two-stage least squares estimator and the k-class estimator

two-stage least squares estimator and the k-class estimator Two-stage least squares has been a widely used method of estimating the parameters of a single structural equation in a system of linear simultaneous equations This article first considers the estimation of a full system of equations This provides a context for understanding the place of two-stage least squares in simultaneous-equation estimation The article concludes with some comments on the lasting contribution of the two-stage least squares approach and more generally the future of the identification and estimation of simultaneous-equations models Two-stage least squares (2SLS) was originally proposed as a method of estimating the parameters of a single structural equation in a system of linear simultaneous equations It was introduced more or less independently by Theil (1953a; 1953b; 1961), Basmann (1957) and Sargan (1958) The early work on simultaneous equations estimation was carried out by a group of econometricians at the Cowles Foundation This work was based on the method of maximum likelihood In particular, Anderson and Rubin (1949; 1950) developed the limited information maximum likelihood (LIML) estimator for the parameters of a single structural equation Anderson (2005) gives the history of 2SLS a revisionist twist by pointing out that Anderson and Rubin (1950) indirectly includes the 2SLS estimator and its asymptotic distribution The notation of that paper is difficult and the exposition is somewhat obscure, which may explain why few econometricians are aware of its contents See Farebrother (1999) for additional insights into the precursors of 2SLS 2SLS was by far the most widely used method in the 1960s and the early 1970s The explanation involves both the state of statistical knowledge among applied econometricians and the state of computer technology The classic treatment of maximum likelihood methods of estimation is presented in two Cowles Commission monographs: Koopmans (1950), Statistical Inference in Dynamic Economic Models, and Hood and Koopmans (1953), Studies in Econometric Method, which was directed at a wider audience Among applied econometricians, relatively few had the statistical training to master the papers in these monographs, especially Koopmans (1950) By the end of the 1950s computer programs for ordinary least squares were available These programs were simpler to use and much less costly to run than the programs for calculating LIML estimates Owing to advances in computer technology, and, perhaps, also the statistical background of applied econometricians, the popularity of 2SLS started to wane towards the end of the 1970s In particular, the difficulty of calculating LIML estimates was no longer an important constraint This article first considers the estimation of a full system of equations and then focuses on 2SLS This approach provides a context for understanding the place of 2SLS in simultaneous-equation estimation The article is organized as follows A twoequation structural form model with normal errors and no lagged dependent variables is introduced in section Section reviews the properties of the ordinary least squares estimator of the parameters of a structural equation The indirect least squares estimator is introduced in section In section presents the indirect feasible generalized least squares estimator, and briefly discusses maximum likelihood methods Section develops two rationales for the 2SLS procedure, and the k-class family of estimators is defined in section Finite sample results on the comparisons of estimators are reported in section 7, and the concluding comments are in section (Our exposition of structuralform estimation draws heavily on the treatment by Goldberger, 1991 For the presentation of GMM and more recent methods of simulation-equation estimation, see Mittelhammer, Judge and Miller, 2000.) The model In the spirit of Goldberger (1991), we consider a two-equation demand and supply model to fix ideas and notation The endogenous variables are y1 (quantity) and y2 (price), the exogenous variable is x (income), and the disturbances are u1 (demand shock) and u2 (supply shock) For convenience the intercepts are suppressed in both equations The structural form of the model is Demand y1 = α1 y2 + α x + u1 , Supply y2 = α y1 + u2 With the terms in y1 and y2 transferred to the left-hand side, the matrix representation of structural form is  ( y1 , y2 )   −α1 −α  ÷ = x(α , 0) + (u1 , u2 ),  or ′ =x Β ′ +u ′ yΓ In the structural-form coefficient matrices Γ and Β , the columns refer to equations, while