1.1 Independent Errors

For the simplest possible motivating example, let Equations (1) and (2) hold. Suppose ϵ₁ and ϵ₂ have the standard model error property of being mean zero and are conditionally independent of each other, so ϵ₁ ⊥ ϵ₂∣Z and E(ϵ₁)=0. It would then follow immediately that the key identifying assumption cov(Z, ϵ₁ϵ₂)=0 holds, because then E(ϵ₁ϵ₂ Z)=E(ϵ₁)E(ϵ₂ Z)=0. This, along with ordinary heteroscedasticity of the errors ϵ₁ and ϵ₂, then suffices for identification.

More generally, independence or uncorrelatedness of ϵ₁ and ϵ₂ is not required, for example, it is shown below that the identifying assumptions still hold if ϵ₁ and ϵ₂ are correlated with each other through a factor structure, and they hold in a classical measurement error framework.

1.2 Classical Measurement Error

Consider a standard linear regression model with a classically mismeasured regressor. Suppose we do not have an outside instrument that correlates with the mismeasured regressor, which is the usual method of identifying this model. It is shown here that we can identify the coefficients in this model just based on heteroscedasticity. The only nonstandard assumption that will be needed for identification is the assumption that the errors in a linear projection of the mismeasured regressor on the other regressors be heteroscedastic, which is more plausible than homoskedasticity in most applications.

The goal is estimation of the coefficients β₁ and γ₁ in where the regression error V ₁ is mean zero and independent of the covariates X, Y*₂. However, the scalar regressor Y*₂ is mismeasured, and we instead observe Y ₂, where Here, U is classical measurement error, so U is mean zero and independent of the true model components X, Y*₂, and V ₁, or equivalently, independent of X, Y*₂, and Y ₁. So far, all of these assumptions are exactly those of the classical linear regression mismeasured regressor model.

Define V ₂ as the residual from a linear projection of Y*₂ on X, so by construction Substituting out the unobservable Y*₂ yields the familiar triangular system associated with measurement error models where the Y ₁ equation is the structural equation to be estimated, the Y ₂ equation is the instrument equation, and ϵ₁ and ϵ₂ are unobserved errors.

The standard way to obtain identification in this model is by an exclusion restriction, that is, by assuming that one or more elements of β₁ equal zero and that the corresponding elements of β₂ are nonzero. The corresponding elements of X are then instruments, and the model is estimated by linear two-stage least squares, with Y ₂=X′β₂+ϵ₂ being the first-stage regression and the second stage is the regression of Y ₁ on and the subset of X that has nonzero coefficients.

Assume now that we have no exclusion restriction and hence no instrument, so there is no covariate that affects Y ₂ without also affecting Y ₁. In that case, the structural model coefficients cannot be identified in the usual way and so, for example, are not identified when U, V ₁, and V ₂ are jointly normal and independent of X.

However, in this mismeasured regressor model, there is no reason to believe that V ₂, the error in the Y ₂ equation, would be independent of X, because the Y ₂ equation (what would be the first-stage regression in two-stage least squares) is just the linear projection of Y ₂ on X, not a structural model motivated by any economic theory.

The perhaps surprising result, which follows from Theorem 1 below, is that if V ₂ is heteroscedastic (and hence not independent of X, as expected), then the structural model coefficients in this model are identified and can be easily estimated. The above assumptions yield a triangular model with E(Xϵ)=0, cov(X, ϵ² ₂)≠0, and cov(X, ϵ₁ϵ₂)=0 and hence satisfy this article's required conditions for identification.

The classical measurement error assumptions are used here by way of illustration. They are much stronger than necessary to apply this article's methodology. For example, identification is still possible when the measurement error U is correlated with some of the elements X and the error independence assumptions given above can be relaxed to restrictions on just a few low-order moments.

1.3 Unobserved Single-Factor Models

A general class of models that satisfy this article's assumptions are systems in which the correlation of errors across equations are due to the presence of an unobserved common factor U, that is: where U, V ₁, and V ₂ are unobserved variables that are uncorrelated with X and are conditionally uncorrelated with each other, conditioning on X. Here, V ₁ and V ₂ are idiosyncratic errors in the equations for Y ₁ and Y ₂, respectively, while U is an omitted variable or other unobserved factor that may directly influence both Y ₁ and Y ₂.

Examples:

Measurement Error: The mismeasured regressor model described above yields Equation (3) with α₁=−γ₁ and Equation (4) with γ₂=0 and α₂=1. The unobserved common factor U is the measurement error in Y ₂.

