Adjusting Median and Trimmed-Mean Inflation Rates for Bias Based on Skewness

As shown, the median and trimmed-mean inflation rates are less volatile than headline inflation, a finding that is consistent with the intended design of these limited-influence estimators to remove noise in the price data. Another noticeable feature of the data is that median PCE inflation and trimmed-mean PCE inflation can depart from PCE inflation for extended periods of time. As we will explain is typically the case, we also see that median PCE inflation lies above trimmed-mean PCE inflation. In terms of this divergent behavior, median PCE inflation has averaged about 0.3 percentage points higher than trimmed-mean PCE inflation since 2000.

Why is median PCE inflation typically higher than trimmed-mean PCE inflation, and why does the gap vary over time? The answer lies in the interaction of the construction of each measure—in particular, the way that trimming of outliers is undertaken—and in the cross-sectional distribution of price changes in the underlying components. It is well known that the distribution of price changes across components is highly kurtotic; that is, it has a relatively high presence of outliers. This means that almost every month, a small number of components experience extreme price changes, both positive and negative. Importantly, those components are not always food and energy so that core inflation measures (which exclude food and energy prices) are often buffeted by the very same forces that buffet headline measures. This fact motivated the development of the median and trimmed-mean inflation rates that remove extreme price changes regardless of their source (see Bryan and Cecchetti, 1994; and Bryan, Cecchetti, and Wiggins, 1997).³

What is somewhat less well-known is that the monthly cross-sectional distribution of component price changes is not symmetric and features notable skewness (see Dolmas, 2005, and Verbrugge and Zaman, 2021). There are several ways to measure skewness in a data distribution. Because extreme price changes distort conventional skewness statistics, we focus on two robust measures of skewness: the Bowley coefficient of skewness and the Kelly coefficient of skewness.⁴ Both measures are bounded between -1 and +1, with negative and positive estimates indicating left- and right-skew, respectively. Our skewness measure is the average of these two.

Figure 2 plots the one-sided 12-month moving average of monthly skewness for the period January 1978–December 2021, a timespan covering the entire history of the median PCE and trimmed-mean PCE inflation rates. As shown, the monthly distribution historically has been moderately left-skewed most of the time. The left-skew of the distribution principally reflects the behavior of goods prices that not only tend to increase at a slower pace compared with services prices, but also exhibit ongoing deflation in some components.⁵ Recently, however, the distribution has become right-skewed because of the pandemic’s effect on prices. Specifically, goods prices have exhibited large and sustained increases because of a combination of strong demand, bottlenecks, and supply chain disruptions, while some services prices have shown large increases as they rebound from the steep declines in demand resulting from lockdowns and social distancing.

In general, the presence of skewness in the price-change distribution has important implications. When a distribution is left-skewed, the mean lies below the median, and, hence, the median PCE inflation rate will be upward-biased with respect to its headline counterpart. For the same reason, a symmetric trimmed-mean inflation rate will be upward biased relative to the mean, with the degree of bias increasing in the amount of trimming.⁶ This upward bias motivates the asymmetric trimming used in the Dallas Fed’s trimmed-mean PCE inflation rate. Specifically, that measure trims 24 percent of the weight from the lower tail but 31 percent of the weight from the upper tail.⁷ Because the trimming removes more of the upper tail and retains more of the lower tail, this combination lowers the estimate. Dolmas (2009) demonstrates that this degree of asymmetric trimming historically has resulted in an unbiased estimate of headline PCE inflation over longer periods. This implicit bias adjustment in the trimmed-mean PCE inflation rate gives it an advantage relative to the median PCE inflation rate in “naïve” forecasting exercises in which inflation outcomes are simply compared to readings from the two measures.

The shift in the skewness of the cross-sectional price-change distribution from left (negative) skew to right (positive) skew during the pandemic has relevance for recent readings of median PCE inflation and trimmed-mean PCE inflation and their interpretation for movements in trend inflation. When the price-change distribution is right-skewed, a median or symmetric trimmed-mean will be downward biased with respect to the mean. Consequently, the median PCE inflation rate has now become a downward-biased estimate of the mean. Compared to the median PCE inflation rate, the Dallas Fed’s trimmed-mean PCE inflation rate currently suffers from more downward bias owing to its asymmetric trimming. This occurs because the current configuration of the price-change distribution would call for more trimming of the weight from the lower tail and less trimming of the weight from the upper tail.

Motivated by the previous discussion, we take a closer look at median and trimmed-mean estimators of trend inflation and the bias that can arise when the distribution of price changes is skewed. An issue of particular interest is the relationship between bias and skewness. If there is a significant correlation, then skewness can provide the basis for a bias-adjustment procedure to improve estimates of trend inflation. The bias-adjusted estimates could afford new insights into the historical behaviors of median PCE inflation and trimmed-mean PCE inflation and allow us to assess the extent to which these measures may be understating the trend in PCE inflation today.

Figure 3 plots the resulting estimates of bias for the period June 1986–May 2020.⁹ Bias clearly persists for extended periods of time, underscoring the desirability for some type of adjustment procedure. As described in Carroll and Verbrugge (2019), median PCE inflation typically has an upward bias because the price-change distribution is typically left-skewed. On the other hand, trimmed-mean PCE inflation is designed to minimize long-term bias, and this explains why the bias term for this measure fluctuates around zero over the entire period, although there are still extended departures of the series from this value.

