Divorce Law Reforms and Divorce Rates in the U.S.:
An Interactive Fixed Effects Approach∗
Dukpa Kim
University of Virginia†
Tatsushi Oka
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
22903 ([email protected])
‡
Department of Economics, National University of Singapore, AS2 Level 6, Singapore 117570
([email protected])
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
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-inDifferences 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
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, 1966-1988, is too short to distinguish pre-existing statespecific trends and time-varying treatment effects. Wolfers (2006) uses a sample from 1956
2
Rasul (2006) and Matouschek and Rasul (2008) provide alternative theories predicting positive effects
of unilateral divorce laws on divorce rates.
3
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 Rates,t = T reatmentt (Ts ) + vs,t + us,t ,
where Divorce Rates,t represents the number of divorces per 1,000 population for state
s and year t (s = 1, . . . , S and t = 1, . . . , T ); T reatmentt (·) is a function of the reform
year in state s, Ts , and measures the effects of divorce law reforms on divorce rates; vs,t
represents unobserved heterogeneity possibly correlated with divorce law reforms; us,t is
the error term.
T reatmentt (Ts ) 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 reatmentt (Ts ) =
8
X
βk U nilateralt (Ts , k),
k=1
where U nilateralt (Ts , k) takes one if 2k − 1 ≤ t − Ts ≤ 2k for k = 1, . . . , 7, one if 15 ≤ t − Ts
for k = 8 and zero otherwise, among those states which reformed divorce laws from 1956
to 1988, while U nilateralt (Ts , 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:
vs,t = αs + δt + φs · t + γs · t2 ,
where αs is a state fixed effect; δt is a time fixed effect; φs and γs 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
ws,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., Xs,t
= ws,t · Xs,t for an arbitrary variable Xs,t ). Then
the weighted model is
∗
Divorce Rate∗s,t = T reatment∗t (Ts ) + vs,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 0.29-0.35 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 heteroskedas5
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
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
we consider:
u∗s,t = λ0s ft + s,t ,
where λs is an r × 1 vector to capture state-specific reactions to the common shocks, ft is
an r × 1 vector of unobserved common shocks give the number of factors r, and s,t are
idiosyncratic errors. λs and ft 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 (αs , 1, φs , γs )0 and a vector of time-varying common shocks
(1, δt , t, t2 )0 . Yet, interactive fixed effects can control for a more general form of unobserved
heterogeneity. The common factors, ft , 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. λs
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
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 N T
consistent.
For the estimation, we adopt the method in Bai (2009) and the estimation procedure
is as follows. For notational simplicity, let Ys be a vector of the divorce rates in state s,
Xs be a matrix of the regressors for state s which includes dummies for treatment effects
fixed effects and individual trends, Λ = (λ1 , λ2 , . . . , λS ) be a matrix of factor loadings and
F = (f10 , f20 , . . . , fT0 ) be a matrix of common factors. Then, the regression coefficients are
obtained from the minimization problem that
SSR(β, F, Λ) =
S
X
(Ys − Xs β − F λs )0 (Ys − Xs β − F λs )
s=1
Since F and Λ are unobservable, Bai (2009) suggests an iterative method. That is,
given F and Λ, to compute
β̂(F, Λ) =
S
X
!−1
Xs0 Xs
s=1
S
X
Xs0 (Ys − F λs ),
s=1
and given β, to compute F as the eigenvectors corresponding to the r largest eigenvalues
of W W 0 where W = [W1 , . . . , WS ] and Ws = Ys − Xs β. Upon the normalization that
F 0 F/T = Ir , Λ = W 0 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 10−9 .
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
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 0.272-0.340,
which are comparable to the corresponding values in Table 1 (0.289-0.351). 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
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 0.090.19 increase in the divorce rates for the first eight years, which is much smaller than the
corresponding estimates of 0.29-0.35 in Table 1 and 0.27-0.34 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
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 (Ts ) + vs,t
+ u∗s,t
(2)
∗
Divorce Rate∗s,t = T reatment∗t (Ts ) + vs,t
+ λ0s ft + s,t
(3)
∗
where vs,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, 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
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
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
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
References
Ahn, S., Y. Lee, and P. Schmidt (2007): “Stochastic frontier models with multiple
time-varying individual effects,” Journal of Productivity Analysis, 27(1), 1–12.