Supply and Demand: Equations (3) and (4) are supply and (inverse) demand functions, with Y ₁ being quantity and Y ₂ price. V ₁ and V ₂ are unobservables that only affect supply and demand, respectively, while U denotes an unobserved factor that affects both sides of the market, such as the price of an imperfect substitute.

Returns to Schooling: Equations (3) and (4) with γ₂=0 are models of wages Y ₁ and schooling Y ₂, with U representing an individual's unobserved ability or drive (or more precisely, the residual after projecting unobserved ability on X), which affects both her schooling and her productivity (Heckman 1974, 1979).

In each of these examples, some or all of the structural parameters are not identified without additional information. Typically, identification is obtained by imposing equality constraints on the coefficients of X. In the measurement error and returns to schooling examples, assuming that one or more elements of β₁ equal zero permits estimation of the Y ₁ equation using two-stage least squares with instruments X. For supply and demand, the typical identification restriction is that each equation possess this kind of exclusion assumption.

Assume we have no ordinary instruments and no equality constraints on the parameters. Let Z be a vector of observed exogenous variables, in particular, Z could be a subvector of X, or Z could equal X. Assume X is uncorrelated with (U, V ₁, V ₂). Assume also that Z is uncorrelated with (U ², UV_j , V ₁ V ₂) and that Z is correlated with V ² ₂. If the model is simultaneous, assume that Z is also correlated with V ² ₁. An alternative set of stronger but more easily interpreted sufficient conditions are that one or both of the idiosyncratic errors V_j   be heteroscedastic, cov(Z, V ₁ V ₂)=0, and that the common factor U be conditionally independent of Z. These are all standard assumptions, except that one usually either imposes homoscedasticity or allows for heteroscedasticity, rather than requiring heteroscedasticity.

Given these assumptions, which are the requirements for applying this article's identification theorems and associated estimators.

To apply this article's estimators, it is not necessary to assume that the errors are actually given by a factor model such as ϵ _j =α _j U+V_j . In particular, third and higher moment implications of factor model or classical measurement error constructions are not imposed. All that is required for identification and estimation are the moments along with some heteroscedasticity of ϵ _j . The moments (5) provide identification whether or not Z is subvector of X.

1.4 Empirical Examples

Based on earlier working versions of this article, a number of researchers apply this article's identification strategy and associated estimators to a variety of settings where ordinary instruments are either weak or difficult to obtain.

Giambona and Schwienbacher (2007) applied the method in a model relating the debt and leverage ratios of firms to the tangibility of their assets. Emran and Hou (2008) applied it to a model of household consumption in China based on distance to domestic and international markets. Sabia (2007) used the method to estimate equations relating body weight to academic performance, and Rashad and Markowitz (2007) used it in a similar application involving body weight and health insurance. Finally, in a later section of this article, I report results for a model of food Engel curves where total expenditures may be mismeasured. All of these studies report that using this article's estimator yields results that are close to estimates based on traditional instruments (though Sabia 2007 also noted that his estimates are closer to ordinary least squares). Taken together, these studies provide evidence that the methodology proposed in this article may be reliably applied in a variety of real data settings where traditional instrumental variables are not available.

1.5 Literature Review

Surveys of methods of identification in simultaneous systems include Hsiao (1983), Hausman (1983), and Fuller (1987). Roehrig (1988) provided a useful general characterization of identification in situations where nonlinearities contribute to identification, as is the case here. Particularly relevant for this article is previous work that obtains identification based on variance and covariance constraints. With multiple-equation systems, various homoscedastic factor model covariance restrictions are used along with exclusion assumptions in the LISREL class of models (Joreskog and Sorbom 1984). The idea of using heteroscedasticity in some way to help estimation appears in Wright (1928) and so is virtually as old as the method of instrumental variables itself. Recent articles that use general restrictions on higher moments instead of outside instruments as a source of identification include Dagenais and Dagenais (1997), Lewbel (1997), Cragg (1997), and Erickson and Whited (2002).

A closely related result to this article's is Rigobon (2002, 2003), which uses heteroscedasticity based on discrete, multiple regimes instead of regressors. Some of Rigobon's identification results can be interpreted as special cases of this article's models in which Z is a vector of binary dummy variables that index regimes and are not included among the regressors X. Sentana (1992) and Sentana and Fiorentini (2001) employed a similar idea for identification in factor models. Hogan and Rigobon (2003) propose a model that, like this article's, involves decomposing the error term into components, some of which are heteroscedastic.

