Median inflation (or other trimmed means) work best because of high variation in price changes

The standard inflation indexes track the average change in prices and one of the problems with using any average that is disproportionately influenced by outliers that are highly volatile due to market forces that have nothing to do with monetary policy and central banks need to measure how much the things that they control are affecting inflation so they can adjust what they can directly control: the money supply and short-term interest rates.

Kevin Drum show that the median price change is one of the best measures for adjusting monetary policy:

Here it is:

This chart shows three measures of inflation that are designed to avoid the variability of headline CPI and give a better look at the real level of inflationary pressures:

Core CPI omits food and fuel because they jump around a lot and don’t really tell us anything about underlying inflation.

Trimmed mean CPI cuts off the biggest gainers and losers so that a few outliers don’t affect the bulk of the items in the CPI basket.

Median CPI looks at the inflation rate of the median item in the CPI basket.

For those of you who want evidence that inflation isn’t all that bad right now, median CPI is your hot ticket. Not only is it relatively low at 3.5%, but it’s only 36% above its average 2016-2019 level. The others are 116% and 135% above their 2016-2019 averages.

Here is the conclusion of a recent paper that studied the accuracy of various measures of core inflation:

The last two years have been highly informative about the behavior of alternative core measures. Headline inflation has fluctuated erratically….XFE [eXcluding and Food Energy] inflation has performed quite poorly….Fixed-exclusion measures of core that exclude a wider set of industries, such as the Atlanta Fed’s sticky-price inflation rate, have performed better, but the most successful measures have been weighted medians and trimmed means.

When the inflation rate rises from 2% to 7%, a lot of Americans get very upset and it seems to be as upsetting as an equivalent rise in unemployment even though economic statistics show that the real harm to society of 7% unemployment is far higher than 7% inflation. I’ve always been puzzled why people hate such a modest increase in inflation, but one reason is probably that most Americans pay much more attention to some prices than to others. In particular, they pay attention to the prices of necessities like energy and food which are also the most volatile prices. So the misery of high inflation isn’t really the average inflation, but the inflation in prices that have a bigger impact on lifestyle like food and gas and used cars which are all soaring currently. So perhaps we should create a psychological price index that is weighted according to how strongly people feel about the price of each good. That wouldn’t be as good for the central bank to know how to direct monetary policy, but it would help explain when inflation is most upsetting to people.

Of course, there are always some sort of prices that upset someone. When real wages rise, that makes more people happy because most of us get most of our income from work, but it upsets people who get most of their money from owning stuff (the capitalists). And according to Jonathan Nitzan and Shimshon Bichler, inflation increases typically does the reverse of this and makes the capitalists richer and workers poorer. That is probably a big real reason why ordinary Americans hate inflation.

Blair Fix is an interesting thinker who argues that the amount of variation in prices is so huge that estimating inflation is useless. I don’t go that far, but his analysis is fascinating and ground-breakingly creative.

I can calculate the average of any conceivable set of numbers. But that doesn’t mean my calculation will be informative. That’s because averages define a central tendency, yet do not indicate if this tendency actual exists.

Here’s an example. Suppose two people have an average net worth of $100 billion. Is this a central tendency? Perhaps … if our two people are Warren Buffet (net worth $104 billion) and Mukesh Ambani (net worth $96 billion). Both have close to the same wealth. But what if the two people are Jeff Bezos (net worth $200 billion) and me (net worth $0 billion)? In this case, the average misleads more than it informs.

Of course, scientists are aware of this problem. That’s why they are trained to report averages together with a measure of variation. Doing so gives a sense for the meaningfulness of the average.

Any measure of variation will do, but the most popular is the ‘standard deviation’, which measures the average deviation away from the mean.3 Returning to my wealth example, reporting the standard deviation of wealth tells us when the average measures a real central tendency, and when it does not.

For instance, Warren Buffet and Mukesh Ambani have an average net worth of $100 billion, with a net-worth standard deviation of $5.7 billion. The fact that the variation is small (about 0.06 times the average) indicates that there is a real central tendency. In contrast, Jeff Bezos and I have an average net worth of $100 billion, with a standard deviation of $141 billion. This enormous variation (1.4 times the average) indicates that there is no central tendency in the raw data. So the average is uninformative…

Figure 3 shows the price change of every commodity tracked by the consumer price index. Instead of clustering tightly around the average price level (the ‘official CPI’), real-world commodities have a mind of their own. Their prices head in all sorts of directions — often in ways that seem unrelated to the movement of the average price.