Ahn, S. C., Y. H. Lee, and P. Schmidt (2001): “GMM estimation of linear panel data
models with time-varying individual effects,” Journal of Econometrics, 101(2), 219–255.
Allen, D. W. (1992): “Marriage and Divorce: Comment,” American Economic Review,
82(3), 679–85.
Bai, J. (2009): “Panel Data Models With Interactive Fixed Effects,” Econometrica, 77(4),
1229–1279.
Bai, J., and S. Ng (2002): “Determining the Number of Factors in Approximate Factor
Models,” Econometrica, 70(1), 191–221.
Becker, G. S. (1981): A Treatise on the Family. Cambridge, MA:.Harvard University
Press.
Bertrand, M., E. Duflo, and S. Mullainathan (2004): “How Much Should
We Trust Differences-in-Differences Estimates?,” The Quarterly Journal of Economics,
119(1), 249–275.
Brinig, M. F., and F. H. Buckley (1998): “No-fault laws and at-fault people,” International Review of Law and Economics, 18(3), 325–340.
Carneiro, P., K. T. Hansen, and J. J. Heckman (2003): “2001 Lawrence R. Klein
Lecture Estimating Distributions of Treatment Effects with an Application to the Returns to Schooling and Measurement of the Effects of Uncertainty on College Choice,”
International Economic Review, 44(2), 361–422.
Chamberlain, G. (1977): “Education, income, and ability revisited,” Journal of Econometrics, 5(2), 241–257.
Chamberlain, G., and Z. Griliches (1975): “Unobservables with a VarianceComponents Structure: Ability, Schooling, and the Economic Success of Brothers,” International Economic Review, 16(2), 422–49.
Dickens, W. T. (1990): “Error Components in Grouped Data: Is It Ever Worth Weighting?,” The Review of Economics and Statistics, 72(2), 328–33.
Droes, M. I., and R. C. van Lamoen (2010): “Did Unilateral Divorce Laws Raise
Divorce Rates? A Reconciliation and New Results: Comment,” Working Papers 10-11,
Utrecht School of Economics.
DuMouchel, W. H., and G. J. Duncan (1983): “Using Sample Survey Weights in
Multiple Regression Analyses of Stratified Samples,” Journal of the American Statistical
Association, 78(383), 535–543.
Ellman, I. M. (2000): “Divorce Rates, Marriage Rates, and the Problematic Persistence
of Traditional Marital Roles,” Family Law Quarterly, 34(1), 1–42.
16
Ellman, I. M., and S. L. Lohr (1998): “Dissolving the relationship between divorce
laws and divorce rates,” International Review of Law and Economics, 18(3), 341–359.
Fenelon, B. (1971): “State Variations in United States Divorce Rates,” Journal of Marriage and the Family, 33, 321–327.
Friedberg, L. (1998): “Did Unilateral Divorce Raise Divorce Rates? Evidence from
Panel Data,” American Economic Review, 88(3), 608–627.
Goldberger, A. S. (1972): “Structural Equation Methods in the Social Sciences,” Econometrica, 40(6), 979–1001.
Goldin, C., and L. F. Katz (2002): “The Power of the Pill: Oral Contraceptives and
Women’s Career and Marriage Decisions,” Journal of Political Economy, 110(4), 730–
770.
Gray, J. S. (1998): “Divorce-Law Changes, Household Bargaining, and Married Women’s
Labor Supply,” American Economic Review, 88(3), 628–42.
Gruber, J. (2004): “Is Making Divorce Easier Bad for Children? The Long-Run Implications of Unilateral Divorce,” Journal of Labor Economics, 22(4), 799–834.
Holtz-Eakin, D., W. Newey, and H. S. Rosen (1988): “Estimating Vector Autoregressions with Panel Data,” Econometrica, 56(6), 1371–95.
Johnson, J. H., and C. J. Mazingo (2000): “The Economic Consequences of Unilateral
Divorce for Children,” mimeo.
