Maximum Likelihood Estimation of Stochastic Frontier Models with Interval Dependent Variables
Wen-Jen Tsay** Institute of Economics Academia Sinica Taipei, Taiwan Email:
[email protected] Tsu-Tan Fu Department of Economics Soochow University Taipei, Taiwan Email:
[email protected]
September 2016
__________________________________ * We are thankful for the excellent research assistance of Shih-Yang Lin. ** Corresponding author.
1
Abstract This paper considers the maximum likelihood estimation of a stochastic frontier production function with an interval outcome. We derive an analytical formula for calculating the likelihood function of the interval stochastic frontier models. Monte Carlo experiments reveal that the finite sample performance of our method is promising even though the sample size is relatively moderate. We apply our method to measure the information inefficiency in the labor market for newly graduated college students in Taiwan.
Key words: Stochastic frontier production functions; interval dependent variable
2
1 Introduction Consider a stochastic frontier production function with the following matrix form: y = 𝛸𝛸 𝑇𝑇 𝛽𝛽 + 𝜀𝜀,
(1)
where y and ε are 𝛮𝛮 × 1 vectors of observations on the dependent variable and the random
disturbance, respectively; 𝛸𝛸 is an 𝛮𝛮 × 𝛫𝛫 matrix of observations on a constant term and 𝛫𝛫 −
1 regressors; and 𝛽𝛽 is a 𝛫𝛫 × 1 vector of unknown regression coefficients to be estimated and tested. The error specification is:
𝜀𝜀 = 𝑣𝑣 − 𝑢𝑢,
(2)
where the elements of 𝑣𝑣 are iid as 𝑁𝑁(0, 𝜎𝜎𝑣𝑣2 ), and the elements of 𝑢𝑢 are the absolute value of the variables that are iid as 𝑁𝑁(0, 𝜎𝜎𝑢𝑢2 ). All 𝑣𝑣′s and 𝑢𝑢′s are independent of each other and are also
independent of 𝛸𝛸 . We follow the reparameterization of Aigner, Lovell, and Schmidt (ALS hereafter,1977):
𝜎𝜎 2 = 𝜎𝜎𝑢𝑢2 + 𝜎𝜎𝑣𝑣2 ,
𝜆𝜆 = 𝜎𝜎𝑢𝑢 /𝜎𝜎𝑣𝑣 .
(3)
ALS (1977) show that the log-likelihood function for the aforementioned stochastic frontier model is: 𝐿𝐿0 =
𝑁𝑁
2
−𝜆𝜆
1
𝑁𝑁 2 ln �𝜋𝜋� − 𝑁𝑁ln(𝜎𝜎) + ∑𝑁𝑁 𝑖𝑖=1 ln �Φ � 𝜎𝜎 𝜀𝜀�� − 2𝜎𝜎2 ∑𝑖𝑖=1 𝜀𝜀𝑖𝑖 , 2
(4)
where 𝜀𝜀𝑖𝑖 = 𝑦𝑦𝑖𝑖 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽, 𝓍𝓍𝑖𝑖𝑇𝑇 is the 𝒾𝒾 𝑡𝑡ℎ row of 𝛸𝛸, and Φ(∙) is the standard normal cumulative
distribution function (cdf). The maximum likelihood estimator (MLE) is obtained by maximizing (4) with respect to the parameters (β, 𝜎𝜎𝑢𝑢 , 𝜎𝜎𝑣𝑣 )𝑇𝑇 .
This paper executes stochastic frontier analysis (SFA) where the dependent variable 𝑦𝑦𝑖𝑖∗ is an
ordered interval variable located in 𝐽𝐽 + 1 regimes:
𝑦𝑦 ∗ = 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽 + 𝜀𝜀𝑖𝑖 , 𝑖𝑖 = 1,2, … , 𝑁𝑁; ⎧ 𝑖𝑖 ⎪ ⎪𝑦𝑦𝑖𝑖 = 0, if 𝑦𝑦𝑖𝑖∗ ≤ 𝑤𝑤0 ;
(5)
⎨𝑦𝑦 = 1, if 𝑤𝑤 < 𝑦𝑦 ∗ < 𝑤𝑤 ; 𝑖𝑖 0 1 𝑖𝑖 ⎪ ⋮ ⎪ ∗ ⎩ 𝑦𝑦𝑖𝑖 = 𝐽𝐽, if 𝑦𝑦𝑖𝑖 > 𝑤𝑤𝐽𝐽−1 .
