## 4.1–4.4 Results on Stock Returns, Inflation, Monetary Policy, and the ZLB

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### 4.1 Initial Empirical Results on Real Stock Returns — Methodology and Baseline Findings
- Methodology
  - Real stock index: nominal stock index deflated by the consumer price index; real stock return is year-on-year difference of real stock index in natural logarithm.
  - Baseline regression (panel regressions with fixed effects):
    - Y_{i,t} = β0 + β1 π^e_{i,t} + β2 π^u_{i,t} + Xβ + u_i + ε_{i,t}
    - Y_{i,t}: real return on equity index for country i at time t.
    - π^e and π^u: expected and unexpected inflation.
    - X: controls including industrial production growth rate, change in financial risk, the U.S. three-month Treasury bill yield, and the VIX.
  - Panel diagnostics and econometric concerns:
    - Unit root tests: Augmented Dickey–Fuller, DF-GLS, Phillips–Perron, Im-Pesaran-Shin and Fisher-type tests; panel is unbalanced.
    - Serial correlation: Wooldridge Test rejected no-first-order-autocorrelation at the 1% significance level.
    - Heteroskedasticity: modified Wald Test rejected homoskedasticity at the 1% significance level.
    - Potential cross-sectional dependence assumed; Driscoll-Kraay standard error estimators applied.

- Baseline regression results (summary of Table 2)
  - Estimators: FE = Panel Regressions with Fixed Effects; DK = Fixed Effects with Driscoll-Kraay standard errors.
  - Samples and counts:
    - Observations: 4,573 (full); 2,557 (AM); 2,016 (EM).
    - Number of countries: 63 (full); 31 (AM); 32 (EM).
    - R-squared: 0.288 (full, columns 1 and 4); 0.405 (AM, columns 2 and 5); 0.288 or 0.289 (EM, columns 3 and 6).
  - Key coefficient estimates (point estimate (standard error); significance notation as in source):
    - Expected inflation:
      - full FE: -0.141 (0.111)
      - AM FE: -6.236*** (1.180)
      - EM FE: -0.111 (0.103)
      - full DK: -0.141 (0.0984)
      - AM DK: -6.236*** (1.155)
      - EM DK: -0.111 (0.0957)
    - Unexpected inflation:
      - full FE: 0.450** (0.193)
      - AM FE: -1.323 (1.249)
      - EM FE: 0.522*** (0.161)
      - full DK: 0.450* (0.263)
      - AM DK: -1.323 (1.249)
      - EM DK: 0.522** (0.257)
    - Industrial production growth rate:
      - full FE: 1.288*** (0.162)
      - AM FE: 0.993*** (0.197)
      - EM FE: 1.533*** (0.196)
      - full DK: 1.288*** (0.158)
      - AM DK: 0.993*** (0.183)
      - EM DK: 1.533*** (0.180)
    - Improvement in financial risk rating:
      - full FE: 0.00973*** (0.00327)
      - AM FE: 0.00135 (0.00371)
      - EM FE: 0.0198*** (0.00443)
      - full DK: 0.00973 (0.00597)
      - AM DK: 0.00135 (0.00624)
      - EM DK: 0.0198*** (0.00596)
    - U.S. 3-month Treasury bill yield rate:
      - full FE: 0.00209 (0.00413)
      - AM FE: 0.0219*** (0.00616)
      - EM FE: -0.00551 (0.00649)
      - full DK: 0.00209 (0.0107)
      - AM DK: 0.0219** (0.00925)
      - EM DK: -0.00551 (0.0138)
    - VIX:
      - full FE: -0.0168*** (0.00100)
      - AM FE: -0.0149*** (0.000835)
      - EM FE: -0.0164*** (0.00146)
      - full DK: -0.0168*** (0.00340)
      - AM DK: -0.0149*** (0.00245)
      - EM DK: -0.0164*** (0.00392)
    - Constant:
      - full FE: 0.340*** (0.0214)
      - AM FE: 0.395*** (0.0323)
      - EM FE: 0.346*** (0.0344)
      - full DK: 0.340*** (0.0750)
      - AM DK: 0.395*** (0.0637)
      - EM DK: 0.346*** (0.0856)

- Key quantitative findings (magnitudes preserved)
  - Full sample averages:
    - One percentage point increase in the growth rate of expected inflation is correlated with a 0.14 percentage point decrease in the growth rate of real stock returns.
    - One percentage point increase in the growth rate of unexpected inflation is correlated with a 0.45 percentage point increase in the growth rate of real stock returns.
    - One percentage point increase in the growth rate of industrial production is associated with a 1.29 percentage point increase in the growth rate of real stock returns.
    - One unit increase in the market volatility, as indicated by the VIX index, lowers the growth rate of real stock returns by 1.7 percentage points.
  - Significance notes:
    - Improvement in financial risk rating and the U.S. three-month Treasury bill yield rate are not significant for the full sample (FE results), though significance varies by subsample and estimator.

- Advanced Markets (AM) versus Emerging Markets (EM) differences
  - Advanced Markets:
    - Expected inflation: a one percentage point increase lowers real stock returns by 6.24 percentage points (AM FE).
    - U.S. three-month Treasury bill yield: positively correlated with real stock returns in AM (0.0219*** under FE).
    - Changes in financial risk ratings: not significant determinants in AM.
  - Emerging Markets:
    - Unexpected inflation: one percentage point increase boosts real stock returns by 0.52 percentage points (EM FE).
    - Improvements in financial risk ratings: positively drive real stock returns (0.0198*** under FE).
    - U.S. three-month Treasury bill yield: insignificant in EM.
  - Interpretation: tighter control of inflation in AM implies greater sensitivity to a given inflation shock; differences linked to financial sector vulnerabilities and financial development.

- Robustness and econometric considerations
  - Driscoll-Kraay standard errors used to correct for serial correlation, heteroskedasticity, and cross-sectional dependence; results largely consistent across estimators.
  - Clustered standard errors yield similar results but do not account for cross-sectional dependence.

### 4.2 Augmented Regressions on Stock Returns with Monetary Policy Considerations — Hypotheses and Main Results
- Monetary policy hypothesis and empirical strategy
  - Hypothesis: negative stock return–inflation correlation may be driven by monetary policy actions (central bank reaction function). Countercyclical policy (policy rate rises when output/inflation rises) tightens liquidity and can depress stock returns; acyclical or procyclical policies alter this relation.
  - Empirical tests:
    - Allow various degrees of policy rate cyclicality across countries.
    - Examine monetary policy frameworks (inflation targeting vs exchange rate anchor).
    - Study Zero Lower Bound (ZLB) episodes.

- Augmented regression setup
  - Yi,t = β0 + β1 πe i,t + β2 πu i,t + β3 πe i,t Ci + β4 πu i,t Ci + ZΓ + ui + εi,t
  - Ci: policy rate cyclicality (correlation between cyclical components of real output and policy rate, HP-filtered with smoothing parameter 6.25 for annual data).
  - Countercyclical policy dummy = 1 if correlation > 0.2, zero otherwise.
  - Introducing interaction terms implies effects of expected and unexpected inflation on stock returns become β1 + β3 * Ci and β2 + β4 * Ci.

- Descriptive statistics on cyclicality and sample composition
  - Average correlation between cyclical components of real output and policy rate = 0.26.
  - Among 61 sample countries, 45 have positive correlations and the rest negative.
  - By countercyclical dummy (> 0.2):
    - Advanced markets: 31 out of 33 are countercyclical (exceptions: Norway and Israel).
    - Emerging markets: 12 out of 31 are countercyclical.

