## 14. Frequency of exclusions – CPI-T(10,10)

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---

### I. Introduction — scope and core claim
- Sample and focus:
  - Euro area, 1995 up to the first quarter 2013.
- Main result:
  - Comovement of output and inflation is rather strong at business cycle frequency when using a measure of trimmed mean (underlying) inflation and accounting for a sluggish response of inflation to demand.
  - The underlying inflation lags output by roughly one quarter.
- Interpretation:
  - Strong commonality suggests a dominant role of demand pressures as a single dynamic factor for both nominal and real macro variables.
  - Positive comovement of output and inflation casts doubt on technology-shock-driven business cycles (RBC theory) for the euro area.
- Methodological framing:
  - Objective is to search for the Phillips correlation at business cycle frequencies.
  - Emphasis on frequency-domain analysis and careful treatment of inflation (distinguishing long-run, cyclical, and short-term variation).
  - Trimmed mean inflation (referred to as underlying inflation) is the preferred core-inflation measure.

### II. Underlying inflation for the Euro Area — construction and properties
- Measures evaluated:
  - Trimmed mean (CPI-T)
  - Weighted median inflation (CPI-MED)
  - CPI excluding food and energy (CPI-X)
- Data and computation:
  - EA HICP data at level three disaggregation: 94 subcomponents.
  - Sample length: 72 periods, 1995:1–2013:1.
  - Trimmed mean constructed by excluding left and right percentiles of the cross-section distribution; trims may be asymmetric.
  - Criterion for choosing trim percentiles: maximize cross-correlation (coherence) of trimmed mean inflation with output at cyclical frequencies.
  - Output cyclical component identified jointly with trimming percentiles; band-pass filter used for cyclical-frequency identification (rectangular band pass).
- Optimal trimming and sensitivity:
  - 31 output cycles evaluated at frequencies from 0 up to 20 or 60 quarters; percentiles varied in steps of size one; 2341 core inflation measures evaluated.
  - Optimal trimming percentage (whole sample 1995:1–2013:1): [48; 28].
  - Optimal trimming for reduced sample up to 2007:1: [37; 21].
  - Optimal symmetric trim: 38.
  - Gains after the 10th percentile are very modest; median inflation is a relatively robust measure.
- Statistical properties of price-change distribution:
  - Cross-section distribution is highly non-Gaussian with very thick tails.
  - Kurtosis in excess of Gaussian distribution averages around 15.
  - Skewness of the weighted price distribution is volatile and positive on average.
- Comparison of measures and biases (2000Q1–2012Q):
  - CPI-X (ex food & energy) has an average bias of -0.54 with respect to headline CPI.
  - Median has an average bias of -0.37 with respect to headline CPI.
  - Baseline trimmed mean has an average bias of 0.14 with respect to headline CPI.
  - CPI-T displays greater comovement with the cyclical component of output for trimming percentiles larger than 10 and appears superior to CPI-X, which persistently understates demand effects.

### III. Inflation–Output comovement — measurement and evidence
- Measurement approach:
  - Use of coherence (spectral analogue to correlation):
    - ρ^2_{x,y}(λ) = |S_{x,y}(λ)|^2 / [S_x(λ) S_y(λ)] ∈ [0,1] for 0 < λ ≤ π.
  - Two approaches to handle nonstationarity of output:
    1. Compute coherence using band-pass-filtered output series and inflation (time-domain band-pass; subject to end-sample inaccuracies).
    2. Compute bivariate spectrum of GDP growth and inflation, then apply integration filter to obtain cross-spectrum of levels (frequency-domain exact filtering). Coherence is invariant to linear, time-invariant, invertible filters (ρ^2_{x,y} = ρ^2_{Δx,y}).
  - Spectral estimation:
    - Parametric estimation via VAR(p) (results reported use VAR(2)); non-parametric Bartlett lag-window estimator used as robustness check. Both approaches yield similar results.
    - Wild bootstrap used for confidence intervals (small sample consideration).
- Key empirical findings:
  - Spectral densities of output and underlying-inflation cycles are similar; variance of the output cycle roughly three times larger.
  - Sample estimate of coherence peaks at a value of 0.9 at business cycle frequencies.
  - Phase evidence: at business cycle frequency with greatest power, inflation lags output roughly by one quarter (phase expressed in periods).
  - Median inflation also shows strong coherence with output; coherence remains strong despite larger confidence intervals.
- Visual and time-domain alignment:
  - Output cycle computed using Christiano and Fitzgerald (1999) band-pass filter; underlying inflation scaled to output variance and phase-shifted by one quarter for alignment.
  - Inflation, output, and unemployment (cyclical components) display close comovement; unemployment cycle (with sign reversed for display) comoves with underlying inflation deviation from target.
- Robustness of spectral estimation:
  - Monte Carlo experiments indicate parametric VAR(2) approach yields unbiased results if data generating process close to VAR(2); Bartlett estimator on average slightly underestimates coherence peak.

