Divorce Law Reforms and Divorce Rates in the U.S.: An Interactive Fixed Effects Approach

Transcription

1 Divorce Law Reforms and Divorce Rates in the U.S.: An Interactive Fixed Effects Approach Dukpa Kim Tatsushi Oka University of Virginia National University of Singapore July 7, 2011 Abstract This paper estimates the effects of unilateral divorce laws on divorce rates from a panel of state-level divorce rates. We use the interactive fixed effects model to address the issue of endogeneity due to the association between cross-state unobserved heterogeneity and divorce law reforms. We document that earlier studies in the literature do not fully control for unobserved heterogeneity and result in mixed empirical evidence on the effects of divorce law reforms. Our results, while reconciling these conflicting results, suggest that divorce law reforms have positive effects on divorce rates but the magnitude and duration of the effects are smaller than the findings in Wolfers (2006). Keywords: Unilateral divorce law, No-fault divorce, Difference-in-Differences estimation Interactive fixed effects, Common factors JEL Code: C12, J12, K36 We are grateful to Justin Wolfers for providing his data and codes on his website. We also thank Steven Stern, Gary Solon, Shinsuke Tanaka and Ken Yamada for their helpful comments. Department of Economics, University of Virginia, Monroe Hall, McCormick Road, Charlottesville, VA (dk4p@virginia.edu) Department of Economics, National University of Singapore, AS2 Level 6, Singapore (oka@nus.edu.sg)

2 1 Introduction Marital dissolution in the U.S. has shown dramatic changes over the past half century. The divorce rate in the U.S. sharply rose in the 1960s and 1970s, and has been stable or declining since the 1990s. What accounts for these drastic changes in the divorce rate is of interest to social scientists. The association between divorce rates and divorce law reforms has been considered one potential causal link to explain the weakening of traditional family. The increase in divorce rates in the 1970s coincided with the liberalization of divorce laws socalled the no-fault revolution, during which about three quarters of states liberalized their divorce system. Yet, analyzing casual relationships between divorce rates and adaptation of unilateral laws is not simple because of the possibility that regional attributes affected both divorce rates and divorce law reforms. As such, there is mixed empirical evidence regarding the effects of divorce law reforms In this article, we estimate the effects of divorce law reforms on state-level divorce rates in the U.S. by using the interactive fixed effects model in Bai (2009). In earlier studies, there is a premise that state effects, time effects and state-specific trends can capture the unobserved heterogeneity properly. However, the unobserved heterogeneity might have evolved in a more complex way, because of changes in a number of social and cultural factors, such as the stigma of divorce, religious belief, family size, female participation in the work force, population age structure and even contraceptive use. 1 These factors have changed over time and possibly affected both divorce rates and divorce law reforms differently across states. The interactive fixed effects approach captures these time varying common factors with individual loadings representing different degrees of impacts of the common factors across states. Another benefit of the interactive fixed effects model is that it accounts for source of serial and cross sectional correlations and provides more reliable statistical inferences. We reach two main conclusions. First, the divorce law reforms in the U.S. contributed 1 Goldin and Katz (2002) analyze the effects of the birth control pill on the professional career and marriage decision among females. 1

3 to statistically significant increases in the divorce rates, but the magnitude and duration of the increases are smaller than what are reported in an earlier study by Wolfers (2006). Our estimates suggest that the adaptation of unilateral or no-fault divorce system led to about 0.1 more divorce per 1,000 population for the first two years since the adaptation and at most 0.2 more divorce per 1,000 population for the next six years, whereas Wolfers (2006) estimates about 0.3 more divorce for the eight years after the adaptation. Second, the interactive fixed effects approach we adopt here resolves conflicting results between Wolfers (2006) and recent empirical studies that cast doubt on the robustness of his results. These studies report almost the opposite effects of the divorce law reforms depending on the weighting scheme they use. We argue that the model specifications employed in these studies do not fully control for the unobserved heterogeneity across states and thus they can induce omitted variable bias whose direction depends on the weighting scheme. However, once the unobserved heterogeneity is better controlled via the interactive fixed effects approach, the estimated effects of the divorce law reforms are robust to different weighting schemes. We also provide some simulation evidence to support the robustness of the interactive fixed effects approach to the presence of unobserved heterogeneity in a Difference-in- Differences framework. We consider our findings to have general implications to a wide variety of empirical studies. The remainder of the paper is organized as follows. The next section reviews the existing literature and provides our replications of earlier empirical studies. Section 3 describes our econometric method with interactive fixed effects. Section 4 presents the estimation results on the effect of divorce law reforms on divorce rates. Section 5 provides some simulation results to examine the robustness of our approach. Section 6 presents the estimation results using alternative codes of divorce law reforms. Section 7 provides summary and concluding remarks. 2

4 2 Literature Review and Replication 2.1 Early contributions A theoretical model presented in Becker (1981) predicts that divorce rates remain unchanged between mutual consent and unilateral divorce laws. This is because the liberalization of divorce law, as in the Coase theorem, would only redistribute property rights between spouses. 2 Earlier studies examine this prediction using cross-sectional data. Peters (1986) finds that divorce rates were not significantly different between no-fault and fault divorce states. Allen (1992) points out that the results in Peters (1986) rely on inclusion of regional dummies and finds statistically significant effects of divorce law changes on divorce rates if the regional dummies are excluded. In her reply, Peters (1992) argues that the exclusion of regional dummies introduces omitted variable bias because of possible correlation between unobserved regional heterogeneity and divorce law reforms. To control for unobserved cross-state heterogeneity, state-level panel data has been used to estimate the effect of unilateral divorce laws. By using repeated cross-sectional data, Gray (1998) deals with state and time fixed effects and shows insignificant effects of divorce law changes on divorce rates. Using state-level panel data, Friedberg (1998) demonstrates the importance of flexible treatment of state-specific heterogeneity. What Friedberg (1998) shows is that it is not enough to control only state and year fixed effects, but that the model should also include state-specific time trends. After accounting for cross-state heterogeneity, Friedberg (1998) finds that the adaptation of unilateral divorce laws contributed to increases in divorce rates. 2.2 Time-varying effects of divorce law reforms on divorce rates Wolfers (2006) argues that the results in Friedberg (1998) are misleading because the sample period used in Friedberg s analysis, , is too short to distinguish pre-existing statespecific trends and time-varying treatment effects. Wolfers (2006) uses a sample from Rasul (2006) and Matouschek and Rasul (2008) provide alternative theories predicting positive effects of unilateral divorce laws on divorce rates. 3

5 to 1988 and considers a variant of the Difference-in-Differences model, where divorce rates depend on treatment effects possibly varying over the years following the liberalization of divorce laws, unobserved heterogeneity and independent errors. Specifically, the model is: Divorce Rate s,t = T reatment t (T s ) + v s,t + u s,t, where Divorce Rate s,t represents the number of divorces per 1,000 population for state s and year t (s = 1,..., S and t = 1,..., T ); T reatment t ( ) is a function of the reform year in state s, T s, and measures the effects of divorce law reforms on divorce rates; v s,t represents unobserved heterogeneity possibly correlated with divorce law reforms; u s,t is the error term. T reatment t (T s ) traces out the effects of divorce law reforms for every two years after the adaptation and 15 years and later. The dynamic effects of divorce law reforms are represented by the summation of dummy variables. That is: T reatment t (T s ) = 8 β k Unilateral t (T s, k), k=1 where Unilateral t (T s, k) takes one if 2k 1 t T s 2k for k = 1,..., 7, one if 15 t T s for k = 8 and zero otherwise, among those states which reformed divorce laws from 1956 to 1988, while Unilateral t (T s, k) always takes zero if state s did not liberalize divorce laws during the period. In addition, Wolfers (2006) assumes that the unobserved heterogeneity can be captured by state effects, time effects and state-specific quadratic time trends: v s,t = α s + δ t + φ s t + γ s t 2, where α s is a state fixed effect; δ t is a time fixed effect; φ s and γ s are coefficients of statespecific quadratic trends. Wolfers (2006) estimates the model by the Weighted Least Squares (WLS), where the regression equation is weighted by the square root of state population. More precisely, let 4

6 w s,t be the square root of population in state s for time t. We use to denote a variable weighted by the state population (i.e., X s,t = w s,t X s,t for an arbitrary variable X s,t ). Then the weighted model is Divorce Rate s,t = T reatment t (T s ) + v s,t + u s,t. (1) We replicate the estimation results in Wolfers (2006) and report them in the first three columns of Table 1, where all models include state and time fixed effects. The models in columns (2) and (3) also include state specific linear and quadratic trends, respectively. The estimates in the three columns confirm Wolfers findings that while the shift from mutual consent to unilateral divorce has temporal positive effects on divorce rates, the effects are either significantly negative or insignificant in the long run. The results in column (3) suggests that divorce law reforms contribute to more divorce per 1,000 population for the first eight years since the adaptation and have statistically insignificant effects afterwards. 2.3 Conflicting findings Recent studies cast doubt on the robustness of the findings in Wolfers (2006) and draw the conclusion that the effects of divorce law reforms on divorce rates are unclear. These studies point out several issues in Wolfers approach and provide conflicting evidence. First, the results in Wolfers (2006) heavily rely on whether or not the regression model is weighted by state population. In fact, one might conclude that the divorce law reforms did not affect the divorce rates by using unweighted equation or the Ordinary Least Squares (OLS). Droes and van Lamoen (2010) and Lee and Solon (2011) document that the estimates from the OLS estimates are considerably different from the WLS estimates. The last three columns in Table 1 present the OLS estimates. The specifications are otherwise identical to those in the first three columns. In particular, the statistically insignificant estimates in the last column indicate no effects of divorce law reforms. Second, Wolfers (2006) employs the weighting scheme in order to mitigate heteroskedas- 5

7 ticity across states, while Lee and Solon (2011) argue that the efficiency gain from the weighting scheme only appears under a restrictive assumption that the error terms for individuals before state-level aggregation are homoskedastic and independent within states. This point is made in a different context by Dickens (1990). Lastly, Wolfers (2006) uses standard errors for homoscedastic and serially uncorrelated errors, which are inappropriate in the presence of serial correlation as argued in Bertrand, Duflo, and Mullainathan (2004). Lee and Solon (2011) show the presence of serial correlation in the errors. Vogelsang (2008) stresses the importance of cross-sectional dependence of error terms in statistical inferences and reports robust standard errors for the estimates in Wolfers (2006). Taken together, the above issues articulate the potential efficiency loss from the use of WLS and the inappropriateness of the standard errors used in Wolfers (2006), but they do not totally invalidate the WLS estimates because the WLS estimates can be consistent. In other words, those issues suggest that the OLS is preferred to the WLS, but they do not answer why the WLS and OLS estimates are so different. We view the conflicting results as due to a violation of the exogeneity assumption. 3 The premise in the literature is that state effects, time effects and state-specific trends can capture the cross-state unobserved heterogeneity, which is possibly correlated with divorce law reforms. But, the presence of more complex unobserved heterogeneity can cause biases in both the WLS and OLS estimates. The different directions of biases may arise from the weighting scheme using state population. The positive association between population inflows and divorces, perceived by Fenelon (1971), Wright and Stetson (1978) and Ellman (2000) among others, suggests that the weighting scheme using state population gives leverage to the states with high divorce rates. Since the states with high divorce rates are more likely to have adapted unilateral divorce laws, the WLS may overestimate the effects of divorce law reforms. 3 Following the arguments in DuMouchel and Duncan (1983), Lee and Solon (2011) argue that the discrepancy is due to the misspecified functional form of the regression equation, and estimate a model where the dependent variable is the logarithm of divorce rates. 6

8 In effect, the weighting scheme systematically gives leverage to a few states, which can be seen from the plots of state-level divorce rates. Figure 1 separately shows divorce rates from 1956 to 1988 according to whether to have liberalized divorce systems. Panel A plots original divorce rates and shows rather similar patterns of divorce rates across states. Striking heterogeneity across states can be found in Panel B, where divorce rates are multiplied by the square root of state population. It is clear that the weighted divorce rates in California, Texas and Florida show exceptional patterns, which is consistent with the view provided by Lee and Solon (2011), who argue that California has leverage under the weighting scheme. We further experiment with the application of the WLS to samples that do not include at least one of California, Texas and Florida and present the results in Table 2, where all results are obtained using state effects, time effects and state-specific quadratic trends. The estimates suggest that the findings in Wolfers (2006) rely on the weighted observations of those three states. When all three states are excluded from the sample as in column (6), all estimates are statistically insignificant. 3 Empirical Framework As we explained in the previous section, there is a concern that the standard approach in the literature fails to control for unobserved heterogeneity correlated with divorce law reforms, because the specification is not flexible enough to capture factors varying across time and state. Friedberg (1998), Ellman and Lohr (1998) and Ellman (2000) discuss several important factors which change across states and time and may affect both divorce rates and divorce law reform. For instance, the endogeneity problem arises in omitting social and cultural factors, such as the stigma of divorce, religious belief, family size, female participation in the work force and contraceptive use, for most of which we do not have data or appropriate proxy variables. To resolve the issue of the endogeneity due to omitted variables, we incorporate interactive fixed effects into the model in (1). Interactive fixed effects are considered to play a role in controlling for the remaining unobserved heterogeneity in the error term. Specifically, 7

