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Methodology

The main problem in attempting an objective study of stock prices is the identification of trends. What seems like a downtrend to one observer is just a minor correction in an overall uptrend to another. As an objective criterion, I used moving average crossovers.

If the current value of a stock index is larger than the moving average, the trend is up (and vice versa). Points where the trend changes are called crossovers. An upwave is the interval from the minimum that lies between a negative and a positive crossover to the maximum that occurs before the next crossover. A downwave is defined correspondingly (Figure 2).

I used stock price data for the Dow Jones Industrial Average from May 13, 15185 to January 30, 1986 because it contained one of the most distinctly pronounced advances in recent years (from September 18, 1985 to January 7, 1986) and any hypothesis developed from this data could be tested on more recent data.

My program determined the upwaves and downwaves and, for each wave, calculated the number of advancing days, the number of declining days, total amount of advances, total amount of declines and, where these values were not zero, the average advance, the average decline, number of advancing over declining days, total advances over declines, average advances over declines, and total advance (decline) over new advance (decline).

I ran the program for moving average cycle lengths of 5,10, 15,20, 25, 30,35,40, 45,50, 60,70, 80, 90 and 100 days. In addition, I manually compared the lengths of subwaves in the upwave from September 18, 1985 to January 7, 1986 and determined the ratios of the total number of advancing days over declining days in this wave. I examined the neutral period of June 20,1985 to September 18,1985 using the same ratios. Figure 3 are observations for the 10-,20- and 30-day moving averages. Moving averages of cycle lengths from 5 to 100 days gave similar results.

Assuming the market behaves similarly whether the trend is up or down, I combined observations for upand downwaves, using reciprocal ratios for downwaves (declining Г· advancing days instead of advancing Г· declining days). There is no obvious consistency in these values. I did a two-tailed t-test on each of the ratios to statistically test the null-hypothesis that the mean value of the ratio is 1.618.

The null hypothesis says there is no validity to the claim that two variations of the same thing can be

FIGURE 1:

FIGURE 2:

FIGURE 3: NA = number of advancing days, ND = number of declining days, A = total advances, D = total declines, A = average advance, D = average decline, D= net advance [decline].

distinguished by a specific procedure. In hypothesis testing, the null-hypothesis is never accepted, but can be rejected if warranted by the data. A small level of significance is desirable because it represents the probability of mistakenly rejecting the null-hypothesis when the hypothesis is, indeed, true.

Hence, rejecting the null-hypothesis in these tests means that, with only a slight probability of error, the tested ratio is not 1.618. For the ratio of total advances over net advances, the null-hypothesis could be rejected at the 1% level of significance for 10-day moving averages. For 20- and 30-day moving averages, the null-hypothesis could not be rejected at this significance level, but it could be rejected at the 10% level for 20-day moving averages.

For the ratio of advancing days to declining days, the null-hypothesis could be rejected at the 1% level for 10-day moving averages, but could not be rejected for the 20- or 30 day moving averages.

For total advances vs. total declines, the null-hypothesis could be rejected at the 5% level for 10-day moving averages, but could not be rejected for the 20- or 30-day averages.

Finally, for the average advance vs. the average decline, the null-hypothesis could be rejected at the 10% level for 10-day moving averages, but could not be rejected for the 20- or 30 day moving averages.