# Endogeneity, Time-Varying Coefficients, and Incorrect vs. Correct Ways of Specifying the Error Terms of Econometric Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Simultaneous Equations Model

#### 2.1. Conventional Practice

#### 2.1.1. Constant Coefficients

**X**is a T $\times $ K matrix of T observations on K exogenous and predetermined variables,

**B**is a K $\times $ M matrix of constant coefficients, and $\mathit{U}$ is a T $\times $ M matrix of structural disturbances that are assumed to be serially independent. The above notation implies that model (1) is a system of M linear simultaneous equations. Suppose that the identities have already been removed from (1). The linearity assumption in Equation (1) and implied constancy of the elements of $\mathit{\Gamma}$ and

**B**are unduly restrictive and are relaxed in the next section.3

**Interpretation**

**of**

**$\mathit{U}$:**

**The findings**

**of**

**PS**

**(1988)**

**[2]**

**(p. 34):**

**Assumption**

**1.**

**Y**|

**X**) does not always exist, but sufficient conditions for its existence are given in Rao (1973) [5] (p. 97), although not all economists and statisticians interpret the condition E($\mathit{U}|\mathit{X}$) = 0 in the same way. For example, Greene (2012) [4] (p. 223) interpreted it to mean that

**X**is exogenous in model (1) in the sense that

**X**is determined outside of the model. Engle et al. (1983) [6] listed four distinct concepts of exogeneity corresponding to different notions of what is “determined outside the model under consideration” according to the purposes of the inferences being conducted. In Friedman and Schwartz’s (1991) [7] (pp. 41–42) view, it may be appropriate to regard a variable as exogenous for some purposes and as endogenous for others. In this respect, the assumptions of one of Lehmann and Casella’s (1988) [8] (Theorem 4.12, p. 184) theorems are the same as our Assumption 1. Finally, PS (1988) [2] (p. 34) showed that the stronger version of E($\mathit{U}|\mathit{X}$) = 0, i.e., $\mathit{X}$ independent of $\mathit{U}$, is meaningless if the error term of each equation in (1) is made up of relevant regressors omitted from the equation. We show below that Assumption 1(i) does not hold if the coefficients and error term of each equation in (1) are non-unique.

**Reduced**

**form:**

**Normalization**

**rule:**

**Identification:**

**B**, such that only the identity matrix of order M is the admissible value for an M $\times $ M nonsingular matrix

**P**in $-\mathit{B}\mathit{P}{\mathit{P}}^{-1}{\mathit{\Gamma}}^{-1}=\mathit{\Pi}$. The econometric literature has evolved a necessary order and a sufficient rank condition for obtaining unique solutions for the unknown coefficients of the equations in (1) using equations $\mathit{\Pi}\mathit{\Gamma}=-\mathit{B}$ where, under Assumption 1(i), the conditional mean of $\mathit{Y}$ given

**X**is5

**X**has the interpretation of a partial derivative, since

**X**is determined outside model (1).6 However, in the case of endogenous variables, the ratio of a change in one of them to a change in another cannot have a partial derivative interpretation and is therefore meaningless without first determining what caused the change in the denominator (see Greene (2012) [4] (p. 320)).

**Specific**

**Example:**

#### 2.1.2. Conflict between the Exogeneity Assumption about Certain Regressors in a Model and Non-Uniqueness of Its Coefficients and Error Term

**Assumption**

**2.**

**Π**as a result of this deletion. These omitted regressors and omitted-regressor biases are different from omitted regressors constituting ${\mathit{u}}_{\mathit{j}}$ and the biases they introduce, respectively. A less confusing definition of uniqueness is the following:

**Definition**

**(Uniqueness):**

**Theorem**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**U**|

**X**) ≠ 0 which proves the corollary.

**Corollary**

**2.**

#### 2.2. New Practice

#### 2.2.1. Time-Varying Coefficients

**Assumption**

**3.**

#### 2.2.2. Unique Coefficients and Error Term

**Assumption**

**4.**

**Theorem**

**2.**

**Proof.**

**Noteworthy**

**features**

**of**

**Equation**

**(12):**

**The**

**error**

**term**

**of**

**(12):**

**Omitted-regressor**

**biases**

**of**

**the**

**coefficients**

**of**

**(12):**

**Corollary**

**3.**

**Proof.**

**Measurement**

**errors:**

**Measurement-error**

**biases:**

**Components**

**of**

**the**

**coefficients**

**of**

**model**

**(13):**

#### 2.2.3. Comparison of Conventional and New Practices

#### 2.3. Estimation

#### 2.3.1. Parameterization of Model (13)

**Admissibility**

**Condition:**

#### 2.3.2. Choice of Dependent Variable and Regressors to be Included in (13) and Choice of Coefficient Drivers to Be Included in (15) and (16)