- Main empirical findings (with M2 growth controlled)
  - Average associations:
    - A one percentage point increase in the growth rate of expected inflation is correlated with a 0.69 percentage point decrease in the growth rate of real stock returns.
    - A one percentage point increase in the growth rate of unexpected inflation is correlated with a 0.70 percentage point decrease in the growth rate of real stock returns.
    - A one percentage point increase in the growth rate of industrial production is associated with a 1.19 percentage point increase in the growth rate of real stock returns.
    - One unit of improvement in financial risk rating increases the growth rate of real stock return by 1.2 percentage points.
    - One unit increase in the VIX index lowers the growth rate of real stock returns by 1.7 percentage points.
    - A one percentage point increase in the growth rate of monetary aggregate (M2) is associated with a 0.63 percentage point increase in the growth rate of real stock returns.
  - Monetary aggregate growth (M2) dampens responsiveness of real stock returns to inflation when introduced as a control.

- Role of monetary policy cyclicality (interaction results)
  - Interaction coefficient: expected inflation × policy rate cyclicality = -5.47 (statistically significant).
    - Interpretation: switching from acyclical to perfectly countercyclical policy makes stock market–inflation responsiveness more negative by 5.47 units.
  - Expected inflation when monetary policy is acyclical: -1.9 (a one percentage point increase in expected inflation correlated with a 1.9 percentage point decrease in real stock returns).
  - Interaction with countercyclical policy dummy:
    - expected inflation × countercyclical policy dummy coefficient = -4.445*** in Table 3 specifications.
  - Effects larger in advanced markets than in emerging markets; reasons include better policy transmission and lower baseline inflation in advanced markets.

- Cross-country institutional interpretation
  - Advanced markets: most pursue countercyclical policies and inflation targeting; monetary aggregates less informative.
  - Emerging markets: many use monetary aggregates targets; monetary aggregate growth significant in EM but not AM.
  - Conclusion: monetary policy cyclicality and transmission differences partly explain AM–EM differences in stock return–inflation dynamics.

### 4.3 Results by Monetary Policy Framework — Exchange Rate Anchor vs Inflation Targeting and Heterogeneity among Inflation Targeters
- Monetary policy frameworks classified: exchange rate anchor, monetary aggregate target, inflation targeting, other (AREAER).
- Hypothesis: nominal anchors shape stock return–inflation relation; exchange rate anchor implies little direct policy response to inflation, inflation targeting implies direct response and thus stronger policy cyclicality effects.

- Exchange Rate Anchor: main findings
  - Real stock returns do not respond to monetary policy cyclicality under exchange rate anchor regime for both AM and EM (interaction terms not statistically significant).
  - Expected and unexpected inflation are negative and statistically significant under exchange rate anchor.
    - Table 4, Column (1) (full sample): Expected inflation = -1.017*** (0.256); Unexpected inflation = -1.220*** (0.389).
  - Interpretation: central bank prioritizes exchange rate stabilization; nevertheless, rising inflation is associated with decreases in real stock returns, implying other frictions operate.

- Inflation Targeting: main findings
  - Effects larger and more statistically significant for inflation targeters than baseline.
    - Table 5, Column (1) (full sample): Expected inflation = -4.289*** (1.000); Unexpected inflation = -3.680*** (0.694).
    - Table 5, Observations (full sample) = 1,214; AM = 564; EM = 650. R-squared (full) = 0.409.
  - Interaction of inflation with policy rate cyclicality:
    - Interaction terms extremely negative and statistically significant for advanced markets, but not for emerging markets.
    - Example: Expected inflation × policy rate cyclicality = -4.510** (2.219); Unexpected inflation × policy rate cyclicality = -3.833 (2.337) in different specifications.
  - Interpretation: credibility and effectiveness of inflation-targeting central banks differ across AM and EM; AM inflation targeters’ policy cyclicality strongly amplifies negative stock return response to inflation.

- Credibility, announced bands, and heterogeneity among inflation targeters
  - Among 30 inflation targeting countries:
    - 24 have more than 5 years of experience.
    - 20 have announced explicit inflation bands to the public.
  - Top vs bottom targeters split by percent of time inflation is within announced range (threshold = 45%).
  - Table 6 (percent of time inflation within announced range) lists top and bottom targeters (examples):
    - Canada 70% / Peru 44%
    - Thailand 65% / Australia 40%
    - Brazil 64% / Czech Republic 40%
    - New Zealand 61% / Indonesia 38%
    - Colombia 60% / Philippines 38%
    - Chile 57% / Israel 35%
    - South Africa 50% / Turkey 28%
    - Mexico 48% / Poland 27%
    - South Korea 46% / Serbia 18%
    - Iceland 46% / Romania 18%
  - Findings:
    - EM inflation targeters with more than 10 years of experience are generally not successful in shaping market perceptions; historical record matters more than mere length.
    - Announcing an inflation band helps a little.
    - Markets react to monetary policy cyclicality in the top group, but not in the bottom group.

- Top inflation targeters: within / below / above band scenarios
  - In the top group, inflation is above the band three times more often than below: 153 observations above vs. 57 below in regressions.
  - Interaction terms are extremely significant when inflation is above the range, but not when within or below the range.
  - Interpretation: central banks react asymmetrically, more concerned with inflation above the band; markets learn this asymmetry.

- Additional control coefficient examples (from Tables 4 and 5)
  - Industrial production growth rate: positive and significant (e.g., Table 4, Column (1) = 1.408*** (0.197)).
  - VIX: negative and highly significant (e.g., Table 4, Column (1) = -0.0138*** (0.00361)).
  - Improvement in financial risk rating: often positive and sometimes significant (e.g., Table 5, Column (1) = 0.0125** (0.00478)).
  - Driscoll-Kraay standard errors used; significance denoted as *** p < 0.01; ** p < 0.05; * p < 0.1.

### 4.4 Results with Respect to the Zero Lower Bound (ZLB)
- Regression specification
  - Y_i,t = β0 + β1 π_i,t (1−ZLB) + β2 π_i,t ZLB + β3 π_i,t C_i (1−ZLB) + β4 π_i,t C_i ZLB + ZΓ + u_i + ε_i,t.
  - ZLB indicator: ZLB = 1 when policy rate ≤ 25 basis points.
  - Actual inflation used and interacted with ZLB dummy; Ci interacted with inflation and ZLB dummy.

- Key empirical findings (Table 8 summary)
  - Under normal times (ZLB does not bind):
    - Real stock returns respond negatively to inflation.
    - Negative response stronger when monetary policy is more countercyclical.
    - Example coefficients:
      - Inflation × (1- ZLB): -1.459*** (0.339) in column (1).
      - Inflation × (1- ZLB): -5.808*** (0.905) in column (2).
      - Inflation × (1- ZLB) × policy rate cyclicality: -4.871*** (0.879) in column (3).
      - Inflation × (1- ZLB) × countercyclical policy dummy: -4.044*** (0.659) in column (5).
  - Under ZLB episodes (policy rate ≤ 25 basis points):
    - Coefficients on inflation and interactions are generally not statistically significant.
    - Example coefficients:
      - Inflation × ZLB: -1.431 (1.511) in column (1).
      - Inflation × ZLB: -0.365 (2.174) in column (2).
      - Inflation × ZLB × policy rate cyclicality: 2.034 (3.423) in column (3).
      - Inflation × ZLB × countercyclical policy dummy: 3.601 (3.099) in column (5).

