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APPLICATION OF T-TECHNIQUE FACTOR ANALYSIS TO THE STOCK MARKET JOHN W. FRENCH Published online: 10 Jun 2010.

To cite this article: JOHN W. FRENCH (1972) APPLICATION OF T-TECHNIQUE FACTOR ANALYSIS TO THE STOCK MARKET, Multivariate Behavioral Research, 7:3, 279-286, DOI: 10.1207/s15327906mbr0703_1 To link to this article: http://dx.doi.org/10.1207/s15327906mbr0703_1

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APPLICATION OF’ T-TECHNIQUE FACTOR ANALYSIS TO THE STOCK MARKET JOHN W. FRENCH Sarasota, Florida

ABSTRACT

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.o

Proportional changes in the prices of 425 individual common stocks were recorded for each of 44 quarter-year periods. After normalization of the data f o r each stock, sums of cross-products were computed among the time periods across stocks. Factor analysis of the cross-product matrix yielded seven factors. By cumulating the factor loadings successively over the 44 quarters, so as to simulate absolute values rather than change scores, six of these factors appear to be generated by the simplicial nature of the cross-product matrix, while the seventh was an expected factor representing general stock market strengths and weaknesses. Some possibilities f o r prediction are discussed, but little in the way of direct prediction of stock market fluctuations is supported by these findings.

Many thousands of books have been written on how to “beat” the stock market, and, by now, dozens of investment services use computers t o grind out oceans of information for their bewildered subscribers. It is perhaps fitting for us to try the success of some of our own multivariate methods in solving some aspects of this problem. Let us consider the several “techniques” or models of factor analysis. For example, the phrase “a factor analysis of stocks” seems to imply an analysis based on the intercorrelations of the price movements of individual stocks across occasions or time periods. This will give rise t o factors representing groups of stocks that behave in similar fashion. It will display the stocks in logical groups: foods, oils, railroads, etc. Members of each of these groups tend to move in a parallel way, because particular economic pressures tend to have simultaneous effects on all of the stocks in one industry. However, a welter of charts and analyses are already concerned with the special movements of groups of stocks such as these. Nothing new would be revealed by proving that stockholders of U. S. Steel and of Bethlehem Steel usually have the same joys or exasperations as they look at the financial page on a particular day. A more interesting question seemed to be: are there any patterns of strengths and weaknesses among individual stocks that represent good times to buy or good times to sell? In order to identify different characteristic patterns, interrelationships were computed among t i m e periods and across stocks. Thus two time periods having a high correlation with one another would be charJULY, 1972.

279

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acterized by having the same pattern of strong stocks and weak stocks. If a particular pattern or factor characterized good buying or good selling periods, it would make sense to watch for a recurrence of this same pattern in the future. It is interesting here to consider here the letter labels that have been proposed for the various factorial “techniques” used in psychometric research (Cattell, 1966, page 70). The most usual one, widely designated as R technique, makes use of the intercorrelations of tests taken across subjects. Strictly speaking, the stock market has neither subjects nor tests. However, let us regard the individual stocks as subjects and regard price change during a specified time as the one and only test being used throughout this study. It follows, then, that the best description of the present study is one that employs the interrelationships of time periods or occasions across individual stocks or subjects. This is T technique.

DATA With a sigh of relief that he did not have to arrange for hundreds of subjects and did not have to pay these subjects for working conscientiously on a long battery of tests, the author collected all of the data for this study from a chartbook (Securities Research Co., 1970). An even easier but more expensive way to obtain data might have been to use the data bank of some financial house. This chartbook, however, was ideal. It gives very clear charts of the monthly ranges and closing prices of 975 stocks plotted on a logarithmic scale. On these charts price changes were measured by ruler in units of .5 mm. A 10% change equals six of 44; these units. A stock that doubled would receive a score of one that halved itself would be scored - 44. The time periods used were long enough to represent important changes; they comprised the 44 quarters in the years 1959-1969. The stocks selected for study were the first 381 in the chartbook that were fully charted for the pertinent 11-year period plus an appropriate proportion of railroad and utility stocks, which appeared in a separate section of the book. A total of 425 stocks was used.

