The long-run impact of age demographics on the U.S. divorce rate.I. Introduction
The steady rise in the United States (U.S.) divorce rate from the mid-1960s to the mid-1970s has been a topic of much debate among demographers and economists (Friedberg 1998; Goldstein 1999; Michael 1978, 1988; Oppenheimer 1997; Preston 1997; Ruggles 1997; Wolfers 2006). Researchers have focused much of their attention on explaining the rise in the divorce rate by examining changes in divorce laws during the early-1970s (Friedberg 1998; Wolfers 2006) and the economic empowerment of women (Bremmer and Kesselring 2004; Nunley 2010; Ruggles 1997). The consensus from the literature on divorce laws suggests that the reforms led to a small, transitory rise in the divorce rate. (1) The literature on the impact of the economic empowerment of women on the divorce rate has produced mixed results, which is likely due to the fact that women began participating in the labor market and in higher education at increasing rates long before the sharp rise in the divorce rate in the mid-1960s.
As research has recently shifted to panel data studies, earlier time-series evidence on the role played by age-specific divorce rates, such as Michael (1978, 1988) and South (1985), appears to have been largely bypassed, although it clearly suggests that growth in the fraction of young adults in the population, a group with a disproportionately high divorce risk, contributed very significantly to the sharp rise in the divorce rate during the 1960s and early 1970s. For example, Michael (1978) finds that women in their 20s contributed to approximately 60 percent of the rise in the divorce rate observed between 1960 and 1974. Carlson (1979) also provides evidence that a higher rate of divorce is present for individuals in their 20s over the same period.
The purpose of this study is to use more recent time-series techniques and a longer time horizon than previous studies to reexamine the extent to which the large variation in the divorce rate can be attributed to a simple demographic change, the rise and decline in the share of young adults in the population. As the divorce probability is higher among young adults and also during the first years of marriage, the divorce rate should change ceteris paribus with the fraction of the population in their twenties. Figure 1 suggests that the rise in the fraction of young adults is indeed closely tied to the divorce rate per marriage in the U.S., in particular over the period from the mid-1960s to the mid-1970s, which has been the focus of numerous divorce studies in recent years. This study can be considered a reminder that simple explanations of long-run changes in the divorce rate should not be dismissed simply because they do not show up during the relatively short time horizons typically covered by panel data sets.
[FIGURE 1 OMITTED]
Previous time-series studies of the divorce rate, such as South (1985), Michael (1988), Bremmer and Kesselring (2004) or Nunley (2010), not only consider shorter time horizons than this study (2) but also use rather different methodologies and variables from study to study, which makes it difficult at times to compare results. (3) Therefore, a secondary objective of this study is to help in making previous studies and their results easier to understand and compare.
Our primary finding is that the results of South (1985) and Michael (1988) that show the importance of the percentage of young adults in the population as a key driver of long-run changes in the divorce rate can be replicated in our analysis, with estimated elasticities ranging from 1.0 to 1.3. The age-composition variable is by far the most robust of all variables that we include to explain the divorce rate. The participation rate of females in higher education is used as a proxy for female economic independence and tends to be positive, which is in line with the previous literature. We identify a rise in the underlying trend of the divorce rate from 1969 to 1972 that is not explained by our included variables. Based on previous research, we attribute this increase to the "pill effect," as discussed for example in Michael (1988), and the temporary impact of divorce law changes (Wolfers 2006). The underlying trend in the divorce rate moves up again between 1988 and 1998 because the observed divorce rate does not decline as fast as predicted by its fundamental driving forces. However, as of now, there appears to be no obvious reason for this sluggish decline in the divorce rate.
We use time-series data from 1932 to 2006, which goes significantly beyond the years covered by panel-data studies (Charles and Stephens 2004; Nunley and Seals 2010; Weiss and Willis 1997) and other studies that make use of aggregate time-series data (South 1985, Michael 1988, Bremmer and Kesselring 2004, Nunley 2010).
