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Table 2.8 reports autocorrelation matrices Y(/f) of the vector of weekly returns of live size-sorted portfolios, formed from the sample of stocks using weekly returns from July 10, 1002, to December 27, 1904 (1,095 observations). 1 .el X, denote the vector j /7 R2l llu ft ], where is the return on Ihe equal-weighted portfolio of stocks in the itli quintile. Then the Alh order autocorrelation m /n.rolX, is given by T{k) = D l/-l .t(X, i -

)(X, - /t)]D l/-, where D = diagfrrf.....a;) and /* = F.(X,). Ну this

convention, the i, element of Y(/<) is the correlation of R k with . The estimator Y(/t) is the usual sample autocorrelation matrix.

An interesting pattern emerges from Table 2.8: The entries below the diagonals of Y(/r) are almost always larger than those above the diagonals. For example, the fust-order autocorrelation between last weeks return on large stocks (Rr -\) with this weeks return on small stocks (/{ ) is 20.5%, whereas Ihe first-order autocorrelation between last weeks return on small stocks (/?n-i) with this weeks return on large stocks (Rr ) is only 2.4%. Similar patterns may be seen in the higher-order autocorrelation matrices, although Ihe magnitudes are smaller since the higher-order cross-autocorrelations decay. The asymmetry ol the Y(/c) matrices implies that the autocovariance matrix estimators V(h) are also asymmetric.

This intriguing lead-lag pattern, where larger capitalization stocks lead and smaller capitalization stocks lag, is more apparent in fable 2.9 which reports the difference of the autocorrelation matrices and their transposes. Kvery lower-diagonal entry is positive (hence every upper-diagonal entry is negative), implying thai the correlation between current returns of smaller slocks and past returns ol larger stocks is always larger than the correlation between current returns of larger stocks and past returns of smaller stocks.

Of course, the nontrading model of Chapter 3 also yields an asvm-melric autocorrelation matrix. However, we shall see in that chapter that unrcalistically high probabilities of nontrading are required to generate cross-autocorrelations ol the magnitude reported in Table 2.8.

The results in fables 2.8 and 2.9 point to the complex patterns of cross-effects among securities as significant sources of positive index autocorrelation. Indeed, l.o and MacKinlay (1990c) show that over half of the positive index autocorrelation is attributable to positive cross-effects. Thev also observe that positive cross-effects can explain the apparent profitability of contrarian investment strategies, strategies thai are contrary to the general market direction. These strategies, predicated on the notion that investors lend to overreact to information, consist of selling winners and buying losers. Selling the winners and buying the losers will earn positive expected profits in the presence ol negative serial correlation because current losers are likely to become future winners and current winners are likelv lo become future losers.

Table 2.9. Asymmetry of cross-autocorrelation matrices.

T(1)-T(l) =

Y(2) - Y (2) =

Y(3)-T(3) =

Y(4)-Y(4) =

 10.000 -0.104 -0.153 -0.195 -0.241 \ 0.104 0.000 -0.061 -0.113 -0.181 0.153 0.061 0.000 -0.054 -0.134 0.195 0.113 0.054 0.000 -0.088 ft, \ 0.241 0.181 0.134 0.088 0.000 / /0.000 -0.052 -0.079 -0.089 -0.094 \ 0.052 0.000 -0.029 -0.042 -0.055 0.079 0.029 0.000 -0.014 -0.029 0.089 0.042 0.014 0.000 -0.018 \ 0.094 0.055 0.029 0.018 0.000 / rt, /0.000 -0.035 -0.069 -0.087 -0.0934 0.035 0.000 -0.024 -0.054 -0.062 0.069 0.034 0.000 -0.022 -0.035 0.087 0.054 0.022 0.000 -0.018 V 0.093 0.062 0.035 0.018 0.000 / /0.000 -0.033 -0.059 -0.084 -0.1024 0.033 0.000 -0.024 -0.050 -0.070 0.059 0.024 0.000 -0.023 -0.049 0.084 0.050 0.023 0.000 -0.030 \0.102 0.070 0.049 0.030 0.000 У

Differences between autocorrelation matrices and their transposes for the vector of siic-sorted portfolio returns X, = [Я Л 3i Rv Л5, ) where R is the week-/ return on

the equal-weighted portfolio оГslocks in the ilh quintile, i=l.....5 (quintile 1 contains the

smallest stocks), for the sample of NYSE-AMEX stocks from July 10, 1962 to December 27, 1994 (l,fi9!T observations). Note that Y(A) s D- 2El(X,-> - ц)(К, - ir))D- a, where D = diaglaf.....a*].

