ESTIMATION OF THE TWO-TIERED STOCHASTIC FRONTIER MODEL WITH THE SCALING PROPERTY. 1. Introduction

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1 ESTIMATION OF THE TWO-TIERED STOCHASTIC FRONTIER MODEL WITH THE SCALING PROPERTY CHRISTOPHER F. PARMETER Abstract. The two-tiered stochastic frontier model has enjoyed success across a range of application domains where it is believed that incomplete information on both sides of the market leads to surplus which buyers and sellers can extract. Currently, this model is hindered by the fact that estimation relies on very restrictive distributional assumptions on the behavior of incomplete information on both sides of the market. However, this reliance on specific parametric distributional assumptions can be eschewed if the scaling property is invoked. The scaling property has been well studied in the stochastic frontier literature, but as of yet, has not been used in the two-tier frontier setting. 1. Introduction Despite the popularity of the stochastic frontier model, a common criticism (and concern) is its reliance on distributional assumptions to separately identify two-sided noise from inefficiency. In the presence of observable characteristics which can potentially influence the level of inefficiency, it is possible to dispense with distributional assumptions, either in a parametric setting, by assuming the scaling property (Wang & Schmidt 2002) and estimating the model via nonlinear least squares, or in a semiparametric fashion (Tran & Tsionas 2009, Parmeter, Wang & Kumbhakar 2016), deploying the partially linear estimator. The scaling property has many attractive features (Alvarez, Amsler, Orea & Schmidt 2006), and is satisfied by any single parameter distribution, such as the commonly called upon half-normal and exponential distributions (for more background see Parmeter & Kumbhakar 2014). Concerns over distributional specification carry over with equal force to the two-tier stochastic frontier model (Polachek & Yoon 1987). This model is called upon in settings where University of Miami Date: May 22, Key words and phrases. Incomplete Information, Nonlinear Least Squares, Heteroskedasticity. Christopher F. Parmeter, Department of Economics, University of Miami; cparmeter@bus.miami.edu. All R code used in this paper is available upon request. 1

2 2 TWO-TIER it is likely that incomplete information exists on both sides of the market. For example, Polachek & Yoon (1987) describe a labor market where search frictions can lead to suboptimal placement of workers in firms, leading to imperfect pairings. These information gaps are assumed to exist on both sides of the labor market, where firms incomplete information leads to paying wages which are larger than need be and incomplete information leads workers to accept wages that are smaller than need be. These two one-sided impacts on wages are assumed to come from exponential distributions, and a tractable likelihood function is available, from which all of the parameters of the model can be estimated. Here it could be argued that distributional misspecification is even more important since it is less obvious the impact that misspecification will have on the corresponding maximum likelihood estimator. In the presence of observable variables which influence incomplete information the scaling property can be invoked to estimate the two-tier stochastic frontier model eschewing distributional assumptions. With the scaling property a simple nonlinear least squares estimator can recover all pertinent information of the model without requiring distributional assumptions. It appears that this feature has gone unnoticed in applications where determinants of incomplete information exist; see Groot & Oosterbeek (1994), Kumbhakar & Parmeter, (2009, 2010), and Tomini, Groot & Pavlova (2012) as a few examples who use the exponential distributional assumptions suggested by Polachek & Yoon (1987). While not nearly as popular as the stochastic frontier model proposed by Aigner, Lovell & Schmidt (1977) and Meeusen & van den Broeck (1977), the two-tier stochastic frontier model has recently seen increased interest in applied work. The application of the two-tier stochastic frontier model, while continuing to be used to investigate incomplete information in labor markets (Sharif & Dar 2007, Murphy & Strobl 2008), has seen repeated application across a wide array of scientific milieus. Beginning with Gaynor & Polachek (1994) (and further developed by Chawla (2002) and Tomini et al. (2012)), the model has been used to study the impact of incomplete information in the market for medical services while Groot & van den Brink (2007) studied quality of life using the two-tier methodology. The two-tier approach has also by used by Lian & Chung (2008) to investigate the effects of financing constraints and agency costs on investment behavior and Kumbhakar & Parmeter (2010) to study bargaining and its subsequent effect on housing prices. Kinukawa & Motohashi (2010), following Kumbhakar & Parmeter (2010), studied bargaining in technology markets. Poggi (2010) used the model to investigate job satisfaction, Ferona & Tsionas (2012) modeled over and underbidding in timber auctions, Yu & Liang (2012), in a similar vein as Lian & Chung

