Factors predicting the US birth rate

Hamid Baghestani [email protected] American University of Sharjah Michael Malcolm [email protected] West Chester University

June 2015

ABSTRACT This paper takes a forecasting approach to examine the relationship between the US birth rate, marriage rate and economic conditions. Utilizing monthly data, we specify several forecasting models and use recursive estimation to generate multi-period forecasts of the birth rate. We then employ standard evaluation methods to compare the predictive content of the forecasts. We find that the birth rate is pro-cyclical and that, while both realized and expected unemployment contain useful information for predicting the birth rate, expected future unemployment is a more informative indicator. Exogenous changes to marriage also impact the birth rate. JEL Classification: J12, C32 Keywords: Demographics; Marital formation; Expected unemployment; Michigan Surveys of Consumers; Forecast evaluation

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1. Introduction As early as Malthus, economists have been interested in the relationship between childbearing and economic conditions. For economies in the developing stage, it has generally been the case that economic growth is accompanied by declines in the birth rate (Jones and Vollrath 2013). By contrast, in developed economies, low birth rates are increasingly challenging policymakers on a number of fronts. Data from the 2013 CIA Factbook indicate that France is the only country in Europe with a birth rate that exceeds replacement. Australia, Canada and all of the developed East Asian economies also have birth rates lower than replacement. The birth rate in the United States is hovering around replacement, but only because of immigrants. A low birth rate brings with it a whole host of public policy obstacles. For example, the Social Security Administration reports that, in concert with higher life expectancy, declining birth rates have reduced the ratio of workers to social security beneficiaries in the United States from 16.5 to 2.9 over the past 60 years. In light of these challenges, policymakers increasingly have their eye on governmental action that can manage the birth rate. In Japan, where the birth rate has fallen to 1.4 births per woman, the government has resorted to cash payments in excess of $3000 per year for each child under the age of 16 as an incentive for child-bearing (Wakabayashi and Inada 2009). In this context, one place that policymakers can look is to the relationship between economic conditions and childbearing decisions. While it is generally understood that economic conditions and prospects can influence the birth rate, the nature of this relationship is an open question in the literature, and much of what exists provides conflicting answers. The foundation of much of the theoretical work on the matter is provided by Becker et al. (1960), who assert that family income is a key determinant of child-bearing decisions. Importantly, their model suggests that children should be normal goods. By contrast, competing models and

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casual observation from developed countries have suggested that children might be inferior. Empirically, this question has been addressed in the literature by analyzing whether the birth rate displays pro-cyclical or counter-cyclical tendency, which would provide evidence for the normal and inferior hypotheses, respectively. There is a significant body of work on the relationship between child-bearing and income/employment, but there is no consensus on an answer. This question is critical for policymakers who seek to understand the complex relationship between child-bearing and the performance of the economy – not only for purposes of forecasting and planning, but also in the context of policy that actively targets management of the birth rate. With this in mind, we take a new approach to the question by utilizing forecasting techniques to offer a straightforward and direct way to examine the nature and robustness of the relationship between birth rates, marriage rates and economic conditions. In a basic sense, we first seek to know whether economic fluctuations provide information that can help predict future birth rates. To do this, we use monthly data from 1975 to 2008 to specify several time-series models that attempt to predict the monthly birth rate using lagged information on birth, marriage and economic conditions. By comparing the accuracy of forecasts generated by these alternative models, we can learn something about the fundamental drivers of child-bearing decisions. We begin with a univariate ARIMA model that uses past information on birth rates only. We then construct three augmented ARIMA (ARIMA-A) models that add additional variables, so that we can test the relationship between these variables and the birth rate. The first ARIMA-A model includes past information on marriage rates in addition to birth rates. The second ARIMA-A model adds past information on the unemployment rate, as well as birth and marriage rates. The third ARIMA-A model is similar to the second except that it includes past information on expected unemployment (derived from the Michigan Survey of Consumers) rather than the actual

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unemployment rate. We use the data from January 1975 to December 2001 (1975.01-2001.12) to generate estimates of the model parameters, and then use the remaining years to evaluate the goodness of our forecasts. It is worth noting first that all of our forecasts are directionally accurate, free of systematic bias and efficient. These results help us to address a number of economic questions. First, our forecasts confirm the assertion by Becker, et al. (1960) that economic conditions are a factor for households in making child-bearing decisions. Our results show systematically that models incorporating economic fluctuations (measured either by realized or expected unemployment) provide stronger predictive power than models using only marriage and birth histories. Additionally, our results provide evidence for the pro-cyclical hypothesis, which suggests that children are normal goods. Second, we find that expected future unemployment is more informative than realized unemployment for predicting the birth rate. From a theoretical perspective, this finding provides evidence that families make life-cycle decisions with respect to child-bearing, with an eye to future economic conditions even more strongly than current conditions. The theoretical basis of this lifecycle perspective on child-bearing decisions was pioneered by Heckman and Willis (1976). From a policy perspective, this result and the previous one both strongly suggest that counter-cyclical macroeconomic policy and automatic stabilizers that shield families from macroeconomic shocks are helpful in promoting childbirth. Anchoring expectations is especially critical. Third, even after controlling for trends in the birth rate, fluctuations in marriage rates independently add useful predictive content for the birth rate. Our results therefore add empirical rigor to the analysis of the impact that changes to marriage are likely to have on birth rates, and confirm the assertion of Becker et al. (1960) that children are still a marriage-specific investment. From a policy perspective, Lundberg and Pollak (2007) argue that institutional changes to the