But the presence of positive cross-effects provides another channel through which contrarian strategies can be profitable. If, for example, a high return for security A today implies that security Bs return will probably be high tomorrow, then a contrarian investment strategy will be profitable even if each securitys returns are unforecastable using past returns of thajl security alone. To see how, suppose the market consists of only the two stocks, A and B; if As return is higher than the market today, a contrarian sells it and buys B. But if A and В are positively cross-autocorrelated, a higher return for A today implies a higher return for В tomorrow on average, and thus the contrarian will have profited from his long position in If on average.

2. The Predictability of Asset Returns

Nowhere is it required that the slock market overreacts, i.e., that individual returns are negatively aulocorrelated. Therefore, the fact that some contrarian strategies have positive expected profits need not imply stock market overreaction. In fact, for the particular contrarian strategy that I.о and MacKinlay (1990c) examine, over half of the expected profits is due to cross-effects and not to negative autocorrelation in individual security returns.

These cross-effects may also explain the apparent profitability ol several other trading strategies that have recently become popular in the financial community. For example, long/short or market-neutral strategies in which Ion positions arc offset dollar-for-dollar by short positions can earn superior retijrns in exactly the fashion described above, despite the fact that they are designed to take advantage of own-effects, i.e., positive and negative forecasts of individual securities expected returns. The performance of mntched-boolt or flairs trading strategies can also be attributed to cross-effects as well as own-effects.

Although several studies have attempted to explain these striking lead-lag qffecls (see, for example, Badrinaih, Kale, and Noe [ 1995], Boudoukh, RicWardson, and Whitelaw [ 1994], jegadecsh and Swaniinathan [199:1], Conrad, Kaul, and Nimalendran [1991], Brennan, Jegadecsh, and Swami-nathjtn [1993], Jegadecsh and Titman [1995], and Mech [1993]), we are still far from having a complete understanding of their nature and sources.

smt-

j 2.8.4 Tests Using I.ong-IIorizon Returns

Several recent studies have employed longer-horizon returns-multi-year re-turiisjii most cases-in examining the random walk hypothesis, predictability, and the profitability of contrarian strategies, with some surprising results. Distinguishing between short and long return-horizons can be important because-it is now well known that weekly fluctuations in stock returns differ in many ways from movements in three- lo five-year returns. We consider the econometric trade-offs between short-and long-horizon returns in more detail in Chapter 7, and provide only a brief discussion here of the long-horizon implications for the random walk hypotheses.

In contrast to the positive serial correlation in daily, weekly, and monthly index returns documented by l.o and MacKinlay (1988) and others, Fama and French (1988b) and Ioteiba and Summers (1988) find negative serial correlation in multi-year index returns. For example, Potcrba and Summers (1988) report a variance ratio of 0.575 for 90-month returns of the value-weighted CRSP NYSE index from 1926 to 1985, implying negative- serial correlation at some return horizons (recall that the variance ratio is a specific linear combination of autocorrelation coefficients). Both Fama and French (1988b) and Potcrba and Summers (1988) conclude that there is

2.8. Recent Empirical Evidence

substantial mean-reversion in stock market prices at longer horizons, which they aiiribule lo the presence of a transitory component such as the y, component in (2.5.1).

There is, however, good reason to be wary of such inferences when they are based on long-horizon returns. Perhaps the most obvious concern is the extremely small sample size: From 1926 to 1985, there are only 12nonover-lapping live-year returns. While overlapping returns do provide some incremental information, the results in Boudoukh and Richardson (1994), l.o and MacKinlay (1989), Richardson and Smith (1991), and Richardson and Slock (1989) suggest that this increment is modest at best and misleading at worst. In particular, Richardson and Slock (1989) propose an asymptotic approximation which captures the spirit of overlapping long-horizon return calculations-they allow the return horizon (/ to increase with the sample size T so that ij/T converges lo a finite value S between zero and one- which shows that variance ratios can be severely biased when the return horizon is a significant fraction of the total sample period. For example, using their asymptotic approximation (2.5.10), discussed in Section 2.5.1, the expected value for the variance ratio with overlapping returns is given by i2.5.12) under RWl. This expression implies that with a return horizon of 96 mouths and a sample period of 60 years, <5=8/60=0.133 hence the expected variance ratio is (1-5)2=0.751, despite the fact that RWl is assumed to nold. Under RW2 and RW3, even more dramatic biases can occur (see, fov example, Romano and Thombs [ 1996]).

These difficulties are reflected in the magnitudes of the standard errors associated with long-horizon return autocorrelations and variance ratios (see, for example, Richardson and Stock (1989, Table 5), which are typically so large as lo yield z-stalistics close lo zero regardless of the point estimates. Richardson (1993) and Richardson and Slock (1989) show thai properly adjusting for the small sample sizes, and for oilier statistical issues associated with long-horizon returns, reverses many of the inferences of Fama and French (1988b) and Potcrba and Summers (1988).

