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/. liilnitluiiiuti market participants and measuring die reaction of security prices. II prices do rot move when information is revealed, then the market is efficient with respect to that information. Although this is clear conceptually, it is hard to сап у out such a test in practice (except perhaps in a laboratory). jMalkiels third sentence suggests an alternative way to judge the efficiency of a market, by measuring the profits that can be made by trading on information. This idea is the foundation of almost all the empirical work on market efficiency. It has been used in two main ways. First, many researchers have tried to measure the profits earned by market professionals such as mutual fund managers. If these managers achieve superior returns (after adjustment for risk) then the market is not efficient with respect to the information possessed by the managers. This approach has the advantage that it concentrates on real trading by real market participants, but it has the disadvantage that one cannot directly observe the information used by the managers in their trading strategies (see Fama [1970, 1991] for a thorough review of this literature). As an alternative, one can ask whether hypothetical trading based on an explicitly specified information set would earn superior returns. To implement this approach, one must first choose an information set. The classic taxonomy of information sets, due to Roberts (1967), distinguishes among Weak-form Efficiency: The information set includes only the history of prices or returns themselves. Semistrong-Form Efficiency: The information set includes all information known to all market participants {publicly available information). Strong-Form Efficiency: The information set includes all information known to any market participant (private information). The next step is to specify a model of normal returns. Here the classic assumption is that the normal returns on a security are constant over time, hut in recent years there has been increased interest in equilibrium models with time-varying normal security returns. Finally, abnormal security returns are computed as the difference be-tweeii)lhe return on a security and its normal return, and forecasts of the abriorjmal returns arc constructed using the chosen information set. If the abnormal security return is unforecastablc, and in this sense random, then the hypothesis of market efficiency is not rejected. 1.5.1 Efficient Markets and the Law of Iterated Expectations The idea that efficient security returns should be random has often caused confiujion. Many people seem to think that an efficient security price should 1.5. Market Efficiency be smooth rather than random. Black (1971) has attacked this idea rather effectively: Л perfect market for a stock is one in which there are no profits to be made by people who have no special information about the company, and in which it is difficult even for people who do have special information lo make profits, because the price adjusts so rapidly as the information becomes available Thus we would like lo see randomness in the prices of successive transactions, rather than great continuity---- Randomness means that a series of small upward movements (or small downward movements) is very unlikely. If the price is going to move up, it should move up all at once, rather than in a series of small steps---- 1 .arge price movements are desirable, so long as they are not consistently followed by price movements in die opposite direction. Underlying this confusion may be a belief dial returns cannot be random if security prices are determined by discounting future cash flows. Smith 1968), for example, writes: I suspect that even if die random walkers announced a perfect niathematic proof of randomness, I would go on believing thai, in the long run future earnings influence present value. In fact, the discounted present-value model of a security price is entirely consistent with randomness in security returns. The key to understanding this is the so-called Law of Iterated Expectations. To stale this result we define information sets /, and / where /, С J, so all the information in /, is also in /, bul J, is superior because it contains some extra information. We consider expectations of a random variable X conditional on these information sets, written E[X I /,] or E[X 1 J,]. The Law of Iterated Expectations says that E[X I /,] = E[E[X 1 J,] I /,]. ln words, if one has limited information the best forecast one can make of a random variable X is the forecast of the forecast one would make of X if one bad superior information J,. This can be rewritten as E[X - E[X /,] /,] = 0, which has an intuitive interpretation: One cannot use limited information I, to predict the forecast error one would make if one had superior information Jt. Samuelson (1965) was the first to show the relevance of the Law of Iterated Expectations for security market analysis; 1-е Roy (1989) gives a lucid review of the argument. We discuss the point in detail in Chapter 7, but a brief summary may be helpful here. Suppose lhat a security price at time /, , can be written as the rational expectation of some fundamental value V, conditional on information /, available at lime /. Then we have p, = E[V* I /,] = E,V\ (1.5.1) The same equation holds one period ahead, so (1.5.2) Bui then the expectation of the change in (lie price over die next period is K,lI = K,[1 (V*l ~E,fV ]] = 0, (1.5.3) because /, С so ЕДЕ,411 \ ]] - E, V*] by tlie Law of Iterated Expectations. Thus realized changes in prices are unlorecastable given information in the set /,. /. 