Our outcome variable is divorces per 1,000 married persons, (4) and our key explanatory variable is the percentage of the population in the 20-29 year-old age group. We motivate this age-composition variable along the lines of Michael (1988). Marriages are less stable when individuals marry at younger ages (Becker et al. 1977; Castro-Martin and Bumpass 1989), and they are less stable during the early years of marriage (Sweet and Bumpass 1987). Because the median age at first marriage ranges from the low-20s to the mid-20s, (5) the 20-29 year age group has a disproportionate number of short-duration marriages. As a consequence, a rise or fall in the fraction of 20-29 year-olds in the total population should increase or decrease the aggregate divorce rate irrespective of other causes of the divorce rate. Figure 1 illustrates this point. The sharp rise in the divorce rate from the mid1960s to the mid-1970s happens to coincide with a perceptible increase in the 20-29 age group as a fraction of the total population. Likewise, the steady decline in the divorce rate from 1980 onwards coincides with a reduction in the fraction of 20-29 year-olds in the population. (6) Figure 1 also suggests that the age composition variable alone cannot explain the long-run swings in the divorce rate over the last 70 years. There are also other forces at work.
We try to capture these other forces by including the appropriate control variables. Female economic empowerment is considered an important driver of the divorce rate in the long run (Ruggles 1997). The majority of the empirical literature reports that the increasing economic independence of women led to a rise in divorces. Economic empowerment is typically associated with increases in the female labor force participation rate (FLFPR) or in the participation of females in higher education. (7) For example, Bremmer and Kesselring (2004) find a positive, long-run relationship between the FLFPR and the divorce rate. Nunley (2010) identifies a positive relationship between the divorce rate and changes in female participation in higher education, which is taken as a proxy for female economic empowerment.
As in Nunley (2010), we also employ the percentage of females enrolled in higher education as a proxy for female economic independence. (8) However, we note that, similar to female labor force participation, the female participation in higher education is not directly measuring economic empowerment since enrolling in higher education does not necessarily translate into actual earnings. (9) As our data cover the years of World War II, we follow Michael (1988) in employing a variable (GI) that identifies the percentage of the population older than 18 years of age that is in the military.
In order to understand the results of the previous time-series literature on the aggregate divorce rate, we also try two macroeconomic variables, the inflation rate and the unemployment rate. Nunley (2010) finds a positive and significant impact of inflation on the divorce rate, but only weak evidence for an impact of unemployment. The weak significance also emerges from Michael's (1988) models if one corrects his results for autocorrelation and from South's (1985).
More recently, researchers have examined the divorce rates of marriages that formed at different times. For example, Stevenson and Wolfers (2007a, 2007b) find higher levels of marital instability for marriages that formed prior to the 1970s and more marital stability for those that formed after the 1970s. Because much of the literature on the age-divorce relationship focuses on the divorce rates of different marriage cohorts, we check the robustness of our results to the inclusion of a proxy for cohort effects. Our proxy for the cohort effect comes from Stevenson and Wolfers (2007b). The authors construct it by calculating the probability of a couple divorcing by the 5th wedding anniversary for each marriage cohort.
Using a long sample requires some solutions to apparent data problems. For example, the Center for Disease Control (CDC) stopped collecting data on divorces per 1,000 married persons in 1997. However, we are able to calculate the number of divorces per 1,000 married couples from 1998 to 2006 from generally available data sources. (10) In addition, there are missing years of data for the variable measuring the participation of females in higher education. Only odd years are reported before 1945. We replace the missing years of data with the average of the odd years. For example, female participation in higher education in 1934 is taken to equal the average of the 1933 and 1935 values. Some averaging is also done for the variable capturing the divorce risk facing different marriage cohorts. In addition, missing values for the marriage-cohort variable are replaced with the known values of the cohort subsample at either end. For the other variables used in the analysis, data are available for the entire sample period.
Table 1 provides variable names, definitions, and summary statistics for the variables used in the analysis. For the cointegration part of the empirical analysis, we test each of the variables used in our analysis for the presence of a unit root using Augmented Dickey-Fuller tests (ADF) with the null hypothesis of a unit root. The ADF tests cannot reject a unit root for any of the variables at the five percent level except for the unemployment rate. We conclude that the unemployment rate should only be added into the cointegrating equation together with at least two other variables to avoid spurious results.
