Algunas objeciones a la metafísica de Bradley

Fig. 4.30a shows the size-variance relationship for firms. Although the slope β(S) increases with S, it does not display a strong crossover predicted by the preferential attachment model described here. This discrepancy may arise from the inaccuracy of two assumptions made in the model. The first assumption is that the new units attached to a class are taken at random from a general distribution of unit sizes which does not depend on the size of a particular class. The second assumption is that the

growth rates of units are taken at random from a general distribution of unit growth rates which do not depend on a size of a unit.

To test the first possibility, the products are randomly reassigned to the firms keeping the number of the products in each firm unchanged, and keeping the history of the fluctuation of each product sales unchanged.

This randomization practically does not change the size-variance dependence of the firms. (Fig. 4.30a).

This test demonstrates that although there is a small correlation between the num- ber of products in the firm and their sizes (see Fig. 4.30b), this correlation is not responsible for the origin of the power-law size-variance relationship observed for the empirical data.

To test the second possibility, the sizes of products ξi and their number Kαat year t

for each firm are kept the same as in the original data, so St=

PKα

i=1ξi is the same

as in the empirical data. However, to compute the sales of a firm in the following year eSt+1=

PKα

i=1ξ0i, it is assumed that ξ0i= ξiηi, where ηi is an annual growth rate

of a randomly selected product. The surrogate growth rate eg = lnSet+1

St obtained in

this way does not display any size-variance relationship (Fig. 4.30c). The second test shows that the size-variance relationship for the firm growth rates in the phar- maceutical data base is generated on the level of products. Indeed, the size-variance relationship of the growth rates of products gξ = ln(ξt+1_ξt ) shows a large range of

approximate power law behavior with β = 0.096 (Fig. 4.30b). The origin of this size- variance relationship cannot be determined from the present data set. It can come from the fact that small experimental products prescribed by few physicians to few patients are less stable then well established products prescribed to large number of patients.

Comparing the size-variance relationship for firms and products, for small firms this two distribution almost coincide, however for large firms,the exponent β is much larger than for the products.

Figure 4.31 shows the survivor function for ρt

i. ρti is the number of products that

represent the 50% of the whole company i size at time t. Figure 4.31 tells how many products give the 50% of the whole company size. This curve is the survivor function, and for ρ < 10 the curve is a power law with exponent around 1.7. In only few cases the company size is determined by more than 10 products. In the most cases the firms size is only due at 2 or 3 larger products. The fluctuations of the firms are due to the fluctuations of the largest or the few larger products. This data set tells only that the distribution of the number of products is power law, and there is no more information. So the shape of the firms growth rate is not due to the preferential attachment model, that only supports this evidence, but to the shape of the products growth rate. The mystery is why products have such kind of distribution, but this data set has not enough information to understand this behavior.

4.4 Size -variance relationship 113 size-variance relationship of the products and not from the preferential attachment model. To test this, the data have been left almost unchanged, randomizing only one parameter of the data at a time. In the first case, the products are reassigned to the firms in random order (keeping the number of products in each firm and the history of each products unchanged). Figure 4.30a shows β for firms left almost unchanged. This lets to conclude that β is not only due to the allocation process of products to the firms. In the other case, the product growth rate is reassigned in a random order, keeping the number of the products in each firm, K, and the size of each product

ξi in the year t the same as in original data. The firm size at t + 1 is computed as

P_K

i=1ξiηk where ηk is the growth rate of a randomly selected product. Figure 4.30b

and Figure 4.30c show the results. In this case beta becomes −0.028. There is no size-variance relationship. These two analysis clearly show that β originates already on the level of products but not due to the crossover in the preferential attachment. This database has not enough information to understand more deeply the origin of β.

0 5 10 15 20 −3 −2 −1 0 ln( σ (S)) ln(S) (a) 0 2 4 6 8 10 12 14 16 −3 −2.5 −2 −1.5 −1 −0.5 0 log( σ (g/s)) and log( σ ( η / ξ ))

log(s) and log(ξ)

(b) 0 5 10 15 20 −3 −2 −1 0 ln( σ (S)) ln(S) (c)

Figure 4.30: (a) Size-variance relationship at the firm level. Continuous line with circle represents the empirical data. The slope of the linear fit is -0.18 (β = −0.18). Dashed line with asterisks shows the size-variance relationship for the surrogate data set in which it is randomly allocated the products to the firms, keeping the number of products in each firms and the historical records for each product unchanged. In this case the slope is -0.14 (β = −0.14). (b) Size-variance relationship at the product and firms level. The continuous line with circle plots the empirical data for firms, whereas the dashed line with cross the data for products.The slope for the firms data is β = −0.18 and for products is β = −0.096. (c) Size-variance relationship at the firm level. The continuous line with circle plots the empirical data, whereas the dashed line with asterisks the surrogate data t but after random reassignment of products growth rates. The slope for the empirical data is β = −0.18, but after random reassignment of the products growth rates explained in the text the slope becomes β = −0.028

4.4 Size -variance relationship 115 100 101 102 10−5 10−4 10−3 10−2 10−1 100 Survivor, 1−CDF( ρ ) ρ Figure 4.31: Survivor function of ρ. ρt

i is the number of products that represent the

50% of the whole company i size at time t. .

Chapter 5

Remarks and Conclusions

5.1

Single asset model and empirical results

Empirical analysis and computational experiments of high-frequency data for a double-auction (book) market have been presented. The results of Raberto et al. 2004 [161] are confirmed and strengthened. Indeed, the return distribution is leptokurtic not only when the order generation process is Poisson but also in presence of memory, as in the case of Weibull-distributed waiting times. Moreover, with memory, return tails are fatter.

This result deserves attention because previous empirical analysis [132, 164, 175] has shown that the distribution of trade waiting times is non-exponential. Conversely, exponentially distributed trade waiting times only result from a finite thinning of a Poisson order process. Consequently, the distribution of order waiting times should be a more general distribution, e.g., a Weibull, than an exponential distribution un- derlying a simple memoryless Poisson process.

The hypothesis of non exponentially distributed order waiting times is empirically accessible and can be directly checked if full book information were available. More- over, one can try to solve the inverse problem: given a trade waiting time distribution, which is the originating order waiting time distribution?

Empirical analysis confirmed that trading waiting times are not exponentially dis- tributed, whereas a Weibull process cannot be rejected. This result deserves attention because exponentially distributed trade waiting times only result from a finite thin- ning of a Poisson order process. Consequently, the distribution of order waiting times should be a more general distribution than an exponential distribution underlying a simple memoryless Poisson process.

In order to better understand the relationship between order and trading waiting times, computer experiments on the Genoa Artificial Stock Market have been consid- ered. In particular, a double-auction mechanism (i.e., limit order book) with order waiting times characterized by a mixture of Poisson process has been modeled and implemented. The characteristics of Poisson process in the mixture have been prop- erly estimated by real data. It has been shown that, both order and trading waiting

times, reject hypothesis of exponentially distributed point process. Conversely, the hypothesis of Weibull distribution cannot be rejected and estimation of the corre- sponding Weibull parameter β pointed out values quite close to those obtained in the case of 30 DJIA stocks traded at NYSE in October 1999. Finally, the influence of the number of exponential distribution of the mixture appears almost negligible as far as survivals of order and trading waiting times and β do not change significantly. This allows one to conclude that already a mixture of two Poisson process can be sufficient in order to reproduce the behavior of real stock market.