9 we consider: u s,t = λ sf t + ɛ s,t, where λ s is an r 1 vector to capture state-specific reactions to the common shocks, f t is an r 1 vector of unobserved common shocks give the number of factors r, and ɛ s,t are idiosyncratic errors. λ s and f t are called factor loadings and common factors, respectively. The same specification can apply to the error terms in the unweighted regression model. In the next section, we provide the estimation results from both the weighted and unweighted regression model with interactive fixed effects. Interactive fixed effects can be regarded as an extension of state effects, time effects and state-specific trends, because these can be written as an inner product of a vector of statespecific time-invariant entries (α s, 1, φ s, γ s ) and a vector of time-varying common shocks (1, δ t, t, t 2 ). Yet, interactive fixed effects can control for a more general form of unobserved heterogeneity. The common factors, f t, would be considered as time-varying common shocks such as social norms or ideologies concerning marriage and divorce. The common shocks affect all states in the U.S., but do not necessarily affect them homogeneously. λ s would represent the heterogeneous effects of the common shocks across states. The common factor structure is used to capture unobserved heterogeneity in panel data by Goldberger (1972), Joreskog and Goldberger (1975), Chamberlain and Griliches (1975) and Chamberlain (1977) among others. The early studies consider the case for which the size of cross-section N is large, while the time dimension T is small. MaCurdy (1982) considers the case where the error terms have a common factor structure that is not correlated with regressors. Holtz-Eakin, Newey, and Rosen (1988) analyze a panel of vector autoregressioins with interactive fixed effects. Ahn, Lee, and Schmidt (2001) analyze the generalized method of moments (GMM) estimation of a panel data model with interactive fixed effects. Carneiro, Hansen, and Heckman (2003) incorporate a common factor structure into a dynamic treatment effects model in the context of the returns to schooling. Ahn, Lee, and Schmidt (2007) use interactive fixed effects to estimate individual firms technical inefficiency, which is unobservable and time varying. 8

10 Recent developments in the literature provide theoretical guidance in accommodating interactive fixed effects when both N and T are large. Pesaran (2006) augments the regression equation with the cross sectional averages of the dependent and independent variables. On the other hand, the method in Bai (2009) is based on the principal component analysis. The estimator in Pesaran (2006) is N consistent while the one in Bai (2009) is NT consistent. For the estimation, we adopt the method in Bai (2009) and the estimation procedure is as follows. For notational simplicity, let Y s be a vector of the divorce rates in state s, X s be a matrix of the regressors for state s which includes dummies for treatment effects fixed effects and individual trends, Λ = (λ 1, λ 2,..., λ S ) be a matrix of factor loadings and F = (f 1, f 2,..., f T ) be a matrix of common factors. Then, the regression coefficients are obtained from the minimization problem that SSR(β, F, Λ) = S (Y s X s β F λ s ) (Y s X s β F λ s ) s=1 Since F and Λ are unobservable, Bai (2009) suggests an iterative method. given F and Λ, to compute That is, ( S S ˆβ(F, Λ) = X sx s) 1 X s(y s F λ s ), s=1 s=1 and given β, to compute F as the eigenvectors corresponding to the r largest eigenvalues of W W where W = [W 1,..., W S ] and W s = Y s X s β. Upon the normalization that F F/T = I r, Λ = W F/T. Bai (2009) also suggests to take the minimum SSR of two iterations, the first starting from the least squares estimate of β obtained ignoring the common factor structure and the second starting from the pure factor structure assuming β = 0. Our iteration stopped when the percent change in the SSR is smaller than Once we obtained an estimate for β, we corrected the bias using the correction terms in (23) and (24) in Bai (2009). The standard errors for this bias corrected estimate are obtained using Theorem 4 in Bai (2009). 9

11 4 Estimation Results The existing methods incorporating interactive fixed effects can be applied only to balanced panel data, while the data on divorce rates used in Wolfers (2006) and the previous subsection have missing values in nine states. For our analysis, we conduct interpolation based on the forth order polynomial trends for each of those states. 4 We exclude Indiana and New Mexico because there are missing observations around years of adapting unilateral divorce law, and also drop Louisiana from our sample because only eight observations are available out of thirty three years. As a result, the interpolation creates a balanced panel which contains 48 states and 33 years. (See Appendix for more details on interpolation.) Before estimating models that add interactive fixed effects, we investigate whether the balanced panel data can reproduce the estimates in Table 1, obtained from the unbalanced panel. Table 3 presents the estimates from the balanced panel and the specification of each column parallels that of each column of Table 1. The estimates show similar patterns to those from the original data. Furthermore, the standard errors keep similar magnitudes. The WLS estimates in the first three columns take positive and statistically significant values for the first eight years and become negative or statistically insignificant after the ninth year. In column (3), the estimates for the first eight years are between , which are comparable to the corresponding values in Table 1 ( ). The last three columns also confirm the finding in Table 1 that the OLS estimates do not show strong positive association between divorce rates and divorce law reforms. Overall, the results suggest that the balanced panel data remains very close to the unbalanced panel data. We now turn to the model that accounts for interactive fixed effects. Table 4 presents our main results for the time-varying treatment effects of divorce law reforms on divorce rates after incorporating seven common factors. The specification of each column follows that in Table 1, except that interactive fixed effects are included. There are several important differences between Tables 1 and 4. First, the comparison between columns (1)-(3) and 4 For states in the treatment group, our interpolation separates observations before and after the divorce reforms to estimate forth order polynomial trends. See the details in Appendix. 10

