Stochastic frontier models with network selectivity

Abstract

Worker peer-effects and managerial selection have received limited attention in the stochastic frontier analysis literature. We develop a parametric production function model that allows for worker peer-effects in output and worker-level inefficiency that is correlated with a manager’s selection of worker teams. The model is the usual “composed error” specification of the stochastic frontier model, but we allow for managerial selectivity (endogeneity) that works through the worker-level inefficiency term. The new specification captures both worker-level inefficiency and the manager’s ability to efficiently select teams to produce output. As the correlation between the manager’s selection equation and worker inefficiency goes to zero, our parametric model reduces to the normal-exponential stochastic frontier model of Aigner et al. (1977) with peer-effects. A comprehensive application to the NBA is provided.

Appendices

Appendix A: Mathematical derivations

• 1. $h\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)$

  $$\begin{array}{*{20}{l}} {h\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)} \hfill & = \hfill & {\frac{{{\int}_0^{Z _{s}{t}} {\kern 1pt} h\left( {\epsilon _{st},u_{ist}} \right)d\epsilon }}{{{\int}_0^{Z _{s}{t}} {\int}_0^\infty h\left( {\epsilon _{st},u_{ist}} \right)dud\epsilon }}} \hfill \\ {} \hfill & = \hfill & {\frac{1}{{1 - e^{ - \delta _{st}}}}{\int}_0^{Z _{s}{t}} h\left( {\epsilon _{st},u_{ist}} \right)d\epsilon .} \hfill \end{array}$$

  (11)
  $${\mathrm{Because}}{\kern 1pt} {\int}_0^{Z _{s}{t}} {\kern 1pt} h\left( {\epsilon _{st},u_{ist}} \right)d\epsilon = \frac{1}{{\sigma _u}}\left[ {e^{ - \frac{{u_{ist}}}{{\sigma _u}}} - \kappa _{st}e^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \delta _{st}}} \right],{\mathrm{where}}{\kern 1pt} \kappa _{st} = {\mathrm{1}} + \alpha \delta _{st},$$

  (12)
  $$h\left( {u_{ist}|\epsilon _{st} < \delta _{st}} \right) = \frac{1}{{\sigma _uP_{st}}}\left[ {e^{ - \frac{{u_{ist}}}{{\sigma _u}}} - \kappa _{st}e^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \delta _{st}}} \right],{\mathrm{where}}{\kern 1pt} P_{st} = 1 - e^{ - \delta _{st}}.$$

  (13)

• 2. $E\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)$

  $$\begin{array}{*{20}{l}} {E\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)} \hfill & = \hfill & {{\int}_0^\infty {\kern 1pt} u_{ist}h\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)du} \hfill \\ {} \hfill & = \hfill & {\frac{1}{{\sigma _uP_{st}}}{\int}_0^\infty {\kern 1pt} u_{ist}\left[ {e^{ - \frac{{u_{ist}}}{{\sigma _u}}} - \kappa _{st}e^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \delta _{st}}} \right]du} \hfill \\ {} \hfill & = \hfill & {\frac{{\sigma _u}}{{P_{st}}}\left[ {1 - \frac{1}{{\kappa _{st}}}e^{ - \delta _{st}}} \right].} \hfill \end{array}$$

  (14)

• 3. $h\left( {\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)$, where μ_ist = v_ist − u_ist

  $$\begin{array}{l}v_{ist}{\kern 1pt} {\mathrm{is}}{\kern 1pt} {\mathrm{independent}}{\kern 1pt} {\mathrm{of}}{\kern 1pt} u_{ist}{\kern 1pt} {\mathrm{and}}{\kern 1pt} \epsilon _{st}{\mathrm{,}}\\ {\mathrm{so}}{\kern 1pt} h\left( {u_{ist},v_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right) = h\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)\frac{1}{{\sigma _v}}\phi \left( {\frac{{v_{ist}}}{{\sigma _v}}} \right),\\ {\mathrm{where}}{\kern 1pt} \phi {\kern 1pt} {\mathrm{(}} \cdot {\mathrm{)}}{\kern 1pt} {\mathrm{is}}{\kern 1pt} {\mathrm{standard}}{\kern 1pt} {\mathrm{normal}}{\kern 1pt} {\mathrm{pdf}},\\ {\mathrm{then}}{\kern 1pt} {\mathrm{the}}{\kern 1pt} h\left( {\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right){\mathrm{will}}{\kern 1pt} {\mathrm{be}}{\kern 1pt} {\mathrm{given}}{\kern 1pt} {\mathrm{by}},\end{array}$$

