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31 mere should be a weak positive relationship between socioeconomic status and later fertility in traditional settings. This hypothesis derives from the positive health effects implied by relatively advantaged social positions. Husband characteristics may prove vital in measuring family well-being (B11 16, B12,16 > 0), especially nutritional levels and overall health. Healthy women are more fecund. AFB may also reflect social standing (63,16 < 0), as argued earlier. However, there is no basis for positing a curvilinear relationship between AFB and LF in traditional societies; hence in these settings, we would set 816,16 = 0. Because of their correlations with health and fecundity, respondent char- acteristics should also affect LF (~l,16, 82,16, 89,16, 610,16 > 0). Each individual relationship should be weak both because there are so many, and because they imperfectly approximate the underlying, unmeasured variables. 1 .4 REDUCED AND SE==DUCED FOR IMPLICATIONS OF THE MODEL His section discusses some implications of the model, first explaining why its complexity needs to be reduced. Next, certain implications of the model for three familiar variables--children ever born, contraceptive use, and infant and child mortality--are traced. This leads to develop- ment of implied and actual semireduced form equations for these variables, followed by a discussion of inferences that can be drawn about the coefficient signs in these equations. The Need for Reduction As shown above, the threefold distinction between onset, early fertility, and later fertility permits development of an intricate and comprehensive micro model embodying a number of untested hypotheses. This micro model is not the total model, which would also incorporate macro variables to explain coefficient variability across settings. We intend to continue the theoretical development of the model begun in this paper elsewhere. Our present focus is on testing the implications of the micro model. As a starting point, Table 1.2 presented the predicted signs of the structural coefficients in traditional and transitional settings implied by the micro model. How is the model to be tested? Its estimation for a single country would permit one kind of test. In particular, where we have hypothesized that a coefficient should have a positive sign in all settings, the hypothesis could be rejected empirically if the sign were found to be in the opposite direction. However, this would be a weak test. Among other things, it applies only to those coefficients for which a single sign was hypothesized. How can hypotheses be tested which relate to coefficients for which sign variation across settings is possible? There will be variation not only in the sign, but also in the magnitudes of the coefficients. His variability is theoretically admissible and is implicitly assumed to exist in the above discussion, since for a coefficient to Travels from negative to positive, it must take on a
32 range of values. More strongly, the magnitudes of the coefficients should increase or decrease as a function of the "distance. from various ideal-typic anchor points of the traditional/transitional continuum. Because the model postulates contextual variability in a large proportion of the micro structural coefficients, its testing must involve estimation of its micro and macro components. If the empirical reali- zation of the model is precisely as summarized in the preceding sections, there w~11 be 99 micro structural coefficients. Of these, some 60 are hypothesized to vary in sign as well as magnitude. Although others can vary similarly, we cannot specify the nature of their variability as a function of traditional/transitional setting differences. Hence, at the macro level there is a set of at least 60 potential structural equations, one for each micro coefficient of interest. Suppose that this set is estimable (i.e. Of does not encounter the problem of negative degrees of freedom), as will be the case if the micro-macro model is envisioned as consisting of a set of equations that includes micro regressors and macro regressors, and interactions between them. We then have the practical problem of data exploration which accompanies theory development for each of the 60 macro equations. Of course, equations will not be conceptually independent since they involve the same underlying hypotheses. Neverthe- less, a great deal of complexity must be entertained, especially during the data exploration stage. It would be helpful if this complexity could be reduced, even if that reduction were unnecessary for testing the model To do this we examine the model to determine if it contains implica- tions that can be tested directly in a simple way. Although the results of this examination, described below, will be mixed, the process itself proves enlightening. Connections With and Between Familiar Variables Some of the implications explored below relate to equations for children ever born (CEB) and infant and child mortality (CM). m