Measuring inflation

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.

Month-to-month inflation (annual rate) as measured by median CPI and conventional CPI-U

Median CPI


Aug 04



Sep 04



Oct 04



Nov 04



Dec 04



Jan 05



Feb 05



Mar 05



Apr 05



May 05



Jun 05



Jul 05



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.

16 thoughts on “Measuring inflation

  1. FTX

    Ooh. Controversial topic, and I think you know it. I’m betting you’re now sitting in front of your monitor wearing a Kevlar vest and helmet.
    There are all sorts of conspiracy theories regarding the US CPI calculations, so I’ll give this Bill Gross link as a starter:
    As for the oft-cited ‘chicken for steak’ gripe, have a look at the last paragraph of this document:
    It appears to dismiss such concerns regarding the current method of CPI calculation.

  2. odograph

    Are there any independant inflation metrics that compete with the CPI? I imagine it would be a pretty huge job …
    FWIW, I’ve seen the “conspiracy” articles on the CPI and don’t really buy them, but they always leave me with the feeling that it would be a little to easy to “put one’s thumb on the scale” – if one wanted to.
    [not an economist]

  3. TCO

    How about just looking at interest rates demanded by the market as a useful proxy? I mean, if I construct a CPI with hedonic deflators that is vastly out of wack with “reality”, couldn’t one get an indication from the market that it is not correct? I know this is not foolproof since there is competition for capital and risk premiums and the like, but still one would think that grossly out of wack calculations could be shown as such and also, one could try extracting some useful signal out of interest rates.
    I mean, if inflation is “really 6%, but hedonics is making it 3%”, the markets will demand that 6% plus a risk premium. (Of course many of the hedonic kvetchers also beleive in incredibly inefficient markets…) Sigh.

  4. FTX

    “…if I construct a CPI with hedonic deflators that is vastly out of wack with “reality”, couldn’t one get an indication from the market that it is not correct?”
    I’m sure this would be the case if the computed CPI was “vastly out of whack”, but I’m not so sure the market could or would efficiently adjust for an understatement of say 1.0% because of the huge complexity and subjectivity in calculating the index. And yet 1.0% compounded over a couple of decades…
    In any case, many of those affected by any understatement (e.g. those on social benefits) would have no market in which to express their distrust. Unless you count the ballot box.
    I have no particular axe to grind regarding CPI calculations, but nevertheless I would point out that we are in an era where government ‘spin’ of official statistics for political purposes has become an accepted part of life. In the UK for example there is currently great distrust of educational exam results, which appear to show pass rates constantly improving, to the incredulity of those few who gained top grades a couple of decades ago. It wouldn’t surprise me to hear that there are similar concerns in the US.
    Since suspicion of politically inspired statistical manipulation is now manifest, and since it is very much in the interest of governments to understate inflation figures, we should not be at all surprised that distrust of official CPI data is growing.
    This, to me, is a dangerous development. You can fool all the people some of the time etc.

  5. odograph

    Should we forgive them because a low statement of inflation can lead to a self-fulfilling prophesy, and lower inflation? It amounts to a form of wage and price controls, I think.

  6. TCO

    Is the point that an “off CPI” would not be discernable by interest rates or that interest rates would actually mirror the rates

  7. FTX

    I don’t really see how we could determine from market interest rates whether the CPI numbers are ‘off’ or not. If you’re right about market efficiency, then presumably the adjustment is already accounted for in current rates, but this leaves us none the wiser.
    I wonder if Prof Hamilton knows of some way that historical CPI could be checked for its ‘accuracy’. Maybe comparing growth in nominal GDP to growth in some monetary aggregate such as M3 over a long time period ought to yield an implied inflation rate to compare against CPI?
    BTW, there was a very balanced article on hedonics/CPI in the WSJ not so long ago. It’s reproduced here:
    The article notes that computers have not been subject to hedonic adjustment since September 2003.

  8. muckdog

    I don’t watch the CPI as much as I do the jobs numbers and wage growth. To me, that’s a better indication whether or not labor tightness is going to cause a rise in wages, which would then cause inflation down the road. And, eventually show up in the CPI.
    As an investor, by the time the numbers show up in the CPI, it’s too late to adjust the portfolio. I need some heads up that the Fed might go on a rate hiking binge.

  9. Anonymous

    1. The median growth in the Personal Consumption Expenditure (PCE) Deflator has properties similar to the median CPI. Also, the median PCE inflation series tends to get revised less than the headline series. There are many reasons the PCE is preferable to the CPI for current and historical analysis, most prominently the allowance of revisions, methodological consistency, and lower substitution-effect skewness.
    2. Bryan and Cecchetti and Todd Clark and Julie Smith were all looking for a best “core” inflation, and one of the major criteria was the degree to which inflation in the “core” measure over a quarter paralleled average inflation over the following 1-12 quarters. While these numbers are informative, and useful for monetary policy (which is too cumbersome to counteract erratic price jumbps), they do lack the “power” of the headline series because they don’t have the same responsiveness to actual price changes.
    3. Wouldn’t it be nice if the BLS computed these statistics by income group?

  10. walker

    FTX, thanks for the link, I was not aware that the hedonic adjustments for computers had been discontinued. Still, hedonics is only one method employed to perform “quality adjustment” and all this means is that they have shifted to another one in the case of computers.
    The larger issue is that quality adjustment, however undertaken, is at some level a subjective exercise. The article did a good job conveying this in the two paragraphs discussing quality adjustment in autos.

  11. TCO

    FTX, I agree it’s not a magic bullet and there is likely lots of signal to noise. Still, it seems that one should think about and attempt to use market valuations in addition to fundamental analysis when possible. I guess that’s why there’s a literature on it.
    To further, the topic, one might also ask what are the effects of an “off CPI”. I can imagine that in some sense it is not catostrophic, if one makes sure that one is not having gross inflation or deflation. And in that sense one might almost think of it as academic (so the econ teachers have a tough life getting good numbers, who cares). Yet, on the other hand, there can be some ill effects: for instance:
    1. Fed misaction (but presumablu moderated by some tendancy to slow action)
    2. Misadjustment of Government benefits and salaries (but in the end salaries will be dictated by the market)
    3. Misreaction by the economy (decision-makers, corporate salaries, stock prices, etc): however, one would think that the market would eventually react to real drivers, so this may not happen so much.
    Just something to think about.

  12. anonymous-but-informed

    I am amazed at the lack of understanding of the CPI displayed in these comments. There is no political manipulation of the CPI, period. Bill Gross is quite misinformed and ignorant of the relevant theory. The PCE is full of imputations, so one can hardly claim it is superior in all respects. Hedonic adjustment is necessary, it is the only game in town.
    See the work of Bob Gordon and Leonard Nakamura to get better informed criticism of the CPI. (The Rudd note in the JEL suffers from some flaws but is, again, mostly informed.) Note also the absences in these EXPERTS’ work of the outlandish kinds of claims one gets from the New York traders.

  13. Jim

    It’s not that Bill Gross is ill-informed, it’s that he can’t help but talk his position.
    “If those Moms are holders of government bonds based upon a benign outlook for inflation, they had better cash some of them in, especially at today’s 4.0% yield for 10-year Treasuries.”
    Yeah, Bill. Get out of them and into something where PIMCO can get a cut.
    (And Bill Gross is in CA, not New York.)

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