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\multicolumn{5}{c}{\Huge \textbf{Teacher Turnover and School Characteristics}} \\
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\LARGE St\'{e}phane Bonhomme & $\qquad$ & \LARGE Gr\'{e}gory Jolivet & $\qquad$ & \LARGE Edwin Leuven \\
\\
\LARGE CEMFI, Madrid & $\qquad$ & \LARGE VU, Amsterdam & $\qquad$ & \LARGE UvA, Amsterdam \\
\\
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&& \LARGE April 2007 && \\ 
\end{tabular}
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\newpage

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\section*{\LARGE Motivation}

\vskip 1cm

\LARGE \noindent We are interested in estimating teachers' preferences for school characteristics.

\vskip 1.5cm

\LARGE \noindent We use an exhaustive matched school/teacher panel of Dutch primary school teachers to address potential bias in previous joint studies of turnover and compensating differentials: \\
\\
$\rightarrow$ correlation between one's current job characteristics and his/her outside opportunities.

\vskip 1.5cm

\noindent Disentangling between preferences and outside opportunities in teacher mobility between schools can then help to address sorting at the teacher labor market entry and/or exit.

\newpage

\section*{\LARGE Estimation of preferences for job characteristics}

\vskip 1cm

\noindent Hedonic regressions: based on the theory of compensating differentials, they aim at estimating wage/amenity correlations in cross-section (Brown, 1981, Duncan and Holmlund, 1983). \\
\\
$\rightarrow$ No strong empirical evidence of marginal willingness to pay (MWP) for amenities.

\vskip 1.5cm

\noindent Job duration approach: derived from on-the-job search theory, these studies estimate the elasticities of the job hazard rate to infer on workers MWP for amenities (Gronberg and Reed, 1994, Hwang et al., 1998). \\
\\
$\rightarrow$ Most studies of teachers preferences for school characteristics derive from this approach (Hanushek et al., 2004, Boyd et al., 2005).

\newpage

\section*{\LARGE Identification of workers preferences for job characteristics}

\noindent Let a job be described by a set of characteristics $A=(a_1,...,a_J)$ and let $T$ be an indicator for job-to-job transitions. \\
\\
\\
Typically, the GR approach consists in regressing $T$ on $A$:
\\
\begin{equation*}
f(T|A) = \Phi\left(\overset{J}{\underset{j=1}{\sum}}\psi_j\cdot a_j\right)
\end{equation*}
\\
\\
$\rightarrow$ Can the $\psi_j$ parameters be interpreted as preference parameters ? \\
\\ 

\newpage

\vskip 1cm

\noindent Assume  that workers value jobs according to $u(A)=\overset{J}{\underset{j=1}{\sum}}\theta_j\cdot a_j$. \\
\\
Let $A^*$ be either an outside job offer or an indicator of the worker's outside opportunity.\\
\\
The worker decides to quit his current job if $u(A^*)-u(A)+\varepsilon>0$. \\
\\
Back to the GR approach, we can now write:
\\
\begin{equation*}
f(T|A)=\int f(T|A,A^*)\cdot f(A^*|A)dA^*
\end{equation*}
\\
$\rightarrow$ If $f(A^*|A)\neq f(A^*)$, regressing $T$ on $A$ leads to estimates, $\psi_j$, that are a ``mix'' of true preference parameters, $\theta_j$, and determinants of job offers. \\
\\
$\rightarrow$ We need a variable that only affects either the mobility decision or job offers.

\newpage

\section*{\LARGE Data}

\vskip 1cm

\noindent Exhaustive panel of Dutch primary school teachers. Each contract between a school and a teacher between 1999 and 2002. We keep one observation per teacher/year (Dec. $31^{\text{st}}$). \\
\\
For each teacher, we observe, at each year $t$:\\
- the school $S$ where he/she works, \\
- a few individuals characteristics $X$ (age, gender), \\
- the job characteristics at $t$ and $t+1$, $A_t$ and $A_{t+1}$, in school $S$, \\
- a dummy $T_t$ equal to one if he/she moves to another school, \\
- the job characteristics $A^*$ in the new school $S^*$. \\
\\
Among job characteristics, we observe the wage $W$, the teaching load $H\in[0,1]$, a dummy for long-term contracts $LTC$, an indicator for urbanity $URB\in\{1,..,5\}$, a dummy for public schools $PUB$ and an indicator for class size:
\begin{equation*}
CSIZE=\frac{\#_S\text{ students}}{\sum_{i\in S} H_i}.
\end{equation*}

