Barriers

Price Barriers in the Dow Jones Industrial AverageAuthor(s): R. Glen Donaldson and Harold Y. KimReviewed work(s):Source: The Journal of Financial and Quantitative Analysis, Vol. 28, No. 3 (Sep., 1993), pp.313-330Published by: University of Washington School of Business AdministrationStable URL: http://www.jstor.org/stable/2331416 .Accessed: 18/01/2012 12:33

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JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS VOL 28, NO 3, SEPTEMBER 1993

Price Barriers in the Dow Jones Industrial Average

R. Glen Donaldson and Harold Y. Kim*

Abstract

This study tests the popular claim that the DJIA's movements around key reference points affect "investor sentiment" and thus price behavior. It is found that the DJIA's rise and fall is indeed restrained by "support" and "resistance" levels at multiples of 100 (e.g., 2800, 2900, 3000, etc.) but that, having broken through a 100-level, the DJIA then moves by more than otherwise warranted. A Monte Carlo study and comparisons with other indices confirm the significance of these findings. This suggests that some agents may trade on the basis of the DJIA but does not necessarily suggest that the market is inefficient.

I. Introduction

The commentaries of financial market analysts frequently imply that certain

values of widely followed asset prices and aggregate market indices hold special

significance for market participants. Perhaps the best example of this can be found

in the unusual amount of attention that the financial press pays to the behavior of

the Dow Jones Industrial Average (DJIA) as it approaches and crosses?or fails

to cross?multiples of 100 and especially 1000. The DJIA's reluctance to cross

"the 3000 barrier" during the latter part of 1990 and early 1991, for example, was seen by some financial commentators as a sign of weakness, with one Wall Street Journal report claiming that"... the inability ofthe Dow Jones industrials to

push above the psychologically important 3000 level was a key sign that the stock market was faltering" (Wall Street Journal, January 2, 1991, p. R3). Conversely, the DJIA's eventual breaking of the 3000 barrier was seen by some traders as a

sign of strength. The large price increase that occurred as the DJIA rose above

the 3000 mark on June 1, 1991, for example, was explained by one analyst as

follows: "Once we crossed the 3000 mark there were a number of investors who

were finally convinced of the bullish status of this market and decided to jump on

the bandwagon" (Toronto Globe andMail, June 2, 1991, p. B10).

* Faculty of Commerce and Business Administration, University of British Columbia, Vancouver,

BC, Canada, V6T 1Z2, and Department of Economics, Princeton University, Princeton, NJ 08544, respectively. This paper is a substantial revision of Donaldson (1990). The comments of Giuseppe Bertola, John Campbell, Ludger Hentschel, Alan Kraus, Eduardo Ley, Rick Mishkin, Richard Quandt, Ken Stewart, Hal Varian, JFQA Referee Wayne Joerding, and seminar participants at the University of British Columbia, Princeton University, the NBER Summer Institute, and the 1991 Canadian Economic Association meetings are thankfully acknowledged, as is the financial support of Princeton University's John M. Olin Research Program.

313

314 Journal of Financial and Quantitative Analysis

Since one would preserve the DJIA's relevant information on relative price movements with a rescaling of the index, and would lose in the rescaling process

only information regarding the particular numerical values assumed by the index, it is unclear why some values of the DJIA should be perceived by the market to be

of more importance than other values. The preceding quotations, however, suggest that some traders nevertheless believe in the special importance of key reference

points in the DJIA. In particular, the preceding assertion that traders may "jump on the bandwagon" of buying (selling) once the DJIA breaks up (down) through a "psychologically important" level suggests that the crossing of one of these

sentiment-induced barriers may push the index up (down) more than otherwise

warranted. Frequently used phrases such as "support level" and "resistance level"

also imply that, until such time as the psychologically important barrier is broken, increases and decreases in the DJIA may be restrained so as to prevent crossing of

the barrier level. A report on this subject in the Wall Street Journal, for example, states that psychologically important "... stock index or average levels become

sentiment signals. As market barometers approach those levels, stock buyers become less aggressive, fearing a turn in the market, while sellers need less coaxing to drop their price a notch or two" (Wall Street Journal, April 15, 1991).1

Given the preceding discussion, the goal in this paper is to test the null hy?

pothesis that the behavior ofthe DJIA is not influenced by its proximity to special levels of the index against the alternative hypothesis that psychologically impor? tant levels in the DJIA form sentiment-induced "barriers" that inhibit the rise or

fall of the index past a barrier support or resistance level but that, having broken

through a barrier, a breakout or bandwagon effect causes the index to move up or

down by more than usual. The results of this study's tests reject the no-barriers

null in favor of the barriers alternative. In particular, it is found that movements

in the DJIA are indeed restrained by barrier support and resistance effects at 100-

levels in the index (e.g., 2000, 2100, 2200, etc.) but that, having broken through a

100-level, the index rises or falls by more than usual. This anomalous behavior is

not present in the less popular Wilshire Associates 5000 index, which was used as

the main control for the DJIA, nor is it present in any of the computer-generated indices that were examined. The results are shown to be robust with respect to the

time period investigated and the statistical significance of the results is confirmed

with a Monte Carlo study based on randomized returns and AR(1)-GARCH(1,1) simulations. The results suggest that some less-than-fully-rational agents may be

basing their buying/selling decisions on the value ofthe widely reported DJIA, but

do not necessarily suggest that the market is inefficient.