Figure 4 provides a scatterplot of bias versus skewness for median PCE inflation and trimmed-mean PCE inflation for the period June 1986–May 2020. As shown, there is clear and compelling evidence of a negative relationship between bias and skewness for each limited-influence estimator, a finding that accords with our previous discussion.

Drawing upon the visual evidence above, we use a simple linear regression to formalize the relationship between bias and skewness and to provide a basis to construct bias adjustment terms. We estimate the following regression for median PCE inflation:

where ${skew}_{m,t}$ is the 12-month one-sided moving average of skewness depicted in Figure 2. The regression equation for trimmed-mean PCE inflation is specified in an analogous manner.

Table 1 reports the regression results for the period June 1986–May 2020, where t-statistics from robust standard error estimation are reported in parentheses.¹⁰ As shown, there is a highly statistically significant negative relationship between bias and skewness for both median PCE inflation and trimmed-mean PCE inflation. In addition, the R² values indicate that skewness explains about 40 percent to 50 percent of the total variation in bias.

The coefficient estimates can now be used to bias-adjust either limited-influence estimator. For example, the bias-adjusted median PCE inflation measure $\left({\tilde{\pi }}_{m,t}^{Median}\right)$ is given by the following:

The corresponding bias-adjusted trimmed-mean PCE inflation measure can be derived in an analogous manner using the relevant coefficient estimates in Table 1.¹¹

Figure 5 plots headline PCE inflation along with the unadjusted and bias-adjusted measures for median PCE inflation and trimmed-mean PCE inflation. Examining the data over the post-2000 period allows us to highlight three takeaways from the figure. First, median PCE inflation and trimmed-mean PCE inflation display very similar behavior after bias adjustment. Second, while bias adjustment shifts down median PCE inflation over almost the entire sample, there is also clear evidence of time variation in the bias adjustment for both median PCE inflation and trimmed-mean PCE inflation. This latter point is particularly evident during the long recovery from the Great Recession and especially between 2014 and 2017, when median PCE inflation and trimmed-mean PCE inflation are both shifted down because skewness was notably negative over this period. Third, the current pandemic period is associated with unusual inflation dynamics, wherein bias adjustments are driven by positive skewness (see Figure 2) and whose sources we discuss below. The magnitude of the recent upward adjustments to trimmed-mean PCE inflation is particularly noteworthy. Since April 2021, the average upward adjustment to trimmed-mean PCE inflation is +0.30 percentage points, while the average upward adjustment to median PCE inflation over this period is +0.08 percentage points. At the end of our sample in December 2021, our bias-adjustment procedure would boost the final reading of median PCE inflation by 14 basis points to 3.75 percent and would boost the final reading of trimmed-mean PCE inflation by 34 basis points to 3.39 percent.

A Closer Look at the Recent Change in Skewness: Goods versus Services

The positive bias adjustments that emerged earlier this year come directly from the shift in the cross-sectional price-change distribution from negative skewness to positive skewness. To help explain this change in skewness, Figure 6 plots the estimated skewness for headline PCE inflation and for inflation across goods and inflation across services. As in Figure 2, we plot these measures over the post-1977 period to establish the salient features of goods inflation versus services inflation.

Three observations are particularly noteworthy. First, the price-change distributions for both goods and services categories are contributing to the positive skewness in headline PCE inflation in the current period.¹² Second, while positive skewness in services inflation is not an uncommon phenomenon, positive skewness in goods inflation is a rare occurrence. Third, although not depicted in Figure 6, the positive skewness in goods inflation is associated with another marked change in the behavior of goods inflation. Specifically, the pandemic period has witnessed strong positive inflation in the goods category, a shift which is a dramatic change from the outright declines in goods prices that have been generally observed in recent decades. Because goods inflation was typically negative and well below average inflation, components classified within the goods category had mostly tended to be in the left tail of the price-change distribution, helping to generate the left (negative) skew. After the onset of the pandemic, goods inflation has surged and reflects the notable shift in consumers’ spending patterns away from services and toward goods, along with the impact of bottlenecks and supply chain disruptions. This movement of goods inflation into the positive region of the price-change distribution suggests that goods components have moved weight from the left tail of the distribution into the right tail of the distribution, contributing to the right (positive) skew.

Conclusion

Median and trimmed-mean inflation rates remain popular estimators of trend inflation. While these estimators tend to move in line with the underlying trend in inflation over long periods, there can be deviations that persist over shorter episodes. This Commentary focuses on time-variation in this bias and presents evidence linking its movements to the degree of skewness in the distribution of price changes. We use the estimated relationship between bias and skewness to construct bias-adjusted median and trimmed-mean measures of inflation. We find median PCE inflation and trimmed-mean PCE inflation display very similar behavior after bias adjustment. In addition, the analysis makes note of a shift in the skewness of the cross-sectional price-change distribution during the pandemic and reports that readings from median and trimmed-mean inflation rates have recently been understating the trend in 12-month PCE inflation by about 15 basis points and 35 basis points, respectively.