Joreskog, K. G., and A. S. Goldberger (1975): “Estimation of a Model with Multiple
Indicators and Multiple Causes of a Single Latent Variable,” Journal of the American
Statistical Association, 70(351), 631–639.
Lee, J. Y., and G. Solon (2011): “The Fragility of Estimated Effects of Unilateral
Divorce Laws on Divorce Rates,” NBER Working Papers 16773, National Bureau of
Economic Research, Inc.
MaCurdy, T. E. (1982): “The use of time series processes to model the error structure
of earnings in a longitudinal data analysis,” Journal of Econometrics, 18(1), 83–114.
Matouschek, N., and I. Rasul (2008): “The Economics of the Marriage Contract:
Theories and Evidence,” Journal of Law & Economics, 51(1), 59–110.
Mechoulan, S. (2006): “Divorce Laws and the Structure of the American Family,” Journal of Legal Studies, 35, 143–174.
Nakonezny, P. A., R. D. Shull, and J. L. Rodgers (1995): “The Effect of NoFault Divorce Law on the Divorce Rate Across the 50 States and Its Relation to Income,
Education, and Religiosity,” Journal of Marriage and Family, 57(2), 477–488.
Pesaran, M. H. (2006): “Estimation and Inference in Large Heterogeneous Panels with
a Multifactor Error Structure,” Econometrica, 74(4), 967–1012.
17
Peters, H. E. (1986): “Marriage and Divorce: Informational Constraints and Private
Contracting,” American Economic Review, 76(3), 437–54.
(1992): “Marriage and Divorce: Reply,” American Economic Review, 82(3), 687–
93.
Rasul, I. (2006): “Marriage Markets and Divorce Laws,” Journal of Law, Economics and
Organization, 22(1), 30–69.
Vogelsang, T. J. (2008): “Heteroskedasticity, Autocorrelation, and Spatial Correlation
Robust Inference in Linear Panel Models with Fixed-Effects,” mimeo, Michigan State
University.
Wolfers, J. (2006): “Did Unilateral Divorce Laws Raise Divorce Rates? A Reconciliation
and New Results,” American Economic Review, 96(5), 1802–1820.
Wright, G. C., and D. M. Stetson (1978): “The Impact of No-Fault Divorce Law
Reform on Divorce in American States,” Journal of Marriage and the Family, 40(3),
575–580.
18
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.
State
IN
LA
NM
Table A1: States Excluded from the Analysis
The number of Missing years
missing values
6
1970-1973, 1975, 1988
25
1956-1975, 1977, 1981-1988
9
1968-1969, 1971-1973, 1981-1982, 1986-1987
Law reform year
1973
2000
1973
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 rates,t = α0 + α1 t + α2 t2 + α3 t3 + α4 t4 + es,t ,
where es,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 1955-1988
before
after
IL
2
1956-1957
2000
31
NA
KY
3
1956-1958
1972
13
17
MA
1
1956
1975
18
14
NC
2
1956-1957
2000
31
NA
NY
2
1956-1957
2000
31
NA
WV
2
1956-1957
2000
31
NA
19
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***
-0.219
0.141
0.050
(0.085)
(0.062)
(0.054)
(0.192)
(0.096)
(0.075)
3-4 years
0.210** 0.319*** 0.289***