II. Data

As the most widely reported of all stock market indices, the DJIA is the

obvious candidate for testing the presence of psychological pricing barriers. As

the primary benchmark against which to compare the DJIA's behavior, the Wilshire

Associates 5000 (WA) index is also examined. The WA is a more obscure index

than the DJIA and therefore a less likely candidate for barriers. The WA was

1 The authors thank Hal Varian for bringing this quotation to their attention.

Donaldson and Kim 315

selected over other possible alternatives (e.g., Standard & Poor's 500, Russell

2000, Value Line, etc.) because only the WA has values of comparable magnitude to the DJIA and has therefore cycled through enough hundred levels to justify a

direct comparison using the frequency distribution methodology described below.

Both the DJIA and WA data are taken from the Dow Jones Tradeline Database. The

limited availability of the WA determines the sample period: October 14, 1974, to May 18, 1990 (the day the data were collected). To test robustness, various

subsamples were also studied and DJIA data from as far back as January 1, 1952, were examined.2

The DJIA begins with a price of 673.5 on October 14, 1974, and ends with

a price of 2819.9 on the last day of the sample. The unconditional mean and

variance of the daily return (log of price relatives) are 0.036 percent and 0.012

percent, respectively. The WA begins at 3581.0 and rises to 5394.0 on August 10, 1981. The index was then rescaled (by Wilshire Associates) to begin at 1399.0 on

August 11, 1981, and end with a value of 3383.7. The WA price series was not

adjusted to remove this discontinuity since the tests are based on the realized price level of the index. (As shown by the Monte Carlo study below, the differences

found between the WA's behavior and the behavior of the DJIA are not a result of

the break in the WA series.) Excluding the discontinuity day, the unconditional

mean and variance of the WA's daily return are 0.033 percent and 0.007 percent,

respectively. For the one test below that uses returns in addition to the price level, the WA's daily return for August 11, 1981, was set equal to the unconditional mean

return over the rest of the sample.

III. Testing the Barriers Hypothesis

In this section, the barriers hypothesis is formalized and three statistical tests

for the presence of pricing barriers in stock indices are presented. Evidence of

barriers at 100-levels in the indices is specifically looked for?e.g., 2800, 2900,

3000, etc.?since these are the price levels to which special significance is most

frequently attached by financial commentators. (Tests for a separate barriers effect at 1000-levels may also be of interest. However, the limited price range of the DJIA

precludes use of the frequency distribution tests employed in this paper.) Results

produced suggest that there may indeed be barriers at 100-levels in the widely followed DJIA but not in the relatively obscure WA. The strength and robustness of the results are confirmed in Section IV.

A. Implications of Barriers in the DJIA

Before barriers in the data can be tested for, the hypotheses to be tested must first be formalized. Unfortunately, there is no abstract economic theory that

says how stock price indices should behave in the presence of sentiment-induced barriers at "psychologically important" reference points. There is, however, a well-

established theoretical literature on the closely related problem of price behavior in the presence of exogenously enforced barriers to price movements in other types

2The authors thank Eduardo Ley for providing the data for 1952-1974, which he collected from Pierce(1986).


of assets, such as central bank-enforced target zones for currency exchange rates.

Along with the implications of the preceding quotations from financial market

analysts, results from this closely related theoretical literature help in formalizing the barriers hypothesis for stock indices.

Central banks frequently announce "target zones" or trading ranges for the

foreign exchange value of their currency. The central banks state that they will enforce these zones by intervening in the market so as to keep the price of their

currency from exceeding the upper barrier of the zone or from falling through the lower barrier of the zone. If the banks really did enforce these zones, then the price barriers would be "imperforate" since they would confine the asset price forever. More often than not, however, the exchange rate is eventually allowed to break through the barriers. When this happens and the barriers are perforated, a

"realignment" typically occurs during which the asset price moves an unusually large amount and then a new set of barriers is established that contains the new

price level. An interesting problem for international economists has been the

modeling of exchange rate behavior as the currency's price fluctuates within, and

occasionally breaks out of, its barriers. Even though these exchange rate barriers exist because of outside intervention, while what barriers (if any) that exist in the DJIA are presumably the product of investors' sentiment-induced trading behavior, results from this literature are nevertheless interesting for this research because they

help in understanding the potential dynamic behavior of the DJIA as it fluctuates

within and breaks out of its supposed barriers.3

Krugman's (1987), (1991) initial investigation of exchange rate barriers, and

subsequent work by Froot and Obstfeld (1989) and Flood and Garber (1991), assume that the barriers are imperforate and thus that the asset price can never take on values outside of the target zone. Given the additional assumption that

there is no anticipated excess profit available from holding the asset as its price fluctuates around various points within its imperforate barriers, Krugman proves that the amount of time the price spends close to the barriers must be greater than

the amount of time it spends away from the edges of the barrier zone. Thus, the

ergodic distribution of price realizations within an imperforate barrier zone should

be U-shaped, with more frequent price realizations close to the edges of the zone and less frequent realizations in the center of the zone. In other words, Krugman proves that an asset price confined by imperforate barriers should, in general, hover

just above its lower barrier (i.e., support level) or just below its upper barrier (i.e., resistance level). In terms of barriers in stock indices, this result implies that if the DJIA were truly constrained to never pass through a 100-level barrier (e.g., if the index were constrained to always assume values between 00 and 99), then DJIA

