Two-stage least squares as minimum distance

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1 Econometrics Journa (2018), voume 0, pp doi: /ectj Two-stage east squares as minimum distance FRANK WINDMEIJER Department of Economics and IEU, University of Bristo, Priory Road, Bristo, BS8 1TU, UK. E-mai: f.windmeijer@risto.ac.uk First version received: Apri 2018; fina version accepted: May 2018 Summary The two-stage east-squares (2SLS) instrumenta-variaes (IV) estimator for the parameters in inear modes with a singe endogenous variae is shown to e identica to an optima minimum-distance (MD) estimator ased on the individua instrument-specific IV estimators. The 2SLS estimator is a inear comination of the individua estimators, with the weights determined y their variances and covariances under conditiona homoskedasticity. It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic. This provides an intuitive interpretation of the Sargan test. The equivaence resuts aso appy to the efficient two-step generaized method of moments and roust optima MD estimators and criterion functions, aowing for genera forms of heteroskedasticity. It is further shown how these resuts extend to the inear overidentified IV mode with mutipe endogenous variaes. Keywords: Instrumenta variaes, Minimum distance, Overidentification test, Two-stage east squares. 1. INTRODUCTION For a singe endogenous-variae inear mode with mutipe instruments, the standard instrumenta variaes (IV) estimator is the two-stage east-squares (2SLS) estimator, which is a consistent and asymptoticay efficient estimator under standard reguarity assumptions and conditiona homoskedasticity; see, e.g. Hayashi (2000, p. 228). This means that the 2SLS estimator comines the information from the mutipe instruments that are asymptoticay optimay under these conditions. An aternative estimator is the optima minimum-distance (MD) estimator, using an estimator of the variance matrix of the individua instrument-specific IV estimators of the parameter of interest. It is shown in the next section, Section 2, that this optima MD estimator, with the variance specified under conditiona homoskedasticity, is identica to the 2SLS estimator. It is further shown that the Sargan test statistic for overidentifying restrictions is the same as the MD criterion test statistic, providing another intuitive interpretation of the Sargan test. Surprisingy, it appears that these equivaence resuts are not avaiae in the iterature and are not discussed in standard textooks. Angrist (1991) derives simiar resuts, ut for the specia case of orthogona inary instruments (see aso the discussion in Angrist and Pischke, 2009, Section 4.2.2), whereas the resuts here are for genera designs. Recenty, Chen et a. (2016)used this setting and the two estimators as an exampe in their much wider-ranging paper, ut they did not reaize that their equivaence and the resuts otained in Section 2 modify the statements of Chen et a. (2016, pp ) The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society. Puished y John Wiey & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Maden, MA, 02148, USA. This is an open access artice under the terms of the Creative Commons Attriution License, which permits use, distriution and reproduction in any medium, provided the origina work is propery cited.