  (15)
  $$\begin{array}{*{20}{l}} {h\left( {\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)} \hfill & = \hfill & {{\int}_0^\infty {\kern 1pt} h\left( {u_{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)\frac{1}{{\sigma _v}}\phi \left( {\frac{{u_{ist} + \mu _{ist}}}{{\sigma _v}}} \right)du} \hfill \\ {} \hfill & = \hfill & {\frac{1}{{\sqrt {2\pi } \sigma _u\sigma _vP_{st}}}{\int}_0^\infty \left( {e^{ - \frac{{u_{ist}}}{{\sigma _u}}} - \kappa _{st}e^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \delta _{st}}} \right)e^{ - \frac{{(\mu _{ist} + u_{ist})^2}}{{2\sigma _v^2}}}du} \hfill \\ {} \hfill & = \hfill & {\frac{1}{{P_{st}}}\left[ {\frac{1}{{\sqrt {2\pi } \sigma _u\sigma _v}}{\int}_0^\infty {\kern 1pt} e^{ - \frac{{u_{ist}}}{{\sigma _u}} - \frac{{(\mu _{ist} + u_{ist})^2}}{{2\sigma _v^2}}}du} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - \frac{1}{{\sqrt {2\pi } \frac{{\sigma _u}}{{\kappa _{st}}}\sigma _v}}{\int}_0^\infty {\kern 1pt} e^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \delta _{st} - \frac{{(\mu _{ist} + u_{ist})^2}}{{2\sigma _v^2}}}du} \right].} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{Using}}{\kern 1pt} {\mathrm{the}}{\kern 1pt} {\mathrm{result}}{\kern 1pt} {\mathrm{from}}{\kern 1pt} {\mathrm{Kumbhakar}}{\kern 1pt} {\mathrm{and}}{\kern 1pt} {\mathrm{Lovell}}{\kern 1pt} {\mathrm{(2000}},{\mathrm{p80)}},} \hfill \\ {} \hfill & = \hfill & {\frac{1}{{\sigma _uP_{st}}}\left[ {{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \frac{{\sigma _v}}{{\sigma _u}}} \right)e^{\frac{{\mu _{ist}}}{{\sigma _u}} + \frac{{\sigma _v^2}}{{2\sigma _u^2}}}} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \frac{{\kappa _{st}\sigma _v}}{{\sigma _u}}} \right)e^{\frac{{\kappa _{st}\mu _{ist}}}{{\sigma _u}} + \frac{{\kappa _{st}^2\sigma _v^2}}{{2\sigma _u^2}}}} \right].} \hfill \end{array}$$

  (16)

• 4. $E\left( {u_{ist}{\mathrm{|}}\mu _{ist},\epsilon _{st} < \delta _{st}} \right)$

