atively the same as for the random walk model, and alsowith the folklore statement that it takes volume to make prices move (Section ¥and lower half of Table 1).
The above three models do not of course exhaust the possibilities. In par-ticular it should be noted that if the specifications on the continuous modelwere relaxed from “at all times”, to “at all times except a discrete number ofpoints,” and these discontinuities of function or derivative were marked bylarge volume, the resulting conclusions could be quite similar to those whichwe reached for our discrete transient model. The discrete transient model wasbuilt around the concept of a totally discontinuous process. The prices did notexist except at discrete instants of time [7].
7. COMPARISON OF THEORY AND OBSERVATION
Let us now examine some data to see in what respects the conclusions fromtheory we have drawn in the preceding discussion are supported by the evi-dence. The five sequences considered, and the criteria defining the events, aregiven at the top of Table 1. They were selected as follows: the first four se-quences were picked out of the supply of semi-log charts (linear charts forDaily DJI), available to the author, primarily so as to have an appreciablenumber of J and S events on them. The existence of sufficient J and S eventswas needed to test adequately for their coincidence or avoidance of the otherevents. Stocks which traded in low volume and price were also avoided. Beyondthese two specifications no attempt was made to pick favorable or unfavorablecases. This selection introduces some bias into our sample, since this choicemeans we have picked sequences, or time intervals, in which the stocks really“moved”.331
QUANTITATIVE TESTS FOR STOCK PRICE MODELS
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The fifth sequence (DJI Weekly, using DJI volume) was chosen to checkon some rather unexpected results which appeared in the DJT daily data, usingtotal market volumes.
The identifications of the events | Ap| , [A@p| , & were simply made with aruler on the semi-log charts. In the case of the monthly charts (Col. 4) thisaccounts for the apparently unsymmetrical specification of S vs J events. Foran S event, a 20% fall (4/5 p) is equal in “distance” to a 25% rise (5/4 p) onasemi-log cbart.
The first three binary coincidences in Table 1 simply check what we expectedfrom the definitions; |Ap|, |A®p| and R events do not occur independently.The observed number of coincidences, Nop, is systematically greater thanNtneor calculated under the assumption of independence. In most cases the excessis significant at the 5% level, denoted by (*). Summaries by sequence appear inthe lower row, and by event in the last columns.
It will be noted that the significance probabilities for the summary rows andcolumns are computed using the fact that the sum of Poisson variables is aPoisson variable. Use of this fact requires the assumption that the individualPoisson variables summed are independent. This seems to be a reasonableassumption for the sum of events for a coincidence type (the last column) anddebatable for the sum of all events for a sequence (the summary row). Obvi-ously the numbers in one sequence of |Ap|, |A®p|, and # events are not in-dependent. If and how much this error makes the significance for the sum ofevents for a sequence overstated (too small a value of P(n>Novs), we donot know.
The second group of binary events in Table 1 check for coincidences of Vwith| Ap|; [Ap], and FR events (groups 2 and 3). An excess is expected byboth the random walk model, the discrete transient model, and folklore (“vol-ume make prices move”). In general there is a significant excess of the observednumber of coincidences, Novs, over Ntneor. So we conclude that a “mix” inunknown proportions of these two models, plus folklore, is probably correct.This is certainly not a startling conclusion.
However, detailed examination of this second group of binary coincidencesin Table 1 shows two rather unexpected results. Column 2 represents the dailydata for the Dow Jones Industrial Average and the total market volume. Thesetwo sequences are more closely studied by the trading public than any other.For this wellspring of folklore, for none of the coincidences, either individuallyor in toto (8 observed vs 6.4 theoretical) is there a significant excess of coin-cidences.
We believe that this negative result can be explained in part by the followingargument. The events in the Dow Jones Average were being compared withevents in the total market volume sequence, and strictly speaking one shouldrather have used just the volume associated with the stocks in the average.This is about 7% of the total market volume.
There is in fact a certain amount of “correlation”, or excess of coincidencesbetween events in the Dow Jones Volume and total market volume, as is shownin Table 5. Nevertheless it would be preferable to compare the DJI with justthe volume associated with the stocks in it. This comparison is made in column
 334 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
TABLE 3. TRIPLE COINCIDENCES INVOLVING I AND 8.COMPUTED AS COMPOUND BINARY EVENTS (EQ. 2)
 