### IV. Implications and interpretation
- Macro interpretation:
  - Strong positive comovement at business cycle frequencies supports demand-driven explanations of euro area business cycles rather than technology-shock-driven RBC explanations.
  - Demand shocks (or supply shocks that raise prices and behave like demand shocks in effect) are likely dominant at business cycle frequencies.
  - Tight comovement implies models centered on technology shocks with inflation driven by markups/cost-push shocks are inconsistent with observed joint behavior of output, unemployment, and inflation.
- Policy and measurement implications:
  - Evolution of inflation should guide the design of measures of excess capacity (output gap) that are relevant for inflation (inflation-relevant output).
  - Allowing greater flexibility in trend components of output can reconcile observed inflation with output developments.
  - Under inflation targeting regimes (ECB target explicit since 1999), focus should be on deviations of inflation from the target (inflation gap), not on price level.
- Additional observations:
  - A limited drop in inflation during deep recessions can reflect firms' limited ability to compress margins and a threshold beyond which prolonged output declines cease to further depress inflation.
  - Evidence suggests potential output and the structural rate of unemployment may have shifted during recessions.

### V. Robustness
- Main robustness points addressed:
  - Sample length concerns: results also hold for longer historical sample (text indicates strength of demand-pull inflation is significant in the period from 1970 to ... — analysis continues beyond supplied excerpt).
  - Comovement is robust to changes in trimming percentiles (particularly once trimming exceeds the 10th percentile) and to alternative trimmed-mean specifications (median).
  - Results robust across parametric and nonparametric spectral estimation methods; wild-bootstrap confidence intervals provided.

### Using the synthetic AWM data (2005) — supporting results and placebo testing
- Key empirical highlights with AWM synthetic euro-area data:
  - Strong and positive comovement of output and the consumption deflator at business cycle frequencies.
  - The coherence of the inflation deviation from its trend with the output cycle peaks around 0.6–0.8, depending on a lag length specification of the VAR used for the spectrum estimation.
  - Underlying inflation, defined as an asymmetric trimmed mean, lags output at business cycle frequencies on average by one quarter.
  - Underlying inflation is roughly twice less volatile than output.
  - The coherence of output and underlying inflation at business cycle frequencies lies in the range 0.6–0.9.
  - Due to unavailability of an explicit inflation target, the trend component is approximated by removal of low-frequency component of inflation.
- Trimmed-mean construction and robustness:
  - Asymmetric trimmed mean formula:
    - π_tm_t(α_l, α_r) = (1 / (1 − α_l − α_r)) ∑_{i ∈ I} w_i,t π_i,t, with I = {i : α_l < W_i,t < (1 − α_r)} and W_i,t = ∑_{j=1}^i ̃w_j,t.
  - Changing benchmark period for estimation of trimming percentiles modestly affects results:
    - Shorter benchmark range 2000:1–2007:1 → optimal trimming percentage changes to [37; 21] from baseline of [48; 28].
  - Once the 10th percentile is removed from both sides of the distribution, the additional gains from further trimming are small.
  - The asymmetric baseline case is higher roughly by 30 basis points, annualized, relative to symmetric measures shown.
- Placebo testing (Appendix B) and protection against data mining:
  - Placebo procedure: replace 94 price components with random samples drawn from an autoregressive model fitted to each series; compute optimal percentiles and coherence; repeat for 500 draws.
  - Placebo test results:
    - Main text: Monte Carlo study median peak coherence is just 0.1.
    - Appendix B: The median peak coherence is 0.15, with the highest outliers attaining value of 0.35.
  - Conclusion: The observed strength of output and underlying inflation coherence cannot be replicated by chance under the placebo design.