$$\begin{array}{*{20}{l}} {E\left( {u_{ist}{\mathrm{|}}\mu _{ist},\epsilon _{st} < \delta _{st}} \right)} \hfill & = \hfill & {{\int}_0^\infty {\kern 1pt} u\frac{{h\left( {u_{ist},\mu _{ist},\epsilon _{st} < \delta _{st}} \right)}}{{h\left( {\mu _{ist},\epsilon _{st} < \delta _{st}} \right)}}du} \hfill \\ {} \hfill & = \hfill & {{\int}_0^\infty u\frac{{h\left( {u_{ist},\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)}}{{h\left( {\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)}}du} \hfill \\ {} \hfill & = \hfill & {\frac{{\frac{1}{{\sqrt {2\pi } \sigma _v}}\left[ {{\int}_0^\infty {\kern 1pt} ue^{ - \frac{{u_{ist}}}{{\sigma _u}} - \frac{{\left( {\mu _{ist} + u_{ist}} \right)^2}}{{2\sigma _v^2}}}du - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}{\int}_0^\infty {\kern 1pt} ue^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \frac{{\left( {\mu _{ist} + u_{ist}} \right)^2}}{{2\sigma _v^2}}}du} \right]}}{{\underbrace {{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \frac{{\sigma _v}}{{\sigma _u}}} \right)e^{\frac{{\mu _{ist}}}{{\sigma _u}} + \frac{{\sigma _v^2}}{{2\sigma _u^2}}}}_{{\mathrm{(1)}}} - \underbrace {\frac{{\kappa _{st}}}{{e^{\delta _{st}}}}{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \frac{{\kappa _{st}\sigma _v}}{{\sigma _u}}} \right)e^{\frac{{\kappa _{st}\mu _{ist}}}{{\sigma _u}} + \frac{{\kappa _{st}^2\sigma _v^2}}{{2\sigma _u^2}}}}_{{\mathrm{(2)}}}}}} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{Let}}\left( {\mathrm{1}} \right) = A_{ist}\left( {\mu _{ist}{\mathrm{|}}\delta _{st},\sigma _u,\sigma _v{\mathrm{,}}\alpha } \right){\mathrm{and}}\left( 2 \right) = B_{ist}\left( {\mu _{ist}{\mathrm{|}}\delta _{st}{\mathrm{,}}\sigma _u{\mathrm{,}}\sigma _v{\mathrm{,}}\alpha } \right),{\mathrm{then}},} \hfill \\ {} \hfill & = \hfill & {\frac{{A_{ist}}}{{A_{ist} - B_{ist}}}\frac{1}{{\sqrt {2\pi } \sigma _vA_{ist}}}{\int}_0^\infty {\kern 1pt} ue^{ - \frac{{u_{ist}}}{{\sigma _u}} - \frac{{\left( {\mu _{ist} + u_{ist}} \right)^2}}{{2\sigma _v^2}}}du} \hfill \\ {} \hfill & {} \hfill & { - \frac{{B_{ist}}}{{A_{ist} - B_{ist}}}\frac{1}{{\sqrt {2\pi } \sigma _v\frac{{e^{\delta _{st}}}}{{\kappa _{st}}}B_{ist}}}{\int}_0^\infty {\kern 1pt} ue^{ - \kappa _{st}\frac{{u_{ist}}}{{\sigma _u}} - \frac{{\left( {\mu _{ist} + u_{ist}} \right)^2}}{{2\sigma _v^2}}}du} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{Using}}{\kern 1pt} {\mathrm{the}}{\kern 1pt} {\mathrm{result}}{\kern 1pt} {\mathrm{from}}{\kern 1pt} {\mathrm{Kumbhakar}}{\kern 1pt} {\mathrm{and}}{\kern 1pt} {\mathrm{Lovell(2000}},{\mathrm{p80)}},{\mathrm{this}}{\kern 1pt} {\mathrm{reduces}}{\kern 1pt} {\mathrm{to}}} \hfill \\ {} \hfill & = \hfill & {\frac{{A_{ist}}}{{A_{ist} - B_{ist}}}\left( {{\mathrm{\Lambda }}_{ist} + \sigma _v\left[ {\frac{{\phi \left( { - {\mathrm{\Lambda }}_{ist}/\sigma _v} \right)}}{{{\mathrm{\Phi }}\left( {\Lambda _{ist}{\mathrm{/}}\sigma _v} \right)}}} \right]} \right)} \hfill \\ {} \hfill & {} \hfill & { - \frac{{B_{ist}}}{{A_{ist} - B_{ist}}}\left( {{\mathrm{\Lambda }}_{ist}^ \ast + \sigma _v\left[ {\frac{{\phi \left( { - {\mathrm{\Lambda }}_{ist}^ \ast {\mathrm{/}}\sigma _v} \right)}}{{{\mathrm{\Phi }}\left( {{\mathrm{\Lambda }}_{ist}^ \ast {\mathrm{/}}\sigma _v} \right)}}} \right]} \right),} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{where}}{\kern 1pt} {\mathrm{\Lambda }}_{ist} = - \mu _{ist} - \frac{{\sigma _v^2}}{{\sigma _u}}{\mathrm{and}}{\kern 1pt} {\mathrm{\Lambda }}_{ist}^ \ast = - \mu _{ist} - \frac{{\kappa _{st}\sigma _v^2}}{{\sigma _u}}.} \hfill \end{array}$$

(17)

Appendix B: Three-step estimation approach

Three-step estimation proceeds as follows.

Step 1: Compute $\hat \delta _{st} = F^{ - 1}\left( {\hat P_{st}} \right)$ as in the two-step approach.

Step 2: Do a within transformation on the production function Eq. (2) to eliminate the selection bias, and then run Conditional Maximum Likelihood estimation due to Lee (2007) or 2SLS to get $\hat \lambda$ and $\hat \beta _1$.²⁰

Using the efficient instruments IV = $\left( {W_{st}X,W_{st}^2X \ldots } \right)$ for the right-hand side endogenous variables, W_stY_st, is a standard method for 2SLS estimation of $\hat \lambda$ in the spatial econometrics literature. See Anselin (1988) or Kelejian and Prucha (1999). However, the applicability of this Fixed Effect-2SLS methodology heavily depends on the topology of the weight matrix, and this is closely related to the identification problem in network models. Since Bramoulle et al. (2009) and Cohen-Cole et al. (2012) extensively studied sufficient identification conditions for various network models, we won’t discuss them again here. Instead, we will show how network topology affects the applicability of Fixed Effect-2SLS. Let’s assume the network of group s in time t is given by $W_{st}^ \ast$ = ${\textstyle{1 \over {N_t - 1}}}\left( {1_{N_t}1_{N_t}^\prime - I_{N_t}} \right)$, and N_t is constant over t, where N_t is the number of agents in a chosen group. This weight matrix has been widely used in spatial autoregressive models due to its simplicity, however, if we apply a within transformation on the network, we get $QW_{st}^ \ast$ = $- {\textstyle{1 \over {N_t - 1}}}Q$, then the set of IVs becomes $\left( {QX,QW_{st}^ \ast X \ldots } \right)$ = $\left( {QX, - {\textstyle{1 \over {N_t - 1}}}QX \ldots } \right)$, which implies the set of IVs is not full rank and suffers from a perfect multicollinearity as long as N_t doesn’t vary over t, which, in turn, leads to a failure of 2SLS. For this reason, Horrace et al. (2016) introduce two heterogeneous weight matrices using exclusion restrictions, $W_{st}^1$ and $W_{st}^2$, the same- and different-type weight matrices, respectively, where $W_{0,st}^1$ = [w_0,ij,st] is an adjacency matrix with w_0,ij,st = 1 if the i^th and j^th workers in the group s in time t are of the same type and w_0,ij,st = 0 otherwise. Similarly construct $W_{0,st}^2$, then row-normalize $W_{0,st}^1$ and $W_{0,st}^2$ to produce $W_{st}^1$ and $W_{st}^2$ such that $W_{st}^11_{N_t}$ = $W_{st}^21_{N_t}$ = $1_{N_t}$.

In short, in order to implement 3-step estimation, N_t should vary over t, or the weight matrices should be different enough from $W_{st}^ \ast$. If those issues are addressed, we may apply CML or 2SLS.

Step 3: Using the estimates $\hat \lambda$ and $\hat \beta _1$ from above steps, compute $\hat r_{st}$ = $Y_{st} - \hat \lambda W_{st}Y_{st} - X_{1,st}\hat \beta _1$, then a consistent estimate for μ_st conditional on the x_2,st and β₂ is given by $\hat \mu _{st}$ = $\hat r_{st} - 1_{N_t}x_{2,st}\beta _2$. Plug $\hat \mu _{st}$ and $\hat \delta _{st}$ into (7) without the term $\mathop {\sum}\nolimits_t {\kern 1pt} {\mathrm{ln}}\left| {I_{N_t} - \lambda W_{ts}} \right|$ because we already removed endogeneity due to the spatial autoregressive component in the right-hand side of the production equation from the second step, and run Pseudo-ML. Here we get $\hat \beta _2$, $\hat \sigma _u$, $\hat \sigma _v$, and $\hat \alpha$. As in the two step method, we use Murphy and Topel (2002) to correct the standard errors.

Appendix C: The partial and the second derivatives of the log-likelihood and adjustment of the standard error

$$\begin{array}{*{20}{l}} {{\mathrm{ln}}{\kern 1pt} L} \hfill & = \hfill & {\mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{ln}}{\kern 1pt} h\left( {\epsilon _{st} < \delta _{st}} \right)h\left( {\mu _{ist}{\mathrm{|}}\epsilon _{st} < \delta _{st}} \right)} \hfill \\ {} \hfill & = \hfill & {\mathop {\sum}\limits_t {\kern 1pt} N_t{\kern 1pt} {\mathrm{ln}}\frac{\eta }{{\sigma _v}} + \mathop {\sum}\limits_t {\kern 1pt} {\mathrm{ln}}\left| {I_{N_t} - \lambda W_{st}} \right|} \hfill \\ {} \hfill & {} \hfill & { + \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{ln}}\left[ {{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \eta } \right)e^{\frac{{\mu _{ist}\eta }}{{\sigma _v}} + \frac{1}{2}\eta ^2}} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}{\mathrm{\Phi }}\left( { - \frac{{\mu _{ist}}}{{\sigma _v}} - \kappa _{st}\eta } \right)e^{\frac{{\kappa _{st}\mu _{ist}\eta }}{{\sigma _v}} + \frac{{\kappa _{st}^2\eta ^2}}{2}}} \right].} \hfill \end{array}$$

• 1. Notation For simplicity, we use the following notation,

  $$\omega _{ist} = - \frac{{\mu _{ist}}}{{\sigma _v}} - \eta ,\quad \omega _{ist}^{\kappa _{st}} = - \frac{{\mu _{ist}}}{{\sigma _v}} - \kappa _{st}\eta ,$$

  (18)
  $$\begin{array}{l}\phi _{ist} = \phi \left( {\omega _{ist}} \right),\quad \phi _{ist}^{\kappa _{st}} = \phi \left( {\omega _{ist}^{\kappa _{st}}} \right),\\ {\mathrm{\Phi }}_{ist} = {\mathrm{\Phi }}\left( {\omega _{ist}} \right),\quad {\mathrm{\Phi }}_{ist}^{\kappa _{st}} = {\mathrm{\Phi }}\left( {\omega _{ist}^{\kappa _{st}}} \right),\end{array}$$

  (19)
  $$\tau _{ist} = e^{\frac{{\mu _{ist}\eta }}{{\sigma _v}} + \frac{1}{2}\eta ^2},\quad \tau _{ist}^{\kappa _{st}} = e^{\frac{{\kappa _{st}\mu _{ist}\eta }}{{\sigma _v}} + \frac{{\kappa _{st}^2\eta ^2}}{2}},$$

  (20)
  $${\mathrm{\Psi }}_{ist} = {\mathrm{\Phi }}_{ist}\tau _{ist} - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}{\mathrm{\Phi }}_{ist}^{\kappa _{st}}\tau _{ist}^{\kappa _{st}}.$$

  (21)

• 2. The first derivatives

  $$\begin{array}{*{20}{l}} {\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda }}} \hfill & = \hfill & { - \mathop {\sum}\limits_t {\kern 1pt} tr\left[ {\left( {I_n - \lambda W_t} \right)^{ - 1}W_t} \right] + \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{jt}y_{jst}} \right)}}{{\sigma _v}}\frac{{\xi _{ist}}}{{{\mathrm{\Psi }}_{ist}}},} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{where}}{\kern 1pt} \xi _{ist} = \tau _{ist}\left( {\phi _{{\mathrm{ist}}} - \eta {\kern 1pt} {\mathrm{\Phi }}_{ist}} \right) - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}\tau _{ist}^{\kappa _{st}}\left( {\phi _{{\mathrm{ist}}}^{{\mathrm{\kappa }}_{{\mathrm{st}}}} - \kappa _{st}\eta {\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)} \hfill \end{array}$$

  (22)
  $$\begin{array}{*{20}{l}} {\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \eta }}} \hfill & = \hfill & {\mathop {\sum}\limits_t \frac{{N_t}}{\eta } - \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{\xi _{ist}^\eta }}{{{\mathrm{\Psi }}_{ist}}},} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{where}}{\kern 1pt} \xi _{ist}^\eta = \tau _{ist}\left( {\omega _{ist}{\mathrm{\Phi }}_{{\mathrm{ist}}} + \phi _{ist}} \right) - \frac{{\kappa _{st}^2}}{{e^{\delta _{st}}}}\tau _{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}}{\mathrm{\Phi }}_{ist}^{\kappa _{st}} + \phi _{ist}^{\kappa _{st}}} \right).} \hfill \end{array}$$

  (23)
  $$\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \sigma _v}} = - \mathop {\sum}\limits_t \frac{{N_t}}{{\sigma _v}} + \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{\mu _{ist}}}{{\sigma _v^2}}\frac{{\xi _{ist}}}{{{\mathrm{\Psi }}_{ist}}}.$$

  (24)
  $$\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \beta }} = \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{x_{ist}}}{{\sigma _v}}\frac{{\xi _{ist}}}{{{\mathrm{\Psi }}_{ist}}}.$$

  (25)
  $$\begin{array}{*{20}{l}} {\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \alpha }}} \hfill & = \hfill & { - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}}\frac{{\xi _{ist}^\alpha }}{{{\mathrm{\Psi }}_{ist}}},} \hfill \\ {} \hfill & {} \hfill & {{\mathrm{where}}{\kern 1pt} \xi _{ist}^\alpha = {\mathrm{\Phi }}_{ist}^{\kappa _{st}} - \kappa _{st}\eta \left( {\omega _{ist}^{\kappa _{st}}{\mathrm{\Phi }}_{ist}^{\kappa _{st}} + \phi _{ist}^{\kappa _{st}}} \right).} \hfill \end{array}$$

  (26)

• 3. The second derivatives

  $$\begin{array}{l}\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda ^2}} = - \mathop {\sum}\limits_t {\kern 1pt} tr\left[ {\left( {I_n - \lambda W_t} \right)^{ - 1}W_t} \right]^2 - \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \left( {\frac{{\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}y_{jst}}}{{\sigma _v}}} \right)^2\\ \underbrace {\left\{ {\left( {{\textstyle{{\xi _{ist}} \over {{\mathrm{\Psi }}_{ist}}}}} \right)^2 - {\textstyle{{\tau _{ist}\left( {\left( {\frac{{\mu _{ist}}}{{\sigma _v}} - \eta } \right)\phi _{ist} + \eta ^2{\mathrm{\Phi }}_{ist}} \right) - \frac{{\kappa _{st}}}{{e^{\delta _{st}}}}\tau _{ist}^{\kappa _{st}}\left( {\left( {\frac{{\mu _{ist}}}{{\sigma _v}} - \kappa _{st}\eta } \right)\phi _{ist}^{\kappa _{st}} + \kappa _{st}^2\eta ^2{\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)} \over {{\mathrm{\Psi }}_{ist}}}}} \right\}}_{\left( {Q_{ist}} \right)}.\end{array}$$