ese two variables are commonly used in analyses of survey data and have broad theoretical appeal because they are appropriate "counts variables for cohorts whose fertility is complete. Although they do not appear explicitly in the structure' equations of the micro model, they are present implicitly, since CEB = EF + LF and CM = ECM + LCM; thus equations (1.4) and (1.16) for fertility, and (1.3) and (1.14) for child mortality, sum to equations for CEB and CM, respectively. m ese summed equations are not structural because there is literally no structure for which they are meaningful, given that equations (1.1) to (1.16) state the causal ordering of the variables in the micro model. However, there is no reason why we cannot consider whether the model allows determinate predictions about the signs of their coefficients. If so, a degree of simplification will have been achieved, and at the same time, we will have related the model to CEB and CM. It will then be meaningful to treat CEB and CM as response variables that depend on variables actually In the model. This simplification would be lost if the model did not allow determinate predictions since there would then be no good theoretical reason to expect either positive or negative
33 relationships between CEB and CM and the variables actually in the model. In spite of this simplification, however, still further reduction would be desirable. A number of predetermined variables are clearly secondary, for example, NMAR, DUR, and FEC. Of much greater interest, for practical as well as theoretical reasons, are the socioeconomic variables. It is primarily, but not exclusively, through these variables that changes in the social system are transmitted. Moreover, these variables define social positions. It is of some interest to describe how a process is endogenously related to itself, as in the connection between early and later fertility; however, only two of the structural equations for reproductive process variables--those of boy and girl Buff iciency--exclude socioeconomic predictors. In addition, we have hypothesized that when the fertility process variables interact with other variables in the determination of later fertility outcomes, these interactions are always with socioeconomic variables. Examination of the relationships between the socioeconomic and reproductive process variables that follows below, requires reduction of the structural equations to a simpler form. Children ever born (CEB) and child mortality (CM) are not, of course, the only endogenous variables of interest. The variable for contraceptive use patterns (LCU) is also crucial. The following subsection formulates a model for examining hypotheses about the relationships between socioeconomic position and CEB, CM, and LCU. Implied and Actual semireduced Form Equations To study the implications of the micro model for the relationships between socioeconomic position and CEB, CM, and LCU, we obtain actual and implied semireduced form equations from the structural equations by a process of substitution to remove endogenous variables from the lists of predictors. These semireduced form equations are something between structural and reduced-form equations. Those of interest here exclude as predictors all but the socioeconomic variables, some of which are endog- enous in the total structural equations model. We term the semireduced form equations for CEB and CM n implied" semireduced form equations. To obtain this for CEB, we carry out the usual substitution process for EF and LF, stopping whenever the variable involved is socioeconomic. mis yields two actual semireduced form equations, one for EF and one for OF, whose sum is the implied semireduced form equation for CEB. A similar process yields the implied semireduced form equation for CM. The semireduced form equation for LCU is obtained by substitution, but for this variable there is no need to sum across equations; direct substitu- tion into a single equation suffices. For this reason, we label the expression that relates LCU to the socioeconomic variables the "actual" semireduced form equation. m e caution noted earlier for the sums of structural equations applies here. The implied semireduced form equations are not part of the equation system. m eir relevance Is simply that, given the model, they permit consideration of the semireduced form implications of the socioeconomic variables for CEB and CM. The three semireduced form equations are as follows:
