A cointegration model of age-specific fertility and female labor supply in the United States.

• Author: McNown, Robert

1. Introduction

   Models of fertility based on economic theories of behavior have been subjected to rigorous conceptual and empirical scrutiny (for surveys, see Olsen 1994 and Macunovich 1996a; for critical reviews, see Murphy 1992 and Smith 1981). However, much empirical analysis of aggregate fertility patterns in the United States has relied on traditional regression methods, with little influence from recent developments in multiple time-series methods appropriate for nonstationary variables.

   Although important theoretical propositions are testable with individual data, understanding of trends and patterns in fertility behavior at the societal level requires aggregate analysis (Ryder 1980). The aggregation of individual effects to make statements about total fertility is problematic, as the composition of the population changes over time. Some effects that are measured at the individual level may reflect changes in individuals' positions within a society, and these effects will not be present at the societal level. Alternatively, social interaction may induce behavioral changes across a population that are not reflected in individual responses. As described by Kohler (2001), changes in societal norms and institutions can create feedback loops between aggregate variables and individual incentives toward childbearing, so that effects between aggregates may exceed substantially the responses measured at the level of the household. (1)

   Analysis of aggregate time-series data has its own considerable challenges. Fertility and its determinants are most likely nonstationary time series that trend or drift persistently away from their initial values. Such nonstationarity may undermine classical estimation and inference with traditional regression procedures, leading to spurious inferences about relations among variables. Furthermore, the principal determinants of fertility, for example, women's wages and education levels, female labor force participation, unemployment rates, and husband's incomes, are quite possibly endogenously determined in conjunction with fertility decisions. This problem of endogenous regressors can undermine the identifiability of the fertility model, rendering the relations unestimable. Even if the relations are identified, the problem of endogenous regressors leads to inconsistent least squares estimators of model parameters.

   The objective of this paper is to revisit a simple economic model of fertility, employing the cointegration model of Johansen (1995) that is appropriate for analyzing relations between nonstationary time series. Johansen's procedure allows the empirical determination of the number of stationary relations and produces maximum likelihood estimators of the parameters of these relations that are consistent and normally distributed, even in the presence of endogenous explanatory variables. Furthermore, to capture information on both the level and the timing of fertility, the analysis is applied to two age-specific fertility rates covering the prime childbearing years of U.S. women.

2. Empirical Studies of Fertility with Aggregate Data

   Economic models of fertility are grounded in either Easterlin's (1980a) relative income hypothesis or the New Home Economics (NHE) of Willis (1973) and Becker (1981). The former theory emphasizes the role of male incomes, relative to economic aspirations, as the driving force behind fertility and female labor force participation. Economic aspirations of young adults are determined by material conditions prevailing in their parental homes during their teenage years, when their parents would be close to their prime in earnings capacity. An increase in relative income shifts preferences in favor of childbearing and away from labor force activity by young adult women.

   In the full Easterlin model, relative income is determined by the size of the young adult cohort relative to that of prime aged adults, both measured contemporaneously (Easterlin 1980b). An unusually large cohort of young adults faces competition from their peers in education and employment opportunities, with adverse consequences for their earnings. At the same time, the earnings of their parents, attached to a smaller birth cohort, may have been unusually high, contributing to the formation of high material aspirations by the younger generation. Therefore, relative cohort size influences both incomes and economic aspirations of each generation as they face decisions concerning fertility and labor market activity in their early adult years.

   Empirical tests of the Easterlin model have been surveyed by Pampel and Peters (1995) and Macunovich (1998). Although Pampel and Peters conclude that the evidence in support of the Easterlin hypothesis is rather weak, Macunovich's assessment is considerably more favorable to this model. Given the continuing controversy, it is useful to test Easterlin's propositions with contemporary time-series methods. Consequently, relative male cohort size and relative male incomes are employed in this study rather than the incomes of younger males as suggested by alternative economic theories.

   The NHE model stresses the role of female wages, representing the opportunity cost of childbearing, as a determinant of fertility. Female wages are seen to have both (positive) income and (negative) substitution effects on fertility, with opposite effects on female labor force participation. Income from sources other than women's wages is expected to have a positive effect on the demand for child services, assuming such services are a normal good. Surveys of empirical studies of the NHE model are provided by Macunovich (1996a) and Hotz, Klerman, and Willis (1997). The following discussion builds on these surveys with a focus on issues arising from the nonstationarity of variables and endogenous regressors that are characteristic of empirical studies of fertility with time-series data.

   Numerous studies of fertility from the NHE or the relative income perspectives employ questionable exogeneity assumptions to "achieve" identification of their models. Female wages are treated as exogeneous, for example, in Butz and Ward (1979), Ermisch (1979), Shapiro (1988), Lee and Gan (1989), and Winegarden (1984), often in interaction terms involving other variables. Wage rates depend on work experience, which is interdependent with fertility. Consequently, the treatment of female wages as exogenous in these regressions raises at a minimum the possibility of simultaneity bias and at worst underidentified models.

   Although Mincer (1963) contends that fertility and female labor market activity should be modeled with two separate equations, many researchers include female labor force participation as an argument in their fertility equations. Butz and Ward (1979) and Ermisch (1979), for example, include this variable in interaction terms with other explanatory variables, treating its endogeneity with instrumental variables procedures. However, female labor force participation appears as an exogenous regressor in the fertility models of Shields and Tracy (1986) and Pampel (1993).

   Other researchers have explicitly dealt with the endogeneity of female labor force participation, women's wages, and fertility with simultaneous equations techniques that produce consistent estimators by use of instrumental variables. Devaney (1983) and Sprague (1988) estimate two-equation systems with fertility and female labor force participation rates as jointly dependent variables while also treating female wage endogeneity through instrumental variables. In Macunovich's (1996b) model, fertility and labor force participation do not appear as regressors in the equation for the other variable, and she handles the problem of wage endogeneity by controlling for education, age, and experience differences in the construction of her variables.

   Although these latter studies move toward a solution to the problem of endogeneity of explanatory variables in the fertility equation, they may not go far enough. The entire system of variables involved in aggregate fertility models is subject to rampant endogeneity. Labor force participation rates, unemployment, women's wages, educational attainment, and fertility rates are joint outcomes reflecting decisions made by men and women throughout their young adult years. In addition, male incomes are affected by female wages and labor force participation as a result of possible substitution between male and female workers in labor markets. None of the traditional explanatory variables in fertility equations provides the exogeneity that is necessary for traditional econometric identification and estimation of structural models.

   A further concern with many aggregate fertility studies is the failure to deal with nonstationary variables. Although unit root tests (section 4) indicate that fertility and its covariates are nonstationary, most studies have ignored this issue. (2) Notable exceptions in the fertility literature include Ermisch (1988); Macunovich and Easterlin (1988); Wright (1989); Mocan (1990); Masih and Masih (1999); Abeysinghe (1991, 1993); Bailey and Chambers (1993); Wang, Yip, and Scotese (1994); Cheng (1996); Poot and Siegers (2001); and Hondroyiannis and Pappetrou (2002). All these studies find that the variables in their models must be differenced to become stationary, a property that undermines the validity of traditional estimation procedures and statistical inferences in regressions involving undifferenced series.

   If there is no stationary linear combination of these nonstationary time series, then all variables must be differenced to stationarity prior to estimation and inference. This is the case for the models of Wang, Yip and Scotese (1994), who investigate the relations among total fertility, total weekly hours of work, and real gross national product (GNP); Cheng (1996), who considers the bivariate relation between the crude birth rate and the...