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Predictive Power of Technical Analysis

The fact that some methods of technical analysis admit a formal defini �tion is important. Yet well-defined sequences of finite Markov times {t,} may still have no predictive value. Thus, the next question is. Under what conditions (if any) would the well-defined procedures of technical analysis be useful in prediction over and above the standard econometric models?

There are two results. The first deals with the usefulness of technical analysis under the assumption that observed data can be characterized as linear processes. I adopt the following definition of linearity.

Definition. A process {X,}, E[X,] < �= is said to be linear, or has the linear regression property, if, for jaO,

E[XI+S\X,_^X,_2, • - • • в  *!-*] = <*l*r-1 + • . - + <*kX,-k-

That is, the process is linear if expectations of X, given finite past A"s are linear in the latter. In particular, Gaussian processes are linear. In fact, the class of processes that have the linearity property is identical to the sub-Gaussian processes (Hardin 1982). However, our definition of linearity is not identical to the one given in Hardin (1982), who does not discriminate between past and future A"s as conditioning factors.

Remark. It is interesting to note that the definition of linearity that we have here is not equivalent to E[X,\X,_{, РҐ,_Рі, X,_$ . . .1 being a linear combination of past X,"s. It is possible to construct finite moving average (MA) processes with infinite autoregressive representations, that are not linear according to the definition used here.5

I now show that nonlinearity of asset prices is a necessary condition for the usefulness of technical analysis.

Proposition 2. If the X, process is linear in the sense above, then no sequence of Markov times obtained from a finite history of {A1,} can be useful in prediction over and above (vector) autoregressions.

Proof. If {t,} are Markov times obtained from a finite history of {A",}, they must be measurable with respect to {X,, X,^„ . . . , X,_t}, some finite A*. This means that

= E[X,^s\{XrX,_t,. . . ,X,_k)\

due to the linearity of {A",}.

This proposition may have important implications for technical anal �ysis. First of all, it can be seen thai one necessary condition tor the usefulness of any technical analysis rule is the requirement that asset prices be nonlinear in the sense of the definition above. For example.

5. An inleresling example provided by the referee is the following: Let t, be i.i.d. and

[ - 1 with probability 2/3 I   2   with probability 1/3.

Construct the process X, using

X, = e, + -5t,_|.

Clearly. X, has an infinite autoregressive representation, hence, is a linear combination of all past Ays. Yet. E[X,\X,.,) cannot be linear in Р›",.,. To sec this, note thai E\X,\X,.t в   0) can be directly calculated to be - 1/2, yet if E[X,\X,.\ = 0| were linear in Р›*,., in the sense of our definition, E\X,\X,.l = 0] would have to equal zero. This contradiction implies that the X, process cannot have a linear regression property as defined here.

if a rule is well defined and yet stock prices are Gaussian, then, due to proposition 2, we immediately know that the rule is useless as a prediction technique. If, however, there is some evidence that stock prices are nonlinear, technical analysis may be useful in prediction— that is, it may be a simple way of taking such nonlincarities into ac �count. Under these conditions, the question of whether technical anal �ysis rules have any predictive power becomes an empirical issue.

Recent work such as Hinich and Patterson (1985) and Diebold (1989) provide evidence on the nonlinearity of data from financial markets. Yet because notions of linearity and nonlinearity used in these papers and here are not identical, these empirical results do not necessarily imply that there are forms of technical analysis useful in prediction. For example, Diebold (1989) shows that taking nonlinearities into con �sideration does not improve forecasts of exchange rates, although there appears to be a great deal of evidence that these latter are non �linear.

Since Martingales are linear processes, a corollary to proposition 2 is the following:

Corollary. If the X, process is a Martingale, then no sequence of finite Markov times {С‚,}, calculated from a finite history of {X,}, can be useful in prediction over and above linear regressions.

The point is that, if some technical analysis rules are indeed useful in prediction, then this should rule out a Martingale representation for the series under consideration. Using these propositions we provide some empirical results.