-0.273
0.211**
0.062
(0.085)
(0.070)
(0.065)
(0.194)
(0.107)
(0.092)
5-6 years
0.164*
0.300*** 0.291***
-0.425**
0.177
-0.036
(0.085)
(0.077)
(0.079)
(0.198)
(0.121)
(0.116)
7-8 years
0.158*
0.322*** 0.351***
-0.452** 0.250*
-0.026
(0.084)
(0.084)
(0.097)
(0.200)
(0.132)
(0.144)
9-10 years
-0.121
0.081
0.161
-0.703*** 0.133
-0.210
(0.084)
(0.091)
(0.117)
(0.203)
(0.143)
(0.177)
11-12 years
-0.324***
-0.102
0.047
-0.741*** 0.144
-0.270
(0.083)
(0.099)
(0.142)
(0.203)
(0.154)
(0.215)
13-14 years
-0.461*** -0.202*
0.031
-0.845*** 0.210
-0.289
(0.084)
(0.107)
(0.167)
(0.212)
(0.168)
(0.257)
15 years +
-0.507*** -0.210*
0.251
-0.776*** 0.311*
-0.226
(0.080)
(0.119)
(0.205)
(0.208)
(0.187)
(0.317)
Adjusted R2
0.931
0.973
0.982
0.852
0.972
0.986
Observations
1,631
1,631
1,631
1,631
1,631
1,631
Notes: Sample period is 1956-1988. 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
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***
0.062
0.084
0.027
(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**
0.092
(0.072)
(0.066)
(0.068)
(0.076)
(0.077)
(0.081)
5-6 years
0.203** 0.257*** 0.255***
0.153
0.140
0.079
(0.090)
(0.081)
(0.083)
(0.094)
(0.095)
(0.100)
7-8 years
0.257** 0.349*** 0.307*** 0.253**
0.175
0.167
(0.111)
(0.099)
(0.101)
(0.117)
(0.118)
(0.125)
9-10 years
0.076
0.141
0.151
0.051
0.023
-0.007
(0.137)
(0.121)
(0.122)
(0.144)
(0.145)
(0.154)
11-12 years
0.006
0.033
0.047
-0.005
-0.043
-0.061
(0.167)
(0.146)
(0.147)
(0.176)
(0.177)
(0.187)
13-14 years
-0.047
0.024
0.070
-0.050
-0.061
-0.065
(0.199)
(0.172)
(0.175)
(0.211)
(0.212)
(0.225)
15 years +
0.074
0.264
0.311
0.083
0.062
0.078
(0.242)
(0.211)
(0.216)
(0.255)
(0.260)
(0.275)
Adjusted R2
0.983
0.982
0.982
0.982
0.982
0.981
Observations
1,598
1,598
1,598
1,565
1,565
1,532
Notes: Sample period is 1956-1988. 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
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***
-0.267
0.095
0.023
(0.086)
(0.064)
(0.055)
(0.194)
(0.096)
(0.074)
3-4 years
0.211**
0.250*** 0.280***
-0.332*
0.159
0.049
(0.086)
(0.072)
(0.066)
(0.198)
(0.108)
(0.091)
5-6 years
0.129
0.179**
0.271***
-0.526*** 0.091
-0.055
(0.087)
(0.080)
(0.081)
(0.203)
(0.121)
(0.115)
7-8 years
0.107
0.169*
0.340***
-0.561*** 0.157
-0.024
(0.086)
(0.087)
(0.099)
(0.205)
(0.132)
(0.142)
9-10 years
-0.120
-0.046
0.224*
-0.747*** 0.067
-0.148
(0.085)
(0.094)
(0.121)
(0.206)
(0.143)
(0.175)
11-12 years
-0.342*** -0.258**
0.137
-0.853*** 0.052
-0.195
(0.085)
(0.101)
(0.145)
(0.208)
(0.153)
(0.212)
13-14 years
-0.494*** -0.398***
0.142
-0.954*** 0.093
-0.191
(0.085)
(0.109)
(0.172)
(0.216)
(0.167)
(0.254)
15 years +
-0.505*** -0.428***
0.450**
-0.818*** 0.222
-0.007
(0.081)
(0.121)
(0.211)
(0.209)
(0.186)
(0.314)
Adjusted R2
1,584
1,584
1,584
1,584
1,584
1,584
Observations
0.931
0.972
0.982
0.854
0.973
0.986
Notes: See notes in Table 1.