3
In other words, the cell limits 𝑤𝑤0 < 𝑤𝑤1 < 𝑤𝑤2 < ⋯ < 𝑤𝑤𝐽𝐽−1 are observable even though 𝑦𝑦𝑖𝑖∗ is
unobservable. In particular, 𝑦𝑦𝑖𝑖∗ might be the natural logarithm of a worker’s wage. We cannot
observe it directly, but we can see the range of it. This interval stochastic frontier model can be tackled with the Bayesian estimation strategy of Griffiths et al. (2014) given that 𝑢𝑢 is assumed to follow an exponential distribution. However, its implementation is not trivial.
The objective of this paper is to suggest an analytic formula for evaluating the likelihood function of the SFA model with an interval dependent variable when the inefficient term follows a half-normal distribution as considered in ALS (1977). Since neither a numerical integration nor simulation-based techniques are encountered in the proposed estimation methodology, this indicates that the computational burden of our strategy is relatively mild. Monte Carlo experiments reveal that the finite sample performance of our method is promising even though the sample size is relatively moderate. Furthermore, our method can easily handle the simulations when the number of observation is way over 5000. We also apply our method to the wage rates of newly graduated college students in Taiwan. Since the dependent variable is interval-coded, we thus propose a simulation-based efficiency measure given that MLE is completed. The empirical results indicate the existence of the phenomenon “school quality matters”, in which a worker graduating from a higher quality school has a higher potential wage. The results also show that the undergraduate major affects the level of the potential wage. Moreover, we find that students of public universities attain the highest efficiency score, implying that public university students perform the best in their job market information search. However, it is plausible that such efficiency differentials could also be explained from the perspective of the screening hypothesis of the labor market. The current paper illustrates an application of the proposed model for measuring labor market information inefficiency when the earning variable is measured by interval outcomes. In fact, the proposed model can be applied to any performance evaluation analysis if the dependent variables are measured in interval form. The interval coding data are commonly collected in surveys to individuals. In such surveys, subjects are often asked to answer questions on their performance by choosing among some ordered interval coding values. For examples, the wage rate in the labor market as in the current study, test/GPA score for evaluating a student’s performance, the potential sales amount of a business performance measurement, etc. Other potential applications can be found in survey studies that adopt contingency valuation methods to examine the values of hypothetical quality improvement of public goods or non-market goods (Carson, 2012). Examples include the willingness to pay (WTP) for quality improvement of environmental goods such as water, air, or any such amenity. Studies on WTP for hypothetical non-market goods such as national security, conservation 4
of public parks or natural resources, and new medical treatments are also abundant in both the environment and health-related literature. Since WTP is often expressed in interval form due to the contingency valuation survey design, our proposed model can therefore be used to measure the inefficiency of WTP reporting. The remaining parts of this paper are arranged as follows. Section 2 presents the analytic formula and the main results. Section 3 verifies the theoretical findings generated from the following Theorem 1 through a Monte Carlo experiment. Section 4 applies the proposed method to data on the wage rates of newly graduated college students in Taiwan. Section 5 provides a conclusion.
2 The models and main results Defining 𝑓𝑓(∙) and 𝐹𝐹(∙) as the density and distribution functions, respectively, of 𝜀𝜀𝑖𝑖 , the
probability of 𝑦𝑦𝑖𝑖∗ in each regime is: ⎧ ⎪
𝐹𝐹𝑖𝑖0 = 𝐹𝐹(𝑤𝑤0 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽), if 𝑦𝑦𝑖𝑖∗ ≤ 𝑤𝑤0 ;
𝐹𝐹𝑖𝑖1 = 𝐹𝐹(𝑤𝑤1 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽) − 𝐹𝐹(𝑤𝑤0 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽), if 𝑤𝑤0 < 𝑦𝑦𝑖𝑖∗ ≤ 𝑤𝑤1 ; ⎨ ⋮ ⎪ 𝑇𝑇 𝐹𝐹 = 1 − 𝐹𝐹�𝑤𝑤 − 𝓍𝓍 𝛽𝛽�, if 𝑦𝑦𝑖𝑖∗ > 𝑤𝑤𝐽𝐽−1. ⎩ 𝑖𝑖𝑖𝑖 𝐽𝐽−1 𝑖𝑖
As a consequence, the log-likelihood function for 𝑁𝑁 independent observations is: 𝐽𝐽 ln 𝐿𝐿 = ∑𝑁𝑁 𝑖𝑖=1 ∑𝑗𝑗=0 1(𝑦𝑦𝑖𝑖 = 𝑗𝑗) ln𝐹𝐹𝑖𝑖𝑖𝑖 ,
(6)
(7)
where 1(∙) is the indicator function.