2 2 F. Windmeijer In Section 2.2, the resut is extended to the equivaence of the two-step generaized method of moments (GMM) estimator and the optima minimum-distance estimator ased on a roust variance covariance estimator of the vector of instrument-specific IV estimates, which is roust to genera forms of heteroskedasticity in the cross-sectiona setting considered here. The two-step Hansen J -test statistic for overidentifying restrictions, Hansen (1982), is aso shown to e the same as the roust MD criterion test statistic. In Section 3, we derive equivaence resuts for the mutipe endogenous-variaes case. The setting considered there can est e characterised y the foowing simpe exampe. Consider a inear mode with two endogenous variaes and there are four instruments avaiae. In principe, there are then six distinct sets of two just-identifying instruments. However, a coection of three sets of two instruments that span a instruments is sufficient to provide a information needed. For exampe, if the instruments are denoted z 1, z 2, z 3, and z 4, then the coection of sets {(z 1,z 2 ), (z 2,z 3 ), (z 3, z 4 )} is sufficient. This resuts in three just-identified IV estimates of the two parameters of interest, and as shown in Section 3, that the per parameter optima MD estimators are identica to the 2SLS estimators. 2. EQUIVALENCE RESULT FOR SINGLE ENDOGENOUS-VARIABLE MODEL We have a sampe {(y i,x i,z i )}n i=1 and we consider the mode y i = x i β + u i, where x i is endogenous, such that E(x i u i ) 0. Note that other exogenous variaes in the mode, incuding the constant, have een partiaed out. The k z > 1 instrument vector z i satisfies E(z i u i ) = 0 and is reated to x i via the inear projection or first-stage mode x i = z i π x + v i. (2.1) Let y and x e the n vectors (y 1,y 2,...,y n ) and (x 1,x 2,...,x n ), and et Z e the n k z matrix with ith row z i and jth coumn z j, i = 1,...,n, j = 1,...,k z. Let P Z = Z(Z Z) 1 Z and S() = (y x) P Z (y x). (2.2) The we-known two-stage east-squares (2SLS) instrumenta-variaes estimator is then defined as β 2SLS = arg min S() and is given y β 2SLS = (x P Z x) 1 x P Z y. (2.3) Next consider the individua instrument-specific IV estimators for β, given y β j = (z j x) 1 z j y for j = 1,...,k z.letthek z vector β ind e defined as β ind = ( β 1, β 2,..., β kz ). Then β ind = D 1 zx Z y, 2018 The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

3 2SLS as minimum distance 3 where D zx = diag(z j x). The matrix diag(a j ) is a diagona matrix with jth diagona eement a j. It foows that the 2SLS estimator is a inear comination of the individua estimators, as with k z j=1 w 2ss,j = 1. β 2SLS = (x P Z x) 1 x Z(Z Z) 1 D zx β ind = w 2ss,j β j, (2.4) Under the assumptions that (1/n) n i=1 z j x p λ j 0forj = 1,...,k z, (1/n) n i=1 z i z i H zz and (1/ n) n d i=1 z i u i N(0,σ 2 u H zz ), the imiting distriution of β ind is given y n( β ind ιβ) = ndzx 1 Z u d N(0,σu 2 ), (2.5) where ι is a k z vector of ones and is given y with D λ = diag(λ j ). Let = Dλ 1 H zzdλ 1, = D 1 zx Z ZD 1 zx, with n p. The optima MD estimator for β is then given y resuting in β md = arg min Q(), k z j=1 Q() = ( β ind ι) 1 ( β ind ι), p (2.6) β md = (ι 1 ι) 1 ι 1 β ind. (2.7) It is cear that the MD estimator is aso a inear comination of the individua instrument-specific estimators, β md = k z j=1 w md,j β j, with k z j=1 w md,j = 1. The foowing proposition states the main equivaence resut, Q() = S(), hence β md = β 2ss and w md,j = w 2ss,j for j = 1,...,k z. PROPOSITION 2.1. Let S(), Q(), β 2ss and β