3 TWO-TIER 3 (2008), applied the two-tier frontier to investigate the impact on dividend policies from firm financing constraints, and Huang (2013), used the model to separate premium and underpricing effects in IPOs of Chinese firms. Lastly, Wang, (2016a, 2016b) has used the two-tier model to investigate bargaining over foreign aid being distributed across countries. Interest from a methodological standpoint has also increased. Polachek & Yoon (1996) extended the basic cross-sectional model to a panel framework while Tsionas (2012) proposed estimation of the model assuming the two one-sided error terms were distributed gamma and deployed a Fourier transform, based on the characteristic function of the composed error, to estimate the parameters of the composed error distribution. Papadopoulos (2015) derived the likelihood function and conditional moments (in the spirit of Jondrow, Lovell, Materov & Schmidt 1982) of the one-sided error terms assuming both were distributed half-normally. Blanco (2016), following Greene s (2010) approach for modelling selection in the stochastic frontier model, introduced selection into the two-tier frontier model. It is clear that the two-tier stochastic frontier remains of interest for both practitioners and academicians. And yet, outside of Tsionas (2012) and Papadopoulos (2015), no attempt at mitigating the impact that the assumption of exponential distributions for the two onesided errors has on the model has been explored. The present paper takes a step in this direction my describing how the scaling property can be deployed to estimate the two-tier stochastic frontier model with nonlinear least squares, obviating the need, in this context, for distributional assumptions. What is required is the presence of observable covariates that influence the level of incomplete information, variables that have appeared in many of the previously listed applications of the two-tier stochastic frontier model. Their are many benefits to the scaling property. As noted by Alvarez et al. (2006), ease of interpretation and the ability to dispense with distributional assumptions are key amongst them. Invoking the scaling property to estimate a two-tier stochastic frontier model, while straightforward, has not appeared in the applied literature, nor does it appear that researchers have recognized this connection. Naturally, imposition of correct distributional assumptions will produce efficient estimators through maximum likelihood, but it is rarely the case that information on these distributions exists. Further, as found in Tsionas (2012), the exponential distribution was rejected in favor of a gamma distribution for one of the two one-sided error terms. While the gamma distribution is more flexible than, and nests, the exponential distribution, because a closed form solution to the likelihood does not exist, it is likely this distributional specification will not be adopted regularly in practice. However,

4 4 TWO-TIER the simplicity of nonlinear least squares does suggest that empirical researchers may be more comfortable arguing for imposition of the scaling property, and avoiding issues of distributional specification altogether. The remainder of the article is setup as follows. Section 2 presents an overview of the two-tier stochastic frontier model. Section 3 describes estimation of the model invoking the scaling property. Section 4 provides a Monte Carlo study. Section 5 contains concluding remarks and avenues for future research. 2. Two-Tier Frontier Estimation The salient feature of the two-tier stochastic frontier model is that the outcome variable has both a lower and an upper bound (Polachek & Yoon 1987). The model can be written for the i th observation (i = 1,..., n) as (1) y i = m(x; β) + ε i = x iβ + ε i, where y i is the outcome variable, x i is a vector of covariates, β is the corresponding parameter vector, and ε i = v i u i + w i represents the composite error term encapsulating both sources of incomplete information and stochastic noise. In a buyer and seller framework m(x; β) is the market value of the good. The lowerboundary (frontier) of price (y) is the minimum that the seller is willing to accept and is given by m(x; β) u, u 0. Similarly, the upper boundary (frontier) indicates the maximum that the buyer is willing to pay and is given by m(x; β)+w, w 0. Furthermore, the frontiers are also likely to be affected by the presence of noise, v, which can take both positive and negative values. 1 The vector of parameters in (1), β, can be obtained using standard regression techniques. For example, ordinary least squares (OLS) will yield unbiased estimators of the slope coefficients. Since u and w are one-sided, E (ε) may not be zero, even if E (v) = 0. Consequently, the OLS estimator of the intercept will in general be biased. Thus, if the objective is to estimate β then the OLS estimator of the slope coefficients will be unbiased and consistent. However, we are interested in not only estimating β but also to disentangle the one-sided 1 Other papers formalizing the two-tier stochastic frontier have used v and w to encapsulate incomplete information on each side of the market while letting u capture two sided noise. Given that v is almost universally used in textbook econometrics as random noise, and that the stochastic frontier literature has commonly used u to capture inefficiency, we have decided to use u and w to capture incomplete information. Moreover, in alphabetical order, you have that u is under v (hence ) and w is over v (hence +).

