4901925
46162824
441681418
971474626
1071745822
109234542
115215242
120974038
97984824
All stocks, 1987 to 1991Entire60 862Sim. GBM60 862 DÐ BÐ
158597958
120574656
502291133
611871930
1202057326
1092446928
2657130100
3127140122
177795089
155886955
All stocks, 1992 to 1996Entire59 088Sim. GBM59 088 DÐ BÐ
211409084
162556471
871982452
1151993156
1432167045
1572329733
2999148113
3618163145
2456894102
199769960
Foundations of technical analysis
73
section 2.3.1. Table 2.1 contains results for the NYSE/AMEX stocks, andTable 2.2 contains corresponding results for Nasdaq stocks.Table 2.1 shows that the most common patterns across all stocks and overthe entire sample period are double tops and bottoms (see the row labelled`Entire'), with over 2000 occurrences of each. The second most commonpatterns are the head-and-shoulders and inverted head-and-shoulders, withover 1600 occurrences of each. These total counts correspond roughly to fourto six occurrences of each of these patterns for each stock during each ®veyear subperiod (divide the total number of occurrences by 7  50), not anunreasonable frequency from the point of view of professional technicalanalysts. Table 2.1 shows that most of the ten patterns are more frequent forlarger stocks than for smaller ones, and that they are relatively evenlydistributed over the ®ve-year subperiods. When volume trend is consideredjointly with the occurrences of the ten patterns, Table 2.1 shows that thefrequency of patterns is not evenly distributed between increasing (the rowlabelled `(B)') and decreasing (the row labelled `(D)') volume-trend cases.For example, for the entire sample of stocks over the 1962 to 1996 sampleperiod, there are 143 occurrences of a broadening top with decreasing volumetrend, but 409 occurrences of a broadening top with increasing volume trend.For purposes of comparison, Table 2.1 also reports frequency counts forthe number of patterns detected in a sample of simulated geometric Brownianmotion, calibrated to match the mean and standard deviation of each stock ineach ®ve-year subperiod. 11 The entries in the row labelled `Sim. GBM' showthat the random walk model yields very dierent implications for thefrequency counts of several technical patterns. For example, the simulatedsample has only 577 head-and-shoulders and 578 inverted-head-andshoulders patterns, whereas the actual data have considerably more, 1611and 1654, respectively. On the other hand, for broadening tops and bottoms,the simulated sample contains many more occurrences than the actual data,1227 and 1028 as compared to 725 and 748, respectively. The number oftriangles is roughly comparable across the two samples, but for rectangles anddouble tops and bottoms, the dierences are dramatic. Of course, thesimulated sample is only one realization of geometric Brownian motion, soit is dicult to draw general conclusions about the relative frequencies.Nevertheless, these simulations point to important dierences between thedata and IID lognormal returns.To develop further intuition for these patterns, Figures 2.8 and 2.9 displaythe cross-sectional and time-series distribution of each of the ten patterns forthe NYSE/AMEX and Nasdaq samples, respectively. Each symbol representsa pattern detected by our algorithm, the vertical axis is divided into the ®vequintiles, the horizontal axis is calendar time, and alternating symbols
74
Advanced Trading Rules
(a)
(b)
Figure 2.8 Distribution of patterns in NYSE/AMEX sample
Foundations of technical analysis
(c)
(d)
Figure 2.8 (continued)
75
76
Advanced Trading Rules
(e)
(f)
Figure 2.8 (continued)
Foundations of technical analysis
(g)
(h)
Figure 2.8 (continued)
77
78
Advanced Trading Rules
(i)
( j)
Figure 2.8 (continued)
Foundations of technical analysis
(a)
(b)
Figure 2.9 Distribution of patterns in Nasdaq sample
79
80
Advanced Trading Rules
(c)
(d)
Figure 2.9 (continued)
Foundations of technical analysis
(e)
(f)
Figure 2.9 (continued)
81
82
Advanced Trading Rules
(g)
(h)
Figure 2.9 (continued)
