#  Extending the Range of Robust PCE Inflation Measures ††thanks: We
appreciate feedback from Christian Bustamante, Stephen Cechetti, Gustavo
Joaquim, Rory McGee, Robert Rich, Randal Verbrugge, Steve Williamson, and
Saeed Zaman. A previous version of this paper circulated under the title “How
Robust Are Robust Measures of PCE Inflation?”

Sergio Ocampo
Western University  Email<EMAIL_ADDRESS>website:
https://sites.google.com/site/sergiocampod/. Raphael Schoenle
Brandeis University  Email<EMAIL_ADDRESS>website:
https://people.brandeis.edu/s̃choenle/.  Dominic A. Smith
Bureau of Labor Statistics  Email<EMAIL_ADDRESS>website:
https://www.bls.gov/pir/authors/smith.htm.

(This draft: June 2023
First draft: May 2022)

###### Abstract

We evaluate the forecasting performance of a wide set of robust inflation
measures between 1960 and 2022, including official median and trimmed-mean
personal-consumption-expenditure inflation. When trimming out different
expenditure categories with the highest and lowest inflation rates, we find
that the optimal trim points vary widely across time and also depend on the
choice of target; optimal trims are higher when targeting future trend
inflation or for a 1970s–1980s subsample. Surprisingly, there are no grounds
to select a single series on the basis of forecasting performance. A wide
range of trims—including those of the official robust measures—have an average
prediction error that makes them statistically indistinguishable from the
best-performing trim. Despite indistinguishable average errors, these trims
imply different predictions for trend inflation in any given month, within a
range of 0.5 to 1 percentage points, suggesting the use of a set of near-
optimal trims.

_JEL Codes:_ E3, E5, E6

_Keywords:_ inflation, robust measures

##  1 Introduction

The return of high inflation after the pandemic of 2020 has renewed attention
to robust measures of inflation such as trimmed-mean measures. These measures
trim out the expenditure categories with the highest and lowest inflation
rates before computing mean inflation. Economists use these series to describe
the behavior of underlying trend inflation and to differentiate it from
transitory movements in inflation in a way that is easy to communicate to the
public.111 Communication about inflation and inflation targets affects
inflation expectations and monetary policy. See, for example, Coibion,
Gorodnichenko, and Weber ((2022), Handlan ((2022), and Gorodnichenko, Pham,
and Talavera ((2023) for recent applications and Blinder, Ehrmann, Fratzscher,
De Haan, and Jansen ((2008) for a review of previous results.  Two series used
for this purpose are the official trimmed-mean inflation measure constructed
by the Federal Reserve Bank of Dallas and the median inflation measure
constructed by the Federal Reserve Bank of Cleveland. These series have
received much attention during the recent high-inflation episode.222 For
instance, the president of the Federal Reserve Bank of Richmond has stated
that “I want to see inflation, and median and trimmed mean, compellingly
headed back to our target” ((Torres, 2023a, b).  In this paper, we evaluate
the forecasting performance of a wide set of robust measures of inflation,
including these two prominent measures used by central banks.

In doing so, our analysis makes two contributions. First, we address a key
data limitation: The behavior of robust measures of inflation in high-
inflation regimes is hard to gauge because official releases of the series
start in 1977, covering only one episode of high inflation before 2020. We
therefore extend the official trimmed-mean and median personal-consumption-
expenditure (PCE) inflation series back to 1960.333 By contrast, extending
back in time the median and trimmed-mean Consumer Price Index series published
by the Federal Reserve Bank of Cleveland is of limited use because the
headline Consumer Price Index data are not revised when the methodology used
to calculate inflation for individual components is changed. However, changes
in methodology can imply significant changes in the volatility of inflation as
shown by Hazell, Herreño, Nakamura, and Steinsson ((2022) and Bolhuis, Cramer,
and Summers ((2022).  Doing so extends the sample size by over 40 percent,
providing us with two additional episodes of rising inflation (1968 and 1973)
and covering a total of 129 months of high inflation (inflation above 5
percent). To construct the extended series, we use the underlying PCE data
release from the Bureau of Economic Analysis (BEA) and replicate the
methodologies of the Federal Reserve Banks of Dallas and Cleveland (see Bryan
and Pike, 1991,Bryan and Cecchetti, 1994, Carroll and Verbrugge, 2019, and
Dolmas, 2005). We make them publicly available for analysis at
https://ocamp020.github.io/Robust_Inflation_Series.xlsx.

Second, based on these data, we evaluate the predictive performance of these
two extended robust inflation series and a wide set of robust measures of
inflation against ex-post measures of current and future trend inflation. Our
analysis follows the methodology used to select the trimming cutoffs of the
official trimmed-mean inflation series, described by Dolmas ((2005).
Specifically, we construct a measure of current trend inflation as a 36-month
centered moving average of headline inflation centered on the current period.
For future trend inflation, we construct a 12-month forward moving average of
headline inflation that starts one year ahead of the current period (using
data between 13 and 24 months ahead).444 We consider an alternative measure of
current trend inflation constructed using Christiano and Fitzgerald’s (2003)
band-pass filter and an alternative measure of future trend inflation
constructed as a 24-month forward moving average. Results for these measures
can be found in the appendix.  We then contrast a wide set of robust inflation
measures with trend inflation over our extended sample, a restricted sample
that captures times of high and volatile inflation (1970–89), and a recent
sample with overall low and stable inflation (2000–2022).

Two main findings emerge from our extended data. First, the official measures
have a similar root-mean-square error (RMSE) against both measures of trend
inflation, with a lower RMSE against current trend inflation than against
future trend inflation. The performance of both series is stable over time,
with the Dallas Fed’s trimmed-mean measure slightly outperforming the
Cleveland Fed’s median inflation measure except for the period of highest and
most volatile inflation: between 1970 and 1989. Moreover, the official
measures perform nearly as well as the best of the alternative robust measures
we construct. In fact, the difference between the RMSEs of the different
series is often not statistically significant.

Second, we find that the similarity in the predictive performance of the
robust inflation measures—according to their RMSE—obscures economically
significant differences in the behavior of the series in any given period and
across time. The average range between the _levels_ of the official robust
inflation series is wide, with an average of 0.70 percentage points before
2020, growing to 1.20 percentage points after 2020. This finding implies that
these measures often provide different signals about current and future
inflation in any given month, while tracking the trend equally as well on
_average_.

This pattern also holds when we broaden the set of robust inflation measures
to include alternative trimmed-mean measures chosen based on their average
predictive performance. Even among the measures whose RMSEs against trend
inflation are statistically equivalent, the range remains wide. The range
between these measures is between 0.59 and 1.16 percentage points on average
depending on the trend inflation series being targeted.