the rows refer to variables Each endogenous variable can be solved for in terms of the exogenous variables and structural shocks to get the reduced form of the model: Quantity y1 = π 11 x + v1 , Price y2 = π12 x + v2 In matrix form, ( y1 , y2 ) = x(π 11 , π 12 ) + (v1 , v2 ), or ′ −1 +u ′Γ −1 = x Π ′ +v ′ y ′ = xΒΓ The reduced form is derived by post-multiplying the structural form by Γ −1 , where ′ −1 is the reduced-form Π = ΒΓ −1 is the reduced-form coefficient matrix and v′ = uΓ disturbance vector Next we consider the statistical specification of a linear simultaneous-equation model for the general case of a m × endogenous-variable vector y , the k × exogenous-variable vector x and the m × 1structural-disturbance vector u The specification is the following: ′ = xΒ ′ + u ′, yΓ (A1) Γ nonsingular, (A2) E (u | x) = 0, V (u | xΣ )= (A3) positive definite (A4) Here Γ is m × m, Β is k × m, Σ is m × m Assumption (A1) gives the system of m structural equations in m endogenous variables Assumption (A2) says that the system is complete in the sense that y is uniquely determined by x and u (A3) says that x is exogenous in the sense that the conditional expectation of u given x is zero for all values of x Assumption (A4) is a homoskedasticity requirement, and positive definiteness rules out exact linear dependency among the structural disturbances The implications of the specification (A1)-(A4) are the following: ′ + v ′, v ′= u ′Γ −1 , y ′ = xΠ E ( v | x) = 0, −1 V ( v | xΓ ) = ( ΣΓ ) −1 Ω = positive definite (B1) (B2) (B3) The reduced-form disturbance vector v is mean-independent of, and homoskedastic with respect to, the exogenous variable vector x Next we turn from the population to the sample We suppose that a sample of n observations from the multivariate distribution of x and y is obtained by stratified sampling: n values of x are selected, forming the rows of the n × k observed matrix X with rank (X) = k For each observation, a random drawing is made from the relevant conditional distribution of y given x, giving the rows of the n × m observed matrix Y, where the successive drawings are independent The statements about asymptotic properties of the estimators rely on the additional assumption that the matrix X′X / n has a positive definite limit If instead sampling is random from the joint distribution of x and y , there is no substantial change in the results Ordinary least squares In simultaneous equations models, the parameters of interest are the structural parameter, the α ' s in the demand-supply example and the elements of Γ and Β and in general case, rather than the reduced form parameters, the π 's or Π Ordinary least squares (OLS) estimation of the structural parameters is not appropriate because the structural parameters are not coefficients of the conditional expectation functions among the observable variables We now illustrate this point for the supply equation of the demand and supply model The reduced-form of the demand and supply model expressed explicitly in terms of the structural parameters: y1 = (α x + α1u2 + u1 ) /(1 − α1α ), y2 = (α 2α x + α 3u1 + u2 ) /(1 − α1α ) For convenience, suppose that x, u1 and u2 are trivariate-normally distributed with zero 2 means, variances σ x , σ , σ , and zero correlations Then y2 and y1 are bivariate normal, so the conditional expectation of y2 given y1 is E ( y2 | y1 ) = α * y1 with α * = C ( y1 , y2 ) / V ( y1 ) If a* = α , then the sample least squares regression of y2 on y1 will provide a unbiased minimum variance estimator of α If a* ≠ α , then least squares is not appropriate for the estimation of α From equations (2.1) and (2.2) we calculate C ( y1 , y2 ) = (α 2α 3σ x2 + α 3σ 12 + α1σ 22 ) /(1 − α1α ) , V ( y1 ) = (α 22σ x2 + σ 12 + α12σ 22 ) /(1 − α1α ) 2 Let θ = α σ x + σ Then θ α1σ 22 α * = α3 + (θ + α12σ 22 ) θ + α12σ 22 Clearly the parameter of interest α is not the slope of the conditional expectation function of y2 given y1 This result is usually described by saying that OLS gives a biased estimator of the structural parameter α Another description is that OLS gives a unbiased estimator of slope of the conditional expectation function, which happens to differ from the slope of the structural equation Observe that a* = α in the special case with α1 = ; in this case, y1 is a function of x and u1 only so that E (u2 | y1 ) = The problem with OLS can be illustrated without relying on normality From (1.2) we get E ( y2 | y1 ) = α y1 + E (u2 | y1 ) From eq (2.2), C ( y1 , u2 ) = C (α x + α1u2 + u1 , u2 ) /(1 − α1α ) = α1σ 22 /(1 − α1α ) Because y1 and u2 are correlated, we see that E (u2 | y1 ) ≠ E (u2 ) = Indirect least squares The next method we consider uses OLS to estimate the reduced-form parameters, and then converts the OLS reduced-form estimates into estimates of the structural-form parameters This method, called ‘indirect least squares’ (ILS), produces estimates that are consistent, although not unbiased Koopmans and Hood (1953) attribute ILS to M A Girshick Again see Farebrother (1999) for precursors The key to ILS is the relation that relates the reduced-form coefficients to the structural-form coefficients, namely, Π = ΒΓ −1 , which can be rewritten as ΠΓ = Β Suppose Π is known along with the prior knowledge that certain elements of Γ and Β are zero The question is whether we can solve ΠΓ = Β uniquely for the remaining unknown elements of Γ and Β When a structural parameter is uniquely determined, we say that the parameter is identified in terms of Π or, more simply, that is identified This suggests that the identified structural-form parameters may be estimated via OLS estimates of the reduced-form coefficients The relation between reduced-form and structural coefficients for the demand and supply model is the following: ( π11  −α  π 12 )  ÷= ( α 0)   −α1 There are two equations in three unknowns: π 11 − α1π 12 = α π 12 − α 3π 11 = On the right-hand-side of (3.1), solve the equation for α = π 12 / π 11 We conclude that the slope coefficient of the supply equation is identified With respect to estimation, the ILS estimate of α is obtained by replacing π 11 and π 12 by their OLS counterparts The ILS estimator of α is consistent since the equation-by-equation OLS estimators of π 11 and π 12 are consistent Moreover, the equation-by-equation OLS estimates are the same as the generalized least squares (GLS) estimates, that is, the OLS and GLS estimates coincide in every sample This is because the explanatory variables are identical in the two reduced-form equations A consequence is that the ILS estimator is asymptotically efficient Indirect feasible generalized least squares For some simultaneous-equation models, prior knowledge that certain elements of Γ and Β are zero implies restrictions on Π In this situation, equation-by-equation OLS estimates of the π’s are not optimal, and ILS does not yield a unique estimate of the structural parameters We now illustrate the case with restrictions on Π using a modification of the original structural model The modified model has three exogenous variables, x1 (income), x2 (wage rate) and x3 (interest rate) The modification consists of allowing the three exogenous variables to enter the supply equation: Demand y1 = α1 y2 + α x1 + u1 , Supply y2 = α y1 + α x1 + α x2 + α x3 + u2 The reduced-form of the modified structural-form system is Quantity y1 = π 11 x1 + π 21 x2 + π 31 x3 + v1 , Price y2 = π 12 x1 + π 22 x2 + π 32 x3 + v2 In the ΠΓ = Β format, the relation between the reduced-form and structural coefficients is:  π 11 π 12   ÷ π π 21 22  ÷ −α π ÷  31 π 32  There are now six equations in six unknowns: α α  −α   ÷ ÷ =  α ÷   ÷  α6  π 11 − α1π 12 = α π 12 − α 3π 11 = α π 21 − α1π 22 = π 22 − α 3π 21 = α π 31 − α1π 32 = π 32 − α 3π 31 = α (4.5) The system on the left of (4.5) determines the parameters of the demand equation Solve either of the equations that has on its right-hand side for α1 = π 31 / π 32 = π 21 / π 22 , and then get α from the remaining equation Clearly, the coefficients of the demand equation are identified in terms of Π Furthermore, there is a restriction on the π’s, namely π 31 / π 32 = π 21 / π 22 , because on the left of (4.5) there are three equations in two unknowns The system on the right-hand side of (4.5), which refers to the supply equation, consists of three equations in four unknowns Once a value is assigned to α , the equations can be solved for α , α , α A different arbitrary value for α generates different values for α , α , α The solution is not unique Hence, the coefficients of the supply equation are not identified in terms of Π With respect to estimation, ILS using the equation-by-equation OLS estimates of Π will not give unique estimates of the structural parameters of the supply equation The result is two different ILS estimates of α1 This problem can be overcome by estimating the reduced-form subject to the restriction π 31 / π 32 = π 21 / π 22 The restricted estimates of the π 's can be converted into unique estimates of the α 's using the sample counterpart of the system (4.5) Suppose there are restrictions on Π Then the fact that the explanatory variables are identical in every reduced-form equation does not imply that the OLS and GLS estimates of the π’s are the same In other words, OLS estimation of the reduced form will not be optimal If the variance matrix of the reduced-form disturbance vector Ω is known, then GLS subject to the restrictions on Π is the natural (nonlinear) estimation procedure The conversion of the GLS estimates of the π’s into estimates of the α’s can be described as ‘indirect GLS’ Since the GLS estimator is consistent and asymptotically efficient, the indirect-GLS estimators of Γ and Β are also consistent and asymptotically efficient When Ω is unknown, as is true in practice, feasible GLS is the natural estimation ˆ is used in procedure for Π Feasible GLS is similar to GLS except that an estimator Ω ˆ comes from the residuals of the equation-by-equation OLS place of Ω The estimator Ω reduced-form regressions The resulting estimates of the α’s are referred to as ‘indirect-FGLS’ estimates because the FGLS estimates of the π’s are converted into estimates of the α’s Because the FGLS estimator of Π is consistent and asymptotically efficient, the indirect-FGLS estimators of Γ and Β are also consistent and asymptotically efficient Indirect GLS and indirect FGLS are referred to as ‘full-information’ methods because they use all the restrictions on Π at once Estimation of a single structural equation using only the restrictions on Π for that equation alone is often called ‘limited information’ estimation If all the restrictions are correctly specified, then fullinformation estimators are more efficient than limited-information estimators In some variants of the simultaneous-equation model it is assumed that u | x is multivariate normal The addition of the normality assumption enables the estimation of Π by maximum likelihood The resulting estimator of the structural parameters is known as ‘full-information maximum likelihood’, or FIML If Ω is known, then FIML coincides with indirect-GLS If Ω is unknown, FIML differs from indirect-FGLS, but the estimators have the same asymptotic distribution The difference between FIML and indirect FGLS can be clarified by briefly ˆ = Y - XP , where P = (X′X)-1 X′Y is the turning from the population to the sample Let V estimator of Π obtained by equation-by-equation OLS The estimator of Ω used in ˆ = (1/ n)VV ˆ −1VV ), where ˆ ˆ The criterion minimized by FGLS is tr(Ω FGLS is Ω ˆ = (1/ n)VV ˆ ˆ (as a conditional solution) into V = Y - XΠ FIML proceeds by inserting Ω the log-likelihood function to obtain the log-likelihood concentrated on | V ′V | The consequence is that the criterion minimized by FIML is | V′V | The difference in the criteria explains the difference in the estimators The maximum likelihood estimation of a single structural-form equation that uses only the restrictions on Π for that equation alone is referred to as ‘limited- information maximum likelihood’, or LIML We next consider another limitedinformation estimation method Two-stage least squares The 2SLS estimator uses the unrestricted reduced-form estimate P, the equation-byequation OLS estimates of the π’s, which accounts for its popularity The mechanics of the 2SLS method can be described simply In the first stage, the right-hand-side endogenous variables of the structural equation are regressed on all the exogenous variables in the reduced form, and the fitted values are obtained In the second stage, the right-hand-side endogenous variables are replaced by their fitted values, and the lefthand-side endogenous variable of the equation is regressed on the right-hand-side fitted values and the exogenous variables included in the equation Two rationales for the 2SLS procedure are now developed using the demand equation of the modified structural model The starting point for the first rationale is the expectation of the demand equation conditional on x1, x2, and x3 Taking expectations gives E ( y1 | x1 , x2 , x3 ) = α1 E ( y2 | x1 , x2 , x3 ) + α x1 + E (u1 | x1 , x2 , x3 ), or E ( y1 | x1 , x2 , x3 ) = α1 y2* + α x1 From the reduced-form eq (4.4), y2* = E ( y2 | x1 , x2 , x3 , ) = π 12 x1 + π 22 x2 + π 32 x3 * * Because y2 is linear function of the exogenous variables, it is exogenous If y2 were * observed, then y1 could be regressed on y2 and x1 to get unbiased estimates of α1 and α * But y2 is unobservable because π 12 , π 22 , and π 32 are unknown However, an unbiased and consistent estimate yˆ can be obtained by replacing the unknown π’s by their OLS 10 * estimates Then, making the replacement of yˆ for y2 produces consistent estimates of the structural parameters The second rationale exploits the fact that in the population the following moment conditions hold: E (u1 ) = 0, C ( x1 , u1 ) = C ( x2 , u1 ) = C ( x3 , u1 ) = These