## wpiea2023044-print-pdf

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---

### I. Introduction — purpose and context
- Objective: construct weighted-median core inflation series for 38 countries and compare properties to headline inflation and standard core (XFE) measures.
- Motivation:
  - XFE (inflation excluding food and energy) commonly used but can miss large sectoral shocks outside food and energy.
  - Outlier-exclusion measures (weighted median, trimmed means) remove unusually large price changes across any industry; weighted median is the 50th-percentile industry inflation rate.
- Prior evidence and benchmarks:
  - Ball et al. (2021) U.S. monthly CPI (1985–2019) standard deviations: median 1.05 and XFE 1.42.
  - Pandemic example (January 2020–November 2021) standard deviations: median 1.51 and XFE 3.86.
  - Ball and Mazumder (2020) Phillips-curve fit example: adjusted R2 = 0.48 (weighted median) vs 0.22 (XFE) for 1985–2017 U.S. quarterly data.
- Contribution: compute weighted-median core inflation for 38 countries (advanced and emerging), analyze volatility, relationship to economic slack (Phillips curve), and forecast performance for 12-month ahead headline inflation; monthly results in Appendix A.

### II. Data and methodology — scope and construction
- Sample:
  - 38 countries across advanced economies and emerging markets.
  - Inflation data spans from 1990 to 2021 (unbalanced availability by country).
  - Industry-level CPI at the fourth (“subclass”) level with about 150 to 250 industries per country.
- Definitions:
  - Headline inflation: all-items CPI as reported by national agencies.
  - Standard core (XFE): headline excluding food and energy.
  - Weighted median: industry inflation rate at the 50th percentile of the weighted distribution.
- Frequencies and annualization:
  - Monthly median: month t vs t-1 industry inflation; annualized by multiplying monthly median by 12.
  - Quarterly median: construct quarterly industry price by averaging three months; quarter t vs t-1 inflation; annualized by multiplying quarterly median by 4.
  - 12-month median: industry inflation from t-12 to t; weighted median of those annual rates.
- Seasonal adjustment: U.S. Census Bureau’s X-11 version of X-13ARIMA-SEATS applied to unadjusted monthly and quarterly inflation rates (also applied to headline and standard core).
- Phillips curve baseline specification (quarterly, country-by-country OLS with HAC standard errors):
  - π_t = α + π_t^e + β y_t + ε_t
  - π_t^e is five-year ahead inflation expectations (Consensus Forecasts; robustness uses IMF WEO five-year expectations).
  - y_t is output gap: seasonally adjusted real GDP detrended with Hodrick-Prescott filter (λ=1600).
  - Estimation: OLS with heteroskedasticity and autocorrelation consistent standard errors (Newey & West 1987, Andrews 1991); robustness includes hybrid (lagged inflation) and panel with country fixed effects.

### III. Key results and empirical findings
- Volatility
  - Main finding: weighted median inflation is generally less volatile than both headline and standard core for most countries at quarterly frequency, and much less volatile at the monthly frequency.
  - Result especially strong for advanced economies.
  - More mixed results for emerging markets; for Brazil, Costa Rica, Mexico, and Peru weighted median volatility is close to or higher than standard core.
  - Robustness: similar conclusions when volatility measured by variance of changes rather than levels.
- Average levels and bias
  - Both standard core (XFE) and weighted median typically have lower mean levels than headline inflation — a “negative bias” relative to headline.
  - The negative bias is sometimes larger for standard core, sometimes for weighted median.
  - A group of eight countries (mostly emerging economies) show a considerably larger negative bias in the weighted median.
  - Monthly frequency: bias of median inflation is much larger, related in part to frequency of zeros in monthly industry inflation rates.
  - Exception: United States shows a positive rather than negative bias of weighted median relative to headline.
- Forecasting one-year-ahead headline inflation
  - Approach: compare current headline, standard core, and median inflation as predictors of future headline inflation measured over twelve months from t to t+12.
  - Metric: root mean squared error (RMSE) for monthly, quarterly, and 12-month measures.
  - Main statement: current level of median inflation is a better predictor of headline inflation over the next twelve months than standard core, with the strongest results using current monthly levels of the core measures.

