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Although there have been many tests of technical analysis over the years, most of these tests have focused on the profitability of technical trading rules.9 Although some of these studies do find that technical indicators can generate statistically significant trading profits, but they beg the question of whether or not such profits are merely the equilibrium rents that accrue to investors willing to bear the risks associated with such strategies. Without specifying a fully articulated dynamic general equilibrium asset-pricing model, it is impossible to determine the economic source of trading profits.

Instead, we propose a more fundamental test in this section, one that attempts to gauge the information content in the technical patterns of Section II.A by comparing the unconditional empirical distribution of returns with the corresponding conditional empirical distribution, conditioned on the occurrence of a technical pattern. If technical patterns are informative, conditioning on them should alter the empirical distribution of returns; if the information contained in such patterns has already been incorporated into returns, the conditional and unconditional distribution of returns should be close. Although this is a weaker test of the effectiveness of technical analy-sis—informativeness does not guarantee a profitable trading strategy—it is, nevertheless, a natural first step in a quantitative assessment of technical analysis.

To measure the distance between the two distributions, we propose two goodness-of-fit measures in Section III.A. We apply these diagnostics to the daily returns of individual stocks from 1962 to 1996 using a procedure described in Sections III.B to III.D, and the results are reported in Sections III.E and III.F.

A simple diagnostic to test the informativeness of the 10 technical patterns is to compare the quantiles of the conditional returns with their unconditional counterparts. If conditioning on these technical patterns provides no incremental information, the quantiles of the conditional returns should be similar to those of unconditional returns. In particular, we compute the

9 For example, Chang and Osler (1994) and Osler and Chang (1995) propose an algorithm for automatically detecting head-and-shoulders patterns in foreign exchange data by looking at properly defined local extrema. To assess the efficacy of a head-and-shoulders trading rule, they take a stand on a class of trading strategies and compute the profitability of these across a sample of exchange rates against the U.S. dollar. The null return distribution is computed by a bootstrap that samples returns randomly from the original data so as to induce temporal independence in the bootstrapped time series. By comparing the actual returns from trading strategies to the bootstrapped distribution, the authors find that for two of the six currencies in their sample (the yen and the Deutsche mark), trading strategies based on a head-and-shoulders pattern can lead to statistically significant profits. See, also, Neftci and Policano (1984), Pruitt and White (1988), and Brock et al. (1992).

deciles of unconditional returns and tabulate the relative frequency Sj of conditional returns falling into decile j of the unconditional returns, j = 1,..., 10:

number of conditional returns in decile j 1 total number of conditional returns

Under the null hypothesis that the returns are independently and identically distributed (IID) and the conditional and unconditional distributions are identical, the asymptotic distributions of Sj and the corresponding goodness-of-fit test statistic Q are given by

j=1 0.10n where nj is the number of observations that fall in decile j and n is the total number of observations (see, e.g., DeGroot (1986)).

Another comparison of the conditional and unconditional distributions of returns is provided by the Kolmogorov-Smirnov test. Denote by {Z1t}n==1 and {Z21}n=21 two samples that are each IID with cumulative distribution functions F1(z) and F2(z), respectively. The Kolmogorov-Smirnov statistic is designed to test the null hypothesis that F1 = F2 and is based on the empirical cumulative distribution functions Fi of both samples:

where 1(-) is the indicator function. The statistic is given by the expression n1 n2

Under the null hypothesis F1 = F2, the statistic gn1,n2 should be small. Moreover, Smirnov (1939a, 1939b) derives the limiting distribution of the statistic to be lim Prob(gnin2 < x) = ( (-1)k exp(-2k2x2), x > 0. (20)

An approximate a-level test of the null hypothesis can be performed by computing the statistic and rejecting the null if it exceeds the upper 100ath percentile for the null distribution given by equation (20) (see Hollander and Wolfe (1973, Table A.23), Csáki (1984), and Press et al. (1986, Chap. 13.5)).

Note that the sampling distributions of both the goodness-of-fit and Kolmogorov-Smirnov statistics are derived under the assumption that returns are IID, which is not plausible for financial data. We attempt to address this problem by normalizing the returns of each security, that is, by subtracting its mean and dividing by its standard deviation (see Sec. III.C), but this does not eliminate the dependence or heterogeneity. We hope to extend our analysis to the more general non-IID case in future research.