Notably, the set of such near-optimal trimmed-mean measures is more
informative about the state of current trend inflation than about future trend
inflation. The RMSE of the best trimmed mean measure is at least twice as
large when predicting future trend inflation, relative to current trend
inflation. Moreover, during high inflation periods at the beginning and end of
our extended sample, the set of best trimmed mean measures of inflation are
slow to react and lag behind in their prediction of future trend inflation.
This insight from our extended sample aligns with the finding that trimmed
mean measures appear better at nowcasting than forecasting without further
adjustments ((Verbrugge and Zaman, 2023).

Overall, while these results may appear negative, in that _no single_ best
measure emerges, a positive and practically important message also arises:
Policymakers trying to communicate inflation may prefer to rely on a set of
trimmed-mean measures that provide a range of inflation, as there are no
grounds to select a single series on the basis of predictive performance. In
doing so, the range implied by the set of trimmed-mean measures with the
lowest prediction errors provides a naturally interpretable picture of the
implied behavior of trend inflation that accounts for the uncertainty in
selecting predictors.

#### Relationship to the literature.

Our analysis contributes to a large literature that focuses on which measures
of inflation provide the best signal about the underlying inflation trend. We
add to the literature in a simple but important way by extending robust PCE
inflation measures to periods of high inflation and by focusing on the
performance of a wide set of robust measures of inflation in this extended
sample. Related work has focused on the forecasting properties of different
series, but has used samples in which inflation has been consistently low and
stable ((see, for example, Dolmas, 2005; Rich and Steindel, 2007; Crane,
Khettry, Mester, and Novak, 2013).

Other robust measures of inflation have been proposed, such as median
inflation excluding owners’ equivalent rent ((Carroll and Verbrugge, 2019) and
a version of the Consumer Price Index (CPI) that excludes the eight most
volatile components ((Clark, 2001). Much research has focused on core
inflation, as summarized in a report by the Bank for International Settlements
((1999) and recently evaluated by Verbrugge ((2022). Prominently, the Federal
Reserve Bank of New York’s measure of underlying inflation presents an
alternative measure of trend inflation that takes into account the time-series
dimension in the distribution of inflation rates based on a large dynamic
factor model as in Forni, Hallin, Lippi, and Reichlin ((2000). While such
statistical measures of underlying inflation are not the focus of our
analysis, Amstad, Potter, and Rich ((2017) discuss them and their advantages
relative to trimmed-mean measures of inflation, which we analyze based on a
new, extended data set. In line with a stable time-series factor loading, we
find that many identical components are consistently included across time in
our robust measures of inflation.

##  2 Long Series for Robust Inflation Measures

This section details our computation of headline, core, median, and trimmed-
mean PCE inflation series beginning in 1959 and ending in October 2022,
extending the official median and trimmed-mean inflation series to cover
1960–77. This extension increases the number of months in the sample by almost
40 percent relative to the conventionally available official series published
by the Federal Reserve Banks of Cleveland and Dallas.

Extending the official robust inflation measures to the 1960–77 period is of
interest for two reasons: First, it is of independent interest because it
provides a more consistent view of the patterns for these robust inflation
measures in periods of rising and high inflation. The 1960–77 period provides
us with two additional episodes of rising inflation (1968 and 1973), adding to
the four episodes of rising inflation in the post-1977 period. There are only
three episodes between 1960 and 2022 for which headline PCE inflation is above
5 percent, covering a total of 129 months, 44 of which are in the 1960–77
period. Second, we can use the series as reference points when we evaluate the
predictive performance of a wide set of inflation series in the next section.

We proceed by describing the data we use, how the series are computed, and
their properties.

###  2.1 PCE Inflation Data

We use the underlying data supplements of the National Income and Product
Accounts PCE data release ((Bureau of Economic Analysis, 2022) between January
1959 and October 2022 to construct series for robust inflation measures.555
The main input for BEA price indexes is the CPI, but unlike the CPI, the PCE
is revised when methods change.  The PCE data provide disaggregated price
indexes and expenditure weights that cover US consumer spending.

Two main issues accompany the use of these data and the replication of
official robust inflation measures. First, new expenditure categories are
introduced over time, as is the case with expenditure on personal computers, a
category introduced in 1977. The changes in the set of expenditure categories
are reflected in differences between the set of categories used in the
construction of official trimmed-mean inflation and that used for median
inflation. We harmonize the set of expenditure categories by establishing a
set of 183 series that are available either for the entire period or are
available as soon as a new good is introduced.666 When new expenditure
categories are introduced, they often have almost-zero spending, as they
represent new goods. In those cases, we assume that the goods they represent
were not available before their introduction in the PCE, and we assign to them
a retroactive weight of zero.  We compare the behavior of trimmed-mean and
median inflation using this consistent set of categories.777 The set of
categories we use differs from the 177 categories used by the Dallas Fed and
200 categories used by the Cleveland Fed. The complete list of series included
in each inflation measure can be found in http://dominic-
smith.com/data/category_definitions.xlsx. In a previous version of this paper,
we established that these differences do not impact the results ((Ocampo,
Schoenle, and Smith, 2022).  Second, the price series for multiple expenditure
categories were not updated on a monthly basis before 1970, including owner’s
equivalent rent, the category with the highest weight. Thus, measures of
monthly inflation rates before 1970 contain multiple series with zero
inflation. This issue is mostly reflected in the median inflation series as we
show in Figure 1. We therefore focus most of our analysis on the 1970–2022
period.

###  2.2 Headline and Core PCE Inflation

Headline inflation is calculated as a Fisher index of the underlying inflation
components at the lowest level of aggregation. A Fisher index is the geometric
mean of a Laspeyres and a Paasche index, which are calculated respectively as

$\displaystyle\pi_{t}^{L}=\frac{\sum_{i}q_{t-1}^{i}p_{t}^{i}}{\sum_{i}q_{t-1}^{i}p_{t-1}^{i}};\quad\pi_{t}^{P}=\frac{\sum_{i}q_{t}^{i}p_{t}^{i}}{\sum_{i}q_{t}^{i}p_{t-1}^{i}},$
(1)

where $p_{t}^{i}$ and $q_{t}^{i}$ are, respectively, the price level and
quantity of expenditure category $i$ at time $t$. Core inflation is computed
in the same way but excludes all series under food and beverages purchased for
off-premises consumption, gasoline and other energy goods, and electricity and
gas. We take headline and core PCE inflation directly from the tables
published by the BEA.888 We use series DPCERG for headline inflation and
series DPCCRG for core inflation ((Bureau of Economic Analysis, 2022).