22
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
0.078
0.074
0.046
0.105
0.047
0.056
(0.082)
(0.096)
(0.093)
(0.079)
(0.094)
(0.096)
11-12 years
0.074
0.028
0.053
0.088
0.040
0.035
(0.086)
(0.11)
(0.105)
(0.085)
(0.111)
(0.113)
13-14 years
-0.022
-0.088
-0.032
0.059
0.005
-0.001
(0.092)
(0.119)
(0.116)
(0.089)
(0.124)
(0.131)
15 years +
0.056
-0.086
0.037
0.198**
0.112
0.125
(0.101)
(0.13)
(0.129)
(0.099)
(0.143)
(0.157)
Notes: Sample period is 1956-1988. 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
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
0.272
0.272
0.167 0.373
0.271
0.135 0.408
3-4 years
0.280
0.280
0.154 0.399
0.278
0.112 0.446
5-6 years
0.271
0.271
0.120 0.420
0.269
0.064 0.476
7-8 years
0.340
0.341
0.158 0.521
0.340
0.096 0.582
9-10 years
0.224
0.227
-0.003 0.444
0.222
-0.077 0.522
11-12 years
0.137
0.139
-0.141 0.407
0.136
-0.228 0.497
13-14 years
0.142
0.143
-0.179 0.458
0.142
-0.291 0.569
15 years +
0.450
0.449
0.059 0.837
0.448
-0.072 0.965
Panel B: The DGP has 7 common factors
First 2 years
0.090
0.283
0.241 0.327
0.096
0.015 0.178
3-4 years
0.140
0.304
0.254 0.355
0.149
0.025 0.273
5-6 years
0.112
0.293
0.230 0.355
0.119
-0.042 0.280
7-8 years
0.187
0.347
0.271 0.424
0.187
-0.014 0.386
9-10 years
0.046
0.223
0.128 0.316
0.042
-0.193 0.269
11-12 years
0.053
0.125
0.011 0.239
0.041
-0.220 0.296
13-14 years
-0.032
0.129
-0.004 0.264
-0.041
-0.333 0.246
15 years +
0.037
0.455
0.290 0.620
0.036
-0.290 0.357
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
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***
0.071
0.075*
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***
0.072
0.278***
(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*** -0.044
0.328***
(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*** -0.013
0.431***
(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*** -0.197
0.437***
(0.114)
(0.119)
(0.093)
(0.094)
(0.083)
(0.126)
(0.079)
11-12 years
0.462*** 0.333**
0.336***
0.156
0.198** -0.287*
0.460***
(0.136)
(0.143)
(0.114)
(0.114)
(0.097)
(0.150)
(0.094)
13-14 years
0.541*** 0.409**
0.398***
0.147
0.207* -0.401** 0.507***
(0.162)
(0.171)
(0.136)
(0.137)
(0.112)
(0.180)
(0.109)
15 years +
0.905*** 0.728*** 0.676***
0.328* 0.373*** -0.227
0.722***
(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*
0.032
0.015
0.029
0.021
0.060
(0.068)
(0.072)
(0.060)
(0.064)
(0.062)
(0.093)
(0.056)
3-4 years
0.314***
0.068
0.172**
0.181**
0.142*
-0.080
0.119*
(0.083)
(0.087)
(0.074)
(0.080)
(0.074)
(0.115)
(0.068)
5-6 years
0.352***
0.155
0.092
0.145
0.082
-0.155
0.113
(0.102)
(0.109)
(0.093)
(0.100)
(0.091)
(0.142)
(0.084)
7-8 years
0.385***
0.150
0.063
0.098
0.101
-0.210
0.195*
(0.125)
(0.133)
(0.117)
(0.126)
(0.111)
(0.179)
(0.102)
9-10 years
0.370**
0.136
-0.040
0.021
0.019
-0.509**
0.206*
(0.151)
(0.162)
(0.144)
(0.155)
(0.133)
(0.220)
(0.121)
11-12 years
0.428**
0.159
-0.120
-0.030
-0.024
-0.523*
0.216
(0.181)
(0.193)
(0.176)
(0.191)
(0.157)
(0.270)
(0.144)
13-14 years
0.597***
0.306
-0.148
0.003
0.042
-0.566*
0.247
(0.216)
(0.231)
(0.215)
(0.234)
(0.186)
(0.329)
(0.170)
15 years +
0.988*** 0.614**
-0.051
0.186
0.154
-0.552
0.507**
(0.266)
(0.283)
(0.272)
(0.294)
(0.228)
(0.416)
(0.206)
