# Identifying Latent Structures in Maternal Employment: Evidence on the German
Parental Benefit Reform

Sophie-Charlotte Klose Department of Management, University of Duisburg-Essen,
Germany, e-mail<EMAIL_ADDRESS>I am grateful to my
supervisor Prof. Dr. Marie Paul for the support and encouragement. Her
numerous comments and suggestions have greatly improved the paper. I thank
Prof. Dr. Bernd Fitzenberger, Prof. Dr. Marie Paul, and Arnim Seidlitz for the
fruitful collaboration in the data preparation. Moreover, I gratefully
acknowledge financial support by the German Science Foundation (DFG) through
the priority program 1764 “The German Labor Market in a Globalized World”.

(August 27, 2024)

###### Abstract

This paper identifies latent group structures in the effect of motherhood on
employment by employing the C-Lasso, a recently developed, purely data-driven
classification method. Moreover, I assess how the introduction of the generous
German parental benefit reform in 2007 affects the different cluster groups by
taking advantage of an identification strategy that combines the sharp
regression discontinuity design and hypothesis testing of predicted employment
probabilities. The C-Lasso approach enables heterogeneous employment effects
across mothers, which are classified into an a priori unknown number of
cluster groups, each with its own group-specific effect. Using novel German
administrative data, the C-Lasso identifies three different cluster groups
pre- and post-reform. My findings reveal marked unobserved heterogeneity in
maternal employment and that the reform affects the identified cluster groups’
employment patterns differently.

Keywords: female employment, fertility, parental benefits, C-Lasso, latent
group structures, dynamic nonlinear panel
JEL codes: J13, J18, J21, C33, C38

##  1 Introduction

As women’s labor market participation has increased over the past several
decades while fertility rates have simultaneously decreased, policymakers face
the challenge of encouraging fertility, and trying to keep women after
childbirth close to the labor market (Raute (2019)). In almost all European
economies, paid parental leave policies try to mitigate this challenge by
providing employment protection and some extent of earnings replacement, which
reduces child-rearing opportunity costs and facilitates the return to
employment after childbearing.111For an overview of parental leaves in Europe
see, e.g., Pronzato (2009). The efficient design of these policies hinges on
an accurate estimate of child age’s effect on maternal labor supply.

Providing an accurate estimate of maternal employment is sophisticated because
mothers’ labor supply reactions following a child’s birth are very
heterogeneous. However, researchers usually disregard the heterogeneity and
only estimate the average effect of child age on employment. Sometimes effect
heterogeneity regarding observable characteristics like education or age is
investigated (e.g., Angrist and Evans (1998); Fitzenberger et al. (2013)).
Some studies show that the impact of child age on maternal labor supply
differs between high-earning (highly educated) and less-earning (low-educated)
women (e.g., Troske and Voicu (2010); Troske and Voicu (2013)). Due to higher
opportunity costs of childbearing, highly educated and high-earning women are
typically more labor market responsive than their less-educated and lower-
earning peers (Lundborg et al. (2018)). This pattern shows up at the so-called
“baby gap” (Raute (2019)): the negative relationship between education and
completed fertility, which leads to a fertility differential between highly
educated and less-educated women.

Nevertheless, the employment effects may also differ regarding unobservable
characteristics: having a child may affect different groups of women based on
the data’s latent structures very differently. Estimating average effects may
hide important and potentially policy-relevant differences in the employment
trajectories. Reforms may also change the composition of groups and alter how
a child influences a mother’s employment.

In this paper, I focus specifically on identifying latent structures in the
effect of motherhood on employment through data-determined grouping. This is
an important issue because policymakers require the most accurate information
on this effect to react appropriately. A machine learning approach is
appealing for examining latent heterogeneity in maternal employment decisions
because it does not require the specification of any modeling mechanism for
the latent group structure.

My goal in this paper is twofold. First, I examine how the heterogeneous
patterns of mothers’ return behavior to employment up to six years after first
childbirth look like. For this purpose, I classify mothers into a priori
number of unknown cluster-groups. The group membership of each mother is also
a priori unknown. Within each group, the effects of child age on employment
are the same, whereas, across the groups, the effects of child age on
employment differ. The econometric approach I use is a recently developed,
purely data-driven classification method called C-Lasso proposed by Su et al.
(2016). It isolates three different clusters of mothers with similar effects
of child age on employment: a group of mothers who return to employment
quickly after first childbirth, a cluster of mothers who are on average less
employed, and a cluster of mothers who take an extended family break after
birth before returning to employment. This finding reveals pronounced
unobserved heterogeneity in the impact of child age on employment across
mothers. In contrast to the recently used, purely descriptive classification
analysis of maternal career trajectories after first birth conducted by
Frühwirth-Schnatter et al. (2016), the C-Lasso technique achieves simultaneous
group classification and oracle-efficient estimation of the coefficients in
each group. Moreover, the C-Lasso method allows for the heterogeneity in the
slope coefficients when the extent of the heterogeneity is unknown. To the
best of my knowledge, the C-Lasso method has been used only once in the
context of estimating heterogeneous effects of the minimum wage on employment
within a hierarchical model (see Wange et al. (2019)).

Second, I want to find out how the most recent German parental leave reform,
as one of the most significant reforms in German family policies in the last
decades, has affected mothers’ heterogeneous employment patterns after the
first birth. This reform substantially changed parental compensation for time
absent from employment following childbirth. Implemented in 2007, the new
parental benefit offers a generous earnings-dependent income replacement up to
14 months after childbirth or a 3,600 EUR basic cash transfer for women not in
the labor force in the pre-birth period. In contrast, the benefit eligibility
under the old scheme was means-tested and only targeted to low-income
families. Nonetheless, although the new benefits scheme raised the financial
incentives by up to 21,000 EUR, there are huge differences in the benefit
transfers across mothers from different education and earning groups.

To measure the extent to which the different cluster-groups have reacted to
this reform and offer a new look to the empirical evidence of policy impacts
on maternal employment, I apply a sharp regression discontinuity (RD) design
in the spirit of Kluve and Schmitz (2018). To identify the reform effects
within the C-Lasso methodology’s panel structure model, I use an
identification strategy that exploits the hypothesis testing of predicted
employment probabilities. As my data set, I use a novel custom-shaped
administrative data set for which precise information on childbirths from the
German Pension Registry have been matched to female employment biographies in
Germany.

If the parental leave reform generates heterogeneous reactions to how mothers’
labor supply responds to the child’s age, I should observe different changes
in predicted employment probabilities within the three cluster-groups. My
empirical analysis does uncover marked latent heterogeneity in the reform
reactions across groups: The reform generates strong negative and long-
standing effects for the medium-sized group, whereas it slightly encourages
the largest group to take up employment immediately after benefit exhaustion.

My findings contribute to a developing body of dynamic structural life cycle
models and quasi-experimental literature on labor supply, fertility, and
family policies. Most of the previous papers in the dynamic structural life
cycle literature (e.g., Carrasco (2001); Hyslop (1999); Michaud and Tatsiramos
(2011); Moffitt (1984)) analyze female labor supply and fertility jointly and
find that labor supply decisions are characterized by significant state
dependence and unobserved heterogeneity but do not directly link their
evidence to parental leave policies. The literature using dynamic life-cycle
models traditionally specifies labor supply decisions as a threshold-crossing
model and estimates a random-effects version of this model. In contrast, I
apply the C-Lasso to a dynamic fixed effects probit model because this avoids
the distribution of unobserved heterogeneity from being arbitrarily restricted
and the correlation with the covariates, and it avoids the initial conditions
problem. More recent papers (e.g., Troske and Voicu (2010); Troske and Voicu
(2013)) estimate the causal effects of birth on subsequent employment of
mothers by extending the aforementioned dynamic structural life-cycle model
towards a framework that takes into account endogeneity of labor market and
fertility decisions, heterogeneity of the effects of children and their
correlation with fertility decisions, and correlation of sequential labor
market decisions. Both papers concentrate on the heterogeneity in the timing
and spacing of births around the average career path. Whereas the major part
of the literature has assessed the effect heterogeneity of children on labor
supply within a tight framing, Frühwirth-Schnatter et al. (2016) provide to my
knowledge the only structural model evidence for more general latent
heterogeneity in the labor market outcomes of mothers by modeling separate
cluster-group-specific career trajectories. Using Bayesian Markov chain
clustering analysis, the authors classify women into five cluster groups with
very different long-run trajectories. Their approach is purely descriptive
because the authors focus on clustering transition processes (in and out of
parental leave, non-employment, and different forms of employment) rather than
credible isolate causal pathways. Compared to this study, I can provide
simultaneous group classification and estimation of the effect of child age on
employment. Moreover, my identification strategy that combines the sharp RD
design and hypothesis testing of predicted employment probabilities can
isolate causal pathways of the most recent parental benefit reform – albeit
under strong assumptions that need to be imposed.

The quasi-experimental literature on paid parental leave in Germany focuses
largely on policy effects on maternal employment (see particularly Bergemann
and Riphahn (2011); Greyer et al. (2015); Kluve and Tamm (2013); Kluve and
Schmitz (2018); Schönberg and Ludsteck (2014)). All papers adopt a sharp RD
design or a differences-in-differences design by exploiting a natural
experiment to compare mothers who gave birth to children shortly before the
parental leave reform (control group) with mothers who gave birth to children
shortly after the reform (treatment group). The mothers from the control and
treatment groups face different paid maternity leaves, both in terms of leave
duration and transfers. Overall, there is a large consensus across the
studies: The generous paid parental leave reform in 2007 generates a strong
disincentive-to-work effect in the short run. Schönberg and Ludsteck (2014)
analyze the impact of five expansions in maternity leave coverage on mother’s
labor market outcomes after childbirth and find that each expansion reduces
mothers’ postbirth employment rates in the short-run, whereas the long-run
effects are small. Greyer et al. (2015) find that especially mothers with
higher pre-birth earnings after the reform have more incentives to stay at
home in the first year after giving birth than before the reform. However,
they find stronger incentives to work after benefit exhaustion for low-income
mothers. Kluve and Schmitz (2018) also find beneficial medium-run employment
effects of the parental benefit reform. The authors find considerable positive
and statistically significant effects on maternal employment, ranging up to
10%. The positive impacts are centered on mothers from the medium and high
terciles of earning distribution, whereas low-income mothers do not benefit.

Most recent literature also assesses the impact on fertility (see Raute
(2019)). Using the large differential changes in maternity leave benefits of
the most recent parental leave reform in 2007 across education and income
groups in a differences-in-differences design, Raute (2019) finds a positive
and statistically significant impact of increased benefits on fertility,
driven mainly by women at the middle and upper end of the earnings
distribution.

I add to the quasi-experimental literature on parental leave policies along
several important dimensions. Most importantly, I provide evidence on how the
most recent German parental benefit reform affects the different isolated
cluster-groups’ employment patterns. These group-specific policy impacts on
maternal employment are potentially more policy-relevant than estimating only
the average policy impact on employment. Another limitation of the previous
literature is that it mostly concentrates on effect heterogeneity of the
policy change based on observables. In contrast to the aforementioned
literature, on the one hand, I cannot decipher the main determinants of
heterogeneity as it would be feasible in an analysis based on observables.
However, on the other hand, my approach provides a far more complete picture
of how policy change affects maternal employment by taking into account latent
structures in employment decisions across mothers. Finally, my evidence on
heterogeneous parental leave effects is especially relevant for policymakers
to design efficient parental leave legislation.

The remainder of this paper is structured as follows. Section 2 explains the
institutional background of the parental leave legislation change in Germany
and describes the mechanisms through which it can affect the heterogenous
employment effects of mothers. Section 3 introduces the panel structure model
and the C-Lasso technique of Su et al. (2016). Section 4 describes the data
set employed in this study. Section 5 reports the main findings and Section 6
concludes the paper. The main tables are stated in the Appendix. Readers who
are interested in additional insights on the data’s sample restriction and the
methodology are referred to the Online Appendix.

##  2 Institutional background

###  2.1 The German parental benefit reform

In Germany, mothers have been entitled to government-provided paid leave six
weeks before and eight weeks after childbirth since the mid-1950s, and
parental leave regulations have continuously extended in a series of reforms
starting in the late 1970s.222For an overview of the developments of the
parental leave regulations in Germany, see Kluve and Schmitz (2018), Kluve and
Tamm (2013), Raute (2019) and Schönberg and Ludsteck (2014). Since 1979,
mothers have been granted six months of post-birth job protection, and it has
been expanded up to a maximum of 36 months after birth since 1992. During this
period, government-provided financial transfers have been extended to two
years while on leave and targeted to low-income families. In 2007, however,
the German parental benefit reform came into force that compensated women much
more generous for forgone earnings by taking into account the opportunity
costs of child-rearing.

The previous benefit program, ‘child-raising allowance’ (Erziehungsgeld), was
targeted to low-income households and was much less generous. It provided two
options: Option 1 comprised a maximum of 300 EUR per month for up to 24 months
after birth (total 7200 EUR), and Option 2 comprises a monthly payment of 450
EUR over 12 months after birth (total 5400 EUR). Benefit eligibility under
both options was means-tested on household income, and the eligibility
criteria did not permit to work more than 30 h per week during benefit
receipt. Of the parents entitled to the parental allowance, the majority (66%)
chose Option 1, and only 10% chose Option 2.33324% of parents were not
eligible at all (Kluve and Tamm (2013)).

Table 1: Overview over Changes in the Benefit Structure Induced by the Reform |  Parental money (Post-2007 benefit) |  | Child-rearing allowance (Pre-2007 benefit)
---|---|---|---
|  |  |  Option 1 |  Option 2
Monthly benefit |  67% of ave. monthly pre-birth net income; min. 300 EUR, max. 1800 EUR. Mothers with no pre-birth income entitled to 300 EUR. |  |  300 EUR |  450 EUR
Means tested |  no |  |  yes |  yes
Max. duration |  14 months (2 month for the father) |  |  24 months |  12 months
Proportion of parents covered |  almost 100% |  |  66% |  10%
Requirements |  not working more than 30 h per week during receipt |  | not working more than 30 h per week during receipt
Total max. benefit |  3,600-21,600 EUR (+ 600-3,600 EUR for the father) |  |  7,200 EUR |  5,400 EUR
Notes: See Raute (2019) for further details of the reform.

On January 1, 2007, a new parental leave benefit, ‘parental money’
(Elterngeld), replaced the previous benefit regulation. Table 1 summarizes the
changes in the benefits system induced by the new parental benefit reform. The
objective of the reform is “to promote the effective and sustainable economic
security of families after childbirth through a parental allowance: to avoid
income drops, $\displaystyle\dots$ to promote the economic independence of
both parents, and to allow a fair compensation of opportunity costs of
childbearing” BMFSFJ (2008). Whereas the old benefit scheme was means-tested
and thus targeted to low-income families, the coverage of the new ‘parental
money’ is effectively 100% of parents. Most importantly, the ‘parental money’
is an earnings-dependent leave benefit, which compensates women according to
their childbearing opportunity costs. In contrast to the old means-tested
benefits, the new regulation is much more generous: For most mothers, the new
benefit replaces 67% of pre-child birth net labor earnings for up to 14 months
after childbirth (12 months if one parent only takes up the benefit). The
maximum amount is truncated at 1,800 EUR per month for households at the top
of the earnings distribution, and a flat minimum of 300 EUR per month is paid
to parents with no previous labor earnings. The monthly benefits translate
into a total benefit ranging between 3,600 EUR and 21,000 EUR, depending on
where the household is on the earnings distribution. Like the old regulation,
benefit eligibility depends on not working more than 30 h a week during
benefit receipt. Mothers who choose to work below 30 h during their benefit
entitlement face a reduction in benefits with increasing labor earnings.