###  2.3 Trimmed-Mean PCE Inflation Measures

Trimmed-mean PCE inflation measures select a sample of expenditure categories
in each month by removing the categories with the lowest inflation rates
accounting for $\alpha$ percent of expenditure and the categories with the
highest inflation rates accounting for $\beta$ percent of expenditure. The
remaining categories are assigned weights using an average of the expenditure
on each category at current-period quantities and previous-period quantities,
which approximates the weights used in the PCE index formula,

$\displaystyle\omega_{t}^{i}=\frac{1}{2}\frac{p_{t-1}^{i}q_{t-1}^{i}}{\sum_{i}p_{t-1}^{i}q_{t-1}^{i}}+\frac{1}{2}\frac{p_{t-1}^{i}q_{t}^{i}}{\sum_{i}p_{t}^{i}q_{t}^{i}}.$
(2)

The trimmed-mean PCE inflation series is the expenditure-weighted mean across
the selected categories, where the weights are adjusted to reflect the fact
that $\alpha+\beta$ percent of expenditure has been trimmed out. Once the
monthly rates, $\pi_{t}^{tm,mo}$, are constructed, they are chained to form a
yearly inflation index,

$\displaystyle\pi_{t}^{tm}=\prod_{s=0}^{11}\pi_{t-s}^{tm,mo},\quad\pi_{t}^{tm,mo}=\sum_{i}\omega_{t}^{i}\frac{p_{t}^{i}}{p_{t-1}^{i}}.$
(3)

The official trimmed-mean inflation measure sets $\alpha=24$ and $\beta=31$,
it is published monthly by the Federal Reserve Bank of Dallas ((Dolmas, 2005),
and it is available from 1977 onward ((Federal Reserve Bank of Dallas, 2021).

The median inflation series is calculated by trimming out all categories
except the one with the median inflation rate (that is, $\alpha=\beta=50$).
The chained index is then

$\displaystyle\pi_{t}^{m}=\prod_{s=0}^{11}\frac{p_{t-s}^{i(m,t-s)}}{p_{t-s-1}^{i(m,t-s)}},$
(4)

where $i(m,t-s)$ is the index of the series with the median inflation at time
$t-s$. The official median PCE inflation is published monthly by the Federal
Reserve Bank of Cleveland ((Carroll and Verbrugge, 2019; Bryan and Cecchetti,
1994; Bryan and Pike, 1991) and is also available from 1977 onwards ((Federal
Reserve Bank of Cleveland, 2021).999 The official series for median PCE and
trimmed-mean PCE use different weights and categories. A previous version of
this paper found those differences have small effects ((Ocampo, Schoenle, and
Smith, 2022).

###  2.4 Inflation Series

We replicate the official trimmed-mean and median inflation series after 1977
and extend them back to 1960.101010 See Figures A.1 and A.2 in Appendix A.
Figure 1 plots the series together with headline and core PCE inflation. The
three robust inflation measures track the behavior of headline inflation
despite significant disagreement between them, which translates into similar
predictive performance as we show in the next section. The average range
between the levels of the robust inflation series is 0.77 percentage points.
These differences between the series increase at the end of the sample, with
the average range growing to 1.20 percentage points between 2021 and 2022 (see
Figure B.1 in Appendix B).111111 There are other disagreements. For instance,
the sign of the change (increasing or decreasing) of core PCE inflation
matches that of headline PCE in 74 percent of the months in our sample; the
values for median and trimmed-mean PCE inflation are 67 and 57 percent,
respectively.  As the next section makes clear this pattern of a similar
predictive performance (over time) paired with economically significant
differences in month-to-month levels is common across a wide range of trimmed-
mean measures.

Figure 1: Robust Measures of Inflation, 1960–2022 Notes: The figure shows
trimmed-mean personal consumption expenditure (PCE) and median PCE as
calculated by the authors using the methodologies of Federal Reserve Bank of
Cleveland ((2021) and Federal Reserve Bank of Dallas ((2021). Appendix A shows
that these measures match those produced by the relevant Federal Reserve Banks
after 1977. The vertical line in January 1977 indicates that the official
trimmed-mean and median measures are available starting in 1977. Headline and
core inflation are taken directly from the PCE data published by the Bureau of
Economic Analysis.

The differences between the inflation measures come from the underlying set of
expenditure categories used to compute them. Trimming out categories narrows
the range of inflation rates used to construct the robust inflation measures.
The distribution of inflation rates across categories is remarkably wide, with
a range between the 10th and 90th percentiles of 0.46 percentage points on
average, which makes trimming potentially very consequential. Despite this,
the distribution of month-to-month inflation rates is concentrated across the
median, as reflected by the interquartile range (the range between the 25th
and 75th percentiles of the distribution), which is 1.33 percentage point on
average. That most categories exhibit relatively small price changes and some
categories exhibit large changes is reflected in the high kurtosis of price
changes.121212 Figure B.3 in Appendix B illustrates this by plotting the 10th,
24th, 50th, 69th, and 90th percentiles of month-to-month inflation across
expenditure categories. The 24th and 69th percentiles correspond to the trims
of the official trimmed-mean inflation series, and the 50th percentile to the
trim of the official mean inflation series.

The set of categories varies over time, as it depends on the contemporaneous
distribution of inflation across expenditure categories. However, there are
certain regularities. The categories most commonly excluded from the
construction of robust inflation measures are food and fuel categories (for
example, eggs, vegetables, gasoline), which are the same categories excluded
from core PCE. This set is much larger for median inflation. In fact, 71 of
the 183 categories we consider never coincide with the median inflation
category, while every category has been included in the trimmed mean at some
point. Table B.2 lists the categories most commonly excluded and included in
the construction of median and trimmed-mean inflation.

##  3 Evaluating Official Robust Inflation Measures

How well do the official trimmed-mean and median inflation measures perform in
matching current and future inflation trends across multiple samples? Our
analysis shows that their performance is indeed robust across the samples we
consider, with the official trimmed-mean series slightly outperforming median
inflation in terms of capturing both current and future trend inflation.
However, the similar performance of the series obscures significant
differences in their monthly levels.

