The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
IF ®u@ll ®cd1 0 June 3, 1977 Statistical Mal practice The Wall Street Journal does it. The San Francisco Examiner does it. The Los Angeles Times does it. And so do the Christian Science Monitor, the Atlanta Constitution and the Philadelphia Inquirer, for that matter. But what they do is incorrect and should not go unnoticed. These and probably many other publications incorrectly calculate the annual rates of change in the various economic indicators that government agencies report. Their mistake is not in converting these monthly or quarterly figures to equivalent annual rates, though some economists question the value of paying close attention to what happens in short time periods. Indeed, the public tends to think in terms of annual rates of economic growth and inflation, so journalists rightfully render the monthly (or quarterly) figures more meaningful by presenting them in an annual context. The only problem is that they usually do it incorrectly. Ruie of thumb The problem lies in the use of a simple rule of thumb, which per'forms quite nicely when dealing with small numbers but not when dealing with large numbers. Unfortunately, the inflation rate-to take the obvious example-is not small today. The rule of thumb involves multiplying monthly rates of change by 12 (or quarterly rates by 4) to get the equivalent annual rates of change. The consumer-price index, for example, rose by 0.8 percent (seasonally adjusted) in April. Well, 0.8 times 12 is 9.6, so many newspapers reported that consumer prices had risen at an annual rate of 9.6 percent. This is not too far from the correct figure of 10.0, but it is off and the error increases with the magnitude of the monthly price change. Thus, when the wholesale price index increased by 1.1 percent last month, this was equivalent to an annual rate of 14.0 percent, not 13.2 percent as reported by some members of the press. Altogether, the rule of thumb applied to the consumer price index would have resulted in at least a smaIi error in 20 of the past 23 months. Compounding-the key Converting a monthly rate to an annual rate involves finding the annual rate of change that would occur if the index rose at that same monthly rate for 12 months straight. The rule of thumb goes wrong because it does not capture the compounding effect. It ignores the fact that the base to which that constant monthly percentage is applied grows somewhat each month. Thus, the annual rate is always larger than the rule-of-thumb rate, although this difference may be less than a tenth of a percentage point when (continued on page 2) ID®}p>@rrttmru®[ffilt IF I8)@mllk §@I TIl IF Til If(illI Opinions expressed in this newsletter do not necessarilv reflect the views of the management of the Federal ReServe Bank of San Francisco, nor of. the Board of Governors of the Federai Reserve System. the monthly rate is relatively low. But compounding is the only way to go, and without it one has only a rough approximation. Frequently, the rule-of-thumb conversion will do the job. There is no error to a tenth of a percent (which is all we usually worry about) when the monthly rate is less than 0.3 and the quarterly rate is less than 1.0. When inflation rates were below 3 percent in the 1950's and early 1960's,the rule of thumb was usually a pretty safe bet. But ever since severe inflation reared its ugly head, that rough approximation has consistently understated the situation. Come the revolution when inflation and other evils are banished from the land, the ru Ie of thumb can be resurrected. In the meantime, let us go forth and compound. Deflating statistics A second common error involves the calculation of real" or deflated values of such economic phenomena as wages or GNP. During last year's California cannery strike, for example, a union leader claimed that the real wages of cannery workers had fallen by 13 percent over the preceding three years. This occurred, he said, because while the average cannery worker's wage increased by only 22.4 percent, the CPI rose by 35.3 percent. The rule of thumb, in this case, involves simply subtracting the percentage change in prices from the percentage change in wages. While any decline in real wages is something to complain about, this particular calculation is off by several percentage points. The purchasing power or real wages of cannery workers actually declined by 9.5 percent, not 13.0 percent. With this rule of thumb, the roughnessof the approximation increases with both the difference between the percentage changes in wages and prices and the general level of the percentage changes. Latin American examples might be instructive, for those are countries where changes in price and wage levels historically have been quite large. For example, in a country where wages increased at 150 percent and prices at 145 percent, the rule of thumb would yield a rise in