index, a QE communication index, and a Brexit-related communication index.
  - Narrative approach (Romer and Romer, 2004; Cloyne and Hürtgen, 2016) to extract policy shocks from indices, orthogonalizing to contemporaneous macro information using real-time forecasts.
  - Local projections (Jordà, 2005) to estimate dynamic causal effects of communication shocks on yield components up to a 12-month horizon.

### Theoretical model: debt supply, liquidity, and term premium
- Households allocate wealth among money, one-period nominal bonds (T-bills), and long-term consols; liquidity preference for T-bills is captured by parameter 휙.
- Government issues T-bills and consols; consolidated monetary-fiscal balance sheet adjustments enter lump-sum transfers.
- Cash-in-advance constraint binds consumption to real liquid assets: consumption ≤ real money balances + 휙 × T-bills.
- Key pricing results:
  - Consol hypothetical price: q_t ≡ β E_t [u′(c_{t+1})/u′(c_t) × p_t/p_{t+1}].
  - T-bill price: x_t = (1−휙) q_t + 휙 (liquidity premium term).
- No-arbitrage with trading frictions: q'_t = q_t e^{ψ(b_t − b̄)} where ψ captures nonlinear costs; larger debt stock b_t above steady-state b̄ reduces market price of consols relative to T-bills.
- Propositions:
  - If 휙 = 0 and b = b̄, T-bill and consol prices equal the hypothetical one-period bond price.
  - The market price of consols relative to T-bills falls with liquidity preference (휙) and with issuance of bonds (b).
- Interpretation: reductions in net supply of long-term bonds (e.g., QE) are expected to lower term premia; higher demand for liquidity or increases in net supply raise term premia.

### Text-mining of BoE communications and indices constructed
- Corpus: BoE monetary policy summaries and MPC minutes (January 1999–December 2019).
- Pre-processing steps: PDF→plain text, strip cover and voting sections, tokenization, remove whitespace/punctuation/numbers/capitalization, stopwords removal, stemming.
- Indices constructed:
  - Tone (hawkishness vs dovishness) via dictionary method:
    - Tone_{(H−D)}_m = [sum_{h∈H} (h_m w_h) − sum_{d∈D} (d_m w_d)] / [sum_{h∈H} (h_m w_h) + sum_{d∈D} (d_m w_d)]
    - w_h and w_d are tf-idf weights; index bounded between +1 (hawkish) and −1 (dovish).
  - QE communication index via Latent Dirichlet Allocation (LDA):
    - LDA with K = 44 topics (robustness K ∈ [40,60]); Topic 19 interpreted as QE-related with top tokens including “purchase”, “asset”, “program”, “bond”, “corporate”, “gilt”, “yield”, “stimulus”, “reserve”.
    - QE topic proportion spikes correspond to QE announcements (first QE 2009; second QE October 2011; third QE August 2016).
  - Brexit-related communication index via LDA:
    - Topic 16 interpreted as Brexit-related with tokens including “referendum”, “uncertainty”, “risk”, “capital”, “invest”, “leave”, “exchange”.
    - Brexit index peaks around 14 July 2016 (first MPC meeting after referendum).
- Examples of minutes passages that LDA/dictionary methods pick up are provided (e.g., May 2016 minutes noting “The outcome of the referendum on EU membership continued to be the largest domestic risk.”; Aug 2016 minutes on asset purchase programme).

### Identification of communication shocks
- Central bank communication measures orthogonalized to contemporaneous macro information using real-time NIESR forecasts for GDP growth, inflation, and unemployment (Romer and Romer (2004) style regressions).
- Policy shocks defined:
  - Policy rate shock (ξ_m^{MP}) = component of Δ Bank Rate not predicted by real-time macro information.
  - Tone shock, QE shock, Brexit shock (ξ_m^{TONE}, ξ_m^{QE}, ξ_m^{Brexit}) extracted from regressions that control for Bank Rate shock and forecast variables.
- Notable extracted shock magnitudes/events:
  - Policy rate shock index falls by more than 50 basis points during the Great Recession of 2009 (reductions in Bank Rate exceeded what real-time macro information implied).
  - QE communications shock rises by more than five standard deviations in 2009 at first QE announcement.
  - Brexit-related communication increased ahead of the June 2016 referendum and spikes after the vote.

### Data used
- Monthly frequency sample January 1999–December 2019 (irregular from late 2016 to accommodate MPC meeting frequency change).
- Yield decomposition: UK 10-year bond yields from NIESR Sovereign Bond Term Premia Series, decomposed into risk-free rate expectations and term premium (Adrian et al., 2013; see Annex I).
- Macroeconomic and financial series: monthly GDP growth (NIESR), exchange rate, FTSE100, VIX (Datastream), CPI inflation (ONS).
- Sample excludes late transition period past December 2019 to avoid confounding events like COVID-19; formal UK exit occurred on 31 January 2020 but analysis focuses on market reactions anticipated before that date.

### Empirical results: effects on yield curve components
- Validation: GDP growth shocks
  - GDP growth increase → rise in risk-free rate expectations (consistent with monetary policy reaction functions).
  - GDP growth increase → decrease in term premium (agents demand less compensation for long-term bonds).
- Policy rate shock
  - Increase in policy rate → increase in risk-free rate expectations.
  - Policy rate shock → reduction in risk premium, but not statistically significant (suggesting unconventional policy channels may be more influential on term premium).
- QE communication shock
  - QE-related communication had a noticeable and statistically significant effect in reducing the term premium, particularly over a 1 to 5 month horizon.
  - QE communication influenced expectations of risk-free rates only marginally (noting near-zero nominal rates caveat).
- Tone (hawkish vs dovish) shock
  - Tone shocks contributed to reduction in term premium, with effects becoming statistically significant beyond month 11 of the projection horizon.
  - Hawkish tone shocks also raised future short-term interest rate expectations.
- Brexit communication shock
  - Brexit-related communication significantly reduced the term premium, most strongly over months 4–8 after shocks.
  - Brexit communication did not appear to have a strong relationship with short-term rate expectations.
- Complementarity and timing
  - QE communication: most effective over 1–5 months.
  - Brexit-related communication: most effective over 4–8 months.
  - Tone: most effective at horizons greater than 11 months.
  - Cumulative quantitative effect: the combined influence of QE, Brexit communication, and tone decreased the risk premium by about 30 basis points (within a 95% confidence interval), with each channel contributing broadly similar magnitudes but at different horizons.

### Robustness checks and identification notes
- QE shock variable constructed from March 2009 when minutes began including substantial purchases-related content.
- Shocks orthogonalized to contemporaneous macro forecasts to isolate exogenous communication.
- Exogenous dummies included for Global Financial Crisis (December 2007–June 2009), Euro area sovereign crisis (May 2010–July 2012), and Brexit (starting July 2016) to control large residual outliers.
- Robustness: cutting sample at end-2016 (pre-Brexit referendum) removes significance of Brexit-related communications as expected; QE and forward guidance effects persist.

### Policy implications and interpretation
- Immediate post-referendum movements in bond markets were driven largely by revised expectations about the monetary policy stance (anticipated loosening), which offset Brexit-related increases in risk premia.
- The decline in risk-free rate expectations on 24 June 2016 is interpreted as market pricing of expected accommodative policy and the subsequent policy rate cut on 4 August 2016.
- The significant fall in the term premium following the referendum appears primarily due to market expectations of a new asset purchase programme (QE) that reduced net supply of long-term debt.
- The BoE’s combination of conventional and unconventional measures, alongside communication strategies (QE communication, Brexit-focused messaging, and tone), acted complementarily to reduce the term premium and stabilize markets during Brexit-induced uncertainty.
- Communications improved transparency and helped signal commitment to manage Brexit-induced risks; different tools operated on different horizons and reinforced one another (e.g., QE signalling strengthened forward guidance credibility).