Foundations of technical analysis
(i)
( j)
Figure 2.9 (continued)
83
84
Advanced Trading Rules
(diamonds and asterisks) represent distinct subperiods. These graphs showthat the distribution of patterns is not clustered in time or among a subset ofsecurities.Table 2.2 provides the same frequency counts for Nasdaq stocks, anddespite the fact that we have the same number of stocks in this sample (50 persubperiod over seven subperiods), there are considerably fewer patternsdetected than in the NYSE/AMEX case. For example, the Nasdaq sampleyields only 919 head-and-shoulders patterns, whereas the NYSE/AMEXsample contains 1611. Not surprisingly, the frequency counts for the sampleof simulated geometric Brownian motion are similar to those in Table 2.1.Tables 2.3 and 2.4 report summary statistics ± means, standard deviations,skewness, and excess kurtosis ± of unconditional and conditional normalizedreturns of NYSE/AMEX and Nasdaq stocks, respectively. These statisticsshow considerable variation in the dierent return populations. For example,in Table 2.3 the ®rst four moments of normalized raw returns are 0:000,1:000, 0:345, and 8:122, respectively. The same four moments of post-BTOPreturns are 0:005, 1:035, 1:151, and 16:701, respectively, and those of postDTOP returns are 0:017, 0:910, 0:206, and 3:386, respectively. The dierencesin these statistics among the ten conditional return populations, and thedierences between the conditional and unconditional return populations,suggest that conditioning on the ten technical indicators does have some eecton the distribution of returns.2.4.6 Empirical resultsTables 2.5 and 2.6 report the results of the goodness-of-®t test (Equations(2.16) and (2.17)) for our sample of NYSE and AMEX (Table 2.5) andNasdaq (Table 2.6) stocks, respectively, from 1962 to 1996 for each of the tentechnical patterns. Table 2.5 shows that in the NYSE/AMEX sample, therelative frequencies of the conditional returns are signi®cantly dierent fromthose of the unconditional returns for seven of the ten patterns considered.The three exceptions are the conditional returns from the BBOT, TTOP, andDBOT patterns, for which the p-values of the test statistics Q are 5.1 per cent,21.2 per cent, and 16.6 per cent respectively. These results yield mixed supportfor the overall ecacy of technical indicators. However, the results of Table2.6 tell a dierent story: there is overwhelming signi®cance for all tenindicators in the Nasdaq sample, with p-values that are zero to threesigni®cant digits, and test statistics Q that range from 34.12 to 92.09. Incontrast, the test statistics in Table 2.5 range from 12.03 to 50.97.One possible explanation for the dierence between the NYSE/AMEX andNasdaq samples is a dierence in the power of the test because of dierentsample sizes. If the NYSE/AMEX sample contained fewer conditional
Table 2.3 Summary statistics (mean, standard deviation, skewness, and excess kurtosis) of raw and conditional one-day normalized returns of NYSE/AMEXstocks from 1962 to 1996, in ®ve-year subperiods, and in size quintiles. Conditional returns are de®ned as the daily return three days following the conclusion of anoccurrence of one of ten technical indicators: head-and-shoulders (HS), inverted head-and-shoulders (IHS), broadening top (BTOP), broadening bottom (BBOT),triangle top (TTOP), triangle bottom (TBOT), rectangle top (RTOP), rectangle bottom (RBOT), double top (DTOP), and double bottom (DBOT). All returnshave been normalized by subtraction of their means and division by their standard deviations
Moment
Raw
HS
IHS
BTOP
BBOT
TTOP
TBOT
RTOP
RBOT
DTOP
DBOT
0.0380.8670.1352.428
0.0400.9370.6604.527
0.0051.0351.15116.701
0.0620.9790.0903.169
0.0210.9550.1373.293
0.0090.9590.6437.061
0.0090.8650.4207.360
0.0140.8830.1104.194
0.0170.9100.2063.386
0.0010.9990.4607.374
Smallest quintile, 1962 to 1996Mean0.0000.014S.D.1.0000.854Skew.0.6970.802Kurt.10.8733.870
0.0361.0021.3377.143
0.0930.9401.7716.701
0.1880.8500.3670.575
0.0360.9370.8614.185
