Friday, May 8, 2009

Societal Wealth and the "Freedom Fraction:" A Brief Statistical Study of the Relationship Between per-Capita GDP and Libertarian Economic Attitudes

Introduction Every year, The Heritage Foundation publishes the Index of Economic Freedom, a ranking of all the nations of the world on the basis of economic freedom. The ranking system is fairly complex, looking at everything from monetary policy and inflation, to government expenditure as a fraction of GDP, to ease of opening a business, and everything in between. There are ten main scoring categories, each with its own subdivisions and complex evaluation systems and rules. Though some of the evaluation systems become perhaps overly sophisticated, dredging up controversies of interpretation a bit unnecessarily, nevertheless, it provides a comprehensive look at virtually every national economy across the globe, and its authors probably deserve some leeway in their methodology as they have chosen quite a formidable task. Invariably, attempts are made at statistical correlations between the freedom ratings and societal wealth, which most agree is best measured by per-capita GDP. Undoubtedly, the trend is there: the top ten most-free nations are always some of the wealthier nations of the world, and the usual suspects are always lurking at the bottom of the curve: North Korea, Burma, Cuba, and an assortment of African economic basketcases. But from a quantitative point of view, the data provide a less-than-straightforward interpretation, as a graph of the 2009 data clearly shows: Speaking as a natural scientist, this scatterplot leaves much to be desired if the notion that societal wealth is a function of economic libertarianism is to be taken more seriously than some airy, idealistic notion. A true “natural law” ought to leave a mathematically visible fingerprint, and the haphazard spray of data points generated by this survey leaves enough wiggle room for considerable doubt, especially when there is not much suggestion of a specific, theoretically reasonable, mathematical model that might be tested for fit against the data. After all, a lot of scenarios could lead to such a graph which might have little to nothing to do with economic freedom per se. On the whole, it leaves the reader not already predisposed to the libertarian point of view asking the question: is there really something to the notion that freedom makes societies wealthy, or is it more an exercise in relatively stable, well managed democratic societies that only tend to be more free wagging their fingers at geopolitical bad-boys, who have managed to land themselves in poverty simply as a result of their generally bad behavior? Am I seeing a real effect, or am I being fed some ideologically driven line? Is there really an important natural law lurking somewhere in that data, or are some overly enthusiastic libertarians straining just a little too hard to find a pattern that is just not there? The question is an important one, and without a concrete theory capable of withstanding quantitative scrutiny, the case is less than watertight and graphs like this one tend to leave a lingering shadow of doubt hanging over the libertarian's case. In this brief study, I present a statistical manipulation that may help shed some light on the matter and suggests that a somewhat different interpretation may be more appropriate, with some surprising implications. Data Normalization and the Threshold Model The writer at La Griffe du Lion has come up with a powerful method for comparison of normal distributions and teasing out mathematical relationships based on variances in these distributions, which he applies in an effort to explain a variety of social phenomena, some of them quite politically controversial. Because the normal distribution is so ubiquitous in nature, the method finds useful application across a wide range of questions. In particular, La Griffe himself addresses the relationship between IQ and per-capita GDP, finding that there is a very strong statistical case to be made that the IQ distribution of a population is the main determinant of per-capita GDP. Even more impressive, he generates a mathematical equation to describe the relationship that closely fits the observed data, which he terms the “Smart Fraction Theory.” In summary, he postulates that per-capita GDP is directly proportional to the fraction of the population which is above a certain IQ threshold. He then develops the equations necessary to describe such a phenomenon and shows that the data closely fit the predicted pattern. A more detailed description of the method and its derivation can be found here. Could something similar be going on with the data compiled by the Index of Economic Freedom? It may be that the correlation the economists are looking for isn’t necessarily proportional to the freedom score itself, but is convoluted by a similar effect of population heterogeneity. Could a similar “liberty thresholding” of societies explain the bizarre data pattern seen here? Treatment of Data Unfortunately, the data presented by the Index of Economic Freedom lack standardized, normalized descriptions of the societies they describe, or a scale similar to the IQ scale for measuring human intelligence which has a statistically well characterized distribution. Fortunately, the second hurdle can be overcome by constructing an ad-hoc scale based on the distributions of the various nations. The first must be dealt with by making a few approximations and assumptions, which is clearly suboptimal, but the assumptions are not without substantial grounding in reality. Construction of the normalized scoring scale. Looking at a histogram of national scores, we can see that the dataset produces a roughly normal distribution, skewed