 
Sum over Sequences (1) through (4)Coincident Event
 
 
 
 
Nobs N theor P(n=Novs)
(S, (|p|, |A®p|)) 2 0.97 0.26(S, (| Ap], R)) 1 0.98 0.63(S, (jap, R)) 2 1.00 0.27(S, (jAp|, 7)) 0 0.73 1.0
(S, ({A@pl, V)) 0 0.8 1.0
(S, (R, V)) 0 1.14 1.0Sum 5 5.38 0.57Sequence Sum over 6 triple coincidences involving S(1) Magnavox, daily 5 1.56 0.02*(2) DJI Daily, Mkt., Vol. 0 75 1.0
(3) United Airlines, weekly 0 2.29 1.0
(4) Johns Manville, monthly 0 0.80 1.0Sum 5 5.38 0.57
Sum over sequences (1)—(4)Coincident Event
 
 
 
 
 
Novs N theor P(n> Nos)
(, (JAp|, |AMp|) 2 1.05 0.27(I, (/Ap|, R)) 3 1.09 0.10(1, (|A®p|, B)) 8 1.11 <0.01*(, (jap|, V)) 2 0.79 0.2
Z, ({a@p|, V)) 3 0.65 0.03*(1, (R, V)) 7 1.26 <0.01*Sum 30 11.33 <0.01*Sequence Sum over 6 triple coincidences involving I(1) Magnavox, daily 13 1.79 <0.01*(2) DJI Daily, Market Vol. 5 0.73 <0.01*(3) United Airlines, weekly 6 2.64 0.04*(4) Johns Manville, monthly 1 0.97 0.63Sum 30 11.33 <0.01*
 
