## A Note of Caution on the Relation between Money Growth and Inflation

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---

### Introduction
- Purpose: Assess the bivariate relation between money growth and inflation in the euro area and the United States using hybrid time-varying parameter Bayesian VAR models.
- Key motivation: Recent surge in inflation following non-standard monetary policy measures revived debate on the role of money growth for inflation.
- Core caution: Results warn against assuming a simple, time-invariant relationship when modeling monetary aggregates and consumer prices.

### Data and model
- Data series:
  - CPI inflation: πt = 100 (Pt / Pt−4 − 1) (consumer price index).
  - Money growth: μt = 100 (Mt / Mt−4 − 1) (M3).
  - Sample ranges:
    - Euro area: 1971Q1 to 2022Q4.
    - United States: 1961Q1 to 2022Q4.
- Model framework:
  - Hybrid time-varying parameter bivariate Bayesian VAR with stochastic volatility (framework of Chan and Eisenstat (2018)).
  - Dependent vector: yt = (πt, μt)′.
  - General form includes a 2x2 lower triangular B0t with ones on the diagonal, time-varying intercepts δt, time-varying coefficient matrices Bit (i = 1,..,p), orthogonal disturbances εt with stochastic volatility.
  - Log volatilities follow random walks: ht = ht−1 + ζt, ζt ~ N(0, Vh).
  - Regression-parameter vector θt = θt−1 + ηt, ηt ~ N(0, Vθ); θt split into θ1,t (inflation equation) and θ2,t (money-growth equation).
  - Four model configurations considered:
    - Both equations constant (no time variation in θ1,t and θ2,t).
    - Time variation in inflation equation only (θ1,t time varying; θ2,t constant).
    - Time variation in money-growth equation only (θ2,t time varying; θ1,t constant).
    - Both equations time varying (θ1,t and θ2,t time varying).
- Estimation details and priors:
  - Lag length set to 4.
  - Uninformative normal prior on initial states of regression parameters: θ1,0 and θ2,0 ~ N(0, 5I).
  - Initial log volatilities prior: hi,0 ~ N(γi, 0.25), where γi set to match prior mean of exp(hi,0) with residual variance of a constant-parameter univariate AR(4).
  - Diagonal elements of Vθ have inverse Gamma priors: di (5, 0.08) for constants and di (5, 0.0004) for other parameters.
  - Diagonal elements of Vh have di (5, 0.4) priors.

### Results — model comparison and impulse responses
- Model comparison (log marginal likelihoods; highest marginal likelihood in bold):
  - Euro area:
    - Both equations are constant: -458.2
    - Time variation in equation for πt: -455.7
    - Time variation in equation for μt: -452.5
    - Both equations are time varying: -449.9 (preferred)
  - United States:
    - Both equations are constant: -622.2
    - Time variation in equation for πt: -616.4
    - Time variation in equation for μt: -620.6
    - Both equations are time varying: -614.6 (preferred)
- Strength of evidence:
  - Comparing the model with time-varying parameters in both equations to the constant-parameter model yields evidence characterized as “very strong” using two times the difference in log marginal likelihood and Kass and Raftery (1995) terminology.
  - Evidence in favor of both-equations time variation relative to single-equation time variation models is “positive” or “very strong”.
- Time variation in the effect of money growth on inflation:
  - Dynamics changed substantially over the sample in both regions.
  - Money growth shocks had a clear positive impact on inflation in the 1970s and 1980s (euro area) and in the 1980s (United States).
  - Relationship became very weak or non-existent starting in the 1990s.
  - Relationship strengthened markedly after 2020 (early 2020s), showing a dramatic comeback.
- Impulse-response findings (preferred model: both equations time varying; impulse size = one standard deviation unless otherwise noted):
  - 1982Q4: shocks to money growth had an inverted u-shape effect on inflation.
  - 2015Q4: response of inflation to money growth was indistinguishable from zero.
  - 2022Q4: inflation increases again with a shock to money growth, with more of a delay relative to earlier periods.
  - The United States shows a quantitatively larger effect in the early-2020s period related to larger underlying shock magnitudes (see appendix standard deviation figures referenced).
- Sensitivity to stochastic volatility:
  - Because the model includes stochastic volatility, the size of the impulse is time varying.

### Link between trend inflation and the impact of money growth
- Trend inflation measure:
  - Calculated as the local mean (first element of ξt = (I − At)−1 α˜t) from the model’s reduced-form companion representation.
  - Trend inflation represents the value to which inflation forecasts from the model would converge.
- Relation examined:
  - Maximum effect on inflation of a unit shock to money growth is related to trend inflation (shock size of unity used to make effects comparable over time).
- Findings:
  - Maximum effect of money growth on inflation tends to be higher at higher levels of trend inflation.
  - Co-movement strength:
    - Euro area correlation between trend inflation and maximum effect: 0.95.
    - United States correlation between trend inflation and maximum effect: 0.64.
  - Co-movement appears stronger for the euro area than for the United States but is present in both.