  (27a)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda _1\partial \lambda _2}}} \hfill & = \hfill & { - \mathop {\sum}\limits_t {\kern 1pt} tr\left[ {\left( {I_n - \lambda _1W_t^1 - \lambda _2W_t^2} \right)^{ - 1}W_t^1\left( {I_n - \lambda _1W_t^1 - \lambda _2W_t^2} \right)^{ - 1}W_t^2} \right]} \hfill \\ {} \hfill & {} \hfill & { - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \left( {\frac{{\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}^1y_{jst}\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}^2y_{jst}}}{{\sigma _v^2}}} \right) \times Q_{ist}.} \hfill \end{array}$$

  (27b)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \eta ^2}}} \hfill & = \hfill & { - \mathop {\sum}\limits_t \frac{{N_t}}{{\eta ^2}} - \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \left\{ {\left( {\frac{{\xi _{ist}^\eta }}{{{\mathrm{\Psi }}_{ist}}}} \right)^2} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - {\textstyle{{\tau _{ist}\left( {\omega _{ist}\phi _{ist} + \left( {\omega _{ist}^2 + 1} \right){\mathrm{\Phi }}_{ist}} \right) - \frac{{\kappa _{st}^3}}{{e^{\delta _{st}}}}\tau _{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}}\phi _{ist}^{\kappa _{st}} + \left( {\left( {\omega _{ist}^{\kappa _{st}}} \right)^2 + 1} \right){\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)} \over {{\mathrm{\Psi }}_{ist}}}}} \right\}.} \hfill \end{array}$$

  (28)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \sigma _v^2}} = - \mathop {\sum}\limits_t \frac{{N_t}}{{\sigma _v^2}} - \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \left( {\frac{{\mu _{ist}}}{{\sigma _v^2}}} \right)^2 \times Q_{ist}.} \hfill \end{array}$$

  (29)
  $$\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} nL}}{{\partial \beta \beta \prime }} = - \frac{1}{{\sigma _v^2}}\mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} {\kern 1pt} x_{ist}x_{ist}^\prime \times Q_{ist}.$$

  (30)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} nL}}{{\partial \alpha ^2}}} \hfill & = \hfill & { - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \left\{ {\left( {\frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}}\frac{{\xi _{ist}^\alpha }}{{{\mathrm{\Psi }}_{ist}}}} \right)^2} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - \frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}^2\eta }}{{e^{\delta _{st}}}}\frac{{\omega _{ist}^{\kappa _{st}}\xi _{ist}^\alpha + 2\phi _{ist}^{\kappa _{st}} + {\mathrm{\Phi }}_{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}} - \kappa _{st}\eta } \right)}}{{{\mathrm{\Psi }}_{ist}}}} \right\}} \hfill \end{array}$$

  (31)
  $$\begin{array}{l}\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda \partial \eta }} = \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} w_{tj}y_{jst}} \right)}}{{\sigma _v}}\\ \underbrace {\left\{ {{\textstyle{{\xi _{ist}\xi _{ist}^\eta } \over {\Psi _{ist}^2}}} + {\textstyle{{\tau _{ist}\left( {\omega _{ist}\eta {\mathrm{\Phi }}_{ist} - {\mathrm{\Phi }}_{ist} + \eta \phi _{ist}} \right) - \frac{{\kappa _{st}^2}}{{e^{\delta _{st}}}}\tau _{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}}\kappa _{st}\eta {\mathrm{\Phi }}_{ist}^{\kappa _{st}} - {\mathrm{\Phi }}_{ist}^{\kappa _{st}} + \kappa _{st}\eta \phi _{ist}^{\kappa _{st}}} \right)} \over {{\mathrm{\Psi }}_{tsi}}}}} \right\}}_{\left( {R_{ist}} \right)}.\end{array}$$

  (32)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda \partial \sigma _v}} = - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}y_{jst}} \right)}}{{\sigma _v^2}}\left\{ {\frac{{\xi _{ist}}}{{{\mathrm{\Psi }}_{ist}}} + \frac{{\mu _{ist}}}{{\sigma _v}} \times Q_{ist}} \right\}.} \hfill \end{array}$$

  (33)
  $$\frac{{\partial ^2lnL}}{{\partial \lambda \partial \beta \prime }} = - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}y_{jst}} \right)x_{ist}^\prime }}{{\sigma _v^2}} \times Q_{ist}.$$

  (34)
  $$\begin{array}{l}\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \lambda \partial \alpha }} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}y_{jst}} \right)}}{{\sigma _v}}\frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}}\\ \underbrace {\left\{ {{\textstyle{{\xi _{ist}\xi _{ist}^\alpha } \over {{\mathrm{\Psi }}_{ist}^2}}} - {\textstyle{{\phi _{ist}^{\kappa _{st}}\left( {1 + \kappa _{st}\eta ^2} \right) + {\mathrm{\Phi }}_{ist}^{\kappa _{st}}\left( {\kappa _{st}^2\omega _{ist}^{\kappa _{st}}\eta ^2 - 2\kappa _{st}\eta } \right)} \over {{\mathrm{\Psi }}_{ist}}}}} \right\}}_{\left( {S_{ist}} \right)}.\end{array}$$

