## 3.1 Data, 3.2 Initial Tightening, and 3.6 Robustness checks

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---

### Data: minute-level identification, volatility measures, sources, and sample coverage
- Minute-level event-time identification
  - Event times are hand-collected from English newspapers, local language newspapers, tweets of reporters, videos of the actual announcements, and government websites.
  - The minute when a COVID-related lockdown or reopening was announced is manually identified; when the precise minute cannot be identified, the authors make their best estimate based on all available information.
  - For France’s reopening announcement, the minute is estimated using a three-step procedure (Appendix 1).
  - Announcements outside trading hours are treated as occurring at the next opening minute.
  - Counterfactual models explicitly introduce a dummy variable for the first 30 minutes after market opening and apply the same treatment to the last 30 minutes to account for potentially higher fluctuations.
- Volatility measures and horizons
  - The response variable is medium-term expected volatility proxied by options-based expected volatility indices.
  - Horizons used: one-month, three-month, and six-month; focus is especially on three-month and six-month horizons because these represent the “medium-term” and data for longer horizons are not available in all countries (only the US has a one-year-ahead volatility index).
  - For US events: CBOE S&P 500 three-month and six-month expected volatility indices (six-month is the primary medium-term proxy).
  - For Italy, Germany, and France events: Euro STOXX 50 expected volatility index used as Europe’s equivalent of the VIX.
  - For Germany events: in addition to the eurozone-wide volatility index, a Germany-specific expected volatility index based on the DAX stock price index is used.
  - Limitations: no intraday expected volatility data are available for Italy; no data beyond one month are available for France. Consequently, events in Italy and France are studied only using the eurozone-wide volatility index.
- Data sources and auxiliary variables
  - Minute-level data on expected volatility and on the underlying stock price indices are from Bloomberg.
  - Underlying stock price indices are used in constructing counterfactual models.
  - Counterfactual models include GARCH and EGARCH variants and control for additional variables (e.g., fiscal stimulus) that may be active at announcement times.
- Sample coverage and timing
  - Minute-level data cover business days from January 2, 2020 to October 29, 2020, around 210 days in total (the specific number of days varies slightly by country/region and by maturity).
  - Table 1 reports summary statistics of the volatility and stock price index data and provides the mean of daily changes defined as the highest volatility minus the lowest volatility observed during the day.

