Removing specification errors from the usual formulation of binary choice models

Swamy, Paravastu A. V. B. et al. Article Removing specification errors from the usual formulation of binary choice models Econometrics Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Swamy, Paravastu A. V. B. et al. (2016) : Removing specification errors from the usual formulation of binary choice models, Econometrics, ISSN 2225-1146, MDPI, Basel, Vol. 4, Iss. 2, pp. 1-21, https://doi.org/10.3390/econometrics4020026 This Version is available at: https://hdl.handle.net/10419/171877 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/ econometrics Article Removing Specification Errors from the Usual Formulation of Binary Choice Models P.A.V.B. Swamy 1,†, I-Lok Chang 2, Jatinder S. Mehta 3, William H. Greene 4, Stephen G. Hall 5,6 and George S. Tavlas 5,7,* 1Federal Reserve Board (Retired), Washington, DC 20551, USA; [email protected] 2Department of Mathematics (Retired), American University, Washington, DC 20016, USA; [email protected] 3Department of Mathematics (Retired), Temple University, Philadelphia, PA 19122, USA; [email protected] 4Department of Economics, New York University, 44 West Fourth Street, 7-90 New York, NY 10012, USA; [email protected] 5Leicester University, Room Astley Clarke 116, University Road, Leicester LEI 7RH, UK; [email protected] 6Bank of Greece, 21 El. Venizelos Ave., 10250 Athens, Greece 7Monetary Policy Council, Bank of Greece, 21 El. Venizelos Ave., Athens 10250, Greece *Correspondence: [email protected]; Tel.: +30-210-320-2370 † Current Address: 6333 Brocketts Crossing, Kingstowne, VA 22315, USA Academic Editor: Kerry Patterson Received: 22 December 2015; Accepted: 19 May 2016; Published: 3 June 2016 Abstract: We develop a procedure for removing four major specification errors from the usual formulation of binary choice models. The model that results from this procedure is different from the conventional probit and logit models. This difference arises as a direct consequence of our relaxation of the usual assumption that omitted regressors constituting the error term of a latent linear regression model do not introduce omitted regressor biases into the coefficients of the included regressors. Keywords: binary choice models; specification errors; stochastic coefficients JEL: C13; C51 1. Introduction It is well-known that binary choice models are subject to certain specification errors. It can be shown that the usual approach of adding an error term to a mathematical function leads to a model with nonunique coefficients and error term. In this model, the conditional expectation of the dependent variable given the included regressors does not always exist. Even when it exists, its functional form may be unknown. The nonunique error term is interpreted as representing the net effect of omitted regressors on the dependent variable. Pratt and Schlaifer (1988, p. 34) [ 1 ] pointed out that omitted regressors are not unique and as a result, the condition that the included regressors be independent of “the” excluded variables themselves is “meaningless”. There are cases where the correlation between the nonunique error term and the included regressors can be made to appear and disappear at the whim of an arbitrary choice between two observationally equivalent models. To avoid these problems, we specify models with unique coefficients and error terms without misspecifying their correct functional forms. The unique error term of a model is a function of certain “sufficient sets” of omitted regressors. We derive these sufficient sets for a binary choice model in this paper. In the usual approach, omitted regressors constituting the error term of a model do not introduce omitted-regressor biases into the coefficients of the included regressors. In our approach, they do so. Econometrics 2016,4, 26; doi:10.3390/econometrics4020026 www.mdpi.com/journal/econometrics Econometrics 2016,4, 26 2 of 21 Following the usual approach, Yatchew and Griliches (1984) [ 2 ] 1 showed that if one of two uncorrelated regressors included in a simple binary choice model is omitted, then the estimator of the coefficient on the remaining regressor will be inconsistent. They also showed that, if the disturbances in a latent regression model are heteroscedastic, then the maximum likelihood estimators that assume homoscedasticity are inconsistent and the covariance matrix is inappropriate. In this paper, we show that the use of a latent regression model with unique coefficients and error term changes their results. Our binary choice model is different from those of such researchers as Yatchew and Griliches [ 2 ], Cramer (2006) [ 5 ], and Wooldridge (2002, Chapter 15) [ 5 ]. The concept of unique coefficients and error term is distinctive to our work. Specifically, we do not assume any incorrect functional form, and we account for relevant omitted regressors, measurement errors, and correlations between excluded and the included regressors. Our model features varying coefficients (VCs) in which we interpret the VC on a continuous regressor as a function of three quantities: (i) bias-free partial derivative of the dependent variable with respect to the continuous regressor; (ii) omitted-regressor biases; and (iii) measurement-error biases. This interpretation of the VCs is unique to our work and allows us to focus on the bias-free (i.e., partial derivatives) parts of the VCs. The remainder of this paper is comprised of three sections. Section 2summarizes a recent derivation of Swamy, Mehta, Tavlas and Hall (2014) [ 6 ] of all the terms involved in a binary choice model with unique coefficients and error term. The section also provides the conditions under which such a model can be consistently estimated. Section 3presents an empirical example. Section 4 concludes. An Appendix at the end of the paper has two sections. The first section compares the relative generality of assumptions underlying different linear and nonlinear models. The second section derives the information matrix for a binary choice model with unique coefficients and error term. 2. Methods of Correctly Specifying Binary Choice Models and Their Estimation 2.1. Model for a Cross-Section of Individuals Greene (2012, pp. 681–683) [ 3 ] described various situations under which the use of discrete choice models is called for. In what follows, we develop a discrete choice model that is free of several specification errors. To explain, we begin with the following specification: y˚ i“ψipx˚ i1, ..., x˚ iLiq(1) where iindexes nindividuals, ψipx˚ i1 , ..., x˚ iLiq is a mathematical function, and its arguments are mathematical variables. Let ψip . q be short hand for this function. We do not observe y˚ i but view the outcome of a discrete choice as a reflection of the underlying mathematical function in Equation (1). We only observe whether a choice is made or not (see Greene (2012, p. 686) [3]). Therefore, our observation is yi“1 if y˚ ią0 (2) yi“0 if y˚ iď0 where the choice “not made” is indicated by the value 0 and the choice “made” is indicated by the value 1, i.e., yi takes either 0 or 1. An example of model (2), provided in Greene (2012, Example 17.1, pp. 683–684) [ 3 ], is a situation involving labor force participation where a respondent either works or seeks work (yi= 1) or does not (yi= 0) in the period in which a survey was taken.2 In Equation (1), x˚ i“ px˚ i1 , ..., x˚ i,Liq1 is Liˆ 1, Li denotes the total number of the arguments of ψipx˚ i1 , ..., x˚ i,Liq “ ψip . q ; there are no omitted arguments needing an error term. Equation (1) is a 1See, also, Greene (2012, Chapter 17, p. 713) [3]. 