  (35)
  $$\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \eta \partial \sigma _v}} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\mu _{ist}}}{{\sigma _v^2}} \times R_{ist}.$$

  (36)
  $$\frac{{\partial ^2lnL}}{{\partial \eta \partial \beta \prime }} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{x_{ist}^\prime }}{{\sigma _v}} \times R_{ist}.$$

  (37)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \eta \partial \alpha }}} \hfill & = \hfill & { - \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} \frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}}\left\{ {\frac{{\xi _{ist}^\alpha \xi _{ist}^\eta }}{{{\mathrm{\Psi }}_{ist}^2}}} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { - \kappa _{st}\frac{{\omega _{ist}^{\kappa _{st}}\xi _{ist}^\alpha + 2\phi _{ist}^{\kappa _{st}} + {\mathrm{\Phi }}_{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}} - \kappa _{st}\eta } \right)}}{{{\mathrm{\Psi }}_{ist}}}} \right\}} \hfill \end{array}$$

  (38)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\mathrm{ln}}{\kern 1pt} L}}{{\partial \sigma _v\partial \beta }} = - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{x_{ist}}}{{\sigma _v^2}}\left\{ {\frac{{\xi _{ist}}}{{{\mathrm{\Psi }}_{ist}}} + \frac{{\mu _{ist}}}{{\sigma _v}} \times Q_{ist}} \right\}.} \hfill \end{array}$$

  (39)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \sigma _v\partial \alpha }} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{\mu _{ist}}}{{\sigma _v^2}}\frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}} \times S_{ist}} \hfill \end{array}$$

  (40)
  $$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L}}{{\partial \beta \partial \alpha }} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} \frac{{x_{ist}}}{{\sigma _v}}\frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}} \times S_{ist}.} \hfill \end{array}$$

  (41)

• 4. Adjustment of the Hessian for multi-step estimation

Following Murphy and Topel (2002), the adjusted standard error is given by,

$$\Sigma = R_2^{ - 1} + R_2^{ - 1}\left[ {R_3^\prime R_1^{ - 1}R_3 - R_4^\prime R_1^{ - 1}R_3 - R_3^\prime R_1^{ - 1}R_4} \right]R_2^{ - 1},$$

(42)

$$\begin{array}{l}R_1 = - E\frac{{\partial ^2lnL_1}}{{\partial \theta _1\partial \theta _1^\prime }},\\ R_2 = - E\frac{{\partial ^2lnL_2}}{{\partial \theta _2\partial \theta _2^\prime }},\\ R_3 = - E\frac{{\partial ^2lnL_2}}{{\partial \theta _1\partial \theta _2^\prime }},\\ R_4 = E\frac{{\partial lnL_1}}{{\partial \theta _1}}\left( {\frac{{\partial lnL_2}}{{\partial \theta _2}}} \right)^\prime ,\end{array}$$

(43)

where ln L₁, and θ₁ are the log-likelihood and the parameters from the first step, and the ln L₂, and θ₂ are the log-likelihood and the parameters from the second step. R₁ and R₂ can be obtained from the Hessian matrix from the conditional logit model and the second derivatives in Appendix C. As the log-likelihood for the first step is given by,

$${\mathrm{ln}}{\kern 1pt} L_1 = \mathop {\sum}\limits_t \left( {Z_{st}\gamma - {\mathrm{ln}}\mathop {\sum}\limits_{j \in At} {\kern 1pt} e^{Z_{jt}\gamma }} \right) + C,$$

(44)

where γ is the parameter in the first step, Z_st is a set of explanatory variables for line-up s in time t, A_t is a set of possible line-ups in t, and C is the constant which does not depend on γ, the first derivative of the first selection model is given by,

$$\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_1}}{{\partial \gamma }} = \mathop {\sum}\limits_t \left( {Z_{st} - \mathop {\sum}\limits_{j \in At} {\kern 1pt} P_{jt}Z_{jt}} \right),$$

(45)

where P_jt is given by $P_{jt} = {\textstyle{{e^{Z_{jt}\gamma }} \over {\mathop {\sum}\nolimits_{k \in At} {\kern 1pt} e^{m_{kt}\gamma }}}}$. Then R₄ can be computed from the equation above and the first derivatives in Appendix C. For R₃, using ${\textstyle{{\partial \delta _{st}} \over {\partial \gamma }}} = {\textstyle{{\partial F^{ - 1}\left( {P_{st}} \right)} \over {\partial \gamma }}} = {\textstyle{1 \over {e^{ - \delta _{st}}}}}{\mathrm{{\Pi}}}_{st}$ where ${\mathrm{{\Pi}}}_{st} = P_{ts}\left( {m_{ts} - \mathop {\sum}\nolimits_{j \in At} {\kern 1pt} P_{tj}m_{tj}} \right)$, we have,

$$\frac{{\partial {\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_2}}{{\partial \gamma }} = - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{st}\frac{{\tau _{ist}^{\kappa _{st}}\left( {\alpha \xi _{ist}^\alpha - \kappa _{st}{\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)}}{{{\mathrm{\Psi }}_{ist}}}.$$