\newpage

\section*{\LARGE Pupil population and school budget}

\vskip 1cm

The budget of a school is a function of the weighted number of pupils. The weights correspond to five different categories: \\
\\
5 - Children from non-Dutch cultural background - $p_5 = 1.9$ \\
4 - Children with traveller parents - $p_4 = 1.7$ \\
3 - Children living in a foster family - $p_3 = 1.4$ \\
2 - Children whose parents have a low level of education - $p_2 = 1.25$ \\
1 - Any other type of children - $p_1 = 1$ \\
\\
We compute an indicator of the school's budget as $BUD_t=\sum_{k=1}^5\#_{kt}\cdot p_k$ \\
\\
We also compute two other school characteristics: $PIMM$ is the proportion of children in categories 4 and 5, and $PNEDLE$ the proportion of children in categories 2 and 3.

\newpage

\begin{center}
\begin{tabular}{lr}
\multicolumn{2}{c}{\textbf{Sample description - Teachers}} \\
\\
\hline\hline
Number of teachers & 86 891 \\
women & $82.6\%$\\
movers & $10.7\%$ \\
mean age & 42\\
\hline
\multicolumn{2}{l}{Proportion of movers among...} \\
women & $10.9\%$ \\
men & $9.6\%$ \\
\hline
\multicolumn{2}{l}{Mean age of...} \\
women & 41 \\
men & 45 \\
movers & 38 \\
stayers & 42 \\
\hline
Average teaching load & .78 \\
Proportion of long-term contracts & 87\% \\
\hline\hline
\end{tabular}
\end{center}

\newpage

\begin{center}
\begin{tabular}{lr}
\multicolumn{2}{c}{\textbf{Sample description - Schools}} \\
\\
\hline\hline
Number of schools & 6 716 \\
Average teacher population & 15 \\
Average full-time teacher population & 11 \\
Average pupil population & 223 \\
Average class size & 20 \\
\hline
Proportion of schools with... & \\
no school leaver & $23\%$ \\
no school entrant & $21\%$ \\
\hline
Among positive teacher flows: & \\
average inflow & 3.9 \\
average outflow & 4.0\\
\hline
Mean of... &   \\
PIMM   &  11\% \\
PNEDLE &  16\% \\
URB    &  4.4  \\
PUB    &  33\% \\
\hline\hline
\end{tabular}
\end{center}

\newpage

\section*{\LARGE Using demographic shocks on the pupil population as an instrument for teacher turnover}

\vskip 1cm

\noindent $BUD$ is a very strong predictor of the number of teachers per school ($R^2=.91$) and of the overall teaching load ($R^2=.95$). We thus use $BUD_{t+1}$ to predict the school's demand of teachers $pPROF$ and full-time teachers $pTEACH$ at $t+1$. \\
\\
We then compute the following variable: \\
\begin{equation*}
Z = \frac{pTEACH - TEACH_t}{TEACH_t},
\end{equation*}
\\
and use it as an instrument in teacher's mobility decisions. \\
\begin{equation*}
f(T=1,A^*|A,Z) = f(T=1|A,Z,A^*)\cdot f(A^*|A).
\end{equation*}

\newpage

\begin{minipage}[c]{\linewidth}
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\begin{tabular}{c}
Distribution of $\Delta TEACH = pTEACH - TEACH_t$ \\
\\
\includegraphics[width=0.8\textwidth,clip]{grdteach.eps}
\end{tabular}
\end{minipage}

\newpage

\noindent Define the dummy $VAC_{t+1} = 1\{E(TEACH_{t+1}) + 1 < pTEACH\}$ and regress it on school's characteristics.