2 We will show below that the inconsistency problems Yatchew and Griliches (1984, p. 713) [ 2 ] pointed out with the probit and logit models are eliminated by replacing these models by the model in (1) and (2). Econometrics 2016,4, 26 3 of 21 mathematical equation that holds exactly. Inexactness and stochastic error term enters into (1) when we derive the appropriate error term and make distributional assumption about it. The reasons for this are explained below. The number Li may depend on i. This dependence occurs when the number of arguments of ψip . q is different for different individuals. For example, before deciding whether or not to make a large purchase, each consumer makes a marginal benefit/marginal cost calculations based on the utilities achieved by making the purchase and by not making the purchase, and by using the available funds for the purchase of something else. The difference between benefit and cost as an unobserved variable y˚ i can vary across consumers if they have different utility functions with different arguments. These variations can show up in the form of different arguments of ψip . q for different consumers. It should be noted that (1) represents our departure from the usual approach of adding a nonunique error term to a mathematical function and making a “meaningless” assumption about the error term. Pratt and Schlaifer (1988, p. 34) [ 1 ] severely criticized this approach. To avoid this criticism, what we have done in (1) is that we have taken all the x - variables and the variables constituting the error term ε in Cramer’s (2006) [ 4 ] (or e in Wooldridge’s (2002, p. 457) [ 5 ]) latent regression model, and included them in ψip . q as its arguments. In addition, we also included in ψip . q all relevant pre-existing conditions as its arguments.3. The problem that researchers face is that of uncovering the correct functional form of ψip . q . 4 However, any false relations can be shown to have been eliminated when we control for all relevant pre-existing conditions. To make use of this observation due to Skyrms (1988, p. 59) [ 7 ], we incorporate these pre-existing conditions into ψip . q by letting some of the elements of x˚ i represent these conditions. 5 Clearly, we have no way of knowing what these pre-existing conditions might be, how to measure them (if we knew them), or how many there may be. To control for these conditions, we use the following approach. We assume that all relevant pre-existing conditions appear as arguments of the function ψip . q in Equation (1). This is a straightforward approach. Therefore, when we take the partial derivatives of ψip . q with respect to x˚ ij , a determinant of y˚ i included in ψip . q as its argument, the values of all pre-existing conditions are automatically held constant. This action is important because it sets the partial derivative By˚ i{Bx˚ ij equal to zero whenever the relation of y˚ i to x˚ ij , an element x˚ i , is false (see Skyrms (1988, p. 59) [7]). The function ψip . q in (1) is exact and mathematical in nature, without any relevant omitted arguments. Moreover, its unknown functional form is not misspecified. Therefore, it does not require an error term; indeed, it would be incorrect to add an error term to ψip . q . We refer to (1) as “a minimally restricted mathematical equation,” the reason being that no restriction other than the normalization rule, that the coefficient of y˚ i is unity, is imposed on (1). Without this restriction, the function ψip . q is difficult to identify. The reason why no other restriction is imposed on it is that we want (1) to be a real-world relationship. With such a relationship we can estimate the causal effects of a treatment. Basmann (1988, pp. 73, 99) [ 8 ] argued that causality is a property of the real world. We define that real-world relations are those that are not misspecified. Causal relations are unique in the real world. This is the reason why we insist that the coefficients and error term of our model be unique. From Basmann’s (1988, p. 98) [ 8 ] definition it follows that (1) is free of the most serious objection, i.e., non-uniqueness, which occurs when stationarity producing transformation of observable variables are used.6We do not use such variables in (1). 3We explain in the next paragraph why we have included these conditions. 4 Some researchers may believe that there is no such thing as the true functional form of (1). Whenever we talk of the correct functional form of (1), we mean the functional form of (1) that is appropriate to the particular binary choice in (2). 5Here we are using Skyrms’ (1988, p. 59) [7] definition of the term “all relevant pre-existing conditions.” 