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structure of marriage, like no-fault divorce and changes to how alimony are calculated, reduce the credible commitment aspect of marriage, compromise the gains from trade, and in turn reduce investment in household public goods like children. Our paper proceeds as follows. Section 2 reviews the relevant theoretical and empirical literature on the relationship between economic conditions and the birth rate. Section 3 discusses the data and the forecasts. Section 4 reports the forecast evaluation results. Section 5 concludes. 2. Related literature Early contributions in estimating the responsiveness of the birth rate to various economic variables include Drakatos (1969) for Greece, using cross-section data and Ben-Porath (1973) for Israel, using time-series data. They reach opposite conclusions as to whether birth rates are pro- or counter-cyclical, but Ben-Porath is notable for examining the time-series links between childbearing and a number of different economic variables, including per-capita income, unemployment, and industrial output. Later, Schultz (1985) analyzed county-level wage shocks in Sweden to show that birth rates are pro-cyclical for younger men, but counter-cyclical for older men – the two effects are of approximately equal magnitude and basically wash each other out. Jones and Tertilt (2008) use US cross-sectional data to show that birth rates have declined with income increases across many generational cohorts. In contrast, Black et al. (2013) use exogenous income shocks related to energy prices to argue that birth rates are pro-cyclical. A number of more recent studies analyze the relationship between employment shocks and child-bearing, again with mixed results. Female employment, generally, has been associated with declines in birth rates in developed countries (Adsera 2005), but this result is difficult to interpret causally. Dehejia and Lleras-Muney (2004) find no statistically significant effect of maternal or paternal unemployment on birth rates, although the sign of the estimated coefficient supports the

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pro-cyclical hypothesis. Adsera (2004) finds that both unemployment and job instability do depress birth rates, especially among young women. Adsera (2005) argues that the gap between female and male unemployment is an important determinant of child-bearing propensity among European women. Del Bono et al. (2012), using data from Austria, find that temporary job displacements reduce birth rates, but that the effect is mainly attributable to women in skilled occupations. Using data from Finland, Huttunen and Kellokumpu (2012) show that only female job displacements are important. Lindo (2010) finds that male job displacements do reduce total child-bearing in the long-run, but that they accelerate child-bearing in the short-run. Lindo’s dynamic model suggests that the expected future trajectory of economic outcomes may be more important than the present state. This insight is particularly informative in light of our result that expected future unemployment is more informative than the current unemployment rate in predicting birth rates. The theoretical picture of this relationship is no clearer. Becker, et al. (1960) suggest that children should be normal goods. But they also note that, if we extend our index of children to incorporate both the quantity and the quality of children, then the quantity of children may well fall when income rises (accounting for the observed inferiority of quantity of children to income and employment shocks). The picture is especially murky when we account for the opportunity cost of parents’ time. Furthermore, in understanding the time-series structure of birth rates, timing matters, as well as total number of children desired. Heckman and Willis (1976) provide the seminal contribution on incorporating child-bearing timing into life-cycle models. As Lindo (2010) suggests, these dynamic considerations can lead to counterintuitive findings in the data. To summarize, the literature does not offer a clear picture of the relationship between childbearing and economic fluctuations. There is existing research that supports positive, negative, and

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zero correlation. Given the current state of the literature, our forecasting approach is indeed a new and important contribution, since it directly investigates the predictive information content of economic indicators for child-bearing choices and outcomes. 3. Data and alternative forecasts The birth rate is the number of live births per 1000 population, and the marriage rate is the total marriages per 1000 population. The monthly birth and marriage rates reflect enumerated counts from official records rather than survey data, and they come from various issues of the National Vital Statistics Report (published by the National Center for Health Statistics only up to December 2008). The unemployment rate data come from the Bureau of Labor Statistics, and the expected unemployment index is available on the Michigan Surveys of Consumers (MSC) website. This survey probes consumer sentiment on personal finances, buying and business conditions, and expectations.1 Utilizing a randomly selected sample of at least 500 US households, the survey collects individual responses to approximately 50 core questions. One question asks, “How about people out of work during the coming 12 months -- do you think that there will be more unemployment than now, about the same, or less?” Using the individual responses, the survey calculates the index values (= less – more + 100). We utilize this index as the measure of expected unemployment (EUt).2 With the way the index is constructed, a positive (negative) relationship between EUt and the birth rate leads to the conclusion that the birth rate is pro-cyclical (countercyclical) with respect to expected future economic conditions.

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See Ludvigson (2004), among others, on examining the MSC indicators.

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The MSC expected unemployment data are available on a monthly basis beginning January 1978.

Prior to this date the survey was conducted every February, May, August, and November. For the years prior to 1978, we have used the February (May) data for both January and March (April and June). Similarly, we have used the August (November) data for both July and September (October and December).

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Figure 1 depicts the time plots of the series for January 1975 to December 2008 (1975.012008.12). The birth rate (BRt), which has been trending down since the mid-1980’s, has a mean value of 15.1 and high and low of 17.7 and 13.0. The marriage rate (Mt), which has also been trending downward, especially since the 1980’s, displays a mean value of 9.2 and high and low of 15.8 and 4.2. The unemployment rate (Ut) has a mean value of 6.2 and high and low of 10.8 and 5.9%, and expected unemployment (EUt) has a mean value of 81.92 and high and low of 129 and 33 index points. Our initial estimation period is 1975.01-2001.12. In examining the stochastic properties of the series for this period, we employ the KPSS (Kwiatkowski et al., 1992) test using the Bartlett window approach with the optimal bandwidth value automatically determined according to Andrews’ (1999) method. Rows 1-3 (column 1) of Table 1 report the KPSS test statistics for BRt, Mt, and Ut. For these series, we reject the null hypothesis of stationarity in favor of a unit root alternative. Further inspection indicates that these series each have a unit root since, as reported in rows 1-3 (column 2), the null hypothesis of stationarity cannot be rejected for ∆BRt, ∆Mt, and ∆Ut. Finally, for expected unemployment (EUt) in row 4, we cannot reject the null hypothesis of stationarity. In constructing the univariate ARIMA model, we utilize the simple and partial autocorrelation functions of ∆BRt. Based on these function estimates for 1975.01-2001.12 along with the Akaike and Schwarz information criteria, we specify the following ARIMA model for the birth rate, (1 – Ø12B12)∆BRt = (1+ ∑12 i=1 θi B)εt

(1)

where B is a backward operator and εt is the error term. We then exclude the insignificant movingaverage terms with the results reported in column 1 of Table 2. As can be seen, the Ljung–Box Q test (which detects serial correlation up to the 24th order) yields a p-value well above 0.10, indicating

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that the residual series is white noise, and thus the model is correctly specified. With the autoregressive and moving-average parameter estimates significant, we utilize the model to generate the three-, six-, and nine-month-ahead forecasts of the birth rate as follows. The 1975.012001.12 parameter estimates of the ARIMA model in column 1 are used to generate the forecasts of the birth rate for 2002.01-2002.09. The forecasts for 2002.03, 2002.06, and 2002.09 are, respectively, the three-, six-, and nine-month-ahead forecasts of the birth rate. Re-estimating the model for 1975.01-2002.01, we use the updated parameter estimates to generate the forecasts for 2002.022002.10. The forecasts for 2002.04, 2002.07, and 2002.10 are, respectively, the three-, six-, and ninemonth-ahead forecasts of the birth rate. This procedure is repeated until the last set of forecasts is generated for 2008.10-2009.06 using the 1975.01-2008.09 parameter estimates. The forecasts for 2008.12, 2009.03, and 2009.06 are, respectively, the three-, six-, and nine-month-ahead forecasts of the birth rate. As such, the sample periods for the three-, six-, and nine-month-ahead forecasts are, respectively, 2002.03-2008.12, 2002.06-2009.03, and 2002.09-2009.06. In this study, however, we use a single period (2002.09-2008.12) for every forecast horizon, with 76 observations. Our first augmented ARIMA model of the birth rate (denoted as ARIMA-A1) includes past information on the marriage rate (∆Mt-10 through ∆Mt-12) as follows, (1 – Ø12B12)∆BRt = (1+ ∑i θi Bi)εt + ∑12 j=10 bj ∆Mt-j

(2)

where i = 1, 4, 5, and 12. Since decisions to have children are made approximately nine months prior to birth, the lag on marriage starts at t-10. This, in turn, allows us to obtain the birth rate forecasts up to nine months ahead using a single equation. The estimates of Equation (2) for 1975.01-2001.12 are reported in column 2 of Table 2. As can be seen, the calculated Ljung–Box