The major difficulty for the maximum likelihood estimation centers on computing 𝐹𝐹𝑖𝑖𝑖𝑖 in (7),
for example:
−𝑥𝑥 𝑇𝑇 𝛽𝛽
2
𝑄𝑄
𝑎𝑎𝜀𝜀
2
𝐹𝐹𝑖𝑖0 (−𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽) = ∫−∞𝑖𝑖 𝑓𝑓(𝜀𝜀𝑖𝑖 )𝑑𝑑𝑑𝑑𝑖𝑖 = 𝜎𝜎 ∫−∞𝑖𝑖 (∫−∞𝑖𝑖 𝜙𝜙(𝜍𝜍)𝑑𝑑𝑑𝑑)𝜙𝜙(𝑏𝑏𝜀𝜀𝑖𝑖 )𝑑𝑑𝜀𝜀𝑖𝑖 = 𝜎𝜎 𝐼𝐼(𝑄𝑄𝑖𝑖 , 𝑎𝑎, 𝑏𝑏), where
𝜆𝜆
1
𝑄𝑄𝑖𝑖 = −𝑥𝑥𝑖𝑖𝑇𝑇 𝛽𝛽, 𝑎𝑎 = − 𝜎𝜎 , 𝑏𝑏 = 𝜎𝜎,
(8)
(9)
and 𝜙𝜙(∙) stands for the standard normal density function. In other words, provided that we have a good approximation formula for the value of 𝐼𝐼(𝑄𝑄, 𝑎𝑎, 𝑏𝑏) in (8) for various choices of 𝑄𝑄, we can
accurately evaluate the likelihood function of the ordered SFA models. The first contribution of this 5
paper is to derive such an approximate formula 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 by extending the methodology of Tsay et al.
(2013) where they only deal with the cases a > 0 in (9). Since the formula developed in this paper does not impose a constraint on the value of 𝑎𝑎, the analytic-based methodology can be used for both cost and production function scenarios encountered in interval stochastic frontier models.
Before presenting 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 in the following Formula 1, we define an error function as: 𝐸𝐸𝐸𝐸𝐸𝐸(𝑧𝑧) =
2 𝑧𝑧 ∫0 𝑒𝑒 −𝑡𝑡 𝑑𝑑𝑑𝑑 𝜋𝜋 √
2
√2𝑧𝑧
= 2 ∫0
𝜙𝜙(𝑡𝑡)𝑑𝑑𝑑𝑑.
(10)
Formula 1. 𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈𝑈 (𝑄𝑄, 𝑎𝑎, 𝑏𝑏) 𝜖𝜖 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑅𝑅 𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏 ≠0, 𝐼𝐼(𝑄𝑄, 𝑎𝑎, 𝑏𝑏) 𝑖𝑖𝑖𝑖 (8) 𝑖𝑖𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 𝑏𝑏𝑏𝑏 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 : 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 =
⎡ ⎤ �𝑎𝑎2 𝑐𝑐1 𝑎𝑎2 𝑐𝑐2 1 ⎞⎥𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑎𝑎) ⎢1+𝐸𝐸𝐸𝐸𝐸𝐸⎛ 4𝑏𝑏2 −4𝑎𝑎2 𝑐𝑐2 ⎢ ⎥ 2�𝑏𝑏2 −𝑎𝑎2 𝑐𝑐2 ⎣ ⎝ ⎠⎦ 𝑒𝑒
4�𝑏𝑏 2 −𝑎𝑎2 𝑐𝑐2
√𝑏𝑏2 1−𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑎𝑎)
+ 2𝑏𝑏2
2
−
⎡ ⎤ 2 2 𝑎𝑎2 𝑐𝑐2 𝑎𝑎𝑐𝑐1 1 ⎞+𝐸𝐸𝐸𝐸𝐸𝐸⎛−𝑎𝑎𝑐𝑐1 +√2𝑄𝑄�𝑏𝑏 −𝑎𝑎 𝑐𝑐2 �∙𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑎𝑎∙𝑄𝑄)⎞⎥ ⎢𝐸𝐸𝐸𝐸𝐸𝐸⎛ 2 2 4𝑏𝑏 −4𝑎𝑎 𝑐𝑐2 ⎢ ⎥ 2�𝑏𝑏2 −𝑎𝑎2 𝑐𝑐2 2�𝑏𝑏2 −𝑎𝑎2 𝑐𝑐2 ⎣ ⎝ ⎠ ⎝ ⎠⎦ 𝑒𝑒
4�𝑏𝑏 2 −𝑎𝑎2 𝑐𝑐2
+
where 𝑐𝑐1 = -1.0950081470333 and 𝑐𝑐2 = -0.75651138383854.