Klein and Vella (2010) also used heteroscedasticity restrictions to obtain identification in linear models without exclusion restrictions (an application of their method is Rummery, Vella, and Verbeek 1999), and their model also implies restrictions on how ϵ² ₁, ϵ² ₂, and ϵ₁ϵ₂ depend on regressors, but not the same restrictions as those used in the present article. The method proposed here exploits a different set of heteroscedasticity restrictions from theirs, and as a result, this article's estimators have many features that estimators in Klein and Vella (2010) do not have, including the following: This article's assumptions nest standard mismeasured regressor models and unobserved factor models, unlike theirs. This article's estimator extends to fully simultaneous systems, not just triangular systems, and extends to a class of semiparametric models. Klein and Vella (2003) assumed a multiplicative form of heteroscedasticity that imposes strong restrictions on how all higher moments of errors depend on regressors, while this article's model places no restrictions on third and higher moments of ϵ _j conditional on X, Z. Finally, this article provides some set identification results, yielding bounds on parameters, that hold when point-identifying assumptions are violated.

The assumption used here that a product of errors be uncorrelated with covariates has occasionally been exploited in other contexts as well, for example, to aid identification in a correlated random coefficients model, Heckman and Vytlacil (1998) assumed covariates are uncorrelated with the product of a random coefficient and a regression model error.

Some articles have exploited GARCH system heteroscedastic specifications to obtain identification, including King, Sentana, and Wadhwani (1994) and Prono (2008). Other articles that exploit heteroscedasticity in some way to aid identification include Leamer (1981) and Feenstra (1994).

Variables that in past empirical applications have been proposed as instruments for identification might more plausibly be used as this article's Z. For example, in the returns to schooling model, Card (1995) and others propose using measures of access to schooling, such as distance to or cost of colleges in one's area, as wage equation instruments. Access measures may be independent of unobserved ability (though see Carneiro and Heckman 2002) and may affect the schooling decision. However, access may not be appropriate as an excluded variable in wage (or other outcome) equations because access may correlate with the type or quality of education one actually receives or may be correlated with proximity to locations where good jobs are available (see, e.g., Hogan and Rigobon 2003). Therefore, instead of excluding measures of access to schooling or other proposed instruments from the outcome equation, it may be more appropriate to include them as regressors in both equations and use them as this article's Z to identify returns to schooling, given by γ₁ in the triangular model, where Y ₁ is wages and Y ₂ is schooling.

The next section describes this article's main identification results for triangular and then fully simultaneous systems. This is followed by a description of associated estimators and an empirical application to Engel curve estimation. Later sections provide extensions, including set identification (bounds) for when the point-identifying assumptions do not hold and identification results for nonlinear and semiparametric systems of equations.

2.1 Triangular Model Identification

First, consider the linear triangular model: Here, β₁₀ indicates the true value of β₁, and similarly for the other parameters. Traditionally, this model would be identified by imposing equality constraints on β₁₀. Alternatively, if the errors ϵ₁ and ϵ₂ were uncorrelated, this would be a recursive system and so the parameters would be identified. Identification conditions are given here that do not require uncorrelated errors or restrictions on β₁₀. Example applications include unobserved factor models such as the mismeasured regressor model and the returns to schooling model described in the Introduction.

Assumption A1.  Y=(Y ₁, Y ₂)′ and X are random vectors. E(XY′), E(XY ₁ Y′), E(XY ₂ Y′), and E(XX′) are finite and identified from data. E(XX′) is nonsingular.

Assumption A2. E(Xϵ₁)=0, E(Xϵ₂)=0, and, for some random vector Z, cov(Z, ϵ₁ϵ₂)=0.

The elements of Z can be discrete or continuous, and Z can be a vector or a scalar. Some or all of the elements of Z can also be elements of X. Sections 1.1, 1.2, and 1.3 provide examples of models satisfying these assumptions.

Define matrices Ψ _ZX and Ψ _ZZ by and let Ψ be any positive definite matrix that has the same dimensions as Ψ _ZZ .