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Q test statistic p-value indicates that the residual series is white noise. In addition, the sum of the parameter estimates on ∆M is significant and has the expected positive sign. Our second augmented ARIMA model (denoted as ARIMA-A2) contains past information on both the marriage rate and the unemployment rate (∆Ut-10 through ∆Ut-12) as follows, 12 (1 – Ø12B12)∆BRt = (1+ ∑i θi Bi)εt + ∑12 j=10 bj ∆Mt-j + ∑j=10 cj ∆Ut-j

(3)

where i = 1, 4, 5, and 12. The estimates of Equation (3) for 1975.01-2001.12 are reported in column 3 of Table 2. As can be seen, the sum of the parameter estimates on ∆M is significant and has the expected positive sign. In addition, the sum of the parameter estimates on ∆U is significant with a negative sign. This indicates that the birth rate is pro-cyclical. Our third and last augmented ARIMA model of the birth rate (denoted as ARIMA-A3) contains past information on both the marriage rate and expected unemployment (EUt-10 through EUt-12) as follows, 12 (1 – Ø12B12)∆BRt = (1+ ∑i θi Bi)εt + ∑12 j=10 bj ∆Mt-j + ∑j=10 dj EUt-j

(4)

where i = 1, 4, 5, and 12. The estimates of Equation (4) for 1975.01-2001.12 are reported in column 4 of Table 2. As can be seen, the sum of the parameter estimates on ∆M is insignificant, but with the expected positive sign. In addition, the sum of the parameter estimates on EU is significant with a positive sign, indicating that the birth rate is pro-cyclical with respect to expected future economic conditions. In the same manner described above for generating the ARIMA forecasts, we have utilized the above three augmented ARIMA models to obtain three additional sets of the three-, six-, and nine-month-ahead forecasts of the birth rate for the period from 2002.09 to 2008.12.

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4. Forecast evaluation test results Our evaluation focuses on answering the following four questions: 1. Are the forecasts directionally accurate? 2. Are the forecasts unbiased? 3. Are the forecasts efficient? 4. Which augmented ARIMA forecast is more informative? In what follows, At+f is the actual birth rate in month t+f, and Pt+f is a general notation for both univariate and augmented ARIMA forecasts of At+f made in month t (the forecast horizon f = 3, 6, ) and 9 months). More specific notations are Pt (Af) and Pt (AA which, respectively, denote the f

univariate ARIMA and augmented ARIMA forecasts. 4.1. Are the forecasts directionally accurate? With At-1 denoting the actual birth rate in month t-1, we define the actual change in the birth rate as (At+f – At-1) and the predicted change as (Pt+f – At-1). The directional accuracy rate is the proportion of correct sign predictions, and it is calculated as the number of observations for which (At+f – At-1) and (Pt+f – At-1) have the same sign divided by the sample size. As reported in Table 3, the directional accuracy rates (ranging from 87 to 92%) for both the univariate and augmented ARIMA forecasts are quite high. In testing the null hypothesis of no (directional) association between actual and predicted changes, we utilize the chi-square tests with and without Yate’s continuity correction and Fisher’s exact test (Sinclair et al., 2010). These tests are based on a 2×2 contingency table whose elements consist of the numbers of correct and incorrect sign forecasts. As indicated by superscript b in Table 3, we reject the null hypothesis of no (directional) association and, thus, the univariate and augmented ARIMA forecasts are all directionally accurate.

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4.2. Are the forecasts unbiased? We examine unbiasedness by estimating At+f =  +  Pt+f + vt+f

(5)

(At+f – Pt+f) = ’ + vt+f

(6)

where At+f is the actual birth rate, Pt+f is the forecast of At+f, and (At+f – Pt+f) is the forecast error. The forecast is unbiased if we cannot reject the null hypotheses that  = 0,  = 1, and ’ = 0 (Holden and Peel, 1990). Since Pt+f is made in month t, the forecast error follows an fth-order moving-average process under the null hypothesis of rationality. With the forecast errors generally heteroscedastic, we utilize the Newey-West (1987) procedure to correct for both heteroscedasticity and the inherent f th-order serial correlation. Table 4 reports the OLS estimates of Equations (5) and (6) with the correct (Newey-West) standard errors for the ARIMA forecasts in rows 1-3, and for the augmented ARIMA forecasts in rows 4-12. As can be seen, the forecasts in rows 1-8 and 10-12 are all unbiased since, for these forecasts, we cannot reject the null hypotheses that  = 0,  = 1, and’ = 0. The forecast in row 9 is not unbiased since we reject the null hypotheses that  = 0 and  = 1. However, this forecast is still free of systematic bias since we cannot reject the null hypothesis that ’ = 0. To augment these findings, the last column of Table 4 reports Theil’s U coefficient calculated as the mean squared error (MSE) of Pt+f divided by the MSE of the no-change forecast (which is the birth rate in month t-1). As can be seen, the U coefficient estimates (ranging from 0.111 to 0.182) are far below one. We use the Diebold-Mariano (1995) test to examine the null hypothesis of equal forecast accuracy (i.e., the MSE of Pt+f equals the MSE of the no-change forecast). As indicated by superscript d in Table 4, the p-value of this test for every forecast in rows 1-12 is below 0.10, indicating that the ARIMA and augmented ARIMA forecasts produce significantly lower MSEs than the no-change forecast. 12

