An Approximate Divisia Index of Total Factor Productivity - Star, Spencer; Hall, Robert E. (Econometrica, 1976)
derived and discussed by these authors, with the exception of Diewert, treats time as continuous; the index is an integral over time. Empirical work has proceeded by calculating discrete annual changes, which are then used to obtain an approximation to the continuous index. Even when the object of the calculation is the estimation of total factor productivity over a very long time, annual data on output, inputs, and factor shares are used. However, detailed data are frequently not available annually. For example, much of the detailed data on labor inputs are available only from the Census of Population, which is taken every 10 years. The question studied in this paper is whether or not a reasonable approximation to the continuous Divisia index can be calculated using data from only the beginning and end of a long period of time. Our answer is favorable. We derive a suitable approximation, calculate bounds on its errors, and suggest that in the usual cases these errors are likely to be small. We then calculate the growth in total factor productivity in the American economy from 1909 to 1958 using the conventional method based on annual changes and compare it to our approximation based on data for 1909 and 1958 only. The two calculations give almost exactly the same result. We conclude that in most cases the data for intervening years are superfluous. This conclusion makes it possible to make detailed and accurate calculations of the growth in total factor productivity from one decennial census