III. Econometric Methodology
Our focus is to identify the long-run relationship between the divorce rate and the size of the age group most likely to divorce. We consider two newer time-series techniques that have been used before to study the time-series behavior of the aggregate divorce rate, cointegration and unobserved component modeling. (11)
Given that our explanatory variable of primary interest and most control variables are plausibly exogenous, we employ the single-equation cointegration approach introduced by Engle and Granger (1987) rather than the multivariate one of the Johansen (1988) type. (12) Using the residuals from the Engle-Granger cointegrating regression, we employ an ADF test with a testing-down procedure for the optimal number of lags of the dependent variable; p-values for the cointegration test statistic come from MacKinnon (1996).
Following the approach taken by Nunley (2010), we also employ an unobserved component model to identify the impact of the age-composition variable on the divorce rate. Intuitively, this approach can be thought of as a modified regression, where the constant term is replaced by stochastic components that capture any underlying trend or seasonal or cyclical variation. The variation of the dependent variable is decomposed into an observable part, the contribution of the regression variables, and an unobservable part, which is captured by the stochastic components. This decomposition approach has a number of advantages. The most relevant from an economic perspective is that it delivers reliable estimates of the impact of the observed variables, such as the age-composition variable in our case, even if important driving forces of the dependent variable are not explicitly entering the equation, either because theory is silent about them or because they are difficult to measure in practice. Leaving out important variables of this type or not capturing their impact fully with the included variables has the potential to cause a well-known omitted variables problem in least squares regression. The UCM modeling approach delivers useful coefficient estimates even under these circumstances because the impact of the missing variables is picked up by the unobserved components. (13) Which unobserved components need to be added to the model depends on the nature of the dependent variable and the observed explanatory variables. In our application, the model structure is given as
[[gamma].sub.t] = [[mu].sub.t] + [summation.sub.i] [[beta].sub.i] [x.sub.i,t] + [[epsilon].sub.t],
where [[gamma].sub.t] is the dependent variable and [[mu].sub.t] a stochastic trend component, which absorbs variation in the dependent variable that is not captured by the observed explanatory variables [x.sub.i,t] and their associated coefficients [[beta].sub.i;] [[epsilon].sub.t] is a random error with zero mean and constant variance. The stochastic trend [[mu].sub.t] is modeled itself as a random walk,
[[mu].sub.t] = [[mu].sub.t-1] + [[eta].sub.t],
where [eta] is an random error term with zero mean and constant variance. (14)
IV. Estimation Results
Table 2 presents the results from the cointegration analysis. For each model, we conduct a test for cointegration. In cointegration space, we include a constant and a time trend, along with the explanatory variables of interest in logarithmic form. (15) The cointegration tests reveal at the five-percent level the presence of a common trend among the divorce rate and the explanatory variables tested in Models 1 through 3 and in Model 5. The evidence for cointegration is less strong for Models 4 and 5 and not present at common levels of statistical significance for Model 7.
The impact of the age-composition variable is positive and larger in size in each model specification than the other explanatory variables. However, the magnitude of the effect varies somewhat depending on the additional explanatory variables included in cointegration space. For example, in Models 1 and 3, a one-percent increase in the percentage of the population in the 20-29 age group results in a 1.38 percent rise in the divorce rate. For all models, for which cointegration can be assumed, the elasticity value lies approximately between 1.2 and 1.5, which suggests a fair amount of robustness to alternative model specifications.
