I"=j8 + ^8111(2^/36 5) + /S 3 cos (2irD/ 36 5) (3) 



+ £ sin(477Z'/36 5) + j8 cos(4tt£/365) + e 



3 4 



The addition of semiannual oscillatory terms to the re- 

 gression equation improves the fit obtained with the annual 

 terms. Tests of significance of sums of squares attributable 

 to annual and semiannual oscillations are performed using 

 the appropriate i^-ratios. 



Computation of the autocorrelation functions of the 

 residual time series after equation (3) has been fitted to the 

 series of sea-surface temperatures yields the plots of fig- 

 ure 3. These residuals are themselves autocorrelated, 

 although no additional oscillatory terms exist. The least 

 squares method is valid if (1) the error between the true 

 regression curve and the observed value is distributed in- 

 dependently of the independent variables with zero mean and 

 constant variance; and (2) ideally, successive errors are 

 distributed independently of one another. Actually, the 

 problem of using the method of least squares when the error 

 terms are autocorrelated has been solved if the e's follow 

 certain autoregressive processes. 4 The autocorrelated 

 residuals should affect the distributions of the regression 

 coefficients. As will be seen later, the effect on the 

 variance of the regression coefficients is negligible. 



4 Anderson, R. L. , "The Problem of Autocorrelation in 

 Regression Analysis," American Statistical Association. 

 Journal, v. 49, p. 113-129, March 1954 



13 