imply two orthogonality conditions: E ( y2*u1 ) = 0, E ( x1u1 ) = * If we let u1 = y1 − (a1 y2 + a2 x1 ), then α1 and α are the values for a1 and a2 that make E ( y2*u1 ) = and E ( x1u1 ) = 2SLS chooses the estimates that make the analogous sample quantities zero, that is, ∑ i yˆ 2i u1i = and ∑ i x1i u1i = 0, (i = 1, , n) This illustrates that 2SLS has an instrumental-variable (IV) interpretation The IV interpretation can be illustrated more explicitly by writing the demand equation in terms of the observations for a sample size of n: y = y 2α1 + x1α + u1 α  x1 )  ÷+ u1 α2  = Zα + u 1, = ( y2 where in the context of the demand equation y and y are the columns of Y and x1 is the first column of X As we have shown, regressing y on Z1 will not give a consistent estimator for α1 Instead replace Z1 by Zˆ = NZ1 = N (y , x1 ) = ( Ny , Nx1 ) = ( yˆ , x1 ) , where N is the idempotent matrix X(X′X −1 )X′ Regressing y on Zˆ gives the normal equations, ˆ ′y , Zˆ 1′Zˆ 1a*1 = Z 1 the solution to which is the 2SLS estimator The 2SLS normal equations are equivalent to a set of orthogonality conditions: * Zˆ 1′u1 = 0, where u1 = y - Z1a1 The equivalence follows from an algebraic fact: 11 Zˆ 1′Z1 = Z1′N′Z1 = Z1′N′NZ1 = Zˆ 1′Zˆ The variables in Zˆ are legitimate instruments because they are, at least asymptotically, uncorrelated with the disturbance The IV interpretation implies that the 2SLS estimator is consistent In fact, it is the optimal feasible IV estimator We also note that the 2SLS estimator can be interpreted as a general-method-of* moments (GMM) estimator In the above example, this follows from the fact that a1 minimizes the quadratic form (y - Z1a1* )′X(X′X) −1 X′(y - Z1a1* ) It can shown that 2SLS is the optimal feasible GMM estimator An advantage of the GMM approach is that heteroskedasticity and autocorrelation can be accommodated by an appropriate redefinition of the optimal weighting matrix in the definition of the GMM estimator (see Ruud, 2000, pp 718–21) We conclude this section with some remarks on estimation in the simultaneousequation model • If all the structural equations are identified, and there are no restrictions on Π , then ILS, indirect-FGLS, FIML, LIML and 2SLS all produce the same estimates • If there are restrictions on Π , then LIML and 2SLS produce different estimates in the sample, but the estimators have the same asymptotic distribution, and similarly for indirect FGLS and FIML • If a parameter is not identified, then there is no method to estimate it consistently • We have confined our attention to the case in which the prior information used for identification consists of normalizations and exclusions (zero restrictions) If other information is available (for example, Σ is diagonal, or a coefficient in one structural equation is equal to a coefficient in another), then some modifications are needed in the description of the estimators and their statistical properties The k-class family 12 The k-class family of estimators of the coefficients of a single structural equation is illustrated for the demand equation of the modified structural model For this equation, the estimator is ( a*k = Z1′ (I - kM)Z1 ) −1 Z1′ (I - kM )y , where M = I - N This family was introduced by Theil (1953b; 1961) It includes the OLS estimator (k = 0) and the 2SLS estimator (k = 1) A remarkable fact is that the k-class family includes LIML The LIML estimator is obtained by setting k = λ * , where λ * the smallest root of | W1 - λ W | = In the determinantal equation, W1 is the cross-product matrix of residuals from the OLS regression of ( y , y ) on x1 (the included endogenous variables on the included exogenous variable), and W is the cross-product matrix of the residuals from the OLS regression of ( y , y ) on X (the included endogenous on all the exogenous variables) Moreover, the LIML estimator of α1 is the value of a1 that minimizes the variance ratio: (1, − a1 ) W1 (1, − a1 )′ , (1, −a1 ) W (1, −a1 )′ Accordingly, the LIML estimator is also sometimes referred to as the ‘least-variance ratio’ estimator The k-class estimator is consistent if (k – 1) converges in probability to and has the same limiting distribution as 2SLS if n1/ (k − 1) converges in probability to These conditions are clearly not satisfied when k = and hence by OLS A proof that these conditions are satisfied for LIML is given in Amemiya (1985, pp 237-238) Zellner (1998) shows that a Bayesian method-of-moments approach justifies certain other members of the k-class (and double k-class) family of estimators Finite sample distributions There has been debate about whether the LIML estimator is better than 2SLS The