###  3.1 Evaluation Criteria

To evaluate the performance of inflation series, we construct ex-post measures
of current and future trend inflation, which aim to smooth out the transitory
components of inflation. Then, we compare each inflation measure to these
target trend measures and calculate the root mean squared error (RMSE) over
three different samples, following the criteria used by Dolmas ((2005) to
select the trimming cutoffs of the official trimmed mean inflation
series.131313 We do not include core inflation in our exercise because it is
outperformed by trimmed-mean and median inflation (see Verbrugge 2022).
Additionally, focusing on the official trimmed-mean and median inflation
series allows us to extend our analysis to more general trimmed-mean measures
that include the official series as special cases (see Section 4).  We
consider a long sample (1970–2022), a restricted sample (1977–2000), and a
recent sample (starting in 2000).141414 We conduct the exercise starting in
1970 to avoid the period in which the prices of several PCE categories were
not regularly updated. See Figure A.4 in Appendix A.

Our main measure of current trend inflation is constructed a 36-month centered
moving average of headline inflation centered on the current period. Thus, it
includes data from 18 months before to 18 months after the current period.
This measure was proposed by Bryan, Cecchetti, and Wiggins ((1997) and has
been used as a benchmark since. It provides an ex-post proxy for what
underlying inflation was at a point in time.

Our main measure of future trend inflation is constructed as a 12-month
forward moving average of headline inflation that starts one year ahead of the
current period. Thus, it includes data between 12 and 24 months ahead. Future
trend inflation is particularly useful as decision makers, like central
bankers, are forward-looking and make decisions based on the future behavior
of inflation.

We also construct an alternative measure of current trend inflation using the
trend component of Christiano and Fitzgerald’s (2003) band-pass filter
(removing frequencies below 39 months) and an alternative measure of future
trend inflation (constructed as the 24-month forward-looking average rate of
inflation as in Dolmas ((2005)).151515 The trend component of the band pass
filter removes high frequency movement in inflation. Dolmas ((2005) found that
removing frequencies below 29 months maximizes the correlation between the
resulting trend series and the Federal Funds rate target set by the Federal
Open Market Committee.  We plot all these series in Figure B.4 of Appendix
B.2, in which we also provide the results of our evaluation of the official
series against the alternative trend inflation measures.

Given a target $\bar{\pi}$ (for current or future trend inflation), we
evaluate how well an inflation measure $\pi^{i}$, $i\in\\{\text{trimmed mean,
median}\\}$, tracks it over a given sample. We do so by calculating the RMSE
for each candidate robust measure $i$ as161616 We use the level of inflation
$\pi^{i}$ without transforming it. Transformations such as linear regressions
improve the fit of robust inflation measures such as median inflation by
removing a consistent bias in certain measures ((Rich, Verbrugge, and Zaman,
2022); however, these measures are most often communicated and used in their
direct levels.

$\displaystyle
rmse^{i}=\sqrt{\frac{1}{T}\sum_{t}(\pi^{i}_{t}-\bar{\pi}_{t})^{2}}.$ (5)

###  3.2 Performance of Official Measures

The official trimmed-mean inflation series slightly outperforms the official
median inflation series in capturing the behavior of current and future trend
inflation, except for the restricted sample (1970–89), for which the pattern
reverses (see Table 1).171717 The table also reports the performance of
headline PCE inflation as a benchmark for the exercise. Headline inflation
corresponds to the no-trimming limit. The RMSE that the official trimmed-mean
and median inflation series produce is between 50 and 75 percent of the RMSE
of headline inflation, verifying that trimming does improve the series’
predictive performance.  However, the RMSEs produced by the two official
measures are not meaningfully different in most cases. We test the difference
between the predictions of official series using Diebold and Mariano’s (1995)
test and report the p-value in the last column of Table 1 under the null
hypothesis of no difference in the prediction error between the official
trimmed-mean and median inflation series. The differences between the
performances of trimmed-mean and median inflation are higher when predicting
current trend inflation than when predicting the future trend, but the null
hypothesis is only strongly rejected for the recent sample (starting in 2000).
These same patterns hold when looking at alternative trend inflation measures
as we show in Table B.3 of Appendix B.2.

Despite the official trimmed-mean and median inflation series’ similar
prediction errors, their levels differ substantially in most months (as
explored above). This reflects the main pattern of the alternative robust
measures studied below; namely, relying on the average prediction performance
of different robust inflation series obscures underlying differences in their
predictions for any given month and provides little guidance in judging these
differences.

Table 1: Predicting Performance of Official Measures

Target | Sample | PCE Inflation Measure | DM Test
---|---|---|---
Headline | Trimmed Mean | Median | $\Pr\left(z>\left|\text{DM}\right|\right)$
| 1970-2022 | $2.20$ | $1.10$ | $1.16$ | $0.066$
Current Trend | 1970-1989 | $2.28$ | $1.62$ | $1.51$ | $0.047$
| 2000-2022 | $2.47$ | $0.75$ | $0.95$ | $0.000$
| 1970-2022 | $2.93$ | $2.12$ | $2.14$ | $0.476$
Future Trend | 1970-1989 | $3.48$ | $3.02$ | $3.00$ | $0.841$
| 2000-2022 | $2.93$ | $1.59$ | $1.61$ | $0.561$

  * •

Notes: The table presents the predictive performance of different personal-
consumption-expenditure (PCE) inflation measures with respect to different
trend inflation targets for different samples. The performance is measured
with the series’ root-mean-square error (RMSE) with respect to trend
inflation. The table reports the RMSE for headline, trimmed-mean, and median
inflation. The last column reports the p-value of the Diebold and Mariano
((1995) test for the difference between the RMSEs of the trimmed-mean and
median inflation series.

##  4 Optimal Trimmed-Mean Measures

We now systematically vary the trim cutoffs used to construct trimmed-mean
measures by considering all integer combinations of trims for
$\alpha,\beta\in[0,50]$. Therefore, the resulting set of measures includes the
official trimmed-mean, median, and headline inflation series as special cases.
As before, we evaluate their performance based on their RMSEs against current
and future trend inflation. Then, we contrast their predictive performances
with differences in the levels of trend inflation implied by the different
measures. We extend these results to alternative measures of trend inflation
in Appendix B.3.

We find large differences between the optimal trimming cutoffs and the trims
of the official measures ($\alpha=24$ and $\beta=36$, or $\alpha=\beta=50$).
Table 2 presents the results. The optimal cutoffs are asymmetric, usually with
more trimmed off the top than the bottom $(\beta>\alpha)$, and they vary
significantly across time and target. When we target current trend inflation,
the optimal trims have $\alpha$’s between 18 and 21 and $\beta$’s between 16
and 27, varying across the four sample periods we consider. When we target
future trend inflation, the trims are slightly higher, with $\alpha$’s between
15 and 28 and $\beta$’s between 17 and 33.

