Alternative Investments. Hossein Kazemi

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      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 R_t against R_t−1. Positive autocorrelation causes more observations in the northeast and southwest quadrants of the scatter plot, where R_t and R_t−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 R_t as the dependent variable and R_t−1, R_t−2, R_t−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 e_t is the value in time period t of the series being analyzed for autocorrelation.

      In alternative investments, the series being analyzed (e_t) 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 than 3, which is negative autocorrelation, or less than 1, which is positive autocorrelation. The DW statistics for the market index and the real estate fund are reported in the bottom left-hand corner of Exhibit 4.2. Note that the reported DW statistics for both of the return series fail to reject zero autocorrelation, even though the estimated autocorrelation coefficients appear quite positive.