5 TWO-TIER 5 error terms from the composed error term ε. For this reason, the model is estimated using maximum likelihood (ML) based on the following distributional assumptions of the error components, viz., u, v, and w. It is commonly assumed that: (i) v i i.i.d. N(0, σ 2 v), (ii) u i i.i.d. Exp(σ u ), (iii) w i i.i.d. Exp(σ w ), and (iv) the error components are distributed independently of each other and from the regressors, x. 2 The use of an exponential distribution is commonplace in standard single-tier stochastic frontier studies when ML is used. Based on these distributional assumptions, it is straightforward (but tedious) to derive the probability density function (pdf) of ε i, f(ε i ), which is 3 (2) f (ε i ) = ea 1i ea2i Φ (b 1i ) + Φ (b 2i ), σ u + σ w σ u + σ w ( ) where a 1i = ε i σ u + σ2 v, a 2σu 2 2i = σ2 v ε 2σw 2 i ε σ w, b 1i = i σ v + σv σ u, and b 2i = ε i σ v likelihood function for a sample of n observations is n (3) ln L (x; θ) = n ln (σ u + σ w ) + ln [e a 1i Φ (b 1i ) + e a 2i Φ (b 2i )] where θ = {β, σ v, σ u, σ w }. maximizing the above log likelihood function. i=1 σv σ w. The log The ML estimates of all the parameters can be obtained by Alternatively, if both u and w were assumed to be distributed half-normal, then the density of ε i would be (See Papadopoulos (2015) for the full derivation): (4) f (ε i ) = 2 s φ(ε i/s) [G(ε i ; 0, ω 1, λ 1 ) G(ε i ; 0, ω 2, λ 2 )] where s = σ 2 v + σ 2 u + σ 2 w, ω 1 = s σv 2+σ2 w σ u G(ε i ; 0, ω, λ) = 2B, ω 2 = s σv 2+σ2 u σ w, λ 1 = σwσvs σ u ( (ε i /ω, 0), ρ = λ ), 1 + λ 2, λ 2 = σuσvs σ w and where B(, ) is the cumulative distribution function of a bivariate standard normal random deviate. Now suppose that we wished to model σ u and σ w as functions of sets of covariates, z u and z w, respectively. In this case, defining σ u σ u (z u ; δ u ) and σ w σ w (z w ; δ w ), our optimization 2 Here Exp (σ z ) denotes a random variable z that is exponentially distributed with mean σ z and variance σ 2 z. 3 See Kumbhakar & Parmeter (2009) for the full derivation.

6 6 TWO-TIER problem for the exponential-exponential setting becomes n (5) arg max ln L (x, z; θ) = n ln (σ u + σ w ) + θ ln [e a 1i Φ (b 1i ) + e a 2i Φ (b 2i )] where θ = {β, δ u, δ w, σ v }. To allow incomplete information to depend on observable characteristics, Groot & Oosterbeek (1994) proposed modeling the parameters of the exponential distributions of u and w, similar to the determinants of efficiency approach that is common in the stochastic frontier literature. i=1 One caution with the specification used in Groot & Oosterbeek (1994) (as well as Groot & van den Brink 2007, Tomini et al. 2012), is that the means of both u and w are modeled linearly, which cannot mimic the true mean of the distribution since both error terms are one-sided. The more common approach to modeling the mean of the distribution, initially proposed by Kumbhakar & Parmeter (2009, 2010), is as an exponential function of the observable characteristics. 3. Estimation with the Scaling Assumption Whether the assumption of half-normal or exponential is used for the two one-sided error components, because these distributions belong to the class of one parameter distributions, they naturally possess the scaling property. 4 A benefit of the scaling property is that it permits estimation of the stochastic frontier model without explicit distributional assumptions (Wang & Schmidt 2002, Alvarez et al. 2006, Parmeter & Kumbhakar 2014). This same feature carries over to the two-tier stochastic frontier model. To see this, start with (1), assuming that the distributions of w and u depend upon the level of observable characteristics z u and z w. In this case, taking expectations we have: (6) E[y x, z u, z w ] = m(x) E[u x, z u, z w ] + E[w x, z u, z w ]. If both u and w are exponentially distributed, then E[u x, z u, z w ] = σ u (z u ; δ u ) and E[w x, z u, z w ] = σ w (z w ; δ w ). Given that these expectations need to be positive, the common parameterizations, e δ uz u and e δ wz w can be used. Moreover, the nonlinearity of these expectations naturally leads to identification of both δ u and δ w, provided that if z u = z w, that δ u δ w. 5 4 Other examples include the Gamma or Weibull distributions with fixed shape parameters. In this case the shape parameter does not need to be known, but it has to be a constant w.r.t. (x, z). 5 Alternatively, different functional forms for σ u (z u ) and σ w (z w ) could be used to achieve identification when z u = z w.