As in Michael (1988), a rise in the percentage of the population in the military is associated with a rise in the divorce rate. The elasticity is relatively low, however, with a value around 0.10. The economic empowerment of women, which is proxied in this study by their participation in higher education (fem_educ), is positively related to the divorce rate in the long run, but with a relatively low elasticity between 0.10 and 0.20. These results are consistent with earlier work by Nunley (2010), Bremmer and Kesselring (2004), and South (1985), who also find a positive impact of their proxies for female economic independence. If either the inflation rate or the unemployment rate is included as an additional variable, as in Models 5 or 7, the models lose their cointegration property. Model 6, however, demonstrates that a model with only the age-composition variable and the unemployment and inflation rates passes the contegration test. The inflation rate has a positive effect on the divorce rate, while the unemployment rate turns out to have a negative influence on the divorce rate. Our finding for a positive relationship between the inflation rate and the divorce rate is consistent with previous work (Nunley 2010). The negative relationship found for the unemployment rate and the divorce rates is at odds with some studies (e.g., South 1985) but is consistent with others (e.g., Nunley 2010). (16)
The relative sensitivity of the cointegration property to the inclusion of the macroeconomic variables is partly explained by the correlation coefficients of the explanatory variables presented in Table 3: the coefficients of the macroeconomic variables tend to be larger than those of the demographic variables. Table 3 also reveals that the GI variable is negatively correlated with the unemployment rate, which is largely driven by the period around WWII.
An important result of the cointegration analysis for our study, which is at least implicit in Table 2, is that there exists no combination of variables that can reject the null hypothesis of no-cointegration unless the age-composition variable is included among the variables under investigation. That means, the age-composition variable is of crucial importance; no other variable is. The age-composition variable continues to be a decisive variable for cointegration if the sample is restricted to the years after WWII (1948-2006). In that case, cointegration can be established, at the 10 percent level, only if both the age-composition variable and female participation in higher education are included in the cointegration equation. Inclusion of any macroeconomic variable eliminates any cointegration property.
Adding the cohort-effect variable to any of the cointegration regressions reported in Table 2 results in a loss of the cointegration property. The correlation coefficients of Table 3 reveal a potential reason: the cohort variable is highly correlated with a number of other variables, in particular the participation of females in higher education. This suggests that the cohort variable is not identifying an independent causal explanation of movements in the divorce rate over time. In fact, one may speculate that the increase in the cohort variable in the 1960s and 1970s is causally related to the changes in society happening at that time, of which the increase in the proportion of the population in the 20-29 year-old age group is likely to be key. The sheer size of the Baby Boom generation that triggered the divorce rate changes of the 1960s and 1970s may be thought of as also being the key factor to bringing about the societal changes that characterized the 1960s and 1970s as "radically different" from the past. Hence, we can think of the demographic change as the most likely cause also of the observed cohort changes of the 1960s and 1970s. This is borne out if we try to predict the cohort variable for the period 1958 to 1995 with the age-composition variable. It turns out that the proportion of the population in the 20-29 year-old age group can explain 78 percent of all the variation in the cohort variable over that time period. (17)
Unobserved Component Model
An important difference between the cointegration analysis of the last section and the unobserved component modeling lies in the fact that the latter depends much less on asymptotic estimator properties and the assumption of a stable environment. In practice, we replace an underlying deterministic trend term (time trend in Table 2) with a more flexible stochastic trend. In return, we need to pay attention now to the statistical properties of the estimator, as reflected by statistical tests on the model residuals. The issues of outliers and time lags also arise.
Table 4 summarizes the estimation results. (18) The most important aspect is evident from the first row of estimates. The estimated elasticity of the age-composition variable has about the same range (1 to 1.3) as in Table 2, and it remains highly statistically significant regardless of the model specification. This suggests our conclusion that the age-composition variable is a key driver of the divorce rate is very robust.
We note that the GI variable now enters with a lag of one year. The estimated elasticities are rather similar to those of the cointegrating equations of Table 2. The elasticity of the participation rate of females in higher education tends to be slightly larger on average in Table 4 compared to Table 2. However, by far the largest differences compared to the cointegration results arise in the elasticities of the two macroeconomic variables. The inflation elasticity is significantly smaller than the ones reported in Table 2 and the elasticity for the unemployment rate changes sign. This supports our earlier conclusion that the impact of the macroeconomic variables is much less certain than that of our age-composition variable or our proxy for female economic independence.
Some of the models reported in Table 4 suffer from autocorrelation and non-normality in the residuals. Autocorrelation is an issue in Models 1 to 3, and non-normality in Models 1, 2, and 5. Model 3 shows that the inclusion of observation specific dummy variables either among the regressors or in the stochastic trend can remove the non-normality problem. That means non-normality is caused by outliers. As a direct comparison of Models 2 and 3 reveals, removing the non-normality with dummy variables has no material impact on our focus variable, the age-composition variable. By far the best fit is achieved with Model 6, especially considering that no observation specific dummy variables are used. Again, both our age-composition variable and our proxy for female economic independence are in line with the results of Table 2.