6This is Basmann’s (1988, pp. 73, 99) [8] statement. Econometrics 2016,4, 26 4 of 21 2.2. Unique Coefficients and Error Terms of Models 2.2.1. Causal Relations Basmann (1988, pp. 73, 99) [ 8 ] emphasized that the word “causality” designates a property of the real world. Hence we work only with appropriate real-world relationships to evaluate causal effects. We define that the real-world relationships are those that are not subject to any specification errors. It is possible to avoid some of those errors, as we show below. The real-world relationships and their properties are always true and unique. Such relationships cannot be found, however, by imposing severe restrictions because they can be false. Examples of these restrictions are given in the Appendix to the paper. For example, certain separability conditions are imposed on (1) to obtain (A1) in the Appendix. These conditions are so severe that their truth is doubtful. For this reason, (A1) may not be a real-world relationship and may not possess the causality property. Again in the Appendix, (A2) is a general condition of statistical independence which is very strong. Model (A5) of the Appendix with a linear functional form could be misspecified.7 2.2.2. Derivation of a Model from (1) without Committing a Single Specification Error To avoid misspecifications of the unknown correct functional form of (1), we change the problem of estimating (1) to the problem of estimating some of its partial derivatives in y˚ i“α˚ i0`x˚ i1α˚ i1` ¨ ¨ ¨ ` x˚ iLiα˚ iLi(3) where, for `“ 1, . . . , Li , α˚ i`“Bψip.q Bx˚ i` if x˚ i` is continuous and = ∆ψip.q ∆x˚ i` with the right sign if x˚ i` is discrete having zero as one of its possible values, ∆ is the first-difference operator, and the intercept α˚ i0“y˚ i´řLi `“1x˚ i`α˚ i` . In words, this intercept is the error of approximation due to approximating y˚ it by ( řLi `“1x˚ i`α˚ i` ). Therefore, model (3) with zero intercept does not misspecify the unknown functional form of (1) if the error of approximation is truly zero and model (3) with nonzero intercept is the same as (1) with no misspecifications of its functional form, since y˚ i´řLi `“1x˚ i`α˚ i``řLi `“1x˚ i`α˚ i`“y˚ i . This is how we deal with the problem of the unknown functional form of (1). Note that no separability conditions need to be imposed on (1) to write it in the form of (3). This is the advantage of (3) over (A1). Note that in the above definition of the partial derivative ( α˚ i` ), the values of all the arguments of ψit p.q (including all relevant pre-existing conditions) other than x˚ i` are held constant. These partial derivatives are different from those that can be derived from (A1) with εi suppressed. This is because in taking the latter derivatives the values of x˚ i,K`1, ..., x˚ iLtare not held constant. Equation (3) is not a false relationship, since we held the values of all relevant pre-exiting conditions constant in deriving its coefficients. The regression in (3) has the minimally restricted equation in (1) as its basis. The coefficients of (3) are constants if (1) is linear and are variables otherwise. In the latter case, the coefficients of (3) can be the functions of all of the arguments of ψit p.q . Any function of the form (1) with unknown functional form can be written as linear in variables and nonlinear in coefficients, as in (3). We have already established that this linear-in-variables and nonlinear-in-coefficients model has the correct functional form if its intercept is zero and is the same as (1) otherwise. In either case, (3) does not have a misspecified functional form. In this paper, we take advantage of this procedure. Not all elements of x˚ i are measured; suppose that the first Kof them are measured. This assumption needs the restriction that min pL1 , ..., Lnq ą K . Even these measurements may contain errors so that the observed argument xij is equal to the sum of the true value x˚ ij and a measurement 7 Cramer (2006, p. 2) [ 4 ] and Wooldridge (2002, p. 457) [ 5 ] assume that their latent regression models are linear. This is a usual assumption. We are simply justifying our unusual assumptions without criticizing the usual assumptions. Econometrics 2016,4, 26 5 of 21 error, denoted by ν˚ ij . 8 The arguments, x˚ ig , g =K + 1, . . . , Li , for which no data are available, are treated as regressors omitted from (3). 9 These are of two types: (i) unobserved determinants of y˚ i and (ii) all relevant pre-existing conditions. We know nothing of these two types of variables. With these variables being present in (3), we cannot estimate it. Again without misspecifying (1) these variables should be eliminated from (3). To do so, we consider the “auxiliary” relations of each x˚ ig to x˚ i1 , ..., x˚ iK . Such relations are: For g=K+ 1, . . . , Li, x˚ ig “λ˚ ig0`x˚ i1λ˚ ig1` ¨ ¨ ¨ ` x˚ iKλ˚ igK (4) where λ˚ igj “Bx˚ ig Bx˚ ij if x˚ ij is continuous holding the values of all the regressors of (A8) other than that of x˚ ij constant and = ∆x˚ ig ∆x˚ ij with the right sign if x˚ ij is discrete taking zero as one of its possible values and λ˚ ig0“x˚ ig ´řK j“1x˚ tjλ˚ igj. This intercept is the portion of x˚ ig remaining after the effect (řK j“1x˚ ijλ˚ igj) of x˚ i1, ..., x˚ iK on x˚ ig has been subtracted from it. In (4), there are Li´K relationships. The intercept λ˚ ig0 is the error due to approximating the relationship between the gth omitted regressor and all the included regressors (“the correct relationship”) by řK j“1x˚ tjλ˚ igj . If this error of approximation is truly zero, then Equation (4) with zero intercept has the same functional form as the correct relationship. In the alternative case where the error of approximation is not zero, (4) is the same as the correct relationship, i.e., x˚ ig ´řK j“1x˚ tjλ˚ igj `řK j“1x˚ tjλ˚ igj . In either case (4) does not misspecify the correct functional form. According to Pratt and Schlaifer (1988, p. 34) [ 1 ], the condition that the included regressors be independent of “the” omitted regressors themselves is meaningless. This statement supports (4) but not the usual assumption that the error term of a latent regression model is uncorrelated with or independent of the included regressors. 10 The problem is that omitted regressors are not unique, as Pratt and Schlaifer (1988, p. 34) [1] proved. 2.2.3. A Latent Regression Model with Unique Coefficients and Error Terms Substituting the right-hand side of Equation (4) for x˚ ig in (3) gives y˚ i“α˚ i0` Li ÿ g“K`1 λ˚ tg0α˚ ig ` K ÿ j“1 x˚ ijpα˚ ij ` Li ÿ g“K`1 λ˚ igjα˚ igq(5) The error term, the intercept, and the coefficients of the nonconstant regressors of this model are řLit g“K`1λ˚ ig0α˚ ig,α˚ i0, and pα˚ ij `řLi g“K`1λ˚ igjα˚ igq,j = 1, . . . ,K, respectively. Bias-free partial derivatives: α˚ ij “Bψip.q Bx˚ ij or “∆ψip.q ∆x˚ ij ,j“1, . . . , K(6) These partial derivatives have the correct functional form if the x˚ ij’s are continuous. “Sufficient” sets of omitted regressors : The regressors, x˚ i,K`1 , ..., x˚ iLi , are called “omitted regressors” because they are included in (3) but not in Equation (5). The regressors x˚ i1 , ..., x˚ iK are called “the included regressors.” It can be seen from (5) that the portions λ˚ i,K`1,0 , ..., λ˚ iLi0 of omitted regressors, x˚ i,K`1 , ..., x˚ iLi , respectively, in conjunction with the included regressors x˚ i1 , ..., x˚ iK are sufficient to determine the value of y˚ i exactly. For this reason, Pratt and Schlaifer (1988, p. 34) [ 1 ] called 8We postpone making stochastic assumptions about measurement errors. 9The label “omitted” means that we would remove them from (3). 