7 TWO-TIER 7 More specifically, define u i = u(z u,i, δ u ) and w i = w(z w,i, δ w ). The random variable u, say, possesses the scaling property if u(z u,i, δ u ) = g(z u,i, δ u )u i, where g(z u,i, δ u ) 0 and u i is not dependent upon z. As noted by Wang & Schmidt (2002), g(z u,i, δ u ) is termed the scaling function and the distribution of u is the basic distribution. All single parameter distributions, such as the exponential and half-normal, possess the scaling property by default. However, other common distributions do not satisfy the scaling property without some type of restriction. For example, the truncated normal distribution, N(µ, σ 2 ) + with µ 0 does not directly satisfy the scaling property; it can be made to satisfy the scaling property by assuming that the basic distribution is N(µ, σ 2 ) +. Another boon to invoking the scaling property for u and w is that the distributions of each can differ. The important point to recognize is that regardless of the basic distribution, if a random variable possesses the scaling property, then the conditional mean of this random variable can be estimated without distributional assumptions. To estimate the essential features of the two-tier stochastic frontier model with the scaling property, nonlinear least squares is required (given the necessary nonlinearity in the assumed functional forms for the conditional means of u and w): ( (7) β, δu, δ ) n [ w, µ u, µ w = min n 1 y i x iβ + µ ue z u,i δu µ we z δw] 2 w,i. β,δ u,δ w,µ u,µ w i=1 The parameters µ w and µ u are to set the scale of each of the individual means and can be subsumed inside the exponential functions by including an intercept in both z u and z w. The parameterizations e z u,i δu and e z w,i δw are to ensure that the components of the conditional mean of y characterized by incomplete information are of the correct sign. Given the presence of heteroskedasticity in the composed error term it is recommended that either a generalized nonlinear least squares procedure is deployed (though this requires distributional assumptions to disentangle σv 2 from σu 2 and σw 2 ) or heteroskedasticity robust standard errors are constructed to conduct proper inference. Heteroskedasticity robust errors can be easily calculated by noting that the nonlinear least squares estimator is simply an M-estimator. Consider an objective function Ψ(y, x, z, θ), the derivative of which is ψ(y, x, z, θ). An M-estimator of θ, θ, is defined as n (8) ψ(y i, x i, z i, θ) = 0. i=1