Figure 2 gives a graphical representation of the estimates implied by Model 6. The top panel shows the logarithm of the divorce rate, which is the dependent variable in all our models. The panel in the middle presents the predictions of the explanatory variables, the so-called regression component. The third panel shows the evolution over time of the underlying trend, which is the unobserved model component. Except for the residual, which is not shown, the regression component and the unobserved component add up to the dependent variable.
We see that the regression component closely tracks the dependent variable. However, a few deviations are notable. These are evident in the behavior of the unobserved component in the bottom panel. The unobserved component stays around 5.0 for much of the time to around 1968, except for some ups and downs during the war years, which are not fully captured by the GI variable. We note a distinct upward drift from 1968 to about 1972, which is presumably a result of the "pill effect" and the changes in divorce laws at this time. The unobserved component stays flat for numerous years until the late 1980s, when it rises again for about 10 years in the decade of the 1990s. This last rise, however, appears to be temporary, unlike the increase around 1970. Another difference compared to 1970 is that the rise in the unobserved component is not reflected by an increase in the divorce rate but comes about because the divorce rate falls more slowly than predicted by the included regression variables. What may be the cause of this temporary rise in the unobserved component of the divorce rate is not clear.
As a robustness check, we also estimate unobserved component models for the period 19482006. The results can be summarized as follows. The age-composition variable remains stable, with an estimated elasticity at around unity. The elasticity of the female participation rate in higher education stays at around 0.30. The estimation results for the unemployment rate and the inflation rate again tend to be variable and not uniformly significant. (19)
V. Summary and Conclusions
The sharp increase in the aggregate divorce rate from the mid-1960s to the mid-1970s and its decline beginning around 1980 has been a topic of much debate. The sharp rise and steady decline in the divorce rate happens to coincide with a perceptible increase and decrease in the 20-29 age group as a fraction of the total population, which is largely the result of the Baby Boom generation passing through this age group at that time. This has been noted before in the literature but seems to have been pushed into the background as research into the causes of changes in the divorce rate has shifted almost exclusively toward questions that are amenable to panel data studies.
[FIGURE 2 OMITTED]
The purpose of this study has been to bring back to the forefront the centrality of the age composition of the population for questions surrounding the aggregate divorce rate. We confirm earlier results by South (1985) and Michael (1988) that the percentage of the 20-29 year-olds in the population is a very robust predictor of the divorce rate in the long run, with estimated elasticities ranging from 1.0 to 1.3. These elasticities are consistently larger than for any other determinant of the divorce rate and are not sensitive to the inclusion of other explanatory variables that have been used in the literature. This applies not only to the estimates on the complete sample period from 1932 to 2006, but is true also for estimates limited to the time period after WWII.
We find that our proxy for the economic independence of women, the female participation rate in higher education, is positive and has an elasticity between 0.2 to 0.3, depending on the model specification. We show that the macroeconomic variables we use, the unemployment rate and the inflation rate, are far more fragile in their impact on the aggregate divorce rate than our age composition variable or the variable capturing female participation in higher education.
We find some evidence of a positive "pill effect" and/or a temporary impact of the divorce law changes from the late 1960s to the early 1970s. We also identify another unpredicted hump in the divorce rate during the 1990s, which arises because the divorce rate falls far more slowly over that time period than predicted. According to our estimates, there is little evidence to support the notion that there is a cohort effect present around the 1970s. What has been identified as a cohort effect by Stevenson and Wolfers (2007b) is largely predictable by our age-composition variable in conjunction with the female participation rate in higher education.
The authors thank Charles Baum, Gregory Givens, Mark Owens, and Alan Seals for helpful comments and suggestions.