*Source: IMF Working Paper (Introduction section of "Deciphering Delphic Guidance: The Bank of England and Brexit", wpiea2024160-print-pdf).*

### 2016. Financial market risk measures did not move substantially on other days in our sample, suggesting that

### wpiea2024160-print-pdf - 2016. Financial market risk measures did not move substantially on other days in our sample, suggesting that

### Response of financial markets and main findings
- Since 2008, the Bank of England’s (BoE) announcements about its asset purchase programmes led to a persistent reduction in the term premium, which continued to drop given the BoE’s Brexit-related communication and use of the tone.
- Relating these findings to movements in bond yield components on days when news about Brexit emerged:
  - The decline in the bond yield after the referendum most likely resulted from a deterioration in the economic outlook and a change in expectations about monetary policy towards a more expansionary policy stance.
  - Brexit-related uncertainty added to pressure on sovereign bond yields.
  - Anticipation of expansionary monetary policy measures in response to Brexit appear to have offset any risk premia, possibly explaining an overall negative response of long-term gilt yields.
- The BoE implemented a mixed strategy involving both conventional and unconventional monetary policy measures:
  - Quantitative easing, use of the tone, and communication surrounding Brexit uncertainty were key elements of the mixed strategy.
  - These measures impacted the term premium at different points in time, indicating varying degrees of effectiveness and a complementary role in reducing the term premium at the long end of the curve.
- The study is presented as the first explicit investigation of the relationship between the BoE’s monetary policy strategy, its Brexit-related communication, and implications for ten-year UK bond yield.
- The evidence highlights the criticality of effective communication at times of high market volatility and suggests strategic communication can augment the effectiveness of monetary policy tools like Quantitative Easing and use of the tone (forward guidance) as a means for risk premium reduction.

### Conclusions and policy implications
- Strategic, well-timed central bank communication can materially complement conventional and unconventional monetary policy tools in reducing term premia and managing market expectations during episodes of elevated uncertainty (e.g., Brexit).
- A multipronged response that combines quantitative easing, tone and communication can produce clear complementarity in shaping market expectations and lowering risk premia.
- Effective communication is critical to navigate the economy through challenging times and can serve as an additional tool to monetary policy operations for risk-premium reduction.

### Annex I — Long-term bond yield decomposition (method and key observations)
- Decomposition objective:
  - Gilt yields are decomposed into expectations about future risk-free rates and a term premium compensating investors for liquidity, uncertainty around monetary policy stance, and general market risks.
- Estimation method:
  - A three-step estimation method proposed by Adrian et al. (2013) is applied to estimates of zero-coupon gilt yields at different maturities provided by the Bank of England.
  - From cross-sectional dispersion in yields across maturities, five pricing factors are extracted using principal components analysis and assumed to follow dynamic processes:
    - X_{t+1} = μ + Φ X_t + v_{t+1}  (equation (16))
  - With affine market prices of risk λ_t = Σ^{−1/2}(λ_0 + λ_1 X_t) and an exponentially affine pricing kernel M_{t+1}:
    - M_{t+1} = exp(−r_t − 1/2 λ_t′ λ_t − λ_t′ v_{t+1})  (equation (17))
  - Continuously compounded risk-free rate r_t is used to obtain log excess holding returns:
    - r^x_{t+1}(n−1) = ln P_{t+1}(n−1) − ln P_t(n) − r_t  (equation (18))
  - Excess returns written as:
    - r^x_{t+1}(n−1) = β^{(n−1)′}(λ_0 + λ_1 X_t) − 1/2(β^{(n−1)′}(n−1) + σ^2) + β^{(n−1)′} v_{t+1} + e_{t+1}(n−1)  (equation (19))
  - Estimation steps:
    1. Estimate equation (16) by ordinary least squares.
    2. Regress excess returns on a constant term, lagged pricing factors and factor innovations (stacked into matrix V̂_t):
       - r^x_{t+1}(n−1) = a I_T′ + β′ V̂_t + c X_t + E_{t+1}  (equation (20)); yields estimate of β and residuals Ê_{t+1} used to estimate σ^2.
    3. Estimate price of risk parameters λ_0 and λ_1 by cross-sectional regression across yields at different maturities.
    4. Calculate expectations of risk-free short-term rates by setting λ_0 and λ_1 to zero; term premium obtained as difference between short-term rate estimates and observed yields.
  - Bond yield components estimated at monthly frequency; then daily data and estimated parameters used to construct daily-frequency time series.
- Historical observations (sample focus and dynamics):
  - To avoid non-stationarity concerns over a longer horizon (1970s in particular), the analysis focuses on the period since the Zero Lower Bound (ZLB), which is from 2009 onward.
  - Historical decomposition shows expectations about risk-free rates declined since 2009, particularly as the financial crisis hit and the Bank of England cut their policy rate to historically low levels, hitting the nominal zero.
  - The term premium, after increasing sharply at the height of the financial crisis, has since continued a downward trend.
  - Since 2016, expectations decoupled from the path in the term premium and increased again, explaining much of the observed spike in sovereign bond yields.
  - After the referendum, markets started to anticipate a tapering phase, leading to increased interest rate expectations.
  - The years 2018-2019 saw a substantial surge in the 10-year expectations.
  - The arrival of the COVID-19 pandemic prompted another round of bond-buying by the BoE, realigning market expectations.