4.3. Are the forecasts efficient? A forecast is said to be efficient if it contains past information in the target variable. By default, the univariate forecast is efficient since the ARIMA model efficiently employs past information in the birth rate. We thus examine the efficiency of the augmented ARIMA forecasts by estimating,

) At+f = γ0 + γ1 Pt (Af) + γ2 Pt (AA + vt+f f

(7)

) where Pt (Af) and Pt (AA denote, respectively, the univariate ARIMA and augmented ARIMA f

) forecasts. Pt (Af) lacks the useful information contained in Pt (AA f , when the estimate of γ1 is

insignificant (or significant but negative) and the estimate of γ2 is positive and significant. The ) converse is also true. Pt (Af) and Pt (AA contain distinct information when the estimates of γ1and γ2 are f ) both positive and significant. Pt (Af) and Pt (AA contain similar information when the estimates of γ1 f

and γ2 are both insignificant. Table 5 reports the OLS estimates of Equation (7) along with the correct standard errors for the ARIMA-A1, ARIMA-A2, and ARIMA-A3 in, respectively, rows 1-3, 4-6, and 7-9. As can be seen, the estimates of γ1 are insignificant but the estimates of γ2 are all significant, indicating that the augmented ARIMA forecasts contain useful predictive information above and beyond that contained in the univariate ARIMA forecasts. Therefore, we conclude that the augmented ARIMA forecasts are all efficient. Furthermore, the test results in rows 1-3 indicate that, controlling for past information in the birth rate, the marriage rate contains useful information for predicting the birth rate. 4.4. Which augmented ARIMA forecast is more informative? We examine this question by estimating,

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At+f = δ0 + δ1 Pt (Af1) + δ2 Pt (Af2) + vt+f

(8)

At+f = η0 + η1 Pt (Af1) + η2 Pt (Af3) + vt+f

(9)

At+f = ω0 + ω1 Pt (Af2) + ω2 Pt (Af3) + vt+f

(10)

where Pt (Af1) , Pt (Af2) , and Pt (Af3) denote, respectively, the ARIMA-A1, ARIMA-A2, and ARIMA-A3 forecasts. Rows 1-3 of Table 6 report the OLS estimates of Equation (8) along with the correct standard errors. As can be seen, the estimates of δ1 are generally insignificant but the estimates of δ2 are all positive and significant, indicating that the ARIMA-A2 forecasts are more informative than the ARIMA-A1 forecasts. As such, controlling for past information in birth and marriage rates, the unemployment rate has useful information for predicting the birth rate. Rows 4-6 of Table 6 report the OLS estimates of Equation (9) along with the correct standard errors. As can be seen, the estimates of η1 are insignificant and the estimates of η2 are significant, indicating that the ARIMA-A3 forecasts are more informative than the ARIMA-A1 forecasts. Again, this means that, controlling for past information in birth and marriage rates, expected unemployment has useful information for predicting the birth rate. Finally, rows 7-9 of Table 6 report the OLS estimates of Equation (10) along with the correct standard errors. As can be seen, the estimates of ω1 are insignificant and the estimates of ω2 are all significant, indicating that the ARIMA-A3 forecasts are more informative than the ARIMA-A2 forecasts. We thus conclude that expected unemployment has richer predictive information for the birth rate than realized unemployment. Figure 2 plots both the ARIMA-A2 and ARIMA-A3 ninemonth-ahead forecasts against the actual birth rate. As can be seen, both forecasts are free of systematic bias and closely replicate the variability in the actual birth rate. The ARIMA-A3 forecast generally displays a tighter relationship with the actual rate, implying that expectations

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about the future economic climate are more informative than realized economic conditions in predicting the birth rate. 5. Interpretation and Conclusion Declining birth rates are at the forefront of many policy discussions, from pension insolvency to declining crime rates. Thus, understanding the factors that underlie the time-series fluctuations in birth rates is important for policymakers both for predictive and for management purposes. In this paper, we have taken a new econometric approach to studying these fluctuations. Our results confirm in a new and robust way the assertion by Becker, et al. (1960) that fertility decisions made by families are inextricably linked to economic conditions. Thus, in a forecasting sense, our results confirm the hypothesis that economic conditions can help predict birth rates. Using our estimates, we have learned two things about the relationship. First, the data provide support for the pro-cyclical hypothesis that high incomes and low unemployment are associated with increases in the birth rate. Second, the data provide support for the assertion that families make life-cycle decisions with respect to fertility, as suggested by Heckman and Willis (1976) and Lindo (2010). Specifically, expectations about future economic conditions are a better predictor of the birth rate than current economic conditions. Furthermore, our results inform the discussion on the relationship between institutional changes to the structure of marriage, such as changes to divorce laws, and their effect on childbearing. Fluctuations in marriage rates are independently useful in predicting birth rates, even after incorporating information already contained in birth rate fluctuations. This finding reinforces the point from Becker et al. (1960) that children are a marriage-specific commodity and that changes in propensity to marry are independently informative in predicting changes in childbearing, even when marriage rates fluctuate for reasons exogenous to birth rates. Lundberg and Pollak (2007) have argued that institutional changes to the structure of marriage have been a causative factor underlying declining birth rates, and indeed fluctuating marriage rates do appear 15