34 11.17) CM a0,17 + al,l7RESC + a2,1tWED + as ltWBM + ad 17RES + ~5,1}WSM + ~6,l7HED + ~7,17HOCC. ) LCU a0,18 + a1,18RESC + ~2,1§wED + ~3 1§WBM + Ad 1§RES + as, 1~SM + a6,1~ED + a7,1dHOCC + ~8,18(RESC) ~ a9 16RESC*WED + ~1o~l8(WED) + all,l8WED WBM + a12'l8REsC HOCC ~ a:3,l§WED*HOCC + a14,1§WBM*HOCC. ( ) CEB ~0,19 + al,lgREsc + ~2,1}WED + ~3 19WBM + ~4 l9RES + as 1§WSM + a6 lgHED + a7,l9HOCC + a8,19(RESC) + As 1gRESC*WED + alo~lgtWED) + all,l9WED WBM + a12,1,RESC*HOCC ~ a13,1~WED*HOCC + al4,l}WBM*HOCC. It is immediately clear that although the equation for CM is linear and additive, those for LCU and CEB are not. Equations ~1.18) and ~1.19) contain quadratic terms in the two ultimate exogenous variables (RESC and WED), as well as a number of product terms. These terms occur because of several hypotheses associated with the structural equations: 1) an extreme age at first birth (young or old) dampens contraceptive use and increases later fertility among older cohorts of women; 2) the positive effect of high early fertility on later contraceptive use, and the negative effect of high early fertility on later fertility, are both enhanced by greater than average respondent education; 3) the positive effect of high early fertility on later contraceptive use, and the corresponding negative effect of high early fertility on later fertility, are both enhanced by the degree of modernity of husband's occupation; and 4) the effects of boy sufficiency on subsequent fertility and contra- ceptive use may vary with socioeconomic status. Because of the nonlinearities and interactions in equations (1. 18) and (1. 19), the actual and implied semireduced form effects of RESC, WED, WBM, and HOCC cannot be read directly from the equations. Instead, first partial derivatives must be evaluated. These partials have the same form for the LOU and CEB equations, and we present them below only for LCU. In so doing, we assume that RESC is continuous; as we defined it operationally, it is not, although conceptually there is every reason to treat it so: aLCU/aRESC = ~: :8 + 2a~,~RESC + OWED + a,2,:8HOCC LCU/ WED a2,~8 + OWED + ~9,~8RESC + a::,~8WBM + am :8HOCC aLCO/aWBM ~ ~3,18 + ~11,18WE~ aLCU/aHOCC = ~7~18 + al2,18RESC + al3,1gWED + al4,l8wBM
35 As these derivatives show, the effects of RESC, WED, WBM, and HOCC depend on each other to different degrees . Thus far, we have discussed functional form without mention of a country's location on the traditional/transitional continuum, which has implications for the functional forms of the LOU and CEB equations. For the LOU equation, there should be no semireduced form socioeconomic effects in ideal-typic traditional settings. In ideal-typic transitional settings, on the other hand, all of the effects hypothesized should be found. For the CEB equation, there should be no interactions and no nonlinearities in a traditional setting. In transitional settings, all hypothesized interactions and nonlinearities should be present. The basis of these expectations will be made explicit in the next subsection. Since ideal-typ~c societies cannot be observed, data for a given country can always deviate somewhat from expectations without necessarily disconfirming an underlying hypothesis. This might be summarized as follows: where interactions have been posited, the likelihood that those interactions actually exist should increase as the degree of transition- ality increases, and decrease later as all socioeconomic groups come to participate in the fertility decline. The derivation of equations (1.17), {1.18), and (l.l9) provides the initial implications of the model, which concern functional form. First, the child mortality equation should be a linear and additive function of the designated socioeconomic variables. In the modeling of child mortality as a function of designated socioeconomic variables, replicated demonstration of nonlinearities (in the sense of sign reversals over the range of a predetermined variable) and interactions would constitute evidence against the model. The second implication is that the functional forms for the contraceptive use and children ever born equations involve specific nonlinearities (in the sense of sign reversals over the range of a predetermined variable) and interactions, depending on a society's position on the traditional/transitional continuum and other macro variables. Empirical failure to detect these interactions as a function of context and other macro influences would constitute evidence against the model. The anticipated departures from linearity and additivity are unusual and would probably not be hypothesized and similarly justified in direct, single-equation modeling of the effects of the socioeconomic variables. Chidambar~m and Mastropoalo {1982), for example, specify a linear and additive single-equation model of contraceptive use. The work of Cochrane (1979), to take another example, includes the hypothesis of a nonlinear effect of education on fertility, but for reasons which differ considerably from ours. Moreover, Cochrane's model does not imply the existence of interactions between respondent's education and other socioeconomic characteristics. Expectations of variability in functional form by setting are bound up with those for the structural coefficients summarized in Table 1.2, as well as those about the linear combinations of these coefficients that comprise the actual and implied semireduced form coefficients of equations (1.17), (1.18), and (1.19~. In the next subsection, these anticipations are discussed in summary form. This will clarify hypotheses about the variability of functional form by context, and it will also indicate whether it is possible to derive implications about the directions of