8 8 TWO-TIER Nonlinear least squares and maximum likelihood are both types of M-estimators. As shown in White (1994, Thm. 6.10), (9) n ( θ θ ) d N(0, S(θ)), where S(θ) = H(θ)D(θ)H(θ), with H(θ) = ( E[ ψ (y, x, z, θ)] ) 1 and D(θ) = V ar[ψ(y, x, z, θ)]. H(θ) is the inverse of the expected value of the second derivative of the objective function, Ψ(y, x, z, θ), and can be thought of as the Hessian matrix. A natural estimator for D(θ) is the outer product of the empirical first derivatives of the objective function (or the outer product of gradients estimator): n (10) D = n 1 ψ(y i, x i, z i, θ)ψ(y i, x i, z i, θ). i=1 These matrices can be easily calculated for the scaling form of the two-tier stochastic frontier model. For example, in R, the sandwich package (Zeileis 2004, 2006) can easily construct robust standard errors for a model estimated via nonlinear least squares through the nls() command Identification. The δ u and δ w parameters can be difficult, if not impossible, to identify in certain settings. Consider the case where δ u = δ w and the same zs influence the one-sided error distributions, z u = z w. In this case E[w z] E[u z] = 0 z. In practice there will need to be sufficient differences in either the influence of specific variables on the one-sided errors (through δ u and δ w ), or different variables need to influence u and w all together (through z u and z w ), or in the functional specifications of the one-sided means. One important reason to model the scaling function as nonlinear, as opposed to the linear specification of say Groot & Oosterbeek (1994), is that the effect of z on either u or w can be identified, provided the effects are not identical. In the linear framework, only the joint effect can be identified, i.e. E[w u z]. Again, given that the means must be positive, the linear specification is not appropriate, but this provides further insight into why the linear specification should not be adopted in applications if the scaling assumption is imposed. One consequence of the linear specification, is that if distributional assumptions were to be eschewed, then identification is not possible unless z u z w. Further still, as shown in Harding, Rosenthal & Sirmans (2003), unraveling the impacts of incomplete information on

9 TWO-TIER 9 market outcomes requires assumptions on different effects in the model. Here, because we do not use linear specifications for the conditional means of u and w, it becomes easier to tease out impacts from the data. 4. Finite Sample Performance In this section we present the results of a series of simulations designed to demonstrate that nonlinear least squares estimation of the two-tier stochastic frontier model performs reasonably well when incomplete information depends on variables z u and z w Design of the Simulation Experiment. We will generate data from a simple twotier stochastic frontier model: (11) y i = βx i + v i u i + w i. The stochastic noise v i, is generated as i.i.d. N(0, σv), 2 while we allow the distribution of u i and w i to be either half-normal or exponential random variables, scaled by exponential functions of z u and z w, respectively: (12) u i = e δu 0 +δu 1 z u,i u i ; w i = e δw 0 +δw 1 z w,i wi where both u i and wi are i.i.d random variables. We consider three separate cases. First, we focus on the setting where both u and w are distributed exponential, second we have both u and w to be distributed half-normal, and finally, we have u as an exponential and w as half-normal. For all simulations we generate the vector (x i, z u,i, z w,i ) as i.i.d standard trivariate normal with correlation ρ. For all simulations x i, z i, v i, u i and wi are mutually independent from one another. 10,000 simulations were conducted for each scenario considered. For all settings we fix, β = 1, δ0 u = 0, δ0 w = 0, and σv 2 = 1. Samples sizes vary over n {100, 400, 1600}, while δ u {0.6, 0.8, 1.1} and δ w {0.5, 1.2, 1.4}. The correlation amongst the covariates is fixed at ρ = 0.1. We report the bias and mean square error of the nonlinear least squares estimates of the key model parameters. All calculations were carried out in R and used Levenberg-Marquardt optimization, which is available in the minpack.lm library. The bias and mean square error for the simulations appear in Tables 1-3. The results are as expected, regardless of the distribution for u and w. As the sample size increases both

10 10 TWO-TIER the bias and mean square error of the main parameters decays toward zero. In general we see that estimation of β is more precise than estimation of either δ u or δ w. More specifically, consider the results when u and w are both distributed exponentially (Table 1). We see that when δ u is nearly equal to δ w that the nonlinear least squares estimator has similar bias and mean square error; however, when δ u and δ w are quite different, the bias and mean square error also reflect this. Consider the setting where δ u = 1.1 and δ w = 0.5. In this case δ u is more than double δ w. For n = 1600 we see that the bias of δ u is less than half that of δ w, while the opposite holds for the mean square error. As δ w increases to 1.1, these disparities in the bias/mean square error dissipate. [Table 1 about here.] Almost identical results hold when we generate u and w from a half-normal distribution (Table 2). In the setting where δ u = 1.1 and δ w = 0.5 with n = 1600 we see that the bias of δ u is less than half that of δ w, while the opposite holds for the mean square error. As δ w increases to 1.1, these disparities in the bias/mean square error dissipate. It seems that regardless of the distribution, when u and w have the same distribution, the ability of the nonlinear least squares estimator to reliably estimate the main parameters of the model is not impeded. [Table 2 about here.] Table 3 presents an interesting insight for nonlinear least squares estimation of the two-tier frontier. Here u is distributed exponential while w is distributed half-normal. Currently, outside of the Fourier estimator of Tsionas (2012), no closed form expression for the likelihood function exists to estimate the parameters of the two-tier frontier and the two scaling functions. And yet, we see similar performance to the two cases where analytic expressions for the likelihood functions exist. This is reassuring as the distribution of u and w have no reason to be equivalent in applied settings. Thus, provided the scaling function is a realistic assumption, nonlinear least squares offers an attractive avenue for estimating the parameters of the two-tier frontier. [Table 3 about here.] 5. Concluding Remarks Invoking the scaling property has proven useful in many situations when deploying a stochastic frontier model. This paper has taken this simple idea and applied it to the two-tier