3Since the barrier (or target) zone model is both well known and somewhat involved, the full formal model is not replaced here. The basic nature ofthe model, however, is easily summarized. The model is based on the assumption that there is an asset whose price is influenced by two factors: a fundamental (usually a Brownian motion with drift) and a term that captures the market's expectations about future price changes in the presence of barriers. To derive the equilibrium behavior of the price, one begins by using Ito's Lemma to solve for the expected rate of change in the price as a function of the fundamental. The result is then plugged back into the pricing equation to produce the second order differential equation that governs the behavior of the asset's price as it fluctuates within, and breaks out of, the barrier zone. The solution to this equation, and thus price behavior, depends on several integration constants, the values of which depend on whether the barrier zone that constrains the asset's price movements is perforate or imperforate. For an excellent review, see Bertola and Caballero (1992).


prices in the single digits and nineties would be observed more often than prices in

the middle range between the barriers. Over this study's sample period, however, the DJIA has assumed values from below 600 to over 2900 and has therefore

obviously passed through, or perforated, several 100-levels. Thus, the theoretical

results for imperforate barrier zones should probably not apply to the behavior of

the DJIA around 100-levels.

More recently, Bertola and Caballero (1992) and Pesenti (1990) consider the

converse case of a perforate barrier zone in which a support level or resistance level can be crossed, though perhaps with some difficulty. Once a barrier is breached in

this environment, a "breakout" or realignment occurs during which the price ofthe

asset moves an unusually large amount and then a new barrier zone is established

that contains the new price level. This case appears closer to that of pricing barriers in stock indices. Bertola-Caballero prove that, given the sudden change in prices associated with the barrier-crossing breakout effect (in their case, an

exchange rate realignment), the absence of unexploited excess profit opportunities

implies that the amount of time the price spends close to the barriers must be less

than the amount of time it spends away from the edges of the zone. Thus, the

ergodic distribution of price realizations within a perforate barrier zone should

be hump-shaped, with less frequent price realizations close to the edges of the

zone and more frequent realizations in the center of the zone. Empirically, this

hump-shaped distribution is frequently observed in exchange rate data. In terms

of barriers in the DJIA, this result implies that, if there are in fact perforate barriers

at 100-levels in the DJIA, then the DJIA should exhibit prices near these 100-level

barriers less frequently than it exhibits prices around other levels in the index and,

indeed, this is precisely what the DJIA data reveal below.

The intuition behind this result can be seen in the following example. Suppose that the DJIA is currently at 2980 and that fundamentals will call for a 19-point increase with probability 0.5 and a 21-point increase with probability 0.5. Further,

suppose that if the 21 fundamental is observed then, as suggested by the preceding

quotations from market analysts, the market's excitement at crossing the 3000

barrier will produce a "breakout effect" during which excited investors "jump on

the bandwagon" and produce a 30-point increase in the index.4 Given the breakout

effect, for there to be no expected excess profit from holding the asset at the price 2980 (i.e., for the expected increase to still be 20 points) it must be true that, if the

19-point fundamental is observed and the barrier is not broken, then (ignoring the

small rounding error) the market will produce only a 10-point rise in the index.

This restraining ofthe fundamental increase might be called the "resistance effect," as explained by one ofthe preceding quotations in which it was stated: "How can a 'resistance level' exist? Because traders believe it is there_Stock index or

average levels become sentiment signals. As market barometers approach those

levels, stock buyers become less aggressive, fearing a turn in the market, while

4Recall the quotation from the introduction, for example, in which the 27-point increase that brought the DJIA over the 3000 mark on June 1, 1991, was explained by one analyst as follows: "Once we crossed the 3000 mark there were a number of investors who were finally convinced of the bullish status of this market and decided to jump on the bandwagon". {Toronto Globe and Mail, June 2, 1991, p. B10.)


sellers need less coaxing to drop their price a notch or two" (Wall Street Journal,

April 15, 1991). This type of behavior, the rationality of which is discussed below, has two

important implications. First, note that the final price in the preceding example with barriers is either 3020 or 2990, instead of the no-barriers fundamental prices 3001 or 2999. Thus, as suggested by the preceding quotations and the formal

theoretical results of Bertola-Caballero, an index with perforate barriers at 100-

levels will exhibit prices less frequently around 100-levels and more frequently

away from 100-levels than would an index without pricing barriers at 100-levels.

Second, since a "breakout effect" occurs when a barrier is crossed, pushing the

price to 3020 instead of 3001 in the preceding example, the percentage rate of

change?i.e., return?on the DJIA as it rises up through (fails down through) a

100-level barrier should be greater than (less than) the return conditional on the

DJIA rising up through (falling down through) other points in the index. These

two implications of barriers in stock indices are tested formally below.