###  2.2 Mechanisms: the heterogeneous effects of the reform on employment

The current literature on the parental benefit reform’s employment effects
documents heterogeneous patterns after childbirth based on observables.
Whereas previous studies confirm that the reform encourages mothers to remain
at home during the first year following childbirth and thus reduces maternal
employment during benefit receipt, especially for groups who benefit strongly
(e.g., Bergemann and Riphahn (2011), Greyer et al. (2015), Kluve and Tamm
(2013), Kluve and Schmitz (2018)), the study of Kluve and Schmitz finds
pronounced beneficial patterns for the medium run (2-5 years after
childbirth). In particular, the authors provide evidence that the effects on
mothers’ employment probability are positive, significant, and large, ranging
up to 10%.

Figure 1: Recent Findings of the Literature

$\displaystyle\Delta$ employment rateMonths since birth$\displaystyle
24$$\displaystyle 48$$\displaystyle
72$$\displaystyle-10\%$$\displaystyle-20\%$$\displaystyle-30\%$$\displaystyle-40\%$$\displaystyle-50\%$$\displaystyle
10\%$$\displaystyle 20\%$$\displaystyle 30\%$$\displaystyle
40\%$$\displaystyle 50\%$Phase IPhase IIPhase III14260

Notes: Changes in Employment Rates induced by the Reform.
Filled gray: All mothers. Shaded blue: Upper tercentile. Shaded lightgray:
Middle tercentile.

Given the earnings-dependent design of the reform, quite heterogeneous labor
supply reactions of mothers following the birth of a first child can be
expected. The existing literature confirms that the reform impact on maternal
employment differs across socioeconomic groups (Kluve and Tamm (2013), Kluve
and Schmitz (2018)). Figure 1 summarizes the findings of Kluve and Schmitz
(2018), and shows the different changes in employment rates across income
terciles induced by the reform. For all mothers, the authors report a 25%
decrease in maternal employment over the control mean during Phase I, mainly
driven by mothers from the upper tercile of the income distribution (41%) and
first-time mothers (32%, not reported in Fig. 1), which are the groups most
strongly incentivized by the reform. In Phase II, they find no significant
impacts on employment for most socioeconomic groups. Only treatment group
mothers in the upper tercile of the income distribution display a 6% increase
over the control mean. For Phase III, the authors find that the reform
substantially increased maternal employment, especially for mothers from the
middle and upper terciles of the income distribution, partially offsetting the
negative short-run employment effects.

In sum, the literature on reform’s employment effects documents heterogeneous
patterns after birth concerning observables (mostly across income and
education groups). However, the 2007 reform in paid leave may affect
employment decisions also concerning unobservables: e.g., mothers who want to
stay longer at home to spend more time with their child - and pre-reform were
not able to take up longer leave because of financial constraints - may
decrease their labor supply under the more generous regulation. On the other
hand, as paid leave benefits are paid conditional on pre-birth earnings, the
reform may positively impact employment. E.g., Career-oriented mothers may
directly return to work with the end of the benefit receipt to smooth the
household’s income and minimize childraising opportunity costs.

##  3 Model and Methodology

In this section, I formulate the panel structure model and describe the
methodology introduced by Su et al. (2016).

###  3.1 Model

I consider data in form of a real-valued deterministic design
$\displaystyle\bm{X}=[\bm{X}_{11}\ \bm{X}_{12}]\in\mathbb{R}^{(NT)\times p}$
and a binary response $\displaystyle\bm{y}\in\mathbb{R}^{NT}$. The design
$\displaystyle\bm{X}$ is partitioned in two blockmatrices
$\displaystyle\bm{X}_{11}\in\mathbb{R}^{(NT)\times q}$ and
$\displaystyle\bm{X}_{12}\in\mathbb{R}^{(NT)\times(p-q)}$, with $\displaystyle
q\leq p$. I denote the rows of $\displaystyle\bm{X}$ by
$\displaystyle\bm{x}_{it}\in\mathbb{R}^{p}$ where $\displaystyle
i=1,\dots,N,t=1,\dots,T$, and the columns of $\displaystyle\bm{X}$ by
$\displaystyle\bm{x}^{j}\in\mathbb{R}^{NT}$ where $\displaystyle j=1,\dots p$.
Similarly, I denote the rows of $\displaystyle\bm{X}_{11}$ and
$\displaystyle\bm{X}_{12}$ by $\displaystyle\bm{x}_{11_{it}}\in\mathbb{R}^{q}$
and $\displaystyle\bm{x}_{12_{it}}\in\mathbb{R}^{p-q}$, respectively. The
columns of $\displaystyle\bm{X}_{11}$ and $\displaystyle\bm{X}_{12}$ are
denoted by $\displaystyle\bm{x}_{11}^{j_{1}}\in\mathbb{R}^{NT}$ and
$\displaystyle\bm{x}_{12}^{j_{2}}\in\mathbb{R}^{NT}$ with $\displaystyle
j_{1}=1,\dots,q$ and $\displaystyle j_{2}=q+1,\dots,p$, respectively.
The econometric model takes the form of a dynamic binary choice model

$\displaystyle\displaystyle
y_{it}=\mathbf{1}\\{\bm{x}_{11_{it}}^{\top}\bm{\gamma^{*}}+\bm{x}_{12_{it}}^{\top}\bm{\beta^{*}}_{i}+\mu_{i}^{*}-\epsilon_{it}>0\\},\quad
i=1,\dots,N;t=1,\dots,T,$ (1)

where $\displaystyle i$ and $\displaystyle t$ denote mother $\displaystyle i$
and month after childbirth $\displaystyle t$, respectively,
$\displaystyle\bm{x}_{it}=[\bm{x}_{11_{it}}\
\bm{x}_{12_{it}}]\in\mathbb{R}^{p}$ is the vector of covariates containing
also the dynamic component $\displaystyle y_{i,t-1}$,
$\displaystyle\bm{\gamma^{*}}\in\mathbb{R}^{q}$ is the unknown regression
vector that is common across all $\displaystyle i$,
$\displaystyle\bm{\beta^{*}}_{i}\in\mathbb{R}^{p-q}$ is the unknown individual
specific regression vector, $\displaystyle\mu_{i}^{*}$ denotes the unknown
individual fixed effect, and $\displaystyle\epsilon_{it}$ is the idiosyncratic
error term. Note, when $\displaystyle q=0$, all parameters of interest are
assumed to be individual-specific, and when $\displaystyle q=p$, all
parameters of interest are assumed to be common across $\displaystyle i$. In
case of $\displaystyle 0<q<p$, I refer to the mixed panel structure model of
Su et al. (2016) where some parameters of interest are common across all
individuals whereas others are individual-specific.

A latent group structure is imposed on the $\displaystyle\bm{\beta^{*}}_{i}$.
For this purpose, I consider a partition of the index set
$\displaystyle\\{1,\dots,N\\}$, that is, a collection of disjoint sets
$\displaystyle\mathcal{G}^{1},\dots,\mathcal{G}^{K^{*}}$ that satisfy
$\displaystyle\cup_{k=1}^{K^{*}}\mathcal{G}^{k}=\\{1,\dots,N\\}$.
$\displaystyle K^{*}$ denotes the true a priori unknown number of groups. In
particular, I assume for $\displaystyle\bm{\beta^{*}}_{i}$ a sparsity inducing
group structure of the following form

$\displaystyle\displaystyle\bm{\beta^{*}}_{i}=\left\\{\begin{array}[]{ll}\bm{\alpha^{*}}_{\mathcal{G}^{1}}&\quad\quad\text{if}\
i\in\mathcal{G}^{1}\\\ \vdots&\quad\quad\quad\ \ \vdots\\\
\bm{\alpha^{*}}_{\mathcal{G}^{K^{*}}}&\quad\quad\text{if}\
i\in\mathcal{G}^{K^{*}},\end{array}\right.$ (5)

where $\displaystyle K^{*}\ll N$ and
$\displaystyle\bm{\alpha^{*}}_{\mathcal{G}^{k}}\neq\bm{\alpha^{*}}_{\mathcal{G}^{l}}$
for $\displaystyle k\neq l$. Intuitively, the above model says that mothers in
the same Group $\displaystyle\mathcal{G}^{k}$ share the same parameter vector
$\displaystyle\bm{\alpha^{*}}_{\mathcal{G}^{k}}$, and mothers in different
groups have parameter vectors that differ from each other.

###  3.2 Methodology

Following Su et al. (2016), the goal is to obtain classifier-Lasso (C-Lasso)
estimates $\displaystyle\bm{\hat{\gamma}}$,
$\displaystyle\bm{\hat{\beta}}_{i}$,
$\displaystyle\bm{\hat{\alpha}}_{\hat{\mathcal{G}}^{k}}$ with the family of
regularized estimators

$\displaystyle\displaystyle\bm{\hat{\gamma}},\bm{\hat{\beta}}_{i},\bm{\hat{\alpha}}_{\hat{\mathcal{G}}^{k}}\in\operatorname*{arg\,min}_{\begin{subarray}{c}\bm{\gamma}\in\mathbb{R}^{q},\\\
\bm{\beta}_{i},\bm{\alpha}_{\mathcal{G}^{k}}\in\mathbb{R}^{p-q}\end{subarray}}\\{Q(\bm{\gamma},\bm{\beta}_{i})+rh[\bm{\beta}_{i},\bm{\alpha}_{\mathcal{G}^{k}}]\\},$
(6)

for an objective function $\displaystyle Q(\cdot)$, a tuning parameter
$\displaystyle r\in[0,\infty)$, and with prior function $\displaystyle
h:\mathbb{R}^{p-q}\rightarrow[0,\infty]$ which is specified here as

$\displaystyle\displaystyle
h[\bm{\beta}_{i},\bm{\alpha}_{\mathcal{G}^{k}}]:=\frac{1}{N}\sum_{i=1}^{N}\prod_{k=1}^{K}||\bm{\beta}_{i}-\bm{\alpha}_{\mathcal{G}^{k}}||_{2},$
(7)

with the remainder of the notation being defined as above. Following Su et al.
(2016), I use fixed effects penalized profile likelihood (PPL) to estimate the
unknown parameter vectors in eq. 6. Therefore, the first term of eq. 6

$\displaystyle\displaystyle Q(\bm{\gamma},\bm{\beta}_{i})=$
$\displaystyle\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}y_{it}\log\Phi(y_{it}-\bm{x}_{11_{it}}^{\top}\bm{\gamma}-\bm{x}_{12_{it}}^{\top}\bm{\beta}_{i}-\hat{\mu}_{i})$
$\displaystyle\displaystyle+(1-y_{it})\log(1-\Phi(y_{it}-\bm{x}_{11_{it}}^{\top}\bm{\gamma}-\bm{x}_{12_{it}}^{\top}\bm{\beta}_{i}-\hat{\mu}_{i}))$
(8)

is the negative profile log-likelihood function and $\displaystyle\Phi(\cdot)$
denotes the conditional normal cumulative distribution function (CDF).444 The
individual fixed effect estimates are obtained by
$\displaystyle\displaystyle\hat{\mu_{i}}\in\underset{\mu_{i}\in\mathbb{R}}{\arg\min}$
$\displaystyle\displaystyle\frac{1}{T}\sum_{t=1}^{T}y_{it}\log\Phi(y_{it}-\bm{x}_{11_{it}}^{\top}\bm{\gamma}-\bm{x}_{12_{it}}^{\top}\bm{\beta}_{i}-\mu_{i})+(1-y_{it})\log(1-\Phi(y_{it}-\bm{x}_{11_{it}}^{\top}\bm{\gamma}-\bm{x}_{12_{it}}^{\top}\bm{\beta}_{i}-\mu_{i})).$
Based on the estimated group structure
$\displaystyle\hat{\mathcal{G}}^{1},\dots,\hat{\mathcal{G}}^{K}$ in eq. 6 and
7, post C-Lasso estimates
$\displaystyle\bm{\hat{\hat{\gamma}}}=(\bm{\hat{\hat{\gamma}}}_{1}^{\top},\dots,\bm{\hat{\hat{\gamma}}}_{K}^{\top})^{\top}\in\mathbb{R}^{qK}$,555Note
that each $\displaystyle\bm{\hat{\hat{\gamma}}}_{k}$ contains the same
estimates because $\displaystyle\bm{\hat{\hat{\gamma}}}_{k}$ is not group-
specific.
$\displaystyle\bm{\hat{\hat{\beta}}}=(\bm{\hat{\hat{\beta}}}_{1}^{\top},\dots,\bm{\hat{\hat{\beta}}}_{N}^{\top})^{\top}\in\mathbb{R}^{(p-q)N}$
and
$\displaystyle\bm{\hat{\hat{\alpha}}}=(\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{1}}^{\top},\dots,\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{K}}^{\top})^{\top}\in\mathbb{R}^{(p-q)K}$
are obtained. In particular, I use fixed effects quasi-maximum likelihood to
estimate the common slope parameters
$\displaystyle\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}$ and set
$\displaystyle\bm{\hat{\hat{\beta}}}_{i}=\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}$
for all $\displaystyle i\in\hat{\mathcal{G}}^{k}$ with
$\displaystyle\hat{\mathcal{G}}^{k}:=\\{i\in\\{1,\dots,N\\}:\bm{\hat{\beta}}_{i}=\bm{\hat{\alpha}}_{\hat{\mathcal{G}^{k}}}\\}$
for $\displaystyle k=1,\dots,K$.

As the true number of groups in eq. 7 is a priori unknown, I follow Su et al.
(2016) and make the dependence of $\displaystyle\bm{\hat{\beta}}_{i}$ and
$\displaystyle\bm{\hat{\alpha}}_{\hat{\mathcal{G}}^{k}}$ on
$\displaystyle[K,r]$ explicit. I choose $\displaystyle K$ and $\displaystyle
r$ in eq. 6 and 7 to minimize the following BIC-type information criterion

$\displaystyle\displaystyle\hat{K}[r]\in\underset{1\leq K\leq
K_{\max}}{\arg\min}\\{\tilde{Q}(\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}[K,r]})+spK\\},$
(9)

where $\displaystyle r$, $\displaystyle p$ and $\displaystyle K$ are defined
as above,
$\displaystyle\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}[K,r]}=[\bm{\hat{\hat{\gamma}}}_{k}\
\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}]\in\mathbb{R}^{p}$ denotes the
vector of post C-Lasso estimates, and $\displaystyle s$ is the tuning
parameter with $\displaystyle s:=1/4\log(\log(T))/T$. I assume that the true
number of groups $\displaystyle K^{*}$ is bounded from above by $\displaystyle
K_{\max}=4$. The first term of eq. 9

$\displaystyle\displaystyle\tilde{Q}(\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}[K,r]})$
$\displaystyle\displaystyle=\frac{2}{NT}\sum_{k=1}^{K}\sum_{i\in\hat{\mathcal{G}}^{k}[K,r]}\sum_{t=1}^{T}y_{it}\log\Phi(y_{it}-\bm{x}_{it}^{\top}\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}[K,r]}-\hat{\mu}_{i}^{\hat{\mathcal{G}}^{k}[K,r]})$
$\displaystyle\displaystyle+(1-y_{it})\log(1-\Phi(y_{it}-\bm{x}_{it}^{\top}\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}[K,r]}-\hat{\mu}_{i}^{\hat{\mathcal{G}}^{k}[K,r]}))$
(10)

denotes again the neagative profile log-likelihood function and
$\displaystyle\hat{\mu}_{i}^{\hat{\mathcal{G}}^{k}[K,r]}$ is a group-specific
individual fixed effect.