5.2 /.v Market Efficiency Testable? Although the empirical methodology summarized here is well-established, there are some serious difficulties in interpreting its results. First, any test of efficiency must assume an equilibrium model that defines normal security returns. If efficiency is rejected, this could be because the market is truly inefficient or because an incorrect equilibrium model has been assumed. T\m joint liyfiolhesis problem means that market efficiency as such can never be rejected. Second, perfect efficiency is an unrealistic benchmark that is unlikely to hold in practice. Even in theory, as Grossman and Stiglitz (1980) have shown, abnormal returns will exist if there are costs of gathering and processing information. These returns are necessary lo compensate investors for their information-gathering and information-processing expenses, and arc no longer abnormal when these expenses are properly accounted for. In a large and liquid market, information costs are likely to justify only small abnormal returns, but it is difficult to say how small, even ifsuch costs could be measured precisely. The notion of relative efficiency-the efficiency of one market measured against another, e.g., the New York Stock Exchange vs. the Paris Bourse, futures markets vs. spot markets, or auction vs. dealer markets-may be a more useful concept than the all-or-nothing view taken by much of the traditional market-efficiency literature. The advantages of relative efficiency over absolute efficiency are easy to see by way of an analogy. Physical systems arc often given an efficiency rating based on the relative proportion of energy or fuel converted to useful work. Therefore, a piston engine may he rated at 00% efficiency, meaning that on average 00% of the energy contained in the engines fuel is used to turn the crankshaft, with the remaining 40% lost to other forms of work such as heat, light, or noise. Few engineers would ever consider performing a statistical test to determine whether or not a given engine is perfectly efficient-such an engine exists only in the idealized frictiouless world of the imagination. But measuring relative efficiency-relative to the frictiouless ideal-is commonplace. Indeed, we have come in expect such measurements for many household products: air conditioners, hot water heaters, refrigerators, etc. Similarly, market efficiency is an idealization that is economically unrealizable, but that serves as a useful benchmark for measuring relative efficiency. For these reasons, in this book we do not take a stand on market efficiency itself, but focus instead on the statistical methods that can be used to test the joint hypothesis of market efficiency and market equilibrium. Although many of the techniques covered in these pages are central to the market-efficiency debate-tests of variance bounds, Euler equations, thfe CAPM and the APT-we feel that they can be more profitably applied lb measuring efficiency rather than to testing it. And if some markets turn out to be particularly inefficient, the diligent reader of this text will be well-prepared to take advantage of the opportunity. The Predictability of Asset Returns ONE OK THE EARLIEST and most enduring questions of financial econometrics is whether financial asset prices are forecastable. Perhaps because of the obvious analogy between financial investments and games of chance, mathematical models of asset prices have an unusually rich history that predates virtually every other aspect of economic analysis. The fact that many prominent mathematicians and scientists have applied their considerable skills to forecasting financial securities prices is a testament lo the fascination and the challenges of this problem. Indeed, modern financial economics is firmly rooted in early attempts to beat the market, an endeavor that is still of current interest, discussed and debated in journal articles, conferences, and at cocktail parties! In this chapter, we consider the problem of forecasting future price changes, using only past price changes to construct our forecasts. Although restricting our forecasts to be functions of past price changes may seem too restrictive to be of any interest-after all, investors are constantly bombarded with vast quantities of diverse information-nevertheless, even as simple a problem as this can yield surprisingly rich insights into the behavior of asset prices. We shall see that the martingale and the random walk, two of the most important ideas in probability theory and financial economics, grew out of this relatively elementary exercise. Moreover, despite the fact that we shall present more sophisticated models of asset prices in Chapters 4-9, where additional economic variables are used to construct forecasts, whether future price changes can be predicted by past price changes alone is still a subject of controversy and empirical investigation. In Section 2.1 we review the various versions of the random walk hypothesis and develop lests for each of these versions in Sections 2.2-2.4. I.ong-hori/.on returns play a special role in detecting certain violations of the random walk and we explore some of their advantages and disadvantages in Section 2.5. Focusing on long-horizon returns leads naturally to 1 2 3 4 [ 5 ] 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 |