− 1 tends to be followed by a below-average (above-average) return in time period t. Zero autocorrelation indicates that the returns are linearly independent through time. Positive autocorrelation is seen in trending markets; negative autocorrelation is seen in markets with price reversal tendencies.
We start here by assuming the simplest scenario: The returns on an investment are statistically independent through time, which means there is no autocorrelation. Further, we assume that the return distribution is stationary (i.e., the probability distribution of the return at each point in time is identical). Under these strict assumptions, the distribution of log returns over longer periods of time will tend toward being a normal distribution, even if the very short-term log returns are not normally distributed.
How do we know that log returns will be roughly normally distributed over reasonably long periods of time if the returns have no autocorrelation and if very short-term returns have a stationary distribution? One explanation is that the log return on any asset over a long time period such as a month is the sum of the log returns of the sub-periods. Even if the returns over extremely small units of time are not normally distributed, the central limit theorem indicates that the returns formed over longer periods of time by summing the independent returns of the sub-periods will tend toward being normally distributed.
Why might we think that returns would be uncorrelated through time? If a security trades in a highly transparent, competitive market with low transaction costs, the actions of arbitrageurs and other participants tend to remove pronounced patterns in security returns, such as autocorrelation. If this were not true, then arbitrageurs could make unlimited profits by recognizing and exploiting the patterns at the expense of other traders.
However, markets for securities have transaction costs and other barriers to arbitrage, such as restrictions on short selling. Especially in the case of alternative investments, arbitrage activity may not be sufficient to prevent nontrivial price patterns such as autocorrelation. The extent to which returns reflect nonzero autocorrelation is important because autocorrelation can impact the shape of return distributions. The following material discusses the relationships between the degree of autocorrelation and the shapes of long-period returns relative to short-period returns.
Autocorrelation of returns can be used as a general term to describe possible relationships or as a term to describe a specific correlation measure. Equation 4.20 describes autocorrelation in the context of a return series with constant mean:
where Rt is the return of the asset at time t with mean μ and standard deviation σt, Rt−k is the return of the asset at time t − k with mean μ and standard deviation σt − k, and k is the number of time periods between the two returns.
Equation 4.20 is the same equation used to define the Pearson correlation coefficient in Equation 4.17) with substitution of Equation 4.15 for covariance) except that Equation 4.20 specifies that the two returns are from the same asset and are separated by k periods of time. Thus, autocorrelations, like correlation coefficients, range between −1 and +1, with +1 representing perfect correlation.
There are unlimited combinations of autocorrelations that could theoretically be nonzero in a time series; thus, in practice, it is usually necessary to specify the time lags separating the correlations between variables. One of the simplest and most popular specifications of the autocorrelation of a time series is first-order autocorrelation. The first-order autocorrelation coefficient is the case of k = 1 from Equation 4.20, which is shown in Equation 4.21:
Thus, first-order autocorrelation refers to the correlation between the return in time period t and the return in the immediately previous time period, t − 1. Note that in the case of first-order autocorrelation, the returns in time period t − 1 would also be correlated with the returns in time period t − 2; thus, the returns in time period t would also generally be correlated with the returns in time period t − 2, as well as those of earlier time periods. Because first-order autocorrelation is generally less than 1, the idea is that the autocorrelation between returns diminishes as the time distance between them increases.
While autocorrelation would be zero in a perfectly efficient market, substantial autocorrelation in returns can occur when there is a lack of competition, when there are substantial transaction costs or other barriers to trade, or when there are returns that are calculated based on nonmarket values, such as appraisals. Autocorrelation of reported returns due to the use of appraised valuations or valuations based on the discretion of fund managers raises important issues, especially in the analysis of alternative investments.
Autocorrelation in returns has implications for the relationship between the standard deviations of a return series computed over different time lengths. Specifically, if autocorrelation is positive (i.e., returns are trending), then the standard deviation of returns over T periods will be larger than the single-period standard deviation multiplied by the square root of T. If autocorrelation is zero, then the standard deviation of returns over T periods will be equal to the single-period standard deviation multiplied by the square root of T. Finally, if autocorrelation is negative (i.e., returns are mean-reverting), then the standard deviation of returns over T periods will be less than the single-period standard deviation multiplied by the square root of T.
An important task in the analysis of the returns of an investment is the search for autocorrelation. An informal approach to the analysis of the potential autocorrelation of a return series is through visual inspection of a scatter plot of Rt against Rt−1. Positive autocorrelation causes more observations in the northeast and southwest quadrants of the scatter plot, where Rt and Rt−1 share the same sign. Negative autocorrelation causes the southeast and northwest quadrants to have more observations, and zero autocorrelation causes balance among all four quadrants.
Another common approach when searching for autocorrelation is to estimate the first-order autocorrelation measure of Equation 4.20 directly, using sample data. Exhibit 4.2 shows the estimated autocorrelation coefficients for the two return series. For autocorrelations beyond first-order autocorrelation, an analyst can use a linear regression with Rt as the dependent variable and Rt−1, Rt−2, Rt−3, and so forth as independent variables.
4.3.7 The Durbin-Watson Test for Autocorrelation
A formal approach in searching for the presence of first-order autocorrelation in a time series is through the Durbin-Watson test. To test the hypothesis that there is no autocorrelation in a series involves calculating the Durbin-Watson statistic:
(4.22)
where et is the value in time period t of the series being analyzed for autocorrelation.
In alternative investments, the series being analyzed (et) may be returns or a portion of returns, such as the estimated active return. A DW value of 2 indicates no significant autocorrelation (i.e., fails to reject the hypothesis of zero autocorrelation). If DW is statistically greater than 2, then the null hypothesis may be rejected in favor of negative autocorrelation; and if DW is statistically less than 2, then the null hypothesis may be rejected in favor of positive autocorrelation. The magnitude of the difference from 2 required to reject zero autocorrelation is complex, but a rule of thumb is that zero autocorrelation is rejected when DW is greater