#### 2.3.3. Identification

#### 2.3.4. Vector Formulation of Equations (13), (15) and (16)

**Assumption**

**5.**

**Assumption**

**6.**

#### 2.3.5. Estimation of the Bias-Free Components of the Coefficients of (13)

## 3. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

- J.W. Pratt, and R. Schlaifer. “On the Nature and Discovery of Structure (with discussion).” J. Am. Stat. Assoc. 79 (1984): 9–21. [Google Scholar] [CrossRef]
- J.W. Pratt, and R. Schlaifer. “On the Interpretation and Observation of Laws.” J. Econom. 39 (1988): 23–52. [Google Scholar] [CrossRef]
- P.A.V.B. Swamy, I. Chang, J.S. Mehta, W.H. Greene, S.G. Hall, and G.S. Tavlas. “Removing Specification Errors from the Usual Formulation of Binary Choice Models.” Econometrics 4 (2016): 26. [Google Scholar] [CrossRef]
- W.H. Greene. Econometric Analysis, 7th ed. Upper Saddle River, NJ, USA: Pearson, Prentice Hall, 2012. [Google Scholar]
- C.R. Rao. Linear Statistical Inference and Its Applications, 2nd ed. New York, NY, USA: John Wiley & Sons, 1973. [Google Scholar]
- R.D. Engle, D. Hendry, and J. Richard. “Exogeneity.” Econometrica 51 (1983): 277–304. [Google Scholar] [CrossRef]
- M. Friedman, and A.J. Schwartz. “Alternative Approaches to Analyzing Economic Data.” Am. Econ. Rev. 81 (1991): 39–49. [Google Scholar]
- E.L. Lehmann, and G. Casella. Theory of Point Estimation, 2nd ed. New York, NY, USA: Springer, 1998. [Google Scholar]
- A. Kreuger, and S. Dale. Estimating the Payoff to Attending a More Selective College. Working Paper 7322; Cambridge, MA, USA: National Bureau of Economic Research (NBER), 1999. [Google Scholar]
- P.A.V.B. Swamy, H.G. Hall, G.S. Tavlas, I. Chang, H.D. Gibson, W.H. Greene, and J.S. Mehta. “A Method for Measuring Treatment Effects on the Treated without Randomization.” Econometrics 4 (2016): 19. [Google Scholar] [CrossRef]
- P.A.V.B. Swamy, J.S. Mehta, G.S. Tavlas, and S.G. Hall. “Two Applications of the Random Coefficient Procedure: Correcting for Misspecifications in a Small Area Level Model and Resolving Simpson’s Paradox.” Econ. Model. 45 (2015): 93–98. [Google Scholar] [CrossRef]
- P.A.V.B. Swamy, J.S. Mehta, G.S. Tavlas, and S.G. Hall. “Small Area Estimation with Correctly Specified Linking Models.” In Recent Advances in Estimating Nonlinear Models, with Applications in Economics and Finance. Edited by J. Ma and M. Wohar. New York, NY, USA: Springer, 2014, pp. 193–228. [Google Scholar]
- J. Pearl. Causality. Cambridge, UK: Cambridge University Press, 2000. [Google Scholar]
- C.L. Cavanagh, and T.J. Rothenberg. “Generalized Least Squares with Nonnormal Errors.” In Advances in Econometrics and Quantitative Econometrics. Edited by G.S. Maddala, P.C.B. Phillips and T.N. Srinivasan. Cambridge, MA, USA: Blackwell Publishers, Inc., 1995, pp. 276–290. [Google Scholar]