\vskip 1.5cm

\begin{center}
\begin{tabular}{lcc}
\multicolumn{3}{c}{$VAC_{t+1}$ (mean = $28\%$)} \\
\\
\hline\hline
\\
$PIMM_{t+1}$   && $\underset{(.13)}{6.18}$   \\
$PNEDLE_{t+1}$ && $\underset{(.12)}{2.63}$   \\
$CSIZE_{t+1}$  && $\underset{(.007)}{.51}$   \\
$URB_{t+1}$    && $\underset{(.03)}{-.118}$  \\
$PUB_{t+1}$    && $\underset{(.03)}{-.0009}$ \\
\\
\hline\hline
\end{tabular}
\end{center}

\newpage
\section*{\LARGE First approach: selection model}

\vskip 1cm

\noindent We assume that the binary regression \textit{\`{a} la} Gronberg et Reed (1994) is the first stage of a selection model where the outcomes (offers) are correlated with the current job's characteristics. \\
\begin{eqnarray*}
\text{Mobility:} && T=\textbf{1}\left\{\theta_0+ (A^*-A)'\theta+X'\theta_X+\theta_Z Z+\varepsilon_T>0\right\}, \\
&& \\
\text{Offers:} && a_j^*=\alpha^0_j+X'\alpha^X_j+\alpha^j_j a_j+\underset{k\neq j}{\sum}\alpha^k_j a_k+\varepsilon_j, \quad j\in[1,J]. 
\end{eqnarray*}
Then the first step regression is: \\
\begin{equation*}
T=\textbf{1}\left\{\psi_0+A'\psi+X'\psi_X+\theta_Z Z+U_T>0\right\},
\end{equation*}
where: \\
\begin{equation*}
\psi_j = (\alpha^j_j-1)\theta_j+\underset{k\neq j}{\sum}\alpha^j_k \theta_k.
\end{equation*}

\newpage

\begin{center}
\LARGE \textbf{Preferences for job characteristics - First estimates}
\end{center}

\vskip 1cm

\begin{center}
\begin{tabular}{l|crcrcr}
\hline\hline
&& $\theta_j$ && $-\psi_j$ && $\alpha_j^j$ \\
\hline
\\
$PIMM$   && -1.17 && $-\underset{(.048)}{.870}$ && $\underset{(.019)}{.239}$ \\
$PNEDLE$ && -.208 && $-\underset{(.049)}{.305}$ && $\underset{(.010)}{.223}$ \\
$CSIZE$  && -.074 && $-\underset{(.003)}{.061}$ && $\underset{(.012)}{.124}$ \\
$URB$    &&  .053 && $\underset{(.011)}{ .035}$ && $\underset{(.019)}{.053}$ \\
$PUB$    &&  .046 && $-\underset{(.011)}{.036}$ && $\underset{(.008)}{.625}$ \\
$LTC$    &&  .621 && $\underset{(.013)}{ .447}$ && $\underset{(.019)}{.339}$ \\
$H$      &&  .534 && $\underset{(.023)}{ .164}$ && $\underset{(.008)}{.593}$ \\
\hline\hline
\end{tabular}
\end{center}
\begin{equation*}
\widehat{\theta}_Z : -\underset{(.045)}{1.38}
\end{equation*}

\newpage

\section*{\LARGE Issues}

\vskip 1cm

\noindent - Some binary outcomes ($PUB$, $LTC$) are treated as continuous variables. \\
\\
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\noindent - Unobserved heterogeneity: mobility decision ? outcomes ? \\
\\
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\noindent - This method does not allow for non linear effects of job characteristics on utility $\rightarrow$ Can we adopt a more ``flexible'' approach ? \\
\\
\\
\noindent - $Z$ can be correlated with outside job offers through local labor market effects $\rightarrow$ Control for the evolution of demand in outside schools due to demographic shocks.

\newpage

\section*{\LARGE Controlling for local labor market conditions}

\vskip 1cm

\noindent We use geographic variables in order to observe which schools are in the same town (``gemeente'') or in the same region (``onderwijsgebied''). \\
\\
Then, for each school S, we compute the following variables:\\
\begin{eqnarray*}
D^+_{St}    & = & \sum_{S'\in\text{town}(S)}(pTEACH_{S'} - TEACH_{S't})\cdot1\{pTEACH_{S'}>TEACH_{S't}\} \\ 
D^-_{St}    & = & \sum_{S'\in\text{town}(S)}-(pTEACH_{S'} - TEACH_{S't})\cdot1\{pTEACH_{S'}<TEACH_{S't}\} \\
PIMM^+_{St} & = & \mathbb{E}\left(PIMM_{S't+1}|S'\in\text{town}(S),pTEACH_{S'}>TEACH_{S't}\right)
\end{eqnarray*}

\vskip 1cm

\noindent We compute similar variables for $PNEDLE$ and $CSIZE$, and we repeat the procedure at the region and at the country level.