### Selected quantitative highlights (preserved exactly)
- Sample length: 72 periods, 1995:1–2013:1.
- EA HICP components used: 94.
- Number of core inflation measures evaluated: 2341.
- Optimal trimming percentiles (baseline whole sample): [48; 28].
- Optimal trimming percentiles (shorter sample 2000:1–2007:1): [37; 21].
- Optimal symmetric trim: 38.
- Variance relation: variance of the output cycle roughly three times larger than underlying-inflation cycle.
- Coherence peak (sample estimate): 0.9 at business cycle frequencies.
- Coherence range reported with AWM data: 0.6–0.9.
- Lag of underlying inflation relative to output: one quarter.
- Relative volatility: underlying inflation roughly twice less volatile than output.
- Asymmetric baseline premium: higher by roughly 30 basis points, annualized.
- Placebo Monte Carlo median peak coherence: 0.1 (main text); 0.15 (Appendix B); highest outliers up to 0.35.
- Placebo draws used: 500.
- Statistical property: kurtosis in excess of Gaussian distribution averages around 15.
- Biases (2000Q1–2012Q): CPI-X average bias -0.54; Median average bias -0.37; Baseline trimmed mean average bias 0.14.

*Source: _wp13192 - 14. Frequency of exclusions – CPI-T(10,10) (excerpt) and related AWM synthetic-data results included in the supplied content.*

### References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### _wp13192 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Appendices
- A.    Trimmed mean methodology and computations.  .  .  .  .  .  .  .  .  .  .  .  .  .  .    22
- B.Placebo test for the matching estimator   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 23
- C.Additional graphs & tables    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    24

### Figures (listed with titles and page numbers)
- 1.Headline (thin) vs. underlying inflation (thick), %, ann.    .  .  .  .  .  .  .  .  .  .  .  .9
- 2.Trimmed mean vs. CPI-X, % ann.  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 10
- 3.Output and inflation cycles .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .12
- 4.Spectral properties – output and inflation cycle  .  .  .  .  .  .  .  .  .  .  .  .  . 14
- 5.GDP level and median inflation coherence   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 14
- 6.Underlying inflation – output and unemployment cycles   .  .  .  .  .  .  .  .  .  .16
- 7.Headline inflation and variety of trimmed means   .  .  .  .  .  .  .  .  .  .  .  .  .18
- 8.Skewness and excess kurtosis   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .24
- 9.Sample distribution of price changes    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .24
- 10.   Effect of asymmetric trimmed percentage on the loss  .  .  .  .  .  .  .  .  .  . 24
- 11.   Baseline trimmed mean vs weighted median   .  .  .  .  .  .  .  .  .  .  .  .  .  . 25
- 12.   Cons. deflator inflation and output cycle .  .  .  .  .  .  .  .  .  .  .  .  .  .  .25
- 13.   Price level implications of underlying inflation measures  .  .  .  .  .  .  .  .25

### Content focus and organization (inferred from headings)
- Methodological material on trimmed mean computation and alternative underlying inflation measures is concentrated in Appendix A and in figures comparing trimmed mean, CPI-X, weighted medians, and a variety of trimmed means.
- Empirical diagnostics on cycles, spectral properties, and coherence between inflation, output, and unemployment are presented in Figures 3, 4, 5, and 6.
- Distributional properties of price changes, including skewness and excess kurtosis, and the sample distribution of price changes, are documented in Figures 8 and 9.
- Robustness and sensitivity analyses, including placebo testing for the matching estimator and the effect of asymmetric trimming on loss, are provided in Appendix B and Figure 10.
- Price-level implications and comparisons with consumption deflator inflation and output cycles are addressed in Figures 12 and 13.

