Dependency Rates and Savings Rates: Further Comment - Goldberger, Arthur S. (American Economic Review, 1973)
The empirical results on dependency rates and savings rates reported by Nathaniel Leff (1969) cannot be correct. For several cross-country samples, Leff estimates pairs of equations of the form (1) ^1 = 00-1- aiXi + CiXi-f (2) ^2 = 60 + bixi + biXi-f 63*3-|- biXi-f- d where yi = ln S/Y = ln domestic savings ratio yi = In S/N = In per capita savings xi = ln Y/N = In per capita income a:2 = | = growth rate of per capita income X3 = ln Di = ln percentage of population aged 14 or less Xi = ln Di = ln percentage of population aged 65 or more, and a and b are least-squares regression co-efl&cients, and e \ \ and e ^ are least-squares resid-uals. As noted by Leff, S/N^iS/Y)iY/N). Consequently, yi = yi-\ \ -Xi. Least-squares re-gression being what it is, a proper com-putation of (2) should produce = flo + (1 + ai)xi, + 0.2*2 +(3) That is, regressing y ^ on the x should give the same coefficients and the same residuals as occur when y ^ is regressed on the *, except for the coefficient of Xi, which should in-crease by exactly 1. Furthermore, if regression coefficients are guaranteed to be equal, their standard errors, and hence their ^-ratios, must be equal. If regression coefficients are guar-* Professor of economics, University of Wisconsin, Madison. anteed to differ by unity, their standard errors must be equal, and hence their t-ratios must be related by bx/si, = (ajAa.)((l-f ai)/ai) But the results Leff reports do not satisfy these arithmetic requirements. For example, consider the upper panel of his Table 1, p. 891, which refers to a sample of 47 under-developed countries. In the present notation we find: