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Scaling by a constant c multiplies the original powerlaw relation by the constant ck. All phenomena that scale have a particular scaling exponent and are equivalent to constant factors, since each is simply a scaled version of the others. This behavior produces a linear relationship when logarithms are taken of both f(x) and x, and the straight line on the log-log plot is often called the "signature" of a power law. With real data, such straightness is a necessary but not sufficient condition for the data to follow a scaling relationship (see Stumpf and Porter [2012] for the necessary and sufficient conditions). In this article, we use Stumpf and Porter's guidelines as well as others' to confirm that our results do (or do not) exhibit scaling.
The possible existence of scaling does not conflict with the existence of market-cap breakpoints. If the breakpoints do exist, there should be a different scaling exponent for each capitalization group. We say this because within each capitalization group one would expect the relationships between the "activity" (such as bid/ask spread) and the stocks to be approximately the same. If they are approximately the same, this generates a scaling exponent that covers most if not all the stock-"activity" relationship vis-à-vis the market-capitalization range. Conversely, the between-group relationship of "activity" and capitalization should differ, e.g., the scaling exponent for the "activity" and midcap stocks should be different than that of small-caps and the same "activity." If this is not the case, then the "activity" is not something that can distinguish small-caps from midcaps.
To make this a little clearer, it is generally accepted that daily stock returns do not follow a normal distribution. It has been established by Bouchaud and Potters [2000] as well as others that stock return series typically follow two if not three distributions, each with its own scaling exponent. To take an example, "extreme" negative returns typically follow a power law whose scaling exponent is approximately 1.5. Extreme positive returns have a slightly larger scaling exponent (closer to 2), and the remainder of the return series follow a power law with a scaling exponent of approximately 2. So, here we have an example of three different scaling exponents in the same time series. We look for tripartite scaling exponents in our market-capitalization breakpoint tests, since they indicate the breakpoints are validated by the "activity" data.
Our tests take the form of plotting the logarithm of the market cap against the logarithm of an "activity," such as the bid/ask spread, that is thought to be related to market cap. A single scaling value (a single line through the data) indicates the "activity" is not confirming the market-cap breakpoints. More than one scaling value could confirm the breakpoints, and we will look at those situations where this occurs. We follow the guidelines established by Stumpf and Porter by using four or five orders of magnitude in our calculations (this can correspond to market capitalizations going from tens of millions or hundreds of millions of U.S. dollars to hundreds of billions of U.S. dollars, based upon the market). We also follow Zumbach in the calculation of the linear fits we make of the data so we can assess the goodness of fit.
We show in the next section that for most of the countries we examined, there is in most cases a single scaling exponent across market capitalizations, not multiple ones. This lack of multiple exponents argues against empirical evidence for market-cap breakpoints, and this appears to be true regardless of the measures ("activities") used, whether in this or in other articles. (As an aside, it should be noted that single exponents—but not multiple exponents— do not exist at different time scales either, as can be seen in the work of Zumbach and Li et al.)
The "activities" we examine are the bid/ask spread, volume (number of shares traded) and the price per share times the volume traded (often referred to as "dollar volume traded" but, since we are working with several different countries, this activity could be "pounds sterling volume traded" or "ringgit volume traded"). As Kertesz and Eisler and others do, we look at the logarithm of the "activity" versus the logarithm of market capitalization. We plot individual data points, since our averaging of the stock data covers a single month versus years. We look at the period from January 2000-February 2012 on a monthby- month basis, and we try to fit a straight line through the data, following the method of Zumbach.3
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