^{1}The concept of “sufficient sets” of omitted regressors is due to PS (1988) [2] (p. 34). The term “bias-free component” means the component free of omitted-regressor and measurement-error biases.^{2}See Greene (2012) [4] (pp. 317,318). We will have an occasion below to discuss the inaccuracy of exogeneity assumption.^{3}The constancy assumption about the coefficients of (1) may mean that this equation system is not the correct specification of the model of $\mathit{Y}$. Here we do not want to use the term “true specification”. Econometricians generally disapprove of the use of the word “true model.” Note that we do not use the econometrician’s term “data-generating process” because it is not informative about omitted-regressors unrepresented by any data in our analysis, preferring instead the term “correct” to “true”.^{4}Some economists and statisticians believe that if model (1) were correctly specified, then the rows of $\mathit{U}$ would be identically and independently distributed (i.i.d.), being free of omitted influences. First of all, one cannot prove that any model is “correctly specified,” and second, the i.i.d. assumption about the rows of $\mathit{U}$ does not mean that each row of $\mathit{U}$ is free of omitted influences.^{5}These order and rank conditions do not hold if the coefficients and error term of each equation in (1) are non-unique, as shown below.^{6}It is shown below that the exogeneity of X does not hold; so analyses based on the reduced form in (4) cannot be carried out if the coefficients and error term of each equation in (1) are non-unique.^{7}Equations (8) and (9) are treated as deterministic.^{8}There is a connection between Theorem 1 and a related theorem in Swamy et al. (2015) [11] that derives uniqueness of the coefficients and error term of a model as a necessary condition for its correct specification.^{9}To avoid a possible misunderstanding, we hasten to point out here that Section 2.1 is written not to criticize econometricians and statisticians in general and Lehmann and Casella [8] in particular but merely to point out the implication of a PS’s result about a meaningless assumption typically made in conventional practice for the consistency of regression coefficient estimators. Note that in proving Theorem 1, only Greene’s (2012) [4] (p. 13) interpretation of the error terms of econometric models was required without resort to further potentially arbitrary assumptions.^{10}For ease of comparison of the derivation in this section with that in the previous section, we do not change the notation ${x}_{tk}^{*}$ to ${y}_{tk}^{*}$.^{11}This result arises as a direct consequence of (11).^{12}The $\gamma $’s in (13) should not be confused with those in (8).^{13}Pearl (2000) [13] (p. 99) elaborated on this condition.^{14}This procedure is different from that of PS (1988) [2] (p. 49). Their method is to search like a non-Bayesian for concomitants that absorb “proxy effects” for omitted regressors. Section 4.2 of their paper shows how they use the concomitants they found.^{15}The rationale for these coefficient drivers is: (i) If we do not make the coefficients of the EE relationship functions of age, then the relationship neglects the fact that most people have higher incomes when they are older than when they are young, regardless of their education. Thus, without the coefficient driver “Age” or without the interaction term between education and age, the coefficient will overstate the marginal effect of education on earnings; (ii) It is often observed that income tends to rise less rapidly in the latter earning years than in the early years. To accommodate this possibility, we enter the square of age to the list of coefficient drivers; (iii) In addition, previous empirical work of ours has shown that the husband’s education and family income are strongly related to the bias-free component and that the other coefficient drivers are strongly related to the omitted-regressor bias component of ${\gamma}_{t1j}$.^{16}The non-sample (prior) values in estimators (21) and (22) can change from user to user and the bias-free components ${\alpha}_{thj}^{*}$ and ${\beta}_{tkj}^{*}$ are not always constants. It is very hard to study the large sample properties of such estimators. Bayesian methods also cannot be used to estimate the ${\alpha}_{thj}^{*}$’s and ${\beta}_{tkj}^{*}$’s because in any Bayesian analysis, it is the knowledge about fixed and unknown parameters that Bayesians model as random and the ${\alpha}_{thj}^{*}$’s and ${\beta}_{tkj}^{*}$’s are unknown but may not be fixed. PS (1988) [2] (p. 49), the Bayesian statisticians, did not really recommend Bayesian analysis of laws but said that “a Bayesian will do much better to search like a non-Bayesian for concomitants that absorb …[‘proxy effects’ for excluded variables]”. We would use this sentence with “concomitants” replaced by “coefficient drivers.”

© 2017 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license ( http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Swamy, P.A.V.B.; Mehta, J.S.; Chang, I.-L. Endogeneity, Time-Varying Coefficients, and Incorrect vs. Correct Ways of Specifying the Error Terms of Econometric Models. *Econometrics* **2017**, *5*, 8.
https://doi.org/10.3390/econometrics5010008

**AMA Style**

Swamy PAVB, Mehta JS, Chang I-L. Endogeneity, Time-Varying Coefficients, and Incorrect vs. Correct Ways of Specifying the Error Terms of Econometric Models. *Econometrics*. 2017; 5(1):8.
https://doi.org/10.3390/econometrics5010008

**Chicago/Turabian Style**

Swamy, P.A.V.B., Jatinder S. Mehta, and I-Lok Chang. 2017. "Endogeneity, Time-Varying Coefficients, and Incorrect vs. Correct Ways of Specifying the Error Terms of Econometric Models" *Econometrics* 5, no. 1: 8.
https://doi.org/10.3390/econometrics5010008