10 Cramer (2006, p. 4) [4] and Wooldridge (2002, p. 457) [5] make the usual assumption. Econometrics 2016,4, 26 6 of 21 λ˚ i,K`1,0 , ..., λ˚ i,Li,0 “certain “sufficient sets” of omitted regressors.” The second term ( řLi g“K`1λ˚ tg0α˚ ig ) on the right-hand side of (5) is called “a function of these sufficient sets of omitted regressors x˚ i,K`1 , ..., x˚ iLi ” Pratt and Schlaifer (1988, 1984) [ 1 , 9 ] pointed out that this function can be taken as the error term of (5). It remains as a mathematical function until we make a distributional assumption about it. Note that the problem with the error terms of the usual latent regression models including those of (A1), Karlsen, Myklebust and Tjøstheim’s (2007) [ 10 ] and White’s (1980, 1982) [ 11 , 12 ] models is that they are not the appropriate functions of the sufficient sets of omitted regressors and hence are not unique and/or are arbitrary.11 Deterministic omitted-regressors bias : The term řLi g“K`1λ˚ igjα˚ ig contained in the coefficient of x˚ tj in (5) measures such a bias. Swamy et al. (2014, pp. 197, 199, 217–219) [ 6 ] proved that the coefficients pα˚ ij `řLit g“K`1λ˚ igjα˚ igq and the error term řLi g“K`1λ˚ ig0α˚ ig of (5) are unique in the following sense: Uniqueness : The coefficients and error term of model (5) are unique if they are invariant under the addition and subtraction of the coefficient of any regressor omitted from (5) times any regressor included in (5) on the right-hand side of (3) (see Swamy et al. 2014, pp. 199, 219) [6]. The equations in (4) play a crucial role in Swamy et al.’s (2014) [ 6 ] proof of the uniqueness of the coefficients and error term of (5). If we had taken the sum x˚ i,K`1α˚ i,K`1 + ¨ ¨ ¨ + x˚ iLiα˚ iLi of the last Li´K terms on the right-hand side of (3) as its error term, then we would have obtained a nonunique error term. The reason why this would have happened is that omitted regressors are not unique. What (4) has done is that it has split each gth omitted regressor into a “sufficient set” and an included-regressors’ effect. This sufficient set times the coefficient of the gth omitted regressor has become a term in the (unique) error term of (5) and the included-regressors’ effect times the coefficient of the gth omitted regressor has become a term in omitted-regressor biases of the coefficients of (5). This is not the usual procedure where the whole of each omitted regressor goes into the formation of an error term and no part of it becomes a term in omitted-regressor biases. The usual procedure leads to nonunique coefficients and error term. In the YG procedure, only some of the included regressors which, when omitted, introduce omitted-regressor biases into the (nonunique) coefficients on the remaining included regressors. Yatchew and Griliches (1984) [ 2 ], Wooldridge (2002) [ 5 ], and Cramer (2006) [ 4 ] followed the usual procedure. Without using (4) it is not possible to derive a model with unique coefficients and error term. 2.2.4. A Correctly Specified Latent Regression Model Substituting the terms on the right-hand sides of equations x˚ ij “xij ´ν˚ ij , j= 1, . . . ,K, for x˚ ij,j= 1, . . . ,K, respectively, in (5) gives a model of the form y˚ i“γi0`xi1γi1` ¨ ¨ ¨ ` xiKγiK (7) where the intercept is defined as γi0“α˚ i0` Li ÿ g“K`1 λ˚ ig0α˚ ig ´ÿ ν˚ ij PS2 ν˚ ijpα˚ ij ` Li ÿ g“K`1 λ˚ igjα˚ igq(8) and the other terms are defined as xijγij “xijp1´ν˚ ij xij qpα˚ ij ` Li ř g“K`1 λ˚ igjα˚ igqif xij PS1 “xijpα˚ ij ` Li ř g“K`1 λ˚ igjα˚ igqif xij PS2 (9) 11 Cramer (2006, p. 4) [ 4 ] and Wooldridge (2002, p. 457) [ 5 ] adopt the usual latent regression models with nonunique coefficients and error terms. Econometrics 2016,4, 26 7 of 21 where the set S1 contains all the regressors of Equation (7) that are continuous, the set S2 contains all the regressors of (7) that can take the value zero with positive probability, the ratio of p 1 ´ν˚ ij xij q in the first line of Equation (9) comes from the equation x˚ ij “xij ´ν˚ ij “ p 1 ´ν˚ ij xij qxij , and this ratio does not appear in the second line of Equation (9) because xij PS2 can take the value zero with positive probability. Equation (7) implies that a model is correctly specified if it is derived by inserting measurement errors at the appropriate places in a model with unique coefficients and error term (see Swamy et al. (2014, p. 199) [6]). Deterministic measurement-error biases : The formula ´ř ν˚ ij PS2 ν˚ ijpα˚ ij ` Li ř g“K`1 λ˚ igjα˚ igq in (8) measures the sum of measurement-error biases in the coefficients of xij PS2 and the formula p´ν˚ ij xij qpα˚ ij ` Li ř g“K`1 λ˚ igjα˚ igqin the first line of (9) measures such a bias in the coefficient γij of xij PS1. Under our approach, measurement errors do not become random variables until distributional assumptions are made about them. 2.2.5. What Specification Errors Are (3)–(8) Free from? (i) The unknown functional form of (1) is not allowed to become the source of a specification error in (3); (ii) The uniqueness of the coefficients and error term of (5) eliminates the specification error resulting from non-unique coefficients and error term; (iii) Pratt and Schlaifer (1988, p. 34) [ 1 ] pointed out that the requirement that the included regressors be independent of the excluded regressors themselves is “meaningless”. The specification error introduced by making this meaningless assumption is avoided by taking a correct function of certain “sufficient sets” of omitted regressors as the error term of (5); (iv) The specification error of ignoring measurement errors when they are present is avoided by placing them at the appropriate places in (5). It should be noted that when we affirm that (7) is free of specification errors, we mean that it is free of specification-errors (i)–(iv). Using (3)–(6) we have derived a real-world relationship in (7) that is free of specification-errors (i)–(iv). Thus, our approach affirms that any relationship suffering from anyone of these specification errors is definitely not a real-world relationship. 