### Initial Tightening: main empirical pattern, examples, models, and interpretation
- Overview and main empirical pattern
  - In Spring 2020, numerous Western countries announced strict nationwide lockdowns as COVID cases grew rapidly.
  - Lockdown announcements often decreased market participants’ perceptions of six-month-ahead uncertainty: actual six-month-ahead volatility frequently fell below mean counterfactual paths and below the 90% confidence interval lower bound.
- Key intraday statistics (panels report sample size, mean, sd, number of days, daily change mean, daily change sd)
  - Panel (A): Six-month volatility
    - Volatility for S&P 500: Number of obs 84,849; Mean 31.1; Sd 8.3; Number of days 210; Daily change mean 1.9; Daily change sd 2.1
    - Volatility for STOXX 50: Number of obs 105,647; Mean 29.5; Sd 9.9; Number of days 213; Daily change mean 1.9; Daily change sd 2.2
    - Volatility for DAX: Number of obs 105,151; Mean 31.0; Sd 10.1; Number of days 212; Daily change mean 1.7; Daily change sd 1.9
  - Panel (B): Three-month volatility
    - Volatility for S&P 500: Number of obs 84,847; Mean 31.6; Sd 10.5; Number of days 210; Daily change mean 2.8; Daily change sd 3.4
    - Volatility for STOXX 50: Number of obs 102,171; Mean 29.8; Sd 11.9; Number of days 206; Daily change mean 2.4; Daily change sd 2.5
    - Volatility for DAX: Number of obs 101,663; Mean 30.0; Sd 11.7; Number of days 205; Daily change mean 2.0; Daily change sd 2.2
  - Panel (C): One-month volatility
    - Volatility for S&P 500: Number of obs 158,843; Mean 30.2; Sd 12.8; Number of days 210; Daily change mean 4.2; Daily change sd 4.2
    - Volatility for STOXX 50: Number of obs 91,203; Mean 31.5; Sd 17.9; Number of days 184; Daily change mean 4.2; Daily change sd 4.2
    - Volatility for DAX: Number of obs 90,889; Mean 33.0; Sd 17.4; Number of days 183; Daily change mean 4.1; Daily change sd 4.0
  - Panel (D): Stock price indices
    - S&P 500: Number of obs 85,050; Mean 3,136.0; Sd 284.7; Number of days 210; Daily change mean 52.3; Daily change sd 38.4
    - STOXX 50: Number of obs 112,854; Mean 355.1; Sd 35.5; Number of days 213; Daily change mean 6.2; Daily change sd 4.4
    - DAX: Number of obs 110,920; Mean 12,182.9; Sd 1,274.3; Number of days 212; Daily change mean 225.0; Daily change sd 136.0
- Minute-level event evidence (examples)
  - Trump’s state-of-emergency declaration (March 13, minute of remarks completion 3:36 pm):
    - Six-month-ahead volatility dropped sharply starting at 3:36 pm and fell by as much as 3.2% during one minute shortly after.
    - In 15 minutes it dropped from a level of 45 at 3:36 pm to 41 at 3:51 pm, more than twice the average daily change (about 1.9).
    - The drop lay below the lower bound of the 90% confidence interval for the counterfactual.
  - Germany’s historic national lockdown announcement:
    - Six-month volatility fell significantly below the mean counterfactual and below the 90% confidence interval lower bound despite the actual (and expected) volatility trend being upward.
- Methodology and model specification
  - Counterfactual volatility constructed with an ARIMA(1,1,1) model augmented with:
    - the stock price index itself, and
    - the GARCH-implied volatility.
  - Rationale:
    - ARIMA component captures persistence of historical volatility patterns.
    - Stock price and GARCH components capture new information associated with the announcement.
  - Comparison to literature:
    - Similar spirit to Engle and Gallo (2006) who add a one-month-ahead MEM-implied volatility forecast to an AR(1) model of VIX; this study adds time t GARCH-implied actual volatility to an ARIMA(1,1,1) model of VIX (ex post GARCH-implied volatility used to enhance counterfactual accuracy).
    - Similar to Fernandes, Medeiros, and Scharth (2014) factor model; here “factors” include stock price, ARIMA components, and GARCH-implied volatility.
  - Two other counterfactual models used in robustness checks.
- Identification choices and confounding controls
  - Stock price included to proxy for four forces tied to a lockdown announcement:
    1. Announcement confirms outbreak severity → tends to decrease stock price and increase short-and-medium-term expected volatility.
    2. Coincident or signaled stimulus policies (fiscal or monetary) → affect both stock price and expected volatility.
    3. Lockdown causes short-term economic disruptions → decreases stock price and increases volatility.
    4. Lockdown may deliver medium-term benefit by containing the outbreak → reduces expected volatility.
  - Controlling for stock price helps proxy Forces (1)-(3), allowing focus on detection of Force (4).
  - Monetary easing ambiguity noted: aggressive easing may reduce volatility or be interpreted as central bank running out of firepower, increasing volatility. Example: on March 15 Sunday, the Fed cut 125 bps to 0 and launched a massive $700b QE; some news articles link this to market fear and the massive stock price declines on March 16 and morning VIX spikes on March 16.
  - Event window: 30-minute window used to control for confounding events; intraday literature considers 30 minutes relatively long but markets digest new information within 5-60 minutes.
- Model validation and prediction error evidence
  - Figure 2: average prediction errors (actual minus counterfactual) for six-month-ahead volatility at each minute across the sample:
    - Mean prediction errors across all minutes during the 30-minute event window are very close to 0 — counterfactual model broadly unbiased.
    - Most negative prediction errors (actual much lower than counterfactual) are concentrated on event days; prediction errors on event days fall below the 10th percentiles of empirical distributions.
    - This pattern reaffirms that events induced significant downward revisions in volatility forecasts relative to counterfactuals.
- Macroeconomic relevance
  - Regressing growth rate of normalized purchasing manager index (PMI) on various (lagged) VIX measures:
    - VIX measures are negatively and significantly correlated with growth rate of the one-month-ahead PMI.
    - Horizon Oct 2010 – Sep 2020; p-values reported; significance marks: *** p<0.01, ** p<0.05, * p<0.1.
- Summary interpretation
  - Empirical takeaway: lockdown announcements often lower medium-term (six-month-ahead) expected volatility even when short-term effects might increase volatility.
  - Suggestive mechanism: intertemporal trade-off — lockdowns disrupt the economy (raising short-term volatility) but contain the outbreak (lowering medium-term volatility). The volatility-decreasing effect tends to dominate in the medium term.
  - Minute-level analysis and counterfactual validation mitigate endogeneity concerns, though such endogeneity cannot be fully ruled out.
  - Monotonicity across maturities: effects are strongest for six-month-ahead volatility, weaker for three-month, and generally absent for one-month volatility in aggregate.