12 columns (4)-(6) of Table 4 indicates that our approach resolves the conflicting results from the weighted and unweighted equation models in Table 1. All columns of Table 4 show similar patterns. The adaption of unilateral divorce laws has statistically significant positive impacts on divorce rates for the first eight years since the adaptation. Second, columns (1)-(3) suggest that Wolfers (2006) overestimates the effects of divorce law reforms, while columns (4)-(6) indicate that the unweighted model underestimates them. The most general model in column (3) of Table 4 indicates that the divorce law reforms contribute to increase in the divorce rates for the first eight years, which is much smaller than the corresponding estimates of in Table 1 and in Table 3. Third, both columns (3) and (6) indicate that divorce law reforms do not have immediate impacts on divorce rates. The estimated effects for the first two years are about one third of the one reported in Wolfers (2006). We estimate models with different numbers of common factors to examine the robustness of our findings. It is known that the over-specification of the number of common factors does not cause inconsistency, but the under-specification does. We change the number of common factors from one to nine in the model, where state effects, time effects and statespecific quadratic trends are always included. 5 In Figure 2, we plot the estimates. Panels A.1 and A.2 show the estimates from the weighted model and Panels B.1 and B.2 from the unweighted model. Panel A.1 presents the case for which the number of factors is from zero to four and shows some variation across different numbers of common factors. In Panel A.2, however, once more than six common factors are included, the estimates become very similar. Likewise, Panels B.1 and B.2 show that once enough numbers of common factors are included, the estimates for the reform effects are stable. The results suggest that our selection of seven factors is reasonable. 5 In this particular application, the information criteria in Bai and Ng (2002) give little guidance. Two criteria pick the maximum number of factors and the other criterion picks the minimum. 11

13 5 Simulation Results In this section, we assess the extent to which interactive fixed effects can mitigate policy effects in the Difference-in-Differences framework, and the extent to which the presence of interactive fixed effects can cause bias in the least squares estimate. We use the simulated data to evaluate the performance of the following two models: Divorce Rate s,t = T reatment t (T s ) + v s,t + u s,t (2) Divorce Rate s,t = T reatment t (T s ) + v s,t + λ sf t + ɛ s,t (3) where v s,t controls state fixed effects, time fixed effect state-specific quadratic time trends. The first model in (2) is the weighted model in Wolfers (2006) and estimated under the assumption that u s,t is uncorrelated with the rest of the model. For the second model in (3), it is assumed that the error term, ɛ s,t, is uncorrelated with the rest of the model after the inclusion of the common factors. The first simulation serves as an experiment to see if the estimates controlling for interactive fixed effects have biases when the true model does not have any common factors. That is, we take the model in (2) as the data generating process (DGP), where we use the coefficients values and the variance of the error of the estimation results in column (3) of Table 3. 6 The errors are generated from independent normal distribution whose variance is obtained from the estimated residuals. We estimate reform effects both with and without including the interactive fixed effects. The number of replications is 5,000. Panel A of Table 5 presents the relevant results. Column (1) shows the true reform effects. Columns (2)-(4) and columns (5)-(7) correspond to the estimates without and with controlling for interactive fixed effects, respectively. The median values in both columns (2) and (5) are very close to the true values. The 95% range from the interactive fixed effects model in columns (6) and (7) are slightly wider than those in columns (3) and (4) but does not show systematic underestimation. It is worth noting that the 95% range 6 The simulation results are very similar and our conclusion does not change, when the DGP is based on the estimates from the original unbalanced panel. 12

14 is obtained separately for each parameter and the entries in the same columns are not necessarily obtained from the same experiment. Thus, we explore how likely the estimates for the first eight years simultaneously take values below the corresponding estimates in column (3) of Table 4. The chance of such an event in our simulation is only with 0.02% (8 times out of 5,000). Hence, it is very unlikely that the common factor estimates capture all of policy effects and leave little to the policy dummies. In the second experiment, we instead consider the case where the true model has seven common factors in (3), otherwise keeping the same structure as in the first experiment. The parameter values are taken from the estimates in column (3) of Table 4. The estimated common factors and factor loadings from the specification of column (3) of Table 4 are kept for all replications. Panel B of Table 5 presents the relevant results. Column (1) shows the true treatment effects in the DGP. Column (2) clearly indicates that the simple WLS overestimates the true treatment effects and the median values of the estimates are very similar to those in column (3) in Table 3, especially for the first eight years. Columns (2) and (3) of Panel B confirm the upward bias of the WLS estimates. The probability that the WLS estimates for the first eight years are simultaneous larger than those in column (3) in Table 3 is 46.1% (2,304 out of 5,000). In contrast, columns (5)-(7) of Table 5 indicate that the interactive fixed effects model shows an excellent performance. In summary, the simulation studies here suggest that the estimation model with the common factors structure is robust to the presence of common factors in the true model at the expense of a slightly wider confidence intervals when the true model does not include the common factors. On the other hand, when the true model includes the common factors, the lack of controlling common factors could cause serious bias. 6 Alternative Years of Divorce Law Reforms There are disagreements in defining what constitute a change in divorce laws. We examine a variety of definitions of divorce reforms, which are proposed in either the economics or law 13