### IV. P*: An Unbiased Core Measure (Section III.D)
- Definition and construction
  - P* is defined for each country as the inflation rate at the percentile of the sectoral inflation distribution that minimizes the average difference between its level and that of headline (h) inflation over the sample T.
  - For country j, p_j minimizes 1/T ∑ (π_{jt}^h − 1/T ∑ π_{jt}^{p*}), where π_{jt}^{p*} = π_{i,j,t} for industry i such that ℛ_{i,j,t} = p_j; the ranking operator ℛ_{i,j,t} orders sectoral inflation rates.
  - The industry at percentile p_j can change every month, analogous to the industry at the median.
- Empirical characterization of the percentile p_j
  - For most countries the percentile p_j that minimizes bias is above the median (50th percentile), and usually between the 50th and the 60th percentile for inflation at the quarterly frequency.
  - The US is an exception: the percentile that minimizes bias for the US is below the 50th percentile, slightly below the 40th percentile.
  - The percentile that defines P* differs across countries, across data frequencies (monthly, quarterly, and year-on-year), and depends on the sample period.
- Volatility and bias
  - By construction P* minimizes the average bias relative to headline inflation; empirical results show P* is close to zero bias on average (finite-series mean bias can remain).
  - For most countries, the volatility of P* is close to the volatility of the weighted median and lower than the volatility of headline inflation and standard core.
  - Exceptions: Korea and Slovakia, where P* is more volatile than standard core (median is less volatile than standard core, but P* is not).
  - Conclusion: moving from weighted median to P* fixes the median’s bias without sacrificing much stability of the core measure in most cases.
- Phillips curve performance
  - Success judged by R^2 of Phillips curve regressing inflation on expected inflation and the output gap (quarterly).
  - P* R^2 is generally close to the R^2 for median inflation and noticeably higher than headline and standard core for several countries.
  - P* and weighted median outperform headline and standard core for about three quarters of the countries in country-specific and panel regressions.
  - Higher R^2 when inflation is measured by weighted median or P* implies more of inflation’s variance is explained by domestic economic slack.
- Predictive ability for one-year-ahead headline inflation
  - Metric: ratio = std(one-year-ahead headline − current core) / std(one-year-ahead headline − current headline). Ratio < 1 implies core measure is a better predictor than current headline inflation.
  - All three core measures (standard core, weighted median, and P*) almost always yield ratios less than one.
  - For most countries the ratio is lower for the weighted median than for standard core.
  - P* usually performs worse than the weighted median in this predictive exercise; correcting the median’s bias appears counterproductive for predicting one-year-ahead headline inflation.
- Performance in the 2021–22 inflation event
  - Time-series percentiles across countries show weighted median and P* started capturing the increase in inflation almost immediately in 2021.
  - Standard core failed to capture the pickup as early as weighted median or P* in low-inflation countries: e.g., the 10th percentile of 12-month weighted median inflation began increasing consistently from March 2021, whereas standard core did not until August 2021.
- Caveats and limitations
  - The percentile that defines P* is sample-dependent (varies by country, frequency, and sample period).
  - Finite numbers of inflation series per country prevent eliminating bias completely.
  - In some countries (Korea and Slovakia) P* may be more volatile than standard core.
  - P* usually does not improve one-year-ahead headline inflation prediction relative to the weighted median.

### V. Additional methodological and empirical notes
- Industry-level price data are not seasonally adjusted prior to X-13ARIMA-SEATS.
- Quarterly median inflation differs from aggregating monthly medians because distributions of industry price changes can differ across frequencies.
- Heteroskedasticity and autocorrelation consistent standard errors computed with R function vcovHAC using Andrews (1991) default weights; see Zeileis (2004, 2006) for implementation notes.
- Frequencies analyzed: monthly, quarterly, and 12-month inflation metrics.
- Comparative statistics used: volatility, volatility of the difference in inflation, mean, bias to headline, RMSE versus a random walk, goodness of fit (R2) for Phillips curve specifications, and significance of output gap coefficients.
- Models tested include forward-looking Phillips curve (main body) and Hybrid Phillips curve (appendix) with theoretical justification from Galí and Gertler (1999).
- Robustness: results hold across monthly and quarterly metrics, with some metrics (bias) more acute at monthly frequency.

