Various Methods of Calculating Inflation

Why are There So Many Different Ways to Calculate Inflation?

We’ve all heard the old saying, “Figures don’t lie but liars figure” or perhaps “You can make numbers say anything you want“. Both of these sayings contain the underlying assumption (or at least possibility) of malicious intent. But even if you have the raw data and just want to get at the truth, number calculations can present difficulties because the answer you get can depend on how you analyze it.

For instance, you would think that if you want to know the “average” income of a group of 10 people it would be easy to calculate. And if all the incomes are relatively closely grouped it is easy… simply add them all up and divide by ten. This method produces a result commonly called an “average” but technically called a “mean”.

But suppose your ten people’s incomes are widely divergent. For example, suppose one person earns $10,000, eight people earn $15,000 and one person earns $100,000. Without calculating you would say most people earn about $15,000 right? Or you might even say the average person earns $15,000. But if you take the average (mean) you will get (1 x $10,000) + (8 x $15,000) + (1 x $100,000) = $230,000 / 10 = $23,000.

This doesn’t really represent what the average person earns because the single high value skews the result. This is why often in school when they are “grading on a curve” they will throw out the highest and the lowest values. This is called a “trimmed mean”. In the above example, if we eliminated the highest and the lowest values, the trimmed mean would be $15,000 or exactly what you’d expect to be the “average.”

Another method is called the median or the number in the middle. If you have an odd number of items in the list it is easy simply put them in order from lowest to highest and take the middle number. If you have an even number of items in the list, you would take the average (mean) of the two middle numbers. In this case, however the middle two numbers are both $15,000 so the median is $15,000.

The Consumer Price Index

The most widely used inflation calculation is the Consumer Price Index for all Urban Consumers (CPI-U) produced by the U.S. Bureau of Labor Statistics (BLS). The CPI-U is a weighted average. In other words it takes all the items the average urban consumer would buy and gives each item a relative weight based on what percentage of their budget this item would comprise. Of course each consumer is different so the weighting itself is an estimate and does not necessarily represent any individual or family.

The weighting is based on what percentage of an average budget is spent on food and what percentage is spent on clothing or gasoline or housing etc. The problem arises when a single component rises much more (or less) than all the other items. Then it experiences the same problem we saw with the income example above, where one item skews the results.

As we’ve said numerous times the primary cause of price inflation is an increase in the money supply and theoretically it should affect all items relatively equally but if for instance a freeze in Florida causes a massive increase in the price of orange juice that would not be the result of an increase in the money supply. Economists call that “relative-price” change and not “inflation” in the monetary sense. So if we want to track only inflation caused by an increase in the money supply you have to do some numerical manipulation.

The Core CPI

During the 1970’s the Arab Oil embargo had a drastic effect on inflation as measured by the CPI so economists decided to begin tracking just the “core inflation rate” i.e. this allowed them to follow the less volatile other items that were not affected by external forces and gauge only the effects of the money supply based inflation. But the core CPI had some problems of its own, namely food and energy are a big portion of the overall cost of living and so they should not be totally ignored. What if gas or food is actually trending up due to monetary inflation?

Trimmed Mean CPI

The next method economists came up with to try to determine (and forecast) monetary based inflation is called “trimmed mean” this is similar to the “grading on a curve” example where the highest and lowest values are thrown out.

The method of calculating a trimmed mean is simple. Sort all the individual components by their inflation rate in ascending order from “fell the most” through “fell the least” and finally to “rose the most” and then eliminate a certain percentage of the highest and lowest numbers. The trimmed mean tends to provide better data than either the straight CPI or the core CPI.

The Cleveland FED uses a 16% trimmed mean which is simply eliminating 8% from the top and 8% from the bottom. The Dallas FED on the other hand believes that the best results for a “trimmed mean” result from trimming 24% from the lower tail and 31% from the upper tail, which at first glance seems to bias the result to the down side but an argument can be made that bigger numbers have more effect than smaller numbers. The Cleveland FED also recommends a 49.5% trimmed mean, which results in trimming almost 25% from each end, basically using the middle 50%.

What is the Median CPI and Why Does the FED Use It?

Another measurement used by the Cleveland FED is the “median CPI” which takes the same list of items sorted by percentage and simply picks the middle number. This is very similar to a 99% trimmed mean. So if you trim off all of the top tail and all of the bottom tail you will be left with a single number which will be the median (or middle). This works if you have a large enough list of numbers. The actual method they use is to take the monthly inflation rate and annualize it (i.e. multiply it by 12) and then sort by this column.

Over the last 12 months, the CPI rose 2.0%, and the CPI less food and energy rose 1.8%, the 16% trimmed-mean CPI rose 1.8% and the median CPI rose 2.2%.

This is the data for the median CPI published by the Cleveland FED. Note that because housing was such a big component of the CPI the Cleveland FED actually breaks it down into four regional components to lend a bit more accuracy to the median calculation. An interesting anomaly in this particular list is that at the top (fell the most) is Fuel oil and other fuels and at the bottom (rose the most) is motor fuel. So if we were looking at the core CPI (minus food and energy) both would be eliminated but from opposite ends of the spectrum.