Notes: Sample period is 1956-1988. 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
Table 7: Dynamic Effects of Divorce Law Reforms
with Controlling for Interactive Fixed Effects
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 0.171*** 0.105***
-0.089**
-0.122*** -0.182*** -0.083**
-0.018
(0.042)
(0.04)
(0.043)
(0.040)
(0.033)
(0.042)
(0.037)
3-4 years
0.195*** 0.139**
-0.001
-0.042
-0.141*** -0.149**
0.038
(0.056)
(0.058)
(0.054)
(0.056)
(0.045)
(0.061)
(0.049)
5-6 years
0.176** 0.155**
-0.069
-0.126*
-0.154** -0.191**
0.044
(0.073)
(0.073)
(0.067)
(0.071)
(0.060)
(0.078)
(0.062)
7-8 years
0.190**
0.067
-0.122
-0.232*** -0.194*** -0.139
0.069
(0.089)
(0.089)
(0.081)
(0.086)
(0.074)
(0.096)
(0.075)
9-10 years
0.147
0.002
-0.254*** -0.369*** -0.226**
-0.173
-0.001
(0.098)
(0.098)
(0.094)
(0.102)
(0.089)
(0.112)
(0.087)
11-12 years
0.186*
0.006
-0.260**
-0.419*** -0.259*** -0.069
0.007
(0.110)
(0.108)
(0.107)
(0.113)
(0.099)
(0.126)
(0.097)
13-14 years
0.114
-0.005
-0.380*** -0.505*** -0.382*** -0.185
-0.073
(0.117)
(0.117)
(0.115)
(0.123)
(0.107)
(0.134)
(0.105)
15 years +
0.283**
0.174
-0.359*** -0.517*** -0.458*** -0.308**
-0.011
(0.129)
(0.132)
(0.128)
(0.132)
(0.116)
(0.145)
(0.115)
Panel B: Ordinary Least Squares
First 2 years 0.070*
0.084**
0.014
-0.031
-0.042
0.071
0.046
(0.038)
(0.04)
(0.038)
(0.038)
(0.036)
(0.059)
(0.034)
3-4 years
0.148*** 0.165*** 0.145***
0.065
0.030
0.127
0.117**
(0.052)
(0.052)
(0.051)
(0.052)
(0.052)
(0.082)
(0.047)
5-6 years
0.172*** 0.199***
0.128**
0.035
-0.005
0.062
0.114**
(0.065)
(0.069)
(0.059)
(0.062)
(0.064)
(0.093)
(0.056)
7-8 years
0.104
0.100
0.076
-0.088
-0.078
0.110
0.105
(0.083)
(0.082)
(0.075)
(0.081)
(0.078)
(0.123)
(0.070)
9-10 years
0.052
0.017
0.024
-0.176*
-0.133
-0.051
0.066
(0.098)
(0.096)
(0.091)
(0.096)
(0.089)
(0.144)
(0.084)
11-12 years
0.036
-0.011
-0.006
-0.242**
-0.188*
0.016
0.068
(0.115)
(0.111)
(0.105)
(0.109)
(0.105)
(0.169)
(0.098)
13-14 years
-0.021
0.004
-0.061
-0.312**
-0.214*
-0.081
0.020
(0.132)
(0.128)
(0.122)
(0.126)
(0.120)
(0.191)
(0.114)
15 years +
0.160
0.202
0.007
-0.256*
-0.181
-0.027
0.162
(0.159)
(0.150)
(0.144)
(0.145)
(0.132)
(0.232)
(0.137)
Notes: Sample period is 1956-1988. All regressions include state effects, year effects, statespecific quadratic trends and interactive fixed effects. 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.
26
Figure 1: State-Level Divorce Rates from 1956 to 1988
Notes: The sample period is 1956-1988. The treatment group includes states which liberalized divorce laws during the period,
while the control group includes the other sates. Panel A shows divorce rates without weight, and Panel B shows divorce rates
multiplied by the square root of state population. Diamond marks () denote years of divorce law reforms. For this figure, Nevada
is not included because the divorce rates are exceptionally high.
Figure 2: Dynamic Effects of Divorce Law Reforms for Different Numbers of Common Factors
Notes: The sample period is 1956-1988. All models include state fixed effects, time fixed effects and state-specific quadratic
time trends. The regression models in Panels A.1 and A.2 are weighted by state population and those in Panel B.1 and B.2 are
not weighted.