 
5, using weekly data. It will be observed that in general, i.e., for the summarydata for this sequence, (N.ts=19 vs. Ntneor=6.9) there is a significant excess,in agreement with what was concluded from the other sequences. But in par-ticular there is not a significant excess of (| AMp| , V) coincidences (Novs = 4 vs.Ntneor = 2.7). This brings us to the second unexpected result in Table 1.According to the random walk model, the volume and the sequential disper-sion (measured by |A®p|) should increase together. This implies an excessQUANTITATIVE TESTS FOR STOCK PRICE MODELS 835
number of (|A@p| , V) coincidences. The row of data in summary for thisevent in Table 1 (Nobs=19 vs. Ntheor= 13.9) does not quite significantly sup-port this conclusion (P=0.09). This is a slightly embarrassing result, since inaddition to the random walk model it also contradicts the predictions of thediscrete transient model, and also folklore. The one exception to this embarrass-ment is in the first column, for an individual stock for data at daily intervals(Novs = 8 vs. N theor = 3.1).
From this limited data we can guess that folklore and the discrete transientmodel may be correct for daily data on individual stocks in predicting excess( | AMp| , V) coincidences, but not for longer basic intervals (week, month), orfor an average. A better test would be to generate a synthetic random walkwith variable numbers of steps (volume) between observations, and then testthese data for ([A@p| , V) coincidences. We can guess at this stage that theexcess of (| AMp| , V) coincidences which do occur in daily data for individualstocks are primarily due to non-random structure (folklore and the discretetransient model), and that the random walk contribution to (| AMp| , V) coin-cidences is rather small.
Let us now consider the binary coincidences involving J and S with the re-maining four events (Table 2). An excess of these eight different types of coinci-dences (four with J and four witb S) is predicted both by the discrete transientmodel, and also by the folklore statement, “Important moves end climatically,or on large volume”.
First observe the summary figures of J and S separately. For the coincidenceswith S events the totals are Nop, = 22 vs. Ntneor= 15.7, and an excess not quitesignificant at the 5% level (P(N obs > 22| Ntheor= 15.7) = 0.07. Most of the excessis provided by the individual stock at daily intervals (Nons=8 vs. Ntheor = 2.78),and none of the others show a significant excess of coincidences of S with theother four events. So far as these data are concerned, we can say that theconclusion of folklore that “important moves upward (S events) end on largevolume or climactically”, applies to individual stocks and daily data (one ex-ample only), not to longer intervals, or an average.
The case for minima, or J events, is somewhat more convincing, since thedifference of the totals, Nops=50 vs. Ntneor= 16.98 is highly significant. Theindividual cases also separately show a significant excess of coincidences of Ievents with the remaining four, except for the monthly interval data (Nobs=4vs. Ntheor= 2.96), and except for (J, | Ap|) coincidences, where the excess ofcoincidences is also not significant (Nops=5 vs. N theor = 2.54).
In summary, therefore, the conclusions of folklore and the discrete transientmodel hold better at minima than maxima, better in individual stocks than anaverage, and better for intervals of a day or week, than for intervals of a month.
The asymmetry between the behavior at maxima and minima is at least inagreement with one item of folklore. Traditionally stocks tend to rally some-what abruptly from a decline, whereas the passage through a peak is somewhatmore leisurely. The above conclusions are in agreement with this statement.The same statement can be made in a variety of ways. One might say “stocksbounce something like a pingpong ball, with discontinuities at the minima ofits trajectories”. In market terms one can say that bargain hunting is more
 336 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
concentrated in time and price than profit taking. This statement is in agree-ment with the inference that limit orders to buy below the market are moreconcentrated in price than limit orders to sell above the market (ref. 6, p. 377).
There are four binary events marked with a dagger in Table 2. These arethe coincidences which are expected to occur less rather than more frequentlythat given by independence, if a continuous in time price mechanism werecorrect. Examination of the data for these coincidences gives no support tosuch a mechanism, but merely supports our previous conclusions. ComparingNove V8. Ntneor, there is a neglibible departure (an excess) for coincidences withS events, and a considerable excess, rather than deficiency for coincidencewith the J events.
There are twelve possible triple coincidences involving J and S, which mightbe examined, six with J and six with S, These twelve (Table 3) are formed bycombining J or S with the six binary coincidences listed in Table 1. For prac-tically all of these twelve the calculated number of coincidences (using theform of Eq. 2) was considerably less than one. The observed number was zerofor the majority of individual sequences, in the case of coincidences with Sevents, and only one or two in the case of coincidences with I events. Hence wegive only the summary figures for these triple coincidences. Our previous con-clusions from the binary coincidences are confirmed. Except for the individualstock at daily intervals (Nops=5 vs. Nyneor= 1.6), there is no significant excessfor S events but there is for J events, except for intervals of one month.
It should be pointed out here that Ntneor for the triple coincidences in Table3 are computed as though they were binary coincidences, of one single eventwith another single event (which happens to be binary). This method of calcu-lation is indicated by the notation in Table 3, for example ((), (R(), | Apt|)),rather than (I(¢), R(@, |Ap(@|) for the triple coincidence between I, R, and| Ap| events. The reason follows from the discussion of the three groups of oursix events in section 2.
It is known, or suspected, that R and | Ap| events are not independentsimply from their definitions, Thus, the expected, or theoretical number oftriple coincidences as computed in Table 3, is
Novs(1)Nova(R, Ap) T
Neneor = TP (2)Specifically we do not useNovs(I)Novs(R)Novs( | Ap| )TNemes = oa(L) Nova (BR) Nova(| Ap| ) (3)
T8
which would be the expected number of triple coincidences if all three of theevents I, R, | Ap| were assumed under the null hypothesis to be tested, to beindependent of each other.
In Table 4 are given the data for a test for delayed coincidences of volumeevents with J and S events. The effects are small, but the summary column atthe left does give definite evidence that volume events tend to precede J and Sevents (primarily S events) and avoid following J and S events by one unit of337
QUANTITATIVE TESTS FOR STOCK PRICE MODELS
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TABLE 5. DATA ON COINCIDENCES BETWEEN TOTAL MARKET VOLUMEAND VOLUME FOR DOW-JONES INDUSTRIAL COMPONENTS ONLY.WEEKLY DATA FROM JULY 1956 TO OCTOBER 1959, TAKENGRAPHICALLY FROM CHARTS OF SECURITIES RESEARCHCORPORATION, BOSTON, MASS.
No. of intervals 172
 
