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The predictive power of zero intelligence in financial markets

Communicated by Kenneth J. Arrow, Stanford University, Stanford, CA, December 8, 2004 (received for review February 9, 2004)
Article Figures & SI
Figures
Data supplements
Farmer et al. 10.1073/pnas.0409157102.
Supporting Information
Files in this Data Supplement:
Supporting Text
Supporting Table 1
Supporting Table 2
Supporting Figure 5
Supporting Figure 6
Supporting Figure 7
Supporting Figure 8
Supporting Figure 9
Supporting Figure 10
Supporting Figure 11Table 1. Summary statistics for stocks in the data set
Stock ticker
No. events (1,000s)
Average (per day)
Limit (1,000s)
Market (1,000s)
Deletions (1,000s)
Eff. limit (shares)
Eff. market (shares)
No. of days
AZN
608
1,405
292
128
188
4,967
4,921
429
BARC
571
1,318
271
128
172
7,370
6,406
433
CW.
511
1,184
244
134
134
12,671
11,151
432
GLXO
814
1,885
390
200
225
8,927
6,573
434
LLOY
644
1,485
302
184
159
13,846
11,376
434
ORA
314
884
153
57
104
12,097
11,690
432
PRU
422
978
201
94
127
9,502
8,597
354
RTR
408
951
195
100
112
16,433
9,965
431
SB.
665
1,526
319
176
170
13,589
12,157
426
SHEL
592
1,367
277
159
156
44,165
30,133
429
VOD
940
2,161
437
296
207
89,550
71,121
434
Fields from left to right: stock ticker symbol, total number of events (effective market orders + effective limit orders + order cancellations) in thousands, average number of events in a trading day, number of effective limit orders in thousands, number of effective market orders in thousands, number of order deletions in thousands, average limit order size in shares, average market order size in shares, and number of trading days in the sample.
Table 2. A summary of the bootstrap error analysis described in the text
Regression
Estimated
Standard
Bootstrap
Low
High
Spread intercept
0.06
0.21
0.29
0.25
0.33
Spread slope
0.99
0.08
0.10
0.09
0.11
Diffusion intercept
2.43
1.22
1.76
1.57
1.97
Diffusion slope
1.33
0.19
0.25
0.23
0.29
The columns (left to right) are the estimated value of the parameter, the standard error from the crosssectional regression in Fig. 10, the one standard deviation error bar estimated by the bootstrapping method, and the one standard deviation low and high values for the extrapolation, as shown in Figs. 3 e and f and 4 e and f.
Supporting Figure 5Fig. 5. Illustration of the procedure for measuring the price diffusion rate for Vodafone (VOD) on August 4, 1998. On the x axis, we plot the time tin units of ticks, and on the y axis, the variance of midprice diffusion V(t ). According to the hypothesis that midprice diffusion is an uncorrelated Gaussian random walk, the plot should obey V(t) = Dt. To cope with the fact that points with larger values of thave fewer independent intervals and are less statistically significant, we use a weighted regression to compute slope D.
Supporting Figure 6Fig. 6. Time series (Upper) and autocorrelation function (Lower) for daily price diffusion rate D_{t} for Vodafone. Because of longmemory effects and the short length of the series, the longlag coefficients are poorly determined; the figure is simply to demonstrate that the correlations are quite large.
Supporting Figure 7Fig. 7. Subsample analysis of regression of predicted vs. actual spread. To get a better feeling for the true errors in this estimation (as opposed to standard errors, which are certainly too small), we divide the data into subsamples (using the same temporal period for each stock) and apply the regression to each subsample. (a) The results for the intercept; (b) the results for the slope. In both cases, we see that progressing from right to left, as the subsamples increase in size, the estimates become tighter. (c and d) The mean and standard deviation for the intercept and slope. We observe a systematic tendency for the mean to increase as the number of bins decreases. (e and f) The logarithm of the standard deviations of the estimates against log n, the number of each points in the subsample. The line is a regression based on binnings ranging from m = N to m = 10 (lower values of m tend to produce unreliable standard deviations). The estimated error bar is obtained by extrapolating to n = N. To test the accuracy of the error bar, the dashed lines are one standard deviation variation on the regression, whose intercepts with the n = N vertical line produce high and low estimates.
Supporting Figure 8Fig. 8. Subsample analysis of regression of predicted vs. actual price diffusion (see Fig. 10), similar to Fig. 7. The scaling of the errors is much less regular than for the spread, so the error bars are less accurate.
Supporting Figure 9Fig. 9. Market impact collapse under four kinds of axis rescaling. In each case, we plot a normalized version of the order size on the horizontal axis vs. a (possibly normalized) average market impact log(p_{t}+1)  log(p_{t}) on the vertical axis. (a) Collapse using nondimensional units based on the model; (b) order size is normalized by its mean value for the sample. (c) Order size is normalized the average daily volume. (d) Order size is multiplied by the current best midpoint price, making the horizontal axis the monetary value of the trade.
Supporting Figure 10Fig. 10. The variance plot procedure used to determine error bars for mean market impact conditional on order size. The horizontal axis n denotes the number of points in the m different samples, and the vertical axis is the standard deviation of the m sample means. We estimate the error of the full sample mean by extrapolating n to the full sample length.
Supporting Figure 11Fig. 11. The average market impact vs. order size plotted on loglog scale. (Upper Left and Upper Right) Buy and sell orders in nondimensional coordinates; the fitted line has slope b = 0.26 ± 0.02 for buy orders and b = 0.23 ± 0.02 for sell orders. In contrast, Lower Left and Lower Right show the same thing in dimensional units, using British pounds to measure order size. Although the exponents are similar, the scatter among different stocks is much greater.