𝐸𝐸𝐸𝐸𝐸𝐸�
2𝑏𝑏
With 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 , the cdf of 𝜀𝜀 in (8) is approximated by: 2
𝑏𝑏𝑏𝑏 � √2 1+𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠(𝑎𝑎∙𝑄𝑄)
2
𝐹𝐹(𝜀𝜀) = 𝜎𝜎 𝐼𝐼(𝑄𝑄, 𝑎𝑎, 𝑏𝑏) ≈ 𝜎𝜎 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎.
2
,
(11)
Note that 𝐼𝐼𝑎𝑎𝑎𝑎𝑎𝑎 involves the function 𝐸𝐸𝐸𝐸𝐸𝐸 (𝓍𝓍), which can be easily calculated with a standard
statistic package. This implies that the computation of 𝐹𝐹𝑎𝑎𝑎𝑎𝑎𝑎 and that of the associated likelihood function of the interval SFA model are straightforward.
3 Monte Carlo experiment This section considers the finite sample performance of MLE when the data-generating processes (DGP) are the model in (5). Following Olson et al. (1980, p. 76) and Tsay et al. (2013), we consider a set of experiments with a two-regressor model: 𝑦𝑦𝑖𝑖∗𝑙𝑙 = 𝛽𝛽0 + 𝛽𝛽1 𝓍𝓍𝑖𝑖 + 𝜀𝜀𝑖𝑖𝑙𝑙 , 𝑖𝑖 = 1,2, … , 𝑁𝑁, 𝑙𝑙 = 1,2, … ,1000, 6
(12)
where 𝑙𝑙 denotes the 𝑙𝑙-th replication of the data, and the regressors 𝑥𝑥𝑖𝑖 are drawn from 𝑁𝑁(0,1). Without loss of generality, we consider the case where 𝐽𝐽 = 1, i.e., we divide the data into three
regimes depending on whether 𝑦𝑦𝑖𝑖∗𝑙𝑙 is less than 𝑤𝑤0 , or greater than 𝑤𝑤1 , or in between. Three sets of
(𝛽𝛽0 , 𝛽𝛽1 , 𝑤𝑤0 , 𝑤𝑤1 )𝑇𝑇 are considered to check the robustness of the simulation results. Their values are illustrated in the footnotes of Table 1, 2, and 3, respectively.
All the programs are written in GAUSS for four sample sizes, 𝑁𝑁 =100, 400, 1600, and 6400.
Two hundred additional values are generated in order to obtain random starting values. The optimization algorithm used to implement MLE is the quasi-Newton algorithm of Broyden, Fletcher, Goldfarb, and Shanno (BFGS) contained in the GAUSS MAXLIK library. The maximum number of iterations for each replication is 200. The results in Tables 1-3 show the bias and mean squared errors (MSE) of the maximum likelihood estimates. For all 12 combinations of parameters considered in Tables 1-3, we find the bias is really small and the MLE estimates’ MSE always decreases with an increasing sample size. The results again assure the promising performance of the proposed 𝐹𝐹𝑎𝑎𝑎𝑎𝑎𝑎 approximation formula, and enhance our confidence in using 𝐹𝐹𝑎𝑎𝑎𝑎𝑎𝑎 for the interval SFA models.
