It’s easy enough to define inflation as a decline in the purchasing power of a dollar. But the power of a dollar to purchase– exactly what? The devil is in the details.
If you only bought a single good (my students know that I like to use “potatoes” as the example for my lectures), you’d measure inflation as the percentage change in the dollar price of a potato. But if you buy both potatoes and oranges, and their dollar prices go up at different rates, how do you measure inflation?
Calculation of the consumer price index is a quite sophisticated procedure, but the essential idea can be described fairly simply. The Bureau of Labor Statistics has surveyed households to find out what fraction of expenditures a typical urban consumer devotes to various categories. For example, 3.9% of the expenditures of their hypothetical consumer go to buying gasoline and 2.7% to full-service meals away from home. The BLS then surveys various establishments to find out how much you’d pay for gasoline or a Big Mac Meal one month compared to the next. For example, they found average gasoline prices rose 6.4% in July and restaurant meals rose 0.2%. They then calculate if you spent $1,000 among these categories according to these weights ($39 on gasoline, $27 on restaurant meals, and so on), how much more you would have to pay to buy the same things in July. The amount by which that expenditure would go up is summarized by the increase in the consumer price index for urban consumers (CPI-U) between June and July.
Even if in June you bought each of these items in exactly the same portions as the hypothetical BLS consumer (and I know you didn’t), the CPI would still not accurately describe the inflation that you personally experienced in July. The reason is that you didn’t buy your items from the particular outlets that the BLS sampled, and you doubtless changed both the quantity and quality of the items you purchased between June and July. There are also profound challenges in measuring how much you “spent” to live in the house you own, given that, in the process of making your mortgage payments, you are acquiring an asset and earning a capital gain on your equity.
Even under ideal conditions, the CPI should only be viewed as an error-ridden estimate of the object you’re interested in. That raises a statistical question: what is the best way to use the imperfect data collected by the BLS to construct the best estimate of the magnitude of interest?
Suppose that what we’re interested in is the answer to the question, what happened to the purchasing power of a “typical” dollar spent in July? We might think of the BLS data as giving us observations on what happened to 1,000 dollars we happened to sample, 39 of which went to buy gasoline, 27 of which went to buy restaurant meals, and so on. How would we use such data to estimate what happened to the purchasing power of all the dollars people might have spent on anything? One obvious answer would be to take the sample mean, which would be (0.039 x 6.4 + 0.027 x 0.2 + …).
Although we’re accustomed to using the sample mean as a logical estimate of a population’s central tendency, the sample mean is not always the best estimate. A study by Michael Bryan and Stephen Cecchetti suggested that we might want to use an alternative such as the trimmed mean or the median. To calculate either of these estimates, we would first order those 1,000 “sampled dollars”, with those spent on items that experienced the smallest (or most negative) price increases ordered first and those with the biggest price increase ordered last. For the trimmed mean, we would discard the first 75 and last 75 dollars, and calculate the sample mean for those dollars that remain after we in this way “trim” the sample. For example, if gasoline was in the upper 7.5% of all price changes in July, we wouldn’t use it for the July calculation, but instead would calculate (0.027 x 0.2 + …). For the median, we would just look at dollar number 500 during July, and use the price increase on whatever that went to purchase as our measure of inflation for July.
This approach of deliberately ignoring much of the data strikes some people as clearly wrong-headed. In terms of statistical theory, if your original data are well-behaved (for example, drawn from a Normal, also called a Gaussian, distribution), then one can show that the sample mean would be a better estimate than either the trimmed mean or the median. On the other hand, if your data come from other statistical distributions that have a bigger chance of producing very large or very small numbers than those that are usually produced by a Gaussian distribution, the trimmed mean or median can give you a much better inference.
One of the very interesting results that Bryan and Cecchetti came up with was that if your goal is to predict how much the CPI-U, as usually calculated and constructed by the BLS, is going to go up from its current value over the next 12-60 months, you’d actually get a better prediction if you based your forecast of the future CPI-U on the current and past values of either the trimmed-mean CPI or the median CPI than you would obtain if you based your prediction of the future CPI-U on current and past values of the CPI-U. These results were confirmed in subsequent analyses by Todd Clark and Julie K. Smith.
If you were only interested in the very long-run trends, you would get a similar answer no matter which estimate you used. For example, since 1968, calculating year-to-year inflation rates each month using the median CPI, you would conclude that the U.S. has experienced an average annual inflation rate of 4.82%. If you used the conventional CPI-U, you would conclude that the average annual inflation rate over this period has been 4.81%.
Where the measures differ the most is in describing month-to-month fluctuations. The Federal Reserve Bank of Cleveland provides historical and current data on the median CPI, and Macroblog makes regular use of this series in interpreting recent economic developments. Values for the median CPI over the last year are compared with the usual CPI-U in the table at the right. If you were basing your inference about inflation on month-to-month changes in the CPI-U, you would have had the experience of a thrilling amusement park ride this year, being persuaded that the U.S. inflation rate was running above 6% this spring, became negative at the start of the summer, and is now again over 6%. By contrast, the median CPI has sent a fairly clear signal that inflation has consistently been somewhere between 2 and 3%. Using the year-to-year change (rather than month-to-month as in the accompanying table), is probably an even better idea– this suggests an inflation rate of around 2.4% over the last year.
On the other hand, if you’re one of those people who craves excitement– if skydiving and rock-climbing are among your hobbies, for example– then my advice is to completely ignore the median CPI and stick with the good old CPI-U.