*Content derived from the PDF chapter/section listing provided in the source file.*

### 14.   Frequency of exclusions – CPI-T(10,10)    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .    2

### 14.   Frequency of exclusions – CPI-T(10,10)

### I. Introduction — scope and core claim
- Sample and focus:
  - Euro area, 1995 up to the first quarter 2013.
- Main result:
  - Comovement of output and inflation is rather strong at business cycle frequency when using a measure of trimmed mean (underlying) inflation and accounting for a sluggish response of inflation to demand.
  - The underlying inflation lags output by roughly one quarter.
- Interpretation:
  - Strong commonality suggests a dominant role of demand pressures as a single dynamic factor for both nominal and real macro variables.
  - Positive comovement of output and inflation casts doubt on technology-shock-driven business cycles (RBC theory) for the euro area.
- Methodological framing:
  - Objective is to search for the Phillips correlation at business cycle frequencies.
  - Emphasis on frequency-domain analysis and careful treatment of inflation (distinguishing long-run, cyclical, and short-term variation).
  - Trimmed mean inflation (referred to as underlying inflation) is the preferred core-inflation measure.

### II. Underlying inflation for the Euro Area — construction and properties
- Measures evaluated:
  - Trimmed mean (CPI-T)
  - Weighted median inflation (CPI-MED)
  - CPI excluding food and energy (CPI-X)
- Data and computation:
  - EA HICP data at level three disaggregation: 94 subcomponents.
  - Sample length: 72 periods, 1995:1–2013:1.
  - Trimmed mean constructed by excluding left and right percentiles of the cross-section distribution; trims may be asymmetric.
  - Criterion for choosing trim percentiles: maximize cross-correlation (coherence) of trimmed mean inflation with output at cyclical frequencies.
  - Output cyclical component identified jointly with trimming percentiles; band-pass filter used for cyclical-frequency identification (rectangular band pass).
- Optimal trimming and sensitivity:
  - 31 output cycles evaluated at frequencies from 0 up to 20 or 60 quarters; percentiles varied in steps of size one; 2341 core inflation measures evaluated.
  - Optimal trimming percentage (whole sample 1995:1–2013:1): [48; 28].
  - Optimal trimming for reduced sample up to 2007:1: [37; 21].
  - Optimal symmetric trim: 38.
  - Gains after the 10th percentile are very modest; median inflation is a relatively robust measure.
- Statistical properties of price-change distribution:
  - Cross-section distribution is highly non-Gaussian with very thick tails.
  - Kurtosis in excess of Gaussian distribution averages around 15.
  - Skewness of the weighted price distribution is volatile and positive on average.
- Comparison of measures and biases (2000Q1–2012Q):
  - CPI-X (ex food & energy) has an average bias of -0.54 with respect to headline CPI.
  - Median has an average bias of -0.37 with respect to headline CPI.
  - Baseline trimmed mean has an average bias of 0.14 with respect to headline CPI.
  - CPI-T displays greater comovement with the cyclical component of output for trimming percentiles larger than 10 and appears superior to CPI-X, which persistently understates demand effects.