2.3. Comparison of (7) with the Yatchew and Griliches (1985) [ 2 ], Wooldridge (2002) [ 5 ], and Cramer (2006) [ 4 ] Latent Regression Models In Section 2.2.3, we have seen that the relationships between each omitted regressor and the included regressors in (4) introduce omitted-regressor biases into the coefficients on the regressors of (5). We have pointed out in the last paragraph of that section that this is not how Yatchew and Griliches ( YG) derived omitted-regressor biases. They work with models in which the coefficients and error terms do not satisfy our definition of uniqueness. YG considered a simple binary choice model and omitted one of its two included regressors. According to YG, this omission introduces omitted regressor bias into the coefficient on the regressor that is allowed to remain. The results proved by YG are: (i) even if the omitted regressor is uncorrelated with the remaining included regressor, the coefficient on the latter regressor will be inconsistent; (ii) If the errors in the underlying regression are heteroscedastic, then the maximum likelihood estimators that ignore the heteroscedasticity are inconsistent and the covariance matrix is inappropriate (see also Greene (2012, p. 713) [ 3 ]). We do not omit any of the included regressors from (5) to generate omitted-regressor biases. For YG, omitted regressors in (4) are those that generate the error term in their latent regression model. Equation (5)’s error term is a function of those variables that satisfy Pratt and Schlaifer’s definition of “sufficient sets” of our omitted regressors. Thus, YG’ concepts of omitted-regressors, included regressors, and error terms are different from ours. Their model is subject to specification errors (i)–(iv) listed in the previous section. YG’s assumptions about the error term of their model are questionable because Econometrics 2016,4, 26 8 of 21 of its non-uniqueness. Unless its coefficients and error term are unique no model can represent any real-world relationship which is unique. According to YG, Wooldridge, and Cramer, the regressors constituting the error term of a latent regression model do not produce omitted-regressors biases. Their omitted-regressor bias is not the same as those in (5). YG’s results cannot be obtained from our model (7). Their nonunique heteroscedastic error term is different from our unique heteroscedastic error term in (7). It can be shown that the results of YG arose as a direct consequence of ignoring our omitted-regressor and measurement-error biases in (9). Omitted regressors constituting the YG model’s (non-unique) error term also introduce omitted-regressor biases in our sense but not in their sense. Furthermore, the YG model suffers from all the four specification errors (i)–(iv) which equations (3)–(5), (7)–(9) avoid. To recapitulate, misspecifications of the correct functional form of (1) are avoided by expressing it in the form of Equation (3). If the sum, Li ř g“K`1 x˚ igα˚ ig , of the last Li´K terms on the right-hand side of (3) is treated as its error term, then this error term is not unique (see Swamy et al. (2014, p. 197) [ 6 ]). Suppose that the coefficients of (3) are constants. Then the correlation between the nonunique error term and the first Kregressors of (3) can be made uncertain and certain, at the whim of an arbitrary choice between two observationally equivalent forms of (3), as shown by Swamy et al. (2014, pp. 217–218) [ 6 ]. To eliminate this difficulty, a model with unique coefficients and error term is derived by substituting the right-hand side of Equation (4) for the omitted regressor, x˚ ig , in (3) for every g“K` 1, ..., Li . Equation (7) shows how the terms of an equation look like if this equation is made free of specification errors (i)–(iv). For each continuous xij with j> 0 in (7), its coefficient contains the bias-free partial derivative (Bψip.q Bx˚ ij ) and omitted-regressor and measurement-error biases. Parameterization of Model (7) The partial derivative ( By˚ i{Bx˚ ij ) components of the coefficients ( γij ,j= 1, . . . ,K) of (7) are the objects of our estimation. For this purpose, we parameterize (7) using our knowledge of the probability model governing the observations in (7). We assume that for j= 0, 1, . . . , K: γij “zi0πj0`zi1πj1` ¨ ¨ ¨ ` zipπjp `εij (10) where zi0” 1, the π ’s are fixed parameters, the z’s drive the coefficients of (7) and are, therefore, called “coefficient drivers.” These drivers are observed. We will explain below how to select these drivers. The errors ( εij ’s) are included in Equation (10) because the p+ 1 drivers may not be able to explain all variation in γij. Admissible drivers : For j= 0, 1, . . . ,K, the vector Zi“ Zi0”1, Zi1, ..., Zip(1 in (10) is an admissible set of coefficient drivers if given Zi , the value that the vector of the coefficients of (7) would take in unit ihad Xi“ t1, Xi1, ..., XiKu1been xi“ p1, xi1, ..., xiKq1is independent of Xifor all i.12 We use the following matrix notation: xi“ p1, xi1, ..., xiKq 1 is pK`1q ˆ 1, zi“`1, zi1, ..., zip˘1 is pp`1q ˆ 1, π1j“`πj0,πj1, ..., πjp˘ is 1 ˆ pp`1q , γij “π1jzi is a scalar, Π is the (K + 1) ˆ ( p + 1) matrix having π1j as its jth row, and εi“ pεi0, ..., εiKq 1 is pK`1q ˆ 1. Substituting the right-hand side of (10) for γij in (7) gives y˚ i“x1iΠzi`x1iεi(11) Assumption 1. The regressors of Equation (7) are conditionally independent of their coefficients given the coefficient drivers. 12 A similar admissibility condition for covariates is given in Pearl (2000, p. 79) [ 13 ]. Pearl (2000, p. 99) [ 13 ] also gives an equation that forms a connection between the opaque phrase “the value that the coefficient vector of (3) would take in unit i, had Xi“ pXi1, ..., XiKq1been xi“ pxi1, ..., xiKq1” and the physical processes that transfer changes in Xiinto changes in y˚ i. Econometrics 2016,4, 26 15 of 21 belief is that the right sign for this bias-free portion is positive. Now it is appropriate to use the formula pα˚ i1q “ ˆ Ai1“ p1´ˆ Di1q´1pˆ π10 `ř hPG1 zih ˆ π1hq in (25) with j= 1 to estimate α˚ i1“By˚ i Bx˚ i1 . We assume that ˆ Di1 is negligible. We need to choose the terms in the sum pˆ π10 `ř hPG1 zih ˆ π1hq from the terms on the right-hand side of Equation (32). It can be seen from this equation that if we retain the estimate ˆ π10 = ´4.2328 p6.9397q in the sum pˆ π10 `ř hPG1 zih ˆ π1hq , then this sum does not give positive estimate of α˚ i1 for any combination of the six coefficient drivers in (32). Therefore, we remove ˆ π10 from pˆ π10 `ř hPG1 zih ˆ π1hq . We expect the impact of education on earnings to be small. To obtain the smallest possible positive estimate of α˚ i1 , we choose the smallest positive term on the right-hand side of (32). This term is + 0.0702 p0.0676qzi5 . Hence we set the z’s other than zi5 in ř hPG1 zih ˆ π1h equal to zero. Thus, we obtain G1 = 1 and ˆ α˚ i1 = + 0.0702 p0.0676qzi5 . The value of zi5 times 0.0702 gives the estimate of the impact of the ith married woman’s education on her earnings. To conserve space, we present the values of ˆ α˚ i1 = + 0.0702 p0.0676qzi5 only for only i= 1, . . . , 5 in Table 2. The impact estimates for all 753 married women are presented in the form of a histogram or a kernel density function in Figure 1below. We interpret the estimate 0.0702 p0.0676qzi5 to imply that an additional year of schooling is associated with a 0.0702 p0.0676qzi5ˆ 100 percent increase in earnings. This impact of education on earnings is different for different married women. The impact of a wife’s education on her earnings is 0.0702 times her husband’s education. 