\newpage

\section*{\LARGE Three empirical strategies}

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\noindent Once we have computed a satisfying set of variables capturing teacher labor market conditions at the local and national levels, we could follow one of the three following routes: \\
\\
\noindent 1 - Throw these demand variables in the selection model in order to ensure exogeneity of our instrument $Z$ in the mobility decision. We could even try to isolate, among the demand variables, those that solely affect offers and proceed to specification tests. \\
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\noindent 2 - If these demand variables prove sufficient to control for the correlation between current job's characteristics and outside opportunities, we can focus on the first-step binary regression, use a more flexible specification and introduce unobserved heteroegeneity. \\
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\noindent 3 - Ensuring the exogeneity of $Z$ by controlling for teacher demand, we can even turn to more demanding, less parametric methods.

\newpage

\begin{center}
\LARGE \textbf{First-step binary regression using demand variables}
\end{center}

\vskip 1cm

\begin{center}
\begin{tabular}{l|crcrcrc||cl|cr}
\hline\hline
&& (I) && (II) && (III) &&&  && (III ctd) \\
\hline
$PIMM$                && $.870^{***}$  && $.742^{***}$   && $.728^{***}$  &&& $PIMM^+(\text{town})$    && $.42^{***}$\\
$PNEDLE$              && $.305^{***}$  && $.452^{***}$   && $.593^{***}$  &&& $PIMM^+(\text{region})$  && $.86^{*}$  \\
$CSIZE$               && $.061^{***}$  && $.058^{***}$   && $.059^{***}$  &&& $PIMM^+(\text{country})$ && $13.7^{*}$ \\
$URB$                 && $-.035^{***}$ && $-.026^{**}$   && $-.023^{*}$   &&& $PNEDLE^+(\text{town})$  && $-.64^{***}$\\
$PUB$                 && $.036^{***}$  && $.035^{***}$   && $.038^{***}$  &&& $PNEDLE^+(\text{region})$ && $-.87^{**}$\\
$LTC$                 && $-.447^{***}$ && $-.447^{***}$  && $-.452^{***}$ &&& $PNEDLE^+(\text{country})$ && $-13.2^{*}$\\
$H$                   && $-.164^{***}$ && $-.175^{***}$  && $-.176^{***}$ &&& $CSIZE^+(\text{town})$     && $-.004$ \\
$Z$                   && $-1.38^{***}$ && $-1.41^{***}$  && $-1.40^{***}$ &&& $CSIZE^+(\text{region})$   && $-.04$  \\
$D^+(\text{town})$    &&               && $.00086^{***}$ && $.00059$      &&& $CSIZE^+(\text{country})$ && $-1.35^{**}$\\
$D^+(\text{region})$  &&               && $.00049^{***}$ && $-.00078$     &&&    &&  \\
$D^+(\text{country})$ &&               && $.00005$       && $-.0012$      &&&    &&  \\
$D^-(\text{town})$    &&               && $.00028^{*}$   && $-.0032^{*}$  &&&    &&  \\
$D^-(\text{region})$  &&               && $.00017^{*}$   && $-.0024$      &&&    &&  \\
$D^-(\text{country})$ &&               && $.00009$       && $-.0026$      &&&    &&  \\
\hline\hline
\end{tabular}
\end{center}

\newpage

\section*{\LARGE Semi-parametric approach}

\noindent Let $D$ be the vector of demand variables. We are interested in the mean of the instrument $Z$ over the population of school changers.

\vskip .5cm

\begin{eqnarray*}
\mathbb{E}(Z|T=1,A^*,A,D) & = & \int z \cdot f(z|T=1,A^*,A,D)dz\\
                                                    && \\
                          & = & \int z \cdot \frac{f(z|A^*,A,D)\cdot f(T=1|A^*,A,D,z)}{f(T=1|A^*,A,D)}dz\\
                          && \\
                          & = & \frac{\int z\cdot f(z|A,D)\cdot H(u(A^*)-u(A),D,z)dz}
                                     {\int f(z|A,D)\cdot H(u(A^*)-u(A),D,z)dz}\\
                          && \\           
                          & = & K\left(u(A^*),A,D\right).
\end{eqnarray*}

\vskip .5cm

\noindent Then we can estimate $\frac{u'_j}{u'_k}$ as $\frac{\partial}{\partial a^*_j}\mathbb{E}(Z|T=1,A^*,A,D)\Big/\frac{\partial}{\partial a^*_k}\mathbb{E}(Z|T=1,A^*,A,D)$.



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