### VI. Quantitative summaries and selected R2 entries (baseline Phillips curve)
- Selected exact R2 entries from Table 2 (Headline, Standard Core, Weighted Median, P*):
  - AUT 0.04 0.05 0.10 0.07
  - BGR 0.18 0.37 0.41 0.45
  - BRA 0.05 0.10 0.13 0.12
  - CAN 0.00 0.01 0.00 0.00
  - CHE 0.01 0.11 0.15 0.14
  - ESP 0.00 0.03 0.06 0.05
  - EST 0.09 0.24 0.31 0.28
  - GBR 0.02 0.01 0.03 0.02
  - JPN 0.08 0.07 0.03 0.02
  - LTU 0.16 0.38 0.39 0.39
  - LVA 0.32 0.49 0.48 0.52
  - MEX 0.01 0.01 0.00 0.00
  - POL 0.15 0.23 0.26 0.24
  - SWE 0.01 0.01 0.08 0.11
  - USA 0.00 0.06 0.33 0.15

### VII. Key implications and recommendations
- P* minimizes average bias relative to headline inflation while retaining low volatility and good Phillips curve fit in most countries.
- The weighted median generally has superior properties across several metrics:
  - Lower volatility than standard core in most countries.
  - Higher R^2 in forward-looking Phillips curves than standard core for the majority of countries.
  - Better prediction of one-year-ahead headline inflation than standard core in many cases.
- Practical guidance:
  - Policymakers and practitioners should pay more attention to outlier-exclusion core measures such as weighted median inflation and P* when assessing short-run policy trade-offs implied by the Phillips curve and monetary policy reaction.
  - Researchers should continue seeking improvements in the measurement of core inflation and consider theoretical underpinnings of these empirical measures in future work.

*Source: IMF Working Paper — Section III.D, “P*, An Unbiased Core Measure” (wpiea2023044-print-pdf).*

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

### wpiea2023044-print-pdf - References .............................................................................................................

### I. Introduction — purpose and context
- Objective: construct weighted-median core inflation series for 38 countries and compare properties to headline inflation and standard core (XFE) measures.
- Motivation:
  - XFE (inflation excluding food and energy) commonly used but can miss large sectoral shocks outside food and energy.
  - Outlier-exclusion measures (weighted median, trimmed means) remove unusually large price changes across any industry; weighted median is the 50th-percentile industry inflation rate.
- Literature and prior evidence:
  - U.S. evidence: weighted median less volatile than XFE (Ball et al. (2021) — standard deviations: median 1.05 and XFE 1.42 for monthly CPI over 1985-2019).
  - Pandemic example: January 2020-November 2021 standard deviation — median 1.51 and XFE 3.86.
  - Phillips-curve fit example: Ball and Mazumder (2020) find adjusted R2 = 0.48 (weighted median) vs 0.22 (XFE) for 1985-2017 U.S. quarterly data.
- Contribution: compute weighted-median core inflation for 38 countries (advanced and emerging markets), analyze volatility, relationship to economic slack (Phillips curve), and forecast performance for 12-month ahead headline inflation; monthly results in Appendix A.

### II. Data and methodology — scope and construction
- Sample:
  - 38 countries across advanced economies and emerging markets.
  - Inflation data spans from 1990 to 2021 (unbalanced availability by country).
  - Industry-level CPI at the fourth (“subclass”) level with about 150 to 250 industries per country.
- Definitions:
  - Headline inflation: all-items CPI as reported by national agencies.
  - Standard core (XFE): headline excluding food and energy.
  - Weighted median: industry inflation rate at the 50th percentile of the weighted distribution.
- Frequencies and annualization:
  - Monthly median: month t vs t-1 industry inflation; annualized by multiplying monthly median by 12.
  - Quarterly median: construct quarterly industry price by averaging three months; quarter t vs t-1 inflation; annualized by multiplying quarterly median by 4.
  - 12-month median: industry inflation from t-12 to t; weighted median of those annual rates.
- Seasonal adjustment: U.S. Census Bureau’s X-11 version of X-13ARIMA-SEATS applied to unadjusted monthly and quarterly inflation rates (also applied to headline and standard core).
- Phillips curve baseline specification (quarterly, country-by-country OLS with HAC standard errors):
  - π_t = α + π_t^e + β y_t + ε_t
  - π_t^e is five-year ahead inflation expectations (Consensus Forecasts; robustness uses IMF WEO five-year expectations).
  - y_t is output gap: seasonally adjusted real GDP detrended with Hodrick-Prescott filter (λ=1600).
  - Estimation: OLS with heteroskedasticity and autocorrelation consistent standard errors (Newey & West 1987, Andrews 1991); robustness includes hybrid (lagged inflation) and panel with country fixed effects.