36 coeff icients. Put another way, the discussion of functional form involves hypotheses about which coefficients are zero, and the discussion of d irection of association is contingent on knowing which coeff icients will be assumed nonzero. It is thus convenient to discuss both aspects of coef f icient variability together. Inferences About Coeff iciest Signs A number of additional assumptions or hypotheses are necessary if inferences about the signs of the coefficients in the implied and actual semireduced form equations are to be made. This Is clear from structural equations (1.11) to (1.16), which include coefficients whose signs, as suggested, will vary across populations in a way that is essentially orthogonal to the traditional/transitional continuum. In addition, derivation of coefficient signs in the implied and actual semireduced form equations requires assumptions about the intercepts for structural equations (~.1) and (1.3) to (1.6~. These were not discussed earlier for the sake of simplicity in exposition of the structural equations model. To facilitate the development of inferences about reduced-form coefficient signs, the discussion below reviews hypotheses about structural equation intercepts, and then the complex issue of variability in the structural coefficients orthogonal to the traditional/transitional continuum. The intercepts for equations (1.1) and (1.3) to (1.6) are constitu- ents of some of the semireduced form coefficients. For equations (1.1), (1.3), and (1.4), we hypothesize positive intercepts. In fact, if the empirical analysis yields negative intercept estimates after AFB is transformed by subtracting its minimum, the specification of these equations will be in question because the response variables (AFB, EF, and ECM) are nonnegative and because zero values for the predetermined variables are meaningful. We expect the intercepts of equations (1.5) and (1.6) to be zero; no births and no child deaths should correspond to zero values on the boy and girl sufficiency variables. On the issue of coefficient variability (unrelated to the trad~tional/transitional continuum) , and considering equation (1.13) first, we will assume that AFB is inversely related to self-reported fecundability (FEC). As noted earlier, this coefficient could also be positive or zero. The assumption made here is a best guess about the most likely effect of AFB on EEC. The coefficients of equations (1.11) and (1.12) are more problematic. There are reasons for positive, negative, or no SES effects on exposure between age 30 and the time of the survey, likewise for the effect of AFB. Moreover, as already suggested, there is no reason why this variability should be associated with the traditional/transitional continuum. We have therefore made an educated guess about the SES and AFB effects. In particular, the results presented below are derived under the assumption that the SES variables are positively related to marital stability (81,11, 62,11, 64,11, < 0; 61,12, 62,12' 64,12 > 0~, and that.marital stability does not depend on AFB (03,11 = 83,12 = 0) The reasons for this choice of assumptions are not important. Having derived the semireduced form implications of the model with any g iven set
37 of assumptions, it is then possible to return to the exercise using different assumptions. When we in fact did this, the alternative assumptions used did not meaningfully change the conclusions reported below. We consider next the impact of boy and girl sufficiency on later child mortality, contraceptive use patterns, and later fertility, taking the coefficients one equation at a time, with the following assumptions: Child Mortality a) 67,14 = 88,14 = 0; b) 87,14 < 0; 68,14 < 0; c) 67,14 > 0; 08,14 > 0 Assumption a) states that there is no effect of boy or girl sufficiency on later child mortality, assumption b) states that the effect is negative, and assumption c) states that it is positive. Clearly, any of these alternatives is possible, and it is uncertain whether one outcome is more likely than the others. Originally, we expected positive boy and girl sufficiency effects on child mortality. However, there are routes by which the opposite conclusion can be reached. Actually, it turns out not to matter which assumption is used because they all lead to basically the same inferences This point almost, but not quite, holds for the contraceptive use pattern and later fertility equations. Since these two equations are so similar, we will treat them as a pair, making the following alternative assumptions: LCU d) 67~15 = 0; 08,15 = 0; 619,15 = 0; t320,15 = 0; e) 67,15 > 0; 68J15 ~ 0; 019rl5 = 0; 620115 0; f) 67,15 > 0; f38~15 > 0; 8~15 > 0; 820~15 > 0; g) 67,15 > 0; 638,15 > 0; 6319,15 < 0; 6320Ji5 < 0; CEB h) i) i) k) 67,16 = 0; 68,16 = 0; 619,16 = 0; 620,16 = 0; 67,16 ~ 0; 68,16 < 0; 619,16 = 0; 620,16 = 0; 87,16 < 0; 88,16 < 0; 819,16 < 0; 020,16 < 0; 67,16 < 0; 68,16 < 0; 619,16 > 0, 820,16 ~ 0.