0.0201.1572.59212.532
0.0370.8330.1871.793
0.0930.9860.4454.384
0.0430.9500.5112.581
0.0550.9620.0023.989
2nd quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.392Kurt.7.836
0.0690.7720.2230.657
0.1441.0311.1286.734
0.0611.2783.29632.750
0.1131.0040.4853.779
0.0030.9130.5293.024
0.0350.9650.1664.987
0.0180.9791.37517.040
0.0190.8680.4523.914
0.0670.7760.3922.151
0.0111.0691.72815.544
3rd quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.246Kurt.7.466
0.0480.8880.4653.239
0.0430.8560.1071.612
0.0760.8940.0231.024
0.0560.9250.2330.611
0.0360.9730.5382.995
0.0120.7960.1660.586
0.0750.7980.6783.010
0.0280.8920.6184.769
0.0390.9560.0134.517
0.0341.0260.2423.663
All stocks, 1962 to 1996Mean0:000S.D.1.000Skew.0.345Kurt.8.122
(continued)
Table 2.3
Moment
Raw
BTOP
BBOT
TTOP
TBOT
RTOP
RBOT
DTOP
DBOT
0.0220.9030.5921.745
0.1150.9900.4581.251
0.0281.0930.5372.168
0.0220.9860.2174.237
0.0140.9590.4568.324
0.1130.8540.4154.311
0.0650.8210.8203.632
0.0150.8580.5501.719
0.0060.9920.0624.691
Largest quintile, 1962 to 1996Mean0.0000.038S.D.1.0000.843Skew.0.1740.438Kurt.7.9922.621
0.0540.9270.1823.465
0.0810.9970.4703.275
0.0420.9511.0996.603
0.0100.9640.0892.107
0.0490.9650.3572.509
0.0090.8500.1670.816
0.0600.8200.1403.179
0.0180.9710.0113.498
0.0670.9410.5115.035
All stocks, 1962 to 1966Mean0.000S.D.1.000Skew.0.563Kurt.9.161
0.0700.7970.1590.612
0.0900.9250.4621.728
0.1590.8250.3630.657
0.0791.0851.1515.063
0.0331.0680.1582.674
0.0391.0111.2644.826
0.0410.9611.33717.161
0.0190.8140.3411.400
0.0710.8590.4273.416
0.1000.9620.8765.622
All stocks, 1967 to 1971Mean0.000S.D.1.000Skew.0.342Kurt.5.810
0.0440.8090.7543.684
0.0790.9440.6662.725
0.0350.7930.3040.706
0.0560.8500.0850.141
0.0250.8850.6503.099
0.0570.8860.6971.659
0.1010.8311.3938.596
0.1100.8630.3953.254
0.0931.0831.3604.487
0.0790.8350.7011.853
All stocks, 1972 to 1976Mean0.000S.D.1.000Skew.0.316Kurt.6.520
0.0351.0150.3342.286
0.0430.8100.7171.565
0.1010.9850.6996.562
0.1380.9180.2721.453
0.0450.9451.0145.261
0.0100.9220.6764.912
0.0250.8700.2343.627
0.0030.7540.1992.337
0.0510.9140.0563.520
0.1080.9030.3665.047
4th quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.222Kurt.6.452
HS
IHS
0.0120.9640.0551.444
(continued)
All stocks, 1977 to 1981Mean0.000S.D.1.000Skew.0.466Kurt.6.419
0.1380.7860.3041.132
0.0400.8630.0521.048
0.0761.0151.5994.961
0.1140.9890.0330.125
0.1351.0410.7762.964
0.0501.0110.1100.989
0.0040.7550.0841.870
0.0260.9560.5342.184
0.0420.8270.7612.369
0.1781.0952.21415.290
All stocks, 1982 to 1986Mean0.000S.D.1.000Skew.0.460Kurt.6.799
0.0990.8830.4642.280
0.0071.0020.4416.128
0.0111.1090.3722.566
0.0950.9560.1652.735
0.1140.9240.4733.208
0.0670.8011.2495.278
0.0500.8260.2311.108
0.0050.9340.4674.234
0.0110.8500.5281.515
0.0131.0260.8677.400
All stocks, 1987 to 1991Mean0.000S.D.1.000Skew.0.018Kurt.13.478
0.0370.8480.5263.835
0.0330.8950.2724.395
0.0910.9550.1082.247
0.0400.8180.2311.469
0.0530.8570.1654.422
0.0030.9811.2169.586
0.0400.8940.2931.646
0.0200.8330.1243.973
0.0220.8731.1844.808
0.0171.052-0.3684.297
All stocks, 1992 to 1996Mean0.000S.D.1.000Skew.0.308Kurt.8.683
0.0140.9350.5452.249
0.0691.0211.3056.684
0.2311.4063.98827.022
0.2721.1870.5023.947
0.1220.9530.1901.235
0.0411.0782.46012.883
0.0820.8140.1670.506
0.0110.9960.1296.399
0.1020.9600.0911.507
0.0161.0350.3793.358
Table 2.4 Summary statistics (mean, standard deviation, skewness, and excess kurtosis) of raw and conditional one-day normalized returns of Nasdaq stocks from1962 to 1996, in ®ve-year subperiods, and in size quintiles. Conditional returns are de®ned as the daily return three days following the conclusion of an occurrence ofone of 10 technical indicators: head-and-shoulders (HS), inverted head-and-shoulders (IHS), broadening top (BTOP), broadening bottom (BBOT), triangle top(TTOP), triangle bottom (TBOT), rectangle top (RTOP), rectangle bottom (RBOT), double top (DTOP), and double bottom (DBOT). All returns have beennormalized by subtraction of their means and division by their standard deviations.