(46)

Then, the cross-partial derivatives are given by,

$$\begin{array}{l}\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_2}}{{\partial \gamma \partial \lambda }} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{ts}\frac{{\left( {\mathop {\sum}\limits_{j \ne i \in s} {\kern 1pt} w_{tj}y_{tsj}} \right)}}{{\sigma _v}}\tau _{ist}^{\kappa _{st}}\\ \underbrace {\left\{ {{\textstyle{{\left( {\alpha \xi _{ist}^\alpha - \kappa _{st}{\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)\xi _{ist}} \over {{\mathrm{\Psi }}_{tsi}^2}}} + {\textstyle{{\alpha \kappa _{st}\eta \xi _{ist}^\alpha - \kappa _{st}\eta \left( {\kappa _{st}{\mathrm{\Phi }}_{ist}^{\kappa _{st}} - \alpha {\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right) + \kappa _{st}\phi _{ist}^{\kappa _{st}} - \alpha \phi _{ist}^{\kappa _{st}}} \over {{\mathrm{\Psi }}_{tsi}}}}} \right\}}_{\left( {T_{ist}} \right)}.\end{array}$$

(47)

$$\begin{array}{l}\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_2}}{{\partial \gamma \partial \eta }} = - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{ts}\tau _{ist}^{\kappa _{st}}\\ \left\{ {{\textstyle{{\left( {\alpha \xi _{ist}^\alpha - \kappa _{st}\Phi _{ist}^{\kappa _{st}}} \right)\xi _{ist}^\eta } \over {{\mathrm{\Psi }}_{tsi}^2}}} - \kappa _{st}{\textstyle{{\alpha \omega _{ist}^{\kappa _{st}}\xi _{ist}^\alpha - {\mathrm{\Phi }}_{ist}^{\kappa _{st}}\omega _{ist}^{\kappa _{st}}\kappa _{st} - \kappa _{st}\phi _{ist}^{\kappa _{st}} + \alpha \left( {2\phi _{ist}^{\kappa _{st}} + \omega _{ist}^{\kappa _{st}}{\mathrm{\Phi }}_{ist}^{\kappa _{st}} - \eta \kappa _{st}{\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)} \over {{\mathrm{\Psi }}_{tsi}}}}} \right\}\end{array}$$

(48)

$$\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_2}}{{\partial \gamma \partial \sigma _v}} = \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{ts}\frac{{\mu _{ist}}}{{\sigma _v^2}}\tau _{ist}^{\kappa _{st}}T_{ist}.$$

(49)

$$\frac{{\partial ^2lnL_2}}{{\partial \gamma \partial \beta \prime }} = \mathop {\sum}\limits_t {\kern 1pt} \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{ts}\frac{{x_{ist}^\prime }}{{\sigma _v}}\tau _{ist}^{\kappa _{st}}T_{ist}.$$

(50)

$$\begin{array}{*{20}{l}} {\frac{{\partial ^2{\kern 1pt} {\mathrm{ln}}{\kern 1pt} L_2}}{{\partial \gamma \partial \alpha }}} \hfill & = \hfill & { - \mathop {\sum}\limits_t \mathop {\sum}\limits_{i \in s} {\kern 1pt} {\mathrm{{\Pi}}}_{ts}\tau _{ist}^{\kappa _{st}}\left\{ {\frac{{\tau _{ist}^{\kappa _{st}}\delta _{st}}}{{e^{\delta _{st}}}}\frac{{\left( {\alpha \xi _{ist}^\alpha - \kappa _{st}{\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \right)\xi _{ist}^\alpha }}{{{\mathrm{\Psi }}_{tsi}^2}}} \right.} \hfill \\ {} \hfill & {} \hfill & {\left. { + {\textstyle{{\xi _{ist}^\alpha \left( {1 - \alpha \omega _{ist}^{\kappa _{st}}\delta _{st}\eta } \right) + \delta _{st}\eta {\mathrm{\Phi }}_{ist}^{\kappa _{st}}\left( {\omega _{ist}^{\kappa _{st}}\kappa _{st} - \alpha \omega _{ist}^{\kappa _{st}} + \alpha \eta \kappa _{st}} \right) + \delta _{st}\eta \phi _{ist}^{\kappa _{st}}\left( {\kappa _{st} - 2\alpha } \right) - \delta {\mathrm{\Phi }}_{ist}^{\kappa _{st}}} \over {{\mathrm{\Psi }}_{tsi}}}}} \right\}.} \hfill \end{array}$$

(51)