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(1.) The divorce rate began trending upward in the mid-1960s, which predates the widespread adoption of unilateral divorce across states in the 1970s. Wolfers (2006) concludes that the adoption of unilateral divorce laws led to a slight rise in the divorce rate due to pent-up demand; that is, a lot of bad marriages could not end under mutual consent but did so under unilateral divorce. Nevertheless, the effects on the divorce rate of divorce-law reform are small, short-lived, and incapable of explaining its doubling from the mid-1960s to the mid-1970s. Also, as noted for example by Michael (1988), many European countries experienced a similar development of the divorce rate over time as the U.S., yet did not have concurrent divorce law changes. As a consequence, we do not consider divorce law changes in this study.
(2.) All four studies except Michael (1988) focus on post-WWII data; Michael's (1988) estimation sample ends in 1974.
(3.) South (1985) uses an autoregressive distributed lag model that eliminates autocorrelation, but provides only short run effects; Michael (1988) uses least squares without correction for autocorrelation, with potentially biased long-run coefficient estimates; Bremmer and Kesselring (2004) use cointegration methods to identify long-run influences on the divorce rate; and Nunley (2010) uses unobserved component modeling.
(4.) Divorces per 1,000 married couples and divorces per 1,000 persons display similar behavior over the sample period used in this study. In fact, a scatterplot of divorces per 1,000 persons and divorces per 1,000 married couples reveals a clean, linear relationship between the two variables. This suggests that the estimated effects of our explanatory variables would be similar regardless of which divorce measure is used. In fact, we find similar results, regardless of which measure of the divorce rate is used.
(5.) See Carter et al. (2006) for U.S. data from 1850-1999 median ages at first marriage. Note there are differences between males and females. However, the difference between men's and women's median age at first marriage is approximately two to three years, with women typically marrying at younger ages than men. Nevertheless, the median ages at first marriage for both men and women fall into the 20-29 age grouping.
(6.) We also check the robustness of our results to alternative ways of measuring the young population. In particular, we use the fraction of 20-24 year-olds in the population as a robustness check. The results are not materially affected by this substitution. As a result, we present the results using the 20-29 year-old age group measure, as they are most comparable to the measures used by Michael (1988).
(7.) We do not consider the FLFRP in our analysis, as data on this variable are only available back to 1948. Likewise, using the FLFPR as a proxy for women's rising economic empowerment may not be ideal because many women remained secondary earners within households until the late-1960s and 1970s, continued to take their husband's labor-market choices as given, and worked part time with little opportunity for on-the-job advancement (Goldin 2006).
(8.) Similar to the FLFPR, female participation in higher education has grown steadily since the late-1940s. Over this period, Goldin et al. (2006) document how the rate of females taking math and science courses in high school converged to that of men. This better prepared them for college and supplied the necessary skills to sort into professionalized fields of study, such as medical, law, business, and dental schools. As women increased their economic independence through participation in professional jobs, household laborarket decisions became interdependent, perhaps indicating a shift in bargaining power toward women within households (Costa 2000). The gain in bargaining power from increased participation in professionalized fields suggests that female participation in higher education proxies well for the economic empowerment of women.
(9.) Actual earnings of females, as used in a robustness check by Michael (1988), would clearly be a better measure, but that information is not available at the aggregate level and over the long time period we study.
(10.) Multiplying the number of divorces per 1,000 persons by the U.S. population per 1,000 persons gives the total number of divorces in a given year. Dividing this number by the stock of married couples creates the variable of interest: divorces per 1,000 married couples. This measure draws on various U.S. Statistical Abstracts. Data on divorces per 1,000 married couples are available until 1997. Therefore, to check our estimates for the years 1997-2006 use the same calculation method described above for the available years of the divorce rate per 1,000 married couples, and we find that any difference in the estimates are in the decimal places.
(11.) Cointegration methods are applied in Bremmer and Kesselring (2004) and unobserved component modeling in Nunley (2010).
(12.) We note that Johansen type tests for multivariate cointegration are sensible only for models with more than two variables and then only if the variables are jointly determined. In our case, it is unlikely that the divorce rate and the variables used to explain its trend are jointly determined. In particular, Granger causality tests strongly confirm that none of the variables used to explain the divorce rate are themselves Granger-caused by the divorce rate at any reasonable level of statistical significance. That includes but is not limited to our focus variable, the fraction of 20-29 year olds in the population.