to be an independent factor driving changes in childbearing, as evidenced by our result that including marriage rates in the model improves goodness of forecasts relative to incorporating only past information in the birth rate. Overall, despite the lack of clarity in earlier theoretical and empirical work about the underlying nature of the relationship between economic fluctuations and birth rates, our results show in a robust way that there is independent information content in economic fluctuations that can help predict future birth rates. In an econometric sense, it is not just that the two have occurred simultaneous to each other. Exogenous fluctuations in employment and employment expectations definitively impact child-bearing decisions and outcomes, with expectations of future employment being relatively more informative. The policy relevance of these results is manifest. For policymakers trying to predict birth rates, for example in evaluating the solvency of pension and social welfare programs, accurate forecasting models are critical. Economic fluctuations and changes to marriage are important in constructing these forecasts. Furthermore, these results add an additional dimension to evaluating macroeconomic policy generally, especially to the extent that countercyclical policy aims to shield families from major economic shocks and anchor expectations. A stable economy, and especially one where consumers are optimistic about the future, promotes childbearing. Our results also empower policymakers to analyze systematically the impact of changes to the structure of marriage on childbearing. Finally, our results provide empirical support for the efficacy of policies like those in Japan that provide outright financial incentives for childbearing; similar incentives exist in the US through the income tax system, although they are mostly implicit. Declining birth rates will likely continue to challenge policymakers in developed countries. But it is difficult to make headway in addressing the challenges associated with declining birth rates without a systematic understanding of the drivers of birth rates. In this paper, we have developed a new empirical approach that forecasts birth rates with high accuracy. Contrasting 16

across different specifications provides insight into the strongest predictors. These results thus provide an additional dimension for policy analysis of a number of economic and institutional factors that are linked to child-bearing.

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Holden, K. and D.A. Peel (1990) On testing for unbiasedness and efficiency of forecasts. Manchester School 58(2), 120-127. Huttunen, K. and J. Kellokumpu (2012) The effect of job displacement on couples’ fertility decisions. IZA Discussion Study No. 6707. Kwiatkowski, D., P.C.B. Phillips, P. Schmidt, and Y. Shin (1992) Testing the null hypothesis of stationarity against the alternative of a unit root. Journal of Econometrics 54, 159-178. Jones C. and D. Vollrath (2013). Introduction to Economic Growth. W.W. Norton. Jones, L. E. and M. Tertilt (2008) An Economic History of Fertility in the United States: 1826– 1960 (Vol. 1, pp. 165-230). Emerald Group Publishing Limited. Lindo, J.M. (2010) Are children really inferior goods? Evidence from displacement-driven income shocks. Journal of Human Resources 45(2), 301-327. Ludvigson, S. (2004) Consumer confidence and consumer spending. Journal of Economic Perspectives 18(2), 29-50. Lundberg, S. and R. A. Pollak (2007) The American Family and Family Economics. Journal of Economic Perspectives 21(1): 3-26 National Center for Health Statistics. National Vital Statistics Report. Various issues. Newey, W.K. and K.D. West (1987) A simple, positive semi-definite, heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703-708. Schultz, T.P. (1985) Changing world prices, women's wages, and the fertility transition: Sweden, 1860-1910. Journal of Political Economy 93(6), 1126-1154. Sinclair, T.M., H.O. Stekler, and L. Kitzinger (2010) Directional forecasts of GDP and inflation: A joint evaluation with an application to Federal Reserve predictions. Applied Economics 42(18), 2289-2297. Social Security Administration. Ratio of Covered Workers to Beneficiaries. Online. http://www.ssa.gov/history/ratios.html (accessed 6/2/15). Wakabayashi, D. and M. Inada (2009). Baby Bundle: Japan’s Cash Incentive for Parenthood. The Wall Street Journal. 19

Figure 1. Time plots of the series: 1975.01-2008.12

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Figure 2. Actual (solid line) vs. nine-month-ahead forecasts (dotted line): 2002.09-2008.12

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Table 1. KPPS Stationarity test results: 1975.01-2001.12 ______________________________________________________________________________ KPSS test statistics _______________________________ (1) (2) . . Row no. Variable Levels First differences . . 1 BRt 0.416a 0.044 a 2 Mt 0.305 0.007 a 3 Ut 0.209 0.114 4 EUt 0.068 -______________________________________________________________________________ Notes: See the text for the definition of the variables. The KPSS test equations for the variables in levels include a constant and a time trend. The KPSS test equations for the variables in first differences include only a constant. The calculated test statistics (obtained using the Bartlett window approach with the automatically determined bandwidth value) are compared with the critical values in Kwiatkowski et al. (1992). Superscript a indicates that the KPSS test p-value is below 0.10.

