LASSO for Stochastic Frontier Models with Many Efficient Firms

Abstract

We apply the adaptive LASSO to select a set of maximally efficient firms in the panel fixed-effect stochastic frontier model. The adaptively weighted L₁ penalty with sign restrictions allows simultaneous selection of a group of maximally efficient firms and estimation of firm-level inefficiency parameters with a faster rate of convergence than least squares dummy variable estimators. Our estimator possesses the oracle property. We propose a tuning parameter selection criterion and an efficient optimization algorithm based on coordinate descent. We apply the method to estimate a group of efficient police officers who are best at detecting contraband in motor vehicle stops (i.e., search efficiency) in Syracuse, NY.

Keywords:

• Adaptive LASSO
• Fixed-effect stochastic frontier model
• L1 regularization
• Panel data
• Sign restriction
• Zero inefficiency

Supplementary Materials

The online supplementary material includes the proofs of the main results in Sections 3 and 4, and additional Monte Carlo simulation results.

Acknowledgments

We are grateful to the following for helpful comments and suggestions: the associate editor, two anonymous referees, Badi Baltagi, Christopher Parmeter and the participants at the 15th European Workshop on Efficiency and Productivity Analysis, the 28th annual meeting of the Midwest Econometrics Group and the International Association for Applied Econometric 2019 Annual Conference.

Notes

1 Rho and Schmidt (2015) raise an identification issue for this model.

2 Hence, our proposed method is a constrained LASSO (e.g., Hua et al. 2015), which hasn’t yet been applied in economic contexts to the best of our knowledge.

3 In comparison, Park et al. (1998) study the asymptotic properties of the LSDV estimators with iid data.

4 Fixed effects are still widely used to measure agent-specific effects. Examples include measurements of teacher quality (Rothstein 2010; Chetty et al. 2014), school value-added (Angrist et al. 2017) and hospital efficacy (Friedson et al. 2019), where the quantities of interest are calculated by some form of fixed effects. Moreover, if high-frequency data for multiple periods (e.g., years) are available, we can partition the data along the time dimension (e.g., by year) and deploy separate high frequency models for each partition, while allowing inefficiency to vary over time.

5 Note that in the LSDV estimation, the firm with the largest firm fixed effect estimate has a zero inefficiency estimate. For ${\widehat{u}}_i=0$, we use an arbitrarily small value (e.g., $1/N$) to construct the weight.

6 Kutlu et al. (2020) apply the shrinkage technique by Su et al. (2016) to parametric SF models to identify groups of firms sharing the same slope parameters.

7 Alternatively, exponential moment conditions can be employed as in Bonhomme and Manresa (2015).

8 In particular, Assumption 2-(3) implies that λ should decrease as N increases when $N\gg T$.

9 Recall that we impose only finite moment conditions for the errors and covariates and allow for time-series dependence.

10 See the proof of Theorem 1 in the online supplementary material for more details.

11 This reasoning readily applies to the balanced panel case. For an unbalanced panel, the shrinkage effect is $(\lambda /2T_i){\widehat{\pi }}_i$ where T_i is the number of time periods for i, which does not necessarily preserve the ordering of ${\widehat{u}}_i$. In this case, the standard coordinate decent algorithm based on the two equations above can be used.

12 In fact, directly minimizing (5) using any typical method shrinks not only ${\widehat{u}}_i(\lambda )$ but also $\widehat{\alpha }(\lambda )$. However, this is an undesirable shrinkage bias, which may slow down the convergence of $\widehat{\alpha }(\lambda )$, particularly when N is large (Equation (A.6) in the online supplementary material includes the explicit form of this bias). Therefore, in the spirit of post-LASSO estimation (e.g., Belloni and Chernozhukov 2013), our algorithm skips the steps that induce this undesirable shrinkage effect and achieves smaller finite sample bias of $\widehat{\alpha }(\lambda )$. Skipping these steps doesn’t alter any of the asymptotic results in Section 3.

13 A ${\varphi }_{NT}$ that satisfies the rate conditions can take the form of $\nu {(\log N)}^2{c}_{NT}$ where ν is some positive constant that gives flexibility in controlling the degree of penalization in the criterion (similar to ERIC by Hui et al. 2015), and c_NT is a diverging sequence, but its rate of divergence is arbitrarily slow. Note that $c_{NT}=\log \left(\log \left(NT/(N+T)\right)\right)\approx \min \{\log \left(\log N\right),\log \left(\log T\right)\}$ in our case. We also experimented with other types of selection criteria in the simulation study, including ERIC, $I{C}_{p1}$ by Bai and Ng (2002), and LIC${}^{\text{BIC}}$ by Lee and Phillips (2015), and found that (7) performs best in this panel SF model.

14 Additional simulation results for $\delta \in \{0.1,0.9\}$ are in the online supplementary material. As δ decreases, the finite sample performance of the LASSO estimators deteriorates, but we still observe notable improvements from the LASSO estimation compared to the LSDV.

15 The variances of ω_it were chosen so that the overall variance of v_it is approximately one.

16 We are free to choose the value of γ as long as it satisfies the rate conditions in Assumption 2-(3). From the asymptotic analysis, we can see that choosing a larger value for γ ensures the LASSO estimates zero coefficients as zero, but also increases the probability of estimating (small) nonzero coefficients as zero. Therefore, in applications γ should be chosen in light of this tradeoff.

17 When N is very large (e.g., 1000), however, we find that the RMSE of ${\widehat{U}}_{\text{LASSO}}$ starts to increase. This is related to the form of ${\varphi }_{NT}$ in (7). The impact of ${\varphi }_{NT}$ on the selection performance and consequently the estimation of α and u_i is discussed later when we discuss the selection accuracy of the LASSO estimation.

18 Wang and Schmidt (2009) also document the “upward” bias of LSDV estimators using simulations.

19 The degree and pattern of this tradeoff apparently depends on the choice of ${\varphi }_{NT}$, which ultimately affects the estimation of α and u_i . Therefore, similarly as γ, in practice ${\varphi }_{NT}$ should be chosen in the light of this tradeoff. However, we find that ${\varphi }_{NT}$ that is optimal for a wide range of N is difficult to find (e.g., our ${\varphi }_{NT}$ appears to grow rather quickly as N increases, leading to an underestimation of α when $N=1000$). Optimal choice of ${\varphi }_{NT}$ is left for future research.

20 Defining police productivity by success rate has a limitation that officers with higher standard for guilt tend to have a higher success rate since they would only search vehicles with a high probability of carrying contraband. We might consider a composite measure that accounts for both quantity and quality of search, which is left for future research.

21 We use the average number of searches among officers for the T in the BIC criterion (7).

22 Note that our analysis is based on observations within one year and Experience is a yearly variable, so it is time-invariant in this analysis.

23 The between equation is: ${\widehat{y}}_{i.}={\alpha }_0+z_i^{\prime }{\beta }_{0,2}+{\varsigma }_i$, where ${\widehat{y}}_{i.}=(1/T_i){\sum }_{t=1}^{T_i}\left({\text{arrest}}_{it}-x_{it}^{\prime }{\widehat{\beta }}_{1,\text{LSDV}}\right)$ and ${\varsigma }_i$ is the regression error that contains $u_{0,i}$ and the original two-sided error. This regression is valid as long as z_i is exogenous to $u_{0,i}$ and the original two-sided error.

24 The dotted and dashed lines are (kernel smoothed) density functions estimated by the default “ksdensity” command in Matlab.