### Robustness checks: alternative counterfactuals, regressions, decomposition, and panel evidence
- Robustness checks overview
  - Three sets of robustness checks conducted; alternative counterfactual models’ empirical prediction error bands not constructed due to heavy computation burden (construction for each event takes more than 5 hours).
- Event-study robustness checks
  - First check: drop the GARCH-implied volatility predictor; use an ARIMA model augmented only with the stock price index. Results for six-month-ahead volatility are very similar to the main results (Appendix Figures 3-5).
  - Second check: replace GARCH-implied volatility with EGARCH-implied volatility; use ARIMA + stock price index + EGARCH-implied volatility. Results presented in Appendix Figures 6-8 and similar to main results.
  - Third check: decompose overlapping forward-looking volatility indices into non-overlapping components (three-month-ahead and Month 3–6) and repeat event studies; results very similar to main results (Online Appendices).
- Regression-analysis robustness checks
  - Daily regression sample: Weekdays from January 3, 2020 to October 22, 2020; countries/regions: US, Italy, Germany, Euro Area, and the UK.
  - SUR (seemingly unrelated regressions) to account for correlations among countries and volatility products; SUR requires a balanced panel and is conducted for the whole sample only. Appendix Table 3 reports SUR results very similar to full-model results during initial tightening.
  - Decompose six-month-ahead volatility into three-month-ahead and Month 3–6 and repeat regressions for stages; results similar to corresponding stage results (Online Appendices).
- Key regression findings (summarized)
  - Initial tightening stage (full model, Table 3):
    - StringencyIndex: insignificant by itself in the full model; the interaction term (Cases_pct_SI) is significantly negative.
    - Interpretation: containment measures reduce expected volatility mainly through interaction with outbreak dynamics — the more severe the outbreak, the stronger the effect.
    - StockPrice_pct: highly significant and negatively correlated with volatility in all specifications.
    - Lag_StockPrice_pct: positive correlation; StockPrice_pct current term is negative.
    - Observations: 396 (for reported specifications).
    - R-squared: 0.532, 0.526, 0.550 (reported across columns).
    - Selected coefficients (Table 3 examples):
      - StockPrice_pct: -1.849***, -1.883***, -1.919***, -1.874***, -1.894***, -1.933*** (p-values 0.000)
      - Lag_StockPrice_pct: 0.247**, 0.207**, 0.197**, 0.199**, 0.173*, 0.156 (p-values 0.013, 0.039, 0.048, 0.043, 0.082, 0.117)
      - StockPrice_std: -2.165***, -2.385***, -2.007***, -2.780***, -2.930***, -2.639*** (p-values 0.000, 0.000, 0.001, 0.000, 0.000, 0.000)
      - Lag_StockPrice_std: 4.260***, 4.085***, 4.410***, 3.694***, 3.561***, 3.802*** (p-values 0.000)
  - Easing stage (Panel A, Table 4):
    - StockPrice_pct remains highly significant and negatively correlated with expected volatility.
    - Neither StringencyIndex nor Cases_pct nor Cases_pct_SI are statistically significant in the easing stage.
    - Observations: 257.
    - R-squared: 0.598, 0.597, 0.600 (reported across columns).
    - Example coefficients (Panel A):
      - StockPrice_pct: -2.023***, -2.018***, -2.026***, -2.026***, -2.020***, -2.029*** (p-values 0.000)
  - Retightening stage (Panel B, Table 4):
    - StockPrice_pct remains highly significant and negatively correlated with expected volatility.
    - During retightening, more stringent containment measures are again associated with lower volatility; interaction significance lower than in initial tightening.
    - Observations: 289.
    - R-squared: 0.538, 0.537, 0.547 (reported across columns).
    - Example coefficients (Panel B):
      - Cases_pct: 2.299, 128.279**, 3.033, 130.986** (p-values 0.778, 0.022, 0.705, 0.017)
      - Cases_pct_SI: -2.159**, -2.206** (p-values 0.024, 0.018)
      - StockPrice_pct: -1.729***, -1.719***, -1.720***, -1.718***, -1.718***, -1.707*** (p-values 0.000)
  - Magnitude and economic significance:
    - The economic significance of the interaction term has the same order of magnitude as the stock price percent change, indicating stringency is economically relevant in driving expected volatility.
- Overlap decomposition robustness
  - Decomposing six-month-ahead volatility into non-overlapping components (three-month ahead and Month 3–6) and repeating analyses yields results similar to main-stage results (Online Appendices).
- Conclusion from robustness exercises
  - Across alternative counterfactual predictors (excluding GARCH-implied volatility; using EGARCH-implied volatility), SUR estimation, and decomposition into non-overlapping indices, results remain very similar to the main findings:
    - COVID containment measures reduce six-month-ahead expected stock price volatility indices.
    - Effects are weaker or absent for three-month-ahead and one-month-ahead expected volatility.
    - Stock price percent change is a robust and highly significant negative predictor of expected volatility.

*Source: wpiea2021157-print-pdf — Sections 3.1, 3.2, and 3.6*

### 3.1 Data ...............................................................................................................

### 3.1 Data

### Minute-level event-time identification
- Event times are hand-collected by deep dives into multiple information sources: English newspapers, local language newspapers, tweets of reporters, videos of the actual announcements, and government websites.
- The minute when a COVID-related lockdown or reopening was announced is manually identified; when the precise minute cannot be identified, the authors make their best estimate based on all available information.
- For France’s reopening announcement, the minute is estimated using a three-step procedure (Appendix 1).
- Announcements outside trading hours are treated as occurring at the next opening minute.
- Counterfactual models explicitly introduce a dummy variable for the first 30 minutes after market opening and apply the same treatment to the last 30 minutes to account for potentially higher fluctuations.

### Volatility measures and horizons
- The response variable is medium-term expected volatility proxied by options-based expected volatility indices.
- Horizons used: one-month, three-month, and six-month; the paper focuses especially on three-month and six-month horizons because these represent the “medium-term” and data for longer horizons are not available in all countries (only the US has a one-year-ahead volatility index).
- For US events: CBOE S&P 500 three-month and six-month expected volatility indices (six-month is the primary medium-term proxy).
- For Italy, Germany, and France events: Euro STOXX 50 expected volatility index is used as Europe’s equivalent of the VIX.
- For Germany events: in addition to the eurozone-wide volatility index, a Germany-specific expected volatility index based on the DAX stock price index is used.
- Limitations: no intraday expected volatility data are available for Italy; no data beyond one month are available for France. Consequently, events in Italy and France are studied only using the eurozone-wide volatility index.

### Data sources and auxiliary variables
- Minute-level data on expected volatility and on the underlying stock price indices are from Bloomberg.
- Underlying stock price indices (e.g., S&P 500 for the US) are used in constructing counterfactual models.
- Counterfactual models include GARCH and EGARCH variants and control for additional variables (e.g., fiscal stimulus) that may be active at announcement times.