md e as defined in (2.2), (2.6), (2.3) and (2.7), respectivey. Then for R, Q() = S() and hence β md = β 2ss. Proof: As ι D zx = Z x, it foows that, for R, Q() = ( β ind ι) 1 ( β ind ι) = (D 1 zx Z y ι) D zx (Z Z) 1 D zx (D 1 zx Z y ι) = (y x) Z(Z Z) 1 Z (y x) = S(). Note that the equivaence resuts otained in Proposition 2.1, given the choice of, do not rey on any high-eve assumptions. For exampe, whist the imiting distriution of β ind in (2.5) can ony e derived under the assumption that λ j 0 for a j = 1,...,k z, the numerica equivaence resuts hod aso when this assumption is vioated and even if λ j = 0 for a j. Let w j = w md,j = w 2ss,j. Whist k z j=1 w j = 1, the weights w j can e negative, in which case β 2ss is not a weighted average of the β j. From the definition of w j in (2.4) it foows that sign(w j ) = sign( π x,j z j x), where π x,j is the jth eement of the ordinary east 2018 The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

4 4 F. Windmeijer squares (OLS) estimator of π x in (2.1). Without oss of generaity (wog), we can code the instruments such that π x,j 0 for a j and standardise such that z j z j /n = 1. It then foows that sign(w j ) = sign( λ j ), where λ j is the OLS estimator of λ j in the first-stage specification x = z j λ j + v j. Therefore, w j 0 for a j if and ony if (iff) λ j 0 for a j. As λ j = (z j Z/n) π x = π x,j + k z =1, j ρ j π x,, where ρ j = z j z /n = ρ j, it foows that λ j 0if kz =1, j ρ j π x, π x,j. A sufficient condition for w j 0 for a j is then that ρ j 0for a j,, >j, i.e. the instruments are uncorreated or positivey correated with each other. The weights for the minimum-distance estimator are otained from the constrained minimization proem w md = arg min w w w s.t. k z w j = 1. Imposing the constraints w j 0forj = 1,...,k z resuts in a standard quadratic programming proem. If there are negative weights in the origina soution, then imposing non-negativity wi ead to some of the w md,j eing set equa to zero. The resuting estimator for β is then equa to a weighted average of the β j s for a suset of the instruments that minimizes the variance over the susets for which the 2SLS and MD estimators are weighted averages of the instrument-specific estimates. j= Test for overidentifying restrictions The standard test for the nu hypothesis H 0 : E(z i u i ) = 0 is the Sargan test statistic given y Sar( β 2ss ) = σ u 2 S( β 2ss ), where σ u 2 = (1/n) n i=1 (y i x i β 2ss ) 2. Under the nu, standard reguarity assumptions and conditiona homoskedasticity, Sar( β 2ss ) converges in distriution to a χk 2 z 1 distriuted random variae; see, e.g. Hayashi (2000, p. 228). Next consider the MD criterion MD( β md ) = σ 2 u Q( β md ), where we can use σ u 2 ecause β md = β 2ss.Letβ j = pim( β j ). Under the nu hypothesis H 0 : β 1 = β 2 = = β kz = β and the assumptions stated aove for the imiting distriution (2.5) to hod, MD( β md ) converges in distriution to a χk 2 z 1 distriuted random variae; see, e.g. Cameron and Trivedi (2005, p. 203). It foows directy from the resuts of Proposition 2.1 that S( β 2ss ) = Q( β md ) and hence Sar( β 2ss ) = MD( β md ). REMARK 2.1. The equivaence of Sar( β 2ss ) and MD( β md ) estaishes an intuitive interpretation of the Sargan test. It tests whether the individua instrument-specific estimators a estimate the same parameter vaue. For a reated discussion, see Parente and Santos Siva (2012) Efficient two-step estimation The equivaence resuts extend to the efficient two-step GMM estimator. For the cross-sectiona setup considered here, this woud cover the case of genera conditiona heteroskedasticity, 2018 The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