Theorem 1. Let Assumptions A1 and A2 hold for the model of Equations (6) and (7). Assume cov(Z, ϵ² ₂)≠0. Then, the structural parameters β₁₀, β₂₀, γ₁₀, and the errors ϵ are identified, and

Proofs are in the Appendix. For Ψ=Ψ⁻¹ _ZZ , Theorem 1 says that the structural parameters β₁₀ and γ₁₀ are identified by an ordinary linear two-stage least squares regression of Y ₁ on X and Y ₂ using X and [Z−E(Z)]ϵ₂ as instruments. The assumption that Z is uncorrelated with ϵ₁ϵ₂ means that is a valid instrument for Y ₂ in Equation (6) since it is uncorrelated with ϵ₁, with the strength of the instrument (its correlation with Y ₂ after controlling for the other instruments X) being proportional to the covariance of with ϵ₂, which corresponds to the degree of heteroscedasticity of ϵ₂ with respect to Z.

Taking Ψ=Ψ⁻¹ _ZZ corresponds to estimation based on ordinary linear two-stage least squares. Other choices of Ψ may be preferred for increased efficiency, accounting for error heteroscedasticity. Efficient GMM estimation of this model is discussed later.

The requirement that cov(Z, ϵ² ₂) be nonzero can be empirically tested, because this covariance can be estimated as the sample covariance between Z and the squared residuals from linearly regressing Y ₂ on X. For example, we may apply a Breusch and Pagan (1979) test for this form of heteroscedasticity to Equation (7). Also, if cov(Z, ϵ² ₂) is close to or equal to zero, then will be a weak or useless instrument, and this problem will be evident in the form of imprecise estimates with large standard errors. Hansen (1982) type tests of GMM moment restrictions can also be implemented to check validity of the model's assumptions, particularly Assumption A2.

2.2 Fully Simultaneous Linear Model Identification

Now consider the fully simultaneous model where the errors ϵ₁and ϵ₂ may be correlated, and again no equality constraints are imposed on the structural parameters β₁₀, β₂₀, γ₁₀, and γ₂₀.

In some applications, it is standard or convenient to normalize the second equation so that, like the first equation, the coefficient of Y ₁ is set equal to 1 and the coefficient of Y ₂ is to be estimated. An example is supply and demand, with Y ₁ being quantity and Y ₂ price. The identification results derived here immediately extend to handle that case, because identification of γ₂₀ implies identification of 1/γ₂₀ and vice versa when γ₂₀≠0, which is the only case in which one could normalize the coefficient of Y ₁ to equal 1 in the second equation.

Some assumptions in addition to A1 and A2 are required to identify this fully simultaneous model. Given Assumption A2, reduced-form errors W_j are:

Assumption A3. Define W_j by Equation (11) for j=1, 2. The matrix Φ _W , defined as the matrix with columns given by the vectors cov(Z, W ² ₁) and cov(Z, W ² ₂), has rank 2.

Assumption A3 requires Z to contain at least two elements (though sometimes one element of Z can be a constant; see Corollary 1 later). If for some scalar , as would arise if the common unobservable U is independent of , then Assumptions A2 and A3 might be satisfied by letting Z be a vector of different functions of , for example, defining Z as the vector of elements and (as long as is not binary).

Assumption A3 is testable, because one may estimate W_j as the residuals from linearly regressing Y_j on X and then use Z and the estimated W_j to estimate cov(Z, W ² _j ). A Breusch and Pagan (1979) test may be applied to each of these reduced-form regressions. An estimated matrix rank test such as Cragg and Donald (1997) could be applied to the resulting estimated matrix Φ _W , or perhaps more simply test if the determinant of Φ′ _W Φ _W is zero, since rank 2 requires that Φ′ _W Φ _W be nonsingular.

Assumption A4. Let Γ be the set of possible values of (γ₁₀, γ₂₀). If (γ₁, γ₂)∈ Γ, then (γ⁻¹ ₂, γ⁻¹ ₁)∉Γ.

Given any nonzero values of (γ₁₀, γ₂₀), solving Equation (9) for Y ₂ and Equation (10) for Y ₁ yields another representation of the exact same system of equations but having coefficients (γ⁻¹ ₂₀,γ⁻¹ ₁₀) instead of (γ₁₀,γ₂₀). As long as (γ₁₀,γ₂₀)≠(1, 1) and no restrictions are placed on β₁₀ and β₂₀, Assumption A4 simply says that we have chosen (either by arbitrary convenience or external knowledge) one of these two equivalent representations of the system. Assumption A4 is not needed for models that break this symmetry either by being triangular, as in Theorem 1, or through an exclusion assumption, as in Corollary 2. In other models, the choice of Γ may be determined by context, for example, many economic models (like those requiring stationary dynamics or decreasing returns to scale) require coefficients such as γ₁ and γ₂ to be less than 1 in absolute value, which then defines a set Γ that satisfies Assumption A4. In a supply-and-demand model, Γ may be defined by downward sloping demand and upward sloping supply curves, since in that case, Γ only includes elements γ₁,γ₂ where γ₁⩾0 and γ₂⩽0, and any values that violate Assumption A4 would have the wrong signs. This is related to Fisher's (1976) finding that sign constraints in simultaneous systems yield regions of admissible parameter values.

Theorem 2. Let Assumptions A1, A2, A3, and A4 hold in the model of Equations (9) and (10). Then, the structural parameters β₁₀, β₂₀, γ₁₀, and γ₂₀ and the errors ϵ are identified.

2.3 Additional Simultaneous Model Results

Lemma 1. Define Φ_ϵ to be the matrix with columns given by the vectors cov(Z, ϵ² ₁) and cov(Z, ϵ² ₂). Let Assumptions A1 and A2 hold and assume |γ₁₀γ₂₀|≠1. Then, Assumption A3 holds if and only if Φ_ϵ has rank 2.

Lemma 1 assumes γ₁₀γ₂₀≠1 and γ₁₀γ₂₀≠−1. The case γ₁₀γ₂₀=1 is ruled out by Assumption A4 in Theorem 2. This case cannot happen in the returns to schooling or measurement error applications because triangular systems have γ₂₀=0. Having γ₁₀γ₂₀=1 also cannot occur in the supply-and-demand application, because the slopes of supply and demand curves make γ₁₀γ₂₀⩽0. As shown in the proof of Theorem 2, the case of γ₁₀γ₂₀=−1 is ruled out by Assumption A3, because it causes Φ _W to have rank less than 2. However, Theorem 1 can be relaxed to allow γ₁₀γ₂₀=−1, by replacing Assumption A3 with the assumption that Φ_ϵ has rank 2, because then Equation (27) in the proof still holds and identifies γ₁₀/γ₂₀, which along with γ₁₀γ₂₀=−1 and some sign restrictions could identify γ₁₀ and γ₂₀ in this case. However, Assumption A3 has the advantage of being empirically testable.

In either case, Theorem 2 requires both ϵ₁ and ϵ₂ to be heteroscedastic with variances that depend upon Z, since otherwise the vectors cov(Z, ϵ² ₁) and cov(Z, ϵ² ₂) will equal zero. Moreover, the variances of ϵ₁ and ϵ₂ must be different functions of Z for the rank of Φ_ϵ to be 2.

Corollary 1. Let Assumptions A1, A2, A3, and A4 hold in the model of Equations (9) and (10), replacing cov(Z, ϵ₁ϵ₂) in Assumption A2 with E(Zϵ₁ϵ₂) and replacing cov(Z, W ² _j ) with E(ZW ² _j ) in Assumption A3, for j=1, 2. Then, the structural parameters β₁₀, β₂₀, γ₁₀, and γ₂₀ and the errors ϵ are identified.

Corollary 1 can be used in applications where E(ϵ₁ϵ₂)=0. Theorem 2 could also be used in this case, but Corollary 1 provides additional moments. In particular, if only a scalar is known to satisfy , then identification by Theorem 2 will fail because the rank condition in Assumption A3 is violated with , but identification may still be possible using Corollary 1 because there we may let .

Corollary 2. Let Assumptions A1 and A2 hold for the model of Equations (9) and (10). Assume cov(Z, ϵ² ₂)≠0, that some element of β₂₀ is known to equal zero and the corresponding element of β₁₀ is nonzero. Then, the structural parameters β₁₀, β₂₀, γ₁₀, and γ₂₀ and the errors ϵ are identified.

Corollary 2 is like Theorem 1, except that it assumes an element of β₂₀ is zero instead of assuming γ₂₀ is zero to identify Equation (10). Then, as in Theorem 1, Corollary 2 uses cov(Z, ϵ₁ϵ₂)=0 to identify Equation (9) without imposing the rank 2 condition of Assumption A3 and the inequality constraints of Assumption A4. Only a scalar Z is needed for identification using Theorem 1 or Corollary 1 or 2.

3.1 Simultaneous System Estimation

Consider estimation of the structural model of Equations (9) and (10) based on Theorem 2. Define S to be the vector of elements of Y and X, and the elements of Z that are not already contained in X, if any.