### Sample coverage and timing
- For all countries/regions, the minute-level data used in the event studies cover business days from January 2, 2020 to October 29, 2020, around 210 days in total (the specific number of days varies slightly by country/region and by maturity, i.e., six-month or three-month).
- Table 1 (Summary Statistics of Event Study Data) reports summary statistics of the volatility and stock price index data and provides the mean of daily changes defined as the highest volatility minus the lowest volatility observed during the day; Table 1 does not distinguish between initial tightening, easing, or retightening stages because the same data are used to train the volatility prediction models.

*Source: wpiea2021157-print-pdf — Section 3.1 Data*

### 3.2 Initial Tightening

### 3.2 Initial Tightening

### Overview and main empirical pattern
- In Spring 2020, numerous Western countries announced strict nationwide lockdowns as COVID cases grew rapidly.
- Contrary to conventional wisdom, lockdown announcements often decreased market participants’ perceptions of six-month-ahead uncertainty: actual six-month-ahead volatility frequently fell below mean counterfactual paths and below the 90% confidence interval lower bound.

### Key intraday statistics (panels report sample size, mean, sd, number of days, daily change mean, daily change sd)
- Panel (A): Six-month volatility
  - Volatility for S&P 500: Number of obs 84,849; Mean 31.1; Sd 8.3; Number of days 210; Daily change mean 1.9; Daily change sd 2.1
  - Volatility for STOXX 50: Number of obs 105,647; Mean 29.5; Sd 9.9; Number of days 213; Daily change mean 1.9; Daily change sd 2.2
  - Volatility for DAX: Number of obs 105,151; Mean 31.0; Sd 10.1; Number of days 212; Daily change mean 1.7; Daily change sd 1.9
- Panel (B): Three-month volatility
  - Volatility for S&P 500: Number of obs 84,847; Mean 31.6; Sd 10.5; Number of days 210; Daily change mean 2.8; Daily change sd 3.4
  - Volatility for STOXX 50: Number of obs 102,171; Mean 29.8; Sd 11.9; Number of days 206; Daily change mean 2.4; Daily change sd 2.5
  - Volatility for DAX: Number of obs 101,663; Mean 30.0; Sd 11.7; Number of days 205; Daily change mean 2.0; Daily change sd 2.2
- Panel (C): One-month volatility
  - Volatility for S&P 500: Number of obs 158,843; Mean 30.2; Sd 12.8; Number of days 210; Daily change mean 4.2; Daily change sd 4.2
  - Volatility for STOXX 50: Number of obs 91,203; Mean 31.5; Sd 17.9; Number of days 184; Daily change mean 4.2; Daily change sd 4.2
  - Volatility for DAX: Number of obs 90,889; Mean 33.0; Sd 17.4; Number of days 183; Daily change mean 4.1; Daily change sd 4.0
- Panel (D): Stock price indices
  - S&P 500: Number of obs 85,050; Mean 3,136.0; Sd 284.7; Number of days 210; Daily change mean 52.3; Daily change sd 38.4
  - STOXX 50: Number of obs 112,854; Mean 355.1; Sd 35.5; Number of days 213; Daily change mean 6.2; Daily change sd 4.4
  - DAX: Number of obs 110,920; Mean 12,182.9; Sd 1,274.3; Number of days 212; Daily change mean 225.0; Daily change sd 136.0

### Event evidence (minute-level examples)
- Trump’s state-of-emergency declaration (March 13, minute of remarks completion 3:36 pm):
  - Six-month-ahead volatility dropped sharply starting at 3:36 pm and fell by as much as 3.2% during one minute shortly after.
  - In 15 minutes it dropped from a level of 45 at 3:36 pm to 41 at 3:51 pm, more than twice the average daily change (about 1.9).
  - The drop lay below the lower bound of the 90% confidence interval for the counterfactual.
- Germany’s historic national lockdown announcement:
  - Six-month volatility fell significantly below the mean counterfactual and below the 90% confidence interval lower bound despite the actual (and expected) volatility trend being upward.

### Methodology and model specification
- Counterfactual volatility constructed with an ARIMA(1,1,1) model augmented with:
  - the stock price index itself, and
  - the GARCH-implied volatility.
- Rationale:
  - ARIMA component captures persistence of historical volatility patterns.
  - Stock price and GARCH components capture new information associated with the announcement.
- Approach compared to literature:
  - Similar spirit to Engle and Gallo (2006) who add a one-month-ahead MEM-implied volatility forecast to an AR(1) model of VIX; this study adds time t GARCH-implied actual volatility to an ARIMA(1,1,1) model of VIX (ex post GARCH-implied volatility used to enhance counterfactual accuracy).
  - Similar to Fernandes, Medeiros, and Scharth (2014) factor model; here “factors” include stock price, ARIMA components, and GARCH-implied volatility.
- Two other counterfactual models are used in robustness checks (discussed elsewhere in the source).

### Identification choices and confounding controls
- Stock price included to proxy for four forces tied to a lockdown announcement:
  1. Announcement confirms outbreak severity → tends to decrease stock price and increase short-and-medium-term expected volatility.
  2. Coincident or signaled stimulus policies (fiscal or monetary) → affect both stock price and expected volatility.
  3. Lockdown causes short-term economic disruptions → decreases stock price and increases volatility.
  4. Lockdown may deliver medium-term benefit by containing the outbreak → reduces expected volatility.
- Controlling for stock price helps proxy Forces (1)-(3), allowing focus on detection of Force (4).
- Monetary easing ambiguity noted: aggressive easing may reduce volatility or be interpreted as central bank running out of firepower, increasing volatility. Example: on March 15 Sunday, the Fed cut 125 bps to 0 and launched a massive $700b QE; some news articles link this to market fear and the massive stock price declines on March 16 and morning VIX spikes on March 16.
- Event window: 30-minute window used to control for confounding events; intraday literature considers 30 minutes relatively long but markets digest new information within 5-60 minutes.