38 Assumptions d) and h) state that there are no boy or girl sufficiency effects on contraceptive use pattern and children ever born. Assumptions e) and i) state that the effects are additive, and that sufficiency results in an increased likelihood of contracepting, as well as lower later fertility. Although opposite assumptions could be made, they would be implausible given the manifest content of the SCG and SCB variables. Assumptions f) and j) assume interactive effects of SCG and SCB with socioeconomic factors. We expect the impact of boy and girl sufficiency to be stronger for higher- than for lower-status families. Finally, assumptions g) and k) assume the presence of interactions that work in the opposite way: increasing status diminishes the effects of boy and girl sufficiency. We think the counteracting interactions to be empirically unlikely and have included them for the sake of complete- ness. All of these alternatives hold only for transitional settings. The alternative assumptions for the LCU and CEB equations can be considered independently from each other and from the CM equation. To again state the conclusion before discussing the results in detail, assumptions d), e), f), h), i), and j) lead to the same conclusions about the semireduced form LOU and CEB equations. Assumptions g) and k), which include the unlikely interactions, lead to different conclusions for those equations. All of our conclusions about the implied and actual semireduced form effects of the socioeconomic variables on child mortality, contraceptive use, and children ever born are summarized in Table 1.3. The remaining discussion in this section explains how the conclusions reported in Table 1.3 have been reached. To make inferences about the coefficients of equations (1.17) to (1.19), it is necessary to examine the product strings of structural coefficients that comprise each semireduced form coefficient. These product strings are not presented here, since they are very long. To reach the tentative conclusions summarized in Table 1.3, we determined the likely sign of each semireduced form coefficient by inspecting the pattern of signs in the product strings of structural coefficients that comprise the semireduced form coefficient. This determination inevitably involves expectations about the relative magnitudes of terms in a string when not all terms have the same sign. Since an actual numerical outcome will not be known until the full model has been estimated or simulated, we have assumed for present purposes that if the great majority of terms comprising a semireduced form coefficient have the same sign, that coefficient is likely to have that sign, especially if the terms in the minority are mostly those due to adjustment effects (e.g., those associated with FEC, NMAR, and OUR). If, however, the signs of the terms are about equally distributed, we have assumed that no inference about the sign of the semireduced form coefficient is justified. A simple illustration may help to clarify this process. The implied semireduced form equation for CEB is the sum of the actual semireduced form equations for EF and LF. The procedure described above can be illustrated through examination of the actual semireduced form equation for EF. To obtain this equation, we substitute structural equation (1.1) into structural equation (1.4), yielding the following:
39 TABLE 1.3 Derived Signs of Implied and Actual Semi-Reduced Form Socioeconomic Effects on Child Mortality, Contraceptive Use Pattern and Children Ever Born, by Settinga Contraceptive Children Child Mortality Use Pattern Ever Born Socioeconomic Variables Trad. Trans. Trad. Trans. Trad. Trans. Part A. Derived Signs Based on Assumptions a,b,d,e,f,h,i,j~ RESC ? WED ? WBM RES WSM RED HOCC Part B. Derived Signs Based on Assumptions c,g,kb RESC ? ? WED WBM RES WSM HED HOCC ? ? Effects whose signs can be deduced are indicated by the symbol denoting implication. Effects whose signs are inferred are indicated by the absence of this symbol. Effects whose signs cannot be inferred are denoted by a question mark. Effects which are necessarily zero because the variables are undefined for a particular setting are indicated by an axe. Alternative assumptions about the effects of sex preference are labeled as follows: CM a. b. c. LOU ¢7,14 = 0; ¢8,14 = 0 ¢7,14 < 0; h8,14 < 0 h7,14 ~ 0; h8,14 > 0 d ¢7,15 = 0; ¢8,15 = 0; ¢19,15 = 0; 820,15 0 e. b7,15 ~ 0; ¢8,15 > 0; d19,15 = 0; 620,15 = 0 f ¢7,15 > 0; ¢8,15 > 0; ¢19,15 > 0; ¢20 15 ~ 0 g b7,15 > 0; ¢8,15 > 0; Petals < 0; 620 15 < 0 CEB h ¢7,16 = 0; b8,16 = 0; ¢19,26 = 0; ¢2G,16 = 0 ¢7,16~< 0; ¢8,16 < 0; ¢19,26 = 0; 020,16 = 0 ~ t7,16 < 0; b8,16 < 0; b1g~l6 < 0; 620,16 < 0 k ~7,16 < 0; 68,16 < 0; b19,16 ~ 0; 520,16 ~ 0