15 literature. 7 In Table 6, we report the WLS and OLS estimates from the interpolated data. Panels A and B show the WLS and OLS estimates, respectively. The models in all columns include state effects, time effects and state-specific quadratic trends, but do not control for interactive fixed effects. The WLS estimates in Panel A indicate that divorce law reforms raise divorce rates in almost all columns. On the other hand, the OLS estimates in Panel B show small effects except for column (1). The observed pattern in columns (2)-(7) shows the same contrast between the WLS and OLS estimates. Table 7 presents the results from the interactive fixed effects model. The specifications in all columns are parallel to those in Table 6 except that interactive fixed effects are controlled. The difference between the WLS and OLS estimates reported in columns (1) and (2) become small after controlling for interactive fixed effects, and the estimated effects are not very different from those in Table 4. Gruber s and Johnson and Mazingo s reform dates are relatively closer to those used in Wolfers (2006) when compared with the other reform dates. The comparison between the estimates in columns (3)-(7) shows substantial differences between the weighted and unweighted models even after controlling for interactive fixed effects. It is also worth noting that, even within each panel, both the magnitudes and signs of estimates show substantial variations across different reform dates, despite the same weighting scheme. The results in Wolfers (2006) are not sensitive to alternative codes of divorce reforms and he uses them as evidence of the robustness of his findings. In contrast, we consider his results as evidence of the fragility of his findings. The use of different divorce reform years should yield different estimation results, because some of the codes ought to introduce misspecification into the model. Our results highlight that the estimation results are sensitive to different years of divorce law reforms. 7 We use alternative years of divorce law reforms proposed in the following papers: Gruber (2004), Johnson and Mazingo (2000), Mechoulan (2006), Ellman and Lohr (1998), Brinig and Buckley (1998) and Nakonezny, Shull, and Rodgers (1995). Those alternative years are examined in Wolfers (2006). 14

16 7 Summary and Conclusions In this paper we present some evidence on the effects of divorce law reforms on divorce rates in the U.S. Using the interactive fixed effects model to control for cross-state unobserved heterogeneity, we reconcile the contradictory results from the WLS and OLS estimates found in recent empirical studies. We document the inadequacy of the standard specification of unobserved heterogeneity in the literature, which possibly causes omitted variable bias. In effect, we find that the estimation results are unaffected by weighting the regression model, once we control for interactive fixed effects. Our estimation results indicate that the WLS estimates in Wolfers (2006) overestimate the effects, whereas the OLS estimates underestimate. We show the robustness of the model with interactive fixed effects, via simulation experiments. The use of interactive fixed effects allows researchers to implement robust inference in the Difference-in-Differences framework. We consider that our findings have general implications to a wide variety of empirical studies, in particular policy evaluation studies based on the Difference-in-Differences. We also examine alternative codes of divorce law reforms used in the economics and law literature. Our empirical results show that different codes of divorce law reforms could draw substantially different conclusions in contrast to Wolfers (2006) findings that alternative codes of divorce laws yield similar estimates of the reform effects. Our results highlight the importance of properly specifying divorce reforms. 15

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20 Appendix: interpolation to create the balance panel To create a balanced panel data used in Section 4, we conduct an interpolation. Before the interpolation, we exclude three states: Indiana, Louisiana and New Mexico. Indiana and New Mexico are dropped because there are missing observations around years of adapting unilateral divorce law, as in Table A1. We exclude Louisiana from our sample because only eight observations are available out of thirty three years. Table A1: States Excluded from the Analysis State The number of Missing years Law reform year missing values IN , 1975, LA , 1977, NM , , , We interpolate data for six states in Table A2, all of which have a few missing observations. We fit the forth order polynomial trends to the divorce rate data for each state and for each regime (before and after the divorce law reform), separately. More precisely, the model is: Divorce rate s,t = α 0 + α 1 t + α 2 t 2 + α 3 t 3 + α 4 t 4 + e s,t, where e s,t is an approximation error. We impute the fitted values if the observations are missing. Table A2: States Included in the Analysis with Interpolation State The number of Missing years Law reform year The number of obs missing values between before after IL NA KY MA NC NA NY NA WV NA 19

