Xi

Figure 8.3. Geometric interpretation of WLSXY. The lines L0, La and measure the distance from a data point to the fit line for A = 0, 0 <A< to and A = to, respectively.

8.2 for the regression of Y on X.) If the standard deviations are nonzero and 0 < A < to, we measure the distance along the line La (Fig. 8.3). The slope of this line is equal to -A//31 (York 1967).

If heteroscedasticity is in one or both of the noise components, then the ratio A may vary with time (i) and, hence, the line La may vary in its slope. The difficulty of non-identifiability is introduced by unknown A because then it is not unequivocally determined how to measure the distance and minimize the sum of squares.

8.1.3 Wald—Bartlett procedure

A straightforward estimation idea (Draper and Smith 1981: Section 2.14 therein) is to build two groups of the bivariate sample according to the size of the x values, then to take for each group the centres defined by the x and y averages and, finally, to connect the centres using a straight line—defining the estimate of the slope. The intercept estimate is found via the centre of the complete bivariate sample and the slope estimate. This goes back to Wald (1940), who grouped the sample into two halves of same size (if n is even) and Bartlett (1949), who showed that taking three groups improves the accuracy of the regression estimators. (Intuitively, the means of the two groups are further apart for taking thirds than for taking halves, outweighing the deficit of reduced data sizes.) We call this estimation Wald-Bartlett procedure (Fig. 8.4).

The Wald-Bartlett procedure can in principle be applied to any grouping of the set of data points, not only according to the size of the x values. A point to note is that the grouping has to be independent of Xnoise(i) for achieving consistency of the estimators (Wald 1940). This condition is violated when the {Xtrue(i)}™=1 are unknown and the size ordering

0 0