- Control variable estimates (selected, Table 8)
  - Industrial production growth rate: 1.178*** (0.166) in column (1); 0.905*** (0.183) in column (2).
  - M2 growth rate: 0.383** (0.186) in column (1); -0.0220 (0.289) in column (2).
  - Improvement in financial risk rating: 0.0132*** (0.00502) in column (1); 0.000739 (0.00573) in column (2).
  - U.S. 3-month Treasury bill yield rate: 0.00453 (0.0108) in column (1); 0.0220** (0.0109) in column (2).
  - VIX: -0.0169*** (0.00316) in column (1); -0.0165*** (0.00273) in column (2).

- Sample and fit (selected, Table 8)
  - Observations and R-squared (examples):
    - Observations: 3,663 (full, column (1)); 1,945 (AM, column (2)); 3,498 (full, column (3)); 1,944 (AM, column (4)); 3,498 (full, column (5)); 1,944 (AM, column (6)).
    - R-squared: 0.3708 (column (1)); 0.4484 (column (2)); 0.4002 (column (3)); 0.4587 (column (4)); 0.4036 (column (5)); 0.4512 (column (6)).
    - Number of countries: 57 (column (1)); 27 (column (2)); 55 (columns (3)-(6) as reported).

- Interpretation and mechanisms
  - Markets view the ZLB as a different regime:
    - Inflation is usually low under the ZLB; a positive inflation surprise can be good news because the real interest rate is lower and the central bank need not raise rates.
    - A negative inflation surprise at the ZLB can impose costs because the central bank is constrained from lowering rates further.
    - Markets may consider an Effective Lower Bound (ELB) beyond the ZLB.
  - Lack of statistically significant coefficients under ZLB implies inability to reject that real stock returns do not respond to inflation and policy cyclicality in the same way as in normal periods.
  - Analysis focused primarily on advanced markets because of limited ZLB episodes in emerging markets.

- Robustness checks (selected)
  - Re-ran regressions using actual inflation interacted with EM dummy: interaction term positive and statistically significant.
  - Dropped observations when countries adopted the Euro.
  - Excluded hyperinflation periods (inflation > 50% or > 100%).
  - Used Consensus Forecasts data for expected inflation.
  - Constructed a global stock return factor (Systemic-5: US, UK, Germany, Japan, China weighted by nominal GDP); global factor statistically significant.
  - Replaced U.S. Treasury bill yield with Wu-Xia shadow federal funds rate to account for unconventional U.S. policy at the ZLB.
  - Overall: results remain mostly the same; appendix reports additional results.

- Limitations
  - Limited number of ZLB observations and low statistical power; standard errors larger under ZLB.
  - Policy rate cyclicality measured as a constant per country over the sample; may be coarse.
  - Inflation expectations proxied by forecasts (surveys/models); market-based measures limited in coverage.
  - Cross-border listings and multinationals blur national stock return boundaries; concern most relevant for advanced countries.
  - Very few emerging markets experienced ZLB episodes: "Only three emerging markets in Eastern Europe (Bulgaria, Latvia and Lithuania.) have hit the Zero Lower Bound in recent years."

- Policy implications and conclusions (selected)
  - Central bank reaction to inflation (policy cyclicality) is critical in shaping the stock return–inflation relation.
  - More countercyclical monetary policy makes the stock return–inflation relation more negative.
  - Examples of framework changes noted:
    - Federal Reserve announced an Average Inflation Target (AIT) of 2 percent (August 2020).
    - European Central Bank adopted a symmetric target of 2 percent over the medium term.
  - Policy implications for advanced markets:
    - Global Financial Crisis, lower policy rates to the ZLB, and expanded asset purchases changed stock return–inflation patterns because central banks cannot lower policy rates further but can still raise rates when inflation increases.
  - Policy implications for emerging markets:
    - Be cautious about rapid expansions in monetary aggregates as they can create stock price bubbles and threaten financial stability.
    - Differences in cyclicality, frameworks, capacity, and credibility mean EM should be cautious in borrowing AM experience.
  - Central bank communication and credibility crucial for shaping market perceptions; committing to more countercyclical policy can change how stock markets react to inflation.

### Additional Tests, Theory Links, and Robustness (References and Appendix Summaries)
- Data and panel coverage
  - Quarterly data from 1980Q1 to 2015Q2. Sample includes 71 economies (33 Advanced Markets; 38 Emerging Markets) as listed in data appendix.
- Variable definitions (selected, Table 9)
  - stock_index: Stock market index (Bloomberg)
  - CPI: Consumer price index (IMF INS database)
  - IP: Industrial Production (Datastream and IMF IFS databases)
  - M2: M2 (money supply) (Haver Analytics and IMF IFS databases)
  - financial_risk: ICRG financial risk ratings (50 = least risk to 0 = highest risk)
  - TB3MS: U.S. three-month Treasury Bill (Board of Governors of the Federal Reserve System)
  - VIX: VIX (Bloomberg)
  - PR: Central bank’s policy rate
  - WuXiaShadowRate: Wu-Xia shadow federal funds rate
  - Derived series include real_stock_returny, inflationy, inflationy_e, inflationy_u, IP_growthy, M2_growthy, financial_risk_change, MP_correlation_PR, G5_real_stock_returny.
- Tests of Bakshi-Chen monetary policy hypothesis (cross-country regressions)
  - Summary statistics (Table 10, n = 63):
    - Cov(r, pi): Mean = 0.0179, Std. dev. = 0.1497, Min = -0.0176, Max = 1.186
    - Cov(y, m): Mean = 0.0017, Std. dev. = 0.0030, Min = -0.0023, Max = 0.015
    - Var(y): Mean = 0.0015, Std. dev. = 0.0016, Min = 0.00003, Max = 0.0075
  - Cross-country OLS results (Table 11):
    - cov(y, m) coefficients reported around 48.33**, 48.27**, 50.65**, 51.39** (robust SEs ~20–21).
    - var(y) coefficients reported around -58.47**, -50.33**, -62.55**, -57.54** (robust SEs ~23–28).
    - Constants vary (examples: 0.0254* (0.0148)).
    - R-squared examples: 0.5920, 0.5830, 0.1150, 0.3140, 0.6100, 0.617.
    - Interpretation: cov(real equity return, inflation) significantly positively correlated with cov(real output growth, nominal money growth) and negatively with var(real output growth); estimated magnitudes exceed theoretical 1 and −1, especially in EM.
- Robustness checks across multiple tables (12–17)
  - Main patterns robust: expected and unexpected inflation coefficients generally negative; industrial production growth positive; VIX negative; M2 growth generally positive; financial risk improvement positive.
  - Global factor (Systemic-5 real stock return) strongly positively associated with country returns when included (e.g., coefficient 0.784***).
  - Wu-Xia shadow rate used to address U.S. unconventional policy at ZLB; results broadly consistent with baseline.
  - Driscoll-Kraay standard errors for panel regressions; robust SEs for cross-country OLS.