18 Our results in Table 2and Figure 1below show that the greater are the years of schooling of a husband, the larger is the impact of his wife’s education on her earnings. However, the estimates of α˚ i1 appear to be high at least for some married women whose husbands had larger years of schooling. Therefore, they may contain some omitted-regressor biases. Econometrics 2016, 4, 26 18 of 24 Figure 1. Estimates of the “impacts” of education on the earnings for 753 married women. A histogram and a kernel density function presented in Figure 1 are much more revealing than a table containing the values, 5 0 0702 i .z , i = 1,…, 753, and their standard errors would. Also, such a table occupies a lot of space without telling us mush. We are using Figure 1 as a descriptive device. The kernel function in Figure 1 is multimodal. All the estimates in this figure have the correct signs. Greene’s (2012, p. 708) [3] estimate 0.0904 of the coefficient of education in his estimated labor supply model is not comparable to the estimates in Figure 1 because (i) his model is different from our model; (ii) the dependent variable of his labor supply model in Greene (2012, p. 683) [3] is the log of the dependent variable of our model (29); and (iii) our definition of 1 * i * i y x ∂ ∂ in (3) is different from Greene’s (2012, p. 14) [3] definition of 1 i i y x ∂ ∂ . Greene’s estimate is some kind of average estimate applicable to all 753 married women. It is unreasonable to expect his average estimate to be close to the estimate for each married women. We will now show that given the 6 coefficient drivers in (32), it is not possible to reduce the magnitudes of all the estimates in Figure 1 without changing the positive sign of some of these estimates in the left tail end of Figure 1 to the negative sign. This is what has happened in Figure 2. To reduce the magnitudes of the estimates of bias-free parts 11 1 (A ) * *i ii * i y ˆ x α ∂ == ∂ of the 1 ˆ i γ ’s given in Figure 1 for all i, we use the alternative estimates, 5 0 0702 i .z − 0 000013.6i z of the bias-free parts of the 1 ˆ i γ ’s called “modified 1 ˆ A i , i = 1 ,…, 753.” The histogram and kernel density function for the modified 1 ˆ A i is given in Figure 2 below. Figure 1. Estimates of the “impacts” of education on the earnings for 753 married women. 18 According to Geene (2012, p, 708) [ 3 ], it would be natural to assume that all the determinants of a wife’s labor force participation would be correlated with the husband’s hours which is defined as a linear stochastic function of the husband’s age and education and the family income. Our inclusion of husband’s variables in (32) is consistent with this assumption. Econometrics 2016,4, 26 16 of 21 Table 2. Estimates of the bias-free portion of γi1for five married women19. 0.0702zi5 0.8419 (0.8110) 0.6314 (0.6083) 0.8419 (0.8110) 0.7016 (0.6758) 0.8419 (0.8110) . A histogram and a kernel density function presented in Figure 1are much more revealing than a table containing the values, 0.0702 zi5 ,i= 1, . . . , 753, and their standard errors would. Also, such a table occupies a lot of space without telling us mush. We are using Figure 1as a descriptive device. The kernel function in Figure 1is multimodal. All the estimates in this figure have the correct signs. Greene’s (2012, p. 708) [ 3 ] estimate 0.0904 of the coefficient of education in his estimated labor supply model is not comparable to the estimates in Figure 1because (i) his model is different from our model; (ii) the dependent variable of his labor supply model in Greene (2012, p. 683) [ 3 ] is the log of the dependent variable of our model (29); and (iii) our definition of By˚ i Bx˚ i1 in (3) is different from Greene’s (2012, p. 14) [ 3 ] definition of Byi Bxi1 . Greene’s estimate is some kind of average estimate applicable to all 753 married women. It is unreasonable to expect his average estimate to be close to the estimate for each married women. We will now show that given the 6 coefficient drivers in (32), it is not possible to reduce the magnitudes of all the estimates in Figure 1without changing the positive sign of some of these estimates in the left tail end of Figure 1to the negative sign. This is what has happened in Figure 2. To reduce the magnitudes of the estimates of bias-free parts ˆ Ai1“α˚ i1“By˚ i Bx˚ i1 q of the ˆ γi1 ’s given in Figure 1for all i, we use the alternative estimates, 0.0702 zi5´ 0.000013 zi6 of the bias-free parts of the ˆ γi1 ’s called “modified ˆ Ai1 ,i= 1 , . . . , 753.” The histogram and kernel density function for the modified ˆ Ai1is given in Figure 2below. Five of the estimates in the left-tail end of this figure have the wrong (negative) sign. More number of wrong signs will occur if we try to further reduce the magnitudes of the modified estimates. The kernel density function of the modified estimates is unimodal unlike the kernel density function in Figure 1. The range of the modified estimates is smaller than that of the estimates in Figure 1. From these results it is incorrect to conclude that the conventional discrete choice models and their method of estimation give better and unambiguous results than the latent regression model in (11) and (14) and formula (25). The reasons for this circumstance are the following: (i) The conventional models including discrete choice models suffer from four specification errors listed in Section 2.2.5 and the model in (11) and (14) is free of these errors; (ii) The conventional latent regression models have nonunique coefficients and error terms and the model in (11) and (14) is based on model (5) which has unique coefficients and error term. How can a model with nonunique coefficients and error term give unambiguous results? (iii) The conventional method of estimating the discrete choice models appears to be simple because these models are based on the assumption that “the” omitted regressors 19 The standard errors of estimates are given in parentheses below the estimates for five married women. These estimates and standards errors for other married women are available from I-Lok Chang upon request. Econometrics 2016,4, 26 17 of 21 constituting their error terms do not introduce omitted-regressor biases into the coefficients of their included regressors. The model in (11) and (14) is not based on any such assumption; (iv) Pratt and Schlaifer pointed out that