Aplicación al proyecto TESTSCADA

5.2.2. Independence

According to Cont (2001), it is a well-known fact that there is no significant linear correlation in returns. Therefore, the independence assumption assumes that the serial correlation function of returns as denoted by equation (5.1) decays rapidly to zero (Cont, 2001):

( ) ( ( ), ( ) 0

C τ =corr r t r t+ ∆ =t (5.1)

where ( )C τ is the serial correlation coefficient of order τ ^{, ( )}r t is the return on a given series at time t and t∆ is the time scale. The absence of (linear) serial correlation is often cited as evidence in favour of the efficient market hypothesis (Cont, 2001). Campbell et al. (1997) state that if equation (5.1) holds, returns may be considered as serially uncorrelated and therefore, mutually independent. Returns are assumed to show little or no linear serial correlation and if serial correlation is present, it is short-lived. Independence can further be defined from two perspectives. The first relates to statistical independence in returns. The second relates to whether investors can use knowledge of past returns to increase expected profits (Fama, 1965; Mandelbrot, 1967). The independence assumption in this study is of interest primarily from a statistical viewpoint. As much of the discussion relating to the

64 Positive skewness is statistically significant for 26 out of 30 stocks.

validity of the independence assumption centres on testing for serial correlation in returns, the discussion that follows focuses upon this aspect (Giannoplolous, 2000).

Kendall and Hill (1953) conduct an early analysis of the properties of returns and find that the pattern of events in price series is less systematic than generally accepted. Changes from one period to another behave almost like a “wandering series” implying that subsequent returns follow a random walk and are therefore, independent (Kendall & Hill, 1953: 11). Kendall and Hill (1953) first report findings relating to the Chicago Wheat Series.⁶⁵ The series follows a random walk with changes from one period to the other appearing to be independent and thus making serial correlation unlikely. This is confirmed by a finding of small and mostly negligible serial correlation in the series.⁶⁶ An analysis of the serial correlation in series constituting what the authors define as British Industrial Share Prices⁶⁷ yields similar results;

for the most part, changes in prices are independent and where dependence is observed, it is too low to exploit for predictive purposes. These findings suggest that returns are independent from a statistical viewpoint and from a practical perspective.

Fama (1965) states that even though it is difficult to find a series that fully conforms to the assumption of independence, statistical independence holds even if some level of dependence is present but insufficient to account for certain properties of the return distribution. It is proposed that the most basic explanation for the assumption of independence arises from the arrival of new information, which does not follow any consistent pattern.⁶⁸ To test for dependence in returns, Fama (1965) relies upon the serial correlation model and the runs test.

An analysis of the serial correlation structure for the entire sample indicates that overall, the level of serial correlation is low. Only about a third of the series compromising the DJIA show statistically significant serial correlation at the first and second orders, with the proportion of return series showing statistically significant serial correlation decreasing steadily at higher orders.⁶⁹ Fama (1965) notes that even in instances where correlation is statistically significant, the level of dependence implied by a statistically significant serial

65 Basic cash wheat in US cents per bushel in Chicago.

66 This finding is confirmed by low serial correlation up to the 10^th order over the entire sample period from 1883 to 1934 and omitting the period from 1915 to 1920.

67 Each index constituting the sample is an aggregate. For example, one of the series is “Insurance Companies.”

68 According to Fama (1965) these are rather extreme assumptions. Estimates of intrinsic values may be dependent upon the estimates of others and the arrival of information need not be independent; often good news is followed by more good news.

69 11 stocks exhibit statistically significant first order serial correlation and 9 exhibit statistically significant second order serial correlation.

correlation coefficient is so low that it is unimportant from both a statistical and practical perspective. These inferences are applicable when larger differencing intervals are considered; the average size of correlation coefficients decreases with the size of the differencing interval. Fama (1965) states that these findings, based upon the serial correlation model, indicate that dependence is of an extremely low magnitude suggesting that the independence assumption is a valid working assumption. Runs tests support the results of the serial correlation analysis; overall percentage differences between actual and expected runs are small, there is no pattern in the signs of the differences, the lengths of the runs are similar and the number of long runs does not exceed the expected number under the assumption of independence. Fama (1965) concludes that there is little evidence of dependence in returns.