*Source: wpiea2021219-print-pdf - 4.1–4.4 Results on Stock Returns, Inflation, Monetary Policy, and the ZLB*

### 4.1    Initial Empirical Results on Real Stock Returns

### 4.1    Initial Empirical Results on Real Stock Returns

### Methodology
- Real stock index: nominal stock index deflated by the consumer price index; real stock return is year-on-year difference of real stock index in natural logarithm.
- Baseline regression (panel regressions with fixed effects):
  - Y_{i,t} = β0 + β1 π^e_{i,t} + β2 π^u_{i,t} + Xβ + u_i + ε_{i,t}
  - Y_{i,t}: real return on equity index for country i at time t.
  - π^e and π^u: expected and unexpected inflation.
  - X: control variables including industrial production growth rate, change in financial risk, the U.S. three-month Treasury bill yield, and the VIX.
  - Controls interpreted: industrial production growth rate (real economic activity), change in financial risk (financial sector movements), U.S. three-month Treasury bill yield (global liquidity), VIX (global financial market volatility).
- Unit root and panel diagnostics performed (Augmented Dickey–Fuller, DF-GLS, Phillips–Perron, Im-Pesaran-Shin and Fisher-type tests); panel is unbalanced.
- Econometric concerns addressed: serial correlation (Wooldridge Test rejected no-first-order-autocorrelation at the 1% significance level), heteroscedasticity (modified Wald Test rejected homoskedasticity at the 1% significance level), potential cross-sectional dependence.
- To address serial correlation, heteroskedasticity, and potential spatial dependence, Driscoll-Kraay standard error estimators are applied (Driscoll and Kraay (1998)); appropriate given the long time dimension of the quarterly panel dataset.

### Baseline regression results (Table 2 summary)
- Estimation methods reported: FE = Panel Regressions with Fixed Effects; DK = Panel Regressions with Fixed Effects and Driscoll-Kraay standard errors.
- Samples: full, AM = Advanced Markets, EM = Emerging Markets.
- Observations: 4,573 (full); 2,557 (AM); 2,016 (EM).
- Number of countries: 63 (full); 31 (AM); 32 (EM).
- R-squared: 0.288 (full, columns 1 and 4); 0.405 (AM, columns 2 and 5); 0.288 or 0.289 (EM, columns 3 and 6).
- Key coefficient estimates (with robust standard errors in parentheses; significance: *** p < 0.01; ** p < 0.05; * p < 0.1):
  - Expected inflation:
    - full FE: -0.141 (0.111)
    - AM FE: -6.236*** (1.180)
    - EM FE: -0.111 (0.103)
    - full DK: -0.141 (0.0984)
    - AM DK: -6.236*** (1.155)
    - EM DK: -0.111 (0.0957)
  - Unexpected inflation:
    - full FE: 0.450** (0.193)
    - AM FE: -1.323 (1.249)
    - EM FE: 0.522*** (0.161)
    - full DK: 0.450* (0.263)
    - AM DK: -1.323 (1.249)
    - EM DK: 0.522** (0.257)
  - Industrial production growth rate:
    - full FE: 1.288*** (0.162)
    - AM FE: 0.993*** (0.197)
    - EM FE: 1.533*** (0.196)
    - full DK: 1.288*** (0.158)
    - AM DK: 0.993*** (0.183)
    - EM DK: 1.533*** (0.180)
  - Improvement in financial risk rating:
    - full FE: 0.00973*** (0.00327)
    - AM FE: 0.00135 (0.00371)
    - EM FE: 0.0198*** (0.00443)
    - full DK: 0.00973 (0.00597)
    - AM DK: 0.00135 (0.00624)
    - EM DK: 0.0198*** (0.00596)
  - U.S. 3-month Treasury bill yield rate:
    - full FE: 0.00209 (0.00413)
    - AM FE: 0.0219*** (0.00616)
    - EM FE: -0.00551 (0.00649)
    - full DK: 0.00209 (0.0107)
    - AM DK: 0.0219** (0.00925)
    - EM DK: -0.00551 (0.0138)
  - VIX:
    - full FE: -0.0168*** (0.00100)
    - AM FE: -0.0149*** (0.000835)
    - EM FE: -0.0164*** (0.00146)
    - full DK: -0.0168*** (0.00340)
    - AM DK: -0.0149*** (0.00245)
    - EM DK: -0.0164*** (0.00392)
  - Constant:
    - full FE: 0.340*** (0.0214)
    - AM FE: 0.395*** (0.0323)
    - EM FE: 0.346*** (0.0344)
    - full DK: 0.340*** (0.0750)
    - AM DK: 0.395*** (0.0637)
    - EM DK: 0.346*** (0.0856)

### Key quantitative findings (interpreted magnitudes preserved as in text)
- Full sample averages:
  - One percentage point increase in the growth rate of expected inflation is correlated with a 0.14 percentage point decrease in the growth rate of real stock returns.
  - One percentage point increase in the growth rate of unexpected inflation is correlated with a 0.45 percentage point increase in the growth rate of real stock returns.
  - One percentage point increase in the growth rate of industrial production is associated with a 1.29 percentage point increase in the growth rate of real stock returns.
  - One unit increase in the market volatility, as indicated by the VIX index, lowers the growth rate of real stock returns by 1.7 percentage points.
- Statistical significance notes:
  - Improvement in financial risk rating and the U.S. three-month Treasury bill yield rate are not significant for the full sample (FE results), though significance varies by subsample and estimator.

### Advanced Markets (AM) versus Emerging Markets (EM) differences
- Advanced Markets:
  - Real stock returns respond very negatively to expected inflation: a one percentage point increase in the growth rate of expected inflation lowers the growth rate of real stock returns by 6.24 percentage points.
  - U.S. three-month Treasury bill yield appears positively correlated with real stock returns in advanced markets (0.0219*** under FE).
  - Changes in financial risk ratings are not significant determinants.
- Emerging Markets:
  - Unexpected inflation is positively associated with real stock returns: one percentage point increase in the growth rate of unexpected inflation boosts the growth rate of real stock returns by 0.52 percentage points.
  - Improvements in financial risk ratings positively drive real stock returns (0.0198*** under FE).
  - U.S. three-month Treasury bill yield is insignificant in emerging markets.
- Interpretative explanation offered:
  - Inflation is controlled within a much smaller range in advanced markets than in emerging markets; markets are therefore more sensitive to one unit of inflation shock in advanced markets.
  - Differences may be linked to vulnerabilities in financial sectors and levels of financial development; stock markets in advanced countries are more mature and investors may be more rational and have better access to information.

### Robustness and econometric considerations
- Panel diagnostics indicate need to correct standard errors due to serial correlation, heteroskedasticity, and potential cross-sectional dependence.
- Driscoll-Kraay standard errors applied; findings are largely consistent and standard errors do not change dramatically.
- Clustered standard errors yield similar results, but clustered estimator does not account for cross-sectional dependence; Driscoll-Kraay preferred given spatial dependence in stock returns.

*Source: wpiea2021219-print-pdf - 4.1    Initial Empirical Results on Real Stock Returns*

### 4.2    Augmented  Regressions  on  Stock  Returns  with  Mone-

### 4.2    Augmented Regressions on Stock Returns with Monetary Policy Considerations

### Monetary policy hypothesis and empirical strategy
- The literature posits that the negative stock return–inflation correlation may be driven by monetary policy actions: when inflation rises, a central bank that is “leaning against the wind” hikes its policy rate, tightening liquidity and putting downward pressure on stock returns; if monetary policy is acyclical, no such effect arises; if procyclical, policy rate falls with rising inflation and stocks may be boosted.
- The analysis uses a newly compiled cross-country dataset to test three aspects of the monetary policy role:
  - Allowing various degrees of policy rate cyclicality across countries.
  - Examining different monetary policy frameworks (inflation targeting versus exchange rate anchor).
  - Studying Zero Lower Bound episodes when monetary policy is constrained.