40 EF = (80,4 + 63,400,1) + (81,4 + 03,481,1)RESC + (82,4 + 63,482,1)WED + 64,4WBM = aO,4 + al,4RESC + CX2,4WED + a3~4WBM. Next, consider the semireduced form coefficient for RESC, which is l 4 = 51~4 ~ 63,461,1. ~0r~ng few the hypothesized signs in Table 1.], we would expect a sign pattern of (+, +) in an ideal-typic traditional setting (since ~l'4 > 0 and 63~481,1 > 0) and a sign pattern of (+, -) in an ideal-typic transitional setting (since 61,4 > 0 and 63,401 1 < 0~. Thus, given the structural equation hypotheses, the semiretuced form sign of RESC is implied to be positive in traditional settings, while there is no sign implication for transitional settings. Moreover, it would be useless even to make an inference about the sign in transitional settings, since we hypothesize that 61~4 > 0 and 63,401,1 < 0. Of course, we could hypothesize relative magnitudes of these two constituents to resolve the ambiguity. Although we will do so on occasion, such hypotheses must be used sparingly if the overall exercise of sign inference is to be of general interest. Given this procedure for making inferences about the likely signs of the socioeconomic variables, the following discussion focuses on Table 1.3. me three equations are considered separately, first for traditional settings, and then for transitional settings. Child Mortali by tCM) Considering first the equation for child mortality in traditional settings, note first that there are no entries in Table 1.2 for WBM and WSM because in ideal-typic traditional settings, these variables are not defined. Observe next that the effects of current residence (RES) r husband's education (HED), and husband's occupation (HOCC) are implied to be negative. This implication holds because only a single structural coefficient is involved in each instance. The implied semireduced form effects of childhood residence (RESC) and respondent education (WED) are ambiguous. The direct effects of RESC and WED on ECM and LCM are negative (see Table 1.2), as are indirect effects through AFB, NMAR, and ECM. These are offset by positive indirect effects through EF and FEC. It is therefore uncertain whether the implied semireduced form effects of RESC and WED will be positive, zero, or negative. This conclusion is not sensitive to assumptions made about the effects of sex composition, as can be seen in the comparison of panels A and B of Table 1.3. It results from structuring the model to include intermediate outcomes, and probably could not be reached otherwise. The implied semireduced form coeff icients for child mortality in transitional settings share certain similarities with those for tradi- tional settings. In particular, the coefficients for RES, WSM, HED, and HOCC are implied to be negative. This is so because only a single structural coef f icient is involved in each instance.
41 The coefficients for RESC, WED, and WBM are ambiguous in transi- tional settings. If sex sufficiency has no effect on LCM, or if the effect is such that achieving a sufficiency of boys and girls decreases LCM, then the coefficients are likely to be negative. Most terms in the compounds that comprise the implied semireduced form coefficient. for RESC, WED, and WHIM are negative. A few positive terms are present because we hypothesize that RESOf WED, and WBM enhance FEC and EF (see Table 1.2), and should thus have posit zve indirect effects on overall child mortality. In addition, we hypothesize pQS\tiYO relationships between AFB and RESC and WED, implying younger families later on and the possibility of higher LCM. However, since these indirect effects are likely to be minor, the effects of RESC, WED, and WBM will probably be negative. If, on the other hand, a sufficiency of boys and girls increases LCM, the expected signs of the implied semireduced form coefficients for RESC, WED, and WBM are indeterminate; many of the indirect effects of RESC, WED, and WBM on boy and girl sufficiency are positive and translate into positive indirect effects on child mortality, counterbalancing their negative effects via other routes. Panel B of Table 1.3 displays the consequences of this alternative assumption for semireduced form inferences. Without prior knowledge, it is uncertain whether the semireduced form effects of RESC, WED, and WBM are positive, zero, or negative in transitional settings. To summarize the implied semireduced form effects of the socio- economic variables on child mortality in traditional and transitional contexts, out of twelve coefficients, seven have implied signs, two are indeterminate as to sign regardless of sex sufficiency effects, and three are determinate given prior knowledge of particular forms of those effects. Although this may at first seem to be very little result for the effort, the outcome is actually quite helpful. It shows that, given our model, there is no theoretical basis for expecting interpretable coefficient variability in the implied semireduced form socioeconomic effects across the traditional/transitional continuum. In other words, we conclude from this analysis that, since settings differ in their levels of child mortality, it is important to analyze variability in intercepts. For RES, BED, and HOCC, we have deduced that the effects should be negative at both extremes of the traditional/transitiona' continuum. The effect of WSM must also be negative where this negative is defined. AS for RESC and WED, we cannot deduce or even infer that their effects will "travel" from positive to negative as we move across the traditional/transitional continuum. A similar point holds for WBM, although it is not defined at the extreme of the traditional end of the continuum. It appears that the implied semireduced form equations will not differ meaningfully in their structural coefficients, but rather in their intercepts; this is consistent with an additive rather than an interactive analysis-of-covariance model with countries as the basis of classification and the socioeconomic variables as covar~ates. This is the fourth major implication in the present reduction exercise.