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Table 2. Estimates of alternative birth rate models: 1975.01-2001.12 ______________________________________________________________________________ ARIMA:

(1 – Ø12B12)∆BRt = (1+ ∑i θiBi)εt

ARIMA-A1: (1 – Ø12B12)∆BRt = (1+ ∑i θiBi)εt + ∑12 j=10 bj ∆Mt-j 12 ARIMA-A2: (1 – Ø12B12)∆BRt = (1+ ∑i θiBi)εt + ∑12 j=10 bj ∆Mt-j + ∑j=10 cj ∆Ut-j 12 ARIMA-A3: (1 – Ø12B12)∆BRt = (1+ ∑i θiBi)εt + ∑12 j=10 bj ∆Mt-j + ∑j=10 dj EUt-j ______________________________________________________________________________

ARIMA ARIMA-A1 ARIMA-A2 ARIMA-A3 (1) (2) (3) (4) ______________________________________________________________________________ Ø12

0.398 (6.30)

0.382 (6.02)

0.366 (5.64)

0.377 (5.83)

θ1 θ4 θ5 θ12

-0.744 (18.2) -0.199 (3.46) 0.170 (3.04) -0.090 (2.05)

-0.749 (18.6) -0.196 (3.41) 0.174 (3.14) -0.094 (2.19)

-0.751 (18.5) -0.195 (3.37) 0.175 (3.12) -0.078 (1.73)

-0.760 (18.9) -0.189 (3.24) 0.166 (2.97) -0.094 (2.20)

0.067 (2.54)

0.065 (2.45) -0.188 (3.20)

0.023 (0.69)

∑bi ∑ci ∑di

0.001 (1.95)

Adjusted R2 0.536 0.542 0.551 0.547 Q-stat. p-value 0.983 0.970 0.965 0.967 ______________________________________________________________________________ Notes: See the text for the definition of variables. Absolute t-values are in parentheses. The absolute inverted autoregressive and moving-average roots (not reported here) are all below one, indicating that the models in columns 1-4 are all dynamically stable. These models also include the seasonal dummies (for July, October, and November) that are significant; the remaining dummies are insignificant and are thus excluded in order to increase efficiency. The estimation period, after adjusting for lags, is1975.01-2001.12 with 324 observations.

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Table 3. Directional accuracy test results: 2002.09-2008.12 ______________________________________________________________________________ Forecast horizon ( f ) ____________________________________________________________ Model Three-month-ahead Six-month-ahead Nine-month-ahead ______________________________________________________________________________ ARIMA 0.87b 0.89b 0.89b ARIMA-A1 0.91b 0.89b 0.89b ARIMA-A2 0.88b 0.91b 0.92b ARIMA-A3 0.91b 0.91b 0.91b ______________________________________________________________________________ Notes: Numbers are directional accuracy rates. In testing the null hypothesis of no (directional) association between the actual and predicted change in the birth rate, we utilize Fisher’s exact test and the chi-square tests with and without Yate’s continuity correction. Superscript b indicates that the p-values of these tests are all below 0.10.

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Table 4. Unbiasedness test results: 2002.09-2008.12 ______________________________________________________________________________ At+f =  +  Pt+f + vt+f At+f – Pt+f = ’ + vt+f Row ___________________________________________ ___________________ 2 no. f  β R p1 ’ = ME U ______________________________________________________________________________ ARIMA forecasts 1 3 -0.191 (1.113) 1.015c (0.079) 0.74 0.766 0.025 (0.035) 0.130d c 2 6 0.140 (1.092) 0.993 (0.079) 0.67 0.601 0.043 (0.048) 0.137d 3 9 0.893 (0.973) 0.941c (0.072) 0.66 0.075 0.070 (0.058) 0.181d ARIMA-A1 forecasts 4 3 0.125 (1.046) 5 6 0.887 (1.017) 6 9 1.128 (0.927)

0.992c (0.075) 0.938c (0.074) 0.923c (0.068)

0.76 0.69 0.67

0.936 0.432 0.111

0.011 (0.033) 0.021 (0.047) 0.040 (0.057)

0.119d 0.126d 0.170d

ARIMA-A2 forecasts 7 3 0.526 (0.949) 8 6 1.412 (0.902) 9 9 2.085c(0.858)

