Fórmulas jurídicas para regular la dimensión social del comercio

Tables 3.4 (a) to (c) give summary statistics for each of our three main groups of profit-rate measure.

The first row in each table shows the number of missing values, that is, the number of firms for which one or more of the data items needed to compute either s or k was missing;

since the total number of firms in the database is approximately 1×105 _{a broad estimate of}

number of firms for which a given profit rate measure has been calculated can be found by subtraction.

Table 3.4 (a): summary statistics for underlying data, Gillman profit rate measures

gill.1 gill.2 gill.3 gill.4 gill.5s gill.5f gill.6 gill.7s gill.7f

Missing 5.86e4 6.25e4 6.34e4 6.98e4 6.98e4 6.26e4 6.26e4 7.51e4 6.9e4 GRP 2.2451 0.2162 0.5972 0.4518 0.1232 0.0594 0.0526 0.1844 0.088 Mean 15.7 0.731 126 1.83 0.233 0.154 8.58e17 0.424 0.26 10% TM 0.408 0.346 3.81 1.12 0.159 0.0584 0.0451 0.24 0.0861 Median 0.327 0.286 2.19 0.881 0.121 0.0403 0.0326 0.176 0.0591 Mode.nrd 0.1353 0.1261 0.6746 0.4044 0.0719 0.0143 0.0129 0.0923 0.0204 bandwidth 17.09 14.58 225.79 52.13 10.73 4.12 3.19 14.80 5.57 Mode.alt 0.1322 0.1201 0.0957 0.3918 0.0697 0.0143 0.0129 0.0885 0.0204 bandwidth 10.00 10.00 10.00 10.00 10.00 4.12 3.19 10.00 5.57 Skew 221 41.5 115 37.6 24.5 27 174 59.9 55.3

Kurtosis 4.92e4 3090 1.73e4 2640 4670 4870 3.19e4 7420 6820 Min -0.996 -0.996 -2.23e5 -133 -232 -210 -733 -186 -158

Max 6.05e5 381 1.26e6 549 378 355 2.54e22 503 477

Range 6.05e5 382 1.48e6 682 610 565 2.54e22 689 635 1st _{Q 0.161 0.141 0.744 0.425 0.0189 0.00353 0.00304 0.0507 0.0127} 3rd _{Q 0.63 0.535 6.82 1.79 0.299 0.115 0.089 0.425 0.159} SD 2720 3.69 7630 6.07 3.5 3.09 1.32e20 4.12 3.79 IQR 0.469 0.394 6.08 1.36 0.28 0.111 0.086 0.374 0.146 MAD 0.295 0.255 2.75 0.828 0.189 0.071 0.0563 0.238 0.0868 CV 173 5.05 60.4 3.31 15 20.1 154 9.71 14.6 N o t e s

(1) 10% TM: 10 per cent trimmed mean; 1st Q: first quartile; 3rd Q: third quartile; SD: standard deviation; IQR: interquartile range; MAD: median absolute deviation; CV: co-efficient of variation; GRP: general rate of profit.

(2) GRP is Σsi/Σ|ki|

(3) Mode.nrd: bandwidth computed by the function bandwidth.nrd() in S-Plus 2000. Mode.alt: bandwidth as computed by bandwidth.nrd(), but capped at 10

Table 3.4 (b): summary statistics for underlying data, accounting ratios ORE ROCE ATO NPM

Missing 3.26e4 3.88e4 4.84e4 4.39e4 GRP 0.1756 0.0387 0.8903 0.0700 Mean 286 1.22 2.47 0.678 10% TM 0.248 0.0706 1.84 0.0633 Median 0.174 0.0573 1.74 0.0379 Mode.nrd 0.0514 0.0156 0.1233 0.0112 (154.14) bandwidth 14.50 4.15 62.70 3.86 Mode.alt 0.0358 0.0156 0.0832 0.0112 (154.14) bandwidth 10.00 4.15 10.00 3.86 Skew 147 262 159 110

Kurtosis 2.47e4 6.9e4 2.69e4 1.8e4 Min -1.47e5 -84.1 -0.0255 -4160 Max 4.12e6 7.08e4 5700 9600 Range 4.27e6 7.09e4 5700 1.38e4 1st _{Q 0.0284 0.00999 0.937 0.00514} 3rd _{Q 0.462 0.132 2.72 0.117} SD 2.25e4 269 31.1 56.2 IQR 0.434 0.122 1.78 0.112 MAD 0.261 0.0818 1.31 0.0633 CV 78.9 221 12.6 82.8 N o t e

Table 3.4 (c): summary statistics for underlying data, Glick profit rate measures glick.1 glick.2 glick.3 glick.4 glick.5 glick.6 glick.7 glick.8

