Identification and estimation of panel semiparametric conditional heteroskedastic frontiers with dynamic inefficiency

Abstract.

We study a semiparametric panel stochastic frontier model with nonseparable unobserved heterogeneity, which allows for time-varying conditional heteroskedastic productivity components. It does not require distributional assumptions on random noise except conditional symmetry. We utilize conditional characteristic functions from Kotlarski’s Lemma to derive new moment conditions that yield the identification of the heteroskedastic variance parameters of inefficiency and random noise. Identification only requires a panel with three periods for serially correlated inefficiency. A nonparametric estimation procedure is also developed for the conditional variance of inefficiency, and its convergence rate is established. Monte Carlo simulation shows that the estimator is robust to misspecification of inefficiency distributions.

Keywords:

• Conditional heteroskedastic productivity
• Kotlarski’s lemma
• Stochastic Frontier analysis
• semiparametric panel
• time-varying Inefficiency

Acknowledgments

We are grateful to helpful comments by Alfonso Flores-Lagunes, Subal Kumbhakar, Hugo Jales, Yulong Wang, and participants at the 16th EWEPA conference and seminars at Syracuse, Albany, and Binghamton. Jun Cai gratefully acknowledges financial support from the National Natural Science Foundation of China (grant no.72303074).

Notes

1. The scaling property specification possesses some appealing features. See Wang and Ho (2010), Wang and Schmidt (2002), Alvarez et al. (2006) and some others.

2. If one would like to assume that higher central moments exist for U and V, more odd moment conditions can be used to identify the unknown variance parameters ${\sigma }_u^2$ in a similar way.

3. The distribution of e_it may be asymmetric and since $E(U_{it}|X_{it}=x)\ne 0$ we do not have much to say about $E(e_{it}|X_{it}=x)$.

4. One can build a joint multivariate Half-normal distribution on U_it as section 6.3 in Belotti and Ilardi (2018); or construct a joint multivariate Exponential distribution on U_it following Basu (1988) without defining e_it.

5. For the multivariate case (i.e., p > 1), we let ${\omega }_{i,j}(x)=\frac{K(H_j^{-1}(X_{i1}-x))K(H_j^{-1}(X_{i2}-x))}{{\sum }_{i=1}^{n}K(H_j^{-1}(X_{i1}-x))K(H_j^{-1}(X_{i2}-x))}$for $j=A,B$, where K is a nonnegative p-variate kernel function, and H_j is a p×p bandwidth matrix that is symmetric and positive definite. The rest of the discussion holds if we simply consider the product kernel $K(r)={\prod }_{\ell =1}^{p}k(r_{\ell })$ and H_j = h_jI_p for some bandwidth parameter h_j, where I_p is the identity matrix of rank.

6. Here A and following B are generic symbols for A₁₂, A₂₁ and B₁₂, B₂₁.

7. We have at least two periods and X_it and X_iτ are treated as two random variables.

8. Recall the target (distribution) parameters σ_u(x) and σ_v(x) are power functions of the nonparametric covariance A(x) and B(x). This is also necessary for the uniform convergence of the target parameter.

9. Recall that we have two kernels for a univariate X_it as we consider two consecutive periods.

10. Here c is a function of d₁, d₂ and p where d₁ is the maximum continuous derivative of the conditional cumulative distribution function F_m(t|x), d₂ is the maximum continuous derivative of the joint density f(x, x) and p is the dimension of X. For details, see Theorem 4–5, and Theorem 7-8 in Evdokimov (2010).

11. In the case of N = 2500, T = 2, one Monte Carlo simulation is less than 1 second with rule of thumb bandwidth but about one hour implementing the leave-one-out cross validation method to choose the bandwidth. In the application section, we use leave-one-out cross validation to choose the bandwidths.

12. The support trimming procedure ensures that the joint density $f_{X_{i1},X_{i2}}(x,x)>0$, which is a key identification assumption in the nonparametric panel setting.

13. We focus on the impact of unknown model specification on estimation here. We will discuss the misspecification of distributional assumptions in next simulation setting.

14. There is one recent package sftfe for “true” fixed effects models but it cannot estimate “true” random effects models.

15. It takes 6 hours to obtain one simulation with sfpanel in Stata with a sample size of 2500 for one specification with one distribution-FE/RE case.

16. The infeasible “true” fixed effects estimators do not perform as expected in the both homogeneous specification (in pair column 1 and 3) due to the implementation issues of the Stata package sfpanel.

17. For serially correlated U_it, there are some technical problems to generate AR(1) inefficiency following Exponential distribution. So we only consider normal-half normal cases here.

18. We have good reason to believe that results from random effects models are very similar based on previous simulations.

19. Recall that Exp(σ_u) is Gamma(1,σ_u).

20. Misspecify Half Normal as Exponential or vice versa is not appropriate here as Half Normal and Exponential belongs to different distribution families: super smooth and ordinary smooth. Details could be referred to Cai et al. (2021). We recommend applying some diagnostic methods as suggested by Cai et al. (2021) to determine which distribution family is more likely and proceed with our method. Or one could do both distributions.

21. Recall that $E(u)=\mu +\frac{\varphi (-0.5)}{1-\mathrm{\Phi }(-0.5)}\ast {\sigma }_u$.

22. Note the slight notational abuse: the subscript i is an index, while the bold 𝕚 is the imaginary number.

23. E(u) = σ_u, $Var(u)={\sigma }_u^2$, Skewness(u) = 2.

24. The second equality holds as m₁(X), m₂(X), m₃(x) and σ₁₂(X), σ₁₃(X) are constants given $X_{i1}=X_{i2}=x$.

25. Note that c, c(x) and C(x) are different notations. Similarly b, b(x) and B(x) are different notations.