No. of total market volume events Vx 33No. of D. J. volume events Vp,z 23Expected no. of (Va(t), Vps(t)) coincidences33.23N theor = = 4.4172Observed no. of (Vu(t), Vos(¢)) coincidences, Nobs = 14
P(n>Novs) 0.01
time. The former is expressed by an excess of coincidences, the latter by adeficiency. Note that for the latter case the significance probability is for fewerthan the observed number of coincidences, rather than for an excess, as inTables 1 and 2. The slight evidence that volume events tend to precede andavoid following J and S events is an imperfect expression of the complicatednD esoteric rules for volume trading signals (cf. “head and shoulders” pattern{1)).
Table 5 has already been referred to as showing a certain ‘‘correlation’”’ (asmeasured by excess coincidence of volumes events) between the Dow JonesIndustrial volume and the total market volume. This excess expresses a prop-erty of these two volume sequences which is quite analogous to the relationbetween a single stock sequence and a market average, or between two stocksequences in the same industry, say railroads or steels. In other words, thesetwo volume sequences contain a random variable in common. The relationbetween correlation due to a common random variable, and excess coincidenceof events due to a common random variable, would be an interesting questionto explore.
DISCUSSION AND SUMMARY
The methods of this paper were devised, and the data assembled for theexplicit purpose of testing for the correctness of the discrete transient method.In so doing we also tested for the random walk model. The discrete transientmodel was confirmed for price minima, poorly or not at all for price maxima.The volume sequence tended slightly to lead, and avoid lag in the sense ofcoincidences, the price sequence, but in a slightly asymmetric way. In generalshorter time interval data (daily, weekly) tended to show more “non-randomwalk” properties than for longer intervals (monthly). The most puzzling con-clusion from the data is the failure to find, except for daily data on a singlestock, and excess of (Ap, V) coincidences. This is more puzzling becausean excess is predicted by the random walk model, the discrete transient model,and folklore.
In view of the imperfect agreement between theory and observation, both forrandom walk and discrete transient model, at the moment we can only suggestQUANTITATIVE TESTS FOR STOCK PRICE MODELS 339
what further investigations might be undertaken to clarify these matters.Prime targets for further examination are the daily sequences on individualstocks, since our single example of this showed the most “non-random” struc-ture.
It would also be helpful if some synthetic random walks were constructed inwhich the volume (number of steps per day) were varied according to either alognormal or Pareto-Levy distribution [3]. and these synthetic sequencesexamined by the method of coincident events. Such a test would be helpful inshowing what can be expected from such a “extended” random walk model,with variable number of steps between observations. All of this could be muchbetter done by machine than by graphical methods, which the author findsexceedingly tedious. A computer could be readily programmed to: (a) analyzea real price and volume sequence, and (b) at the same time generate and analyzea random walk with the number of steps each day exactly equal, or proportionalto the real volume sequence. The difference in the statistzcal properties of thesynthetic vs. real price and volume sequences would be a good test of the non-random walk properties actually present.
APPENDIX I
The Exact and Poisson Approximation for the Occurrence of k Coincidencesin T Intervals
Let us assume that nz events R, and ny events V occur independently ofeach other and the calendar date (assumptions 1-4) in a span of T days, oneor none of either type of event per day. The total number of ways this mayoccur is
T! T!
———_—. . A-lny (T — ny)! ne(T — ne)!
Out of this total number of ways, exactly k pairs, or (R(t), V(t)) coincidencescan occur in
a AQ(nn — (nay — BENT — (ne + rv — Bi)!
different ways. The ratio of Equation A-2 to A-1 is then the probability P(k)of exactly k comcidences. Rearranged, this ratio is
oe la = (ne rm —®) il mei ll = ot]
(SPS: Ms
This is the hypergeometric distribution (Ref. 8, p. 42, with n=T, m=ne,r=ny)
 340 AMERICAN STATISTICAL ASSOCIATION JOURNAL, JUNE 1967
We repeatedly apply to the bracketed factors in A-3 the general formula
 
T! a—= Ts ex( log. (1 — s/T)) A-4(T-j! zExpand the logarithm and obtain, for j smaller than ./7,T!mo Ti exp(—?/2T + 9/27) A-5(T-