##  4 Data

My analysis draws on a novel custom shaped administrative data set based on
social security records made to investigate female employment biographies in
Germany.666This data set (FEMPSO BIRTH) was created by the project 7 (Female
Employment Patterns, Fertility, Labor Market Reforms, and Social Norms: A
Dynamic Treatment Approach) and project 16 (Custom Shaped Administrative Data
for the Analysis of Labour Market (CADAL)) of the priority program 1764 of the
German Science Foundation (DFG). FEMPSO BIRTH became available in 2018. 777The
same data set (FEMPSO BIRTH) is used in another joint project in Klose and
Paul (2020), where we predict the incidence of future first births based on
employment records within a machine learning approach. Reference is also made
to this paper for the data. It is based on two different data sets: employment
records from the Integrated Employment Biographies (IEB) prepared by the
Federal Employment Agency’s Research Data Centre (RDC) at the Institute for
Employment Research and on a subsample of microdata of actively insured
persons compiled by the German Federal Pension Insurance’s RDC. Both data sets
stem from the same source, the registration procedure for social insurance.

The version of the IEB used in this paper comprises all individuals in Germany
that contributed to the social security system between 1975 and 2014 or
obtained transfer payments from the labor agency or were registered as job
seekers.888Not included are civil servants including the majority of teachers
as well as the self-employed. Anonymous weekly subsamples of the data are
accessible for scientific research (e.g., the “SIAB”) and have been used in
several studies on the employment of mothers and the effects of maternity
leaves (see, e.g., Adda et al. (2017); Ejrnæs and Kunze (2013); Schönberg and
Ludsteck (2014)). The IEB provide detailed daily information on wages,
employment, occupations, periods of unemployment, receipt of unemployment
benefit as well as basic sociodemographics. An important advantage of the IEB
data is that, because of the administrative data’s longitudinal nature,
employment histories are measured precisely. The main disadvantage of the IEB
data is that it does not contain direct information on childbirths. The data
only provides information on whether and when a woman takes leave of absence.
This causes potential misspecification errors because not all leave-taking may
be due to maternity leave, but due to other reasons like illness. Moreover,
the child’s birthday has to be approximated as six weeks after the mother
takes leave since mandatory maternity leave entitlements cover six weeks
before and two months after the due date. This may lead to measurement errors
in the child’s month of birth. Most importantly, this identification strategy
cannot capture births by mothers who are out of the workforce at the timepoint
of childbirth.

To address these drawbacks, for the custom-shaped data set, precise
information on childbirths from the German Pension Insurance has been merged
to the IEB. The pension registry gathers all employed persons in the private
and public sectors because for those it is mandatory to make a contribution to
the statutory pension insurance.999Excluded from the data are also the civil
servants (including teachers), most self-employed, and the economically
inactive because they are not covered by statutory pension insurance. Most
importantly, pension accounts provide a fertility record for any woman who has
been ever enrolled in the pension insurance because women receive an automatic
contribution for child-rearing years. This is basically equivalent to having
appeared in the IEB. The fertility information and some information on
employment, wages, and basic sociodemographics are available for a particular
sample conducted by the German Pension Insurance’s RDC.101010Various data
products of the German Pension Insurance’s RDC rely on this sample, which is
referred to as the “Versicherungskontenstichprobe”. It is a representative
stratified sample from all actively insured individuals, drawn in 1983 and
kept up and complemented with younger cohorts since then. See, e.g.,
Kreyenfeld and Mika, 2008. The merge of the two data sets relies on
statistical matching principles since merging based on social security numbers
or names was not possible due to the unavailability in the RDC at the
Institute for Employment Research. For each woman in the pension data, a
statistical twin has been identified in the IEB data based on her year and
month of birth, regional information, and employment records. To achieve an
accurate match, the data is limited to women without a GDR-employment-
biography and with a certain labor market attachment.111111Readers who are
interested in additional details on the match of the two data sets are
referred to the Online Appendix of Klose and Paul (2020). In this paper, I
only focus on the matches that are almost certainly assessed to be correct.
The resulting data set covers information on around 56 000 mothers and the
years 1975 and 2014.

The sample is further restricted to mothers aged at least 17 before giving
birth to their first child. Before age 17, many mothers have not yet entered
the labor market because they are still in education and thus have no
employment records. Finally, to identify the reform effect, I consider mothers
with a first child born between October 1 and December 31, 2006 (control
sample), and mothers with children born between January 1 and March 31, 2007
(treatment sample), respectively. The overall number of observations available
for the analysis equals 785 mothers. The treatment group comprises 387
mothers, and the control group has 398 mothers. The final data set is a
balanced monthly panel that collects maternal employment information up to 72
(+ 1 for the month of birth) months after birth. It consists of West German
mothers who have some attachment to the labor market and have children born
between the last quarter of 2006 and the first quarter of 2007.

Table 2: Descriptives | Pre-Reform Sample |  | Post-Reform Sample
---|---|---|---
Variable |  Mean |  Std. Dev. |  Min |  Max |  |  Mean |  Std. Dev. |  Min |  Max
Age |  27.98 |  7.07 |  17 |  56 |  |  27.83 |  6.98 |  17 |  51
Mother’s age at $\displaystyle 1st$ birth |  30.10 |  4.35 |  19 |  48 |  |  30.80 |  4.29 |  20 |  44
Mother’s age at $\displaystyle 2nd$ birth∗ |  32.87 |  3.95 |  22 |  42 |  |  32.84 |  4.06 |  22 |  42
Mother’s age at $\displaystyle 3rd$ birth+ |  34.52 |  3.98 |  23 |  41 |  |  34.48 |  3.91 |  26 |  42
Employed |  0.4953 |  0.5000 |  0 |  1 |  |  0.4928 |  0.5000 |  0 |  1
Full-time employed |  0.3155 |  0.4647 |  0 |  1 |  |  0.3152 |  0.4646 |  0 |  1
Part-time employed |  0.1194 |  0.3242 |  0 |  1 |  |  0.1270 |  0.3330 |  0 |  1
Marginally employed |  0.0626 |  0.2423 |  0 |  1 |  |  0.0515 |  0.2211 |  0 |  1
Daily wage |  29.07 |  38.96 |  0 |  201.5 |  |  29.86 |  39.49 |  0 |  201.5
Low educated |  0.1964 |  0.3972 |  0 |  1 |  |  0.2203 |  0.4144 |  0 |  1
Middle educated |  0.6485 |  0.4775 |  0 |  1 |  |  0.6270 |  0.4836 |  0 |  1
High educated |  0.1347 |  0.3414 |  0 |  1 |  |  0.1305 |  0.3369 |  0 |  1
Notes: Control sample (2006m10-m12): $\displaystyle N_{C}=398$. Treatment
sample (2007m1-m3): $\displaystyle N_{T}=387$.
Prebirth period included.
∗ 237 (60%) and 245 (63%) mothers have a $\displaystyle 2nd$ child in the
control and treatment samples, respectively.
+ 48 (12%) and 44 (11%) mothers have a $\displaystyle 3rd$ child in the control and treatment samples, respectively.

Table 9 provides descriptive statistics for some key variables by sample. The
monthly information relates to the pre- and post-birth periods, i.e., the
entire available employment biographies of the considered mothers are taken
into account. The summary statistics show that no systematic differences – at
least for the covariates that are agnostic to post-reform reactions – emerge
in the sociodemographic characteristics between the pre- and post-reform
samples.121212Balancing tests on the pre-birth covariates, similar to those of
Kluve and Schmitz (2018), are omitted because the sample size in the present
study is much smaller than the mentioned study and thus more prone to random
variations, which would vanish with increasing sample size. This pattern
confirms the quasi-random assignment identification strategy suggested by
Kluve and Schmitz (2018). In the majority of cases, mothers have a vocational
degree (women with a birth in 2006 Q4/2007 Q1: 65%/63%), in around 13%/13% of
cases women with a birth in 2006 Q4/2007 Q1 are tertiary educated, and, in
20%/22% of cases, women have no postsecondary degree. The average employment
rate over the considered time span is 49%/50% for women with a birth in 2006
Q4/2007 Q1. The employment rate is highest for full-time work (women with a
birth in 2006 Q4/2007 Q1: 32%/32%), followed by part-time work (women with a
birth in 2006 Q4/2007 Q1: 12%/13%) and marginal employment (women with a birth
in 2006 Q4/2007 Q1: 6%/5%). The average age in both samples is between 27.8
and 28 years, and between 30.1 and 30.8 years at first birth. 60%/63% of the
mothers have a second child, and 12%/11% of the mothers have a third child in
the pre-/post-reform sample. The average daily wage among working and non-
working women is around 29 EUR/30 EUR for the pre-/post-reform sample.

Figure 2: Changes in the Employment Structure induced by the Reform

Average employment rateMonths since birth$\displaystyle 24$$\displaystyle
48$$\displaystyle 72$$\displaystyle 10\%$$\displaystyle 20\%$$\displaystyle
30\%$$\displaystyle 40\%$$\displaystyle 50\%$$\displaystyle
60\%$$\displaystyle 70\%$$\displaystyle 80\%$$\displaystyle
90\%$$\displaystyle 100\%$Phase 1Phase 2Phase 3Phase
414601460$\displaystyle-\Delta$$\displaystyle-\Delta$$\displaystyle+\Delta$$\displaystyle-\Delta$

Notes: Light gray: Control sample. Blue: Treatment sample.

Figure 2 descriptively summarizes the changes in the employment structure
induced by the reform. The $\displaystyle x$-axis indicates the time since
childbirth provided in months. Similar to Kluve and Schmitz (2018), four
phases emerge: the first phase (from 1 to 14 months) covers the periods of the
mandatory maternity leave and the parental benefit receipt. The second phase
(up to 24 months) covers the second year after childbirth. The third phase
(from 25 to 60 months) covers years three to five after childbirth. The last
phase (up to 72 months) comprises the sixth year after childbirth, which
denotes the reference group in the estimations later on. The $\displaystyle
y$-axis denotes the average employment rate in percent. The employment
situations before and after the reform are depicted in light gray and blue,
respectively. Figure 2 confirms the evidence from previous research (e.g.,
Greyer et al. (2015), Kluve and Tamm (2013), Kluve and Schmitz (2018)): Phase
1 leads to a significant drop in maternal employment rates because of the
presence of the parental benefit, which induces on average a large delta in
benefit levels during the first 14 months after childbirth. Phase 2 shows only
a slight, almost negligible decrease in the average employment rate of
mothers. In Phase 3, however, the average maternal employment rate increases.
The last phase again shows a decrease in the average employment rate.

Finally, as a data preprocessing step, I drop all samples from the data that
do not have any variation in the dependent variable (either
$\displaystyle\frac{1}{72}\sum_{t=0}^{72}y_{it}=0$ or
$\displaystyle\frac{1}{72}\sum_{t=0}^{72}y_{it}=1$ for $\displaystyle
i\in\\{1,\dots,398\ (387)\\}$), as they would lead to infinite
($\displaystyle\pm\infty$) fixed effects in the fixed effects probit model
considered here.131313Refer to the Online Appendix for more information on
this specification problem. Overall, this results in sample sizes of
$\displaystyle N_{C}=329$ (control group) and $\displaystyle N_{T}=333$
(treatment group) available for the analysis.

##  5 Empirical Results

As previously explained, I provide estimates of the heterogeneous employment
effects up to six years after childbirth by exploiting the latent group
structures in mothers’ return behavior to employment. Moreover, I estimate the
policy impacts of the parental benefit within the different groups of mothers
by identifying the reform effect similar to Kluve and Schmitz (2018). Before
turning to my main analysis, however, I first introduce the empirical
approach, and in a second step, I also report empirical estimates from the
traditional homogenous panel data model similar to that reported by Hyslop
(1999).

###  5.1 Empirical approach

To investigate maternal employment after the first birth, I split the child
age effect into four phases. The first phase comprises the period of mandatory
maternal protection immediately after childbirth (up to 2 months after birth)
and the parental benefit receipt (3.-14. month after birth). The second phase
(up to 24 months) covers the second year after childbirth and is the short-run
perspective. The medium-run perspective contains years three to five after
childbirth and denotes the third phase. The fourth phase covers the sixth year
after childbirth.

The empirical benchmark model has the following form:

$\displaystyle\displaystyle\text{employ}_{it}$
$\displaystyle\displaystyle=\mathbf{1}\\{\alpha\ \text{employ}_{i,t-1}+\beta\
\text{child}_{\text{2nd}_{it}}+\gamma\
\text{child}_{\text{m1-m14}_{it}}+\delta\
\text{child}_{\text{m15-m24}_{it}}+\eta\ \text{child}_{\text{m25-m59}_{it}}$
$\displaystyle\displaystyle+\zeta_{i}-\epsilon_{it}\geq 0\\},$ (11)

where $\displaystyle employ_{it}$ is the employment indicator, $\displaystyle
employ_{i,t-1}$ is the employment indicator of the previous period,
$\displaystyle child_{2nd_{it}}$ is a dummy variable controlling for a second
child, $\displaystyle child_{m1-m14_{it}}$, $\displaystyle
child_{m15-m24_{it}}$ and $\displaystyle child_{m25-m59_{it}}$ are the child
age indicators for the first child, $\displaystyle\zeta_{i}$ is an individual
fixed effect and $\displaystyle\epsilon_{it}$ is an idiosyncratic error term.
$\displaystyle child_{m60-m72_{it}}$ is the omitted reference group. The three
child age indicators’ estimates measure the effect during the benefit receipt,
the short-run effect immediately after benefit exhaustion, and the medium-run
effect three to five years after childbirth, respectively. I do not include
time-constant covariates such as education level as they are absorbed in the
fixed effects. By combining this baseline specification with the panel
structure formulation of Su et al. (2016) that allows individual-specific
coefficients, I consider the following two specifications:

$\displaystyle\displaystyle\text{employ}_{it}$
$\displaystyle\displaystyle=\mathbf{1}\\{\alpha\ \text{employ}_{i,t-1}+\beta\
\text{child}_{\text{2nd}_{it}}+\gamma_{i}\
\text{child}_{\text{m1-m14}_{it}}+\delta_{i}\
\text{child}_{\text{m15-m24}_{it}}+\eta_{i}\
\text{child}_{\text{m25-m59}_{it}}$
$\displaystyle\displaystyle+\zeta_{i}-\epsilon_{it}\geq 0\\},$ (12)

and

$\displaystyle\displaystyle\text{employ}_{it}$
$\displaystyle\displaystyle=\mathbf{1}\\{\alpha_{i}\
\text{employ}_{i,t-1}+\beta_{i}\ \text{child}_{\text{2nd}_{it}}+\gamma_{i}\
\text{child}_{\text{m1-m14}_{it}}+\delta_{i}\
\text{child}_{\text{m15-m24}_{it}}+\eta_{i}\
\text{child}_{\text{m25-m59}_{it}}$
$\displaystyle\displaystyle+\zeta_{i}-\epsilon_{it}\geq 0\\},$ (13)

which allow for individual specific slope coefficients
($\displaystyle\alpha_{i},\beta_{i},\gamma_{i},\delta_{i},\eta_{i}$).
Following Su et al. (2016), I allow the parameters
($\displaystyle\gamma_{i},\delta_{i},\eta_{i}$) in 5.1 and
($\displaystyle\alpha_{i},\beta_{i},\gamma_{i},\delta_{i},\eta_{i}$) in 5.1 to
follow certain latent group structures. Equation 5.1 is an extension to mixed
panel structure models, where the parameters of lagged employment and the
second child are constrained to be common across all mothers, whereas the
child age parameters of the first child are assumed to be group-specific.

To identify the policy impacts of the parental leave reform for all mothers
and the identified groups, I use a design idea that builds on previous work
(Kluve and Schmitz (2018), Kluve and Tamm (2013)). Kluve and Schmitz (2018)
adopt a sharp RD design that compares the subsequent labor supply behavior of
mothers who gave childbirth shortly before the reform with mothers who gave
birth shortly after the reform. Because the authors argue that the quasi-
random assignment to treatment and control groups is achieved within a range
of three months before and after the discontinuity point of the 1st January
2007, I consider all mothers with first children born between the 1st October
2006 and 31st March 2007. Mothers with children born between the 1st October
and 31st December 2006 belong to the control sample, and mothers with children
born between the 1st January and 31st March 2007 belong to the treatment
sample.