### Model validation and prediction error evidence
- Figure 2: average prediction errors (actual minus counterfactual) for six-month-ahead volatility at each minute across the sample:
  - Mean prediction errors across all minutes during the 30-minute event window are very close to 0 — counterfactual model broadly unbiased.
  - Most negative prediction errors (actual much lower than counterfactual) are concentrated on event days; prediction errors on event days fall below the 10th percentiles of empirical distributions.
  - This pattern reaffirms that events induced significant downward revisions in volatility forecasts relative to counterfactuals.

### Macroeconomic relevance
- Simple test: regressing growth rate of normalized purchasing manager index (PMI) on various (lagged) VIX measures.
  - Result: VIX measures are negatively and significantly correlated with growth rate of the one-month-ahead PMI.
  - Interpretation: the forward-looking volatility indices used are relevant to the macroeconomy, not just financial markets.
  - Notes: Horizon Oct 2010 – Sep 2020; p-values in parentheses; significance marks: *** p<0.01, ** p<0.05, * p<0.1.

### Summary interpretation
- Empirical takeaway: lockdown announcements often lower medium-term (six-month-ahead) expected volatility even when short-term effects might increase volatility.
- Suggestive mechanism: intertemporal trade-off — lockdowns disrupt the economy (raising short-term volatility) but contain the outbreak (lowering medium-term volatility). The volatility-decreasing effect tends to dominate in the medium term.
- The minute-level analysis and counterfactual validation mitigate endogeneity concerns stemming from common shocks (e.g., rising COVID cases driving both policy and volatility expectations), though such endogeneity cannot be fully ruled out.
- Monotonicity across maturities (expanded on in later sections): effects are strongest for six-month-ahead volatility, weaker for three-month, and generally absent for one-month volatility in aggregate, consistent with the intertemporal trade-off explanation.

*Source: wpiea2021157-print-pdf — Section 3.2 Initial Tightening*

### 3.6 Robustness checks

### 3.6 Robustness checks

### Robustness checks overview
- Three sets of robustness checks are conducted to validate the main results.
- Computation note: the alternative counterfactual models’ empirical prediction error bands are not constructed due to the heavy computation burden (the construction for each event takes more than 5 hours).

### Event-study robustness checks
- First check:
  - Counterfactual volatility constructed by dropping the GARCH-implied volatility predictor.
  - Use an ARIMA model augmented only with the stock price index as an additional predictor.
  - Results for six-month-ahead volatility are presented in Appendix Figures 3-5 (one figure per stage: initial tightening, easing, retightening).
  - All results are very similar to the main results.
- Second check:
  - Replace GARCH-implied volatility with EGARCH-implied volatility in the counterfactual models.
  - Use an ARIMA model augmented with the stock price index and EGARCH-implied volatility as two additional predictors.
  - Results are presented in Appendix Figures 6-8 (one figure per stage).
- Third check:
  - Decompose overlapping forward-looking volatility indices to obtain non-overlapping indices.
  - Decompose the six-month-ahead volatility index into (a) the three-month-ahead volatility and (b) volatility from Month 3 to Month 6.
  - Repeat event studies for initial tightening, easing, and retightening using non-overlapping indices.
  - Results are available in the Online Appendices and are very similar to the main results.

### Regression-analysis robustness checks
- Daily regression sample:
  - Weekdays from January 3, 2020 to October 22, 2020.
  - Countries/regions covered due to expected volatility data limitations: the US, Italy, Germany, Euro Area, and the UK (five countries/regions).
- Additional robustness exercises:
  - SUR (seemingly unrelated regressions) to account for correlations among countries and volatility products; SUR requires a balanced panel so it is conducted for the whole sample only (no stage distinctions).
    - Appendix Table 3 reports SUR results which are very similar to the full-model results during the initial tightening stage.
  - Decompose six-month-ahead volatility into the three-month-ahead volatility and Month 3–6 volatility and repeat regressions for initial tightening, easing, and retightening (full models).
    - Results are available in the Online Appendices and are similar to the corresponding stage results.