slightly to the right, with an average score of 60.1 and a standard deviation of 10.4. Data for a few nations were thrown out, due to either a lack of data or to manipulations in getting the per-capita GDP data to line up with scoring data. This is not expected to substantially affect results, as the deletions were not systematic and the remaining data are more than sufficient to accept or reject a pattern. Subtracting the average score from each data point and dividing by the standard deviation, we arrive at a normalized distribution: which is obviously the same graph plotted on a different scale of axes. The x-axis is plotted in units of standard deviation (SD). This transformation makes later manipulations simpler. Likewise, plotting our per-capita GDP data on the new scale gives us the same graph using units of standard deviation: Approximating Individual Populations. Using the global distribution of freedom scores, we now have a normalized scale for quantifying proclivity towards economic freedom. Unfortunately, we have no direct measurements of individual populations. However, we do have the individual nation scores, which provide some insight. If we accept that government policies are a reflection of the beliefs and attitudes of their respective populations, as various and sundry worldly political machinations do their work bringing about at least the minimal necessary harmony such that the masses are quiescent enough that the powerful get to remain so, we can make the approximation that the individual national scores provide a rough measure of the likely median score of the respective nations’ populations. On the other hand, we have virtually no knowledge of the distribution of individual attitudes within nations and their respective scoring distributions. Here, we must make a larger statistical leap. I will use the approximation that the individual distributions have the same standard deviation as the global standard deviation. This is not entirely unreasonable. The global distribution is the aggregate of these same attitudes, so it stands to reason that subpopulations should not have distributions radically different from the aggregate, otherwise they would perterb the graph. The greater is the aggregate of the lesser, and thus should reflect its properties, so the two distributions should be within the same order of magnitude. Furthermore, excessively broad divergence in attitudes within nations would be expected to lead to civil strife, dividing said nation into smaller subunits with distributions more amenable to social cohesion. On the other hand, it cannot be handily rejected that population distributions might be somewhat narrower than the global distribution. But in the absence of such information, I will stick with the original approximation, keeping in mind that both the median and standard deviation approximations are rough and adjust our expectations of the precision of this study to reflect this. Determining the Threshold and Calculating the Approximation Curve The equation we will be fitting to the data takes the form
Per-Capita GDP = cf + d
where c is a proportionality constant (measured in dollars), f is the “freedom fraction,” the fraction of the population with freedom scores that exceed a specified threshold, and d is another constant representing baseline GDP in the absence of any “freedom fraction” at all, expected to be close to zero. Unfortunately, we do not know that threshold a priori, so we will try a number of thresholds to find best fit to the data. Using the Excel worksheetfunction NORMDIST calculates the integral of a normal distribution below a threshold expressed in standard deviations. The “freedom fraction” would therefore be the remainder of the curve, the region above the threshold, or the quantity returned by the NORMDIST calculation subtracted from one. The equation looks like this:
freedom fraction = 1-NORMDIST(threshold, median population score, SD of population, TRUE)
since we have normalized the national scores already, threshold and median scores are expressed in units of standard deviation (SD), and the standard deviation is by definition one, since the scores are normalized. The TRUE argument ensures that the function returns the integral, not the center of mass, and is just an artifact of Excel. Plotting “freedom fractions” vs. per-capita GDP with a range of thresholds generates the following series of graphs: Plotting per-capita GDP versus freedom fraction appears to linearize the scatterplot quite nicely, particularly for threshold values in the range of 1.0, giving an R2 vale of .54. While the data is still clearly scattered, the fact that it has been linearized suggests that our proposed equation, GDP = cf + d is a reasonably good mathematical approximation of a curve that would describe the pattern of the points in the original scatterplot. Using the linear equation of the trendline generated by Excel, we can extract our two constants c and d for the original equation, which now reads:
Per-Capita GDP = $42751*f + $2201.3