+

.J

ANALYSIS AND RESULTS It was considered that, for the purposes of this study, irrelevant variance was represented by (1)the secular trend or general strength of a stock as measured by its mean change score over the 44 quarters, and (2) the volatility of a stock as measured by the a80

M U LTlVARIATE BEHAVIORAL RESEARCH

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J

standard deviation of its change score during the 44 quarters. Therefore, the data matrix was submitted to row scaling (but not column scaling), so that the data for each stock would have a mean of zero and a standard deviation of 1.00.l It should be emphasized here that this procedure permitted the columns or occasions, which were t o be interrelated, t o vary in both mean and standard deviation. In particular, for a quarter containing a general rise in the market, the mean price change for individual stocks was strongly positive and had high variability. Quarters containing a market decline had a negative mean and also had high variability. The matrix of sums of cross-products of the columns in the data matrix was computed. Entries in this matrix have a mean near zero. About one in 20 of the entries reach A.100, and about one in 50 reach k.200. The larger relationships are recorded for quarters that include unusually sharp market moves. Variations in column means produced by general market fluctuations might have been eliminated by the use of intercorrelations instead of sums of cross-products. However, it was not only felt that the elimination from the factor analysis of such important differences among the quarters would be misleading, but it was believed that the equivalent of statistical elimination of these effects would be carried out anyway, if, as expected, they were to fall neatly into one or more of the factors, leaving the other factors independent of the market fluctuations. Principal components factor analysis was based on this matrix. There was no clear criterion for the number of factors that should be rotated. The Eigenroot for the first factor was 3.017. They then dropped abruptly to 317. Seven factors were rotated, the last having a root of .099. Table 1 gives the factor pattern after rotation by maxplane (Eber, 1966). Table 2 gives the intercorrelations of factors. Since the table of loadings lists only dates rather than meaningful names of tests, it is not easy to project meaning into these results. The only meaning that the writer could see on first inspection was that, for quarters with strong market rises, Factor 7 has large positive loadings, while, for quarters including major declines, Factor 7 has large negative loadings. Factor 7, then, contains the variance of means that was allowed to remain in the columns of the data table. 1. Scaling, computation of sums of cross-products, and factor analysis

were carried out by Herbert W. Eber, Psychological Resources Support Systems, Atlanta, Ga. JULY, 1972

281

Of particular practical interest would be a factor that yielded extreme loadings just prior to market declines, or extreme loadings of opposite sign just prior to market rises, or both. No such factor appears. Careful inspection and tabulations of the appropriate entries in the table of sums of cross-products seemed to confirm that no such €actor existed, and so would be unlikely to Table 1 Factor Pattern

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~~

Quarters Factors: 1 1969-1 -.03 2 -.01 3 .oo 4 .10 1960-1 -.03 2 -.02 3 -.03 4 .10 1961-1 .08 2 -.06 3 -.06 4 .08 1962-1 -.14 2 -.02 3 -.09 4 -.04 1963-1 -.08 2 .01 3 -.09 -.lo 4 -.07 1964-1 2 .04 .oo 3 4 -.06 -.06 1966-1 -.01 2 3 .02 4 .06 1966-1 -.03 2 -.07 3 .06 4 -.06 .26 1967-1 .22 2 3 .oo 4 .12 -.14 1968-1 .06 2 3 -.04 4 .06 .06 1969-1 -.07 2 -.02 3 .06 4 ~~