However, we also find that this large variation in the optimal trims does not
translate into a significantly lower RMSE such that the official measures
perform almost as well as those with the optimal trimming cutoffs. The RMSE of
the optimal trims is at most 4.6 percent lower, and the differences are almost
never statistically significant. We verify this result in the last column of
Table 2, in which we report the p-value of the Diebold and Mariano ((1995)
test when comparing the predictive performance of the best trim with the
better of the two official robust inflation measures (see Table 1).

Table 2: Best Trims for Trimmed-Mean Inflation

Target | Sample | Best Trims | Official Trims | DM Test
---|---|---|---|---
Lower | Upper | RMSE | min(RMSE) | $\Pr\left(z>\left|\text{DM}\right|\right)$
| 1970-2022 | $20$ | $22$ | $1.06$ | $1.10$ | $0.014$
Current Trend | 1970-1989 | $18$ | $16$ | $1.44$ | $1.51$ | $0.238$
| 2000-2022 | $21$ | $27$ | $0.74$ | $0.75$ | $0.474$
| 1970-2022 | $28$ | $33$ | $2.09$ | $2.12$ | $0.192$
Future Trend | 1970-1989 | $15$ | $17$ | $2.91$ | $3.00$ | $0.560$
| 2000-2022 | $28$ | $32$ | $1.55$ | $1.59$ | $0.302$

  * •

Notes: The table reports the best trim as determined by the predictive
performance across trims for different targets of trend inflation and
different samples. The table also reports the root-mean-square error (RMSE)
for the best trims and the lower RMSE of the official trimmed and median
inflation series. The last column reports the p-value of the Diebold and
Mariano ((1995) test for the difference between these RMSEs.

Figure 2: RMSE across Trims

(a) Current Trend: 1970-2022

(b) Future Trend: 1970-2022

(c) Current Trend: 1970-1989

(d) Future Trend: 1970-1989

(e) Current Trend: 2000-2022

(f) Future Trend: 2000-2022

Notes: The figures show heat maps of the root-mean-square error (RMSE) when
targeting current and future trend inflation with different trimmed-mean
inflation measures. To ensure comparability across plots, the RMSEs are
reported relative to the RMSE of the best trim reported in Table 2.

The similarity in the performance of the official robust inflation measures
and the optimal trimmed measures is part of a larger pattern: for every target
and sample, a wide range of trims perform similarly. These trims are slightly
asymmetric—once again trimming more of the high-inflation categories (that is,
$\beta>\alpha$)—and the range of trims with similar prediction errors is much
larger when targeting future trend inflation, reflecting in part the added
difficulty of this exercise. Figure 2 presents these results by plotting the
RMSEs of all trim combinations for the two target trends and the three
samples. In Figures B.6 and B.7 of Appendix B.3 we report the sets of trims
whose RMSE is not significantly different from the RMSE of the best trim
according to the Diebold and Mariano ((1995) test. The shapes of these sets
mirror the dark blue areas of Figure 2.181818 These results are consistent
with previous work on the difference between trimmed-mean inflation series
(see, for example, Meyer, Venkatu, and Zaman 2013).

This pattern is explained by effect of trimming on the underlying distribution
of monthly inflation rates. As trimming cutoffs increase, the range of
inflation rates taken into account is reduced, affecting the variability and
the mean prediction of the resulting trimmed mean series. Take, for instance,
an increase in the upper trim. This reduces the average and the standard
deviation of the prediction implied by the trimmed mean series. However, the
reduction in variability is empirically smaller than the reduction in the
average prediction, actually making the trimmed mean series _relatively_ more
variable as the upper trim increases. In fact, the coefficient of variation
(the ratio of the standard deviation to the mean) increases rapidly with the
trimming cutoff, moving from 0.6 to 0.8. As a result, the RMSE of the
resulting series does not change much, while, at the same time, similar RMSEs
can be obtained from different trim combinations that imply different average
predictions.191919 Figure B.8 makes the effect of trimming precise by plotting
the coefficient of variation and the RMSE for trimmed mean series that target
current trend inflation. Near the optimal trim, the RMSE does not change much,
but the slope of the coefficient of variation is positive.

#### Performance beyond prediction errors.

A naive reading of the results above might suggest that a wide range of trims
is functionally equivalent, with the only change being the adjustment of the
range of trims when targeting the behavior of current or future trend
inflation. However, the small differences in the RMSEs of different trims hide
significant variation in the prediction levels of the implied trimmed-mean
series. This makes it complicated to select a single optimal series, as there
are other, near-optimal, series with different implications for trend
inflation in any given month. Take for instance the predicted levels of trend
inflation from the best 50 trim combinations. These trims vary between 22 and
45 percent. In October 2022 the predicted inflation levels are between 7.3 and
7.9 percent, a range of 59 basis points (see Figure B.9).

Figure 3: Prediction Range across Best Trims across Time

(a) Centered Trend

(b) Future Trend

Notes: The figures plot the range of predictions for the current and future
trends between 1970 and 2022 of the set of trimmed-mean measures of inflation
that are statistically equivalent at the 5 percent level according to the
Diebold and Mariano ((1995) test of the difference of their root-mean-square
errors (RMSEs) with respect to the RMSE of the best trim for each target (see
Table 2). The start- and end-dates for each figure reflect the data
requirements for the ex-post measures of trend inflation.

This pattern of large discrepancies between predictions holds throughout the
sample when looking at the set of trimmed mean inflation measures with the
lowest RMSE, that is, those with an RMSE statistically equivalent to that of
the best trim, and is even larger when targeting future trend inflation.202020
We determine the set of statistically equivalent trims with the Diebold and
Mariano ((1995) test using a significance level of 5 percent. Figure B.10
plots the range between the lowest and highest predictions for the current and
future trend between 1970 and 2022 for the set of statistically equivalent
trims and the best 50 and 100 trims according to their RMSE. Figure B.11
complements this information by presenting the average range of inflation
implied by each trim combination.  Figure 3 plots the range of inflation
predictions implied by these sets for 1970–2022 sample for trims targeting
current and future trend inflation. The average range is 0.59 percentage
points when targeting current trend inflation and 1.16 percentage points when
targeting future trend inflation. Even when we restrict attention to the best
50 trims (those whose prediction error is closest to the best trim’s error),
the average prediction range is 42 basis points between 1970 and 2022 and as
high as 1 percentage point.