We apply the goodness-of-fit and Kolmogorov-Smirnov tests to the daily returns of individual NYSE/AMEX and Nasdaq stocks from 1962 to 1996 using data from the Center for Research in Securities Prices (CRSP). To ameliorate the effects of nonstationarities induced by changing market structure and institutions, we split the data into NYSE/AMEX stocks and Nasdaq stocks and into seven five-year periods: 1962 to 1966, 1967 to 1971, and so on. To obtain a broad cross section of securities, in each five-year subperiod, we randomly select 10 stocks from each of five market-capitalization quintiles (using mean market capitalization over the subperi-od), with the further restriction that at least 75 percent of the price observations must be nonmissing during the subperiod.10 This procedure yields a sample of 50 stocks for each subperiod across seven subperiods (note that we sample with replacement; hence there may be names in common across subperiods).

As a check on the robustness of our inferences, we perform this sampling procedure twice to construct two samples, and we apply our empirical analysis to both. Although we report results only from the first sample to conserve space, the results of the second sample are qualitatively consistent with the first and are available upon request.

For each stock in each subperiod, we apply the procedure outlined in Section II to identify all occurrences of the 10 patterns defined in Section II.A. For each pattern detected, we compute the one-day continuously compounded return d days after the pattern has completed. Specifically, consider a window of prices {Pt% from t to t + l + d - 1 and suppose that the

10 If the first price observation of a stock is missing, we set it equal to the first nonmissing price in the series. If the tth price observation is missing, we set it equal to the first nonmissing price prior to t.

identified pattern p is completed at t + l - 1. Then we take the conditional return Rp as log(1 + Rt+l+d+1). Therefore, for each stock, we have 10 sets of such conditional returns, each conditioned on one of the 10 patterns of Section II.A.

For each stock, we construct a sample of unconditional continuously compounded returns using nonoverlapping intervals of length t, and we compare the empirical distribution functions of these returns with those of the conditional returns. To facilitate such comparisons, we standardize all returns— both conditional and unconditional—by subtracting means and dividing by standard deviations, hence:

where the means and standard deviations are computed for each individual stock within each subperiod. Therefore, by construction, each normalized return series has zero mean and unit variance.

Finally, to increase the power of our goodness-of-fit tests, we combine the normalized returns of all 50 stocks within each subperiod; hence for each subperiod we have two samples—unconditional and conditional returns— and from these we compute two empirical distribution functions that we compare using our diagnostic test statistics.

Given the prominent role that volume plays in technical analysis, we also construct returns conditioned on increasing or decreasing volume. Specifically, for each stock in each subperiod, we compute its average share turnover during the first and second halves of each subperiod, t1 and t2, respectively. If t1 > 1.2 X t2, we categorize this as a "decreasing volume" event; if t2 > 1.2 X t1, we categorize this as an "increasing volume" event. If neither of these conditions holds, then neither event is considered to have occurred.

Using these events, we can construct conditional returns conditioned on two pieces of information: the occurrence of a technical pattern and the occurrence of increasing or decreasing volume. Therefore, we shall compare the empirical distribution of unconditional returns with three conditional-return distributions: the distribution of returns conditioned on technical patterns, the distribution conditioned on technical patterns and increasing volume, and the distribution conditioned on technical patterns and decreasing volume.

Of course, other conditioning variables can easily be incorporated into this procedure, though the "curse of dimensionality" imposes certain practical limits on the ability to estimate multivariate conditional distributions nonparametrically.

In Tables I and II, we report frequency counts for the number of patterns detected over the entire 1962 to 1996 sample, and within each subperiod and each market-capitalization quintile, for the 10 patterns defined in Section II.A. Table I contains results for the NYSE/AMEX stocks, and Table II contains corresponding results for Nasdaq stocks.