(13.) It is important for the outcome that the unobserved components are part of the model specification, rather than part of the residual term.
(14.) We note that the variances of [[eta].sub.t] is the only estimable parameter of the stochastic trend [[mu].sub.t]. This variance is estimated jointly with the regression parameters [[beta].sub.i] and the variance of the error term [[epsilon].sub.t], using maximum likelihood in combination with the Kalman Filter. Details are discussed at a nontechnical level in Commandeur and Koopman (2007) and at a more advanced level in Harvey (1989).
(15.) The log form allows us to interpret the coefficients as elasticities.
(16.) From a theoretical perspective, divorce and inflation should be unambiguously positively related as inflation acts as a tax on the household. By contrast, the relationship between divorce and unemployment is ambiguous. Unemployment should raise the divorce rate because it reduces the returns from marriage. But unemployment can also reduce the divorce rate by forcing spouses to rely each other's income as a source of consumption insurance.
(17.) The age composition variable in combination with the variable capturing the rate of female participation in higher education explains even more, 87 percent of the variation in the cohort variable.
(18.) As in the cointegration analysis, we leave out the cohort effects because they are strongly correlated with our age variable and the participation rate of females in higher education.
(19.) We generated a figure analogous to Figure 2 for the 1948-2006 time period. The figure is based on a model that contains only two explanatory variables, the age-composition variable and the female participation rate in higher education. The model explains 51 percent of the variation around the stochastic trend, has normally distributed residuals, no autocorrelation, and no observation specific dummy variables to absorb the effect of outliers. The unobserved component is rather similar to that for the complete sample model depicted in Figure 2. It is relatively stable to the late 1960s, then increases to the early 1970s, and then changes little until a second increase in 1990. Again, this last increase in the unobserved component is temporary, as the unobserved component falls back to its mid 1980s level by the end of the sample. We note that adding the variable for the "pill effect" from Michael (1988) or our cohort effect variable does not change the unobserved component perceptively. In the interest of brevity, we have omitted this figure from the paper. However, it is available upon request.
John M. Nunley, Assistant Professor, Department of Economics, College of Business Administration, University of Wisconsin--La Crosse, La Crosse, WI 54601, phone: 608-785-5145, email: firstname.lastname@example.org, website: www.uwlax.edu/faculty/nunley.
Joachim Zietz, Professor of Economics, Department of Economics and Finance, Jennings A. Jones College of Business, Middle Tennessee State University, Murfreesboro, TN 37132; and EBS Business School, EBS Universitat fur Wirtschaft und Recht, Wiesbaden, Germany; phone: 615-898- 5619, email: email@example.com, website: www.mtsu.edu/~jzietz.
TABLE 1. Variable Names, Definitions, and Basic Statistics Name Definition Divorce Number of divorces per 1,000 marriages per2029 Percentage of the U.S. population in the 20-29 age group fem_educ Women enrolled in higher education as a percentage of the population Inflation Inflation rate; the logarithm is constructed as ln(1+inflation/100) unemployment Unemployment rate GI Percentage of the population older than 18 years in the military cohort effect 100 minus the average rate for men and women of surviving in the marriage (in percent) until the 5th wedding anniversary; taken from Table 3 of Stevenson and Wolfers (2007b); averages formed from adjoining years for missing values; 4 used for missing values at beginning of