Perhaps the most straightforward way to model the reform effect within a sharp
regression discontinuity design is to include an indicator for the treatment
group and to run a linear regression on the entire sample. The treatment
indicator captures the average causal impact of the reform on employment.
However, in the context of a fixed effect approach, it is not possible to
include time-constant covariates as they are absorbed in the fixed effects.
Since the entitlement to parental allowance is constant over time (mothers are
either eligible or not), such a modeling strategy seems
inappropriate.141414One possibility to solve this problem is the interaction
of the reform indicator with the child age indicators. The interaction terms
capture the employment effects of the reform within the different child age
phases. Therefore, I estimate the reform effect indirectly by separately
running C-Lasso estimation for the treatment and control samples. Although I
cannot provide direct causal evidence on how an increase in paid parental
leave for a post-reform baby can affect maternal employment, I can apply an
identification strategy that compares the treatment and control samples in
terms of differences in the average predicted employment probabilities. To
exploit these differences, I compute predictions of the average employment
probabilities:

$\displaystyle\displaystyle\overline{\mbox{$\displaystyle\text{employ}$\raisebox{5.69054pt}{}}}_{j}=1/(N[\hat{\mathcal{G}}^{k}]T[p^{j}])\
G(\bm{x}_{i[\hat{\mathcal{G}}^{k}]t[p^{j}]}^{\top}\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}+\hat{\mu}_{i}[\hat{\mathcal{G}}^{k}]),$
(14)

where $\displaystyle G(\cdot)$ denotes the CDF of the standard normal,
$\displaystyle p^{j},\ j=1,\dots,4$ denotes the four different phases of child
age, $\displaystyle[\cdot]$ denotes the restriction to the group-specific
subspace, and the remainder is defined as in the methodology
section.151515Note that for computing the predictions of the average
employment probabilities, I do not take into account the effect of the second
child. Then, I take the differences in the control- and treatment-specific
average employment probability predictions. These differences
$\displaystyle\Delta_{\text{T-C}_{j}}$ capture the impact of the reform on
maternal employment within the different child age phases. Finally, I test if
the differences in the predicted probabilities are statistically different
from zero:161616cf. Long (2009) for performing group comparisons in nonlinear
models using predicted probabilities.

$\displaystyle\displaystyle
H_{0}:\frac{\Delta_{T-C}}{\sqrt{\frac{\sigma^{2}_{T}}{N[\hat{\mathcal{G}}^{k}]_{T}}+\frac{\sigma^{2}_{C}}{N[\hat{\mathcal{G}}^{k}]_{C}}}}=0,$
(15)

where the plug-in principle is used to estimate the variances of the predicted
probabilities. Since I test the employment probability predictions of two
independent samples, the covariance term drops out in the denominator of eq.
15.

The advantage of this identification strategy is that I can disentangle the
latent effect heterogeneity of the reform utilizing data-determined grouping.
To test the differences in the predicted employment probabilities of the
respective control and treatment pairs and to give them an interpretation in
causal direction, strong assumptions are required. First, it is assumed that
the C-Lasso identifies the same number of groups for the control and treatment
samples. Second, it is assumed that the C-Lasso identifies the same drivers of
unobserved heterogeneity in the respective control and treatment pairs. Only
in this case, the respective control group constitutes a valid counterfactual
situation for the treatment group. Finally, it is assumed that the
statistically significant differences in the predicted employment
probabilities can be interpreted in a causal direction. In the following
sections, I work under these assumptions. Moreover, in the Online Appendix, I
develop a distance-based criterion that should identify the associated control
and treatment pairs under which the assumptions are most likely to hold.

###  5.2 Baseline estimation

Before reporting my estimation results of Eq. 5.1, in Table 5, I provide
baseline estimates for maternal employment rates from the standard Linear
Probability Model (LPM) with fixed effects.171717In the Appendix in Table 6, I
also provide baseline estimates from standard LPM without fixed effects.

[Insert Table 5 here.]

As columns (1), (2), (4), and (5) of Table 5 show, the results for the Phases
I, II, and III show the same pattern identified in previous research (e.g.,
Kluve and Schmitz (2018), Schönberg and Ludsteck, 2014): once for individual-
specific effects, lagged employment and a potential second child is
controlled, mothers in the treatment sample display a statistically
significant lower average probability of being employed during the first 14
months after childbirth compared to the control sample.181818Testing H0:
child${}_{{\text{m1-m14}_{it}}_{T}}-$ child${}_{{\text{m1-m14}_{it}}_{C}}=0$
results in a small $\displaystyle p$-value. cf. columns (4) and (5) of the
lower part of Table 5. Indeed, having a child aged between 1 and 14 months
instead of a child aged between 60 and 72 months reduces the probability of
maternal employment on average by 11.4 percentage points for the treatment
sample holding constant all other factors. In contrast, the control sample
displays an 8.4 percentage point lower employment probability on average. For
Phases II and III, I cannot find statistically significant differences between
the control and treatment samples. In fact, for mothers of the treatment
sample, having a child aged between 25 and 59 months instead of a child aged
between 60 and 72 months reduces the probability of employment on average by
2.2 percentage points holding constant all other factors. I find a 2.3
percentage points lower employment probability on average for corresponding
mothers of the control sample. For an ‘in-group’ comparison, I additionally
test whether the three child age estimates within the pre-/post-reform sample
are statistically different from each other. In the upper part of Table 5,
columns (3)-(5) provide the null hypothesis and the respective $\displaystyle
p$-value for testing the linear combinations. The $\displaystyle p$-values
show that all estimates are statistically different from each other. Although
my results confirm the evidence from previous studies in the short run, I do
not get evidence of pronounced beneficial medium-run labor market effects of
the parental benefit reform identified by Kluve and Schmitz (2018).

Next, I estimate Eq. 5.1 and report the results for the coefficient estimates
and the corresponding marginal effects in columns (1)-(2) and (3)-(4) of
Tables 7 and 9.

[Insert Table 7 and Table 9 here.]

Columns (1) and (2) of Table 7 provide coefficient estimates from a dynamic
probit FE model without bias correction. The estimates in columns (3) and (4)
are bias-corrected by the half-panel jackknife Dhaene and Jochmans (2015). All
standard errors are clustered at the individual level. In line with the
evidence from previous research (e.g., Carrasco (2001), Hyslop (1999),Michaud
and Tatsiramos (2011)), employment is highly persistent, and its association
with the child age effects remains robust across LPM FE, Probit FE, and Probit
FE JK. For both the control and treatment samples, the marginal effects on
lagged employment show that mothers, who were employed the previous month
instead of not employed, are, on average, about 80 percentage points more
likely to be employed this month when holding constant all other factors. The
marginal effects derived from the FE probit estimates are all statistically
different from zero and similar to those obtained by classical LPM with fixed
effects. My results for the control sample show that having a first child in
Phases I, II, and III instead of having a first child in Phase IV decreases
maternal employment probability on average by 13.8, 8.8, and 3.7 percentage
points holding constant all other factors. The corresponding estimates for the
treatment sample are -18.1, -9.0, and -3.7. The $\displaystyle t$-statistics
in column (1) of Table 8 also show that all post-Lasso-PPL estimates of the
child age phases are statistically different from each other.

[Insert Table 8 here.]

Besides, I test whether the differences in the child age effects for the
treatment and control samples are statistically different from zero.
Conducting tests for group comparisons like in linear regressions can lead to
incorrect conclusions in nonlinear models such as probit since they mix the
regression coefficients with the residual variation Allison (1999). Allison
(1999) suggests a test that removes the effect of residual variation. However,
the author works under the strong assumption that at least one coefficient is
the same in both groups. Therefore, I follow Long (2009) and conduct tests of
predicted probabilities – as given in eq. 15 – for comparing control and
treatment groups. The advantage is that the predicted probabilities are
unaffected by group differences in residual variation. Columns (1)-(3) of
Table 10 report the predictions of average employment probabilities for the
four different phases of child age for the control and treatment samples,
their respective differences, and the $\displaystyle p$-values for the test in
eq. 15.191919The average employment predictions show the counterfactual
situation in which mothers do not have a second child, i.e., the predictions
for the four child age phases are purely driven by a first child and lagged
employment.

[Insert Table 10 here.]

The results of the average employment predictions are in line with the
expected labor market effects after childbirth: For both the control and
treatment samples, the employment predictions are gradually increasing with
the age of the first child. The parental benefit reform’s direct policy effect
on maternal employment is quite pronounced: mothers in the treatment sample
display a 10 percentage points lower average employment rate (statistically
significant) during the first 14 months after childbirth compared to mothers
of the control sample. There are small differences in the average employment
rates of the pre- and post-reform mothers in Phases II and IV. In Phase III,
there is no statistically significant difference. Overall, for the post-reform
sample, I find a small and statistically significant drop in the average
employment probability over the entire sample period, driven mainly by the
parental benefit reform’s direct policy effect.

One drawback of the baseline estimation from above is that it remains silent
about the latent heterogeneity in maternal labor supply dynamics. However, it
is precisely this latent heterogeneity that opens up a new perspective on the
ongoing debate on the impact of childbirth on maternal employment. Therefore,
I next want to identify the latent structures in the mothers’ return-to-
employment behavior after first birth by using the framework of the panel
structure model in Section 3. This approach allows the first child’s age,
along with the second child and lagged employment, to have heterogeneous
effects across mothers, which are classified into groups with the C-Lasso
technique. Such heterogeneous employment effects across mothers in childbirth
response have useful implications for policymakers in designing efficient
parental leave legislation.

###  5.3 Results for heterogeneous employment effects

#### Mixed Panel Structure model

First, I use the C-Lasso approach suggested by Su et al. (2016) to identify
the latent group structure in eq. 5.1. Regarding the C-Lasso tuning parameter,
I choose $\displaystyle\lambda=c_{\lambda}\times s^{2}_{employ}\times
T^{-1/3}$, where $\displaystyle s^{2}_{employ}$ is the sample variance of the
employment indicator, $\displaystyle T$ is the number of time
periods,202020Throughout the estimations, I consider $\displaystyle T=73$. and
$\displaystyle c_{\lambda}$ is element of a 10 points geometrically increasing
sequence $\displaystyle(0.01,\dots,0.1)$. I allow for a maximum number of 4
groups. I calculate the information criterion value for each combination of
the number of groups $\displaystyle K$ and the tuning parameter $\displaystyle
c_{\lambda}$ according to eq. 9 of the methodology section. The number of
groups and the tuning parameter are chosen by minimizing the information
criterion from the set of all possible combinations. Fig. 3 shows that the
lowest point of the information criterion is obtained when $\displaystyle K=3$
and $\displaystyle c_{\lambda}=0.0129$ for the control group, and
$\displaystyle K=3$ and $\displaystyle c_{\lambda}=0.01$ for the treatment
group.

Figure 3: The horizontal axis depicts the number of groups and the vertical
axis depicts the corresponding information criterion value.

Applying C-Lasso to the control ($\displaystyle 2006m10-m12$) and treatment
samples ($\displaystyle 2007m1-m3$), I identify three latent groups for both
samples. Columns (5)-(10) of Tables 7 and 9 report the post Lasso-PPL
coefficient estimates and the corresponding marginal effects for each group.
All post-Lasso-PPL estimates are bias-corrected by the half-panel jackknife,
and the standard errors are again clustered at the individual level. Tables 7
and 9 suggest that the estimates for the slope coefficient on lagged
employment are relatively stable across the three groups. Moreover, the
estimates on the lagged employment variable are quite large in magnitude,
indicating a strong persistence in labor supply decisions. In contrast, the
estimates of the slope coefficients of the impact of child age on employment
vary substantially across the three identified groups and even alter the
signs. The $\displaystyle t$-statistics reported in Table 8 show that for
Group 3, all post-Lasso estimates of the child age phases are statistically
different from each other. However, for Group 1 and Group 2, I cannot confirm
that all coefficient estimates are statistically different from each other.

Before disentangling the reform effect, I report the group-specific marginal
effects obtained by the C-Lasso. The marginal effects derived from the group-
specific post-Lasso-PPL estimates are all statistically significant. Starting
with the interpretation of the fourth child age phase, I identify a quite
heterogeneous pattern for the control and treatment samples: a group of
mothers with a very low estimated employment probability (Group 1: 9.4%/8.9%),
a group of mothers with a moderate estimated employment probability (Group 2:
33.9%/31%) and a group of mothers with a high estimated employment probability
(Group 3: 59.4%/58%) when having a child aged between 60 and 72 months and is
not employed previous month. My results for the control and treatment samples
of Group 1 show that having a child aged between 1 and 14 months instead of a
child aged between 60 and 72 months decreases, and a child aged between 15 and
24 months (/ between 25 and 59 months) increases maternal employment by on
average 0.59/1.27, 6.2/5.5 and 4.5/8.7 percentage points holding constant all
other factors. The corresponding estimates for Group 2 are -11.9/-18,
-6.1/-5.4, and -6.1/-7. Finally, for the control and treatment samples of
Group 3, I find that having a child aged between 1 and 14 (/ between 15 and 24
/ between 25 and 59) months instead of a child aged between 60 and 72 months
decreases maternal employment by on average 44.3/43, 29.3/24.3 and 11.5/8.9
percentage points holding constant all other factors. Overall, the marginal
effects of the child age phases from the C-Lasso estimation suggest that
mothers’ identified groups show a highly heterogeneous dynamic in the return-
to-employment behavior. I can identify three different cluster groups, which
are sufficiently distinct, both in statistical terms and in terms of allowing
for a meaningful interpretation.

Columns (4)-(12) of Table 10 report the group-specific predictions of the
average employment probabilities for the four child age phases for the control
and treatment samples, their corresponding differences
$\displaystyle\Delta_{T-C}$ and the $\displaystyle p$-values of the test in
square brackets underneath. In an attempt to interpret the cluster-groups
based on the magnitude of the predicted employment probabilities and the
evidence from previous literature, a clear pattern arises: Group 1 consists of
mothers who are, on average, less employed than the mothers of the other two
groups. Moreover, this cluster has a mobile dynamic of employment as it
switches from less to more to the less average employment rate. Group 2 could
consist of work-oriented mothers who decide immediately after benefit
exhaustion to return to employment. Group 3 can be interpreted as a cluster of
mothers who take an extended family break after childbearing, which goes far
beyond the government-subsidized maternity leave period. However, on the other
hand, most mothers of Group 3 have returned to employment until the age of six
of the first child, i.e., most mothers return to employment during the 36
months of post-birth job protection.

Next, I test the group-specific differences of the four child age phases for
statistical significance. I again conduct tests of predicted probabilities to
compare the three identified groups’ respective control and treatment samples.
First, it is noticeable that the parental benefit policy affects the three
groups differently. In fact, policy reform affects Group 2 the most. The
direct and the indirect policy effects of the parental benefit reform on
maternal employment are quite pronounced because I get evidence of significant
average employment probability drops, ranging between 5-23 percentage
points.212121Here, I am careful to interpret these effects as causal, as this
only works under strict identification assumptions. For Group 1, I also find
large negative direct and indirect policy effects, but on the other hand, I
find no drop in predicted employment in the medium-run. This pattern has the
consequence that I find an overall smaller drop in the average employment
probability (4.8 p.p.) compared to the second group (9 p.p.). For Group 3, I
even find a much smaller drop in average employment probability of 3.3
percentage points during Phase I and a pronounced increase in employment
probabilities of 5.1 and 3.1 percentage points in Phases II and III. This
leads to a statistically significant increase in average employment
probability by 1.2 percentage points over the entire period (72 months after
birth). Fig. 4 shows the reform effect for the three identified groups over
the entire period.

