(excerpt of "Reconstructing the Physical Continuity of Events")

  1. Growth Rate Trend of U.S. Economic Activity

The history of U.S. economic product as recorded in the Dept. Of Commerce National Income and Product Accounts provides an example of the potential for surprising new results using derivative reconstruction on familiar and previously well studied data. This same data set is an important focus of study in introductory economics Footnote1 and has been modeled extensively using conventional econometric Footnote2 and dynamic systems Footnote3 models. Yet, according to one of the more pragmatic recent overviews of the econometric methods conventionally applied, little real advance has been made (Zellner 1994) Footnote4 Throughout the literature on the subject there is a clear sense that the investigators feel there is a pattern in the data to be found and equally clear that it has remained invisible.

What continuity analysis provides, based on the assumption of physical continuity and the homeostasis, is a greatly improved direct measure of the whole system growth trend. The use of this work would not be for making quick policy or investment decisions, nor quarterly forecasts directly, but for informing theoretical analysis and pragmatic forecasting methods. Because derivative reconstruction interprets time-series as direct images of unique non-statistical behaviors that are mathematically too complex to model with equations, in direct contrast to the conventional methods Footnote5, it is understandable that interpreting these results will require some adjustment.

One of the suspicions that this analysis confirms is that the US growth rate has been declining. However, the common notion that this trend began in the early 1970's. Footnote6 appears to be an illusion resulting largely from the untreated data's visual appearance. The actual decline in growth rates appears to have been taking place steadily over a much longer period of time.

Figure 7 shows graphs of the Dept. of Commerce figures for annual U.S. GNP and GDP Footnote7 Footnote8 . The measures have been scaled in constant 1958 dollars. The GNP data includes foreign earnings of U.S. citizens and is no longer used as the principle measure of the US domestic economy due to the globalization of the economy. The more recent figures available are for Gross Domestic Product. In order to make a curve covering the whole period these two measures have been cut and spliced at 1960, a point in the middle of a period of when the two measures were nearly coincident. Some of the easily recognizable features of the graph are its general upward sweep, the major fluctuations of the 30's and 40's involving the great depression and World War II, and the staircase of recessions since the 1970's. For comparison a pure exponential (constant growth rate) curve is shown that seems to nearly match the historic trend until the seventies. This is the simple observation from which economists get the impression that the economy has had an underlying 'growth constant' which ceased operating for some reason in the 70's.

Figure 8 shows the construction of a long term trend using derivative smoothing and inflection point bridging. The first feature to note is that the trend curves thread through the fluctuations. The Long Periods Trend (3) was produced in two stages of smoothing and bridging, shown in more detail in the enlargement of the 1920 to 1960 period. The names of the operators used in the construction are listed below the graph. Derivative smoothing with two iterations (ddsm-2x2) was used first to make a more regular curve (2) within the range of error of the data to use as the basis for locating local inflection points. The Short Periods Trend (3) then results from trend bridging (tlin-1x25pts) drawing a curve through each of the inflection points in the dr smoothed data. This operator would have allowed a bridge of up to 25 years between inflection points, though in application the longest period between inflection points on this application was probably 3 or 4 years. Bridging algorithm produces straight line segments between inflection points including intermediate points corresponding to each data point in the bridge period. Continuous curvature is then restored by derivative smoothing to find the points the curve would have had to pass through if its underlying 4th derivative accelerations are minimized.

The Long Periods Trend resulted from a second stage of inflection point bridging to thread through each of the fluctuations in the smoothed Short Periods Trend, this time connecting alternating inflection points. In the enlargement, the inflection points of the Short Periods Trend used to make the Long Periods Trend are indicated by the circled dots. The uncircled dots are the inflection points that were skipped. The Long Periods Trend is smoothed again before differencing.

Using the same method to thread through time-series graphs of a swinging pendulum demonstrates the physical principles of simple gravitational motion. Connecting the inflection points its horizontal displacement (a decaying sin curve centered on the time axis), would give you a horizontal straight line coinciding with the time axis, indicating the invariance of gravity. Connecting the inflection points of a the pendulum's vertical position (a decaying sin curve bounded by its stationary height) would yield a decay curve indicating the rate of energy loss of the system.