reason for this debate is that the finite sample distributions of the estimators differ when there 13 are restrictions on Π Hence we now limit our attention to the case where restrictions are present A key difference between the estimators is the existence of moments The 2SLS estimator has moments up to certain order By contrast, the LIML estimator has no moments This result holds for an arbitrary number of included endogenous variables Mariano (2001) reviews the moment existence results In addition to the moment existence results, closed form expressions for the moments and probability densities of 2SLS, LIML and k-class estimators have been derived These expressions are complicated; see Phillips (1983) for specific references The finite sample results have mostly come from the study of a model with two included endogenous variables Moreover, it is often assumed that the disturbances are normal The finite sample results are obtained using analytical and simulation methods A survey of the results and their practical implications is given in Mariano (2001) One of the results is that the LIML distribution is far more symmetric than 2SLS, though more spread out, and it approaches normality faster Similarly, Anderson (1982) concludes that for many cases that occur in practice the standard the normal theory is inadequate for 2SLS, but provides a fairly good approximation to the actual distribution of the LIML estimator The symmetry result is not surprising because the approximate distribution of the LIML estimator (obtained from large sample asymptotic expansions) is median unbiased Median unbiasedness does not hold in general for 2SLS It is helpful to put the debate over the relative merits of LIML and 2SLS in perspective On the assumption that the model is correctly specified, the presumption is that maximum likelihood will have better properties in some well-defined sense This is because it uses more information than GMM See Anderson, Kunitomo and Morimume (1986) and Takeuchi and Morimume (1985) for results on the second-order efficiency of LIML An advantage of GMM is that it is robust to misspecifications of the disturbance distribution, although this advantage was not the original motivation for introducing 2SLS Epilogue 14 Keynesian economics initially played a key role in propelling research into the estimation of simultaneous-equation models Interest in the estimation of linear simultaneousequation models began to wane from about the late 1970s Historically, this decline paralleled the decline of the Keynesian paradigm as a result of the monetarist counterrevolution and later rational expectations At the same time, there was growing awareness that linear simultaneous-equation models often suffered from potentially serious misspecification On the one hand, even taking for granted the statistical specification of the model (A1–A4), economic theory did not provide a satisfactory basis for deciding what endogenous and exogenous variables should be excluded from individual structural equations As a consequence, the identification of structural parameters was open to question On the other hand, the statistical specification of the linear structural-form model was itself questionable, due to either the nature of the data or considerations of economic theory or both The issue of identification was highlighted by Sims (1980) in a well-known paper aptly titled ‘Macroeconomics and Reality’ Simultaneous-equation models have been generalized by the introduction of nonlinear structural equations Amemiya (1974) generalized the 2SLS method to nonlinear models His nonlinear 2SLS estimator is a GMM estimator However, unlike in the case of linear models, it cannot be thought of as being obtained in two steps where the first consists of running a least squares regression The GMM is a two-step estimator, but the first step consists of choosing a weighting matrix More generally, GMM and IV estimators can be thought of as the descendants of the 2SLS approach They constitute the contemporary basis for much of the estimation of structural parameters macroeconomics It is in this sense that 2SLS lives on as a structural estimation method Does the identification and estimation of simultaneous-equation models have a future? 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models, the parameters of interest are the structural parameter, the α ' s in the demand-supply example and the elements of Γ and. .. certain other members of the k-class (and double k-class) family of estimators Finite sample distributions There has been debate about whether the LIML estimator is better than 2SLS The reason