### III. Key results and empirical findings
A. Volatility
- Main finding: weighted median inflation is generally less volatile than both headline and standard core for most countries at quarterly frequency, and much less volatile at the monthly frequency.
- Sample nuance:
  - Result especially strong for advanced economies.
  - More mixed results for emerging markets; for Brazil, Costa Rica, Mexico, and Peru weighted median volatility is close to or higher than standard core.
- Robustness: similar conclusions when volatility measured by variance of changes rather than levels.

B. Average levels and bias
- Finding: both standard core (XFE) and weighted median typically have lower mean levels than headline inflation — a “negative bias” relative to headline.
- Bias patterns:
  - The negative bias is sometimes larger for standard core, sometimes for weighted median.
  - A group of eight countries (mostly emerging economies) show a considerably larger negative bias in the weighted median.
  - Monthly frequency: bias of median inflation is much larger, related in part to frequency of zeros in monthly industry inflation rates.
  - Exception noted: United States shows a positive rather than negative bias of weighted median relative to headline.

C. Adjusted-percentile measure (P*)
- Shortcoming identified: weighted median tends to be biased downward on average (especially monthly) because many sectoral inflation rates are zero.
- Constructed adjustment: compute the inflation at a sectoral percentile different from 50% (P*) such that the average of that percentile over the sample equals the sample mean of headline inflation.
- Properties of P*:
  - For quarterly data, P* typically ranges from 50% to 60% for most countries.
  - P* eliminates the average-level bias by construction while retaining the low volatility and comovement with economic slack.
  - Caveat: P* is sample-dependent because it matches the sample mean of headline inflation.

D. Relationship with economic slack (Phillips curve)
- Finding: the relationship between core inflation and economic slack (fit of simple Phillips curve) is generally clearer when core is measured by the weighted median rather than by XFE for most countries at quarterly frequency.
- Prior benchmark: Ball and Mazumder (2020) example—adjusted R2 = 0.48 (weighted median) vs 0.22 (XFE) for U.S. quarterly estimates.

E. Forecasting one-year-ahead headline inflation
- Approach: compare current headline, standard core, and median inflation as predictors of future headline inflation measured over twelve months from t to t+12.
- Metric: root mean squared error (RMSE) for monthly, quarterly, and 12-month measures.
- Main statement: current level of median inflation is a better predictor of headline inflation over the next twelve months than standard core, with the strongest results using current monthly levels of the core measures.

### IV. Additional methodological and empirical notes
- Industry-level price data are not seasonally adjusted prior to X-13ARIMA-SEATS.
- Quarterly median inflation differs from aggregating monthly medians because distributions of industry price changes can differ across frequencies (example illustrated in text).
- Estimation details: heteroskedasticity and autocorrelation consistent standard errors computed with R function vcovHAC using Andrews (1991) default weights; see Zeileis (2004, 2006) for implementation notes.

*International Monetary Fund — IMF WORKING PAPERS: Weighted Median Inflation Around the World: A Measure of Core Inflation*

### Section III .D  below, we address this problem by introducing a new measure of core inflation, called P*, which

### D. P*, An Unbiased Core Measure

### Definition and construction
- P* is defined for each country as the inflation rate at the percentile of the sectoral inflation distribution that minimizes the average difference between its level and that of headline (h) inflation over the sample T.
- For a given country j, P* is the inflation rate π of the industry (i) associated with the percentile p_j of the sectoral inflation distribution that minimizes the average inflation bias versus headline inflation:
  - min_{p_j} 1/T ∑ (π_{jt}^h − 1/T ∑ π_{jt}^{p*})
  - where π_{jt}^{p*} = π_{i,j,t} for industry i such that ℛ_{i,j,t} = p_j
  - the ranking operator ℛ_{i,j,t} is defined by ordering sectoral inflation rates as in the paper.
- The industry whose inflation rate is at the percentile associated with P* changes every month, analogous to the industry at the median of the distribution.