### Annex II — The LDA algorithm (corpus and implementation details)
- Corpus and vocabulary:
  - Corpus: Bank of England meeting minutes from January 1999 to August 2018, C = d_1, d_2, d_3, ..., d_C.
  - Vocabulary: V = w_1, w_2, w_3, ..., w_V.
- LDA generative hierarchical process (as applied in the study):
  1. A V-dimensional Dirichlet distribution, β^K, over the dictionary V, with hyperparameter α, is drawn for each topic.
  2. A K-dimensional Dirichlet distribution over the set of topics K, θ_d, with hyperparameter η, is drawn for each document.
  3. A multinomial distribution governed by θ_d is drawn for each word w to draw its assignment into topics, z_{d,w}.
  4. Depending on z_{d,w} and β^K, each w is drawn from a multinomial distribution governed by β_{z_{d,w}}.
- Latent and observed variables:
  - Observed: words defining the documents.
  - Latent: topics β^K, topic distributions θ_d, and word assignments z_{d,w}.
- Posterior approximation:
  - Posterior p(β, θ, z | w) requires computing p(w) in the denominator, which is intractable; the study adopts a Gibbs sampling algorithm proposed by Griffiths and Steyvers (2004) to approximate the posterior.
- Hyperparameter selection:
  - Following Griffiths and Steyvers (2004), hyperparameters set as α = 50/K and η = 0.1.

### Annex III — Uncertainty indices (text-based measures)
- Constructed measures:
  - Text-based measures of Brexit-related uncertainty and general uncertainty are constructed.
  - Correlation between the two measures is 0.65 in the post-Brexit sample.
- Comparisons with Economic Policy Uncertainty (EPU) Index:
  - Correlation of the Brexit-specific uncertainty index with EPU is 0.4 whether the post-Brexit sample is considered or not.
  - Across the entire sample, the correlation between the general uncertainty measure plus the Brexit-related uncertainty measure and the EPU is 0.56.
- Figures referenced:
  - Figure 14: Comparing text-based measures of uncertainty with Economic Policy Uncertainty (EPU) Index.
  - Figure 15: Comparing text-based measures of uncertainty with EPU Index after 2016.

### Key numeric and temporal facts (preserved exactly)
- Years and periods: 2008; 2016; since 2009 onward; 2018-2019; January 1999 to August 2018; post-Brexit sample.
- Correlations: 0.65 (between text-based Brexit-related and general uncertainty in post-Brexit sample); 0.4 (Brexit-specific uncertainty with EPU); 0.56 (combined measures with EPU across entire sample).
- Methodological citations and parameter values:
  - Three-step estimation method proposed by Adrian et al. (2013).
  - Equations (16), (17), (18), (19), (20) as detailed above.
  - LDA hyperparameters: α = 50/K and η = 0.1.

*Source: IMF Working Paper — Deciphering Delphic Guidance: The Bank of England and Brexit (selected excerpts from wpiea2024160-print-pdf).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024160-print-pdf.pdf_