Figure 3: Price change in the real world. The black line shows the change in the US consumer price index since January 1, 2020. The colored lines show the indexed price of all the individual commodities tracked by the CPI. Many commodities are tracked in multiple locations. [Sources and methods]

Notice how plotting the price-change of all CPI commodities alters the inflation story. Looking at Figure 3, no one would conclude that all prices are inflating uniformly. Yet when we looked at the consumer price index alone, this conclusion seemed plausible.

When we study the whole range of price change, we see that inflation is a messy business. The numbers tell us as much. Since January 1, 2020, the consumer price index rose by 7.3%. That value seems significant … until we measure price-change variation. Over the same period, the standard deviation of price change was 10.7%. So the variation in price change was about 1.5 times larger than the price-change average.5

To put this variation in perspective, let’s return to our wealth example. Jeff Bezos and I have an average wealth of $100 billion. But this value does not indicate a real central tendency. Jeff Bezos is worth $200 billion. I’m worth $0 billion. We can tell that the average is misleading by measuring the standard deviation of our wealth, which happens to be $141 billion. So the variation in our wealth is about 1.4 times the average.

This ratio of 1.4, you’ll notice, is actually less than the ratio of 1.5 we found for price-change variation. So if we conclude that it’s rather meaningless to average my wealth with Jeff Bezos’s wealth, we should also conclude that the movement of the consumer price index is quite meaningless. Both averages mislead more than they inform…

The real story of inflation — the one that goes largely unreported — is of wildly divergent price change among different groups of commodities. Figure 4 shows how this inflation has played out across 12 major commodity groups tracked by the US consumer price index.

Figure 4: US price change by commodity group. Box plots show the range of price change between Jan. 2020 and Oct. 2021, for US CPI commodities classified into 12 major groups. Here’s how to read the box plot. The thick vertical line indicates the median value. The ‘box’ shows the middle 50% of the data. And the line shows the range of the data, excluding outliers. [Sources and methods]

…We can see that inflation varies greatly between different commodity groups. Some groups, like ‘men’s apparel’, have experienced little (if any) inflation. Other groups, like ‘private transportation’, have seen massive price hikes. Figure 4 also shows that inflation varies greatly within each commodity group. Inflation often coexists with deflation — a fact that’s evident when the boxes cross the dashed red line.

So the real inflation story, which goes largely undiscussed, is that price change is remarkably non-uniform.

…let’s look at the long-term history of US price-change variation. Figure 6 shows the data. I start by replotting the average inflation rate (blue line). But then I add some much-needed information — the range of price change across all CPI commodities. That’s the blue region, which plots the 95% range for the annual price change of all commodities (in all locations) tracked by the CPI. The price-change range is … rather large.

Figure 6: The history of US price-change variation. The blue line shows the annual change in the consumer price index, replotted from Figure 5. The shaded region shows the 95% range (the range for the middle 95% of the data) in the annual price change of all commodities (in all locations) tracked by the CPI. [Sources and methods]

The evidence in Figure 6 suggests that our current situation is not unusual. Since the CPI data began in 1913, the US inflation rate averaged 2.8%. But over the same period, the standard deviation of annual price change averaged 5.2%. So the inflation variation was historically about 1.8 times larger than the inflation average. To remind you, the variation between Jeff Bezos’s wealth and my wealth was only 1.4 times our average wealth. So looking at the average US inflation rate is even less meaningful than averaging Bezos’s wealth with my own.

To summarize, the data is pretty clear: the historical norm has been for price-change variation to trump the average rate of inflation. So why does this variation go unreported?

It is a good point. Other economic indexes like the S&P500 don’t report variance either and they should. But arguing that a large coefficient of variation makes an average meaningless isn’t right either because suppose inflation is always zero. The coefficient of variation could be infinite even though the standard deviation of prices were only infinitessimally small. Whenever there is data that is both positive and negative as with inflation data, the coefficient of variation is always going to be huge. Blair Fix is right that variation is important, but the high variation of inflation data reduces the usefulness of inflation as a statistic, but it’s still useful and important.

Posted in Macro

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