282

2 -.07

-.07 .ll .06 .08

.17 .07 .23 .02 .03 .16 .lI -.14 -.06

3 .06 -.01 .07 -.01 .22 .13 -.OS -.OS

.03 702 .05

-.17 -.03 -.03

.01 .09 -.04

-.lo

-.03 -.07

-.lo -.06 -.05

-.16

-.07

.08

-.lo -.06 -.09 .ll

.01

.01 .03 -.17 .o 1 -.18 -.22 -.26 -.06 .12 -.06

-.03 -.02 -.02 .06 .06 -.01 -.02

.01 .05 .07 -.05

.06

-.14 -.08

-.OS -.07

.02 .06

.oo

-.03 .03 .08

.02 -.07 .06

.02 .07 -.02 -.03 .06 .04 .31 .23

4

.oo -.18 -.04 .02 .03 .07 .04 .16 .03 .08 -.06 .07 -.03 -.02

-.08 -.06

.06 .07 -.04

.06 -.03 -.12

.oo

~~

6 .04 .15 -.04 .13 .03 .16

6 .07 -.08

-.01 .01 .06

.oo

.04 -.OS

.02 .01

-.01 .25 -.04 .07

.05 4 6 -.11

.08

-.02 .16 -.16 .03

.14

.oo

.oo

-.02 -.02 .07 -.23

-.lo

-.27

.30 .13 .10 -.01 -64 .09 .36

-.08 -.06 -.08

.06 .06 -.03 -.06 .04 .21 -.04 .16 -.14 .03 -.20 .06 -.14 .13 -.06 -.12 -.07

.10

.12 .06

-.14

.oo

.18 .08 -.OS

-.06

-.04 -.06 .03 .06 -.02 4 6 -.04 4 4 -.02 .12

-.04 .04

.OS .OS

-.06 .43 .12 -.06 -.33 -.29 .01 -.lo

-.04 -.07 ~~

.30 -.OS

-.06 -.08 -*11 -.06

.oo

-.lo

.17

.oo

-.02

.oo

.07

.06 -.06 -.06 -.03

-.03 .09 .06 -.04 -.01 .ll .06

-.04

.03

.oo

-.07 -.08

.07

-.86

.oo

-.06

-.11 -.04

.oo -.06 .oo

.48

.13 -.04 .02 .06 .24 -.20 .13 -.07 .41

-.04

-04 .05

7 .31 .13 -.19 .04 -.14

~~~~~

MULTIVARIATE BEHAVIORAL RESEARCH

Table 2 Intercorrelations of Factors

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=

1. 2. 3. 4. 5. 6. 7.

1

2

1.00 .01 -.61 -.04

.01 1.00 -.26 .14 -.09 -.38 .60

.04

-.41 .68

3 -.61

4

6 .04 -.09 .14 .33

-.04 .14 -.30

-.25

1.00 -.30

1.00 .33 -.39

.14

.53 -.74

1.00

-.32 .ll

.48

7 .68 .60 -.74

6 -.41 -.38 .53 -.39 -.32 1.00 -.78

.48 .ll -.78

1.00

appear even if some other factorial of rotational methodology were t o be used. To facilitate a different kind of interpretation of the factors, consideration was given to the source of the data. Since each datum represents a price change, the loadings themselves can be regarded as some function of change. Therefore, the algebraic cumulations of loadings for each successive quarter were expected to generate some kind of absolute measure that could be charted meaningfully against time. Accordingly, the loadings in Table 1 were cumulated and are plotted chronologically in Figures 1 and 2.

DISCUSSION Figure 1 shows the plot of the cumulated loadings for Factor 7. Since positive loadings seem to reflect rises in the market and I

I

1959

I

I

1

I

1960

1961

I

I

1962

I

1

I

1963

1

I

I

1964

1965

I

I

I

I

1966

1

1967

~

I

1968

1969

Fig. 1. Comparison of the chart of cumulated loadings on Factor 7 (solid line) with Standard and Poor’s 600 Stock Average (dashed line). JULY, 1972

283

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1959

1964

1969 1959

1964

1969

Fig. 2. Charts of the cumulated loadings on Factors 1-6.

negative loadings reflect declines, this cumulation should resemble a stock market average. Plotted for comparison in Figure 1 is Standard and Poor’s 500 Stock Average. The resemblance is obvious. However, a very noticeable difference is the fact that normalization of the scores for each stock in the data matrix forced the mean loading on each factor to be close to zero, so that the graph of cumulated loadings for each factor must begin and end at about the same level. In contrast, the stock average displays an upward trend between 1959 and 1969. Other differences between the market average and the cumulation of Factor 7, such as the greater ebullience of stock prices than of Factor 7 in 1963 are not understood., Figure 2 makes it possible to compare Factors 1-6 with one another and with other charts covering the same period of years. Indeed some rhythmic patterns can be easily perceived. Factors 1, 2, and 3 look a little like single-cycle sine waves at somewhat differing phases. Factors 4, 5, and 6 look a little like double-cycle sine waves at somewhat differing phases. Considering some of the comments of Guttman (1955), Humphreys (1960)’ and others, this tendency for high or low loadings to congregate in groups of variables that are adjacent in time suggests that the crossproduct matrix has somewhat the form of a simplex. Indeed this makes psychological (or financial) sense. Under different economic conditions different industries or individual companies do 284

MULTIVARIATE BEHAVIORAL RESEARCH