sample, 10 used for missing values at end of sample. Name Mean Std.Dev. Min. Max. Divorce 14.89 5.34 6.10 22.80 per2029 0.16 0.02 0.12 0.18 fem_educ 0.017 0.011 0.004 0.034 Inflation 3.58 3.79 -10.30 14.65 unemployment 7.05 4.90 1.20 24.75 GI 1.86 2.11 0.29 12.32 cohort effect 7.67 3.17 4.00 11.65 TABLE 2. Engle-Granger Cointegration Estimates, with the Log of the Divorce Rate per 1,000 Married Couples as Dependent Variable (1932-2006) Model 1 Model 2 Model 3 Model 4 Explanatory Variables: ln_percent_2029 1.384 1.518 1.381 1.180 ln_GI 0.111 0.107 ln_fem_educ 0.119 0.176 ln_inflation ln_unemployment time trend 0.016 0.018 0.013 0.010 Constant 4.609 4.773 5.208 5.231 Diagnostic Statistics: R-squared 0.896 0.938 0.941 0.896 P-value for test of HO 0.030 0.004 0.011 0.058 of No-Cointegration Model 5 Model 6 Model 7 Explanatory Variables: ln_percent_2029 1.208 1.337 1.171 ln_GI 0.080 ln_fem_educ 0.097 0.202 ln_inflation 1.477 1.641 1.349 ln_unemployment -0.103 -0.124 time trend 0.013 0.015 0.008 Constant 4.753 4.675 5.585 Diagnostic Statistics: R-squared 0.954 0.946 0.954 P-value for test of HO 0.082 0.025 0.185 of No-Cointegration Notes: In indicates the natural logarithm. Each model is tested for cointegration over the sample spanning from 1932-2006. The cointegration test results at the bottom of the table use the Engle-Granger cointegration tests, with p-values derived from MacKinnon (1996). In all cases we use a test-down procedure with four lags for the augmented Dickey-Fuller test of the residuals. A p-value below 0.05 indicates rejection of the null hypothesis of no cointegration in favor of a long-run relationship among the variables. No t-values or p-values are reported for the individual coefficients because they are not reliable given the autocorrelation present in the residuals. TABLE 3. Correlation Coefficients of Explanatory Variables (1932-2006) ln_percent2029 In_GI ln_fem_educ 1.000 -0.150 -0.052 ln_percent2029 1.000 -0.185 ln-GI 1.000 ln_fem_educ ln_inflation ln_unemployment cohort effect ln_inflation ln_unemployment cohort effect 0.312 0.343 0.132 ln_percent2029 0.279 -0.790 -0.186 ln-GI 0.284 -0.127 0.958 ln_fem_educ 1.000 -0.317 0.353 ln_inflation 1.000 -0.050 ln_unemployment 1.000 cohort effect Notes: In indicates the natural logarithm. TABLE 4. Unobserved Component Model Estimates, with the Log of the Divorce Rate per 1,000 Married Couples as Dependent Variable (1932-2006) Model 1 Model 2 Model 3 Model 4 Explanatory Variables: ln_percent2029 1.322 1.220 1.266 1.094 (0.002) (0.001) (0.000) (0.000) In_GI(-1) 0.124 0.106 0.115 (0.000) (0.000) (0.000) ln_fem_educ 0.355 (0.000) ln_inflation ln_unemployment Level break 1934 0.204 0.184 Level break 1946 0.390 0.285 Level break 1948 -0.107 -0.105 Outlier 1945 0.168 Outlier 1944 -0.088 Diagnostic Statistics: R-squared 0.972 0.980 0.992 0.993 R-squared around trend 0.113 0.393 0.766 0.794 DW test statistic 1.187 1.333 1.343 1.616 P-values for test of no autocorrelation--Q(3) 0.027 0.006 0.006 0.170 Normality 0.000 0.000 0.113 0.349 Model 5 Model 6 Explanatory Variables: ln_percent2029 1.068 1.011 (0.001) (0.001) In_GI(-1) 0.128 0.150 (0.000) (0.000) ln_fem_educ 0.196 0.205 (0.080) (0.054) ln_inflation 0.757 0.889 (0.001) (0.000) ln_unemployment 0.070 (0.005) Level break 1934 Level break 1946 Level break 1948 Outlier 1945 Outlier 1944 Diagnostic Statistics: R-squared 0.984 0.986 R-squared around trend 0.513 0.574 DW test statistic 1.790 1.940 P-values for test of no autocorrelation--Q(3) 0.459 0.763 Normality 0.000 0.039 Notes: In indicates the natural logarithm. P-values are reported in parenthesis. R-squared around trend identifies the explanatory power of the regression component, which includes observation specific dummy variables as for Models 3 and 4. DW stands for the Durbin-Watson statistic. Autocorrelation is tested by the Box-Ljung Q statistic at three lags, Q(3). Normality is tested by the Bowman-Shenton test.