in the conventional model the condition that its regressors be independent of “the” omitted regressors constituting its error term is meaningless. The error terms of the model in (11) and (14) are not the functions of “the” omitted-regressors. Econometrics 2016, 4, 26 19 of 24 Figure 2. Modified estimates of the “impacts” of education on the earnings for 753 married women. Five of the estimates in the left-tail end of this figure have the wrong (negative) sign. More number of wrong signs will occur if we try to further reduce the magnitudes of the modified estimates. The kernel density function of the modified estimates is unimodal unlike the kernel density function in Figure 1. The range of the modified estimates is smaller than that of the estimates in Figure 1. From these results it is incorrect to conclude that the conventional discrete choice models and their method of estimation give better and unambiguous results than the latent regression model in (11) and (14) and formula (25). The reasons for this circumstance are the following: (i) The conventional models including discrete choice models suffer from four specification errors listed in Section 2.2.5 and the model in (11) and (14) is free of these errors; (ii) The conventional latent regression models have nonunique coefficients and error terms and the model in (11) and (14) is based on model (5) which has unique coefficients and error term. How can a model with nonunique coefficients and error term give unambiguous results? (iii) The conventional method of estimating the discrete choice models appears to be simple because these models are based on the assumption that “the” omitted regressors constituting their error terms do not introduce omitted-regressor biases into the coefficients of their included regressors. The model in (11) and (14) is not based on any such assumption; (iv) Pratt and Schlaifer pointed out that in the conventional model the condition that its regressors be independent of “the” omitted regressors constituting its error term is meaningless. The error terms of the model in (11) and (14) are not the functions of “the” omitted-regressors. 4. Conclusions We have removed four major specification errors from the conventional formulation of probit and logit models. A reformulation of Yatchew and Griliches’ probit model so that it is devoid of these specification errors changes their results. We also find that their model has nonunique coefficients and error term. YG make the assumption that omitted regressors constituting the error term of their model do not introduce omitted-regressor biases into the coefficients of the included regressors. We have developed a method of calculating the bias-free partial derivative portions of the coefficients of a correctly specified probit model. Acknowledgments: We are grateful to four referees for their thoughtful comments. Author Contributions: All authors contributed equally to the paper. Conflicts of Interest: The authors declare no conflict of interest. Figure 2. Modified estimates of the “impacts” of education on the earnings for 753 married women. 4. Conclusions We have removed four major specification errors from the conventional formulation of probit and logit models. A reformulation of Yatchew and Griliches’ probit model so that it is devoid of these specification errors changes their results. We also find that their model has nonunique coefficients and error term. YG make the assumption that omitted regressors constituting the error term of their model do not introduce omitted-regressor biases into the coefficients of the included regressors. We have developed a method of calculating the bias-free partial derivative portions of the coefficients of a correctly specified probit model. Acknowledgments: We are grateful to four referees for their thoughtful comments. Author Contributions: All authors contributed equally to the paper. Conflicts of Interest: The authors declare no conflict of interest. Appendix In this Appendix, we show that any of the models estimated in the econometric literature is more restrictive than (1). We also show that these restrictions, when imposed on (1), lead to several specification errors. A. Derivation of Linear and Nonlinear Regressions with Additive Error Terms A.1. Nonunique Coefficients and Error Terms A.1.1. Beginning Problems—Rigorous Derivation of Models with Additive Error Terms It is widely assumed that the error term in an econometric model arises because of omitted regressors influencing the dependent variable. We can use appropriate Felipe and Fisher’s (2003) [ 17 ] Econometrics 2016,4, 26 18 of 21 separability and other conditions to separate the included regressors, x˚ i1 , ..., x˚ iK , from omitted regressors, x˚ i,K`1, ..., x˚ iLi, so that (1) can be written as y˚ i“ψi1px˚ i1, ..., x˚ iKq ` ψi2px˚ i,K`1, ..., x˚ i,Liq “ ψi1px˚ i1, ..., x˚ iK;β1, ..., βpq ` εi(A1) where εi = ψi2px˚ i,K`1 , ..., x˚ i,Liq is a function of omitted regressors. Let εi be the random error term and let ψi1px˚ i1 , ..., x˚ iKq be equal to ψi1px˚ i1 , ..., x˚ iK ; β1 , ..., βpq which is an unknown function of x˚ i1 , ..., x˚ iK20 . Let β1 , ..., βp be the fixed parameters representing the constant features of model (A1). From the above derivation we know what type of conditions which, when imposed on (1), give exactly the model in Greene (2012, p. 181, (7-3)) [3]. The separability conditions used to rewrite (1) in the form of (A1) are very restrictive, as shown by Felipe and Fisher (2003) [ 17 ]. Furthermore, in his scrutiny of the Rotterdam School demand models, Goldberger (1987) [ 19 ] pointed out that the treatment of any features of (1) as constant parameters such as β1 , ..., βp may be questioned and these parameters are not unique 21 . Use of non-unique parameters is a specification error. Therefore, the functional form of (A1) is most probably misspecified. Skyrms (1988, p. 59) [ 7 ] made the important point that spurious correlations disappear when we control for all relevant pre-existing conditions 22 . Even though some of the regressors, x˚ t,K`1 , ..., x˚ t,Lt , represent all relevant pre-existing conditions in our formulation of (A1), they cannot be controlled for, as we should to eliminate false (spurious) correlations, since they are included in the error term of (A1). Therefore, in (A1), the correlations between y˚ tand some of x˚ t1, ..., x˚ tK can be spurious. Karlsen, Myklebust and Tjøstheim (KMT) (2007) [ 10 ] considered a model of the type (A1) for time series data. They assumed that tεtu is an unobserved stationary process and X˚ t1, ..., X˚ tK( and tY˚ tu are both observed nonstationary processes