### Empirical characterization of the percentile p_j
- For most countries the percentile p_j that minimizes bias is above the median (50th percentile), and usually between the 50th and the 60th percentile for inflation at the quarterly frequency.
- The US is an exception: the percentile that minimizes bias for the US is below the 50th percentile, slightly below the 40th percentile.
- The percentile that defines P* differs across countries, across data frequencies (monthly, quarterly, and year-on-year), and depends on the sample period.

### Volatility and bias
- By construction P* minimizes the average bias relative to headline inflation; empirical results show the black dots for P* are close to the red dots indicating zero bias (finite series mean bias can remain).
- Volatility:
  - For most countries, the volatility of P* is close to the volatility of the weighted median and lower than the volatility of headline inflation and standard core.
  - Exceptions: Korea and Slovakia, where P* is more volatile than standard core (median is less volatile than standard core, but P* is not).
- Conclusion: moving from weighted median to P* fixes the median’s bias without sacrificing much stability of the core measure in most cases.

### Phillips curve performance
- The success of an inflation measure is judged by the R-squared (R^2) of a Phillips curve where inflation depends on expected inflation and the output gap (estimated at quarterly frequency).
- P* performs well in terms of Phillips curve fit:
  - The R^2 for Phillips curve regressions with P* (black dots) is generally close to the R^2 for median inflation (green dots) and is noticeably higher for several countries.
  - Like the weighted median, P* outperforms both headline and standard core inflation for about three quarters of the countries in the sample, both in country-specific and panel regressions.
  - A higher R^2 when inflation is measured by weighted median or P* implies more of inflation’s variance is explained by domestic economic slack.

### Predictive ability for one-year-ahead headline inflation
- Method: compute standard deviation of (one-year-ahead headline inflation − current core measure) and divide by standard deviation of (one-year-ahead headline inflation − current headline inflation). Ratio < 1 implies the core measure is a better predictor of one-year-ahead headline inflation than current headline inflation.
- Results:
  - All three core measures (standard core, weighted median, and P*) almost always yield ratios less than one, so each is generally a better predictor of one-year-ahead headline inflation than current headline inflation.
  - For most countries the ratio is lower for the weighted median than for standard core, indicating the weighted median is a good predictor.
  - P* usually performs worse than the weighted median in this predictive exercise; correcting the median’s bias appears counterproductive for predicting one-year-ahead headline inflation.

### Performance in the 2021–22 inflation event
- Time-series distributions (10th, 25th, 50th, 75th, 90th percentiles across countries) show weighted median and P* started capturing the increase in inflation almost immediately in 2021.
- Standard core failed to capture the pickup as early as weighted median or P* in low-inflation countries: for example, the 10th percentile of 12-month weighted median inflation began increasing consistently from March 2021, whereas standard core did not until August 2021.

### Caveats and limitations
- The percentile that defines P* is sample-dependent (varies by country, frequency, and sample period).
- Finite numbers of inflation series per country prevent eliminating bias completely.
- In some countries (Korea and Slovakia) P* may be more volatile than standard core.
- P* usually does not improve one-year-ahead headline inflation prediction relative to the weighted median.

### Key implications and recommendations
- P* has the desirable property of minimizing average bias relative to headline inflation while retaining low volatility and good Phillips curve fit in most countries.
- The weighted median generally has superior properties across several metrics (lower volatility than standard core, higher R^2 in forward-looking Phillips curves, and better prediction of one-year-ahead headline inflation).
- Policymakers and practitioners should pay more attention to outlier-exclusion core measures such as weighted median inflation and P* when assessing short-run policy trade-offs implied by the Phillips curve and monetary policy reaction.
- Researchers should continue seeking improvements in the measurement of core inflation and consider theoretical underpinnings of these empirical measures in future work.