42 ontracePtive Use (LCU) For those variables defined in traditional settings, the semireduced form socioeconomic effects on LCU are all zero. mis follows from the hypothesized zeros for the structural coefficients of equation (1.15~. The inferences concerning transitional settings are more complex. Equation (~l 5} contains hypothesized interactions and a nonlinearity; the result is a semireduced form equation also containing interactions and nonlinearities, different from those in the structural equation. There is further complexity here because the inferred sign of some product strings depends on assumptions about the effect of the interaction between boy Buff iciency (SCB) and HOCC and WED in the structural equation. As mentioned earlier, four scenarios can be constructed from reasonable alternative assumptions about the direct and interactive effects of sex composition (assumptions d) to gig. Three yield similar conclusions, shown in Table 1.3, Part A. The fourth, based on the assumption that SCB in combination with high socioeconomic status dampens contraceptive use and increases later fertility, weakens considerably the conclusions drawn about semireduced form effects of RESC, WED, and WBM, as shown in Table 1.3, Part B. This latter scenario seems the least likely of the four to occur empirically. me three scenarios that lead to the conclusions summarized in Table 1.3, Part A are as follows: sex composition is a matter of indifference; boy and girl sufficiency have positive additive effects on LCU and negative additive effects on LF; and the interaction of SCB and SES enhances LCU and reduces LF. Considering first those variables which do not interact in the semireduced form equation, the semireduced form effects of RES, WSM, and HED are unambiguously positive in transitional settings under these three sets of assumptions. Among those var tables whose semireduced form effects are interactive, the ef feet of WBM is also unambiguously positive. The effects of the remaining interactive variables (RESC, WED, and HOCC) in the semireduced form are not directly deducible, but are probably positive. In the unlikely event that the fourth scenario prevails, the semireduced form effects of RESC, WED, IBM, and HOCC become ambiguous , although conclusions about HED, WSM, and RES are unchanged. In general, the structural hypotheses summarized in Table 1.3 lead to the expectation that the effects of socioeconomic variables on LCU are positive in most cultures, and become larger In value the higher the socioeconomic status. m e principal sources of ambiguity in the analysis of signs of product strings are that AFB contributes expected negative values to various components of the strings as well as the culture- specific effect of sex composition. Late onset decreases the probability of subsequent contraceptive use both directly, because of the nature of the hypothesized curvilinear relationship between AFB and LCU, and indirectly, by decreasing early fertility and thereby reducing the incentive to contracept. It is expected that, as societies move toward the transitional end of the traditional/transitional continuum, these negative effects should become attenuated relative to the socioeconomic effects. This expectation provides the basis for the inference (but not the deduction) that the partials for RESC, WED, and HOCC will be positive in most transitional settings.