Figure 3 also makes it clear that measures of trimmed mean inflation have a
hard time tracking changes in future trend inflation, instead lagging its
movements. This is particularly evident in the two instances of high inflation
in the pre-1977 sample—that we added to the official series—and at the end of
the sample, when inflation rises again. In fact, the predicted range for
future trend inflation does not start increasing until inflation has already
peaked. By contrast, the range of trimmed mean inflation is more informative
about the behavior of current trend inflation and noticeably less variable
than the headline PCE inflation series. These results suggest that trimmed
mean measures are best used to understand the current inflation situation,
with other methods such as those in Verbrugge and Zaman ((2023) better suited
for predicting future inflation.

##  5 Conclusion

Policymakers often use robust inflation series to communicate with the public
and gauge the behavior of trend inflation. Our evaluation of the performance
of the official series for various targets and across various samples shows
them to be robust across time and comparable with the performance of the best
trimmed-mean inflation measure (the one that selects trimming cutoffs that
minimize prediction error). However, our analysis also finds, perhaps
surprisingly, that results based on the average predictive performance of the
series obscure an underlying pattern of trimmed-mean measures: While a wide
range of measures have the same predictive performance, they produce
substantially different predictions in any given month.

We conclude that following a set of trimmed-mean measures rather than a single
series may provide additional insight about the behavior of inflation. This
information can be effectively communicated by reporting the range of
predictions from the set of measures with the best predictive performance.
This range then informs us about the behavior of trend inflation while
indicating the uncertainty over the selection of predictors in a way that is
easy to communicate to the public.

While we also explored alternative trimmed-mean measures not reported here,
and found similar results,212121 In particular, we constructed trimmed-mean
measures excluding housing, which is the single largest expenditure category
and one of the series most commonly included in the official trimmed-mean and
median inflation measures (see Table B.2). We found that housing does not play
a large role in the behavior of trimmed-mean inflation measures. All of our
results are preserved when excluding housing from the set of expenditure
categories, as we show in Appendix B.4.  future work may consider a twist to
our analysis. Such work may consider evaluating the predictive power of robust
measures of inflation for particular _alternative targets_ —current or future
inflation rates relevant for different sub-groups of the US population. This
differential evaluation may provide valuable new insights into robustness in
the context of heterogeneous effects of monetary policy.

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##  Appendix A Replication of Trimmed-Mean and Median PCE Inflation Series

Figure A.1: Replication of Trimmed-Mean PCE Inflation, 1960–2022 Notes: The
figure shows the authors’ calculation of trimmed-mean PCE inflation along with
the official trimmed-mean PCE series. Figure A.2: Replication of Median PCE
Inflation, 1960–2022 Notes: The figure shows the authors’ calculation of
median PCE inflation along with the official median PCE series.  Figure A.3:
Number of Series with No Monthly Price Changes

(a) Trimmed Mean

(b) Median

(c) Time Consistent

Notes: The lines plot the number of series and fraction of total expenditure
(in percent) with no monthly price change for each of the three sets of series
used in the paper. Figure A.4: Number of Series with Positive Expenditure

(a) Trimmed Mean

(b) Median

(c) Time Consistent

Notes: The lines plot the number of series with positive expenditure in the
PCE series over time from the set of series considered by the trimmed-mean
inflation, median inflation, and the time consistent set constructed in the
paper.

##  Appendix B Additional Results

###  B.1 Time Series Properties

The level of agreement between the series is also captured by the range of
values they cover, shown in Figure B.1 along with the level of headline PCE
inflation. The range is 0.77 percentage points on average over the whole
sample, 0.80 percentage points when inflation is less than 2.5 percent, and
0.91 percentage points when inflation is above 5 percent. Thus, the range
values covered by robust inflation measures is disproportionately wider when
inflation is low than when inflation is high. This again shows that there is
substantially more agreement between the signals provided by the different
inflation measures when inflation is high.

Figure B.1: Range of Robust Inflation Measures, 1960–2022 Notes: The figure
shows the authors’ calculations of the range of robust inflation measures
(core inflation, median inflation, and trimmed-mean inflation) from 1960 to
2022. The range is shown in the shaded area. The blue line corresponds to
headline inflation.  Table B.1: Summary Statistics: Inflation Measures

| Inflation Measures
---|---
| Headline | Core | Median | Trimmed
| Mean
| Full Sample (748 months)
Mean | 3.27 | 3.21 | 3.33 | 2.96
Std. Dev. | 2.42 | 2.13 | 2.01 | 1.86
Coeff. Var. | 0.74 | 0.66 | 0.60 | 0.63
|  $\pi<2.5\%$ (373 months)
Mean | 1.55 | 1.73 | 2.01 | 1.72
Std. Dev. | 0.67 | 0.53 | 0.95 | 0.70
Coeff. Var. | 0.43 | 0.31 | 0.47 | 0.41
|  $2.5\%\leq\pi<5\%$ (252 months)
Mean | 3.61 | 3.51 | 3.55 | 3.17
Std. Dev. | 0.71 | 1.06 | 0.84 | 0.77
Coeff. Var. | 0.20 | 0.30 | 0.24 | 0.24
|  $5\%\leq\pi$ (123 months)
Mean | 7.76 | 7.09 | 6.85 | 6.31
Std. Dev. | 2.00 | 1.59 | 1.60 | 1.57
Coeff. Var. | 0.26 | 0.22 | 0.23 | 0.25

  * •

Notes: The numbers are mean, standard deviation, and coefficient of variation
of the different inflation measures for different samples determined by the
level of PCE inflation. All numbers are in percentage points.

The variability of the robust inflation measures is also higher during low-
inflation episodes despite these measures being constructed to be less
responsive to transitory movements in inflation. Even though the robust
inflation measures are overall less volatile than headline inflation, this
pattern does not hold throughout the whole sample. Table B.1 reports the mean,
standard deviation, and coefficient of variation of the four inflation series
for different samples that depend on the level of headline inflation, Figure
B.2 plots the time series of the standard deviations.

When inflation is below 2.5 percent, median and trimmed mean inflation are
more volatile than headline inflation, and when inflation is between 2.5 and 5
percent all three robust inflation measures are more variable than headline
inflation. Moreover, the coefficient of variation is highest when headline
inflation is below 2.5 percent. The robust inflation measures also change
their ranking in terms of how volatile they are. Core inflation is the most
volatile in the complete sample, but median inflation is more volatile when
inflation is low (below 2.5 percent) and trimmed mean inflation is more
volatile when inflation is high (above 5 percent).

Figure B.2: Time-Series Variability of Measures of Inflation, 1960–2022 Notes:
The figure shows the authors’ calculations of the standard deviations of
headline inflation, core inflation, median inflation, and trimmed-mean
inflation for a rolling window of 24 months.  Figure B.3: Range of Underlying
Inflation, 1960–2022 Notes: The figure shows the authors’ calculations of the
range of inflation series used for different inflation measures from 1960 to
2022. The lines correspond to the 10th and 90th percentiles of the cross
section of monthly inflation rates in the 177 series considered for the
trimmed-mean measure, the 24th and 69th percentiles of the PCE inflation
series that correspond to the range used for trimmed-mean inflation, and the
median inflation series. Percentiles are weighted using the average real
expenditure on each category in months $t$ and $t-1$.  Table B.2: Most
Commonly Excluded and Included Expenditure Categories

| Median | Trimmed Mean | Middle 80%
---|---|---|---
|  |  | (10, 10) Trim
Most Commonly Excluded
1 |  | Eggs | Eggs
2 | 71 series are | Food on farms | Vegetables
3 | never median | Vegetables | Food on farms
4 |  | Fruit | Used automobile margin
5 |  | Gasoline | Fuel oil
Most Commonly Included
1 | Owner-occ homes | Owner-occ homes | Owner-occ homes
2 | Other purchased meals | Other purchased meals | Other purch meals
3 | Tenant-occ homes | Owner-occ mobile homes | Tenant-occ homes
4 | Nonprofit hospitals | Casino gambling | Casino gambling
5 | Physician services | Tenant-occ homes | Lotteries

  * •

Notes: The table reports the five expenditure categories most commonly
excluded and the five most commonly included when computing median and
trimmed-mean inflation as well as those excluded and included when trimming
the middle 90 percent of expenditure, setting trims to $\alpha=\beta=10$. All
the results are from the authors’ calculations of the series reported in
Figure 1. In the case of median inflation (first column), all categories but
one are included in a given month, so we report the number of series that are
never included. A series is considered ”included” if any of the weight of the
series is used in the calculation.

###  B.2 Alternative Measures of Trend Inflation

Figure B.4: Time Series of Trend Inflation

(a) Measures of Current Trend Inflation

(b) Measures of Future Trend Inflation

Notes: The figures show the authors’ calculations of two measures of trend
inflation. The first panel presents two measures of current trend inflation, a
36-month centered inflation trend (current trend), and a band-pass filter
trend (band-pass trend) as described in Section 3, together with the series of
year-on-year headline PCE inflation. The second panel presents two measures of
future trend inflation, a 12-month forward moving average of headline
inflation with data between 12 and 24 months ahead (future trend), and a
24-month forward moving average (forward trend).  Table B.3: Ranking of
Various Methods of Calculating Robust Measures

Target | Sample | PCE Inflation Measure | DM Test
---|---|---|---
Headline | Trimmed Mean | Median | $\Pr\left(z>\left|\text{DM}\right|\right)$
| 1970-2022 | $2.20$ | $1.10$ | $1.16$ | $0.066$
Current Trend | 1970-1989 | $2.28$ | $1.62$ | $1.51$ | $0.047$
| 2000-2022 | $2.47$ | $0.75$ | $0.95$ | $0.000$
| 1970-2022 | $2.11$ | $1.25$ | $1.30$ | $0.066$
Band-Pass Trend | 1970-1989 | $1.98$ | $1.65$ | $1.55$ | $0.035$
| 2000-2022 | $2.42$ | $1.01$ | $1.18$ | $0.000$
| 1970-2022 | $2.93$ | $2.12$ | $2.14$ | $0.476$
Future Trend | 1970-1989 | $3.48$ | $3.02$ | $3.00$ | $0.841$
| 2000-2022 | $2.93$ | $1.59$ | $1.61$ | $0.561$
| 1970-2022 | $2.43$ | $1.62$ | $1.66$ | $0.181$
Forward Trend | 1970-1989 | $2.74$ | $2.38$ | $2.34$ | $0.393$
| 2000-2022 | $2.56$ | $1.09$ | $1.21$ | $0.000$

  * •

Notes: The table presents the predictive performance of different PCE
inflation measures with respect to different trend inflation targets in
different samples. The performance is measured with the series’ root-mean-
square error (RMSE) with respect to trend inflation. The table reports the
RMSEs for headline, trimmed-mean, and median inflation. The last column
reports the p-value of the Diebold and Mariano ((1995) test for the difference
between the RMSEs of the trimmed-mean and median inflation series.

###  B.3 Alternative Trimmed Mean Measures of Inflation

Table B.4: Best Trims for Trimmed-Mean Inflation

Target | Sample | Best Trims | Official Trims | DM Test
---|---|---|---|---
Lower | Upper | RMSE | min(RMSE) | $\Pr\left(z>\left|\text{DM}\right|\right)$
| 1970-2022 | $20$ | $22$ | $1.06$ | $1.10$ | $0.014$
Current Trend | 1970-1989 | $18$ | $16$ | $1.44$ | $1.51$ | $0.238$
| 2000-2022 | $21$ | $27$ | $0.74$ | $0.75$ | $0.474$
| 1970-2022 | $11$ | $11$ | $1.12$ | $1.25$ | $0.000$
Band-Pass Trend | 1970-1989 | $11$ | $9$ | $1.36$ | $1.55$ | $0.003$
| 2000-2022 | $15$ | $18$ | $0.97$ | $1.02$ | $0.050$
| 1970-2022 | $28$ | $33$ | $2.09$ | $2.12$ | $0.192$
Future Trend | 1970-1989 | $15$ | $17$ | $2.91$ | $3.00$ | $0.560$
| 2000-2022 | $28$ | $32$ | $1.55$ | $1.59$ | $0.302$
| 1970-2022 | $15$ | $17$ | $1.59$ | $1.62$ | $0.150$
Forward Trend | 1970-1989 | $13$ | $13$ | $2.26$ | $2.34$ | $0.158$
| 2000-2022 | $23$ | $30$ | $1.09$ | $1.09$ | $0.815$

  * •

Notes: The table reports the best trims for different targets of trend
inflation over different samples as determined by the predictive performance
across trims. The root-mean-square error (RMSE) of the best trim is also
reported along with the lowest RMSE of the official trimmed and median
inflation series. The last column reports the p-value of the Diebold and
Mariano ((1995) test for the difference between the RMSEs of the best trim and
the lowest of the official series.

Figure B.5: RMSE across Trims

(a) Band-Pass Trend: 1970–2022

(b) Forward Trend: 1970–2022

(c) Band Pass Trend: 1970–89

(d) Forward Trend: 1970–89

(e) Band-Pass Trend: 2000–2022

(f) Forward Trend: 2000–2022

Note: The figures show heat maps of the root-mean-square error (RMSE) when
targeting band-pass and forward trend inflation with different trimmed-mean
inflation measures. To ensure comparability across plots, the RMSE numbers are
reported relative to the RMSE of the best trim reported in Table B.4.