Figure 4:

Finally, I discuss some descriptive statistics of the three groups that allow
for further interesting insights from the C-Lasso group classification. Table
3 reports the number of mothers selected in each group. The C-Lasso method
classifies over 50% of the considered mothers in Group 3 for both the control
and treatment samples. Only about 20% of mothers are in the first group and
almost 30% in the second group. My finding confirms the evidence of previous
literature (e.g., Schönberg and Ludsteck (2014)) that most mothers make use of
the generous government-provided partially paid leave in Germany to spend more
time with their infants after childbirth.222222In Germany, women are currently
eligible for three years of partially paid and job-protected leave. Maternal
employment is lowered only in the short- and medium-run. Almost all mothers of
Group 3 return back to employment with the exhaustion of government-provided
job protection. However, there is still a significant proportion of mothers
with a more atypical pattern of return to employment that has not been
identified by the quasi-experimental literature. Indeed, 20% of mothers decide
to work much less on average than the other cluster-groups. On the other hand,
30% of mothers have a permanently high average employment level after the
first child age phase.

Table 3: Number of observations of the heterogenous groups | Number of observations
---|---
| Full sample |  | Group 1 |  | Group 2 |  | Group 3
by |  Control |  Treatment |  |  Control |  Treatment |  |  Control |  Treatment |  |  Control |  Treatment
# obs. |  329 (100 %) |  333 (100 %) |  |  65 (20 %) |  68 (20 %) |  |  95 (29 %) |  83 (25 %) |  |  169 (51 %) |  182 (55 %)
Table 4: Descriptives of $\displaystyle 2nd$ child | Number of observations
---|---
| Total |  | Group 1 |  | Group 2 |  | Group 3
by |  Control |  Treatment |  |  Control |  Treatment |  |  Control |  Treatment |  |  Control |  Treatment
child${}_{\text{m1-m14}_{it}}$ |  11 |  12 |  |  1 |  3 |  |  2 |  0 |  |  8 |  9
child${}_{\text{m15-m24}_{it}}$ |  33 |  45 |  |  7 |  11 |  |  10 |  6 |  |  16 |  28
child${}_{\text{m25-m59}_{it}}$ |  110 |  121 |  |  22 |  33 |  |  43 |  45 |  |  45 |  43
child${}_{\text{m60-m72}_{it}}$ |  18 |  21 |  |  10 |  8 |  |  6 |  8 |  |  2 |  5
$\displaystyle\Sigma$ |  172 (52 %) |  199 (60 %) |  |  40 (62 %) |  55 (81 %) |  |  61 (64 %) |  59 (71 %) |  |  71 (42 %) |  85 (47 %)
* The percentages reported here set the absolute numbers in relation to the number of observations
in each group.

Frühwirth-Schnatter et al. (2016) have also identified cluster groups similar
to those in this paper. The authors obtained five different clusters of
mothers: a ‘high wage’ cluster characterized by a quick return to employment,
a cluster of ‘mobile careers’ switching in and out of employment several
times, an ‘extended family break’ cluster, a ‘drop out of the labor force’
cluster and a ‘part-time working’ cluster. I cannot identify an ‘out of labor
force’ cluster similar to that of Frühwirth-Schnatter et al. (2016) because I
exclude mothers from my sample who are never employed after their first
birth.232323Mothers without employment in the sample period are dropped from
the data since they are associated with infinite ($\displaystyle\infty$ or
$\displaystyle-\infty$) fixed effects in a FE probit model. cf. the Appendix
for more details on this issue. A ‘part-time’ cluster can also not be
identified within my framework because, unlike Frühwirth-Schnatter et al.
(2016), I do not consider a multinomial model. However, in contrast to
Frühwirth-Schnatter et al. (2016), I can determine the group classification
and estimation of the coefficients simultaneously. The approach of Frühwirth-
Schnatter et al. (2016), on the other hand, is purely descriptive, i.e., they
do not obtain coefficient estimates from their Bayesian classification method.

The estimation results of the average employment probability predictions
confirm the parental benefit reform’s expected labor market effects in 2007.
In line with the previous literature (e.g., Kluve and Schmitz (2018), Kluve
and Tamm (2013)), I find that the generous income replacement of the parental
benefit reform generates a strong disincentive-to-work effect up to 14 months
after childbirth. Kluve and Schmitz (2018) has shown that this strong
disincentive-to-work effect is especially driven by those mothers who
financially benefited from the reform. Their results show that the effects on
the employment probability of mothers are positive and significant. Overall, I
can identify three cluster groups with similar group-specific estimates for
both the control and treatment samples. Thus, the parental benefit reform does
not seem to change the composition of groups. Nevertheless, if the estimates
within each group between control and treatment samples are compared, it seems
that the parental benefit reform changes how the age of a child within each
group influences maternal employment. My results also confirm that the
disincentive-to-work effect during Phase I is heterogeneous across groups and
strongest for Group 1 and Group 2. However, for the third group, on the other
hand, I find evidence of a significant increase in employment rates during
Phase II and Phase III. Kluve and Schmitz (2018) also found evidence of
pronounced beneficial medium run labor market effects of the parental benefit,
especially for mothers from the upper tercentile of the income distribution.
My findings suggest that the mothers considered here exhibit enormous
diversities in both the return-to-employment behavior and the response to the
parental benefit reform. In contrast to the recent literature (e.g., Kluve and
Schmitz (2018)), the diversity identified here is generated by unobserved
heterogeneity in the effects of child age on employment. A conclusion on the
main drivers, similar to an analysis based on observable heterogeneity, is
therefore not possible.

If I pool groups 1-3 together and estimate the model in 5.1, I find that the
pooled estimate can be interpreted as a weighted average of the three group-
specific estimates. However, this weighted average remains silent about the
latent heterogeneous pattern in the effects of child age on maternal
employment provided by the C-Lasso method. In particular, estimating average
effects hide important and policy-relevant differences in the employment
trajectories. As we have already seen, the reform generates strong negative
effects for Group 2, whereas it encourages the mothers of Group 3 to take up
employment immediately after benefit exhaustion. My findings confirm the
importance of considering latent structures in the data, which may help design
an efficient parental leave policy.

#### Robustness checks

In this subsection, I present two further specifications exploring the
robustness of the main estimates of the heterogeneous employment effects on
the functional form and panel structure model chosen. First, specification A
runs C-Lasso for a nonlinear fixed effects model without controlling for the
dynamic component and the $\displaystyle 2nd$ child variable. In a second
robustness check, I estimate the mixed panel structure model from above
without controlling for the $\displaystyle 2nd$ child (specification B).

I check for the robustness of my findings using alternative identifying
assumptions because applied researchers using panel data are often confronted
with the challenge of choosing between dynamic and fixed-effects models. They
usually avoid using both identifications in one model like in my approach
since the conditions for consistently estimating causal effects are much more
demanding than those required with lagged dependent variables or fixed effects
alone.242424The challenge of consistent estimation in a dynamic nonlinear
fixed-effect model can be easily seen when taking the first difference to
eliminate the fixed effects and then estimating the differenced model with
OLS. The parameter estimates are inconsistent as the differenced residual is
correlated by construction with the lagged dependent variable, a problem first
noted by Nickell (1981). Even though I apply the split-panel jackknife method
suggested by Dhaene and Jochmans (2015), which is explicitly designed to
handle dynamics in the data and to reduce the bias induced by the incidental
parameter problem (Neyman and Scott (1948)), I would like to find out if I can
find broadly similar results using a plausible alternative specification.

For the first robustness check, I adapt the panel structure model in 5.1 by
dropping $\displaystyle employ_{i,t-1}$ and $\displaystyle child_{2nd_{it}}$
from the model. Then I again use the C-Lasso to identify latent group
structures in the model. By applying C-Lasso to the treatment sample, I still
find three latent groups for the adapted model in 5.1. For the control sample,
even four latent groups are identified.252525For reasons of the empirical
approach’s assumptions, however, only results for $\displaystyle K=3$ are
reported. Figure 5 plots the information criterion function where the vertical
and horizontal axes mark the information criterion values and the number of
groups, respectively. The lowest point of the information criterion is
achieved when $\displaystyle K=4$ and $\displaystyle c_{\lambda}=0.0215$ for
the control sample, and $\displaystyle K=3$ and $\displaystyle
c_{\lambda}=0.0215$ for the treatment sample.

[Insert Figure 5 here.]

The upper panel of Table 11 reports the estimation results of the post-Lasso-
PPL coefficients for each group in model 5.1 and the pooled version of this
model.

[Insert Table 11, Table 12 and Table 13 here.]

The corresponding marginal effects and predicted average employment
probabilities are found in the upper panels of Tables 12 and 13. The tables
show that the identified groups are sufficiently distinct in terms of
estimated child age effects and similar to those obtained by the main mixed
panel structure model reported in the previous subsection. The estimation
results suggest that the group structure is stable when all child age
parameters $\displaystyle\gamma_{i},\delta_{i}$, and $\displaystyle\eta_{i}$
are again allowed to be heterogeneous across mothers, and the dynamic
component and the $\displaystyle 2nd$ child variable are omitted from the
model. In particular, I again identify a group of mothers (Group 1) who are
faced with overall lower predicted employment probabilities than the other two
groups, a group of mothers (Group 2) who timely return to employment after
childbirth, and a group of mothers (Group 3) taking an extended family break
before returning to employment. The only difference I can state is that I get
lower predicted average employment probabilities for Group 1 and larger
negative child age effects, obviously driven by the two omitted variables. The
robustness check confirms that there should be no major concern regarding the
specification of a dynamic nonlinear fixed effects model used in this paper.

The estimation results of specification B are reported in the lower panels of
Table 11, Table 12 and Table 13. As shown in Figure 5, the information
criterion again selects three groups for the control and treatment samples.
The three identified groups are again fairly similar to those obtained in the
main estimation results. This means that there should be no major concern that
subsequent fertility decisions (here for a second child) could be the main
drivers of the C-Lasso group classification.

The results obtained by the mixed panel structure model and the two robustness
checks are in stark contrast to the conclusions of Kluve and Schmitz (2018).
The authors argue that the new parental benefit reform defines an
institutionalized point in time at which mothers go back to work. In Germany,
unpaid parental leave entitlements cover the time period up to three years
after childbirth Schönberg and Ludsteck (2014). According to Kluve and Schmitz
(2018), the exact time point to return to the job was an individual decision
under the previous regulation. However, after the policy change, many working
mothers return to the job at the end of the maximum benefit receipt. The
authors found evidence that the benefit reform induced a strong homogenization
of time spent on paid parental benefit. My results, however, suggest that the
parental benefit reform does not change the composition of groups but only
changes how the age of a child within each group influences maternal
employment. My findings reveal - pre- and post-reform - three highly
heterogeneous groups of maternal return-to-employment behavior. It is
noticeable that the largest group of mothers (Group 3) takes an extended
family break after childbirth, which goes far beyond the government-provided
paid leave.

##  6 Conclusion

In this paper, I identify latent structures in maternal employment using the
C-Lasso methodology, a new data-driven classification approach proposed by Su
et al. (2016). In particular, I assess how the introduction of the generous
German parental benefit reform in 2007 affects mothers’ different clusters. By
exploiting an identification strategy that combines the sharp RD approach and
hypothesis testing of predicted employment probabilities, I can identify the
impacts of the policy reform on employment within the different groups.

Using a new data basis of German custom shaped administrative employment
records, the C-Lasso mechanism identifies three different cluster groups pre-
and post-reform. My results not only reveal marked heterogeneity in the effect
of child age on employment across mothers but also that the German policy
change affects the employment patterns of the three cluster groups
differently. Especially noteworthy is that under the new policy, the largest
group of mothers is faced with comparable small negative employment changes
during the benefit receipt and even exhibits an increase in average employment
rates after exhaustion of government-provided transfers. But the findings also
reveal that under the new policy, one group of mothers suffers from large and
long-lasting negative employment changes after childbirth. The data-determined
grouping results provide some new insights about potential impacts of child
age on maternal employment in general, and more specifically, how the reform
affects these heterogeneous groups. My results suggest that the identified
latent heterogeneity in maternal employment should be taken into account by
policymakers to design efficient parental leave legislation. My evidence also
has important policy implications beyond the German case, as latent group
structures in maternal employment are likely to be found in other countries as
well.

Finally, I note that employment losses of women associated with first
childbirth are still quite pronounced in Germany and that for Group 1 and
Group 2, the parental leave reform is not able to compensate for the strong
disincentive-to-work effect in Phase I by providing a sufficiently strong
incentive-to-work effect after benefit exhaustion. Therefore, a policy should
particularly pay attention to these two groups, which involve around 50% of
mothers.

## References

  * Adda et al. (2017) J. Adda, C. Dustmann, and K. Stevens.  The career costs of children.  _Journal of Political Economy_ , 125(2):293–337, 2017.
  * Allison (1999) P. D. Allison.  Comparing logit and probit coefficients across groups.  _Sociological Methods and Research_ , 28:186–208, 1999\.
  * Angrist and Evans (1998) J. D. Angrist and W. N. Evans.  Children and their parents’ labor supply: evidence from exogenous variation in family size.  _The American Economic Review_ , 88(3):450–477, 1998.
  * Bergemann and Riphahn (2011) A. Bergemann and R. Riphahn.  Female labour supply and parental leave benefits - the causal effect of paying higher transfers for a shorter period of time.  _Applied Economics Letters_ , 18(1):17–20, 2011\.
  * BMFSFJ (2008) BMFSFJ.  Elterngeld als teil nachhaltiger familienpolitik.  _https://www.bmfsfj.de/bmfsfj/service/publikationen/elterngeld-als-teil-nachhaltiger-familienpolitik/73822_ , 2008\.
  * Carrasco (2001) R. Carrasco.  Binary choice with binary endogenous regressors in panel data: estimating the effect of fertility on female labor force participation.  _Journal of Business & Economic Statistics_, 19(4):385–394, 2001.
  * Dhaene and Jochmans (2015) G. Dhaene and K. Jochmans.  Split-panel jackknife estimation of fixed effects models.  _Review of Economic Studies_ , 82:991–1030, 2015.
  * Ejrnæs and Kunze (2013) M. Ejrnæs and A. Kunze.  Work and wage dynamics around childbirth.  _The Scandinavian Journal of Economics_ , 115(3):856–877, 2013.
  * Fitzenberger et al. (2013) B. Fitzenberger, K. Sommerfeld, and S. Steffes.  Causal effects on employment after first birth - a dynamic treatment approach.  _Labour Economics_ , 25:49–62, 2013.
  * Frühwirth-Schnatter et al. (2016) S. Frühwirth-Schnatter, C. Pamminger, and A. Weber-Ebmer.  Mother’s long-run career patterns after first birth.  _Journal of the Royal Statistical Society_ , 179:707–725, 2016.
  * Greyer et al. (2015) J. Greyer, H. Peter, and K. Wrohlich.  The effects of family policy on maternal labor supply: Combining evidence from a structural model and a quasi-experimental approach.  _Labour Economics_ , 36:84–98, 2015.
  * Hyslop (1999) D. R. Hyslop.  State dependence, serial correlation and heterogeneity in intertemporal labor force participation of married women.  _Econometrica_ , 67(6):1255–1294, 1999.
  * Klose and Paul (2020) S.-Ch. Klose and M. Paul.  Predicting the incidence of future first births based on employment records - a machine learning approach.  2020\.
  * Kluve and Schmitz (2018) J. Kluve and S. Schmitz.  Back to work. parental benefits and mother’s labor market outcomes in the medium run.  _ILR Review_ , 71(1):143–173, 2018.
  * Kluve and Tamm (2013) J. Kluve and M. Tamm.  Parental leave regulations, mother’s labor force attachment and father’s childcare involvement: Evidence from a natural experiment.  _Journal of Population Economics_ , 26(3):983–1005, 2013.
  * Kreyenfeld and Mika (2008) M. Kreyenfeld and T. Mika.  Erwerbstätigkeit und fertilität: Analysen mit der versicherungskontenstichprobe der deutschen rentenversicherung.  2008\.
  * Lederer (2018) J. Lederer.  Modern mathematical statistics, 2018.
  * Long (2009) J. S. Long.  Group comparisons in logit and probit using predicted probabilities.  2009\.
  * Lundborg et al. (2018) P. Lundborg, E. Plug, and A. Würtz Rasmussen.  Can women have children and career? iv evidence from ivf treatments.  _American Economic Review_ , 107(6):1611–1637, 2018.
  * Michaud and Tatsiramos (2011) P.-C. Michaud and K. Tatsiramos.  Fertility and female employment dynamics in europe: the effect of using alternative econometric modeling assumptions.  _Journal of Applied Econometrics_ , 26(4):641–668, 2011.
  * Moffitt (1984) R. Moffitt.  Profiles of fertility, labour supply and wages of married women: a complete life cycle model.  _The Review of Economic Studies_ , 51(2):263–278, 1984.
  * Neyman and Scott (1948) J. Neyman and E. Scott.  Consistent estimates based on partially consistent observations.  _Econometrica_ , 16:1–31, 1948.
  * Nickell (1981) S. Nickell.  Biases in dynamic models with fixed effects.  _Econometrica_ , 49(6):1417–1426, 1981.
  * Pronzato (2009) C. D. Pronzato.  Return to work after childbirth: Does parental leave matter in europe?  _Review of Economics of the Household_ , 7(4):341–360, 2009.
  * Raute (2019) A. Raute.  Can financial incentives reduce the baby gap? evidence from a reform in maternity leave benefits.  _Journal of Public Economics_ , 169:203–222, 2019.
  * Schönberg and Ludsteck (2014) U. Schönberg and J. Ludsteck.  Expansions in maternity leave coverage and mother’s labor market outcomes after childbirth.  _Journal of Labor Economics_ , 32(3):469–505, 2014.
  * Su et al. (2016) L. Su, Z. Shi, and P. C. B. Phillips.  Identifying latent structres in panel data.  _Econometrica_ , 84(6):2215–2264, 2016.
  * Troske and Voicu (2010) K. R. Troske and A. Voicu.  Joint estimation of sequential labor force participation and fertility decisions using markov chain monte carlo techniques.  _Labour Economics_ , 17(1):150–169, 2010.
  * Troske and Voicu (2013) K. R. Troske and A. Voicu.  The effects of the timing and spacing of births on the level of labor market involvement of married women.  _Empirical Economics_ , 45(1):483–521, 2013.
  * Wange et al. (2019) W. Wange, P. B. Phillips, and L. Su.  The heterogenous effects of the minimum wage on employment across states.  _Economics Letters_ , 174:179–185, 2019.