4 Efficiency measurement of the earning frontier function
This section applies the proposed method to the earning frontier analysis of newly graduated students in Taiwan. 4.1 Data and variables The data used in the empirical example reported below were drawn from a survey conducted by Peng (2004) on Taiwanese college graduates who graduated in 2003. The survey employed a proportionate stratified two-stage random sampling framework in which schools were used as the stratum.
To serve the purpose of this study, the survey data are further screened to include only
full-time workers, excluding part-time workers, self-employed, family workers, and graduates in military service or in prison.
The above screening of the survey data yields a total of 3973 sample
workers. Following the literature of the human capital theory and the framework of Hofler and Polachek (1985), this section develops a stochastic earning frontier function to empirically measure the labor 7
market inefficiency of recent Taiwanese college graduates. Hofler and Polachek (1985) define the wage differential between the maximum potential wage that any given individual can attain and the wage the person actually earn to be the inefficiency that may be attributed to incomplete worker information. To accommodate the problem that the wage differential may include pure inefficiency as well as unobserved heterogeneity, previous studies have suggested to include demographic strata in the earning function or to use panel data (Hofler and Poacheck, 1985; Polachek and Yoo, 1996). Given our cross-sectional data, several demographic variables are used as wage determinants in the empirical earning function. The dependent variable is the natural logarithm of a worker’s monthly wage. The explanatory variables (X) used in estimating the maximum potential wage are a set of socio-demographic factors. More specifically, the explanatory variables are related to education or work experience, gender, work status, school type, and individual majors in college. Education/Work experience: Experience = 1 if worker ever had a part-time job during undergraduate schooling; Experience = 0, otherwise. Since all subjects are recent graduates (with one year of work experience), they all have the same education level and working year. However, those who had a part-time working experience during college are expected to have more labor market information, which may lead to a better paying job. Gender: Male= 1 if male; and Male = 0, otherwise. Female are usually paid less due to sex discrimination in the job market. Work Status: Hours = Worker’s weekly working hours. Workers with longer working hours are expected to earn more. Knowledge Match = 1 if a worker’s undergraduate major matches the job requirement; Knowledge match = 0, otherwise. The knowledge match is postulated to have a positive impact on the wage rate. Public Sector = 1 if work for the government; and Public Sector = 0, otherwise. Government employees in Taiwan on average are better paid than their private counterparts. School Type: Public University =1 if worker graduated from a public (comprehensive) university; and Public University = 0, otherwise. 8
Public Vocational College =1 if worker graduated from a public vocational college; and Public Vocational College = 0, otherwise. Private Vocational College =1 if worker graduated from a private vocational college; and Private Vocational College = 0, otherwise. Taiwan’s higher education institutions can be classified by types into public/private or comprehensive/vocational. Among the 4 different types, the Private University is used as the base for regression measurement. School type may reflect school quality. Workers who graduated from better quality schools are expected to earn more (Card and Krueger, 1992). In Taiwan, public schools are found to be of better quality than private schools, whereas comprehensive universities are on average better than vocational colleges in school quality (Fu, 2011). Undergraduate Major: Social Science = 1 if worker graduated with a major in social science; Social Science = 0, otherwise. Technology = 1 if worker graduated with a major in engineering and technology; Technology = 0, otherwise. The majors of undergraduates are classified into Social Science, Technology and Engineering, and Humanities.
The Humanities category is used as the base for the regression analysis. In Taiwan’s
labor market, workers with a technology major are generally paid well, followed by Humanities and Social Science.
4.2 Sample statistics Table 4 shows the mean statistics of variables used in the interval SFA model. Among all 3973 subjects of the 4 types of schools, 70% of them (40% from Private University and 30% from Private Vocational College) graduated from private schools.