Let μ=E(Z), and let θ denote the set of parameters {γ₁, γ₂, β₁, β₂, μ}. Define the vector valued functions: Define Q(θ, S) to be the vector obtained by stacking the above four vectors into one long vector.

Corollary 3. Assume Equations (9) and (10) hold. Define θ, S, and Q(θ, S) as above. Let Assumptions A1, A2, A3, and A4 hold. Let θ be the set of all values θ might take on, and let θ₀ denote the true value of θ. Then, the only value of θ∈Θ that satisfies E[Q(θ, S)]=0 is θ=θ₀.

A simple variant of Corollary 3 is that if E(ϵ₁ϵ₂)=0, then μ can be dropped from θ, with Q ₃ dropped from Q, and the Z−μ term in Q ₄ replaced with just Z.

Given Corollary 3, GMM estimation of the model of Equations (9) and (10) is completely straightforward. With a sample of n observations S ₁, …, S_n , the standard Hansen (1982) GMM estimator is for some sequence of positive definite Ω _n . If the observations S_i are independently and identically distributed and if Ω _n is a consistent estimator of Ω₀=E[Q(θ₀, S)Q(θ₀, S)′], then the resulting estimator is efficient GMM with More generally, with dependent data, standard time-series versions of GMM would be directly applicable. Alternative moment-based estimators with possibly better small-sample properties, such as generalized empirical likelihood, could be used instead of GMM (see, e.g., Newey and Smith 2004). Also, if these moment conditions are weak (as might occur if the errors are close to homoskedastic), then alternative limiting distribution theory based on weak instruments, such as Staiger and Stock (1997), would be immediately applicable. See Stock, Wright, and Yogo (2002) for a survey of such estimators.

The standard regularity conditions for the large-sample properties of GMM impose compactness of θ. When γ₂₀≠0, this must be reconciled with Assumption A4 and with Lemma 1. For example, in the supply-and-demand model, we might define θ so that the product of the first two elements of every θ∈Θ is finite, nonpositive, and excludes an open neighborhood of –1. This last constraint could be relaxed, as discussed after Lemma 1.

If one wished to normalize the second equation so that the coefficient of Y ₁ equaled 1, as might be more natural in a supply-and-demand system, then the same GMM estimator could be used just by replacing Y ₂−X′β₂−Y ₁γ₂ in the Q ₂ and Q ₄ functions with Y ₁−X′β₂−Y ₂γ₂, redefining β₂ and γ₂ accordingly.

Based on the proof of Theorem 2, a numerically simpler but possibly less efficient estimator would be the following. First, let be the vector of residuals from linearly regressing Y_j on X. Next, let be the sample covariance of with Z_h , where Z_h is the element of the vector Z. Assume Z has a total of K elements. Based on Equation (27), estimate γ₁ and γ₂ by: where Γ is a compact set satisfying Assumption A4. The above estimator for γ₁ and γ₂ is numerically equivalent to an ordinary nonlinear least squares regression over K observations, where K is the number of elements of Z. Finally, β₁ and β₂ may be estimated by linearly regressing and on X, respectively. The consistency of this procedure follows from the consistency of each step, which in turn is based on the steps of the identification proof of Theorem 1 and the consistency of regressions and sample covariances.

In practice, this simple procedure might be useful for generating consistent starting values for efficient GMM.

3.2 Triangular System Estimation

The GMM estimator used for the fully simultaneous system can be applied to the triangular system of Theorem 1 by setting γ₂=0. Define S and μ as before, and now let θ={γ₁, β₁, β₂, μ} and Let Q(θ, S) be the vector obtained by stacking the above four vectors into one long vector, and we immediately obtain Corollary 4.

Corollary 4. Assume Equations (6) and (7) hold. Define θ, S, and Q(θ, S) as above. Let Assumptions A1 and A2 hold with cov(Z, W ² ₂)≠0. Let θ be the set of all values θ might take on, and let θ₀ denote the true value of θ. Then, the only value of θ∈Θ that satisfies E[Q(θ, S)]=0 is θ=θ₀.

The GMM estimator (12) and limiting distribution (13) then follow immediately.