Table I shows that the most common patterns across all stocks and over the entire sample period are double tops and bottoms (see the row labeled "Entire"), with over 2,000 occurrences of each. The second most common patterns are the head-and-shoulders and inverted head-and-shoulders, with over 1,600 occurrences of each. These total counts correspond roughly to four to six occurrences of each of these patterns for each stock during each five-year subperiod (divide the total number of occurrences by 7 X 50), not an unreasonable frequency from the point of view of professional technical analysts. Table I shows that most of the 10 patterns are more frequent for larger stocks than for smaller ones and that they are relatively evenly distributed over the five-year subperiods. When volume trend is considered jointly with the occurrences of the 10 patterns, Table I shows that the frequency of patterns is not evenly distributed between increasing (the row labeled "t(;)") and decreasing (the row labeled "t(\)") volume-trend cases. For example, for the entire sample of stocks over the 1962 to 1996 sample period, there are 143 occurrences of a broadening top with decreasing volume trend but 409 occurrences of a broadening top with increasing volume trend.

For purposes of comparison, Table I also reports frequency counts for the number of patterns detected in a sample of simulated geometric Brownian motion, calibrated to match the mean and standard deviation of each stock in each five-year subperiod.11 The entries in the row labeled "Sim. GBM" show that the random walk model yields very different implications for the frequency counts of several technical patterns. For example, the simulated sample has only 577 head-and-shoulders and 578 inverted-head-and-shoulders patterns, whereas the actual data have considerably more, 1,611 and 1,654, respectively. On the other hand, for broadening tops and bottoms, the simulated sample contains many more occurrences than the actual data, 1,227 and 1,028, compared to 725 and 748, respectively. The number of triangles is roughly comparable across the two samples, but for rectangles and

11 In particular, let the price process satisfy dP(t) = mP(t) dt + sP(t) dW (t), where W(t) is a standard Brownian motion. To generate simulated prices for a single security in a given period, we estimate the security's drift and diffusion coefficients by maximum likelihood and then simulate prices using the estimated parameter values. An independent price series is simulated for each of the 350 securities in both the NYSE/AMEX and the Nasdaq samples. Finally, we use our pattern-recognition algorithm to detect the occurrence of each of the 10 patterns in the simulated price series.

double tops and bottoms, the differences are dramatic. Of course, the simulated sample is only one realization of geometric Brownian motion, so it is difficult to draw general conclusions about the relative frequencies. Nevertheless, these simulations point to important differences between the data and IID lognormal returns.

To develop further intuition for these patterns, Figures 8 and 9 display the cross-sectional and time-series distribution of each of the 10 patterns for the NYSE/AMEX and Nasdaq samples, respectively. Each symbol represents a pattern detected by our algorithm, the vertical axis is divided into the five size quintiles, the horizontal axis is calendar time, and alternating symbols (diamonds and asterisks) represent distinct subperiods. These graphs show that the distribution of patterns is not clustered in time or among a subset of securities.

Table II provides the same frequency counts for Nasdaq stocks, and despite the fact that we have the same number of stocks in this sample (50 per subperiod over seven subperiods), there are considerably fewer patterns detected than in the NYSE/AMEX case. For example, the Nasdaq sample yields only 919 head-and-shoulders patterns, whereas the NYSE/AMEX sample contains 1,611. Not surprisingly, the frequency counts for the sample of simulated geometric Brownian motion are similar to those in Table I.

Tables III and IV report summary statistics—means, standard deviations, skewness, and excess kurtosis—of unconditional and conditional normalized returns of NYSE/AMEX and Nasdaq stocks, respectively. These statistics show considerable variation in the different return populations. For example, in Table III the first four moments of normalized raw returns are 0.000, 1.000, 0.345, and 8.122, respectively. The same four moments of post-BTOP returns are -0.005, 1.035, —1.151, and 16.701, respectively, and those of post-DTOP returns are 0.017, 0.910, 0.206, and 3.386, respectively. The differences in these statistics among the 10 conditional return populations, and the differences between the conditional and unconditional return populations, suggest that conditioning on the 10 technical indicators does have some effect on the distribution of returns.

