Relative efficiency of count of sign changes for assessing residual autoregression in least squares regression

SUMMARY From a constructed example of 100 random samples of size 40, in conjunction with the author's ACV method for comparing the relative efficiency of different tests of significance, it is found that a simple count of sign changes is nearly as efficient as the familiar DurbinWatson test of autoregression of residuals. Individual decisions based on this test are closely similar to those from the number of runs test. On another actual body of data the three tests seem to be about equally efficient. A table is supplied giving cumulative binomial probabilities for assessing significance for the sign changes test. Probably most workers in multivariate regression of time series, who are computerless, or without a von Neumann subprogram in their computer, adopt the practice of counting the number of sign changes amongst residuals, for the purpose of assessing probable presence of autoregression. If sign changes are few, residual autoregression is inferred, i.e. the regression is not satisfactory because some significant independent variables have been omitted, the linear form assumed is not valid, etc. The practice is rational; if T is the number of sets of observations and if signs, plus or minus, in the sequence are in random order, the frequency of r sign changes will be (T- 1)!/{r!(T- 1 --)!}, the total frequency (2T-1 1) almost the binomial with p = 1. The - I in total frequency arises because a sequence of all T signs the same is inadmissible since the sum of regression residuals is zero. Incidentally, the latter constraint implies that the sequence of values of the residuals or their signs cannot be independent of one another, assumed, as regards number of sign changes, in the use of the binomial. With these, the effect is believed to be negligible when T is not small, and is ignored here. The main object of the present note is to assess the relative efficiency of the count of sign changes r compared with the familiar d statistic developed by Durbin & Watson (1950, 1951). As the writer is unable to cope with the problem of the noncentral frequency of r by algebra, recourse is made to the Monte Carlo method applied to a single particular case.