### Augmented regression setup
- Regression specification (with country fixed effects):
  - Yi,t = β0 + β1 πe i,t + β2 πu i,t + β3 πe i,t Ci + β4 πu i,t Ci + ZΓ + ui + εi,t
  - Ci: measure of monetary policy cyclicality for country i.
  - Z: vector of control variables including monetary aggregate (M2) growth rate.
  - Introducing interaction terms means effects of expected and unexpected inflation on stock returns become β1 + β3 * Ci and β2 + β4 * Ci.
- Monetary policy cyclicality measure:
  - Computed as the correlation between the cyclical components of real output and a central bank’s policy rate (following Vegh and Vuletin (2012)).
  - Hodrick-Prescott (HP) filter applied to derive cyclical components; smoothing parameter set at 6.25 for the annual data.
  - Positive correlation → countercyclical monetary policy; negative correlation → procyclical monetary policy.
  - Countercyclical policy dummy defined as one if correlation > 0.2, zero otherwise.

### Key descriptive statistics on cyclicality and sample composition
- In most countries the correlation between cyclical components of real output and policy rate is mostly positive, with an average of 0.26.
- Among the 61 sample countries, 45 have positive correlations and the rest have negative correlations.
- By countercyclical dummy (> 0.2):
  - Advanced markets: 31 out of 33 are countercyclical (exceptions: Norway and Israel).
  - Emerging markets: 12 out of 31 are countercyclical.

### Main empirical findings
- Including M2 growth as a control, panel regressions show real stock returns react negatively to inflation.
- Average associations reported:
  - A one percentage point increase in the growth rate of expected inflation is correlated with a 0.69 percentage point decrease in the growth rate of real stock returns.
  - A one percentage point increase in the growth rate of unexpected inflation is correlated with a 0.70 percentage point decrease in the growth rate of real stock returns.
  - A one percentage point increase in the growth rate of industrial production is associated with a 1.19 percentage point increase in the growth rate of real stock returns.
  - One unit of improvement in financial risk rating increases the growth rate of real stock return by 1.2 percentage points.
  - One unit increase in the VIX index lowers the growth rate of real stock returns by 1.7 percentage points.
  - A one percentage point increase in the growth rate of monetary aggregate (M2) is associated with a 0.63 percentage point increase in the growth rate of real stock returns.
- Monetary aggregate growth dampens the responsiveness of real stock returns to inflation when introduced as a control.

### Role of monetary policy cyclicality (interaction results)
- Monetary policy cyclicality materially alters stock return–inflation responses:
  - Estimated coefficient on the interaction between expected inflation and policy rate cyclicality: -5.47 (statistically significant).
  - Interpretation: when monetary policy switches from acyclical to perfectly countercyclical, stock market–inflation responsiveness becomes more negative by 5.47 units.
  - Estimated coefficient on expected inflation when monetary policy is acyclical: -1.9 (i.e., if monetary policy is acyclical, a one percentage point increase in the growth rate of expected inflation is correlated with a 1.9 percentage point decrease in the growth rate of real stock returns).
- Results hold when using interaction terms between inflation and the countercyclical policy dummy (e.g., expected inflation × countercyclical policy dummy coefficient -4.445*** in Table 3 specifications).
- Both advanced and emerging markets show negative responses of real stock returns to inflation and to policy rate cyclicality, but:
  - Estimated effects of expected inflation and its interaction with policy rate cyclicality are larger in advanced markets than in emerging markets.
  - Possible reasons: better monetary policy transmission in advanced markets; lower inflation in advanced markets making markets more responsive to a given unit change in inflation.

### Cross‑country interpretation and institutional context
- Advanced markets:
  - Most pursue countercyclical monetary policies and have adopted inflation targeting or implicit inflation targeting.
  - Monetary aggregates have become less informative; central banks rely more on the policy rate.
- Emerging markets:
  - Many still operate under monetary aggregates target frameworks; fewer pursue countercyclical policies.
  - Monetary aggregate growth is significant in emerging markets but not in advanced markets, reflecting continued relevance of M2 in some emerging economies.
- Overall conclusion: differences in stock return–inflation dynamics across advanced and emerging markets are partly explained by monetary policy cyclicality and differences in policy transmission channels.

*Source: wpiea2021219-print-pdf*

### 4.3    Results by Monetary Policy Framework

### 4.3    Results by Monetary Policy Framework

### Overview
- AREAER classifies monetary policy frameworks into: exchange rate anchor, monetary aggregate target, inflation targeting, and other frameworks.
- Hypothesis: the stock return–inflation relation differs across nominal anchors. Exchange rate anchor and inflation targeting are two polar cases: stabilizing the exchange rate implies little direct response of policy to inflation, while inflation targeting implies policy directly responds to inflation and thus policy cyclicality may strongly affect market responses to inflation.
- Sample facts: 30 inflation targeting countries in the sample; 24 have more than 5 years of experience; 20 have announced explicit inflation bands to the public. The threshold to split top vs bottom targeters is 45 percent of the time inflation is within the announced range.

### Exchange Rate Anchor: main findings
- Real stock returns do not respond to monetary policy cyclicality under the exchange rate anchor regime for both advanced and emerging markets (interaction terms between inflation and policy rate cyclicality are not statistically significant).
- Estimated coefficients of expected and unexpected inflation are negative and statistically significant under exchange rate anchor.
  - Table 4, Column (1) (full sample): Expected inflation = -1.017*** (0.256); Unexpected inflation = -1.220*** (0.389).
- Interpretation: markets expect the central bank to prioritize exchange rate stabilization over direct responses to inflation; nevertheless, rising inflation is associated with decreases in real stock returns, implying other frictions are operating.

### Inflation Targeting: main findings
- Effects are larger and more statistically significant for inflation targeting countries than in the baseline.
- On average, real stock returns respond more negatively to expected and unexpected inflation in inflation targeting countries, true for both advanced and emerging markets.
  - Table 5, Column (1) (full sample): Expected inflation = -4.289*** (1.000); Unexpected inflation = -3.680*** (0.694).
  - Table 5, Observations (full sample) = 1,214; AM = 564; EM = 650. R-squared (full) = 0.409.
- Interaction of inflation with policy rate cyclicality:
  - Interaction terms are extremely negative and statistically significant for advanced markets, but not for emerging markets.
  - Example (Table 5): Expected inflation × policy rate cyclicality = -4.510** (2.219) in the relevant specification; Unexpected inflation × policy rate cyclicality = -3.833 (2.337) in another specification.
- Interpretation: credibility and effectiveness of inflation-targeting central banks differ across advanced and emerging markets; advanced-market inflation targeters’ policy cyclicality strongly amplifies the negative stock return response to inflation.

### Credibility, announced bands, and heterogeneity among inflation targeters
- The ability to keep inflation within announced target bands matters for market perceptions and for whether markets pay attention to monetary policy cyclicality.
- Among 30 inflation targeting countries:
  - 24 have more than 5 years of experience.
  - 20 have announced explicit inflation bands to the public.
- Division into top and bottom groups is based on percent of time inflation is within the announced range (threshold = 45%).
- Table 6: Percent of Time Inflation within the Announced Range (Top inflation targeters / Bottom inflation targeters)
  - Canada 70% / Peru 44%
  - Thailand 65% / Australia 40%
  - Brazil 64% / Czech Republic 40%
  - New Zealand 61% / Indonesia 38%
  - Colombia 60% / Philippines 38%
  - Chile 57% / Israel 35%
  - South Africa 50% / Turkey 28%
  - Mexico 48% / Poland 27%
  - South Korea 46% / Serbia 18%
  - Iceland 46% / Romania 18%
- Findings from Table 7 and associated analysis:
  - Emerging-market inflation targeters with more than 10 years of experience are generally not successful in shaping market perceptions; the history/record of inflation targeting matters more than merely the length of the record.
  - Announcing an inflation band helps a little in shaping market perceptions.
  - Markets react to monetary policy cyclicality in the top group, but not in the bottom group—i.e., central banks’ ability to maintain inflation within target bands is key to shaping market perceptions.