real wages of 5.0 percent. But actually, there would be only a 2.4percent rise'in real wages-less than half the rule-of-thumb estimate. So the error rises with the general level of changes being examined. (I 2 If the difference between the wage and price changes is also large, the rule of thumb provides estimates that are essentially worthless. Suppose the' wage gain in our Latin country remained at 150 percent but the price rise was, by wise use of monetary policy, restricted to only 100 percent. A 50 percent gain for workers? No. The gain in real wages is just 25 percent. Nature of bias The bias is always in the same direction. The rule of thumb always overstates the change in real wages. When wages are rising more rapidly than prices, as is generally the case, the rule of thumb makes things appear rosier for workers than they really are. However, when wages happen to be rising less rapidly than prices, as has been the case during much of the 1970's,the rule of thumb overstates the fall in real wages. The difference between the rule of thumb estimate and the actual change can be seen by analyzing a simple formula for the actual change. If the percentage change in wages is wand the percentage change in prices is p, then the percentage change in real wages is 100 times w-p . The rule of thumb 100+p shortcuts the calculation by eliminating the denominator. As the rate of inflation, p, increases, the two calculations diverge. When is the rule of thumb accurate, at least to a tenth of a percent? It's difficult to say, because this depends upon both the level and the relationship between the two percentage changes. Whenever the figures involved exceed 7 percent, or whenever the difference between the two percentage changes exceeds 6 percentage points, the rule of thumb will generate an error. But sometimes errors can result even from smaller numbers. The conservative approach for the amateur statistician is always to use the formula given above, or some variant. At the least, any use of a rule-of-thumb calculation should always be accompanied with a qualification. But when rates of change are of the double (or greater) digit variety, even these qualifications may not be enough. What all this boils down to, unfortunately, is a new rule of thumb, to wit: " Whenever economic rates of change are large, use with caution the trusty old rules of thumb." Michael Gorham NEW PUBLI CATI ON A supplement to the Spring 1977 issue of the Economic Review entitled "The Monetarist Controversy" is presently being distributed to subscribers to the Economic Review, and is available free to readers of this publication by contacting the Public Information Section, Federal Reserve Bank of San Francisco, P.O. Box 7702,San Francisco 94120, phone (415)544-2184. " The Monetarist Controversy)) contains a paper presented by Professor Franco Modigliani at the Federal Reserve Bank of San Francisco's January 1977 Economic Seminar with discussion by Professor Milton Friedman. 3 !! BMBH • • 4Bln • B! UJOHlB:) • • Bpei\8N BUOZPV • "me::> IOJspue.l:l ues (;SL: 'ON .!.IWlBd al Vd 's'n lUVW SSV1::> BANKINGDATA-TWELFTH FEDERAL RESERVE STRICT DI (Dollar amounts in millions) Seleded Assetsand Liabilities large Commell'dallBanks Amount Outstanding Change from 5/18/77 5/11/77 Loans(gross,adjusted) and investments* Loans(gross,adjusted)-total Security loans Commercial and industrial Real estate Consumer instalment U.S.Treasurysecurities Other securities Deposits (lesscashitems)-total* Demand deposits (adjusted) U.S.Government deposits Time deposits-total* Statesand political subdivisions Savingsdeposits Other time deposits:j: Largenegotiable CD's 96,217 74,334 1,962 23,866 22,935 12,938 8,520 13,363 94,247 26,255 425 65,777 5,857 31,929 26,188 9,322 - Weeldy Averages of Daily Figures Weekended Member Bank ReservePosition Excess Reserves(+)/Deficiency (-) Borrowings Net free(+)/Net borrowed (-) Federal Funds-Seven LargeBanks Interbank Federalfund transactions Net purchases(+)/Net sales(-) Transactionswith U.S.security dealers Net loans (+)/Net borrowings (-) + + - + + - - + - + - + + 5/18/77 17 498 136 99 79 42 254 261 67 382 52 6 22 124 95 107 . Change from year ago Dollar Percent + 9.77 + 8,567 + 12.51 + 8,266 .+ 610 + 45.12 + 1,570 + 7.04 + 14.72 + 2,942 + 17.10 + 1,889 755 - 8.14 + 8.58 + 1,056 + 8.42 + 7,320 + 11.34 + 2,674 - 33.49 - 214 + 7.29 + 4,467 - 837 - 12.50 + 21.69 + 5,691 54 - 0.21 - 1,705 - 15.46 - Weekended 5/11177 27 4 31 25 8 33 + 117 216 + 248 + Comparable year-ago period + 214 + 47 16 31 244 + 153 *Includes items not shown separately.:j:lndividuals, partnerships and corporations. lEdito:rial comments may be addressedto the editor (William Burke) or to the author .••• Information on this and other publications can be obtained by calling or writing the Public Information Section, federal Reserve Bank of San Francisco, P.O. Box 7702, San francisco 94120. Phone (415)544-2184. .04 BPI B)ISBIV