NONUNIOUENESS AND DEGENERACY

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Table 4.1 Calculations Used in Finding the Best Line Through the Canada Data Point Country Mexico Nicaragua Dominican Republic Honduras Panama Haiti Jamaica EI Salvador Guatemala Costa Rica Trinidad/Tobago United States Cuba

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Y, - 16.2

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Ix; - 55.01

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x, - 55.0

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-1.5000 -1.0566 -0.9441 -0.7694 -0.6821 -0.6107 -0.5525 -0.5517 -0.5447 -0.5162 -0.1743 -0.1333 -0.0323

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Cumulative Sum of Ix; - 55.01 11.8 38.3 56.2 92.2 109.5 150.6 172.5 216.0 256.8 284.5 332.7 334.2 355.9

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11.8 26.5 17.9 36.0 17.3 41.1 21.9 43.5 40.8 27.7 48.2 1.5 21.7

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The next step is to find the best line passing through the EI Salvador data point. We form the slopes (Yi - 40.2)/(x i - 11.5), put them in increasing order, and construct a table similar to Table 4.1 with a column for Ix i - 11.51 and a column for the cumulative sums. If you construct this table, you will find that the total cumulative sum is 265.5 and that the country whose cumulative sum first exceeds 265.5/2 = 132.75 is the United States. Therefore the best line passing through the EI Salvador data point also passes through the United States data point. The next step is to find the best line passing through the United States data point. Constructing a table similar to Table 4.1 and looking at the cumulative sums, we find that the best line passing through the United States data point also passes through the EI Salvador data point. But this is the same line we obtained at the previous step, and so the algorithm stops. The LAD regression line is the line passing through the data points of EI Salvador and the United States. Its slope is = (40.2 - 16.0)/(11.5 56.5) = - 0.5378 and its intercept is Ii = 40.2 - ( - 0.5378)(11.5) = 46.38. That is, the LAD regression line is Y = 46.38 - 0.5378X. See Figure 1.1.

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NONUNIQUENESS AND DEGENERACY

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The algorithm described in the preceding section will find the LAD regression line for most data sets, but occasionally problems arise with non uniqueness or degeneracy. Nonuniqueness means that there is more than one best

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LEAST-ABSOLUTE-DEVIATIONS REGRESSION

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line passing through a data point. Degeneracy means that the best line through a data point also passes through two or more other data points. Recall that the algorithm proceeds in steps. At each step we find the best line passing through a given data point. The best line always passes through another data point, and this data point is used in the next step. But when there is non uniqueness, there is more than one best line. And when there is degeneracy, the best line passes through more than one other data point. In either case, there is more than one choice for the data point to be used in the next step. By making unlucky choices, the algorithm may go around in circles, or it may stop with a line that is not the LAD regression line. The possibility of such problems is indicated either when equality occurs in condition (4.2) or when the slope {3* = (Yk - yo)/(x k - xo) in (4.3) is equal to (Yk-l - Yo)/(X k - 1 - xo) or (Yk+l - Yo)/(X k + 1 - x o). In such instances we can resort to the algorithm in the following paragraph. Also see the subsection on nonuniqueness and degeneracy at the end of Section 4.6.

A Simpler Algorithm. Another possible approach is to replace the algorithm in Section 4.2 by the following algorithm, which has the advantage of being conceptually simple and the disadvantage of requiring more computation. It is known that the LAD regression line (or at least one of them, in case of non uniqueness) passes through at least two data points. So an LAD regression line can be found among the lines defined by all possible pairs of data points. (Some of these lines may coincide.) We can simply compute the sum of absolute deviations (4.1) for each of these lines and choose the one (or ones) with the smallest sum. The feasibility of this algorithm depends on the sample size n. This algorithm is not disturbed by degeneracy. In case of nonuniqueness, when there are several LAD regression lines, we could arbitrarily choose one, or we could take their average. This average line is also an LAD regression line.

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TESTING {3 = 0

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The method of least absolute deviations estimates the slope of the regression line for the birth rate data to be ~ = - 0.5378, but we do not expect the estimate to be exactly equal to the true value. Even though ~ is negative, it is possible that the true value of {3 may actually be O. That is, what appears to be a negative relationship between birth rate and urban percentage may be due merely to the randomness of the data. Let us test whether or not {3 could be o.

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