### Top inflation targeters: within / below / above band scenarios
- For top inflation targeters, regressions split observations into inflation within, below, or above the target band.
  - In the top group, inflation is above the band three times more often than below: 153 observations above the band vs. 57 observations below the band (in the regressions).
- Regression results:
  - Interaction terms are extremely significant when inflation is above the range, but not significant when inflation is within or below the range.
- Interpretation: central banks are historically more concerned with inflation above the band and react asymmetrically to inflation dynamics; markets learn this asymmetry and are most sensitive when inflation is above the announced band but unresponsive when inflation is within or below the band.

### Additional empirical details and supportive controls (selected)
- Exchange rate anchor sample sizes (Table 4): Observations = 871 (full); AM = 370; EM = 501. R-squared (full) = 0.3052; Number of countries = 40.
- Selected control coefficients (examples):
  - Industrial production growth rate: consistently positive and significant. Example Table 4, Column (1) = 1.408*** (0.197).
  - VIX: consistently negative and highly significant. Example Table 4, Column (1) = -0.0138*** (0.00361).
  - Improvement in financial risk rating: often positive and sometimes significant. Example Table 5, Column (1) = 0.0125** (0.00478).
- Estimation note: Driscoll-Kraay standard errors reported in parentheses. Statistical significance denoted as *** p < 0.01; ** p < 0.05; * p < 0.1.

*Source: 4.3 Results by Monetary Policy Framework (from the supplied IMF content unit).*

### 4.4    Results with Respect to the Zero Lower Bound (ZLB)

### 4.4    Results with Respect to the Zero Lower Bound (ZLB)

### Regression specification
- Main regression estimated (equation (3) in source): Y_i,t = β0 + β1 π_i,t (1−ZLB) + β2 π_i,t ZLB + β3 π_i,t C_i (1−ZLB) + β4 π_i,t C_i ZLB + ZΓ + u_i + ε_i,t.
- ZLB indicator: ZLB = 1 when policy rate is below or equal to 25 basis points.
- Actual inflation is used in the regression and interacted with the ZLB dummy.
- Monetary policy cyclicality measures (C_i) are interacted with inflation and with the ZLB dummy to test asymmetric responses.

### Key empirical findings (summary of Table 8 and narrative)
- Under normal times (ZLB does not bind):
  - Real stock returns respond negatively to inflation.
  - The negative response is stronger when monetary policy is more countercyclical.
  - Example coefficient estimates (selected, from Table 8):
    - Inflation × (1- ZLB): -1.459*** (0.339) in column (1).
    - Inflation × (1- ZLB): -5.808*** (0.905) in column (2).
    - Inflation × (1- ZLB) × policy rate cyclicality: -4.871*** (0.879) in column (3).
    - Inflation × (1- ZLB) × countercyclical policy dummy: -4.044*** (0.659) in column (5).
- Under ZLB episodes (policy rate ≤ 25 basis points):
  - Estimated coefficients on inflation and on interactions between inflation and monetary policy cyclicality measures are generally not statistically significant.
  - Example coefficient estimates (selected, from Table 8):
    - Inflation × ZLB: -1.431 (1.511) in column (1).
    - Inflation × ZLB: -0.365 (2.174) in column (2).
    - Inflation × ZLB × policy rate cyclicality: 2.034 (3.423) in column (3).
    - Inflation × ZLB × countercyclical policy dummy: 3.601 (3.099) in column (5).
- Control variable estimates (selected, from Table 8):
  - Industrial production growth rate: 1.178*** (0.166) in column (1); 0.905*** (0.183) in column (2).
  - M2 growth rate: 0.383** (0.186) in column (1); -0.0220 (0.289) in column (2).
  - Improvement in financial risk rating: 0.0132*** (0.00502) in column (1); 0.000739 (0.00573) in column (2).
  - U.S 3-month Treasury bill yield rate: 0.00453 (0.0108) in column (1); 0.0220** (0.0109) in column (2).
  - VIX: -0.0169*** (0.00316) in column (1); -0.0165*** (0.00273) in column (2).
- Sample and fit (from Table 8, selected):
  - Observations: 3,663 (full sample, column (1)); 1,945 (AM sample, column (2)); 3,498 (full, column (3)); 1,944 (AM, column (4)); 3,498 (full, column (5)); 1,944 (AM, column (6)).
  - R-squared: 0.3708 (column (1)); 0.4484 (column (2)); 0.4002 (column (3)); 0.4587 (column (4)); 0.4036 (column (5)); 0.4512 (column (6)).
  - Number of countries: 57 (column (1)); 27 (column (2)); 55 (columns (3)-(6) as reported).

### Interpretation and mechanisms
- Markets perceive the ZLB as a different regime:
  - Inflation is usually low under the ZLB; a positive inflation surprise can be good news because the real interest rate is lower and the central bank need not respond by raising rates.
  - A negative inflation surprise at the ZLB could impose additional costs because the central bank is constrained in lowering policy rates further; policymakers may be reluctant to lower rates.
  - Markets may also consider the possibility of an Effective Lower Bound (ELB) beyond the ZLB.
- Under ZLB episodes, the lack of statistically significant coefficients implies we cannot reject that real stock returns do not respond to inflation and monetary policy cyclicality in the same way as in normal periods.
- The analysis focuses primarily on advanced markets given the limited number of ZLB episodes in emerging markets.

### Robustness checks (selected)
- Re-ran augmented regressions using actual inflation and interacted with an emerging market dummy: the interaction term is positive and statistically significant.
- Dropped observations when countries adopted the Euro to preserve monetary autonomy.
- Excluded hyperinflation periods (drop observations with inflation higher 50% or 100%).
- Used Consensus Forecasts data as measures of expected inflation.
- Constructed a global stock return factor aggregating Systemic-5 countries (United States, United Kingdom, Germany, Japan and China, weighted by nominal GDP); the global factor is statistically significant.
- Replaced the U.S. Treasury bill yield with the Wu-Xia shadow federal funds rate to account for unconventional U.S. monetary policy at the ZLB.
- Overall: results remain mostly the same; additional results are reported in the appendix.

### Limitations
- Limited number of ZLB observations and low statistical power; standard errors are much larger under ZLB episodes.
- Monetary policy cyclicality measure is computed as a constant for each country over the sample period; this may be coarse because cyclicality could vary over time.
  - Potential improvement: make policy rate cyclicality time-varying by decade.
- Inflation expectations are proxied by forecasts from surveys or models; these forecasts are not identical to true inflation expectations but are the best available long-run measures for many countries.
- Market-based measures of inflation expectations are available only for a limited set of countries in recent years.
- Cross-border listings and multinational firms blur national stock return boundaries; inclusion of such firms means indices may not perfectly represent national economic activity. This concern is most relevant for advanced countries and is judged not to change results substantially in this 71-economy study.
- Very few emerging markets experienced ZLB episodes: "Only three emerging markets in Eastern Europe (Bulgaria, Latvia and Lithuania.) have hit the Zero Lower Bound in recent years."