##  7 Appendix

The appendix consists of two main parts. The first part provides references to
the code material used in this study. The second part contains the main tables
of my estimations.

### Code material

Su et al. (2016) provide documented code material for their simulations and
empirical data applications online. Their coding material used for
implementing the C-Lasso within panel structure models is available for the
Matlab software. I have used their code material, rewritten the code for my
purposes, and extended it to implement the C-Lasso for mixed-panel structure
models, which were also estimated in my study.

### Tables

This subsection lists all the tables that contain the results of the main
estimations.

Table 5: Marginal Effects of Employment Effects | LPM FE |  | Testing linear combinations
---|---|---|---
| Full sample |  |  Null hypothesis |  $\displaystyle p$-Value |  $\displaystyle p$-Value
|  Control mean |  Treatment mean |  |  |  Control sample |  Treatment sample
|  (1) |  (2) |  |  (3) |  (4) |  (5)
employit-1 |  0.8327 (0.0057) |  0.8073 (0.0068) |  |  $\displaystyle\text{child}_{\text{m1-m14}_{it}}-$ child${}_{\text{m15-m24}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{2nd}_{it}}$ |  -0.0367 (0.0059) |  -0.0509 (0.0068) |  |  child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m25-m59}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{m1-m14}_{it}}$ |  -0.0838 (0.0064) |  -0.1144 (0.0076) |  |  child${}_{\text{m15-m24}_{it}}-$ child${}_{\text{m25-m59}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{m15-m24}_{it}}$ |  -0.0499 (0.0056) |  -0.0550 (0.0057) |  |  child${}_{{\text{m1-m14}_{it}}_{T}}-$ child${}_{{\text{m1-m14}_{it}}_{C}}=0$ | 0.0020
child${}_{\text{m25-m59}_{it}}$ |  -0.0225 (0.0035) |  -0.0220 (0.0042) |  |  child${}_{{\text{m15-m24}_{it}}_{T}}-$ child${}_{{\text{m15-m24}_{it}}_{C}}=0$ | 0.5275
|  |  |  |  child${}_{{\text{m25-m59}_{it}}_{T}}-$ child${}_{{\text{m25-m59}_{it}}_{C}}=0$ | 0.9275
Notes: LPM = Linear Probability Model, FE = Fixed Effects.
Standard errors are provided in (parentheses) and are clustered at the
individual level.
Estimation with Stata. I use the lincom command to test whether linear
combinations of parameter estimates within the same sample are statistically
different from each other. I use the suest command to test the equality of two
parameter estimates in the independent treatment and control samples.
I account for individual fixed effects by including a separate dummy for each
sample. Estimation without intercept, because all fixed effects are included.

Table 6: Marginal Effects of Employment Effects | LPM |  | Testing linear combinations
---|---|---|---
| Full sample |  |  Null hypothesis |  $\displaystyle p$-Value |  $\displaystyle p$-Value
|  Control mean |  Treatment mean |  |  |  Control sample |  Treatment sample
|  (1) |  (2) |  |  (3) |  (4) |  (5)
employit-1 |  0.9105 (0.0038) |  0.8858 (0.0053) |  |  child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m15-m24}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{2nd}_{it}}$ |  -0.0191 (0.0038) |  -0.0329 (0.0048) |  |  child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m25-m59}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{m1-m14}_{it}}$ |  -0.0448 (0.0048) |  -0.0693 (0.0062) |  |  child${}_{\text{m15-m24}_{it}}-$ child${}_{\text{m25-m59}_{it}}=0$ |  0.0000 |  0.0000
child${}_{\text{m15-m24}_{it}}$ |  -0.0265 (0.0041) |  -0.0312 (0.0046) |  |  child${}_{{\text{m1-m14}_{it}}_{T}}-$ child${}_{{\text{m1-m14}_{it}}_{C}}=0$ | 0.0017
child${}_{\text{m25-m59}_{it}}$ |  -0.0115 (0.0027) |  -0.0137 (0.0032) |  |  child${}_{{\text{m15-m24}_{it}}_{T}}-$ child${}_{{\text{m15-m24}_{it}}_{C}}=0$ | 0.4419
Intercept |  0.0672 (0.0047) |  0.0880 (0.0063) |  |  child${}_{{\text{m25-m59}_{it}}_{T}}-$ child${}_{{\text{m25-m59}_{it}}_{C}}=0$ | 0.6134
|  |  |  |  cons${}_{T}-$cons${}_{C}=0$ | 0.0082
Notes: LPM = Linear Probability Model.
Standard errors are provided in (parentheses) and are clustered at the
individual level.
Estimation with Stata. I use the lincom command to test whether linear
combinations of parameter estimates within the same sample are statistically
different from each other. I use the suest command to test the equality of two
parameter estimates in the independent treatment and control samples.

Table 7: Effects on Employment Rates | Probit FE |  | Post Lasso-PPL
---|---|---|---
| Full sample |  | Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean
|  (1) |  (2) |  |  (3) |  (4) |  |  (5) |  (6) |  |  (7) |  (8) |  |  (9) |  (10)
employit-1 |  2.9476 (0.0374) |  2.7611 (0.0389) |  |  3.2094 (0.0366) |  3.0192 (0.0383) |  |  2.8172 (0.0734) |  2.5531 (0.0801) |  |  2.8833 (0.0637) |  2.6880 (0.0672) |  |  3.0922 (0.0561) |  3.0225 (0.0571)
child${}_{\text{2nd}_{it}}$ |  -0.5253 (0.0610) |  -0.6059 (0.0549) |  |  -0.2159 (0.0601) |  -0.2835 (0.0541) |  |  -0.4693 (0.1233) |  -0.5731 (0.0972) |  |  -0.2081 (0.0948) |  -0.0123 (0.0926) |  |  -0.3406 (0.1110) |  -0.1742 (0.0944)
child${}_{\text{m1-m14}_{it}}$ |  -1.1139 (0.0732) |  -1.3068 (0.0687) |  |  -1.0069 (0.0715) |  -1.1994 (0.0671) |  |  -0.0396 (0.1231) |  -0.0824 (0.1149) |  |  -0.7882 (0.1253) |  -0.9967 (0.1196) |  |  -2.8898 (0.1543) |  -2.6778 (0.1341)
child${}_{\text{m15-m24}_{it}}$ |  -0.7193 (0.0685) |  -0.7195 (0.0629) |  |  -0.7259 (0.0670) |  -0.7152 (0.0617) |  |  0.3984 (0.1183) |  0.3381 (0.1167) |  |  -0.4544 (0.1183) |  -0.3591 (0.1156) |  |  -2.2118 (0.1297) |  -1.8772 (0.1127)
child${}_{\text{m25-m59}_{it}}$ |  -0.3365 (0.0490) |  -0.3100 (0.0463) |  |  -0.3038 (0.0480) |  -0.2904 (0.0456) |  |  0.2942 (0.0884) |  0.5221 (0.0856) |  |  -0.4473 (0.0894) |  -0.4602 (0.0854) |  |  -1.1117 (0.0953) |  -0.8484 (0.0910)
Notes: FE = Fixed Effects, PPL= Penalized Profile Likelihood. All Post Lasso-
PPL estimates are bias corrected by half-panel jackknife.
Standard errors are provided in (parentheses) and are clustered at the
individual level. Estimation without intercept, because all fixed effects are
included.

Table 8: Supplements to Table 7 | Testing linear combinations ($\displaystyle t$-statistic)
---|---
| Control sample
|  Full sample |  Group 1 |  Group 2 |  Group 3
|  (1) |  (2) |  (3) |  (4)
child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m15-m24}_{it}}$ |  -2.73 |  -2.15 |  -2.06 |  -3.13
child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m25-m59}_{it}}$ |  -3.30 |  -1.70 |  -1.08 |  -8.12
child${}_{\text{m15-m24}_{it}}-$ child${}_{\text{m25-m59}_{it}}$ |  -2.05 |  -0.13 |  0.55 |  -5.74
| Treatment sample
|  Full sample |  Group 1 |  Group 2 |  Group 3
|  (1) |  (2) |  (3) |  (4)
child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m15-m24}_{it}}$ |  -4.36 |  -2.04 |  -4.13 |  -4.02
child${}_{\text{m1-m14}_{it}}-$ child${}_{\text{m25-m59}_{it}}$ |  -4.50 |  -2.66 |  -2.15 |  -8.32
child${}_{\text{m15-m24}_{it}}-$ child${}_{\text{m25-m59}_{it}}$ |  -2.34 |  -1.42 |  0.57 |  -5.37

Table 9: Marginal Effects of Employment Effects | LPM FE |  | Post Lasso-PPL
---|---|---|---
| Full sample |  | Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean
|  (1) |  (2) |  |  (3) |  (4) |  |  (5) |  (6) |  |  (7) |  (8) |  |  (9) |  (10)
employit-1 |  0.8327 (0.0032) |  0.8073 (0.0035) |  |  0.8032 (0.0051) |  0.7810 (0.0040) |  |  0.7744 (0.0074) |  0.7168 (0.0084) |  |  0.7563 (0.0101) |  0.7497 (0.0075) |  |  0.6329 (0.0094) |  0.6214 (0.0085)
child${}_{\text{2nd}_{it}}$ |  -0.0367 (0.0042) |  -0.0509 (0.0045) |  |  -0.0263 (0.0003) |  -0.0368 (0.0003) |  |  -0.0722 (0.0011) |  -0.0939 (0.0016) |  |  -0.0286 (0.0004) |  -0.0019 (0.0000) |  |  -0.0377 (0.0005) |  -0.0198 (0.0003)
child${}_{\text{m1-m14}_{it}}$ |  -0.0838 (0.0045) |  -0.1144 (0.0051) |  |  -0.1381 (0.0012) |  -0.1809 (0.0009) |  |  -0.0059 (0.0001) |  -0.0127 (0.0002) |  |  -0.1194 (0.0016) |  -0.1797 (0.0012) |  |  -0.4430 (0.0030) |  -0.4297 (0.0024)
child${}_{\text{m15-m24}_{it}}$ |  -0.0499 (0.0046) |  -0.0550 (0.0050) |  |  -0.0883 (0.0009) |  -0.0903 (0.0007) |  |  0.0621 (0.0012) |  0.0545 (0.0012) |  |  -0.0614 (0.0009) |  -0.0541 (0.0005) |  |  -0.2928 (0.0027) |  -0.2434 (0.0025)
child${}_{\text{m25-m59}_{it}}$ |  -0.0225 (0.0032) |  -0.0220 (0.0035) |  |  -0.0367 (0.0004) |  -0.0365 (0.0004) |  |  0.0454 (0.0007) |  0.0874 (0.0015) |  |  -0.0611 (0.0008) |  -0.0698 (0.0007) |  |  -0.1154 (0.0010) |  -0.0892 (0.0009)
Intercept |  0.1226 (0.0042) |  0.1430 (0.0047) |  |  0.2638 (0.0091) |  0.2876 (0.0086) |  |  0.0944 (0.0107) |  0.0892 (0.0087) |  |  0.3388 (0.0173) |  0.3099 (0.0148) |  |  0.5939 (0.0208) |  0.5804 (0.0201)
Notes: LPM = Linear Probability Model, FE = Fixed Effects, PPL= Penalized
Profile Likelihood.
Standard errors are provided in (parentheses), and are clustered at the
individual level for the LPM FE (columns (1) and (2)).
All standard errors and average effects derived from the Post Lasso-PPL
estimates are build on a plug-in estimator.
See the Appendix for further information on the plug-in principle used here.
Intercept is calculated ex-post as the average over all individual fixed
effects belonging to the different groups.

Table 10: Prediction of Average Employment Probability | Employment Probability
---|---
| Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p.
|  (1) |  (2) |  (3) |  |  (4) |  (5) |  (6) |  |  (7) |  (8) |  (9) |  |  (10) |  (11) |  (12)
child${}_{\text{m1-m14}_{it}}$ |  0.2704 (0.0162) |  0.1637 (0.0105) |  -10.67 [0.0000] |  |  0.3156 (0.0358) |  0.2184 (0.0247) |  -9.72 [0.0000] |  |  0.4444 (0.0308) |  0.2131 (0.0210) |  -23.13 [0.0000] |  |  0.1437 (0.0194) |  0.1108 (0.0137) |  -3.29 [0.0000]
child${}_{\text{m15-m24}_{it}}$ |  0.4880 (0.0234) |  0.4578 (0.0223) |  -3.02 [0.0071] |  |  0.5681 (0.0497) |  0.4509 (0.0471) |  -11.72 [0.0000] |  |  0.7613 (0.0337) |  0.6869 (0.0385) |  -7.44 [0.0000] |  |  0.2993 (0.0312) |  0.3505 (0.0302) |  5.12 [0.0006]
child${}_{\text{m25-m59}_{it}}$ |  0.6219 (0.0180) |  0.6260 (0.0178) |  0.41 [0.4692] |  |  0.5351 (0.0401) |  0.5444 (0.0389) |  0.93 [0.4589] |  |  0.6516 (0.0293) |  0.6065 (0.0297) |  -4.51 [0.0000] |  |  0.6316 (0.0273) |  0.6624 (0.0261) |  3.08 [0.0000]
child${}_{\text{m60-m72}_{it}}$ |  0.7497 (0.0190) |  0.7132 (0.0193) |  -3.65 [0.0000] |  |  0.2805 (0.0388) |  0.1869 (0.0279) |  -9.36 [0.0000] |  |  0.8108 (0.0266) |  0.7486 (0.0300) |  -6.22 [0.0000] |  |  0.8905 (0.0193) |  0.8731 (0.0199) |  -1.74 [0.0414]
Ave. emp. |  0.5541 (0.0149) |  0.5235 (0.0140) |  -3.06 [0.0000] |  |  0.4492 (0.0333) |  0.4009 (0.0301) |  -4.83 [0.0000] |  |  0.6524 (0.0241) |  0.5620 (0.0227) |  -9.04 [0.0000] |  |  0.5319 (0.0219) |  0.5439 (0.0199) |  1.2 [0.0402]
Notes: Average Employment Probability is calculated as $\displaystyle
1/(N[\hat{\mathcal{G}}^{k}]T[p^{j}])\
G(\bm{x}_{i[\hat{\mathcal{G}}^{k}]t[p^{j}]}^{\top}\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}+\hat{\mu}_{i}[\hat{\mathcal{G}}^{k}]$),
where $\displaystyle G(\cdot)$ denotes the CDF of the standard normal,
$\displaystyle p^{j},\ j=1,\dots,4$ denote the four different phases of
childage, $\displaystyle[\cdot]$ denotes the restiction to a subspace, and the
remainder is defined as above.
Standard errors are given in (parentheses) and are build on a plug-in
estimator.
The $\displaystyle p$-values for the test in eq. 15 are reported in [square
brackets] under the estimated differences of the average employment
predictions.