43 Combining these results, the semireduced form effects of socioeco- nomic position on LCU should be zero in traditional settings and should be positive, either by deduction, reasonable assumption, or plausible inference, in transitional settings. An additional result is that in transitional settings, the semireduced form effects of some of the socio- economic variables are interactive, with those interactions reinforcing the positive socioeconomic effects in most cases. Because the progression is from zero to positive c~effzeients with movement from the traditional to the transitional end of the continuum, we conclude that with actual data, there should be stronger positive socioeconomic effects the more transitional the context; this is the fifth major conclusion of the reduction exercise. Children Ever Born (CEB) The implied semireduced form equation for CEB in traditional settings turns out to be much simpler than indicated by equation (1.19}, which states the equation without respect to context. Because the structural equation for later fertility (equation (1.16)) contains no hypothesized nonlinearities or interactions in traditional settings, it follows that the implied semireduced form equation for CEB contains none as well. This equation is thus additive, with all of its effects implied to be positive, if present. The implied semireduced form equation for CEB in transitional settings contains interactions and nonlinearities, as shown above. The three socioeconomic variables not involved in these interactions and nonlinearities (RES, WSM, and HED) have necessarily negative effects, given their hypothesized structural effects. For no other variables are the signs of the coefficients deducible. The effects of sex preference under one set of assumptions preclude inferences about the semireduced form effects of the remaining socioeconomic variables (Table 1.3, Part B). However, inferences can be derived under any one of the other three sets of assumptions tTable 1.3, Part A). The direct effects of RESC, WED, WBM, and HOCC have all been hypothesized to be negative. Although the effects of onset work in the opposite direction, the logic applied at a corresponding point in the discussion of LCU has equal force here: as the use of contraception becomes more widespread in a society, the effects of AFB should lessen. If this view is correct, the implied semireduced form effects of RESC, WED, WBM, and HOCC in most cultures should become larger in the negative direction as location on the traditional/ transitional continuum shifts toward increased contraception. The results for CEB thus lead to the following implications and inferences concerning the semireduced form model: 1) there should be an inverse relationship between so~zoeconomic effects and degree of transi- tionality such that in most cultures' a shift from traditional to transitional setting is associated with weakening positive and growing negative effects; 2) the more this shift involves the differential participation of social groups, the more likely there are to be non- linearities and interactions in the semireduced form socioeconomic effect; 3) in the unlikely event that intrasocietal cultural differences
44 are such that the interactive effects of boy sufficiency and socioeconomic status are negative for DCU and positive for LF, unambiguous sign expecta- tions for four of seven socioeconomic variables in the semireduced form model cannot be derived. These points comprise the sixth major conclusion of our reduction exercise. Discussion The conclusions we have reached here about the implied and actual semireduced form coefficients depend on certain assumptions, especially those involving the effects of sex preference. Expectations about the signs of many of the semireduced form coefficients depend on the effects of sex preference, especially in transitional settings. It therefore appears that these preferences are crucial to understanding the effects of soazoecenomic variables on CM, LCU, and CEB. m us, if SOB in inter- action with socioeconomic status is expected to decrease LCU and increase OF, there Is no point in regressing LCU and LF on the socioeconomic variables because we could not interpret the effects. On the other hand, if we think alternative assumptions about sex preference are appropriate, we can reach conclusions of varying firmness about the socioeconomic effects. ffl ese conclusions concern functional form, as well as expecta- tions about the nature of covariabilty between implied and actual semireduced form socioeconomic effects and the degree of transitionality. This exercise offers several payoffs. First, we have been able to show what expectations about socioeconomic effects depend on. Second, we have shown that given key assumptions, it is possible to derive expecta- tions for socioeconomic effects that not only are quite specific, but also differ from usual expectations not explicitly based on a structural model such as ours. Third, we have pinpointed areas within the model which deserve highest priority, in particular the boy and girl sufficiency variables. If the effects of these variables turn out to be minimal or follow the patterns leading to the expectations delineated in Table 1.3, Part A, there will be grounds for using the estimation of LCU and CEB based on insights developed in this section to motivate our empirical analyses. Finally, we have shown that contextual differences in the semireduced form model of CM will be concentrated in intercept differ- ences, rather than in variation in socioeconomic effects. 1.5 CONCLUSION We have proposed, justif led, and explored a model that organizes and synthesizes hypotheses about the determinants of fertility and related behaviors . m is model is designed for use with cross-sectional survey data sets, especially those collected by the World Fertility Survey. Although the discussion has focused primarily on the assumptions and hypotheses embodied in the model, the major operational issues have received attention as well. In developing the model, we have concentrated on two main problems: the establishment of a set of micro-level struc- tural equations and the formulation of hypotheses about structural