B. Price Level Tests

Let Pt be the value of the stock price index on date t and Mt be the last two

digits in the integer portion of the price (i.e., Mt = modulo[integer(Pf) : 100]). For

example, if Pt = 1234.56, then Mt = 34. Barriers at 100-levels in the index (e.g.,

2000,2100,2200, etc.) thus become a barrier atM = 00. The preceding discussion

of the formal theory and the discussion of practitioner quotes suggest that, if there

are barriers at M = 00 in the DJIA, then the DJIA should close on prices with

M-values in the neighborhood around M = 00 less frequently than it closes on

prices with M-values in the middle range. Of course, in the absence of barriers

when no level of the index is more important than any other, one would expect no

strong pattern in the distribution of closing prices across M-values (evidence in

support of this claim is provided by a Monte Carlo study presented below). The series of vertical lines in Figure 1 plot the frequency with which the DJIA

closes with its price in each of the 100 M-cells. To provide the eye with a better

indication of the pattern that emerges, this figure also plots as the series of stars

joined by the dark line, the five-cell centered moving average of frequencies. In

similar fashion, Figure 2 plots the individual frequencies for each of the 100 M-

cells and the five-cell centered moving average for the WA index. As expected, the

WA distribution exhibits no strong pattern. However, consistent with the barriers

hypothesis, the DJIA appears to close, on average, less frequently around M = 00

and more frequently away from 00.5

5 Another possible way to observe the barrier effect is to calculate the M-value transition matrix from day t ? 1 to day t; i.e., the probability of the occurrence of Mt given Mt-\. The breakout effect should produce lower frequencies for M-cells just above 00 for a rising index and just below 00 for a falling index. This matrix was computed and some direct evidence of the breakout effect was found. Given Mt-\ = 98-99, for example, the DJIA closes with Mt = 00-01 or 02-03 with only 5-percent probability, but closes with Mt - 04-05 with 11-percent frequency. This result is consistent with the breakout story: having broken from 98-99 up through the 00 barrier, the index shoots through the lower M-values to close in M-cells 04-05. There is also some evidence of a similar effect for downward moves. For example, prob(M? = 98-99 \Mt-i - 02-03) = 4%, whereas

prob(Mf = 96-97 |Mf_i = 02-03) = 10%. Although these results generally support the barriers story, the transition matrix is omitted from this paper because small sample problems prevented the drawing


FIGURE 1

Frequency of DJIA Prices

Frequency %

0.8

0.6

0.4 0 10 20 30 40 50 60 70 80 90

M-Value (last two digits in the price)

Actual Frequency SCell Moving Average

To test the apparent systematic deviation from uniformity in the DJIA dis?

tribution, f(M) is defined to be the frequency with which an index closes with its

last two digits in cell M, minus 1 percent. A first price level test then involves

regressing/(M) for each of the 100 M-cells on a constant and a dummy variable that isolates groups of cells in the neighborhood around M = 00. The regression is

/(Af) = OL + pDij + UM\ M = 00,01,..., 99,

of statistically meaningful conclusions. Note however that, since the unconditional frequency is simply the sum down each column?or across each row?of the conditional transition matrix, Figure 1, which plots the unconditional M-value frequency, completely summarizes the information contained in the DJIA's conditional transition matrix. This information suggests what is observed in Figure 1: less density around M = 00 in the DJIA.


FIGURE 2

Frequency of WA Prices

Frequency %

0.4 0 10 20 30 40 50 60 70 80 90

M-Value (last two digits in the price)

Actual Frequency 5Cell Moving Average

where Dy is a dummy variable that isolates cells in the range from / to j, and Um is a random error. The results of these regressions appear in Table 1, where the

dummies are

?>98-02 = 1 if M > 98 or M < 02, = 0 otherwise;

D95_04 = l ifM>95 or M < 04, = 0 otherwise;

D90-09 = 1 if M > 90 or M < 09, = 0 otherwise.

The preceding discussion suggests that under the no-barriers null hypothesis (3 should be zero, while under the barriers alternative hypothesis (3 should be negative (i.e., less density around the barrier).

The negative coefficients on the DJIA dummy variables in Table 1 reject the

no-barriers null hypothesis and confirm this study's earlier casual observation that

the DJIA closes less frequently on index values whose last two digits are in the


TABLE 1

Price Level Tests for Density around 00

f(M) = a + 0D,j+UM\ M=00,...,99

f(M): the frequency with which the index closes with its last two digits in cell M, minus 1 percent.

Djj. a 1/0 dummy variable that isolates cells in the range from /to j. All entries x100 (standard deviations in parentheses). Test Hq (no barriers): f3 = 0 vs H^ (barriers are present): f3 < 0. Note the significant negative /3s for DJIA but not for WA (for example, D98_02 reveals that DJIA closes 0.2353% - 0.0118% = 0 2235% less than expected around the 00-level). Also note that, as hypothesized, the DJIA barrier effect weakens further away from 00.

neighborhood around the 00. The coefficient on D98_o2> f?r example, implies that the DJIA on average closes 0.2353% - 0.0118% = 0.2235% less frequently than expected in each of the five cells around M = 00. This result is confirmed

in the data directly with the finding that the percentage deviations from 1 percent in the five M-cells around 00 are: 98 = -0.31, 99 = -0.24, 00 = -0.29, 01 =

?0.26, 02 = ?0.01. Standard f-statistics for rejection of the no-barrier null for

the DJIA index range from greater than 99 percent for D98-02 to over 95 percent for ?>95_o4 and D90-09. As implied by the barriers hypothesis, rejection of the no-barriers null becomes weaker as one examines wider ranges around the barrier. Just as important as the DJIA's rejection of the no-barriers null is the WA's failure

to reject; all f-statistics from the WA regressions are insignificant. These results

therefore confirm Figure l's and 2's casual observations of a possible barrier at

M = 00 in the widely-followed DJIA index, but no barrier in the less popular WA

index.