Figure 9 shows some of the difficulty confronted by the conventional approaches to the problem of analyzing the economic data in this way. The plot of annual growth rates (2) suggests a rate of growth wildly fluctuating about a 3.3% norm without any clear trend. Even when the data is strongly smoothed using a running average (3) the growth rate trend (4) fluctuates irregularly about the norm without obvious trend. The running average calculation (dasm) used three point averaging forward and back for a total of five iterations, the first two with proportional center weighted averaging and the last three with unweighted averaging. The growth rates were calculated using the proportional difference {Special Char 68 in Font "Symbol"}y/Y plotting the ratio of the change during each period to the height of the curve. If there really was a growth constant the growth rate curve would presumably have some tendency to parallel the horizontal 3.3%/yr proportional difference curve of the exponential graph shown in figure 7.

Figure 10 shows a derivative reconstruction of the historic trend in growth rates (3) calculated from the Long Periods Trend of figure 8. To help evaluate the reliability of the reconstruction a comparison of the calculation on overlapping periods is shown in figure 11. Here the analysis was done with look back comparisons from 1930, 1960, 1980 and 1993. In all cases the reconstructed trend during the ten year period preceding of the time of calculation diverges somewhat from later calculations. In one case (1960) the direction of the trend in the end period is significantly different from that calculated with the benefit of future data. These effects display the influences of local trends near the end of the analysis period and the degree of error that can be expected if this method were used for extrapolating future trends. Given this qualification, what results is a rather clear pattern continuing to the present. It is a pattern that, while not suggested by the direct and average rate calculations in figure 9, is completely consistent and clearly visible once you look for it. The predominant trend throughout the recorded history of economic growth has been growth rate decline.

No attempt is going to be made to explain why the economy behaved in the way this analysis suggests, but a little further discussion seems in order. Of particular note is that there appears to be no indication whatever of there being a 'growth constant', but rather a series of relatively stable trends. The current period of declining growth rates is shown beginning around 1960 , has been steady and shows no indication of turning. The current trend of decline might be attributed to a drag on productivity growth due to the modern burdens of crowding, complexity, resource scarcities and conflicting environmental impacts, but these conditions did not exist in the previous long period of growth rate decline. Some other source of drag was apparently operating at that time and the current source of drag may not be what we think. One way to identify a common source of drag for the two periods of decline would be to look for other measures with matching trend inflection points.

The two long periods of growth rate decline are interrupted by a period of increasing growth rates shown as spanning from 1920 to 1960. The period of increasing growth rates roughly corresponds to the development of modern heavy industry and the general integration of the sciences, engineering and education with production, along with the compelling and disruptive events of the great depression and World War II. It was also the period when government first took an active and comprehensive interest in economic affairs.

Figure 12 shows a contrasting image of the structure of the growth trend resulting from using a slightly different sequence of derivative smoothing and bridging steps (series B) as compared to the one presented in figure 10 (series A). Here the reconstruction of the growth rate (3), was made without smoothing the raw data prior to the first stage of bridging between inflection points. Otherwise the constructions were the same.

This new image of the long term trend is of interest partly because it displays the possible large scale effect of very subtle differences in the data and analysis procedure. It is also of interest because the new image lends itself to a more event driven view of history than an evolutionary one. The period of increase in the growth rate closely coincides with the period of the great depression and World War II, and seemingly little else.

The possibility that the growth rate rebounded during the 30's and 40's in conjunction with the century's two most disruptive events, and that our current trend of growth rate decline started immediately afterward in 1945, is intriguing. It would support the 'accumulating rigidity & creative destruction' model of economic cycles on a large scale. It might also portend a destruction of the rigidities in our present world order as the culmination of our current long trend of growth rate decline. There are certainly other possibilities. The natural world is full of self-organized systems of all kinds that exhibit long trends of growth and growth rate decline, and which do not become unstable. What would seem suggested, though, is a comparative study of systems that do and do not become unstable in the absence of growth, to see what makes them different.