### III. Inflation–Output comovement — measurement and evidence
- Measurement approach:
  - Use of coherence (spectral analogue to correlation) to measure comovement:
    - ρ^2_{x,y}(λ) = |S_{x,y}(λ)|^2 / [S_x(λ) S_y(λ)] ∈ [0,1] for 0 < λ ≤ π.
  - Two approaches to handle nonstationarity of output:
    1. Compute coherence using band-pass-filtered output series and inflation (time-domain band-pass; subject to end-sample inaccuracies).
    2. Compute bivariate spectrum of GDP growth and inflation, then apply integration filter to obtain cross-spectrum of levels (frequency-domain exact filtering). Coherence is invariant to linear, time-invariant, invertible filters (ρ^2_{x,y} = ρ^2_{Δx,y}).
  - Spectral estimation:
    - Parametric estimation via VAR(p) (results reported use VAR(2)); non-parametric Bartlett lag-window estimator used as robustness check. Both approaches yield similar results.
    - Wild bootstrap used for confidence intervals (small sample consideration).
- Key empirical findings:
  - Spectral densities of output and underlying-inflation cycles are similar; variance of the output cycle roughly three times larger.
  - Sample estimate of coherence peaks at a value of 0.9 at business cycle frequencies.
  - Phase evidence: at business cycle frequency with greatest power, inflation lags output roughly by one quarter (phase expressed in periods).
  - Median inflation also shows strong coherence with output; coherence remains strong despite larger confidence intervals.
- Visual and time-domain alignment:
  - Output cycle computed using Christiano and Fitzgerald (1999) band-pass filter; underlying inflation scaled to output variance and phase-shifted by one quarter for alignment.
  - Inflation, output, and unemployment (cyclical components) display close comovement; unemployment cycle (with sign reversed for display) comoves with underlying inflation deviation from target.
- Robustness checks on spectral estimation:
  - Monte Carlo experiments indicate parametric VAR(2) approach yields unbiased results if data generating process close to VAR(2); Bartlett estimator on average slightly underestimates coherence peak.

### IV. Implications and interpretation
- Macro interpretation:
  - Strong positive comovement at business cycle frequencies supports demand-driven explanations of euro area business cycles rather than technology-shock-driven RBC explanations.
  - Demand shocks (or supply shocks that raise prices and behave like demand shocks in effect) are likely dominant at business cycle frequencies.
  - Tight comovement implies models centered on technology shocks with inflation driven by markups/cost-push shocks are inconsistent with observed joint behavior of output, unemployment, and inflation.
- Policy and measurement implications:
  - Evolution of inflation should guide the design of measures of excess capacity (output gap) that are relevant for inflation (inflation-relevant output).
  - Allowing greater flexibility in trend components of output can reconcile observed inflation with output developments.
  - Under inflation targeting regimes (ECB target explicit since 1999), focus should be on deviations of inflation from the target (inflation gap), not on price level.
- Additional observations:
  - A limited drop in inflation during deep recessions can reflect firms' limited ability to compress margins and a threshold beyond which prolonged output declines cease to further depress inflation.
  - Evidence suggests potential output and the structural rate of unemployment may have shifted during recessions (consistent with external evidence cited).

### V. Robustness
- Main robustness points addressed:
  - Sample length concerns: results also hold for longer historical sample (text indicates strength of demand-pull inflation is significant in the period from 1970 to ... — analysis continues beyond supplied excerpt).
  - Comovement is robust to changes in trimming percentiles (particularly once trimming exceeds the 10th percentile) and to alternative trimmed-mean specifications (median).
  - Results robust across parametric and nonparametric spectral estimation methods; wild-bootstrap confidence intervals provided.

*Source: _wp13192 - 14.   Frequency of exclusions – CPI-T(10,10) (excerpt).*

### 2005. Using the synthetic data for the euro area compiled for the Area Wide Model (AWM)

### 2005. Using the synthetic data for the euro area compiled for the Area Wide Model (AWM)

### Key findings on comovement of output and inflation
- Strong and positive comovement of output and the consumption deflator at business cycle frequencies.
- The coherence of the inflation deviation from its trend with the output cycle peaks around 0.6–0.8, depending on a lag length specification of the VAR used for the spectrum estimation.
- Underlying inflation, defined as an asymmetric trimmed mean, lags output at business cycle frequencies on average by one quarter.
- Underlying inflation is roughly twice less volatile than output.
- The coherence of output and underlying inflation at business cycle frequencies lies in the range 0.6–0.9.
- The close comovement is highly suggestive of the dominance of demand factors in the euro area business cycle; technology-shock–based supply stories imply a negative comovement and so are inconsistent with these findings.
- Note: Due to unavailability of an explicit inflation target, or long-term inflation expectations, the trend component is approximated by removal of low-frequency component of inflation.