21 Table 1: Replications, Dynamic Effects of Divorce Law Reforms Weighted Least Squares Ordinary Least Squares Basic Linear Quadratic Basic Linear Quadratic Trend Trends Trend Trends (1) (2) (3) (4) (5) (6) First 2 years 0.267*** 0.342*** 0.302*** (0.085) (0.062) (0.054) (0.192) (0.096) (0.075) 3-4 years 0.210** 0.319*** 0.289*** ** (0.085) (0.070) (0.065) (0.194) (0.107) (0.092) 5-6 years 0.164* 0.300*** 0.291*** ** (0.085) (0.077) (0.079) (0.198) (0.121) (0.116) 7-8 years 0.158* 0.322*** 0.351*** ** 0.250* (0.084) (0.084) (0.097) (0.200) (0.132) (0.144) 9-10 years *** (0.084) (0.091) (0.117) (0.203) (0.143) (0.177) years *** *** (0.083) (0.099) (0.142) (0.203) (0.154) (0.215) years *** * *** (0.084) (0.107) (0.167) (0.212) (0.168) (0.257) 15 years *** * *** 0.311* (0.080) (0.119) (0.205) (0.208) (0.187) (0.317) Adjusted R Observations 1,631 1,631 1,631 1,631 1,631 1,631 Notes: Sample period is All regressions include state and year fixed effects. Regression models in Columns (1)-(3) are weighted by state population and those in Columns (4)-(6) are not weighted. Standard Errors are in parentheses. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 20

22 Table 2: Dynamic Effects of Divorce Law Reforms Weighted Least Squares, excluding CA, FL and TX Excluded States CA FL TX CA & FL CA & TX CA, TX & FL (1) (2) (3) (4) (5) (6) First 2 years 0.110* 0.277*** 0.305*** (0.059) (0.055) (0.057) (0.061) (0.064) (0.066) 3-4 years 0.198*** 0.247*** 0.272*** 0.134* 0.162** (0.072) (0.066) (0.068) (0.076) (0.077) (0.081) 5-6 years 0.203** 0.257*** 0.255*** (0.090) (0.081) (0.083) (0.094) (0.095) (0.100) 7-8 years 0.257** 0.349*** 0.307*** 0.253** (0.111) (0.099) (0.101) (0.117) (0.118) (0.125) 9-10 years (0.137) (0.121) (0.122) (0.144) (0.145) (0.154) years (0.167) (0.146) (0.147) (0.176) (0.177) (0.187) years (0.199) (0.172) (0.175) (0.211) (0.212) (0.225) 15 years (0.242) (0.211) (0.216) (0.255) (0.260) (0.275) Adjusted R Observations 1,598 1,598 1,598 1,565 1,565 1,532 Notes: Sample period is CA, FL and TX denote California, Florida and Texas, respectively. All regressions include state effects, year effects and state-specific quadratic trends. All regression models are weighted by state population. Standard Errors are in parentheses. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 21

23 Table 3: Dynamic Effects of Divorce Law Reforms Interpolated Balanced Panel Weighted Least Squares Ordinary Least Squares Basic Linear Quadratic Basic Linear Quadratic Trend Trends Trend Trend (1) (2) (3) (4) (5) (6) First 2 years 0.257*** 0.283*** 0.272*** (0.086) (0.064) (0.055) (0.194) (0.096) (0.074) 3-4 years 0.211** 0.250*** 0.280*** * (0.086) (0.072) (0.066) (0.198) (0.108) (0.091) 5-6 years ** 0.271*** *** (0.087) (0.080) (0.081) (0.203) (0.121) (0.115) 7-8 years * 0.340*** *** (0.086) (0.087) (0.099) (0.205) (0.132) (0.142) 9-10 years * *** (0.085) (0.094) (0.121) (0.206) (0.143) (0.175) years *** ** *** (0.085) (0.101) (0.145) (0.208) (0.153) (0.212) years *** *** *** (0.085) (0.109) (0.172) (0.216) (0.167) (0.254) 15 years *** *** 0.450** *** (0.081) (0.121) (0.211) (0.209) (0.186) (0.314) Adjusted R 2 1,584 1,584 1,584 1,584 1,584 1,584 Observations Notes: See notes in Table 1. 22