A second price level test examines not just the tails of the frequency distribu?

tion immediately around the barriers, but the entire shape of the distribution. Here, not only does one need to specify the null hypothesis that the distribution should be

uniform in the absence of barriers, but the alternative that the distribution should

have some particular shape if barriers are present. The work of Bertola and Ca-

ballero (1992) and others suggests that a hump-shape is an appropriate alternative.

One can examine this possibility by running the regression,

f(M) = a + (3M + 8M2 + UM', M = 00,01,... ,99.


Under the null ofno barriers 8 should be zero, while under the barriers alternative

8 will be negative. The results of such a procedure are reported in Table 2. Again, it appears that the DJIA possesses a barrier while the WA does not.6

TABLE 2

Price Level Tests for Hump-Shape

f(M) = a + PM+6M+ UM; M=00,...,99

Index a P 6 R2

DJIA -0.0931 0.0070 -0.00008 9.80 (0.0498) (0.0023) (0.00002)

WA 0.1189 -0 0058 0.00005 1.00 (0.0737) (0.0034) (0.00004)

f(M): the frequency with which the index closes with its last two digits in cell M, minus 1 percent. All entries x100 (standard deviations in parentheses). Test H0 (no barriers): 6 = 0 vs. H<\ (barriers are present): 6 < 0. Note the significant negative 6 for DJIA but not for WA.

C. Conditional Returns Tests

As a third test for barriers, each index's behavior is studied as it progresses through various M-cells from one closing price to the next. To conduct this test, one must

first calculate Rt = Ln(Pt) ? Ln(Pt-\): the index "return" for date t. Second, the

value of Rt is assigned to each of the M-cells implicitly passed by the index on

date t. Thus, if Pt-i = 1492 and Pt = 1497, then the return Rt = 0.0033 would be

assigned to cells M = 93, 94, 95, 96, and 97 since these are the cells through which

the index passes as the price rises from 1492 to 1497. This procedure is repeated for

every day in the sample. Finally, for each of the 100 M-cells (M = 00,01,..., 99) the average (i.e., mean value) of all the returns that were assigned to that cell is

calculated. This average is defined as RM: the average daily return conditional on

having passed through cell M (M = 00,01,..., 99). The behavior of RM across

the various M-cells forms the basis for the conditional returns tests.

Appealing to the barriers story, it is seen that the existence of a barrier at 100-

levels in an index implies a negative correlation between RM and M. This occurs

for four reasons. First, if a "breakout effect" really does occur once the DJIA

crosses a barrier, then the buying pressure associated with traders "jumping on the

bandwagon" of optimism once the DJIA rises up through a 00-level will push the

index well past the 00-level. This will result in less frequent closings of the DJIA

just above the 00-level, as observed in the previous section, and will also result

in larger-than-normal positive returns as the index rises up through low M-values

6If one had a theory about where the peak of the barriers distribution should occur, one could also test the overidentifying restriction that ?/3/28 = X, where X is the peak value. Since there is no truly formal theory that says how investors who believe in psychological price barriers should behave, however, it is difficult to state with certainty where the peak should occur, other than to say that it should be away from M = 00 (the authors thank the referee for bringing this point to their attention). Also note that for a single peak to exist it must be true that there are no minor barriers at intermediate levels, such as 50s (no significant evidence of such a minor barrier in the DJIA is found).


such as 01,02,03, etc. Second, if falling down through a barrier is eonsidered bad news by the market, then the extra selling pressure associated with a downward breakout would push the DJIA down by more than otherwise warranted once a 00 barrier is crossed. This will result in less frequent closings ofthe DJIA just below the 00-level, as observed in the previous section, and will also result in larger-than- normal negative returns as the index falls down through high M- values such as 99, 98, 97, etc. Third if, as suggested above, the barrier restrains movements past a 00 resistance level, then movements up toward high M-values would be restrained to be smaller than normal. Similarly, declines toward low M-values would be restrained to be smaller than normal by the barrier support level. Thus, with a barrier at M = 00, low M-cells will be filled with larger than normal increases and smaller than normal decreases, while high M-cells will be filled with smaller than normal increases and larger than normal decreases. The existence of a barrier at 100-levels (i.e., M = 00) in the index thus implies a negative correlation between

Rm and M. Of course, in the absence of barriers, one would expect no significant correlation between RM and M (evidence in support of this claim is provided by a Monte Carlo study presented below).

Table 3 presents the results from running the regression,

Rk = a + (3M + UM; M = 00,01,...,99,

where Um is a random error. From the preceding paragraph, it is seen that under the no-barriers null hypothesis (3 should be zero, while under the barriers alternative

[3 should be negative. For the DJIA, the /-statistic of -10.805 on (3 strongly rejects the no-barriers null in favor of the barriers alternative, while for the WA, the null cannot be rejected. These results are interpreted as support for the existence of barriers at 100-levels in the popularly tracked DJIA, but not in the more obscure WA index.

TABLE 3

Conditional Returns Tests

RM = a + /3M+UM; M = 00,. ,99

Index

DJIA

WA

0.1899 (0.0055) 0.0654

(0.0029)

P

-0.0010 (0 0001) 0.0002

(0.0001)

R*

53.90

7.21

RM: the average daily return conditional on having passed through cell M(M= 00, All entries x100 (standard deviations in parentheses). Test H0 (no barriers). (3 = 0 vs. H-j (barriers are present): j3 < 0. Note the significant negative (3 for DJIA but not for WA.

,99).