### Trimmed-mean construction, results, and robustness
- The asymmetric trimmed mean is defined as:
  - π_tm_t(α_l, α_r) = (1 / (1 − α_l − α_r)) ∑_{i ∈ I} w_i,t π_i,t, with I = {i : α_l < W_i,t < (1 − α_r)} and W_i,t = ∑_{j=1}^i ̃w_j,t, where ̃w_j,t are weights corresponding to price changes sorted in ascending order.
- Changing the benchmark period for estimation of trimming percentiles modestly affects results:
  - Shorter benchmark range 2000:1–2007:1 → optimal trimming percentage changes to [37; 21] from baseline of [48; 28].
  - Interpretation: In the shorter sample less extreme price decreases are being removed from the headline inflation measure due to absence of year 2009 and a dramatic drop in energy and food related prices.
  - The similarity of both measures is very large, so the scope for error is limited.
- Once the 10th percentile is removed from both sides of the distribution, the additional gains from further trimming are small (inferred from the profile of the loss function in dependence on trimming).
- The results favour asymmetric trimmed means, even for a shorter benchmark range.
- Median inflation and a variety of trimmed mean inflation measures also display cyclical comovements with output.
- The asymmetric baseline case is higher roughly by 30 basis points, annualized, relative to symmetric measures shown.

### Placebo testing and protection against data mining
- A ‘placebo’ sampling test checks whether the matching estimator could generate the comovement by weighting random draws from processes having univariate characteristics of individual price categories.
- Placebo test procedure (Appendix B):
  - Replace individual 94 price components with random samples drawn from an autoregressive model fitted to each series.
  - For each artificial set of HICP components, compute optimal percentiles to trim and the coherence with output.
  - Construct a distribution of coherence from 500 draws.
- Placebo test results:
  - Main text: Monte Carlo study median peak coherence is just 0.1 (rejects the possibility that matching estimator could generate observed comovement by chance).
  - Appendix B: The median peak coherence is 0.15, with the highest outliers attaining value of 0.35.
- Conclusion: One cannot replicate the strength of the output and underlying inflation coherence by ‘accident’ or by manipulating the series.

### Data, treatment, and methodological notes
- Data:
  - Price and weights data are at the level 3 of disaggregation as provided by Eurostat.
  - Immediate source is Haver Analytics database; 94 items are used to replicate headline CPI growth with negligible loss in accuracy.
  - Real output data are seasonally adjusted, as provided by Eurostat.
- Treatment of missing data:
  - Missing data are treated as any data with zero weight in the aggregate.
  - Missing data are mostly pharmaceutical prices in 1995–1996 and affect the aggregate in a negligible way.
- Seasonal adjustment:
  - Baseline computations use seasonally adjusted data provided by Haver Analytics.
  - Authors note a univariate seasonal adjustment can break general equilibrium links between relative prices across seasons.

### Supporting empirical evidence (selected quantitative highlights)
- Coherence peak values: around 0.6–0.8 (VAR lag dependent); overall coherence range 0.6–0.9.
- Lag of underlying inflation relative to output: one quarter.
- Relative volatility: underlying inflation roughly twice less volatile than output.
- Asymmetric baseline premium: higher by roughly 30 basis points, annualized.
- Optimal trimming percentiles: baseline [48; 28]; shorter-sample [37; 21].
- Placebo Monte Carlo median peak coherence: 0.1 (main text); 0.15 (Appendix B); highest outliers up to 0.35.
- Placebo draws used: 500.

### Conclusions and implications for modeling and policy
- The strong degree of comovement between inflation and output indicates demand cycles play a central role in euro area inflation determination at business cycle frequencies.
- Structural models must capture positive comovement of output and inflation at business cycle frequencies to interpret euro area developments plausibly.
- Models that rely primarily on technology shocks or supply-side narratives that imply negative comovement of output and inflation will struggle to explain these empirical regularities.
- Authors acknowledge that supply-side and policy factors still shape dynamics at low and high frequencies, but demand factors dominate at business cycle frequencies.

*Source: 2005. Using the synthetic data for the euro area compiled for the Area Wide Model (AWM)*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13192.pdf_