Least Squares with Unknown Variance

All reduced form parameters are estimable directly by using least squares, so the reduced form is identified in all cases. Now, 71 = n /n2. On is the residual variance in the euqation (yi - yiy2) = Sj, so G must be estimable (identified) if 71 is. Now, with a bit of manipulation, we find that con - ron = -On/A. Therefore, with On and Yi "known" (identified), the only remaining unknown is y2. which is therefore identified. With 71 and y2 in hand, P may be deduced from n2. With y2 and P in hand, ct22 is the residual variance in the equation (y2 - Px -YiVi) 2, which is dfrectly estimable, therefore, identified. 

If the function may be made linear with respect to its unknown parameters by a suitable transformation, then it may be fitted by the Linearized Least Squares method (10) so as to minimize the root mean square error in the original (untransformed) space. The essence of this technique is to use weighted (linear) least squares to effect a non-linear least squares fit. Assume that the equation has been transformed into an equal variance space and let... 

Uniform weighting (known as simple least squares) is appropriate wJien the expected variances of the observations are equal and is commonly used when these values are unknown. GREGPLUS uses this w eighting when called with JWT = 0 the values MwijJ = 1 are then provided automatically. 

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