Moment
Raw
BTOP
BBOT
TTOP
TBOT
RTOP
RBOT
DTOP
DBOT
0.0420.9941.2908.774
0.0090.9600.3973.246
0.0090.9950.5862.783
0.0200.9840.8956.692
0.0170.9320.7163.844
0.0520.9480.7105.173
0.0430.9290.7554.368
0.0030.9330.4054.150
0.0350.8800.1042.052
Smallest quintile, 1962 to 1996Mean0.0000.018S.D.1.0000.845Skew.0.7540.325Kurt.15.8591.096
0.0321.3191.7564.221
0.0870.8740.2391.490
0.1530.8940.1090.571
0.0591.1132.72714.270
0.1081.0442.30010.594
0.1361.1871.7418.670
0.0130.9820.1991.918
0.0400.7730.1260.127
0.0430.9060.3680.730
2nd quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.844Kurt.16.738
0.0640.8480.4062.127
0.0760.9911.89211.561
0.1091.1060.1222.496
0.0931.0260.6353.458
0.0850.8050.0360.689
0.0380.9970.4551.332
0.0660.8980.5792.699
0.0150.8970.4163.871
0.0391.1191.1963.910
0.0340.8210.1900.777
3rd quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.698Kurt.12.161
0.0330.9330.2231.520
0.0280.9060.5291.526
0.0780.9310.6561.003
0.2100.9710.3260.430
0.0300.8250.5391.673
0.0681.0020.4421.038
0.1170.9920.8852.908
0.2100.9700.8204.915
0.1090.9970.1635.266
0.0750.9730.1232.573
All stocks, 1962 to 1996Mean0.000S.D.1.000Skew.0.608Kurt.12.728
HS
IHS
0.0160.9070.0173.039
4th quintile, 1962 to 1996Mean0.000S.D.1.000Skew.0.655Kurt.11.043
0.0790.9110.4562.525
0.0370.9572.67119.593
0.0060.9920.1742.163
0.0440.9750.3851.601
0.0801.0760.5547.723
0.0070.8240.7173.930
0.0840.8900.2901.555
0.0440.8511.0342.982
0.0380.8570.1542.807
0.0480.8190.1492.139
Largest quintile, 1962 to 1996Mean0.0000.026S.D.1.0000.952Skew.0.1000.266Kurt.7.9765.807
0.0581.0020.1444.367
0.0700.8951.6998.371
0.0311.0601.2255.778
0.0521.0760.4091.970
0.0130.8710.0252.696
0.0010.7940.1051.336
0.0240.9581.3007.503
0.0320.8440.3152.091
0.0180.8770.3632.241
All stocks, 1962 to 1966Mean0.000S.D.1.000Skew.0.575Kurt.6.555
0.1160.9120.7111.538
0.0410.9491.7949.115
0.0990.9890.2522.560
0.0901.0391.2586.445
0.0281.0151.6017.974
0.0660.8390.2471.324
0.1000.9252.01613.653
0.0100.8731.0215.603
0.0961.03