### Key regression findings (summarized from full models and panels)
- Initial tightening stage (full model, Table 3):
  - StringencyIndex: insignificant by itself in the full model; the interaction term (Cases_pct_SI) is significantly negative.
  - Interpretation: (a) marginal effect of stringency on expected volatility equals coefficient of interaction term multiplied by COVID case growth rate (COVID case growth rate is positive); (b) containment measures reduce expected volatility mainly through interaction with outbreak dynamics—the more severe the outbreak, the stronger the effect; (c) containment measures mitigate the volatility-increasing effect of COVID case growth.
  - StockPrice_pct: highly significant and negatively correlated with volatility in all specifications.
  - StockPrice_std and Lag_StockPrice_std: lagged term is positively correlated with volatility; current term is negatively correlated.
  - Observations: 396 (for reported specifications).
  - R-squared: 0.532, 0.526, 0.550 (reported across columns).
  - Selected coefficient and significance examples (Table 3):
    - StockPrice_pct: -1.849***, -1.883***, -1.919***, -1.874***, -1.894***, -1.933*** (p-values 0.000).
    - Lag_StockPrice_pct: 0.247**, 0.207**, 0.197**, 0.199**, 0.173*, 0.156 (p-values 0.013, 0.039, 0.048, 0.043, 0.082, 0.117).
    - StockPrice_std: -2.165***, -2.385***, -2.007***, -2.780***, -2.930***, -2.639*** (p-values 0.000, 0.000, 0.001, 0.000, 0.000, 0.000).
    - Lag_StockPrice_std: 4.260***, 4.085***, 4.410***, 3.694***, 3.561***, 3.802*** (p-values 0.000).
- Easing stage (Panel A, Table 4):
  - StockPrice_pct remains highly significant and negatively correlated with expected volatility.
  - Neither StringencyIndex nor Cases_pct nor Cases_pct_SI are statistically significant in the easing stage.
  - Observations: 257.
  - R-squared: 0.598, 0.597, 0.600 (reported across columns).
  - Example coefficients (Panel A):
    - StockPrice_pct: -2.023***, -2.018***, -2.026***, -2.026***, -2.020***, -2.029*** (p-values 0.000).
- Retightening stage (Panel B, Table 4):
  - StockPrice_pct remains highly significant and negatively correlated with expected volatility.
  - During retightening, more stringent containment measures are again associated with lower volatility; statistical significance of interaction is lower than in initial tightening.
  - Observations: 289.
  - R-squared: 0.538, 0.537, 0.547 (reported across columns).
  - Example coefficients (Panel B):
    - Cases_pct: 2.299, 128.279**, 3.033, 130.986** (p-values 0.778, 0.022, 0.705, 0.017).
    - Cases_pct_SI: -2.159**, -2.206** (p-values 0.024, 0.018).
    - StockPrice_pct: -1.729***, -1.719***, -1.720***, -1.718***, -1.718***, -1.707*** (p-values 0.000).
- Magnitude and economic significance:
  - The economic significance of the interaction term has the same order of magnitude as the stock price percent change, indicating stringency is economically relevant in driving expected volatility.

### Overlap decomposition robustness (both event studies and regressions)
- Because six-month-ahead volatility overlaps the three-month-ahead index for the first three months, both event-study and regression approaches decompose the six-month index into non-overlapping components (three-month ahead and Month 3–6).
- Repeating analyses with non-overlapping indices yields results similar to the corresponding main-stage results (Online Appendices).

### Conclusion from robustness exercises
- Across alternative counterfactual predictors (excluding GARCH-implied volatility; using EGARCH-implied volatility), SUR estimation, and decomposition into non-overlapping indices, results remain very similar to the main findings:
  - COVID containment measures reduce six-month-ahead expected stock price volatility indices.
  - Effects are weaker or absent for three-month-ahead and one-month-ahead expected volatility.
  - Stock price percent change is a robust and highly significant negative predictor of expected volatility.

