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Figures 7-9 show the results of all the countries examined (U.S., Great Britain, Japan, Switzerland, Malaysia, Korea, Hong Kong, India, Brazil, Israel, Italy and South Africa). However, plots for only the U.S. and Malaysia are shown in Figures 1-6. The U.S. can serve as a reasonable proxy for most of the countries on the list, while Malaysia can stand in for the few countries where there are either two or more single scaling exponents or no scaling exponent at all.
Scaling Exponents For Market-Caps And Various 'Activities' In Different Countries
In this article's tests, daily data is used; the stocks are those in the Thomson Reuters Country Stock Indexes. Most if not all of these indexes include some micro-cap stocks. The inclusion or exclusion of micro-cap stocks does not add or subtract from our breakpoint examination, since we are looking for small-, mid- and large-cap breakpoints.
In Figures 1 and 2, it is clear that volume and dollar volume traded have a single scaling exponent when plotted against market cap. The lack of multiple exponents means anyone pointing to either of these variables as an empirical justification for the existence of breakpoints in the U.S. would be mistaken. In Figure 3, there is a second scaling exponent, appearing at approximately $103 million, i.e., $1 billion. Given its size, this is a breakpoint that signifies either the existence of a small-cap floor or a micro-cap ceiling. Either way, this cannot be done in terms of separating small-, mid- and large-caps, as there is a single scaling exponent.
In Figure 4, there is evidence of two scaling exponents. One exponent's market-cap starts between 1 million ringgits and 1.5 million ringgits and ends between 2.5 million ringgits and 3 million ringgits. The second exponent's market cap starts at approximately 3 million ringgits and covers the rest of the market-cap range. There is no evidence of three scaling exponents but some evidence of two. And, as can been seen graphically, the spread of points about the fits is wide, suggesting the data may not follow a power law. We note this poor-to-middling fit in Figure 7 and discuss its implications later.
The same issues that appeared in the log-volume plot appear here for the log-spread plot. There is the appearance of two scaling exponents but also clear evidence of a poor-to-middling fit. Please note that the end of one scaling exponent (and the start of the other) is between 2.5 million ringgits and 3 million ringgits, just as occurred in the log-volume plot.
The plot of price times volume has the best linear fit and has an absence of multiple scaling exponents. Here there is clear evidence of price times volume not supporting market-cap breakpoints.
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