and are of unit-root type. White (1980, 1982) [ 11 , 12 ] also considered (A1) for time-series data and assumed that the εt ’s are serially independent and are distributed with mean zero and constant variance. He also assumed that εt is uncorrelated with X˚ t1, ..., X˚ tK( for all t. Pratt and Schlaifer (1988, 1984) [ 1 , 9 ] criticized that these assumptions are meaningless because they are about εt which is not unique and is composed of variables of which we know nothing. Any distributional assumption about a nonunique error term is arbitrary. A.1.2. Full Independence and the Existence of Conditional Expectations Consider (A1) again. Let X= ψi1`x˚ i1, ..., x˚ iK˘ ,Y= y˚ i and M= εi , be three random variables. Then Xand Mare statistically independent if their joint distribution can be expressed as the product of their marginal distributions. It is not possible to verify this condition. Let H(X) and K(M) be the functions of Xand εi , respectively. As Whittle (1976) [ 20 ] pointed out, we must live with the idea that, for the given random variables like Mand X, we may be only able to assert the validity of the condition ErHpXqKpMqs “ ErHpXqsErKpMqs (A2) where the functions Hand Kare such that E[H(X)] < 8 and E[K(M)] < 8 . If condition (A2) holds only for certain functions, Hand K, then we cannot say that Xand Mare independent. Suppose that Equation (A2) holds only for linear K, so that E[H(X)M] = E(M)E[H(X)] for any H for which E[H(X)] < 8 . This equation is equivalent to Epεi|xq “ Epεiq which shows that the disturbance at 20 Another widely cited work that utilized a set of separability conditions is that of Heckman and Schmierer (2010) [ 18 ]. These authors postulated a threshold crossing model which assumes separability between observables Z that affect choice and an unobservable V. They used a function of Z as an instrument and used the distribution of V to define a fundamental treatment parameter known as the marginal treatment effect. 21 The “uniqueness” is defined in Section2.2.3. 22 We have been using the cross-sectional subscript i so far. We change this subscript to the time subscript t wherever the topic under discussion requires the use of the latter subscript. Econometrics 2016,4, 26 19 of 21 observation iis mean independent of xat i. This may be true for all iin the sample. This mean independence implies Greene’s (2012, p. 183) [3] assumption (3). Let us now drop condition (A2) and let us assume instead that HpXqbe a Borel function of X, (A2.1) E|Y| ă 8, (A2.2) E|YHpXq| ă 8. (A2.3) Using these assumptions, Rao (1973, p. 97) [14] proved that ErHpXqY|X“xs “ HpxqEpY|xq(A3) EtHpXqrY´EpY|Xqsu “ 0 (A4) Equations (A3) and (A4) prove that under conditions (A2.1)–(A2.3), E (Y|x) exists23. A.1.3. Linear Conditional Means and Variances Under the necessary and sufficient conditions of Kagan, Linnik and Rao’s (KLR’s) (1973, pp. 11–12) [ 22 ] lemma (reproduced in Swamy and von zur Muehlen (1988, pp. 114–115) [ 23 ]), the following two equations hold almost certainly: EpY˚ i|x˚ i1, ..., x˚ iKq “ K ÿ j“0 x˚ ijβjwith x˚ i0”1 for i“1, ..., n(A5) and Var pY˚ i|x˚ i1, ..., x˚ iKq “ σ2 ε, a finite, positive constant for all i“1, . . . , n(A6) If the conditions of KLR’s lemma are not satisfied, then (A5) and (A6) are not the correct first and second conditional moments of Y˚ i . The problem is that we cannot know a priori whether or not these conditions are satisfied. The conditions of KLR’s Lemma are not satisfied if x˚ t1, ..., x˚ tK( and ty˚ tu are integrated series 24 . Furthermore, ty˚ tu cannot be made stationary by first differencing it once or more than once because of the nonlinearity of ψt1`x˚ t1, ..., x˚ tK˘ .In these cases, we can use, as Berenguer-Rico and Gonzalo (2013) [ 24 ] do, the concepts of summability, cosummability and balanced relationship to analyze model (A1). Clearly the conditions of KLR’s lemma are stronger than White’s assumptions which, in turn, are stronger than KMT’s (2007) [ 10 ] assumptions. It is clear that KMT’s assumptions are not always satisfied. B. Derivation of the Information Matrix for (10) Consider the log likelihood function in (18). For this function, B2lnL BπLongBpπLongq1“B BpπLongq1 n ř i“1”yifi Fi´p1´yiqfi p1´Fiqıpzibxiq bpx1ibx1iqδε “n ř i“1„yipf1i Fi´f2 i F2 i q ´ p1´yiqp f1i p1´Fiq`f2 i p1´Fiq2qs pzibxiqpz1ibx1iq px1ibx1iqδε (A7) 23 This proof is relevant to Heckman’s interpretation that in any of his models, the error term is the deviation of the dependent variable from its conditional expectation (see Heckman and Vytlacil (2005) [ 21 ]. Conditions (A2.1)–(A2.3) do not always hold and hence this conditional expectation does not always exist. 24 A nonstationary series is integrated of order d if it becomes stationary after being first differenced d times (see Greene (2012, p. 943) [ 3 ]). If y˚ t( in (A1) is a nonstationary series of this type, then it cannot be made stationary by first differencing it once or more than once if ψt1`x˚ t1, ..., x˚ tK˘ is nonlinear. Basmann (1988, p. 98) [ 8 ] acknowledged that a model representation is not free of the most serious objection, i.e., nonuniqueness, if stationarity producing transformations of its observable dependent variable are used. Econometrics 2016,4, 26 20 of 21 where f1iis the partial derivative of fiwith respect to πLong. Epyiq “ 1ˆFi`0ˆ p1´Fiq “ Fi. Using this result in (A7) gives E„´B2lnL BπLongBpπLongq1“ n ÿ i“1 f2 i Fip1´Fiq pzibxiq`z1ibx1i˘ ´x1ibx1i¯δε (A8) where the condition that n> (K+ 1)(p+ 1) is needed for the matrix on the right-hand side of this equation to be positive definite. B2ln L BπLongBδ1ε “B Bδ1ε n ř i“1 pzibxiq bpx1ibx1iqδε”yifi Fi´p1´yiqfi p1´Fiqı “n ř i“1 pzibxiq bpx1ibx1iqδε„yipf1i Fi´f2 i F2 i q ´ p1´yiqp f1i p1´Fiq`f2 i p1´Fiq2q ˆp´1 2qpz1ibx1iqπLong´x1ibx1i¯ ´´x1ibx1i¯δε¯3{2`n ř i“1”yifi Fi´p1´yiqfi p1´Fiqıp´1 2qpzibxiq´x1ibx1i¯ ´´x1ibx1i¯δε¯3{2 (A9) E„´B2ln L BπLongBδ1ε“ n ÿ i“1 f2 i Fip1´Fiq pzibxiq c´x1ibx1i¯δε p´1 2q`z1ibx1i˘πLong ´x1ibx1i¯ ´´x1ibx1i¯δε¯3{2(A10) B2ln L BδεBδ1ε“B Bδ1ε n ř i“1”yifi Fi´p1´yiqfi p1´Fiqıpz1ibx1iqπLongp´1 2q„1 px1ibx1iqδε3 2pxi1bxi1q1 “n ř i“1„yipf1i Fi´f2 i F2 i q ´ p1´yiqp f1i p1´Fiq`f2 i p1´Fiq2q ˆrpz1ibx1iqπLongs2p1 4q„1 px1ibx1iqδε3 px1ibx1iq1px1ibx1iq `n ř i“1”yifi Fi´p1´yiqfi p1´Fiqı ˆpz1ibx1iqπLongp´1 2qp´3 2q„1 px1ibx1iqδε5 2pxi1bxi1q1px1ibx1iq (A11) Taking the expectation of both sides of Equation (A11) gives E„´B2ln L BδεBδ1ε“ n ÿ i“1 f2 i Fip1´Fiqp1 4q“`z1ibx1i˘πLong‰2 ”´x1ibxi1¯δεı3px1ibx1iq1px1ibx1iq(A12) where the condition that n> (K+ 1)(K+ 1) is needed for the matrix on the right-hand side of this equation to be positive definite. 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