5 2SLS as minimum distance 5 E(u 2 i z i) = g(z i ). Assume (1/n) n i=1 u 2 i z iz i p.using β 2ss as the initia consistent one-step GMM estimator, the efficient two-step GMM estimator is defined as β gmm = arg min J(), J()= (y x) Z 1 ( β 2ss )Z (y x), n ( β 2ss ) = (y i x i β 2ss ) 2 z i z i. i=1 d The Hansen J -test for overidentifying restrictions is given y J( β gmm ). Under standard assumptions, J( β gmm ) χk 2 z 1 under the nu H 0: E(z i u i ) = 0. Under the assumptions as stated aove, the imiting distriution of β ind is given y n( β ind ιβ) d N(0, r ), where r = Dλ 1 D 1 λ and, as β md = β 2ss, a roust variance estimator for β ind is given y Define the roust MD estimator as r = D 1 zx ( β 2ss )D 1 zx. β md,r = arg min MD r (), 1 MD r () = ( β ind ι) ( β ind ι). Under the nu as specified aove, H 0 : β 1 = β 2 = = β kz = β and the assumptions stated aove, MD r ( β md,r ) converges in distriution to a χk 2 z 1 distriuted random variae. It foows directy from the proof of Proposition 2.1 that, for R, MD r () = J() and hence β gmm = β md,r and J( β gmm ) = MD r ( β md,r ). REMARK 2.2. An aternative one-step roust variance estimator for the MD estimator is given y r,ind = D 1 zx ( β ind )D 1 zx, with the eements of ( β ind ) given y ( β ind ) j, = r n (y i x i β j )(y i x i β )z ij z i i=1 for j, = 1,...,k z. The resuting minimum-distance estimator has the same imiting distriution as β md,r, ut differs in finite sampes. Note that the minimum-distance ojective function we consider here is different from the minimum-distance approach that eads, for exampe, to the imited information maximum ikeihood (LIML) and continuousy updating (CU) GMM estimators. Consider the OLS estimators π x and π y for π x in mode (2.1) and π y in the specification y i = z i π y + ε i = 2018 The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

6 6 F. Windmeijer z i (βπ x) + u i + βv i. Then consider the minimum-distance estimator ) ) ( πy βπ ( β nmd, π x,nmd ) = arg min x ( πy V 1 βπ x, β,π x π x π x π x π x where V = Var(( π y π x ) ).If V is a variance estimator vaid under conditiona homoskedasticity ony, β nmd is equa to the LIML estimator; see Goderger and Okin (1971). If V is a roust variance estimator, β nmd is the CU-GMM estimator; see the discussion in Windmeijer (2018). Other recent approaches to minimum-distance estimation are Søvsten (2017) and Koesár (2018). 3. MULTIPLE ENDOGENOUS VARIABLES Consider next the mutipe endogenous-variaes mode y i = x i β + u i, where x i is a k x vector of endogenous variaes. There are k z >k x instruments z i avaiae. Let X e the n k x matrix of expanatory variaes, with th coumn x. Then the 2SLS estimator is otained as and is given y β 2ss = arg min S(), S() = (y X) P Z (y X) β 2ss = (X P Z X) 1 X P Z y. An MD estimator coud of course e otained here in simiar fashion to the one-variae case aove, from ( k z k x ) sets of just-identifying instruments and a generaized inverse for the now rankdeficient variance matrix. Cacuating the variance matrix under conditiona homoskedasticity eads again to equivaence of the 2SLS and MD estimators. However, more interesting resuts can e derived for the 2SLS and MD estimators of the individua coefficients β, = 1,...,k x. Denote y X the th coumn of X, and et X e the k x 1 coumns of X, excuding X. The 2SLS estimator for β is given y β,2ss = ( X M X X ) 1 X M X y = ( x x ) 1 x y, (3.1) where, for a genera n k matrix A, M A = I n P A, with I n the identity matrix of order n, and x = M X X. (3.2) Let {Z [t] } k z k x +1 t=1 e a coection of k z k x + 1setsofk x instruments Z [t] such that a instruments have een incuded. For exampe, {Z [t] = (z t,...,z t+kx 1)} k z k x +1 t=1 is such a set. From these sets, we get k z k x + 1 just-identified IV estimates β [t] of β. Let β,ind = ( β [t] ) e the (k z k x + 1) vector of the individua estimates of β.let X [t] = P Z [t]x, 2018 The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