11 TWO-TIER 11 stochastic frontier model, showing that many of same insights and simplifications carryover. The model can be easily estimated through nonlinear least squares, while inference can be conducted through the use of robust standard errors due to the presence of heteroskedasticity. An attractive feature of invoking the scaling property in the two-tier frontier setting is that the model parameters can be estimated without requiring distributional assumptions on the two one-sided errors terms. This is also true in the traditional single-tiered stochastic frontier model, yet, in the two-tier frontier setting, it is even more appealing because aside from very specific assumptions on the distribution of the one-sided error terms, a closed form solution for the likelihood function does not exist, making estimation challenging in practice. Given the ease with which nonlinear least squares can be implemented in traditional statistical software, invoking the scaling property should raise the appeal of the two-tier frontier model in applied milieus. The Monte Carlo simulations presented here paint a picture of a readily accessible estimator of a common applied model across a variety of disciplines. While the scaling assumption is an assumption of convenience, when it is correct, the nonlinear least squares estimator estimates the model parameters quite well with decaying bias and mean square error, as expected. Future research should focus on inference related to the scaling assumption and implementing the nonlinear least squares estimator in practice and comparing to standard findings from the exponential-exponential setting.

12 12 TWO-TIER References Aigner, D. J., Lovell, C. A. K. & Schmidt, P. (1977), Formulation and estimation of stochastic frontier production functions, Journal of Econometrics 6(1), Alvarez, A., Amsler, C., Orea, L. & Schmidt, P. (2006), Interpreting and testing the scaling property in models where inefficiency depends on firm characteristics, Journal of Productivity Analysis 25(2), Blanco, G. (2016), Who benefits from job placement services? A two-tiered earnings frontier approach. Unpublished Working Paper. Chawla, M. (2002), Estimating the extent of patient ignorance of the health care market, in S. Devarajan & F. H. Rogers, eds, World Bank Economists Forum, Vol. 2. Ferona, A. & Tsionas, E. G. (2012), Measurement of excess bidding in auctions, Economics Letters 116(2), Gaynor, M. & Polachek, S. W. (1994), Measuring information in the market: an application to physician services, Southern Economic Journal 60(4), Greene, W. H. (2010), A stochastic frontier model with correction for sample selection, Journal of Productivity Analysis 34(1), Groot, W. & Oosterbeek, H. (1994), Stochastic reservation and offer wages, Labour Economics 1(3), Groot, W. & van den Brink, H. M. (2007), Optimism, pessimism and the compensating income variation of cardiovascular disease: a twotiered quality of life stochastic frontier model, Social Science & Medicine 65(7), Harding, J. P., Rosenthal, S. S. & Sirmans, C. F. (2003), Estimating bargaining power in the market for existing homes, The Review of Economics and Statistics 85(1), Huang, Z.-Y. (2013), The study of IPO pricing efficiency in chinese gem market: An empirical measure based on two-tier stochastic frontier model, Journal of Guangdong University of Finance and Economics (2). URL: shtml Jondrow, J., Lovell, C. A. K., Materov, I. S. & Schmidt, P. (1982), On the estimation of technical efficiency in the stochastic frontier production function model, Journal of Econometrics 19(2/3), Kinukawa, S. & Motohashi, K. (2010), Bargaining in technology markets: an empirical study of biotechnology alliances. RIETI Discussion Paper Series 10-E-200. Kumbhakar, S. C. & Parmeter, C. F. (2009), The effects of match uncertainty and bargaining on labor market outcomes: evidence from firm and worker specific estimates, Journal of Productivity Analysis 31(1), Kumbhakar, S. C. & Parmeter, C. F. (2010), Estimation of hedonic price functions with incomplete information, Empirical Economics 39(1), Lian, Y. & Chung, C. F. (2008), Are Chinese listed firms over-investing? Available at Meeusen, W. & van den Broeck, J. (1977), Efficiency estimation from Cobb-Douglas production functions with composed error, International Economic Review 18(2), Murphy, A. & Strobl, E. (2008), Employer and employee ignorance in developing countries: the case of Trinidad and Tobago, Review of Development Economics 12(2), Papadopoulos, A. (2015), The half-normal specification for the two-tier stochastic frontier model, Journal of Productivity Analysis 43(2), Parmeter, C. F. & Kumbhakar, S. C. (2014), Efficiency Analysis: A Primer on Recent Advances, Foundations and Trends in Econometrics 7(3-4), Parmeter, C. F., Wang, H.-J. & Kumbhakar, S. C. (2016), Nonparametric estimation of the determinants of inefficiency, Journal of Productivity Analysis. Forthcoming.