Moreover, the point estimates of autocorrelation coefficients and other time series parameters tend lo exhibit considerable sampling variation for long-horizon returns. For example, simple bias adjustments can change the signs of the autocorrelations, as Kim, Nelson, and Startz (1988) and Richardson and Stock (1989) demonstrate. This is not surprising given the extremely small sample sizes that long-horizon returns produce (see, for example, the magnitude of the bias adjustments in Section 2.4.1).

Finally, Kim, Nelson, and Startz (1988) show that the negative serial correlation in long-horizon returns is extremely sensitive lo the sample period and may be largely due to the first ten years of the 1926 to 1985 sample. Although ten years is a very significant portion of the data and cannot be excluded without careful consideration, nevertheless it is troubling that the

sign ol the si-rial correlation coefficient hinges on (.lata from the (.real Depression. This conundrum-whether to omit data influenced bv a single cataclysmic event, or to include it and argue that such an event is representative of the economic system-underscores the fragility of small-sample statistical inference. Overall, there is little evidence for mean reversion in long-horizon returns, (hough this may be mote of a symptom of small sample sizes rather than conclusive evidence against mean reversion-we simply cannot tell.

These considerations point to short-horizon returns as the more immediate source from which evidence ol predictability might he culled. This is not lo say that a careful investigation of returns over longer time spans will be uninformative. Indeed, it may be only at these lower frequencies that the impact of economic factors such as the business cycle is delectable. Moreover, lo the extent that transaction costs are greater for strategies exploiting short-horizon predictability, long-horizon predictability may be a more genuine lorm of uuexploited profit opportunity. Nevertheless, the econometric challenges posed by long-horizon returns are considerable, and the need for additional economic structure is particularly great in such cases.

2.9 Conclusion

Recent econometric advances and empirical evidence seem to suggest that financial asset returns are predictable to some degree. Thirty years ago this would have been tantamount lo an outright rejection of market efficiency. However, modern financial economics teaches us that other, perfectly rational, factors may account for such predictability. The fine structure of securities markets and frictions in the trading process can generate predictability. Time-varying expected returns due to changing business conditions can generate predictability. Л certain degree of predictability may be necessary to reward investors for bearing certain dynamic risks. Motivated by these considerations, we shall develop many models and techniques to address these and other related issues in the coming chapters.

Problems-Chapter 2

2.1 II /) is a martingale, show that: (1) the minimum mean-squared error forecast of + conditioned on the entire history (/, ...), is simply /; (2) nonoverlapping M\ differences are uncorrelated at all leads and lags for all > 0.

2.2 How ate the RWI. RW2. RW3, and martingale hypotheses related (include a Venn diagram lo illustrate the relations among the lour models); Provide specific examples of each.

2.3 Characterize the set of all two-state Markov chains (2.2.9) that do not satisfy RWI and for which the CJ statistic is one. What are the general properties of such Markov chains, e.g., do they generate sequences, reversals, etc.?

2.4 Derive (2.4.19) for processes with stationary increments. Why do the weights decline linearly? Using this expression, construct examples of non-random-walk processes for which the variance ratio test has very low power.

2.5 Using daily and monthly returns data for ten individual stocks Jmd the equal-and value-weighted CRSP market indexes (EWRETD and VWRETD), perform the following statistical analysis using any statistical packageof your choice. Note that some of the stocks do not have complete return histories, so be sure to use only valid observations. Also, for subsampie analyses, split the available observations into equal subsamples.

2.5.1 Compute the sample mean Д, standard deviation a, and first-order autocorrelation coefficient /5(1) for daily simple returns over the entire 1962 to 1994 sample period for the ten stocks and the two indexes. Split the sample into four equal subperiods and compute the same statistics in each subperiod-are they stable over time?

2.5.2 Compute the sample mean /1, standard deviation a, and first-order autocorrelation coefficient p(l) for continuously compounded daily returns over the entire 1962 to 1994 period, and for each of the four equal subperiods. Compare these to the results for simple returns-can continuous compounding change inferences substantially?

2.5.3 Plot histograms оГ daily simple returns for VWRETD and EWRETD - over the entire 1962 to 1994 sample period. Plot another histogram

of the normal distribution with mean and variance equal to the sample mean and variance of the returns plotted in the first histograms. Do daily simple returns look approximately normal? Which looks closer to normal: VWRETD or EWRETD? Perform the same analysis for continuously compounded daily returns and compare these results to those for simple returns.

2.5.4 Using daily simple returns for the entire 1962 to 1994 sample period, construct 99% confidence intervals for д for VWRETD, EWRETD, and the ten individual stock return series. Divide the sample into four equal subperiods and construct 99% confidence intervals in each of the four subperiods for the twelve series-do they shift a great deal?

2.5.5 Compute the skewness, kurtosis, and studentized range of daily simple returns of VWRETD, EWRETD, and the ten individual stocks over the entire 1962 to 1994 sample period, and in each of the four equal

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