Based on Theorem 1, a simpler estimator of the triangular system of Equations (6) and (7) is as follows. With γ₂₀=0, β₂₀ can be estimated by linearly regressing Y ₂ on X. Then, letting be the residuals from this regression, β₁₀ and γ₁₀ can be estimated by an ordinary linear two-stage least squares regression of Y ₁ on Y ₂ and X, using X and as instruments, where is the sample mean of Z. Letting overbars denote sample averages, the resulting estimators are: where replaces the expectation defining Ψ _ZX with a sample average, and similarly for ; in particular, for ordinary two-stage least squares, would be a consistent estimator of . The limiting distribution for is standard ordinary least squares. The distribution for and is basically that of ordinary two-stage least squares, except account must be taken of the estimation error in the instruments . Using the standard theory of two-step estimators (see, e.g., Newey and McFadden 1994), with independent, identically distributed observations, this gives: where is the influence function associated with .

While numerically simpler, since no numerical searching is required, this two-stage least square estimator could be less efficient than GMM. It will be numerically identical to GMM when the parameters are exactly identified rather than overidentified, that is, when Z is a scalar. More generally, this two-stage least squares estimator could be used for generating consistent starting values for efficient GMM estimation.

3.3 Extension: Additional Endogenous Regressors

We consider two cases here: additional endogenous regressors for which we have ordinary outside instruments, and additional endogenous regressors to be identified using heteroscedasticity.

In the triangular system, the estimator can be described as a linear two-stage least squares regression of Y ₁ on X and on Y ₂, using X and an estimate of [Z−E(Z)]ϵ₂ as instruments. Suppose now that in addition to Y ₂, one or more elements of X are also endogenous. Suppose for now that we also have a set of ordinary instruments P (so P includes all the exogenous elements of X and enough additional outside instruments so that P has at least the same number of elements as X). It then follows that estimation could be done by a linear two-stage least squares regression of Y ₁ on X and on Y ₂, using P and an estimate of [Z−E(Z)]ϵ₂ as instruments. Note, however, that it will now be necessary to also estimate the Y ₂ equation by two-stage least squares, that is, we must first regress Y ₂ on X by two-stage least squares using instruments P to obtain the estimated coefficient , before constructing . Then, as before, the estimate of [Z−E(Z)]ϵ₂ is . Alternatively, the GMM estimator now has Q ₁(θ, S) and Q ₂(θ, S) given by Q ₁(θ, S)=P(Y ₁−X′β₁−Y ₂γ₁) and Q ₂(θ, S)=P(Y ₂−X′β₂), while Q ₃(θ, S) and Q ₄(θ, S) are the same as before.

Similar logic extends to the case where we have more than one endogenous regressor to be identified from heteroscedasticity. For example, suppose we have the model: So, now we have two endogenous regressors, Y ₂ and Y ₃, with no available outside instruments or exclusions. If our assumptions hold both for ϵ₂ and for ϵ₃ in place of ϵ₂, then the model for Y ₁ can be estimated by two-stage least squares, using X and estimates of both [Z−E(Z)]ϵ₂ and [Z−E(Z)]ϵ₃ as instruments.

6.1 Semiparametric Identification

Consider the model where the functions g_j (X) are unknown. In this simultaneous system, each equation is partly linear as in Robinson (1988).

Assumption B1. Y=(Y ₁, Y ₂)′, where Y ₁ and Y ₂ are random variables. For some random vector X, the functions E(Y∣X) and E(YY′∣X) are finite and identified from data.

Given a sample of observations of Y and X, the conditional expectations in Assumption B1 could be estimated by nonparametric regressions, and so would be identified. These conditional expectations are the reduced form of the underlying structural model.

Assumption B2. E(ϵ₁∣X)=0, E(ϵ₂∣X)=0, and for some random vector Z, cov(Z, ϵ₁ϵ₂)=0.

As before, the elements of Z can all be elements of X also, so no outside instruments are required. No exclusion assumptions are imposed, so all of the same regressors X that appear in g ₁ can also appear in g ₂, and vice versa. If ϵ _j =Uα _j +V_j , where U, V ₁, and V ₂ are mutually uncorrelated (conditioning on Z), cov(Z, ϵ₁ϵ₂)=0 if Z is uncorrelated with U ².

Assumption B3. Define W_j =Y_j −E(Y_j ∣X) for j=1, 2. The matrix Φ _W , defined as the matrix with columns given by the vectors cov(Z, W ² ₁) and cov(Z, W ² ₂), has rank 2.

Assumption B3 is analogous to Assumption A3, but employs a different definition of W_j . These definitions will coincide if the conditional expectation of Y given X is linear in X. Lemma 1 continues to hold with this new definition of W_j and hence of Φ _W , and more generally heteroscedasticity of W ₁ and W ₂ implies heteroscedasticity of ϵ.