*Source: wpiea2021157-print-pdf — 3.6 Robustness checks*

### References

### wpiea2021157-print-pdf - References

### References
- Bibliographic list includes empirical and theoretical studies on COVID-19 economic and financial effects, volatility indices, event-study methodology, and policy evaluation. Key cited works include:
  - Alvarez, Fernando, David Argente, Francesco Lippi. 2020. “A Simple Planning Problem for COVID-19 Lockdown,” Manuscript, University of Chicago.
  - Arnon, Alexander, John Ricco, and Kent Smetters. 2020. “Epidemiological and Economic Effects of Lockdown,” Brookings Papers on Economic Activity Conference Drafts.
  - Ashraf, Badar Nadeem. 2020. “Economic Impact of Government Actions to Control COVID-19 Pandemic: Evidence from Financial Markets,” Journal of Behavioral and Experimental Finance.
  - Baek, Seungho, Sunil K. Mohanty, and Mina Glambosky. 2020. “COVID-19 and Stock Market Volatility: An Industry Level Analysis,” Finance Research Letters, https://doi.org/10.1016/j.frl.2020.101748
  - Baker, Scott R., Nicholas Bloom, Steven J. Davis, Kyle Kost, Marco Sammon, and Tasaneeya Viratyosin. 2020. “The Unprecedented Stock Market Reaction to COVID-19,” Review of Asset Pricing Studies, 0: 1-17, doi/10.1093/rapstu/raaa008/5873533
  - Barro, Robert J., José F. Ursúa, and Joanna Weng. 2020. “The Coronavirus and the Great Influenza Pandemic: Lessons from the ‘Spanish Flu’ for the Coronavirus's Potential Effects on Mortality and Economic Activity,” NBER Working Paper No. 26866.
  - Caselli, Francesca, Francesco Grigoli, Weicheng Lian, and Damiano Sandri. 2020. “Protecting Lives and Livelihoods with Early and Tight Lockdowns,” IMF Working Paper No. 20/234.
  - CBOE. 2019. “White Paper: Cboe Volatility Index,” https://cdn.cboe.com/resources/futures/vixwhite.pdf.
  - Chen, Sophia, Deniz Igan, Nicola Pierri, and Andrea F. Presbitero. 2020. “Tracking the Economic Impact of COVID-19 and Mitigation Policies in Europe and the United States,” COVID Economics, 36: 1-24.
  - Correia, Sergio, Stephan Luck, and Emil Verner. 2020. “Pandemics Depress the Economy, Public Health Interventions Do Not: Evidence from the 1918 Flu,” Working Paper.
  - Deb, Pragyan, Davide Furceri, Jonathan D. Ostry, and Nour Tawk. 2020a. “The Economic Effects of COVID-19 Containment Measures,” COVID Economics, 24: 32–75.
  - Deb, Pragyan, Davide Furceri, Jonathan D. Ostry, and Nour Tawk. 2020b. “The Effect of Containment Measures on the COVID-19 Pandemic,” IMF Working Paper No. 20/159.
  - Eichenbaum, Martin, Sergio Rebelo, and Mathias Trabandt. “The Macroeconomics of Epidemics,” March 2020. Northwestern University Working Paper.
  - Goolsbee, Austan, and Chad Syverson. 2020. “Fear, Lockdown, and Diversion: Comparing Drivers of Pandemic Economic Decline,” NBER Working Paper No. 27432.
  - Hall, Robert E., Charles I. Jones, and Peter J. Klenow. 2020. “Trading Off Consumption and COVID-19 Deaths,” Stanford University Working Paper.
  - Jackwerth, Jens. 2020. “What Do Index Options Teach Us About COVID-19?” Review of Asset Pricing Studies, 10: 618-634.
  - Jones, Callum, Thomas Philippon, and Venky Venkateswaran. 2020. “Optimal Mitigation Policies in A Pandemic,” NYU Working Paper.
  - Sheridan, Adam, Asger Lau Andersen, Emil Toft Hansen, and Niels Johannesen. 2020. "Social Distancing Laws Cause Only Small Losses of Economic Activity During The COVID-19 Pandemic in Scandinavia,” PNAS August 25, 2020 117 (34), https://doi.org/10.1073/pnas.2010068117
  - Zaremba, Adam, Renatas Kizys, David Y. Aharond, and Ender Demir. 2020. “Infected Markets: Novel Coronavirus, Government Interventions, and Stock Return Volatility around the Globe,” Finance Research Letters, 35, https://doi.org/10.1016/j.frl.2020.101597
  - Zhang, Dayong, Min Hu, and Qiang Ji. 2020. “Financial Markets under the Global Pandemic of COVID-19,” Finance Research Letters, 35, https://doi.org/10.1016/j.frl.2020.101528
- Additional cited works cover volatility modeling (e.g., Fernandes, Marcelo, Marcelo C. Medeiros, and Marcel Scharth. 2014), market underreaction (Cheng, Ing-Haw. 2020), event-study methodology (McWilliams and Siegel. 1997), and sectoral market behavior (Haroon and Rizvi. 2020).

### Appendix 1 — Estimating the Event Time: An Example
- Purpose: Example procedure to estimate event minute when exact time is not readily available.
- Three-step procedure applied to France’s reopening announcement:
  - Step 1: Identified a France24 article published at 14:38 of July 5, 2020.
  - Step 2: Determined France24 publication times are in French time by cross-checking an article published on November 12, 2020 that showed “11:33” while actual Washington DC time was 8:39 am Eastern time.
  - Step 3: Inferred announcement time from a cited reporter’s tweet showing 10:15 am of May 7, 2020 (in French time). Concluded announcement time was a few minutes before 10:15 am French time and used 10 am French time of May 7 as the event time, which equals 4 am Eastern time.

### Appendix Figures (listed)
- Appendix Figure 1. Prediction Errors for Six-Month Volatility (Easing)
- Appendix Figure 2. Prediction Errors for Six-Month Volatility (Retightening)
- Appendix Figure 3. Volatility Responses Across Maturities: Italy’s Initial Tightening
- Appendix Figure 4. ARIMA Model: Six-Month Volatility Indices (Initial Tightening)
- Appendix Figure 5. ARIMA Model: Six-Month Volatility Indices (Easing)
- Appendix Figure 6. ARIMA Model: Six-Month Volatility Indices (Retightening)
- Appendix Figure 7. EGARCH Model: Six-Month Volatility Indices (Initial Tightening)
- Appendix Figure 8. EGARCH Model: Six-Month Volatility Indices (Easing)
- Appendix Figure 9. EGARCH Model: Six-Month Volatility Indices (Retightening)