### Policy implications and conclusions (selected)
- How central banks react to inflation (policy cyclicality) plays a critical role in shaping the stock return–inflation relation.
- Central banks that pursue more countercyclical monetary policy make the stock return–inflation relation more negative.
- Changes in monetary policy frameworks (examples noted in source):
  - Federal Reserve announced an Average Inflation Target (AIT) of 2 percent (August 2020) to allow periods of above-the-target inflation.
  - European Central Bank adopted a symmetric target of 2 percent over the medium term.
- Policy implications for advanced markets:
  - The Global Financial Crisis and lower policy rates to the ZLB, plus expanded asset purchases, changed stock return–inflation patterns because central banks cannot lower policy rates further but can still raise rates when inflation increases.
- Policy implications for emerging markets:
  - Policymakers should be cautious about rapid expansions in monetary aggregates as they can create stock price bubbles and threaten financial stability.
  - Differences in monetary policy cyclicality, frameworks, capacity, and credibility mean emerging markets should be cautious when borrowing advanced market experience.
- Central bank communication and credibility are crucial for shaping market perception of inflation shocks; committing to more countercyclical policy can change how stock markets react to inflation.
- Suggested areas for future research (selected from source):
  - Examine the relationship across business cycle conditions, including amplitude and duration effects.
  - Study open-economy and institutional angles: exchange rate movements, capital flows, crisis types, institutional quality, stock market openness, and financial liberalization.
  - Analyze stock price responses to inflation at longer horizons to test implications for the Fisher hypothesis.

*Source: 4.4 Results with Respect to the Zero Lower Bound (ZLB), wpiea2021219-print-pdf*

### References

### References (wpiea2021219-print-pdf)

### Major cited works
- Comprehensive bibliography including empirical and theoretical studies on the relations among inflation, monetary policy, and stock returns. Representative citations include:
  - Adrian, Tobias, Shin, Hyun Song, 2010, “The Changing Nature of Financial Intermediation and the Financial Crisis of 2007–2009”, Annual Review of Economics, Vol.2:1-698.
  - Bakshi, G. S. and Chen, Z., 1996, “Inflation, asset prices, and the term structure of interest rates in monetary economics”, Review of Financial Studies, 9, 241-275.
  - Fama, E., 1981. “Stock returns, real activity, inflation, and money”, American Economic Review, 71, 545–65.
  - Kaul, G., 1987 ”Stock returns and inflation: The role of the monetary sector”, Journal of Financial Economics, 18 253-276.
  - Boyle, Glenn W. and Peterson, James D., 1995, ”Monetary Policy, Aggregate Uncertainty, and the Stock Market”, Journal of Money, Credit and Banking, Vol. 27, No. 2, pp. 570-582.
  - Geske, R., Roll, R., 1983, ”The fiscal and monetary linkage between stock returns and inflation”, the Journal of Finance, Volume 38, Issue 1, Pages 1–33.
  - Wu, Jing Cynthia, and Xia, Fan Dora, 2014, “Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound”, NBER Working Paper No. 20117, May 2014.
- The References list also includes methodology and econometric sources (e.g., Driscoll and Kraay 1998; Hoechle 2007) and many country- and market-specific empirical studies.

### Data description (Appendix)
- Quarterly data from 1980Q1 to 2015Q2. The sample includes 71 economies, covering both advanced and emerging markets.
- Country classification by income group:
  - Advanced Markets (33): Australia, Austria, Belgium, Canada, Cyprus, Czech Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hong Kong SAR, Iceland, Ireland, Israel, Italy, Japan, Luxembourg, Malta, Netherlands, New Zealand, Norway, Portugal, Singapore, Slovak Republic, Slovenia, South Korea, Spain, Sweden, Switzerland, United Kingdom, United States.
  - Emerging Markets (38): Argentina, Bosnia and Herzegovina, Brazil, Bulgaria, Chile, China, Colombia, Costa Rica, Croatia, Egypt, Hungary, India, Indonesia, Jamaica, Jordan, Kazakhstan, Latvia, Lebanon, Lithuania, Macedonia, Malaysia, Mexico, Morocco, Pakistan, Panama, Peru, Philippines, Poland, Romania, Russia, Serbia, South Africa, Sri Lanka, Thailand, Tunisia, Turkey, Ukraine, Venezuela.
- Note: the country classification is based on the International Monetary Fund (IMF)’s World Economic Outlook (WEO).

### Panel data tests (Appendix)
- Unit root tests used: Augmented Dickey–Fuller, DF-GLS and Phillips–Perron on each variable by country, as well as Im-Pesaran-Shin and Fisher-type tests. Some panel unit root tests are not applicable due to unbalanced panel.
- Cross-sectional dependence: Tests cannot be performed; assume cross-section dependence and spatial effects exist given dataset nature.
- Wooldridge test for autocorrelation in panel data: reject the null hypothesis → data has first-order autocorrelation.
- Modified Wald test for groupwise heteroscedasticity in fixed effect regression model: reject the null hypothesis → data has heteroscedasticity.

### Variable definitions (Table 9)
- Original series:
  - stock_index: Stock market index (Bloomberg)
  - RGDP: Real GDP (Haver Analytics, IMF IFS and WEO databases)
  - CPI: Consumer price index (IMF INS database)
  - inflation_CF_current: Inflation forecasts, current year (Consensus Forecasts)
  - inflation_CF_next: Inflation forecasts, next year (Consensus Forecasts)
  - IP: Industrial Production (Datastream and IMF IFS databases)
  - M2: M2 (money supply) (Haver Analytics and IMF IFS databases)
  - financial_risk: ICRG financial risk ratings, International Country Risk Guide (ICRG) database (from a high of 50 (least risk) to a low of 0 (highest risk))
  - TB3MS: U.S. three-month Treasury Bill: Secondary Market Rate (Board of Governors of the Federal Reserve System)
  - VIX: VIX (Bloomberg)
  - PR: Central bank’s policy rate (Haver Analytics, IMF IFS and GDS databases)
  - WuXiaShadowRate: Wu-Xia shadow federal funds rate (Federal Reserve Bank of Atlanta)
  - MPF: Monetary policy framework (AREAER database, IMF)
  - inflation_target, inflation_target_lower, inflation_target_upper: Inflation targets and bands (Central banks’ websites)
- Derived series:
  - stock_returny: Nominal stock return (log) = ln[stock_index(t)] – ln[stock_index(t-4)]
  - real_stock_index: Real stock market index = stock_index / CPI
  - real_stock_returny: Real stock return (log) = ln[real_stock_index(t)] – ln[real_stock_index(t-4)]
  - inflationy: Inflation (log) = ln[CPI(t)] – ln[CPI(t-4)]
  - inflationy_e: Expected inflation (predicted inflation from AR(4) model)
  - inflationy_u: Unexpected inflation = Actual inflation – expected inflation
  - IP_growthy: Industrial production growth = ln[IP(t)] – ln[IP(t-4)]
  - M2_growthy: Monetary aggregate growth = ln[M2(t)] – ln[M2(t-4)]
  - financial_risk_change: Improvement in financial risk rating = financial_risk(t) - financial_risk(t-1)
  - MP_correlation_PR: Policy rate cyclicality = Correlation between the cyclical components of real output and policy rate
  - G5_real_stock_returny: Systemic-5 real stock return = Weighted average of real_stock_returny in the US, UK, Germany, Japan and China