### Policy-relevant implications and cautions
- Empirical caution:
  - Assuming constant-parameter models of the relation between money growth and inflation would provide a misleading description of economic circumstances for policymakers given the documented time variation.
- Practical implications:
  - Forecasting and policy analysis that rely on a time-invariant bivariate relation between monetary aggregates and consumer prices risk being misspecified, especially across periods with differing trend inflation.
- Interpretation:
  - The maximum impact of money growth on inflation is increasing in the trend level of inflation, suggesting that the predictive or causal relevance of money growth shocks for inflation depends on prevailing inflationary conditions.

### Section 2 — Key findings and methodological notes
- Key findings on the money growth–inflation relation:
  - The bivariate relation between money growth and inflation in the euro area and the United States has changed over time.
  - Model selection based on marginal likelihoods indicates the relation is statistically unstable across time in both regions.
  - Time-varying parameter models show the dynamic relation between money growth and inflation:
    - Weakened notably after the 1980s.
    - Re-emerged (made a comeback) after 2020.
  - There is evidence that time variation may be related to the pace of inflation: the maximum impact of money growth on inflation is increasing in the trend level of inflation.
- Quantitative and methodological notes:
  - Impulse-response functions report effects in percentage points on the vertical axis and horizon in quarters on the horizontal axes (figures for euro area and United States).
  - Appendix figures include standard deviations of shocks to money growth and impulse-response functions for constant-parameter models (not reproduced here).

*IMF Working Paper: A Note of Caution on the Relation between Money Growth and Inflation, Helge Berger, Sune Karlsson, and Pär Österholm, WP/23/137 (June 2023).*

### Section 1

### A Note of Caution on the Relation between Money Growth and Inflation

### Introduction
- Purpose: Assess the bivariate relation between money growth and inflation in the euro area and the United States using hybrid time-varying parameter Bayesian VAR models.
- Key motivation: Recent surge in inflation following non-standard monetary policy measures revived debate on the role of money growth for inflation.
- Core caution: Results warn against assuming a simple, time-invariant relationship when modeling monetary aggregates and consumer prices.

### Data and model
- Data series:
  - CPI inflation: πt = 100 (Pt / Pt−4 − 1) (consumer price index).
  - Money growth: μt = 100 (Mt / Mt−4 − 1) (M3).
  - Sample ranges:
    - Euro area: 1971Q1 to 2022Q4.
    - United States: 1961Q1 to 2022Q4.
- Model framework:
  - Hybrid time-varying parameter bivariate Bayesian VAR with stochastic volatility (framework of Chan and Eisenstat (2018)).
  - Dependent vector: yt = (πt, μt)′.
  - General form includes a 2x2 lower triangular B0t with ones on the diagonal, time-varying intercepts δt, time-varying coefficient matrices Bit (i = 1,..,p), orthogonal disturbances εt with stochastic volatility.
  - Log volatilities follow random walks: ht = ht−1 + ζt, ζt ~ N(0, Vh).
  - Regression-parameter vector θt = θt−1 + ηt, ηt ~ N(0, Vθ); θt split into θ1,t (inflation equation) and θ2,t (money-growth equation).
  - Four model configurations considered:
    - Both equations constant (no time variation in θ1,t and θ2,t).
    - Time variation in inflation equation only (θ1,t time varying; θ2,t constant).
    - Time variation in money-growth equation only (θ2,t time varying; θ1,t constant).
    - Both equations time varying (θ1,t and θ2,t time varying).
- Estimation details and priors:
  - Lag length set to 4.
  - Uninformative normal prior on initial states of regression parameters: θ1,0 and θ2,0 ~ N(0, 5I).
  - Initial log volatilities prior: hi,0 ~ N(γi, 0.25), where γi set to match prior mean of exp(hi,0) with residual variance of a constant-parameter univariate AR(4).
  - Diagonal elements of Vθ have inverse Gamma priors: di (5, 0.08) for constants and di (5, 0.0004) for other parameters.
  - Diagonal elements of Vh have di (5, 0.4) priors.