### Appendix Table 1 — Summary Statistics of Panel Regression Data (selected exact statistics)
- Panel (A): Initial tightening (Number of obs = 401)
  - V_3M_pct: Mean 108.3, Sd 922.4, Min -4,881.7, Max 4,013.7
  - V_6M_pct: Mean 96.5, Sd 791.8, Min -4,875.3, Max 3,436.8
  - StringencyIndex: Mean 41.8, Sd 35.5, Min 0.0, Max 93.5
  - Cases_pct: Mean 18.2, Sd 37.4, Min 0.0, Max 400.0
  - Cases_pct_SI: Mean 660.3, Sd 1,263.5, Min 0.0, Max 11,174.8
  - StockPrice_pct: Mean -23.4, Sd 296.3, Min -1,692.4, Max 1,097.6
  - StockPrice_std: Mean 45.6, Sd 72.9, Min 0.3, Max 804.3
- Panel (B): Easing tightening (Number of obs = 257)
  - V_3M_pct: Mean -11.3, Sd 625.0, Min -2,487.0, Max 3,672.6
  - V_6M_pct: Mean -16.0, Sd 450.1, Min -1,252.8, Max 2,705.7
  - StringencyIndex: Mean 64.6, Sd 9.6, Min 42.6, Max 93.5
  - Cases_pct: Mean 1.0, Sd 1.1, Min 0.1, Max 7.5
  - Cases_pct_SI: Mean 66.9, Sd 81.9, Min 3.4, Max 519.1
  - StockPrice_pct: Mean 21.8, Sd 169.5, Min -589.4, Max 567.3
  - StockPrice_std: Mean 30.5, Sd 34.4, Min 0.4, Max 153.4
- Panel (C): Retightening stage (Number of obs = 289)
  - V_3M_pct: Mean -1.9, Sd 442.3, Min -2,195.0, Max 2,440.0
  - V_6M_pct: Mean -5.2, Sd 270.1, Min -730.6, Max 1,553.5
  - StringencyIndex: Mean 57.8, Sd 8.6, Min 43.5, Max 69.9
  - Cases_pct: Mean 1.4, Sd 1.4, Min 0.0, Max 9.3
  - Cases_pct_SI: Mean 77.5, Sd 81.6, Min 1.5, Max 628.6
  - StockPrice_pct: Mean 1.2, Sd 114.4, Min -437.5, Max 322.1
  - StockPrice_std: Mean 25.4, Sd 28.3, Min 0.3, Max 175.6
- Notes:
  - (1) pct = percent change; std = standard deviation; Cases_pct_SI is the interaction of COVID case percent change and stringency index.
  - (2) Because of the rescaling, the units of the volatility percent change (e.g., V_3M_pct) and of the stock price percent change are basis point (i.e., 1/100 percent); the units of the Cases_pct and StringencyIndex remain as percent.

### Appendix Table 2 — Initial Tightening Stage Panel Regressions in the Benchmark Model (selected coefficients and stats)
- Models reported: FE and RE specifications, columns (1) to (6) with SI, Cases, Interaction variants.
- Selected coefficients (p-values in parentheses):
  - StringencyIndex: -1.868 (0.101); -1.462 (0.237); -1.996* (0.072); -1.515 (0.213)
  - Cases_pct: 3.092*** (0.004); 3.358** (0.012); 3.173*** (0.002); 3.418*** (0.010)
  - Cases_pct_SI: -0.018 (0.659); -0.019 (0.640)
- Observations: 401 (for all reported columns)
- R-squared: 0.007; 0.021; 0.027 (as reported)

### Appendix Table 3 — Initial Tightening Stage Panel Regressions in the Stock Price Model (selected coefficients and stats)
- Models reported: FE and RE specifications, columns (1) to (6) with SI, Cases, Interaction variants.
- Selected coefficients (p-values in parentheses):
  - StringencyIndex: -0.850 (0.321); 0.425 (0.650); -0.998 (0.233); 0.354 (0.701)
  - Cases_pct: 0.312 (0.706); 2.082** (0.037); 0.436 (0.595); 2.196** (0.025)
  - Cases_pct_SI: -0.099*** (0.002); -0.101*** (0.001)
  - StockPrice_pct: -1.778*** (0.000); -1.777*** (0.000); -1.830*** (0.000); -1.774*** (0.000); -1.771*** (0.000); -1.825*** (0.000)
  - Lag_StockPrice_pct: 0.174* (0.086); 0.173* (0.094); 0.131 (0.206); 0.178* (0.077); 0.178* (0.082); 0.135 (0.188)
- Observations: 396 (for all reported columns)
- R-squared: 0.464; 0.463; 0.478 (as reported)

### Appendix Table 4 — Seemingly Unrelated Regression Results (Full Sample, All Stages)
- Dependent variables include V_3M_pct and V_6M_pct across columns (1) to (4).
- Selected coefficients for V_3M_pct (p-values in parentheses):
  - StringencyIndex: -3.287*** (0.000); -2.825*** (0.004); -1.004 (0.183)
  - Cases_pct: 4.448*** (0.000); 3.452*** (0.004); 1.942** (0.030)
  - Cases_pct_SI: 0.025 (0.468); -0.080*** (0.003)
  - StockPrice_pct: -2.234*** (0.000) (included in column where reported)
  - Lag_StockPrice_pct: -0.027 (0.743) (where reported)
  - (mean) Std_SP: -0.837* (0.069) (where reported)
  - Lag_StockPrice_std: 1.678*** (0.000) (where reported)
- Selected coefficients for V_6M_pct (p-values in parentheses):
  - StringencyIndex: -2.423*** (0.001); -1.756** (0.027); -0.485 (0.400)
  - Cases_pct: 3.500*** (0.000); 3.457*** (0.000); 2.124*** (0.002)
  - Cases_pct_SI: -0.012 (0.660); -0.097*** (0.000)
  - StockPrice_pct: -1.902*** (0.000) (where reported)
  - (mean) Std_SP: -2.060*** (0.000) (where reported)
  - Lag_StockPrice_std: 2.937*** (0.000) (where reported)
- Observations: 924; 924; 924; 919 (columns 1–4 respectively)
- R-squared: 0.014; 0.026; 0.034; 0.469 (columns 1–4 respectively)
- Notes: (1) p-values are in parentheses. (2) FE = fixed effect; RE = random effect. (3) Constant not shown.

*Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021157-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021157-print-pdf.pdf_