Tables V and VI report the results of the goodness-of-fit test (equations (16) and (17)) for our sample of NYSE and AMEX (Table V) and Nasdaq (Table VI) stocks, respectively, from 1962 to 1996 for each of the 10 technical patterns. Table V shows that in the NYSE/AMEX sample, the relative frequencies of the conditional returns are significantly different from those of the unconditional returns for seven of the 10 patterns considered. The three exceptions are the conditional returns from the BBOT, TTOP, and DBOT patterns, for which the p-values of the test statistics Q are 5.1 percent, 21.2 percent, and 16.6 percent, respectively. These results yield mixed support for the overall efficacy of technical indicators. However, the results of Table VI tell a different story: there is overwhelming significance for all 10 indicators

Table I -i

Frequency counts for 10 technical indicators detected among NYSE/AMEX stocks from 1962 to 1996, in five-year subperiods, in size quintiles, In5 and in a sample of simulated geometric Brownian motion. In each five-year subperiod, 10 stocks per quintile are selected at random among stocks with at least 80% nonmissing prices, and each stock's price history is scanned for any occurrence of the following 10 technical indicators within the subperiod: 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). The "Sample" column indicates whether the frequency counts are conditioned on decreasing volume trend (V(\)'), increasing volume trend ('t(/>)'), unconditional ("Entire"), or for a sample of simulated geometric Brownian motion with parameters calibrated to match the data ("Sim. GBM").