Public schools make up only 30% of the
subjects, with 20% from Public University and 10% from Public Vocational College. Public University has a relatively high mean value in wage, knowledge match, and working in the public sector, as compared to the other 3 types of schools. The mean wages of public schools are averagely higher than those in private schools. It is also found that the sample graduates from both Public and Private Vocational Schools have a higher percentage (46%-55%) of obtaining a Technology and Engineering major than those from general comprehensive schools, whereas private schools are found to have a higher percentage of graduates who majored in Social Science than do public schools. 9
[insert Table 4 here] 4.3 The earning frontier estimation Table 5 shows that the sign of the estimates in the earning frontier function matches our expectations. In other words, workers will have a higher potential wage when working in the public sector, taking a job with a knowledge match, and having more working hours. The positive sign of Male indicates the existence of gender discrimination in Taiwan’s labor market. The school type variables show a positive sign for Public University and negative signs for both types of vocational schools, as compared to Private University. Since school type variables are proxies for school quality, in Taiwan it is well recognized that the quality of public schools is on average higher than that of private schools, as are general comprehensive universities versus vocational colleges. Therefore, the empirical results indicate the existence of the phenomenon of “quality matters”, in which a worker graduating from a higher quality school will have a higher potential wage. 1 In our study, the sample graduates from public universities have the highest potential wage. Table 5 also shows that the undergraduate major affects the level of potential wage. Those who majored in technology and engineering (social science) will have a higher (lower) potential wage than those with a humanities major. In addition to our proposed interval SFA estimation, Table 5 also lists the SFA results using the mid-point values of each observed wage interval, which has been applied in the interval regression literature. Despite the estimated parameters for both estimated frontier equations seeming to be similar, the latter one has the problem of arbitrarily assigning mid-point values in censored intervals, and thus could generate a biased estimation.
4.4 Efficiency distribution Provided the parameters are estimated, one important task for the stochastic frontier analysis is to rank the relative efficiency level of different units. The interval nature of the outcome requires a procedure to conduct the ranking practice.
1
Fruitful discussions on the relationship between school quality and students’ earnings can be found in Card and Krueger (1992, 1996). 10
By (6), for 𝑦𝑦𝑖𝑖 ∈ {0, … , 𝐽𝐽 − 1}, i.e., the first J regimes, we observe the upper bound of 𝜀𝜀𝑖𝑖 as: 𝜀𝜀̂𝑖𝑖 ≤ 𝑤𝑤 � 𝑖𝑖 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽̂ ,
(13)
where 𝛽𝛽̂ are the maximum likelihood estimates, and the value of 𝑤𝑤 � 𝑖𝑖 depends on the regime where 𝑦𝑦𝑖𝑖
is located. Note that 𝑤𝑤 � 𝑖𝑖 = 𝑤𝑤0 when y𝑖𝑖 = 0.
On the other hand, for y𝑖𝑖 = 𝐽𝐽, we observe the lower bound of 𝜀𝜀𝑖𝑖 as: 𝜀𝜀̂𝑖𝑖 ≥ 𝑤𝑤 �𝐽𝐽−1 − 𝓍𝓍𝑖𝑖𝑇𝑇 𝛽𝛽̂ .
(14)
Combining the results in (13) and (14), the conditional expectation of 𝑢𝑢 given that 𝑎𝑎1 < 𝜀𝜀 < 𝑏𝑏1 is
evaluated as:
𝑏𝑏
𝑏𝑏
𝐸𝐸(𝑢𝑢|𝑎𝑎1 < 𝜀𝜀 < 𝑏𝑏1) = ∫𝑎𝑎 1 𝐸𝐸(𝑢𝑢|𝜀𝜀)𝑓𝑓(𝜀𝜀)𝑑𝑑𝑑𝑑 = ∫𝑎𝑎 1 �𝑢𝑢∗ + 𝜎𝜎∗ 1
1
𝑓𝑓(−𝑢𝑢∗ /𝜎𝜎∗ )
� 𝑓𝑓(𝜀𝜀)𝑑𝑑𝑑𝑑,
(15)
1−𝐹𝐹(−𝑢𝑢∗ /𝜎𝜎∗ )
where 𝑢𝑢∗ = −𝜎𝜎𝑢𝑢2 𝜀𝜀 ⁄𝜎𝜎 2 , 𝜎𝜎∗2 = 𝜎𝜎𝑢𝑢2 𝜎𝜎𝑣𝑣2 ⁄𝜎𝜎 2 , and the result following the last equal sign is taken from
Jondrow et al. (1982).
It is not easy to solve the integral in (15) analytically, however, it can be evaluated using the simulation method as: 𝐸𝐸(𝑢𝑢|𝑎𝑎1 < 𝜀𝜀 < 𝑏𝑏1) ∼
1
∑𝑅𝑅 1 𝑅𝑅∗ 𝑖𝑖=1 𝑎𝑎1