IV. Evidence from Additional Tests and Indices

The tests of the preceding section indicate the existence of barriers at 100- levels in the widely followed DJIA index. The goal in this section is to examine the strength of the results.


A. Simulation Results

The potentially nonstandard nature of the distributions involved in Section

III's tests leads to the conducting of Monte Carlo simulations to determine the

true statistical significance of the DJIA results. Two types of simulations are run:

one based on randomized actual returns (a quasi-bootstrap method) and another

based on modeling returns as an AR-GARCH process. The methodologies and

the calculated p-values are presented below.

For the return-randomization "bootstrap" simulation, the actual returns of the

DJIA were randomly reordered and then a new sequence of prices was produced

using the true starting value of the DJIA and the subsequent price levels implied

by the reordered returns series. The regressions in Tables 1, 2, and 3 were then

run on the simulated price series to obtain parameter estimates and standard errors

for the key parameters. This procedure was repeated 1000 times to obtain dis?

tributions from which the significance of this paper's results can be determined.

The distributions produced are roughly symmetric but fat-tailed. Although weak-

ened, the coefficients on the DJIA dummy variables D98-02,?)95-04> and D90-09 from Table 1 's regression generally remain significant, with p-values of 0.009,

0.130, and 0.216, respectively. The 8 in Table 2 exhibits a p-value of 0.107. The

DJIA average return regression M-value coefficient from Table 3 remains highly

significant with a p-value of 0.008.

A second Monte Carlo simulation is based on the modeling of the DJIA daily return as an AR(1)-GARCH( 1,1) process.7 Following Akgiray (1989), the process is described as

*,|/,_i ~ F(pt,ht),

pt = (f)o + (f)iRt-i,

ht = a0 + aie2_{ + (3\ht-u

et = Rt- ({)q- (f)\Rt-\,

where a0 > 0,a\ > 0, and f3\ > 0. Conditional on the information set /,_i,

Rt is drawn from the distribution F(-, ?) with conditional mean pt and conditional

variance ht. The parameter vector z = [a0, a\, (3\, <po, <j)\ ] is estimated assuming normality

of the conditional distribution. Maximization of the log-likelihood function was

accomplished using the MAXLIK module in GAUSS 386, version 2.0. Standard

errors of the point estimates were calculated using the estimated information matrix

at the optimum. Parameter estimates are as follows (standard errors in parentheses),

Rt = 0.000415 + 0.0678/?, _ 1,

(0.000142) (0.0173)

7See Engle (1982), Bollerslev (1986), and Akgiray (1989) for discussions of autoregressive con? ditionally heteroskedastic (ARCH) and GARCH processes. The particular specification used in this

paper follows work by Akgiray (1989).


ht = 0.00000329 + 0.0796<_ l + 0.892/z,_ {.

(0.00000073) (0.00835) (0.0133)

All parameter estimates are statistically significant. The unconditional variances

of e and/?, o2 = a0/(l ? a\ ? (3\) and o\ = o2/(l ? 4>2X), respectively, are 0.0114

percent and 0.0115 percent; the unconditional variance of R accords well with the

value estimated from the raw data of 0.012 percent. The near-integrated nature of

the estimated GARCH process is in line with other empirical studies, implying the

presence of a high degree of volatility persistence (see Lumsdaine (1990)); this

paper's sample parameter estimates indicate an a\ + (3\ = 0.971.

As with the randomized-returns "bootstrap" simulation, 1000 simulated price series were constructed based on the initial DJIA price and subsequent prices

implied by the returns processes generated using the estimated AR-GARCH pa? rameters. Again, the Monte Carlo distributions are symmetric but possess vary?

ing degrees of excess kurtosis. The p-values for Table 1 's DJIA dummy coef?

ficients implied by the AR-GARCH simulation are 0.008, 0.090, and 0.164 for

^98-02^95-04. andZ)9o-o9, respectively, which are slightly more significant than

the corresponding p-values from the randomized-returns simulation. The p-value for 8 in Table 2 is 0.069. For Table 3's average return regression, the M-value

coefficient is again highly significant with a p-value of 0.002. The results of this

section thus appear to confirm the statistical significance of Section III's results.

B. Simulated Price Series

To ensure that Section III's results are not simply a product ofthe general shape traced by the DJIA's price sequence over the time period studied, the regression tests are performed on several computer-generated series that resemble the DJIA in

shape. The first series is an exponential series that has the same starting and ending values and mean return as the DJIA. A second series was produced by multiplying the DJIA by the arbitrarily chosen constant 1.40. As stated in the introduction, such a rescaling of the price series retains the DJIA's relevant returns information

(and shape), but loses information on the particular values of the DJIA's digits. A third price series was created by starting on the true value of the DJIA and

using subsequent price levels implied by a returns series constructed as in the

AR-GARCH simulation described above. In order to be sure of reproducing the

general shape of the DJIA, the first AR-GARCH series the computer produced that possessed the following characteristics was chosen: i) an ending value within

a 100-point band around the DJIA's ending value, ii) a minimum value no less

than that of the DJIA, iii) a global maximum value no greater than 50 points above

the DJIA's maximum, and iv) a maximum over the first 2000 observations no

greater than the DJIA's local maximum over this interval. Finally, a fourth series

was produced in the same manner as above except the bootstrap simulated returns

were used instead of the AR-GARCH returns.