Table 11: Effects on Employment Rates | Probit FE |  | Post Lasso-PPL
---|---|---|---
| Full sample |  | Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean
|  (1) |  (2) |  |  (3) |  (4) |  |  (5) |  (6) |  |  (7) |  (8) |  |  (9) |  (10)
child${}_{\text{m1-m14}_{it}}$ |  -1.8453 (0.0969) |  -2.1090 (0.0973) |  |  -1.7696 (0.0954) |  -2.0322 (0.0955) |  |  0.4483 (0.1639) |  0.4588 (0.1484) |  |  -1.1991 (0.1275) |  -1.7069 (0.1547) |  |  -5.1424 (0.2583) |  -4.7176 (0.2004)
child${}_{\text{m15-m24}_{it}}$ |  -0.9534 (0.0951) |  -0.8537 (0.0911) |  |  -1.0065 (0.0943) |  -0.8928 (0.0905) |  |  1.1439 (0.1638) |  1.2409 (0.1407) |  |  -0.0324 (0.1450) |  0.1298 (0.1807) |  |  -3.9962 (0.1868) |  -3.2772 (0.1532)
child${}_{\text{m25-m59}_{it}}$ |  -0.5194 (0.0752) |  -0.3110 (0.0726) |  |  -0.4647 (0.0745) |  -0.3023 (0.0722) |  |  0.9204 (0.1353) |  1.3331 (0.1124) |  |  -0.4053 (0.1144) |  -0.4787 (0.1416) |  |  -2.0476 (0.1501) |  -1.5772 (0.1333)
employit-1 |  3.0269 (0.0351) |  2.8627 (0.0366) |  |  3.2220 (0.0339) |  3.0505 (0.0357) |  |  2.8941 (0.0658) |  2.6518 (0.0709) |  |  2.9531 (0.0598) |  2.7764 (0.0636) |  |  3.1070 (0.0554) |  3.0008 (0.0548)
child${}_{\text{m1-m14}_{it}}$ |  -0.7795 (0.0579) |  -0.8754 (0.0543) |  |  -0.7625 (0.0578) |  -0.7935 (0.0534) |  |  0.3871 (0.0927) |  0.4901 (0.0842) |  |  -0.5653 (0.0933) |  -0.6753 (0.0927) |  |  -2.7104 (0.1471) |  -2.6012 (0.1249)
child${}_{\text{m15-m24}_{it}}$ |  -0.4359 (0.0563) |  -0.3529 (0.0532) |  |  -0.4881 (0.0560) |  -0.3490 (0.0526) |  |  0.7339 (0.0984) |  0.7980 (0.0969) |  |  -0.3177 (0.1007) |  -0.1051 (0.0991) |  |  -2.0103 (0.1242) |  -1.8071 (0.1080)
child${}_{\text{m25-m59}_{it}}$ |  -0.2282 (0.0450) |  -0.1623 (0.0427) |  |  -0.2172 (0.0448) |  -0.1521 (0.0422) |  |  0.4050 (0.0804) |  0.6896 (0.0763) |  |  -0.4256 (0.0848) |  -0.3117 (0.0810) |  |  -0.9816 (0.0936) |  -0.9927 (0.0987)
Notes: Robustness checks
FE = Fixed Effects, PPL= Penalized Profile Likelihood. All Post Lasso-PPL
estimates are bias corrected by half-panel jackknife.
Standard errors are provided in (parentheses) and are clustered at the
individual level. Estimation without intercept, because all fixed effects are
included.

Table 12: Marginal Effects of Employment Effects | LPM FE |  | Post Lasso-PPL
---|---|---|---
| Full sample |  | Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean |  |  Control mean |  Treatment mean
|  (1) |  (2) |  |  (3) |  (4) |  |  (5) |  (6) |  |  (7) |  (8) |  |  (9) |  (10)
child${}_{\text{m1-m14}_{it}}$ |  -0.3872 (0.0068) |  -0.4507 (0.0070) |  |  -0.4270 (0.0090) |  -0.4728 (0.0110) |  |  0.1182 (0.0044) |  0.1065 (0.0047) |  |  -0.3128 (0.0114) |  -0.4658 (0.0171) |  |  -0.6368 (0.0190) |  -0.6694 (0.0181)
child${}_{\text{m15-m24}_{it}}$ |  -0.2002 (0.0076) |  -0.1918 (0.0078) |  |  -0.2441 (0.0048) |  -0.2131 (0.0042) |  |  0.3039 (0.0113) |  0.3006 (0.0105) |  |  -0.0080 (0.0003) |  0.0336 (0.0013) |  |  -0.5093 (0.0145) |  -0.4869 (0.0122)
child${}_{\text{m25-m59}_{it}}$ |  -0.1056 (0.0058) |  -0.0681 (0.0060) |  |  -0.1107 (0.0021) |  -0.0712 (0.0013) |  |  0.2390 (0.0098) |  0.3039 (0.0157) |  |  -0.0986 (0.0041) |  -0.1205 (0.0043) |  |  -0.2274 (0.0093) |  -0.1960 (0.0070)
Intercept |  0.5964 (0.0050) |  0.5742 (0.0051) |  |  0.7186 (0.0146) |  0.6672 (0.0157) |  |  0.2344 (0.0300) |  0.1146 (0.0163) |  |  0.7496 (0.0229) |  0.7536 (0.0252) |  |  0.8964 (0.0185) |  0.8917 (0.0167)
employit-1 |  0.8399 (0.0031) |  0.8187 (0.0034) |  |  0.8254 (0.0044) |  0.8083 (0.0035) |  |  0.8021 (0.0048) |  0.7420 (0.0071) |  |  0.7876 (0.0096) |  0.7913 (0.0062) |  |  0.6347 (0.0094) |  0.6274 (0.0084)
child${}_{\text{m1-m14}_{it}}$ |  -0.0624 (0.0038) |  -0.0817 (0.0043) |  |  -0.1016 (0.0010) |  -0.1153 (0.0008) |  |  0.0561 (0.0009) |  0.0758 (0.0014) |  |  -0.0840 (0.0014) |  -0.1106 (0.0008) |  |  -0.4203 (0.0025) |  -0.4244 (0.0026)
child${}_{\text{m15-m24}_{it}}$ |  -0.0321 (0.0041) |  -0.0295 (0.0045) |  |  -0.0591 (0.0007) |  -0.0450 (0.0004) |  |  0.1164 (0.0019) |  0.1354 (0.0026) |  |  -0.0435 (0.0008) |  -0.0151 (0.0001) |  |  -0.2715 (0.0029) |  -0.2418 (0.0028)
child${}_{\text{m25-m59}_{it}}$ |  -0.0162 (0.0031) |  -0.0127 (0.0034) |  |  -0.0260 (0.0003) |  -0.0194 (0.0002) |  |  0.0618 (0.0007) |  0.1182 (0.0019) |  |  -0.0587 (0.0010) |  -0.0447 (0.0004) |  |  -0.1036 (0.0012) |  -0.1069 (0.0013)
Intercept |  0.0989 (0.0032) |  0.1075 (0.0035) |  |  0.1824 (0.0074) |  0.1689 (0.0066) |  |  0.0413 (0.0047) |  0.0258 (0.0030) |  |  0.2455 (0.0159) |  0.1971 (0.0118) |  |  0.5290 (0.0221) |  0.5545 (0.0212)
Notes: Robustness checks. LPM = Linear Probability Model, FE = Fixed Effects,
PPL= Penalized Profile Likelihood.
Standard errors are provided in (parentheses), and are clustered at the
individual level for the LPM FE (columns (1) and (2)).
All standard errors and average effects derived from the Post Lasso-PPL
estimates are build on a plug-in estimator.
See the Appendix for further information on the plug-in principle used here.
Intercept is calculated ex-post as the average over all individual fixed
effects belonging to the different groups.

Table 13: Prediction of Average Employment Probability | Employment Probability
---|---
| Full sample |  | Group 1 |  | Group 2 |  | Group 3
|  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p. |  |  Control mean |  Treatment mean |  $\displaystyle\Delta_{\text{T-C}}$ in p.p.
|  (1) |  (2) |  (3) |  |  (4) |  (5) |  (6) |  |  (7) |  (8) |  (9) |  |  (10) |  (11) |  (12)
child${}_{\text{m1-m14}_{it}}$ |  0.2689 (0.0145) |  0.1633 (0.0106) |  -10.56 [0.0000] |  |  0.3409 (0.0352) |  0.1928 (0.0226) |  -14.81 [0.0000] |  |  0.4411 (0.0283) |  0.2761 (0.0276) |  -16.50 [0.0000] |  |  0.0948 (0.0145) |  0.0843 (0.0106) |  -1.05 [0.0244]
child${}_{\text{m15-m24}_{it}}$ |  0.4592 (0.0166) |  0.4302 (0.0159) |  -2.9 [0.0001] |  |  0.5314 (0.0370) |  0.3762 (0.0314) |  -15.52 [0.0000] |  |  0.7424 (0.0232) |  0.7833 (0.0239) |  4.09 [0.0001] |  |  0.2401 (0.0247) |  0.3095 (0.0241) |  6.94 [0.0000]
child${}_{\text{m25-m59}_{it}}$ |  0.6041 (0.0162) |  0.5906 (0.0163) |  -1.35 [0.0004] |  |  0.4685 (0.0374) |  0.4006 (0.0320) |  -6.79 [0.0000] |  |  0.6529 (0.0262) |  0.6274 (0.0288) |  -2.55 [0.0001] |  |  0.6223 (0.0280) |  0.6677 (0.0253) |  4.54 [0.0000]
child${}_{\text{m60-m72}_{it}}$ |  0.7186 (0.0146) |  0.6672 (0.0157) |  -5.14 [0.0000] |  |  0.2344 (0.0300) |  0.1146 (0.0163) |  -11.98 [0.0000] |  |  0.7496 (0.0229) |  0.7536 (0.0252) |  0.40 [0.6735] |  |  0.8964 (0.0185) |  0.8917 (0.0167) |  -0.47 [0.4969]
Ave. emp. |  0.5358 (0.0152) |  0.4945 (0.0145) |  -4.13 [0.0000] |  |  0.4092 (0.0353) |  0.3036 (0.0268) |  -10.56 [0.0000] |  |  0.6388 (0.0253) |  0.5990 (0.0264) |  -3.98 [0.0000] |  |  0.5104 (0.0205) |  0.5387 (0.0188) |  2.83 [0.0000]
child${}_{\text{m1-m14}_{it}}$ |  0.2647 (0.0160) |  0.1605 (0.0104) |  -10.42 [0.0000] |  |  0.3225 (0.0330) |  0.1995 (0.0221) |  -12.30 [0.0000] |  |  0.4217 (0.0323) |  0.2116 (0.0223) |  -21.01 [0.0000] |  |  0.1469 (0.0196) |  0.1105 (0.0138) |  -3.64 [0.0000]
child${}_{\text{m15-m24}_{it}}$ |  0.4833 (0.0233) |  0.4508 (0.0224) |  -3.25 [0.0037] |  |  0.5883 (0.0475) |  0.4090 (0.0438) |  -17.93 [0.0000] |  |  0.7213 (0.0365) |  0.6752 (0.0400) |  -4.61 [0.0188] |  |  0.3039 (0.0317) |  0.3616 (0.0311) |  5.77 [0.0001]
child${}_{\text{m25-m59}_{it}}$ |  0.6031 (0.0185) |  0.5966 (0.0184) |  -0.65 [0.2699] |  |  0.5033 (0.0388) |  0.4775 (0.0367) |  -2.58 [0.0347] |  |  0.6075 (0.0318) |  0.5742 (0.0307) |  -3.33 [0.0029] |  |  0.6363 (0.0278) |  0.6550 (0.0274) |  1.87 [0.0212]
child${}_{\text{m60-m72}_{it}}$ |  0.7296 (0.0198) |  0.6755 (0.0208) |  -5.41 [0.0000] |  |  0.2516 (0.0366) |  0.1248 (0.0226) |  -12.68 [0.0000] |  |  0.7992 (0.0287) |  0.7307 (0.0323) |  -6.85 [0.0000] |  |  0.8914 (0.0198) |  0.8888 (0.0197) |  -0.26 [0.7616]
Ave. emp. |  0.5397 (0.0151) |  0.5011 (0.0144) |  -3.86 [0.0000] |  |  0.4330 (0.0319) |  0.3482 (0.0274) |  -8.48 [0.0000] |  |  0.6190 (0.0263) |  0.5414 (0.0236) |  -7.76 [0.0000] |  |  0.5356 (0.0213) |  0.5446 (0.0205) |  0.90 [0.1347]
Notes: Average Employment Probability is calculated as $\displaystyle
1/(N[\hat{\mathcal{G}}^{k}]T[p^{j}])\
G(\bm{x}_{i[\hat{\mathcal{G}}^{k}]t[p^{j}]}^{\top}\bm{\hat{\hat{\alpha}}}_{\hat{\mathcal{G}}^{k}}+\hat{\mu}_{i}[\hat{\mathcal{G}}^{k}]$),
where $\displaystyle G(\cdot)$ denotes the CDF of the standard normal,
$\displaystyle p^{j},\ j=1,\dots,4$ denote the four different phases of
childage, $\displaystyle[\cdot]$ denotes the restiction to a subspace, and the
remainder is defined as above.
Standard errors are given in (parentheses) and are build on a plug-in
estimator.
The $\displaystyle p$-values for the test in eq. 15 are reported under the
estimated differences of the average employment predictions.

Figure 5: The horizontal axis depicts the number of groups and the vertical
axis depicts the corresponding information criterion value. The upper panels
show the information criterion values of the first robustness check and the
lower panels show the information criterion values of the second robstness
check.

## Online Appendix

This online appendix is composed of two main parts. The first part provides
further insights into the sample restriction, the descriptives, and the data
preprocessing. The second part gives further insights into the underpinning
principles of this paper’s methodology and estimation.