Akgiray (1989) investigates whether returns can be represented by a linear white noise process with independent increments. Based upon Fisher’s kappa and Bartlett’s test,⁷⁰ the assumption of independence is rejected for the CRSP value-weighted index over the entire sample period between January 1963 and December 1986. These findings are supported by periodograms and Ljung-Box Q-statistics (see Ljung & Box, 1978). Moreover, the serial correlation function of the return series indicates high first order serial correlation. According to Akgiray (1989), this permits a conclusive rejection of the hypothesis that return series are white noise suggesting that returns do not approximate independent observations. However, the serial correlation function also reveals that dependence is short-lived. It is hypothesized that the presence of a common market factor, thin trading, a day of the week effect and adjustments to the arrival of new information may be responsible for the presence of statistical dependence in returns on the CRSP value-weighted index.

Campbell et al. (1997) argue that the independence assumption is often violated over the long-run with returns exhibiting long-run dependence. To test whether there is long-run dependence coupled with predictability in returns, the serial correlation structure of returns on the CRSP value and equally-weighted indices is investigated over the period from July 1962 to December 1994. Both indices are found to exhibit substantial first order serial correlation in daily returns. Substantial first order serial correlation is also found in weekly and monthly returns on the CRSP equally-weighted index. Returns on the equally-weighted

70 According to Akgiray (1989), both procedures are primarily designed to test whether a series is white noise and in large samples, these procedures are tests of independence. Six year sub-periods are also considered. The independence assumption is rejected for two sub-periods when Fisher’s test is used and for all four sub-periods by Bartlett’s test.

index exhibit a higher level of serial correlation at all differencing intervals and the serial correlation in returns on this index decays at a slower rate relative to the serial correlation in returns on the value-weighted index. For both indices, serial correlation declines rapidly after the first order, again suggesting that dependence is short-lived regardless of the differencing interval.⁷¹ Although, dependence is observed in both series and regardless of the differencing interval, evidence of dependence is weaker for the CRSP value-weighted index and weaker still at larger differencing intervals for both series. Campbell et al.’s (1997) findings are in line with Akgiray’s (1989) findings of short-lived statistical dependence but contrast with those of Kendall and Hill (1953) and Fama (1965).

Lo and MacKinlay (1988) investigate the serial correlation structure of weekly returns on equally and value-weighted CRSP NYSE-AMEX indices, size based portfolios and returns on individual stocks over the October 1962 to December 1985 period. Weekly data as opposed to daily data is used to minimize biases associated with non-trading, the bid-ask spread and asynchronous prices. Based upon the variance ratio test (q=2), Lo and MacKinlay (1988) find evidence of statistically significant positive first order serial correlation in returns on both CRSP indices and this is seen as evidence in favour of rejecting the random walk hypothesis. Results for size based portfolios are similar; statistically significant positive first order serial correlation is present in returns on three portfolios consisting of firms of similar market value.⁷² These findings again imply a rejection of the random walk hypothesis and thus, the rejection of the independence assumption. Unlike the positive serial correlation in returns on indices and size based portfolios, serial correlation in returns on individual stocks is negative and not statistically significant. Lo and MacKinlay (1988) attribute this to company specific noise which complicates the detection of predictable components. These findings suggest that the independence assumption does not hold for returns on aggregates.

However, there is ambiguity relating to serial correlation in returns on individual stocks.

Unlike the assumption of normality, which is widely rejected, the independence assumption continues to be debated. As indicative of Kendall and Hill (1953) and Fama (1965), a body of literature finds support for the independence assumption and limited evidence of dependence in returns. As is indicative of Akgiray (1989) and Campbell et al. (1997), any statistical

71 For example, first order serial correlation for daily returns on the CRSP value-weighted index is 17.6 percent.

Second order serial correlation is -0.7 percent. For the CRSP equally-weighted index, first order serial correlation is 35.0 percent and second order serial correlation is 9.3 percent.

72 The three quintiles are the smallest, central and largest market value quintiles.

dependence in returns is (very) short-lived, although its presence nevertheless challenges the independence assumption. Matters are further complicated by Lo and MacKinlay (1988) who find that while returns on aggregates show statistically significant first order serial correlation, returns on individual stocks appear to be independent. Given these findings, it is impossible to conclusively pronounce upon the validity of the assumption of independence.

Perhaps the best approach is to investigate the independence assumption on a “case-by-case”

basis.