It is possible that either image would stand out as dominant on further study, though it seems more likely that both were operating simultaneously and are each highlighted by slightly different adjustments of the lense through which the data is being viewed. The evolutionary view of the data was presented first based on the likelihood that initial derivative smoothing more precisely located the true inflection points in the raw data, making it more literally accurate. It was also presented first because event driven hypotheses tend to distract attention from the complex contexts of events that actually seem to dominate most processes of self-organization in natural systems.

More germane to this technical discussion is that these contrasting images point out the mathematical sensitivity of the tools used to what you might call focusing. The construction of time-series derivatives and the location of inflection points is useful because it distills and magnifies small consistent differences. Though the present method is disciplined and stable, focusing an image is still partly a matter of judgment and circumstance. The most frequently confronted type of problem came from small fluctuations in the data at the peak or trough of larger fluctuations. Because the fluctuations are what are used to locate the trend, fluctuations far from the trend can confuse the results. This was confronted in the economic data series during the 30's and 40's where the data contains large irregular disturbances far from the trend. In this case the problem was largely overcome by bridging the fluctuations in two stages.

The major difference between the results presented in figures 10 and 12 seems to largely rest as much on the reduction in the number of inflection points resulting from initial smoothing of the data ance between the results presented in figures 10 and 12 seems to largely rest as much on the reduction in the number of inflection points resulting from initial smoothing of the data as on changing their precise locations. A change in the number of inflection points in one period can unpredictably alter the sequence of inflection point selections in other periods when, as in the presented analysis alternating inflection points are connected for the second stage of fluctuation bridging.

The final picture of the economic data to be offered, figure 13, is of the simple first and second differences dy/dt and d2y/dt2 of the long periods trend curve. Reading linear time derivatives is significantly different than reading proportional rate derivatives. Though time derivatives are more true to absolute scale the shapes of the curves also change with the scale of the subject. Curve group (A) is based on the dr interpolation of figure 10 and curve group (B), shown in the background, is based on the dr interpolation of figure 12.

One of the interesting features displayed is the peak of the second derivative in 1960, marking the center of a long period of increasing and decreasing absolute acceleration in economic production. This turning point is similar to the 2nd derivative turning point at the middle of a shorter period of increasing and decreasing absolute acceleration around 1890 (as shown in the enlargement). Though the earlier event looks different in context due to the changed scale of the economy since then it appears similar in scale to the economy of its time. Another interesting feature is a fairly subtle one, that the second derivative since 1960 has been steadily decaying toward zero. Even considering only the most reliable portion of the construction, ignoring the most recent 10 years, there is a curious strong appearance that expansion of the U.S. economy has been steadily approaching a linear rate of expansion.


(Samuelson & Nordhaus 1985) Ch 36 Economic Growth Theory and Evidence pp793 Referring to graphs of the national product accounts, "Figure 6-3 is important. Linger over it.".


see (Kmenta 1986)


see (Winter 1994)(Meadows et al 1972)(Meadows,, Meadows & Randers 1992)


(Zellner 1994) p218 "Most 'one shot' attempts to model this variable.(GNP).have failed...many large-scale model forecasts (are) not as good as those of very simple univariate naive models."


(Kmenta 1986), pp 207 "In econometrics we deal exclusively with stochastic relations"

pp 203 "In fact the entire body of economic theory can be regarded as a collection of relations among variables..(as defined)..by a given equation"


(Samuelson & Nordhaus 1985) pp 799 "the rapid growth in output per worker came to an abrupt halt in 1973" "after growing at 2 1/2% annually from 1948 to 1973, labor productivity grew at the much slower pace of 1/2% annually from 1973 to 1984"


National Product and Income Accounts

Bureau of Economic Analysis, U.S. Dept. of Commerce ;

NIPA 1869-1970 Gross National Product in 1958 Dollars Series F 1-5 , with the initial figure for 1869-79 treated as an average centered on in 1874

NIPA 1929-1982 Gross National Product in 1982 Dollars Table 1.2 (adjust 1958$)

NIPA 1929-1988 Gross Domestic Product in 1987$ (adjusted to 1958$)


World Almanac 1994, Bureau of Economic Analyses, U.S. Dept. of Commerce

NIPA 1990-1993 Gross Domestic Product p104 (adjusted to 1958 $)