13 TWO-TIER 13 Poggi, A. (2010), Job satisfaction, working conditions and aspirations, Journal of Economic Psychology 31(6), Polachek, S. W. & Yoon, B. J. (1987), A two-tiered earnings frontier estimation of employer and employee information in the labor market, The Review of Economics and Statistics 69(2), Polachek, S. W. & Yoon, B. J. (1996), Panel estimates of a two-tiered earnings frontier, Journal of Applied Econometrics 11(2), Sharif, N. R. & Dar, A. A. (2007), An empirical investigation of the impact of imperfect information on wages in Canada, Review of Applied Economics 3(1-2), Tomini, S., Groot, W. & Pavlova, M. (2012), Paying informally in the Albanian health care sector: a two-tiered stochastic frontier model, European Journal of Health Economics 13, Tran, K. C. & Tsionas, E. G. (2009), Estimation of nonparametric inefficiency effects stochastic frontier models with an application to British manufacturing, Economic Modelling 26, Tsionas, E. G. (2012), Maximum likelihood estimation of stochastic frontier models by the Fourier transform, Journal of Econometrics 170(2), Wang, H.-J. & Schmidt, P. (2002), One-step and two-step estimation of the effects of exogenous variables on technical efficiency levels, Journal of Productivity Analysis 18, Wang, Y. (2016a), Bargaining matters: an analysis of bilateral aid to developing countries, Journal of International Relations and Development. URL: Wang, Y. (2016b), The effect of bargaining on US economic aid, International Interactions 42(3), White, H. (1994), Estimation, Inference, and Specification Analysis, Cambridge University Press, Cambridge, England. Yu, L. & Liang, T. (2012), The performance of dividend policy-based on financing constraints and agency cost trade-off, in Proceedings of 2012 international conference on information management, innovation management and industrial engineering (ICIII), Vol. 1, pp Zeileis, A. (2004), Econometric computing with hc and hac covariance matrix estimators, Journal of Statistical Software 11(10), Zeileis, A. (2006), Object-oriented computation of sandwich estimators, Journal of Statistical Software 16(9), 1 16.

14 14 Tables Table 1. Bias and mean square error for nonlinear least squares estimates for two-tier frontier, u and w distributed exponential. 10,000 simulations. δ w = 0.5 δ w = 1.2 δ w = 1.4 β δ u δ w β δ u δ w β δ u δ w δ u = δ u = δ u =

15 Tables 15 Table 2. Bias and mean square error for nonlinear least squares estimates for two-tier frontier, u and w distributed half normal. 10,000 simulations. δ w = 0.5 δ w = 1.2 δ w = 1.4 β δ u δ w β δ u δ w β δ u δ w δ u = δ u = δ u =

16 16 Tables Table 3. Bias and mean square error for nonlinear least squares estimates for two-tier frontier, u distributed exponential and w distributed half normal. 10,000 simulations. δ w = 0.5 δ w = 1.2 δ w = 1.4 β δ u δ w β δ u δ w β δ u δ w δ u = δ u = δ u =