### Additional tests on the role of monetary policy (theory and empirical specification)
- Bakshi and Chen (1996) MIUF model leads to:
  - cov_t(dq_t/q_t, dP_t/P_t) = cov_t(dy_t/y_t, dM_t/M_t) − var_t(dy_t/y_t)   (Equation (4) in text)
  - Interpretation: covariance between real equity return and inflation equals covariance between real output growth and nominal money growth minus variance of real output growth.
  - Implications: If money stock is exogenous (Fama 1981 proxy hypothesis) → zero cov(real output, money) → covariance between real equity return and inflation negative. Monetary policy countercyclical (Geske and Roll 1983) → negative covariance. Procyclical monetary policy (Kaul 1987) can generate positive covariance.
- Boyle and Peterson (1995) CIA model reaction function specified as:
  - μ_t = k λ_t^ε   (Equation (5) in text)
  - With λ_t i.i.d. and constant risk aversion, key relation:
    - cov_t(ln q*_t, ln Π_t) = cov(λ_t, (ε−1) ln λ_t) = (ε−1) var(λ_t)   (Equation (6) in text)
  - Implication: equity returns negatively correlated with inflation when monetary policy is countercyclical (ε < 0) or weakly procyclical (0 < ε < 1); correlation positive when monetary policy is strongly procyclical (ε > 1).
- Cross-sectional linear regression specification used to test the Bakshi-Chen relation:
  - cov_i(r, pi) = α + β1 cov_i(y, m) + β2 var_i(y) + ε_i   (Equation (7))
  - Joint test: α = 0, β1 = 1, β2 = −1.

### Key statistics and empirical findings
- Table 10: Summary statistics to test the Bakshi-Chen hypothesis (n = 63 observations)
  - Cov(r, pi): Mean = 0.0179, Std. dev. = 0.1497, Min = -0.0176, Max = 1.186
  - Cov(y, m): Mean = 0.0017, Std. dev. = 0.0030, Min = -0.0023, Max = 0.015
  - Var(y): Mean = 0.0015, Std. dev. = 0.0016, Min = 0.00003, Max = 0.0075
- Table 11: Cross-country tests of Bakshi-Chen monetary policy hypothesis (OLS results; robust standard errors)
  - cov(y, m): coefficient estimates reported include 48.33**, 48.27**, 50.65**, 51.39** (depending on specification and sample), with robust standard errors shown in parentheses (e.g., (20.77), (21.34), (21.29), (21.51)).
  - var(y): coefficient estimates reported include -58.47**, -50.33**, -62.55**, -57.54** (depending on specification and sample), with robust standard errors shown in parentheses (e.g., (26.21), (23.12), (27.79), (25.85)).
  - Constant: 0.0254* (0.0148) in one specification; other constants shown such as -0.000381 (0.000376) and 0.0228 (0.0241).
  - Estimation method: OLS. Samples: full, AM (advanced markets), EM (emerging markets). Observations vary (63, 33, 32, 31, 31 in table). R-squared values reported: 0.5920, 0.5830, 0.1150, 0.3140, 0.6100, 0.617.
  - Interpretation in text: OLS results indicate cov(real rate of return on equity, inflation) is significantly positively correlated with cov(real output growth, nominal money growth) and negatively correlated with var(real output growth). However, estimated coefficients are much larger than the theoretical values (1 and −1), especially for emerging markets; advanced markets estimates are closer to theoretical predictions. The unrealistic i.i.d. assumption for output and money growth complicates reconciliation of theory and empirical magnitudes.
- Robustness checks (summary of reported regression tables 12–17)
  - Primary panel regressions have dependent variable: real stock return (quarterly, annualized log difference).
  - Expected inflation (or expected inflation measures from Consensus Forecasts) and unexpected inflation coefficients are consistently negative and often statistically significant across samples (full, AM, EM).
    - Examples of point estimates:
      - Table 12 (using actual inflation): Inflation coefficient estimates include -1.785***, -2.200**, -1.312***, -4.553***, -0.807***, -2.391, -0.863***, -3.637*** (different specs).
      - Interaction terms show that inflation × EM dummy has positive and significant coefficients in some specifications (e.g., 3.202***, 2.846**), implying the stock return–inflation relation is less negative in emerging markets in those specifications.
      - Inflation × policy rate cyclicality and Inflation × countercyclical policy dummy often have negative and significant coefficients (e.g., -4.851***, -7.553***, -3.978***), indicating policy cyclicality and countercyclical policy presence affect the inflation–stock return relation.
  - Control variables with consistent signs and significance:
    - Industrial production growth rate: positive and significant in virtually all specifications (e.g., 1.218***, 0.973***, 1.434***).
    - M2 growth rate: generally positive and sometimes significant (e.g., 0.410**, 0.634***).
    - Improvement in financial risk rating: positive and often significant (e.g., 0.0121**, 0.0212***).
    - VIX: negative and highly significant across specifications (e.g., -0.0164***, -0.0168***).
    - U.S. 3-month Treasury bill yield rate: coefficients small and mixed significance (e.g., 0.00880, 0.0194*).
  - Robustness variations reported:
    - Drop observations when countries adopted the euro (Table 13): expected and unexpected inflation remain generally negative; interaction with policy cyclicality and countercyclical policy retain negative coefficients in many specs.
    - Drop inflation above 100% (Table 14): main patterns persist (expected and unexpected inflation negative; interaction terms with policy cyclicality/countercyclical dummies often negative).
    - Use Consensus Forecasts data (Table 15): expected and unexpected inflation (Consensus Forecasts, current and next year) produce negative coefficients; interaction terms with policy rate cyclicality show varying significance.
    - Add global stock market factor (Systemic-5 real stock return) (Table 16): Systemic-5 real stock return positively and strongly associated with country real stock returns (e.g., coefficient 0.784***, 0.748***, 0.770***); expected inflation coefficients remain negative though magnitudes vary.
    - Use Wu-Xia shadow federal funds rate to address ZLB in the United States (Table 17): results broadly consistent with baseline; Wu-Xia shadow rate itself has small positive coefficients (e.g., 0.00682, 0.0133**) with mixed significance.
  - Sample sizes, R-squared, and number of countries vary by table and specification (examples):
    - Table 12: Observations = 3,498 (full), 1,944 (AM), 1,554 (EM); R-squared range ~0.3992–0.4078; Number of countries = 55 (full), 27 (AM), 28 (EM).
    - Table 13 (drop euro adopters): Observations = 2,701; Number of countries = 45 (full), 17 (AM), 28 (EM); R-squared ~0.3774–0.3963.
    - Table 16 (add Systemic-5): Observations = 3,080; Number of countries = 50 (full), 23 (AM), 27 (EM); R-squared up to 0.5941 (AM).
  - Estimation inference:
    - Driscoll-Kraay standard errors used in panel regressions; robust standard errors used in cross-country OLS.
    - Significance notation: *** p < 0.01; ** p < 0.05; * p < 0.1.

### Empirical interpretation (as presented)
- The relationship between stock returns and inflation is robustly negative for expected and unexpected inflation measures across many specifications.
- The magnitude of the covariance relationship in cross-country tests exceeds the theoretical predictions of Bakshi-Chen under the assumed i.i.d. processes; results for advanced markets are closer to theory than for emerging markets.
- Monetary policy cyclicality and countercyclical policy characteristics materially affect the stock return–inflation relationship; interaction terms often reduce (make more negative) the inflation effect.
- Global factor (Systemic-5 real stock return) explains a substantial portion of country-level real stock return variation when included.

*Source: wpiea2021219-print-pdf - References — https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021219-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021219-print-pdf.pdf_