### More on the Data

#### Restriction of the Sample

The C-Lasso method of Su et al. (2016) does not perform well if $\displaystyle
T$ is fixed and $\displaystyle N\rightarrow\infty$, i.e., the number of cross-
sectional units is much larger than the number of time periods. In my dataset,
there are $\displaystyle 57\,453$ women for the considered period. So
$\displaystyle N$ is approximately $\displaystyle T^{3}/7$ ($\displaystyle
T=73$), making it hard for the C-Lasso to work well. Furthermore, I do not
want to unnecessarily complicate the interpretation of group-specific
coefficients by mixing mothers and non-mothers in my analysis. For these
reasons, and because I want to identify the reform effect within a regression
discontinuity framework, I only focus on mothers with a first child born
between October 1, 2006, and March 31, 2007, which drastically reduces my
sample size. In total, I consider sample sizes of $\displaystyle N=329$ and
$\displaystyle N=333$ for pre- and post-reform samples, respectively.262626The
numbers indicate the final sample sizes after preprocessing. At the same time,
I construct a monthly balanced panel to increase the number of time periods
and achieve more precise estimates.272727Like Su et al. (2016) and Wange et
al. (2019), I generate a balanced panel to simplify the coding. For the same
reasons, I run C-Lasso separately for the pre- and post-reform samples.
A next difficulty is that I cannot include mothers who have never worked or
who have always worked over the considered period of $\displaystyle T=73$
months (I consider $\displaystyle T=72$ months after first birth) since they
are associated with infinite ($\displaystyle-\infty$, $\displaystyle\infty$)
fixed effects in a fixed effect probit model. Mothers who have always worked
after first birth are missing in my sample because mothers are not allowed to
work for two months after birth due to mandatory maternity leave. Mothers who
have never worked over the considered period are a bigger issue. My full
sample (pre- and post-reform samples) contains 123 (out of 785) mothers who
have never worked over the 72 months after the first birth, which is 15,7%.
Conversely, this means that I do not look at mothers who completely
concentrate on the family work and are never employed after birth. The mothers
I look for have a certain attachment to the labor market and return to
employment at some point, i.e., I examine the heterogeneous dynamics of when
women return to work after birth.
Descriptives Table 14 shows the average employment rate in % and the
percentage of mothers never employed for the respective phases up to six years
after childbirth. A graphical illustration of the average employment rate can
be found in Section 4. It can be already seen descriptively that a larger
share of mothers is employed in the medium-run
(child${}_{\text{m25-m59}_{it}}$) post-reform. Indeed, Kluve and Schmitz
(2018) have found that the parental benefit reform caused a relatively large
share of mothers to move from nonparticipation to employment in the medium
run. Moreover, the descriptive statistics show that a lower percentage of
women are ‘never employed’ during the period under consideration.

Table 14: Descriptives | Pre-Reform Sample |  | Post-Reform Sample
---|---|---|---
|  Average employment rate in % |  $\displaystyle\%$ of women never employed |  |  Average employment rate in % |  $\displaystyle\%$ of women never employed
child${}_{\text{m1-m14}_{it}}$ |  20.9 |  56.5 |  |  12.4 |  54.5
child${}_{\text{m15-m24}_{it}}$ |  39.6 |  50.8 |  |  38.2 |  49.6
child${}_{\text{m25-m59}_{it}}$ |  49.1 |  25.4 |  |  50.6 |  22.0
child${}_{\text{m60-m72}_{it}}$ |  59.6 |  30.0 |  |  57.4 |  30.0
$\displaystyle\Sigma$ |  43.9 |  17.3 |  |  42.3 |  14.0
Notes: Descriptives based on data without the preprocessing step.

#### Covariates

Stationarity A major concern in the model is the underlying assumption of
stationarity. One may cast doubt that the covariates are stationary.
Obviously, the cross-sectional distribution of the child age dummies of the
first child changes over time, but also the age of the second child is clearly
trending upwards over the sampling period. Controlling for the second child’s
age shows up in the form of diverging estimates across the two subpanels and
leads to a huge bias estimated by the split-panel jackknife. To reduce the
second child age variable’s strong upward trend, I have decided to control for
a second child with a dummy variable.
Another problem is the assumption that the initial observations on employment
are drawn from a steady-state distribution (In my data preparation, all women
give birth to a first child at time $\displaystyle t=0$, that is, no women are
employed at $\displaystyle t=0$.). Therefore it is unlikely that the initial
observations on employment are drawn from a steady-state distribution.
Exogeneity One advantage of my data preparation is that I do not have to worry
about the exogeneity assumption of the first child. Nevertheless, I assume
exogeneity for the second child. However, in his seminal paper on the
interaction between labor-market and fertility decisions, Hyslop (1999) is
unable to reject the exogeneity of fertility decisions once lagged employment
decisions are taken into account. This finding suggests that the exogeneity
assumption should not be a cause for major concern in my model.

### More on the Methodology

#### General

All fixed effects are included in the FE estimation, and therefore the
constant is left out.

#### Group comparisons

Since I run C-Lasso separately for the pre- and post-reform samples, I have to
make assumptions and formulate criteria to compare the different groups
appropriately. Applying C-Lasso on the dataset, I find three latent groups for
both samples. This has the advantage that I can find an associated control
group for each treatment group. I apply a distance-based method to find the
appropriate treatment and control groups for comparison.282828I choose the
Minkowski-distance as a suitable measure $\displaystyle
d(\bm{x},\bm{x}^{\prime})=(\sum_{i=1}^{n}|x_{i}-x_{i}^{\prime}|^{p})^{1/p}=||\bm{x}-\bm{x}^{\prime}||_{p}$
and set $\displaystyle p=1$. The aim is to compare those control and treatment
groups that are most similar in terms of the number of individuals classified
into the different groups and the predicted average employment probabilities.
Thus, the pair
$\displaystyle(\hat{\mathcal{G}}^{k^{T}},\hat{\mathcal{G}}^{k^{C}})$ of
treatment and control groups to be compared obey the following minimization
rule:

$\displaystyle\displaystyle A$
$\displaystyle\displaystyle:=\min_{o_{1},\dots,o_{3!}}\\{||\bm{N}^{T}_{o_{j}[\hat{\mathcal{G}}^{k}]}-\bm{N}^{C}_{o_{j}[\hat{\mathcal{G}}^{k}]}||_{1}\\}\
\land\
B:=\min_{o_{1},\dots,o_{3!}}\\{||\overline{\mbox{$\displaystyle\bm{employ}$\raisebox{5.69054pt}{}}}^{T}_{o_{j}[\hat{\mathcal{G}}^{k}]}-\overline{\mbox{$\displaystyle\bm{employ}$\raisebox{5.69054pt}{}}}^{C}_{o_{j}[\hat{\mathcal{G}}^{k}]}||_{1}\\},$
$\displaystyle\displaystyle k$
$\displaystyle\displaystyle=1,\dots,3,j=1,\dots,3!,$

where $\displaystyle o_{1},\dots,o_{3!}$ are permutations on
$\displaystyle\\{\hat{\mathcal{G}}^{1}_{T},\hat{\mathcal{G}}^{2}_{T},\hat{\mathcal{G}}^{3}_{T},\hat{\mathcal{G}}^{1}_{C},\hat{\mathcal{G}}^{2}_{C},\hat{\mathcal{G}}^{3}_{C}\\}$,
$\displaystyle\bm{N}$ is a vector containing the number of mothers in each
group,
$\displaystyle\overline{\mbox{$\displaystyle\bm{employ}$\raisebox{5.69054pt}{}}}$
is a vector containing the predicted average employment probabilities over the
different phases of child age, and the rest of the notation is defined as
above. Table 15 shows that the first permutation $\displaystyle o_{1}$ is the
optimal choice for finding the treatment and control group pairs. This
permutation also corresponds to the order of the group classification of the
C-Lasso, i.e. we receive the pairs
$\displaystyle(\hat{\mathcal{G}}^{1^{T}},\hat{\mathcal{G}}^{1^{C}}),(\hat{\mathcal{G}}^{2^{T}},\hat{\mathcal{G}}^{2^{C}}),(\hat{\mathcal{G}}^{3^{T}},\hat{\mathcal{G}}^{3^{C}})$.

Table 15:  |  A |  B
---|---|---
$\displaystyle o_{1}$ |  28 |  0.8626
$\displaystyle o_{2}$ |  58 |  2.1606
$\displaystyle o_{3}$ |  230 |  2.5724
$\displaystyle o_{4}$ |  176 |  1.7588
$\displaystyle o_{5}$ |  230 |  3.0338
$\displaystyle o_{6}$ |  206 |  2.5954

#### Split-Panel Jackknife

In this subsection, I introduce the basic idea of the half-panel jackknife
(Dhaene and Jochmans (2015)) method to reduce the bias in fixed-effect models.
It is well known that, in particular, maximum-likelihood estimates of
nonlinear dynamic models with fixed effects are subject to the incidental
parameter problem (Neyman and Scott (1948)). I follow Su et al. (2016) and use
the half-panel jackknife to correct the bias in the post-C-Lasso estimates.
The jackknife-correction method is explicitly designed to handle dynamics in
the data and yields estimators that are simple to implement, requiring only a
few maximum-likelihood estimates.
In my empirical application, I consider $\displaystyle T=73$ and partition the
panel into two half-panels. As $\displaystyle T$ is odd, I split the panel
into non-overlapping half-panels of the form

$\displaystyle\displaystyle\mathcal{S}=\\{S_{1},S_{2}\\},\quad\text{where}\quad
S_{1}:=\\{1,2,\dots,\left\lfloor 73/2\right\rfloor\\},\ S_{2}:=\\{\left\lfloor
73/2\right\rfloor+1,\dots,73\\}.$

The half-panel estimator $\displaystyle\tilde{\theta}_{1/2}$ proposed by
Dhaene and Jochmans (2015) is then defined as follows

$\displaystyle\displaystyle\tilde{\theta}_{1/2}:=2\hat{\theta}-\bar{\theta}_{1/2},\quad\bar{\theta}_{1/2}:=\frac{1}{2}(\bar{\theta}_{\mathcal{S}_{1}}+\bar{\theta}_{\mathcal{S}_{2}}),$

with
$\displaystyle\bar{\theta}_{\mathcal{S}_{1}}:=\sum_{S_{1}\in\mathcal{S}_{1}}\frac{|S_{1}|}{36}\hat{\theta}_{S_{1}}$
and
$\displaystyle\bar{\theta}_{\mathcal{S}_{2}}:=\sum_{S_{2}\in\mathcal{S}_{2}}\frac{|S_{2}|}{37}\hat{\theta}_{S_{2}}$.

#### Plug-in Estimation

In this subsection, I introduce the plug-in estimators for the mean and
variance. Plug-in estimation is a general scheme to estimate parameters. I
follow Dhaene and Jochmans (2015) and use this estimation scheme for computing
average marginal effects (AME) and its standard error estimates, the
parameters of main interest in my empirical approach. An introduction of the
plug-in estimation can be found, for instance, in Lederer (2018).292929cf.
Chapter 3 on Pages 43ff
First, I consider the population mean

$\displaystyle\displaystyle\gamma:=\mathbb{E}x$

of a random variable $\displaystyle
x:(\mathcal{A},\mathfrak{A},\mathbb{P})\rightarrow(\mathbb{R},\mathfrak{B})$.
Setting $\displaystyle f[\mu]:=\int\text{id}\ d\mu,\ \text{id}[a]=a$ the
identity function, for every probability measure $\displaystyle\mu$ on
$\displaystyle\mathfrak{B}$, we find

$\displaystyle\displaystyle\gamma=f[x[\mathbb{P}]],$

which brings us into the plug-in framework. Given data $\displaystyle
x^{1},\dots,x^{n}$, estimating $\displaystyle x[\mathbb{P}]$ by the emprirical
measure yields the empirical mean

$\displaystyle\displaystyle\gamma=f[\hat{\mathbb{P}}_{n}]=\int\text{id}\
d\hat{\mathbb{P}}_{n}=\frac{1}{n}\sum_{i=1}^{n}x^{i}.$

I look on fixed-effect plug-in estimates of the AME for continuous and
discrete variables. Assume that $\displaystyle x_{1_{it}}$ is a binary
variable and $\displaystyle x_{2_{it}}$ is a continuous variable. The AME for
$\displaystyle x_{1_{it}}$ is obtained by

$\displaystyle\displaystyle AME_{x_{1_{it}}}=$
$\displaystyle\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}[G(\hat{\beta}_{1}+x_{2_{it}}\hat{\beta}_{2}+\dots+x_{5_{it}}\hat{\beta}_{5}+\hat{\mu}_{i})-G(x_{2_{it}}\hat{\beta}_{2}+\dots+x_{5_{it}}\hat{\beta}_{5}+\hat{\mu}_{i})],$
(16)

where $\displaystyle G(\cdot)$ denotes the standard normal CDF. The AME for
the continuous variable $\displaystyle x_{2_{it}}$ is obtained by

$\displaystyle\displaystyle AME_{x_{2_{it}}}=$
$\displaystyle\displaystyle\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}g(\hat{\beta}_{1}+x_{2_{it}}\hat{\beta}_{2}+\dots+x_{5_{it}}\hat{\beta}_{5}+\hat{\mu}_{i})\hat{\beta}_{2},$
(17)

where $\displaystyle g(\cdot)$ denotes the standard normal probability density
function (PDF). Thus, for computing AME, I simply use the plug-in estimate of
the population mean.
Second, we consider the population variance

$\displaystyle\displaystyle\sigma^{2}:=\mathbb{E}[(x-\mathbb{E}x)^{2}]$

of a random variable $\displaystyle
x:(\mathcal{A},\mathfrak{A},\mathbb{P})\rightarrow(\mathbb{R},\mathfrak{B})$.
Setting $\displaystyle f[\mu]:=\int\text{id}\ d\mu,\ \text{id}[a]=a$ the
identity function, for every probability measure $\displaystyle\mu$ on
$\displaystyle\mathfrak{B}$, we find

$\displaystyle\displaystyle\sigma^{2}=f[x[\mathbb{P}]],$

which brings us into the plug-in framework. Given data $\displaystyle
x^{1},\dots,x^{n}$, estimating $\displaystyle x[\mathbb{P}]$ by the emprirical
measure yields the empirical variance

$\displaystyle\displaystyle\sigma^{2}=f[\hat{\mathbb{P}}_{n}]=\int\text{id}\
d\hat{\mathbb{P}}_{n}=\frac{1}{n}\sum_{i=1}^{n}(x^{i}-\bar{x})^{2},$

with $\displaystyle\bar{x}=\sum_{i=1}^{n}x^{i}$.
Because the population mean is not known in practice, I replace the population
variance with the sample variance. The standard error estimates are finally
obtained by taking the square root of the sample variance. Thus, I look at
plug-in estimates of the cross-sectional variance of the within-group AME:

$\displaystyle\displaystyle
s^{2}:=\frac{1}{N-1}\sum_{i=1}^{N}(AME_{i}-AME)^{2},\quad\text{with}\quad
AME=\frac{1}{N}\sum_{i=1}^{N}AME_{i}.$ (18)

Dhaene and Jochmans (2015) notice that the estimators in 16 and 17 are subject
to two sources of bias, even if a bias-corrected estimator (here I use the
split-panel jackknife estimator) is used instead of the maximum-likelihood
estimator. The first stems from using $\displaystyle\hat{\mu}_{i}$ instead of
$\displaystyle\mu_{i}$, and the second arises from using
($\displaystyle\bm{\hat{\gamma}},\bm{\hat{\beta}}_{i}$) instead of
($\displaystyle\bm{\gamma},\bm{\beta}_{i}$). Combined with the bias in the
AME, this results in confidence intervals with poor coverage. However, the
authors show both analytically and with simulations that this is a problem
when $\displaystyle T$ is relatively small compared to $\displaystyle N$.
Under rectangular-array asymptotics, when $\displaystyle N$ is fixed and
$\displaystyle T\rightarrow\infty$, the correct inference is obtained even
without bias correction. Since I consider a relatively large $\displaystyle T$
compared to my sample size $\displaystyle N$ in my empirical application, the
bias is almost negligible, and I refer to the asymptotic equivalence of the
estimated AME with and without bias-correction. Therefore, I do not perform
some bias correction when estimating the AME and its standard errors.