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Naive Trading Rules in Financial Markets

Wiener-Kolmogorov Prediction Theory: A Study of "Technical Analysis"

The attention that technical analysis receives from financial markets is somewhat of a puzzle. According to Wiener-Kolmogorov prediction theory, time-varying vector autoregressions (VARs) should yield best forecasts of a stochas �tic process in the mean square error (MSE) sense. Yet, quasitotality of traders use technical analysis in day to day forecasting although it bears no direct relationship to Wiener-Kolmogorov prediction theory. In fact, technical analysis is a broad class of prediction rules with unknown statistical properties, developed by practitioners without reference to any formalism.

This article investigates statistical properties of technical analysis in order to determine if there is any objective basis to the popularity of its methods. Broadly, there are two issues of in �terest. First, can one devise formal algorithms that can generate buy and sell signals identical to the ones given by technical analysis—that is, are any of these rules (mathematically) well defined? The second issue is to what extent well-defined rules of technical analysis are useful in prediction over and above the forecasts generated by Wiener-Kolmogorov prediction theory.

* The author is professor of economics at the Graduate School and University Center of the City University of New York. I would like to thank Chris Sims for his comments. I also would like to thank Jonathan Sampson.

This article attempts a formal study of techni �cal analysis, which is a class of informal prediction rules, often preferred to Wiener-Kolmogorov predic �tion theory by partici �pants of financial markets. Yet Wiener-Kolmogorov predic �tion theory provides optimal linear fore �casts. This article in �vestigates two issues that may explain this contradiction. First, the article attempts to devise formal algo �rithms to represent various forms of tech �nical analysis in order to see if these rules are well defined. Second, the article discusses under which conditions (if any) technical analy �sis might capture those properties of stock prices left uncxploitcd by linear models of Wiener-Kolmogorov theory.

Normally, just the resources spent on using and developing new forms of technical analysis should provide sufficient motivation for this article. However, a series of interesting papers makes such a study more relevant. For example, Brockett. Hinich. and Patterson (1985) and Hinich and Patterson (1985) have argued that several time scries, among them asset prices, arc stochastically nonlinear. Thus any method that can capture the nonlincarity of asset prices can potentially improve forecasts generated by the Wiener-Kolmogorov prediction theory. For example, Wiener-Kolmogorov theory will not utilize the information contained in higher-order moments of nonlinear processes. It is possible that, in developing technical analysis, practitioners have informally attempted to use the information contained in higher-order moments of asset prices.

In fact, it appears that since the October 19, 1987, crash of financial markets, traders have shown more interest in technical analysis—possibly because a crash of that magnitude is a nonlinear event, and the framework provided by the Wiener-Kolmogorov theory would fail to handle it properly.

In particular, linear models are incapable of describing at least two types of plausible stock market activity that are of interest to partici �pants in financial markets. First is the problem of how to issue sporadic buy and sell signals. By nature, this problem is nonlinear. The decision maker observes some indicators, and at random moments, issues sig �nals. VARs cannot explicitly generate such signals. The second exam �ple involves "patterns" that may exist in observed time series. Linear models such as VARs can handle these patterns only if they can be fully characterized by the first- and second-order moments. This basi �cally involves any pattern with smooth curvatures. A speculative bub �ble, which generates a smooth trend and then ends in a sudden crash, cannot be handled easily by linear models.

This article shows that most patterns used by technical analysts need to be characterized by appropriate sequences of local minima and/or maxima and will lead to nonlinear prediction problems. It is well known that the theory of the minima and maxima of stochastic pro �cesses can be very tedious (Leadbetter, Lindgren, and Rootzen 1983). Under these circumstances, technical analysis may serve as a practical way of using the information contained in such statistics. At the least, this is a possibility that needs to be investigated.1

To the best of my knowledge, there is no formal study of the predictive power of technical analysis. Existing studies arc mostly di �rected toward practical applications, informal treatments of which Pring (1980) is a good example. One of the first illustrations of technical analysis is the discussion of Dow theory in Rhea (1932). Although not directly related to any form of technical analysis, the survey by Tong (1983) and the pioneering work of Granger and Andersen (1978) pro �vide some of the tools used here.2

The article is organized as follows. First, I discuss some reasons behind conducting such a study. In the next section I introduce the notion of Markov times and show that a rule of technical analysis has to generate Markov times in order to be well defined. I then discuss results that can help in deciding whether a rule generates Markov times or not. I show under what conditions well-defined forms of technical analysis can be useful over and above the Wiener-Kolmogorov predic �tion theory. Finally, I provide examples using the Dow-Jones industri �als from 1792 to 1976.

I. The popularity of technical analysts admits a second explanation. If markets are efficient, asset prices would behave (approximately) as Martingales. Then. VARs would yield trivial-looking forecasts, such as {X,,, = X,. С‚ = 1,2...). Finding it unattractive to report such forecasts that remain constant over the forecasting horizon, traders might use (irrationally) techniques thai give them nonlrivial-looking forecasts, even though they are suboptimal. This interpretation requires that financial markets continue lo allo �cate significant resources on a practice lhat has negative relurns.

2. A recent example lo the popularity of technical analysis is the following. "Starting today The New York Times will publish a comprehensive three-column market chart every Saturday. . . . History has shown that when the S&P index rises decisively above its (moving) average the market is likely to continue on an upward trend. When it is below the average that is a bearish signal." [New York Times. March II. 1988]

3. Pring (1980). p. 2.