April 2014
Detail for Computation of the Median CPI
Component	Annualized 1-month % change	Relative Importance	Cumulative Relative Importance
Fuel Oil and Other Fuels	-48.71748	0.299297	0.299297
Energy Services	-20.81101	3.891275	4.190572
Car and Truck Rental	-19.43693	0.079926	4.270498
Watches & Jewelry	-15.76154	0.2163169	4.486815
Processed Fruits & Vegetables	-12.6128	0.3067271	4.793542
Leased Cars and Trucks	-5.683114	0.3933697	5.186912
Men’s & Boys’ Apparel	-4.491017	0.8571489	6.044061
Footwear	-2.282645	0.703445	6.747506
Other Food At Home	-2.264376	2.018987	8.766493
Miscellaneous Personal Goods	-2.198694	0.1943952	8.960888
Nonalcoholic Beverages & Beverage Matls	-1.775123	0.9444487	9.905336
Tenants’ and Household Insurance	-1.149148	0.3654688	10.27081
Misc Personal Services	-0.5752687	1.114855	11.38566
Household Furnishings & Operation	-0.4371489	4.210216	15.59588
Cereals & Bakery Products	-0.353738	1.136422	16.7323
Personal Care Services	-0.2138359	0.6317063	17.36401
Tobacco & Smoking Products	0.6246597	0.7069537	18.07096
Motor Vehicle Fees	0.989953	0.5663932	18.63735
Personal Care Products	1.145749	0.7282558	19.36561
Communication	1.466142	3.818088	23.1837
Alcoholic Beverages	1.545735	1.003594	24.18729
Motor Vehicle Parts and Equipment	1.573913	0.4385149	24.6258
Motor Vehicle Maintenance & Repair	1.656998	1.151407	25.77721
Northeast Urban: Owners’ Equivalent Rent of Residences	1.727802	5.405011	31.18222
South Urban: Owners’ Equivalent Rent of Residences	1.946643	7.746981	38.92921
Recreation	2.192346	5.754914	44.68412
Midwest Urban: Owners’ Equivalent Rent of Residences	2.243664	4.457551	49.14167
Food Away From Home	3.235635	5.714143	54.85581
Medical Care Services	3.309752	5.867292	60.7231
West Urban: Owners’ Equivalent Rent of Residences	3.605284	6.364657	67.08776
New Vehicles	3.676942	3.530261	70.61802
Medical Care Commodities	3.874883	1.698333	72.31635
Water/Sewer/Trash Collection Services	4.014398	1.18214	73.49849
Rent of Primary Residence	4.089647	7.004128	80.50262
Education	4.59088	3.260816	83.76344
Lodging Away From Home	4.591441	0.8148859	84.57832
Women’s & Girls’ Apparel	4.598701	1.510254	86.08858
Dairy and Related Products	6.126217	0.8768851	86.96546
Used Cars and Trucks	6.254493	1.658191	88.62365
Motor Vehicle Insurance	10.94646	2.232075	90.85573
Infants’ & Toddlers’ Apparel	14.89104	0.1375694	90.9933
Fresh Fruits & Vegetables	16.24166	1.039211	92.03251
Meats, Poultry, Fish & Eggs	19.11217	1.903383	93.93589
Public Transportation	20.99853	1.120386	95.05627
Motor Fuel	31.70804	4.94372	99.99999

How Accurate are These Methods of Predicting Inflation?

Because of the existence of “relative-price” noise in the data it is important that any system that tries to measure inflation be able to focus on the monetary inflation component. Forecasting accuracy is measured using a system called Root-mean squared error (RMSE) which measures how closely projections matched actual data. The lower the number the better the projection was.

Looking at the table below we can see that when projecting 12 months ahead just using the previous month’s CPI generates an RMSE accuracy of 3.28. Projections get more accurate if you use the percent change of longer periods. The Core CPI was more accurate with a one month RMSE of about ½ of just using the CPI itself at 1.67 and the median CPI was roughly 16 percent more accurate at 1.46.

Looking 12 Months Ahead
Root-Mean Squared Error (RMSE)
Percent change last	CPI	Core CPI	Median CPI
1 month	3.28	1.67	1.46
3 months	2.59	1.44	1.37
6 months	2.09	1.38	1.34
9 months	1.83	1.37	1.33
12 months	1.7	1.37	1.33
24 months	1.49	1.37	1.33

You would think that looking 24 months ahead would be more difficult than looking ahead 12 months. But as you can see from the second table it is actually more accurate to look 24 months ahead based on 24 months of previous data which results in an RMSE of 1.02.

Looking 24 Months Ahead
Root-Mean Squared Error (RMSE)
Percent change last	CPI	Core CPI	Median CPI
1 month	3.23	1.45	1.22
3 months	2.4	1.14	1.09
6 months	1.86	1.06	1.05
9 months	1.6	1.07	1.05
12 months	1.48	1.07	1.05
24 months	1.25	1.08	1.02

The Personal Consumption Expenditures (PCE) Price Index

All of the above indexes are based on the CPI and data gathered by the U.S. Bureau of Labor Statistics but the FED also uses data collected by a division of the U.S. Department of Commerce called the Bureau of Economic Analysis. The PCE is different than the CPI in that it takes into consideration “substitution” so that if consumers decide steak is too expensive they might substitute hamburger. Weighting is also based on different data sources and since the PCE uses expenditure data it is difficult to implement in “real time”. Because of the effects of substitution the PCE tends to show a lower inflation rate than the CPI.

Billion Price Project

Another alternative to the CPI is the Billion Price Project (BPP) developed by Alberto Cavello and Roberto Rigobon of Massachusetts Institute of Technology (MIT). They developed an algorithm that scrapes prices from the internet and is able to incorporate them into a price index. Even though it tends to under-represent services (which tend to rise in price faster than many consumer commodities) it still is indicating a higher level of inflation than either the CPI or the PCE.

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