0.964c (0.068) 0.902c (0.066) 0.856c (0.063)

0.77 0.71 0.67

0.659 0.090 0.008

0.020 (0.032) 0.036 (0.047) 0.061 (0.063)

0.115d 0.125d 0.182d

ARIMA-A3 forecasts 10 3 0.189 (0.964) 0.987c (0.069) 0.77 0.898 0.011 (0.032) 0.111d c 11 6 0.993 (0.920) 0.931 (0.067) 0.71 0.273 0.022 (0.045) 0.119d 12 9 1.090 (0.872) 0.925c (0.064) 0.69 0.100 0.042 (0.054) 0.158d ______________________________________________________________________________ Notes: At+f is the actual birth rate in month t+f, and Pt+f is a general notation for both ARIMA and augmented ARIMA forecasts of At+f made in month t (the forecast horizon f = 3, 6, and 9 months). The correct (Newey-West) standard errors are in parentheses. Superscript c indicates significance at the 0.10 or lower level. p1 is the p-value for testing the joint null hypothesis of unbiasedness ( = 0 and β = 1). ME is the mean forecast error. The last column reports Theil’s U coefficient estimates. Superscript d indicates that the p-value (of the Diebold-Mariano test statistic) is below 0.10.

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Table 5. Efficiency test results: 2002.09-2008.12 ______________________________________________________________________________ ) At+f = γ0 + γ1 Pt (Af) + γ2 Pt (AA + vt+f f

Row ______________________________________________________________ no. f γ0 γ1 γ2 R2 ______________________________________________________________________________ ARIMA-A1 forecasts 1 3 0.177 (1.045) -0.077 (0.451) 1.065c (0.448) 0.76 c 2 6 1.277 (0.961) -0.353 (0.492) 1.263 (0.484) 0.70 c 3 9 1.254 (0.901) -0.287 (0.510) 1.200 (0.511) 0.67 ARIMA-A2 forecasts 4 3 0.331 (1.038) 5 6 1.090 (1.014) 6 9 1.300 (0.872)

0.145 (0.270) 0.157 (0.275) 0.405 (0.257)

0.833c (0.252) 0.768c (0.251) 0.507c (0.238)

0.76 0.70 0.67

ARIMA-A3 forecasts 7 3 0.101 (1.012) 0.083 (0.261) 0.911c (0.251) 0.77 c 8 6 1.087 (0.961) -0.058 (0.271) 0.982 (0.257) 0.71 c 9 9 1.134 (0.886) -0.051 (0.257) 0.973 (0.252) 0.70 ______________________________________________________________________________ ) Notes: Pt (Af) and Pt (AA denote, respectively, the ARIMA and augmented ARIMA forecasts of the f

birth rate. The correct (Newey-West) standard errors are in parentheses. Superscript c indicates significance at the 10% or lower level.

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Table 6. Encompassing test results: 2002.09-2008.12 ______________________________________________________________________________ At+f = δ0 + δ1 Pt (Af1) + δ2 Pt (Af2) + vt+f Row ______________________________________________________________ no. f δ0 δ1 δ2 R2 ______________________________________________________________________________ 1 3 0.334 (0.998) 0.284 (0.317) 0.693c (0.306) 0.77 2 6 1.103 (0.946) 0.309 (0.293) 0.614c (0.280) 0.71 c c 3 9 1.353 (0.848) 0.482 (0.276) 0.426 (0.262) 0.68 ______________________________________________________________________________ At+f = η0 + η1 Pt (Af1) + η2 Pt (Af3) + vt+f ______________________________________________________________ f η0 η1 η2 R2 ______________________________________________________________________________ 4 3 0.185 (0.996) 0.011 (0.414) 0.976c (0.402) 0.77 c 5 6 1.040 (0.939) -0.100 (0.365) 1.027 (0.356) 0.71 c 6 9 1.118 (0.880) -0.071 (0.303) 0.994 (0.301) 0.70 ______________________________________________________________________________ At+f = ω0 + ω1 Pt (Af2) + ω2 Pt (Af3) + vt+f ______________________________________________________________ f ω0 ω1 ω2 R2 ______________________________________________________________________________ 7 3 0.224 (0.947) 0.379 (0.333) 0.606c (0.336) 0.78 c 8 6 1.021 (0.894) 0.344 (0.324) 0.585 (0.332) 0.72 c 9 9 1.172 (0.858) 0.183 (0.290) 0.736 (0.307) 0.70 ______________________________________________________________________________ Notes: Pt (Af1) , Pt (Af2) , and Pt (Af3) denote, respectively, the ARIMA-A1, ARIMA-A2, and ARIMA-A3 forecasts of the birth rate. The correct (Newey-West) standard errors are in parentheses. Superscript c indicates significance at the 10% or lower level.

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