*Source: IMF Working Paper — Section III.D, “P*, An Unbiased Core Measure”*

### References

### References and Appendixes (wpiea2023044-print-pdf)

### Key findings on inflation measures (monthly and quarterly)
- Volatility:
  - The volatility of the weighted median inflation (green) is below that of standard core (blue) and headline (red) for the majority of countries in the sample when using the monthly inflation metric.
  - Exceptions where standard core outperforms weighted median include some emerging market economies such as Mexico, Peru and Costa Rica.
  - For the P* measure (the inflation measure that minimizes the bias), volatility is typically between the volatility of the weighted median and standard core; P* shows robust volatility performance at monthly and quarterly frequencies.
- Mean and bias:
  - At the monthly frequency the mean of weighted median is below standard core and headline inflation for all but a handful of countries.
  - The weighted median shows a negative bias to headline inflation that is more severe at the monthly frequency than at the quarterly frequency. The USA is a noted exception showing a positive bias.
  - By construction the mean of P* is close to headline inflation; P* greatly reduces the bias between the weighted median and headline inflation.
- Forecasting one-year-ahead headline inflation:
  - P* performance in capturing one-year-ahead headline inflation (measured by RMSE of regression versus RMSE of a random walk) is generally between weighted median and standard core.
  - At quarterly and monthly frequencies P* generally performs comparably to the weighted median.
  - Using the 12-month inflation metric yields mixed results: weighted median and P* do not outperform standard core for a majority of countries in the one-year-ahead RMSE comparison.
- Phillips curve evidence:
  - A Hybrid Phillips curve specification (unconstrained backward and forward-looking inflation components plus output gap) shows higher goodness of fit (R2) for weighted median than standard core for the majority of countries.
  - Both Phillips curves that include standard core or weighted median typically have a larger R2 than when headline inflation is included.
  - Results are robust when restricting to hybrid Phillips curves with a positive and significant coefficient on the output gap; weighted median generally has higher R2 and more frequently yields a positive and significant output gap coefficient than standard core.
  - P* also performs well in the hybrid Phillips curve, with R2 close to, and sometimes exceeding, the R2 when using weighted median.
- Share of zeros and bias:
  - The monthly frequency exhibits a higher share of zeros in sectoral price changes compared to the quarterly frequency.
  - A higher share of zeros at monthly frequency is associated with a larger bias of weighted median relative to headline inflation.

### Quantitative summaries and examples (R2 in baseline Phillips curve; Table 2 highlights)
- The appendix provides country-level R2 values from the baseline Phillips curve for four inflation measures: Headline, Standard Core, Weighted Median, and P*. Selected exact entries from Table 2:
  - AUT 0.04 0.05 0.10 0.07
  - BGR 0.18 0.37 0.41 0.45
  - BRA 0.05 0.10 0.13 0.12
  - CAN 0.00 0.01 0.00 0.00
  - CHE 0.01 0.11 0.15 0.14
  - ESP 0.00 0.03 0.06 0.05
  - EST 0.09 0.24 0.31 0.28
  - GBR 0.02 0.01 0.03 0.02
  - JPN 0.08 0.07 0.03 0.02
  - LTU 0.16 0.38 0.39 0.39
  - LVA 0.32 0.49 0.48 0.52
  - MEX 0.01 0.01 0.00 0.00
  - POL 0.15 0.23 0.26 0.24
  - SWE 0.01 0.01 0.08 0.11
  - USA 0.00 0.06 0.33 0.15
- Figures and tables referenced:
  - Figure 14–19: Monthly volatility, volatility of differences, mean, bias to headline, P* volatility and bias (authors’ calculations).
  - Figure 20–27: Monthly RMSE, quarterly volatility of difference, quarterly bias vs mean, Phillips curve R2 (hybrid), P* quarterly volatility of difference, 12-month RMSE comparisons (authors’ calculations).
  - Figure 29–30: Difference in R2 between measures and relationship between R2 and output gap coefficient for baseline Phillips curve (authors’ calculations).

### Methodological notes and robustness checks (as described)
- Frequencies analyzed: monthly, quarterly, and 12-month inflation metrics.
- Comparative statistics used include volatility, volatility of the difference in inflation, mean, bias to headline, RMSE versus a random walk, goodness of fit (R2) for Phillips curve specifications, and the significance of output gap coefficients.
- Models tested include a forward-looking Phillips curve (main body) and a Hybrid Phillips curve (appendix) with theoretical justification from Galí and Gertler (1999).
- Robustness:
  - Results hold across monthly and quarterly metrics, with some metrics (bias) more acute at monthly frequency.
  - Restricting hybrid Phillips curves to specifications with positive and significant output gap coefficients preserves the superior performance of weighted median versus standard core for most countries.

*Content compiled from the References and Appendixes of the source PDF (Authors’ calculations).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2023/english/wpiea2023044-print-pdf.pdf_