Tables 4, 5, and 6 contain the results from regression tests similar to those

conducted in Tables 1, 2, and 3 above. The absence of significantly negative dummy coefficients in Table 4, significantly negative 8s in Table 5, and significantly


f(M): the frequency with which the index closes with its last two digits in cell M.

Djj. a 1/0 dummy variable that isolates cells in the range from /'to j. All entries x100 (standard deviations in parentheses). Note that, unlike the DJIA in Table 1, the (3s in this table are not significantly negative.

negative M-value coefficients in Table 6 reveals that there is no evidence of DJIA-

style barriers in the generated indices. It is therefore concluded that neither the

general shape of the DJIA nor the statistical properties of the DJIA's returns process can account for the DJIA's apparently anomalous behavior around 100-levels.

C. Robustness

The final concern of this study is for the sensitivity of the results with respect to the sample period and the particular starting and ending values of the DJIA

index. To ensure that the results are not dependent on these factors, the begin?

ning and ending prices from the sample were iteratively truncated and the tests

were rerun. This procedure was performed 30 times to produce starting values

between 608 and 674 and ending values between 2645 and 2831: the results were

not altered appreciably. The sample was also shortened by deleting up to 1000

observations, 100 observations at a time. The smaller sample size weakened the

f-statistics somewhat, but in all cases the dummy regression coefficients from Ta?

ble 1 's regressions remained negative, as did the 8 from the regression in Table


TABLE 5

Price Level Hump Tests on Simulated Data

f(M) = a + (3M+SM + UM\ M = 00,..., 99

f(M): the frequency with which the index closes with its last two digits in cell M, minus 1 percent. All entries x100 (standard deviations in parentheses) Note the lack of significantly negative 6s.

TABLE 6

Conditional Returns Tests on Simulated Data

RM = a + /3M+UM; M = 00_99

RM: the average daily return conditional on having passed through cell M(M = 00,. All entries x100 (standard deviations in parentheses) Test H0(no barriers): (3 = 0 vs. /-/-((barriers are present)- (3 < 0. Note that, unlike the DJIA in Table 2, the ps in this table are not significantly negative

99).

2. The M-value coefficient in Table 3's returns regression also remained highly

significant throughout. To examine further the robustness of the results, the DJIA series was extended

back to January 1, 1952, and all tests were rerun. As Tables 7, 8, and 9 reveal, the 1952-1990 results are very similar to the 1974-1990 results. To check the

significance of these results, the following AR(1)-GARCH(1,1) specification is

estimated for the DJIA for 1952-1990,

Rt = 0.000317 + 0.1412/?,_1,

(0.000069) (0.0107)


ht = 0.00000085 + 0.0697<_ { + 0.920/zr_ x.

(0.00000013) (0.00490) (0.0059)

Monte Carlo results based on this specification produced p-values of 0.015,0.117, and 0.344 for Table 7's dummies Dg%_02,?>95-04> and D90-09, respectively, a p- value of 0.020 for Table 8's 8, and a p-value of 0.004 for Table 9's slope parameter. A randomized-return simulation produced p-values of 0.018,0.154, and 0.342 for Table 7's dummies 1)98-02^95-04^ and D90--09, respectively, a p-value of 0.031

for Table 8's 8, and a p-value of 0.005 for Table 9's slope parameter. It is therefore

concluded that the barriers phenomenon is a robust feature of the DJIA.

f(M): the frequency with which the index closes with its last two digits in cell M.

Df.-: a 1/0 dummy variable that isolates cells in the range from / to /. All entries x100 (standard deviations in parentheses). Test H0(no barriers): (3 = 0 vs. H^ (barriers are present): (3 < 0. Note the significant negative /?s for the 1952-1990 DJIA.

f(M): the frequency with which the index closes with its last two digits in cell M, minus 1 percent. All entries x100 (standard deviations in parentheses). Test /-/0(no barriers): 6 = 0 vs. /-^(barriers are present): 6 < 0.

V. Conclusions

The possible existence of barriers to price movements in the daily closing values of the DJIA have been tested for. Consistent with the economic theory of

price behavior in the presence of exogenously imposed barriers to price movements


TABLE 9

Conditional Returns Robustness Test

RM = a + (3M+UM; M = 00,.. ,99

Index a (3 R2

DJIA 1952-1990 0.1339 -0.0009 58.5 (0.0043) (0.0001)

Rm the average daily return conditional on having passed through cell M(M = 00,.. , 99). All entries x100 (standard deviations in parentheses). Test H0(no barriers): (3 = 0 vs. /-/-((barriers are present): (3 < 0. Note the significant negative /3, as suggested by the barriers hypothesis and by the author's earlier DJIA results.

(e.g., Bertola and Caballero (1992)) and consistent with the claims of financial

market commentators, it is found that the DJIA closes fewer times, on average, on

index values in the neighborhood of 100-levels. It is also found that the DJIA's

conditional returns are negatively correlated with associated M-values. Together, these results imply that the DJIA's rise and fall is restrained by barrier support and

resistance levels but that, having broken out through a barrier, the DJIA moves up or down by more than usual. The statistical significance of the results is confirmed

through randomized-returns and AR(1)-GARCH(1,1) Monte Carlo simulations.

The anomalies do not appear in the less popularly-tracked WA index, nor do they

appear in the numerous computer-generated series that were studied. The results

are not dependent on sample size, beginning and ending values, or the general

shape of the index. The DJIA's anomalous behavior is thus interpreted as an

indication that, as suggested by many financial market commentators, there may well be pricing barriers at 100-levels in the widely followed DJIA index.