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well or poorly. This causes a rotation of stocks moving into public favor and their sliding out of favor. For this reason a set of stocks that do well in one quarter are also likely to do better than average in the immediately following quarter but not necessarily i n - a quarter that is more remote in time. Therefore, adjacent quarters can be expected to be more highly correlated with one another than they are with more remote quarters. A direct check was made on the simplicity of the cross-product matrix. The effect was too small to be obvious on inspection, but some machine work showed that the figures in the principal diagonal average .09, while those in successive diagonals moving away from the principal one average -.07, -.14, and -.09. The most remote figures, those relating the first eight quarters with the last eight, average -.01. As Humphreys (1960) lamented, it is difficult t o make predictions with simplicial data. Perhaps this is basically why it is so difficult to beat the stock market. Fruchter and Fleishman’s (1967) solution to prediction with this kind of data is to look for independent predictors that would be useful a t different time periods. This method is likely to be productive where the time periods constitute different “phases of learning,” which can be expected to recur with a new class of students. However, in the case of the stock market, where each time period may represent a set of economic conditions forever lost to history, prediction for these periods becomes obsolete before it is adequately studied. Kow let us consider the possibility that Factors 1-6 actually represent something more than a slightly increased correlation between adjacent quarters. For many hours the writer stared at these charts and compared them with published charts of average prices of stocks in various industries and with charts of business fundamentals. In making meaningful comparisons, it was necessary, of course, to partial out mentally the general market fluctuations from the published charts just as the factor analysis had partialled out the influence of Factor 7 from the other six factors. A few charts seemed to match up. However, over 50 separate charts of industrial averages or of general business conditions were available for comparison to the six factorial charts. This provided lots of opportunity for coincidental matchings. For example, Factor 2 seems to represent the special strengths and weaknesses of food stocks, and Factor 6 seems to represent the same thing for financial stocks. Inspection of the price swings of individual food and financial stocks during quarters that have salient JULY, 1972

285

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loadings on Factors 2 and 6 confirmed these eye-ball comparisons, but it is still highly likely that we are dealing here with mere coincidental resemblances between factors and particular industries. However, let us for the moment suppose that these findings might not be coincidental. Let us consider that the chart of Factor 2 in Figure 2 really does reveal a latent trend in food stocks. By extrapolation of the sine wave, it seems fitting that food stocks are to do much better than the market in 1971 and are to peak out in 1972 or 1973. At least it is possible to generate some testable hypotheses in this way. These hypotheses will indeed be tested by this writer, but his hopes are forlorn, because making the assumptions about coincidence and about the periodicity of the charts was merely a logical exercise rather than a scientific argument. It must indeed be obvious that the period of the sine waves in Figure 2 probably turned out to be 11years or 5.5 years merely because the study happens to use data from an 11-year period. Nevertheless, it may be interesting to follow up the food and financial stocks over the next few years. Possibly these two industries showed up as factors in this study precisely because their periods of strength and weakness match the 11-year period of the study. If so, their periods may continue, and extrapolation will become justified. This is not a conclusion; it is merely a hypothesis for future investigation. One bona fide conclusion is possible. Since we have a situation that yields simplicial results, we can say that, to some small degree at least, the stocks that are strong during one time period are more likely than others to be strong during the next period. Therefore, you should buy the stocks that are strong. This advice is generally held to be sound, but it is not novel.

REFERENCES Cattell, R. B. (Ed.) Handbook of multivariate expe&wntal psychology. Chicago: Rand McNally, 1966. Eber, H. Toward oblique simple structure : Maxplane. Multivu&te Behuviorat Resealrch, 1966,1,112-126. Fruchter, B. & Fleishman, E. A. A lsimplicial design for the analysis of correlational learning data. Multivariate Behavioral Resewch, 1967, 2, 83-88. Guttman, L. A generalized simplex for factor analysis. Psychmetrika, 1965, 20,173-192. Humphreys, L. G. Investigations of the simplex. Psychometl.ika, 1960, 25, 313-323. Securities Research Co. ?)-Trend cycli-graphs, July 1970. Boston : Securities Research Co., 1970. 286

MULTIVARIATE BEHAVIORAL RESEARCH

APPLICATION OF T-TECHNIQUE FACTOR ANALYSIS TO THE STOCK MARKET.

Proportional changes in the prices of 425 individual common stocks were recorded for each of 44 quarter-year periods. After normalization of the data ...
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