Measurement and Interpretation of Productivity (1979) / Chapter Skim
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Measurement Error in Productivity Statistics
Measurement Error in Productivity Statistics
Pages 221-238
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From page 221... ...
In discussing a general analytical approach, however, our aim is to delineate the problem, to illustrate by example the way in which the analysis can be carried out, and thus set the stage for further research into these issues. MEASUREMENT ERROR DEFINED The use of the term "measurement error" is explained by the following: A productivity measure that is produced by the BUS is a function of a set of quantities that can, in principle, be empirically determined.
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From page 222... ...
In the present setting, the standard deviation of the distribution of ei would determine the margin of error in ei and hence the margin of error in the observed quantity, xi. In our analysis we shall assume that the distribution of ei does not depend on the magnitude of Zi.
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From page 223... ...
2 In principle, the probability distribution of E can be derived from the probability distribution for the errors ei. 2 The symbol E used for overall relative error here should not be confused with the expectation operator for random variables.
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From page 224... ...
Method 3, computer simulation, as is well known, is never a totally satisfactory substitute for the exact derivation of a distribution it is difficult to manage and interpret without a thorough scanning of the full ranges of the parameters of the components of the model. Because exact derivation is complicated in this case, simulation is useful for studying the effects of various assumptions concerning the component errors ei on the overall error E
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From page 225... ...
If one believed that this bias were the principal source of error from the CES source, then the standard deviation for the error would be specified to be very small. Case H of the computer simulation reported below illustrates this point.
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From page 226... ...
o · so in is .
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From page 227... ...
indirect business taxes, etc., and (5) capital consumption allowances, with amounts and weights for 1976 shown in Table 1.4 The five categories for the income side, plus farm business, enable us to express the measured numerator of annual productivity for the private business sector as XN = ZN + ZN (W,e, + W2e2 + Whet + W4e4 + WseS + Wield, (3)
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From page 228... ...
Weight Data Source Current employment 122,093 We = 0.92 CES program statistics: All employees covered by unemploy ment insurance Farm employees; farm proprietors; farm un paid family workers; nonfarm proprietors; nonfarm unpaid family workers; private house holds 20,743 Wg = 0.16 cPs Government enterprise 3,156Woo = 0.02 Less not-for-profit institu- 13,425W'' = - 0.10 tions TOTAL 132.5671 .00 Estimates from BEA data on employment and cPs average weekly hours followed by calculation of the implicit deflator for gross product originating is described in Chapter 4. The combination of price relatives for many individual products and materials and services used in production, weighting information from sEA input-output tables, and various other data that are required suggests a complicated error structure and resulting uncertainty about the true magnitude of the deflator.
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From page 229... ...
The simulation estimates of the standard deviations (i.e., values denoted below as s) appear to be accurate to between two and three decimal places, but we have not made a detailed analysis of their accuracy.
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From page 230... ...
In the general case of independence and constant error variance the standard deviation of the overall error is about 1.5 times the common value for component errors. CASE B Conditions for case B are the same as those for case A except that the correlation between en, employee compensation, and en, CES hours, is set at 0.99.
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From page 231... ...
Comment In this model, with the standard deviations for the deflator and the major components of both numerator and denominator all near zero, the standard deviation of total error is considerably reduced. Prom cases A, C, D, and E we may conclude that reduction of uncertainty is required in both the implicit deflator and the major components before reduction of uncertainty in the overall productivity measure can occur.
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From page 232... ...
CASE H For case H the common standard deviation is set at 0.01. The same intercorrelations are used here as in case G
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From page 233... ...
1 6 2 2 6 2 4 · 6 · 4 5 1 2 5 1 9 · 1 2 · 1 NOTE: Frequencies Over 9 Indicated by X X X -3 -2 -1 0 1 2 3 FIGURE 2 Normal probability plot of 200 observations of overall error.
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From page 234... ...
It follows from standard theory for the multivariate normal distribution that the U V, and Y are jointly normally distributed with mean vector t/ v ~ = Lm and covariance matrix (11)
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From page 235... ...
(13) The Taylor expansion method approximates the distribution of E by a distribution with mean value given by E(u.
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From page 236... ...
While this is a more complicated case than those we have considered, it may, in principle, be studied by the same techn~ques. For example, if the productivity measures for successive time periods are unbiased and the relative errors are uncorrelated, and have the same standard deviations, then it can be shown that the error in the percentage change (i.e., the deviation of the measured change from the true change)
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From page 237... ...
We see from these two previous studies that, at least, the idea of persuading experts to make subjective evaluations of errors in reported statistics is not foreign to the agencies involved. It is our recommendation that the Bureau of Labor Statistics and the Bureau of Economic Analysis explore further the possibility of using techniques similar to the technique in our simplified example in order to estimate the overall margin of error in productivity measurement.
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From page 238... ...
(1974) Reliability of the quarterly national income and product accounts of the U.S., 1947-71.
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