This paper closes with a discussion of the implications of price barriers for

market rationality and efficiency. Standard finance theory suggests that it is the

information on relative price movements (i.e., returns) that should be of interest

to rational market participants. The authors know of no theory that states that

special levels of some arbitrarily scaled index should be of particular importance in a totally rational market. The finding of barriers in the DJIA therefore suggests two possibilities. The first possibility is that the market is totally rational and that,

contrary to the understanding of standard finance theory, certain values of widely followed indices are in fact of genuine significance. (Is it possible, for example, that rational agents who find continuous portfolio rebalancing costly might use the

numerical value of the widely reported DJIA as an easily obtainable signal that

they should engage in optimal but infrequent portfolio rebalancing?) A second, and perhaps more likely, possibility is that some traders are in fact less than fully rational and may indeed be using the DJIA's highly publicized crossing of, or

failure to cross, 100-level reference points as sentiment signals on which to base

their buying/selling decisions. Even if this second possibility is true, however, the

finding of a barrier at DJIA 100-levels, does not necessarily imply that the entire

market is irrational. Froot et al. (1992) and DeLong et al. (1990), for example,

present models in which rational traders follow fundamentally irrational trading


rules in order to profit from the existence of noise traders and thereby produce anomalies in the market as a whole.8

Finally, it is important to note that the finding of barriers at DJIA 100-levels

does not necessarily imply that the market is inefficient. In particular, the existence

of a less-than-fully-rational market does not necessarily imply returns predictabil-

ity given the weak-form information set that contains the value of the DJIA.9

Indeed, as proven by Bertola and Caballero (1992), if there truly are barriers at

100-levels in the DJIA then the hump-shaped price distribution present in Figure 1 is in fact implied by the absence of predictable excess returns. It thus appears that, even if some subset of the market is less than perfectly rational, the Efficient

Market Hypothesis remains a useful paradigm.

References

Akgiray, V. "Conditional Heteroscedasticity in Time Series of Stock Returns: Evidence and Forecasts." Journal of Business, 62 (Jan. 1989), 55-80.

Bertola, G., and R. Caballero. "Target Zones and Realignments." American Economic Review, 82 (June 1992), 520-536.

Bollerslev, T. "Generalized Autoregressive Conditional Heteroskedasticity." Journal of Econometrics, 31 (April 1986), 307-327.

DeLong, B.; A. Shleifer; L. Summers; and R. Waldmann. "Positive Feedback Investment Strategies and Destabilizing Rational Speculation." Journal of Finance, 45 (June 1990), 379-395.

Donaldson, R. G. "Psychological Barriers in Asset Prices, Rationality and the Efficient Market Hy? pothesis." Financial Research Center Memorandum #114, Princeton Univ. (1990).

Engle, R. F. "Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom Inflation " Econometrica, 50 (July 1982) 987-1007.

Flood, R., and P. Garber. "The Linkage between Speculative Attack and Target Zone Models of Exchange Rates." Quarterly Journal of Economics, 106 (Nov. 1991), 1367-1372.

Froot, K., and M. Obstfeld. "Exchange Rate Dynamics under Stochastic Regime Shifts: A Unified Approach." Journal of International Finance, 31 (Nov. 1991), 203-230.

Froot, K.; D. Scharfstein; and J. Stein. "Herd on the Street: Informational Inefficiencies in a Market with Short-Term Speculation

" Journal of Finance, 47 (Sept. 1992), 1461-1484. Krugman, P. "Trigger Strategies and Price Dynamics in Equity and Foreign Exchange Markets." NBER

Working Paper 2459 (1987). _. "Target Zones and Exchange Rate Dynamics." Quarterly Journal of Eco?

nomics, 106 (April 1991), 669-682. Ley, E., and H. Varian. "A Note on the Dow Jones Digits." Univ. of Michigan manuscript (1991). Lumsdaine, R. L. "Asymptotic Properties ofthe Quasi-Maximum Likelihood Estimator in GARCH( 1,1)

and IGARCH(1,1) Models " Harvard Univ. manuscript (1990). Pesenti, P. "Perforate and Imperforate Currency Bands," Yale Univ. manuscript (1990). Pierce, P. S. The Dow Jones Averages 1886-1985, Illinois: Dow Jones Irwin (1986). Toronto Globe andMail, various issues. Wall Street Journal, various issues.

8In particular, Froot et al. (1992) prove that, as long as a significant fraction of the market is following a "chartist" rule, it may pay perfectly rational traders to also chart, even though they realize that charting is fundamentally irrational. DeLong et al. (1990) demonstrate a similar result for the case in which noise traders follow a positive feedback trading rule.

9This claim was checked by examining the average return Rt conditional on Mt- \. No significant relationship was found between Rt and Mt-\. In particular, knowing what value the DJIA closed at yesterday (i.e., Mt-\) does not help predict the return Rt for today. (Note that, since Rjy for day / is calculated conditional on cell M having been passed during day / (note past tense), while Rt measures the expected future return calculated conditional on yesterday's value of M, observing a barrier's effects ex post in Rjy is perfectly consistent with not observing effects on expected returns Rf. In particular, the absence of barrier effects in expected returns does not negate this study's previous results, it simply implies that the market is efficient.) Ley and Varian (1991) present additional evidence that suggests that this paper's finding of barriers in the DJIA does not imply market inefficiency.

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