Sample

BBOT

TTOP

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

423,556 423,556

84,363 84,363

83,986 83,986

84,420 84,420

84,780 84,780

86,007 86,007

1611 577 655 553

182 82 90 58

309 108 133 112

361 122 152 125

332 143 131 110

427 122 149 148

1654 578 593 614

181 99 81 76

321 105 126 126

388 120 131 146

317 127 115 126

447 127 140 140

725 1227 143 409

78 279 13 51

146 291 25 90

145 268 20 83

176 249 36 103

180 140 49 82

All Stocks, 748 1028 220

1962 to 1996 1294 1049 666 300

Smallest Quintile, 1962 to 1996

97 203

256 269

42 122

37 41

2nd Quintile, 1962 to 1996

150 255

251 261

48 135 63 55

3rd Quintile, 1962 to 1996

161 291

222 212

49 151 66 67

4th Quintile, 1962 to 1996

173 262

210 183

42 138

89 56

Largest Quintile, 1962 to 1996

167 283

89 124

39 120

82 81

1193 1176 710 222

159 295 119 22

228 278 147

247 249

149 44

255 210 145 55

304 144

150 62

1482 122 582 523

265 18 113 99

299 20 130 104

24 130 121

259 35 85 102

25 124

1616 113 637 552

320 16 131 120

322 17 149 110

399 31 160 142

24 97 96

25 100

2076 535 691 776

261 129 78 124

372 106 113

154 179

116 144 147

561 69 202 173

2075 574 974

271 127 161 64

420 126 211 107

443 125 215 106

420 122 184 118

521 74 203 138

TO TO

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

55,254 55,254

60,299 60,299

59,915 59,915

62,133 62,133

61,984 61,984

61,780 61,780

62,191 62,191

276 56

104 96

179 92 68 71

152 75 64 54

223 83

114 56

240 68 95 81

299 88

278 58 88 112

175 70 64 69

162 85 55 62

206 88 61

256 120

241 79 89 79

336 78 132

144 26 44

167 16 68

183 16 42

245 24 78

188 28 51

168 16 68

132 17 58

All Stocks, 1962 to 1966 103 179

126 129

29 93

39 37

All Stocks, 134 148 45 57

All Stocks, 93 154 23 50

All Stocks, 1977 to 1981 110 188 200 188 39 100

44 35

1967 to 1971 227 150 126 47

1972 to 1976 165 156 88 32

All Stocks, 108 144 30 62

1982 to 1986 182 152 93 46

All Stocks, 1987 to 1991 98 180

132 131

30 86

43 53

All Stocks, 1992 to 1996 102 173

124 143

24 80

42 50

165 139 109 25

316 9

130 130

354 16 141 122

356 60 113 137

352 68 188 88

172 180 111

115 19 42 41

117 16 39 41

239 84 80 87

258 77 143 53

136 178 78 21

171 16 60 61

182 10 64 67

218 70 53 80

223 71 97 59

167 210 97

146 18 54 53

182 12 60 71

274 90 82 113

290 115 140 76

190 169 104 40

169 150 101

182 31 70 73

260 11 103 73

207 23 95 68

259 10 102 87

313 99 109 116

287 76 105 100

299 87 124 85

285 68 137 68

TO3 O

194 150 110 35

292 18 123 92

315 26 136 96

389 56 149 143

368 88 145 104

Frequency counts for 10 technical indicators detected among Nasdaq stocks from 1962 to 1996, in five-year subperiods, in size quintiles, and in a sample of simulated geometric Brownian motion. In each five-year subperiod, 10 stocks per quintile are selected at random among stocks with at least 80% nonmissing prices, and each stock's price history is scanned for any occurrence of the following 10 technical indicators within the subperiod: 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). The "Sample" column indicates whether the frequency counts are conditioned on decreasing volume trend ("t(\,)"), increasing volume trend ("t(/>)"), unconditional ("Entire"), or for a sample of simulated geometric Brownian motion with parameters calibrated to match the data ("Sim. GBM").

Sample

BBOT

TTOP

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

411,010 411,010

81,754 81,754

81,336 81,336

81,772 81,772

82,727 82,727

83,421 83,421

919 434 408 284

85 36 31

191 67 94 66

224 69 108 71

212 104 88 62

208 109 82 54

817 447 268 325

64 84 25 23

138 84 51

186 86 66 79

214 92 68 83

215 101

58 83

All Stocks, 1962 to 1996

414 508 850 789

1297 1139 1169 1309

69 133 429 460

234 209 185 125 Smallest Quintile, 1962 to 1996

41 73 111 93

341 289 334 367

6 20 56 59

31 30 24 15

2nd Quintile, 1962 to 1996

68 88 161 148

243 225 219 229

11 28 86 109 46 38 45 22

3rd Quintile, 1962 to 1996

105 121 183 155

227 210 214 239

23 35 87 91

56 49 39 29

4th Quintile, 1962 to 1996

92 116 187 179

242 219 209 255

12 26 101 101

57 56 34 22

Largest Quintile, 1962 to 1996

108 110 208 214

244 196 193 219 17 24 99 100 44 36 43 37

1134 96 488 391

165 11 77 59

242 24 111 85

235 15 90 84

296 23 127 104

196 23 83 59

1320

91 550 461

218 12 102

305 12 131 120

244 14 84

303 26 141 93

250 27

92 77

1208 567 339 474

113 140 31 46

219 99 69 90

279 105 78 122

289 115 77 118

308 108 84 98

1147

580 229

125 125 81 17

176 124 101 42

267 100 145

297 97 143 66

282 133 110 46

TO TO

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

Entire Sim. GBM

55,969 55,969

60,563 60,563

51,446 51,446

61,972 61,972

61,110 61,110

60,862 60,862

59,088 59,088

69 129 83

115 58 61 24

34 32 5

56 90 7

37 21

158 59 79 58

40 90 84

63 99

120 61

29 57

30 37

64 90 19 25

120 57 46 56

162 55 64 71

72 163 10 48

104 194 15 71

14 115 0 1

236 1 5

46 162 8 24

50 229 11

87 198 24 52

All Stocks, 1962 to 1966

99 182

123 137

23 104

51 37

All Stocks, 1967 to 1971

123 227

184 181

40 127

51 45

All Stocks, 1972 to 1976

30 29

113 107

All Stocks. 1977 to 1981

36 52

165 176

All Stocks, 1982 to 1986

44 97

168 147

14 46

18 26

All Stocks, 1987 to 1991

61 120

187 205

19 73

30 26

All Stock-s. 115 199 31 56

144 149 98 23

288 24 115 101

329 22 136 116

326 77 96 144

342 90 210 64

171 188 123 19

65 9 26 25

83 8 39 16

196 90 49 86

200 83 137 17

28 110 7

58 46 8

73 212 8 0

57 19 12 5

65 19 12 8

89 110 7 7

107 174 58 22

109 244 69 28

109 23 45 42

265 7

130 100

115 21 52 42

312 7

140 122

120 97 40 38

177 79 50 89

98 48 24

155 88 69 55

TO3 O

232 97

299 9

148 113

361 8 163 145

245 68 94 102

199 76 99 60

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