Missing 5.65e4 7.26e4 3.88e4 6.69e4 4.64e4 3.25e4 4.39e4 4.29e4 GRP 0.0429 0.1096 0.0294 0.0794 0.0517 0.1339 0.0500 0.3364 Mean 0.084 -3.92e17 1.18 -3.38e17 0.112 252 0.592 176 10% TM 0.0611 0.168 0.0521 0.132 0.0873 0.182 0.0467 0.684 Median 0.0514 0.125 0.0428 0.0989 0.0783 0.128 0.0279 0.305 Mode.nrd 0.0174 0.0499 0.0118 0.0415 0.054 0.041 0.0082 0.1144 Bandwidth 3.96 11.55 3.12 8.78 3.98 10.75 2.87 37.09 Mode.alt 0.0174 0.0458 0.0118 0.0415 0.054 0.0388 0.0082 0.0884 Bandwidth 3.96 10.00 3.12 8.78 3.98 10.00 2.87 10.00 Skew 212 -148 263 -160 24 160 93.7 88.3

Kurtosis 4.72e4 2.3e4 6.91e4 2.67e4 6470 2.85e4 1.53e4 1.2e4 Min -84.1 -9.01e21 -84.6 -9.01e21 -84.4 -1.6e5 -4160 -4.56e5 Max 786 6190 7.1e4 3840 85.6 4.12e6 7800 1.51e6 Range 870 9.01e21 7.11e4 9.01e21 170 4.28e6 1.2e4 1.97e6 1st Q 0.00767 0.02 0.00677 0.0159 0.0303 0.0191 0.00332 0.081 3rd Q 0.117 0.317 0.0984 0.248 0.145 0.341 0.0864 1.16 SD 3.54 5.44e19 270 5.05e19 0.773 2.17e4 48 9390 IQR 0.109 0.297 0.0916 0.232 0.115 0.322 0.0831 1.08 MAD 0.0753 0.193 0.062 0.152 0.0812 0.197 0.0481 0.458

CV 42.1 -139 229 -150 6.92 86 81.2 53.3

In the second row we show the Marxian general rate of profit (GRP) – the total net income of the corporate sector divided by the total capital it employs – calculated, in accordance with our discussion above of the treatment of negative capital measures, as Σs Σ|k|, where s, k, are the net income and capital respectively of each firm, as defined for the profit rate measure in question. We thus calculate the size-weighted mean profit rate and allow comparison of this with unweighted measures of central tendency of company profit rates.

Next come estimates of central tendency: the mean, the ten per cent trimmed mean (found by ignoring the smallest and largest five per cent of observations and calculating the

mean of the remainder), the median, and two alternative kernel density estimates of the

mode.33

The first estimate (mode.nrd) is found as follows. We use the normal reference density (NRD) method – a rule of thumb presented in Silverman (1986: 45-47) – to select the bandwidth of a Gaussian kernel density estimator. However, Gaussian kernels have infinite support and thus the resulting density estimate is necessarily non-zero at every point. This tends to redistribute the probability mass away from the mode and towards the tails, thus reducing the density estimate at the former point. While in principle this should not matter, given appropriate choice of bandwidth (Härdle 1991:78), given the extreme ranges of our profit rate measures we prefer to use a triangular kernel, which is in principle capable of giving a density estimate of zero, and to multiply the NRD bandwidth by 2.432, the transformation factor given by Härdle (1991:76). For ATO we give estimates of both the main and secondary mode. NRD bandwidths are given in italic.

However, for certain profit rate measures – in particular Gillman 3 and ATO (the primary mode especially) – the NRD method gives estimates which seem clearly wrong, when compared with the histogram estimates. We therefore give alternative estimates (mode.alt) obtained by estimating the bandwidth with the NRD method but capping the NRD at 10 before transformation. It will be seen that the resulting estimates are considerably more plausible in the light of the histograms.

As inspection of the histograms above would lead one to expect, the estimates of central tendency change sharply as one progresses from the mean to the mode – very sharply indeed, in the cases of Gillman 1, 3 and 6, ORE, and Glick 2 to 4, 6 and 8.

33 _{We could estimate the mode simply by taking it to be the bin containing the greatest number of observations,}

but as examination of Figures 3.3 to 3.6 suggests this leaves one vulnerable to noise in the data, which is not likely to produce monotonic changes in density estimate on either side of the highest bin.

The next pair of statistics are the skewness and kurtosis, both of which have very high values for all measures, in consequence of the extreme ranges of the data and the very lengthy tails to the main probability masses.

The latter point is indicated by the next two groups of data, which indicate first the range and then the dispersion. The nature of the tails is particularly strongly brought out if the standard deviation and coefficient of variation are compared with more robust measures of dispersion, the inter-quartile range and median average dispersion.

A particularly problematic feature is that all these distributions have very extended lower tails (with the exceptions only of Gillman 1 and 2, and ATO, though it should be noted that the first two measures mentioned still have minima whose absolute values are quite large in relation to the corresponding general rates of profit; bear in mind also the exceptional status of ATO on account of its bi-modality).

This is important because the two distributions proposed as models for the rate of profit – the log normal and the gamma – both have bounded support (bounded below at zero, in the case of their standard forms). Extended left-hand tails are either inconsistent which such models, or imply parameter values such that the distributions are approximately normal in any case. We will return to this topic in Chapter Four.