Figure B.6: Statistical Difference of RMSE across Trims

(a) Current Trend: 1970–2022

(b) Future Trend: 1970–2022

(c) Current Trend: 1970–89

(d) Future Trend: 1970–89

(e) Current Trend: 2000–2022

(f) Future Trend: 2000–2022

Notes: The figures group trims according to the outcome of the Diebold and
Mariano ((1995) test, which compares the root-mean-square error (RMSE) implied
by the trims with the RMSE of the best trim as presented in Table 2. The trims
are grouped based on the p-value of the test. The darkest region consists of
trims whose RMSE is statistically equivalent to the lowest RMSE across all
trims.

Figure B.7: Statistical Difference of RMSE across Trims - Alternative Trends

(a) Band-Pass Trend: 1970–2022

(b) Forward Trend: 1970–2022

(c) Band-Pass Trend: 1970–89

(d) Forward Trend: 1970–89

(e) Band-Pass Trend: 2000–2022

(f) Forward Trend: 2000–2022

Note: The figures group trims according to the outcome of the Diebold and
Mariano ((1995) test, which compares the root-mean-square error (RMSE) implied
by the trims with the RMSE of the best trim as presented in Table B.4. The
trims are grouped based on the p-value of the test. The darkest region
consists of trims whose RMSE is statistically equivalent to the lowest RMSE
across all trims.

Figure B.8: The Behavior of Trimmed Mean Measures

(a) Coefficient of Variation

(b) RMSE - Current Trend

Notes: The left panel shows the coefficient of variation for different upper
trims for trimmed mean measures with three different levels of lower trim. The
coefficient of variation is the ratio between the standard deviation and the
mean of thee series over the 1970–2022 sample. The right panel shows the RMSE
of the series when targeting current trend inflation. The dotted vertical line
signals the optimal upper trim and the $*$ indicates the optimal lower trim.

Figure B.9: Prediction across Best Trims

(a) Centered Trend

(b) Future Trend

(c) Band-Pass Trend

(d) Forward Trend

Note: The figures show heat maps of the prediction level for October 2022
across the best 50 trim combinations, ranked according to their RMSE when
targeting band-pass or forward trend inflation over the sample 1970–2022. The
best 50 trims vary according to the trend inflation series being targeted.

Figure B.10: Prediction Range across Best Trims across Time

(a) Centered Trend

(b) Future Trend

Notes: The figures plot the range between the lowest and highest predictions
for the current and future trends between 1970 and 2022 given a set of
trimmed-mean measures of inflation. There are three ranges for each inflation
target: first, the range implied by considering the trims that are
statistically equivalent at the 5 percent level according to the Diebold and
Mariano ((1995) test of the difference of their root-mean-square errors
(RMSEs) with respect to the RMSE of the best trim (see Table 2); second, the
range implied by considering the best 100 trims as ranked by their RMSE;
third, the range implied by considering the best 50 trims as ranked by their
RMSE.

Figure B.11: Average Range of Inflation by Trims

Note: The figure shows the average range of inflation rates across individual
expenditure categories implied by each trim combination,
$\pi_{1-\beta}-\pi_{\alpha}$.

###  B.4 Excluding Housing from Trimmed-Mean Inflation

Housing is the single largest expenditure category and may have different
dynamics from the remainder of the consumption basket ((Adams et al., 2022).
It is also one of the series most commonly included in the official trimmed
mean and median inflation measures (see Table B.2). This can raise questions
over housing having too large of a role in the behavior of trimmed mean
inflation measures. That is not the case. We now reproduce our main results
while excluding owner occupied housing from the construction of trimmed mean
measures and re-weighting the remaining categories accordingly. All of our
results are preserved, with the obvious exception of the level of the optimal
trim cutoffs that changes to reflect the exclusion of housing from the set of
expenditure categories.

Table B.5: Best Trims for Trimmed-Mean Inflation without Housing

Target | Sample | Best Trim
---|---|---
Lower Trim | Upper Trim | RMSE
| 1970–2022 | $33$ | $34$ | $1.18$
Centered Trend | 1970–89 | $27$ | $24$ | $1.65$
| 2000–2022 | $33$ | $37$ | $0.74$
| 1970–2022 | $20$ | $19$ | $1.18$
Band-Pass Trend | 1970–89 | $21$ | $18$ | $1.45$
| 2000–2022 | $30$ | $32$ | $0.98$
| 1970–2022 | $45$ | $49$ | $2.13$
Future Trend | 1970–89 | $25$ | $25$ | $3.02$
| 2000–2022 | $48$ | $50$ | $1.39$
| 1970–2022 | $30$ | $32$ | $1.64$
Forward Trend | 1970–89 | $20$ | $19$ | $2.36$
| 2000–2022 | $42$ | $46$ | $0.96$

  * •

Notes: The table reports the best trims for different targets of trend
inflation over different samples as determined by the predictive performance
across trims. The root-mean-square error (RMSE) of the best trim is also
reported along with the lowest RMSE of the official trimmed and median
inflation series.

Figure B.12: RMSE across Trims: 1970–2022 (Sample without Housing)

(a) Centered Trend

(b) Forward Trend

(c) Band-Pass Trend

(d) Future Trend

Note: The figures show heat maps of the RMSE when targeting trend inflation
with different combinations of trimmed-mean inflation measures. Four measures
of trend inflation are considered. To ensure comparability across plots, the
RMSE numbers are reported relative to the RMSE of the best trim reported in
Table B.5.

Figure B.13: RMSE across Trims: 2000–2022 (Sample without Housing)

(a) Centered Trend

(b) Forward Trend

(c) Band-Pass Trend

(d) Prediction

Note: The figures show heat maps of the RMSE when targeting trend inflation
with different combinations of trimmed-mean inflation measures. Four measures
of trend inflation are considered. To ensure comparability across plots, the
RMSE numbers are reported relative to the RMSE of the best trim reported in
Table B.5.

Figure B.14: RMSE across Trims: 1970–89 (Sample without Housing)

(a) Centered Trend

(b) Forward Trend

(c) Band-Pass Trend

(d) Prediction

Note: The figures show heat maps of the RMSE when targeting trend inflation
with different combinations of trimmed-mean inflation measures. Four measures
of trend inflation are considered. To ensure comparability across plots, the
RMSE numbers are reported relative to the RMSE of the best trim reported in
Table B.5.