### Results — model comparison and impulse responses
- Model comparison (log marginal likelihoods; highest marginal likelihood in bold):
  - Euro area:
    - Both equations are constant: -458.2
    - Time variation in equation for πt: -455.7
    - Time variation in equation for μt: -452.5
    - Both equations are time varying: -449.9 (preferred)
  - United States:
    - Both equations are constant: -622.2
    - Time variation in equation for πt: -616.4
    - Time variation in equation for μt: -620.6
    - Both equations are time varying: -614.6 (preferred)
- Strength of evidence:
  - Comparing the model with time-varying parameters in both equations to the constant-parameter model yields evidence characterized as “very strong” using two times the difference in log marginal likelihood and Kass and Raftery (1995) terminology.
  - Evidence in favor of both-equations time variation relative to single-equation time variation models is “positive” or “very strong”.
- Time variation in the effect of money growth on inflation:
  - Dynamics changed substantially over the sample in both regions.
  - Money growth shocks had a clear positive impact on inflation in the 1970s and 1980s (euro area) and in the 1980s (United States).
  - Relationship became very weak or non-existent starting in the 1990s.
  - Relationship strengthened markedly after 2020 (early 2020s), showing a dramatic comeback.
- Impulse-response findings (preferred model: both equations time varying; impulse size = one standard deviation unless otherwise noted):
  - 1982Q4: shocks to money growth had an inverted u-shape effect on inflation.
  - 2015Q4: response of inflation to money growth was indistinguishable from zero.
  - 2022Q4: inflation increases again with a shock to money growth, with more of a delay relative to earlier periods.
  - The United States shows a quantitatively larger effect in the early-2020s period related to larger underlying shock magnitudes (see appendix standard deviation figures referenced).
- Sensitivity to stochastic volatility:
  - Because the model includes stochastic volatility, the size of the impulse is time varying.

### Link between trend inflation and the impact of money growth
- Trend inflation measure:
  - Calculated as the local mean (first element of ξt = (I − At)−1 α˜t) from the model’s reduced-form companion representation.
  - Trend inflation represents the value to which inflation forecasts from the model would converge.
- Relation examined:
  - Maximum effect on inflation of a unit shock to money growth is related to trend inflation (shock size of unity used to make effects comparable over time).
  - Findings:
    - Maximum effect of money growth on inflation tends to be higher at higher levels of trend inflation.
    - Co-movement strength:
      - Euro area correlation between trend inflation and maximum effect: 0.95.
      - United States correlation between trend inflation and maximum effect: 0.64.
    - Co-movement appears stronger for the euro area than for the United States but is present in both.

### Policy-relevant implications and cautions
- Empirical caution:
  - Assuming constant-parameter models of the relation between money growth and inflation would provide a misleading description of economic circumstances for policymakers given the documented time variation.
- Practical implications:
  - Forecasting and policy analysis that rely on a time-invariant bivariate relation between monetary aggregates and consumer prices risk being misspecified, especially across periods with differing trend inflation.
- Interpretation:
  - The maximum impact of money growth on inflation is increasing in the trend level of inflation, suggesting that the predictive or causal relevance of money growth shocks for inflation depends on prevailing inflationary conditions.

*IMF Working Paper: A Note of Caution on the Relation between Money Growth and Inflation, Helge Berger, Sune Karlsson, and Pär Österholm, WP/23/137 (June 2023).*

### Section 2

### wpiea2023137-print-pdf - Section 2

### Key findings on the money growth–inflation relation
- The bivariate relation between money growth and inflation in the euro area and the United States has changed over time.
- Model selection based on marginal likelihoods indicates the relation is statistically unstable across time in both regions.
- Time-varying parameter models show the dynamic relation between money growth and inflation:
  - Weakened notably after the 1980s.
  - Re-emerged (made a comeback) after 2020.
- There is evidence that time variation may be related to the pace of inflation: the maximum impact of money growth on inflation is increasing in the trend level of inflation.

### Quantitative and methodological notes
- Impulse-response functions report effects in percentage points on the vertical axis and horizon in quarters on the horizontal axes (figures for euro area and United States).
- Appendix figures include:
  - Figure A1: Standard deviation of shocks to money growth in the euro area (Note: Estimated standard deviation of structural shock. Shaded band is 68 percent credible interval).
  - Figure A2: Standard deviation of shocks to money growth in the United States (Note: Estimated standard deviation of structural shock. Shaded band is 68 percent credible interval).
  - Figure A3: Impulse-response function for the euro area from a model with constant parameters in both equations (Note: Size of impulse is one standard deviation. Effect in percentage points on vertical axis. Horizon in quarters and dates on horizontal axes).
  - Figure A4: Impulse-response function for the United States from a model with constant parameters in both equations (Note: Size of impulse is one standard deviation. Effect in percentage points on vertical axis. Horizon in quarters and dates on horizontal axes).

### Interpretation and policy implications
- The results caution against asserting a simple, time-invariant relationship between the dynamics of monetary aggregates and consumer prices for:
  - Forecasting purposes.
  - Policy-making purposes.
- The findings are consistent with a strand of literature that documents:
  - A weakening relationship between money growth and inflation over time (Berger and Österholm (2011a, 2011b); Gertler and Hofmann (2018)).
  - A co-movement of money growth and inflation since 2020 that may be reconciled by conditional forecasting power in a high-inflation regime (Borio et al (2023a)).
  - Stronger money-growth and inflation nexus among high-inflation countries (de Grauwe and Polan (2005); Gertler and Hofmann (2018)).
- The present results generalize prior findings by not relying on an arbitrary definition of a “high” inflation regime or country; instead, they document that the maximum impact of money growth on inflation varies with the trend level of inflation.

*Source: Authors’ calculations, as presented in wpiea2023137-print-pdf - Section 2*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023137-print-pdf.pdf_