Theorem 4. Let Equations (19) and (20) hold. If Assumptions B1 and B2 hold, cov(Z, ϵ² ₂)≠0, and γ₂₀=0, then the structural parameter γ₁₀, the functions g ₁(X) and g ₂(X), and the variance of ϵ are identified. If Assumptions B1, B2, B3, and A4 hold, then the structural parameters γ₁₀ and γ₂₀, the functions g ₁(X) and g ₂(X), and the variance of ϵ are identified.

An immediate corollary of Theorem 4 is that the partly linear simultaneous system where X=(X ₁, X ₂) will also be identified, since g_j (X)=h_j (X ₁)+X ₂β _j0 is identified.

6.2 Nonlinear Model Estimation

Consider the model where the functions G_j (X, β₀) are known and the parameter vector β₀, which could include γ₁ and γ₂, is unknown. This generalizes Equations (3) and (4) by allowing nonlinear functions of X. Letting g_j (X)=G_j (X, β₀), Theorem 4 provides sufficient conditions for identification of this model, assuming that β₀ is identified given identification of the functions g_j (X)=G_j (X, β₀). The immediate analog to Corollary 3 is then that β₀, γ₁₀, γ₂₀, and μ₀ can be estimated from the moment conditions: For efficient estimation in this case, where some of the moments are conditional, see, for example, Chamberlain (1987), Newey (1993), and Kitamura, Tripathi, and Ahn (2003). Ordinary GMM can be used for estimation by replacing the first two conditional moments above with unconditional moments: For some chosen vector valued function ζ(X), asymptotic efficiency may be obtained by using an estimated optimal ζ(X); see, for example, Newey (1993) for details.

As in the linear model, some of these moments may be weak, which would suggest the use of weak instrument limiting distributions in the GMM estimation. See Stock, Wright, and Yogo (2002) for a survey of applicable weak moment procedures.

6.3 Semiparametric Estimation

Consider estimation of the partly linear system of Equations (19) and (20), where the functions g_j (X) are not parameterized. We now have identification based on the moments: These are conditional moments containing unknown parameters and unknown functions and so general estimators for these types of models may be applied. Examples include Ai and Chen (2003), Otsu (2003), and Newey and Powell (2003).

Alternatively, the following estimation procedure could be used, analogous to the numerically simple estimator for linear simultaneous models described earlier. Assume we have n independent, identically distributed observations. Let be a uniformly consistent estimator of H_j (X)=E(Y_j ∣X), for example, a kernel or local polynomial nonparametric regression of Y_j on X. Now, as defined by Assumption B3, W_j =Y_j −H_j (X), so let for each observation i. Next, let be the sample covariance of with Z_h , where Z_h is the element of the vector Z. Assume Z has a total of K elements. Based on Equation (27), estimate γ₁ and γ₂ by where Γ is a compact set satisfying Assumption A4. The above estimator for γ₁ and γ₂ is numerically equivalent to an ordinary nonlinear least squares regression over K observations of data, where K is the number of elements of Z. In a triangular system, that is, with γ₂=0, this step reduces to a linear regression for estimating γ₁. Finally, estimates of the functions g ₁(X) and g ₂(X) are obtained by nonparametrically regressing and on X, respectively. The consistency of this procedure follows from the consistency of each step, which in turn is based on the steps of the identification proof of Theorem 4.

This estimator of and is an example of a semiparametric estimator with nonparametric plug-ins (see, e.g., section 8 of Newey and McFadden 1994). Unlike Ai and Chen (2003), this numerically simple procedure might not yield efficient estimates of and . However, assuming that and converge at a faster rate than nonparametric regressions, the limiting distributions of the estimates of the functions g ₁(X) and g ₂(X) will be the same as for ordinary nonparametric regressions of Y ₁−Y ₂γ₁₀ and Y ₂−Y ₁γ₂₀ on X, respectively.

Further extension to estimation of the partly linear system of Equations (21) and (22) is immediate. For this model, the Assumption B2 moments E(ϵ₁∣X)=0, E(ϵ₂∣X)=0, and cov(Z, ϵ₁ϵ₂)=0 are: which could again be consistently estimated by the above described procedure, replacing the nonparametric regression steps with partly linear nonparametric regression estimators, such as Robinson (1988), or by directly applying an estimator, such as Ai and Chen (2003), to these moments.