Finally, if we plot our newly created model equation against the normalized scatterplot, or the original scatterplot if the SD units are converted back to the original scoring system, we obtain the following graph: which, given a few outliers and some spread in the data due to the fact that we are dealing with relatively complex systems, appears to describe the data rather nicely. Results and Discussion The fit of the data to the model tends to support the contention that societal wealth is a function of the degree of embrace of a libertarian economic ethic by a society, at least to a rough first approximation. It is entirely possible that a more sophisticated model which attributes differential contributions of various slices of the “freedom curve” within a population might give a better fit, for example, by steepening the slope of the middle part of the curve and creating a sharper transition between the flat upper and lower regions with the sloping central region, but in any case, in the absence of better, more direct measurements, such an attempt would be even more speculative than this exercise. There are several important implications if this model is correct. The first, and arguably most important, is that societal wealth is not a function of governing economic policy per se. Rather, it appears that both economic policy and societal wealth are functions of the beliefs and attitudes of the population which makes up the body of the society, as these were the explicit assumptions made in the generation of the model. The distinction is important. It implies that top-down policymaking by governing bodies is far less important to improving the economic lot of a society than the actual embrace of libertarian principles by said society at large. Just as “societal wealth” is a measurement of aggregated individual human action, it appears that individual human belief and action at the lowest levels, not the highest, are responsible both for setting the prevailing ethic of the marketplace, and the resulting potential for expanding the productive capabilities and motivations experienced by individual economic actors. In short, human action looks to be the relevant factor here, not government policy. Another surprising set of implications, at least to this author when he decided to take up the subject, is that increasing wealth through embrace of economic liberty is bounded, as the flat, upper region of the curve implies, and that a fair number of real human societies actually appear to have reached the cusp of this plateau, as clearly visible by the data points at the far right of the curve. The threshold which was found to best fit the available data corresponds to a freedom score of only 70.5, which is to say, that individuals whose attitudes reflect this degree of economic freedom or better are part of the “freedom fraction” and contribute to per-capita GDP improvement at the higher level, while those below this threshold correspond to the baseline. This score corresponds approximately to the scores received by Spain, Sweden, Germany, Cyprus, and Norway, not despotic regimes by any means, but certainly not freewheeling capitalist societies either. The rightward skew of the national score histogram, which we noted earlier, coupled with this low threshold results in a few top scoring societies nearly “saturating” the effect of liberty on per-capita GDP, implying, for example, that over 99% of members of the top-scoring society, Hong Kong, would be above this threshold, though their failure to achieve the per-capita GDP predicted by their score would tend to suggest that there is a disparity between government policy and real public attitudes, assuming the model is correct. Is this the case? It is certainly a testable hypothesis. If the model is correct, it implies that the standard for maximization of human productivity is actually not all that high, at least with regard to economic liberty, as a few nations have actually come very near to achieving it. Though it might be more satisfying to libertarians if the upper end of the curve were boundless, continuing on into infinity, this is probably not reasonable. If not bounded by restrictions on liberty, it stands to reason that a man’s productivity is bounded by something, whether present-day technology, or the near satisfaction of desires at the cost of an additional increment of his time and effort, or simple, overwhelming complexity as the division of labor lengthens indefinitely. It is certainly a hopeful finding that it is within human capacity to achieve the minimal libertarian ethic that allows maximization of human potential, at least in one regard. This model produces a variety of testable hypotheses, as well as suggests further studies. It would be very interesting to determine the actual individual “freedom score” distributions of national populations to test the several assumptions of this study and further validate or refute the model. Perhaps this could be done through polling with a well designed questionnaire. It would also be interesting to combine this model with others, particularly doing a multivariate analysis to determine if the deterministic effects seen in this study can account for the “deviations” seen in another, particularly La Griffe’s IQ study, or vice versa. It could be that only a few determinants can effectively predict the observed economic distributions, if the relatively good fit found here and in his study are any indication. Conclusion A theoretical, mathematical model relating per-capita GDP to economic liberty which appears to have predictive value and real world implications is generated by relating the fraction of a population sympathetic to economic liberty, described by a simple normal distribution, with a proportionality constant. The success of this model suggests that increases in GDP are bounded in the limit of near universal individual acceptance of a certain minimal threshold of economic liberty, that the attitudes of individual economic actors towards economic liberty are more important than governing policy, and that several real-world societies actually approach the maximization of productive potential with respect to this variable. This simple approach appears to be a useful model for further investigations. ----Update!---- In case you are interested.... (click on the graph to enlarge)


  1. Scott,

    Great to see the application of solid,public data to back theory. Always needed for solid argumentation. Great work.

    Scott Smith