Empirical Results.
To illustrate how one can test the predictive value of technical
analysis, I select the method claimed to work the best according to participants
in financial markets. "Although techni cal analysts caution that investors
should consider a variety of factors in trying to discern the market's
direction, they say the single, clearest factor is probably the 150-day moving
average. History has shown that when the (Dow-Jones) index rises decisively
above its moving average the market is likely to continue on an upward trend.
When it is below the average it is a bearish signal."6
The moving average method was one of the few rules that generated Markov
times. Also, these Markov times were easy to quantify. This, plus one other
consideration, made me choose stock prices as the X, in (3), and
the 150-day moving average as the Xf. We then use the algorithm
in (4) to obtain a sequence of Markov times {С‚,}. The last
consideration for making these selections was the availability of a long sample
for the Dow-Jones index. In fact, when using Dow-Jones indus trials it is
possible to go all the way to 1792 and work with almost a 200-year-long monthly
data series. This greatly facilitates investigating the predictive power of technical analysis since on-and-off
prediction rules are likely to yield relatively few signals compared to regular
monthly data.
Proposition 2 and the corollary that follows it provide the necessary
framework to do the empirical work. According to these, we need to show that,
given a long autoregression, the addition of Markov times to the right-hand
side does not improve forecasts of Dow-Jones indus trials. If this is the case,
then the rule in question will have no pre dictive value.
Thus I let
С†>0,
n(13)
i= 1
i = 1
where
D, =
I ifX,> Xf has
occurred at t given X, _, < ДГ*_ i, - I if ЛГ, < Xf has occurred at / given X,., > ДГ?_„ 0 otherwise,
and where the disturbances {*,} form an innovation sequence with re spect
to the finite history of X,,
According to this, e,+tl measures all unpredictable events
between / and / + p.. The {(3,-} represents the contribution of the Markov
times {t,} in explaining X,, С† over and above the own past of the series. To the extent the D,'s
are obtained from {A*,.,, . . . , X,_k), they
should have no contribution to forecasting beyond the finite history of X,
if this latter is a linear process.
There is an important point that concerns inference with equation (13).
Note that (13) requires a sufficiently long auto regressive compo nent (i.e., a
large k). Otherwise, if Рє is small, then some D,_/s may become significant simply because they are
calculated from a more distant past of {X,}.
The parameter p. in (13) determines how many periods ahead one is
forecasting. It captures the claim that the moving average method de tects
changes in long-run trends, and that it is not necessarily useful for
1-period-ahead forecasts. Hence the value of p, should be selected as greater
than 1 or 2 months. In the empirical work reported below, I selected p-
(arbitrarily) as 12 months. The results remain qualitatively similar for p.
greater than 12. Inference with the equation shown in (13) appears to be
straightforward at the outset. However, if p, > I, the errors of equation
(13) will be serially correlated, and this needs to be taken into account. In
fact, the error structure in these equations will always be given by a (p. -
I)th-order MA process:
€, = v, + a,v,_, + a2v,_2
+ a,v,_, + .. . + fl^_,v,_p „ (14)
where the {v,} are the innovations in the X, process.
I corrected for the serial correlation shown in (14) using Herman's
efficient procedure. In fact, the {€,} can be consistently estimated by
applying ordinary least squares to (13). The periodogram of these (first-stage)
residuals is then calculated. The Fourier transform of X, is
divided by the corresponding entries of the square root of the peri odogram of
residuals. This series is then transformed back to the time domain. Equation
(13) is estimated with these transformed data.
Empirical results are provided in tables 1-3. The results are interest ing.
The F-tests on the D, are insignificant for the subperiods
1795-1851, and 1852-1910. However, they are highly significant for the pe riod
1911-76. Thus the particular moving average rule of 150 days seems to have a
significant predictive power for the latter part of the sample. It is
interesting to note that any general belief by market participanls thai such a
rule is useful would be self-fulfilling and would lead to significant {С‚,}.
For this last period, all lags of the dummy variable that indicates buy
(+ 1), sell (- 1), and no action (0) signals are significant. Further more, the
signs are in the right direction, in that they are all positive. It is also
interesting to note thai the coefficients of the dummy vari able have a nice
reverse V shape, with the peak occurring at lag 23 (table 1).
Hence, the moving average method does seem to have some pre dictive
value beyond the own lags of Dow-Jones industrials. In fact, the results
displayed in these tables remained qualitatively similar when different values
were used for p., except for p. = 1, where the 150-day moving average turned
out to be insignificant in all equations.7
7.
Estimates of the same equation with С† = I yields no significant lags for the dummy variable
in consideration. This supports ihe conlention (hat (he method predicts
longer-run behavior of the X,.
Conclusions
This article discussed some criteria that one can apply in evaluating
the set of ad hoc prediction rules widely used in financial markets and
generally referred to as technical analysis. I showed that a few of these rules
generate well-defined techniques of forecasting. Under the hypothesis, economic
time series are Gaussian, and even well-defined rules were shown to be useless
in prediction.
At the same time, the discussion indicated that if the processes under
consideration were nonlinear, then the rules of technical analysis might
capture some information ignored by Wiener-Kolmogorov prediction theory.
Tests done using the Dow-Jones industrials for 1911-76 suggested that
this may indeed be the case for the moving average rule.
Category: Methods of technical analysis
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