24 Table 4: Dynamic Effects of Divorce Law Reforms with Controlling for Interactive Fixed Effects Weighted Least Squares Ordinary Least Squares Basic Linear Quadratic Basic Linear Quadratic Trend Trends Trend Trends (1) (2) (3) (4) (5) (6) First 2 years 0.181*** 0.126*** 0.090** 0.101** 0.080** 0.082** (0.042) (0.038) (0.038) (0.040) (0.040) (0.040) 3-4 years 0.192*** 0.153*** 0.140*** 0.280*** 0.205*** 0.228*** (0.053) (0.051) (0.052) (0.053) (0.052) (0.054) 5-6 years 0.220*** 0.197*** 0.112* 0.232*** 0.182*** 0.183*** (0.064) (0.066) (0.067) (0.063) (0.063) (0.066) 7-8 years 0.196*** 0.219*** 0.187** 0.225*** 0.168** 0.177** (0.076) (0.082) (0.082) (0.072) (0.080) (0.081) 9-10 years (0.082) (0.096) (0.093) (0.079) (0.094) (0.096) years (0.086) (0.11) (0.105) (0.085) (0.111) (0.113) years (0.092) (0.119) (0.116) (0.089) (0.124) (0.131) 15 years ** (0.101) (0.13) (0.129) (0.099) (0.143) (0.157) Notes: Sample period is All regressions include state effects, year effects and interactive fixed effects. Seven common factors are controlled. Regression models in Columns (1)-(3) are weighted by state population and those in Columns (4)-(6) are not weighted. Standard Errors are in parentheses. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 23

25 Table 5: Simulation Results, Robustness of the Interactive Fixed Effects Model Without Controlling for With Controlling for Interactive Fixed Effects Interactive Fixed Effects True Values Median 95% range Median 95% range (1) (2) (3) (4) (5) (6) (7) Panel A: The DGP has no common factors First 2 years years years years years years years years Panel B: The DGP has 7 common factors First 2 years years years years years years years years Notes: The number of repetitions is 5,000. True coefficients values are reported in the first column. The columns of Median shows median values of estimated coefficients, and the columns of 95% range report the 5-th and 95-th percentiles. The DGP in Panel A does not have common factors, while that in Panel B have seven common factors. Errors are generated from independent normal distribution whose variances are taken from the estimates in Table 3 and 4. 24

26 Table 6: Dynamic Effects of Divorce Law Reforms Alternative Codes of Reform Years Gruber Johnson Mechoulan Ellman Ellman Brinig Nakonezny Mazingo Lohr 1 Lohr 2 Buckley et al. (1) (2) (3) (4) (5) (6) (7) Panel A: Weighted Least Squares First 2 years 0.260*** 0.318*** 0.126*** * 0.141** 0.165*** (0.054) (0.057) (0.044) (0.045) (0.042) (0.062) (0.039) 3-4 years 0.278*** 0.278*** 0.247*** 0.236*** 0.234*** *** (0.065) (0.068) (0.052) (0.053) (0.049) (0.072) (0.046) 5-6 years 0.342*** 0.360*** 0.259*** 0.208*** 0.199*** *** (0.078) (0.082) (0.062) (0.064) (0.058) (0.086) (0.055) 7-8 years 0.398*** 0.353*** 0.345*** 0.250*** 0.279*** *** (0.095) (0.099) (0.077) (0.078) (0.070) (0.106) (0.067) 9-10 years 0.442*** 0.369*** 0.320*** 0.199** 0.249*** *** (0.114) (0.119) (0.093) (0.094) (0.083) (0.126) (0.079) years 0.462*** 0.333** 0.336*** ** * 0.460*** (0.136) (0.143) (0.114) (0.114) (0.097) (0.150) (0.094) years 0.541*** 0.409** 0.398*** * ** 0.507*** (0.162) (0.171) (0.136) (0.137) (0.112) (0.180) (0.109) 15 years *** 0.728*** 0.676*** 0.328* 0.373*** *** (0.201) (0.212) (0.171) (0.171) (0.136) (0.231) (0.133) Panel B: Ordinary Least Squares First 2 years 0.155** 0.124* (0.068) (0.072) (0.060) (0.064) (0.062) (0.093) (0.056) 3-4 years 0.314*** ** 0.181** 0.142* * (0.083) (0.087) (0.074) (0.080) (0.074) (0.115) (0.068) 5-6 years 0.352*** (0.102) (0.109) (0.093) (0.100) (0.091) (0.142) (0.084) 7-8 years 0.385*** * (0.125) (0.133) (0.117) (0.126) (0.111) (0.179) (0.102) 9-10 years 0.370** ** 0.206* (0.151) (0.162) (0.144) (0.155) (0.133) (0.220) (0.121) years 0.428** * (0.181) (0.193) (0.176) (0.191) (0.157) (0.270) (0.144) years 0.597*** * (0.216) (0.231) (0.215) (0.234) (0.186) (0.329) (0.170) 15 years *** 0.614** ** (0.266) (0.283) (0.272) (0.294) (0.228) (0.416) (0.206) Notes: Sample period is All regressions include state effects, year effects and statespecific quadratic trends. Regression models in Panel A are weighted by state population and those in Panel B are not weighted. Standard Errors are in parentheses. *, **, and *** denote significance at the 10%, 5%, and 1% levels, respectively. 25