7 2SLS as minimum distance 7 and et X [t] and X [t] e defined anaogousy to aove. The eements of β,ind are then given y for t = 1,...,k z k x + 1, where Hence, with β [t] [t],ind = ( X = ( x [t] M X [t] x [t] [t] ) 1 x x [t] X [t] [t] ) 1 X y M X [t] y = X M X [t] [t]. (3.3) β,ind = D 1 X y, (3.4) X = ( x [1],..., x [k z k x +1] ) and D = diag( x [t] x [t] ), t = 1,...,k z k x + 1. From (3.4) and the proof of Proposition 3.1 eow, it foows that n( D 1 X y ιβ) = n D 1 X u. Hence, the variance of β,ind under conditiona homoskedasticity can e specified as Var( β,ind ) = σ 2 u, The MD estimator for β is then otained as = D 1 X X D 1. β,md = arg min Q (), 1 Q () = ( β,ind ι) ( β,ind ι), where ι is here a k z k x + 1 vector of ones. Therefore, β,md is given y β,md = (ι D ( X X ) 1 D ι) 1 ι D ( X X ) 1 D β,ind = (ι D ( X X ) 1 D ι) 1 ι D ( X X ) 1 X y. (3.5) The next proposition estaishes the equivaence of β,2ss and β,md for = 1,...,k x. PROPOSITION 3.1. For = 1,...,k x,et β,2ss, β,ind and β,md e as defined in (3.1), (3.4) and (3.5), respectivey, with β,ind ased on a coection of k z k x + 1 sets of k x instruments {Z [t] } k z k x +1 t=1 that contains a instruments. Then β,2ss = β,md for = 1,...,k x. Proof: From the definitions of x and x [t] in (3.2) and (3.3), respectivey, it foows that x x[t] = x (P Z P Z X (X P ZX ) 1 X P Z) (P Z [t] P Z [t]x (X P Z [t]x ) 1 X P Z [t])x = x (P Z [t] P Z [t]x (X P Z [t]x ) 1 X P Z [t])x = x x[t] = x [t] x [t] for t = 1,...,k z k x + 1, and hence ι D = x X. Therefore, from (3.5), β,md = ( x P X x ) 1 x P X y The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

8 8 F. Windmeijer As the sets of instruments {Z [t] } k z k x +1 t=1 contain a k z instruments, it foows that x is in the coumn space of X, and so P X x = x. Therefore, for = 1,...,k x. β,md = ( x x ) 1 x y = β,2ss Next consider the Sargan test statistic, given y Sar( β 2ss ) = σ 2 u S( β 2ss ), (3.6) where σ 2 u = (1/n) n i=1 (y i x i β 2ss ) 2. Under the nu H 0 : E(z i u i ) = 0, standard reguarity conditions and conditiona homoskedasticity, Sar( β 2ss ) χk 2 z k x. Consider the MD statistics d MD( β,md ) = σ 2 u Q( β,md ) (3.7) for = 1,...,k x.letβ,ind = pim( β,ind ). Then MD( β,md ) χk 2 z k x under the nu H 0 : β,ind = ιβ, ut as σ u 2 has to e a consistent estimator of σ u 2, the maintained assumptions are that β s,ind = ιβ s for s = 1,...,k x, s. The foowing proposition states the equivaence of Sar( β 2ss ) and MD( β,md ) for = 1,...,k x. PROPOSITION 3.2. Let Sar( β 2ss ) and MD( β,md ) e defined as in (3.6) and (3.7). Then Sar( β 2ss ) = MD( β,md ) for = 1,...,k x. Proof: As X (y X β 2ss ) = 0, β,2ss = β,md, and defining ỹ = M X P Z y, it foows that S( β 2ss ) = (y X β 2ss ) P Z (y X β 2ss ) = (y x β,2ss X β,2ss ) P Z M X P Z (y x β,2ss X β,2ss ) = (ỹ x β,2ss ) (ỹ x β,2ss ) = (ỹ x β,2ss ) P X (ỹ x β,2ss ) X ỹ D 1 X x β,md ) D ( X X ) 1 D ( D 1 1 = ( β,ind ι β,md ) ( β,ind ι β,md ) = Q( β,md ). = ( D 1 d X ỹ D 1 X x β,md ) As for the singe endogenous-variae case, these resuts can e extended to the two-step GMM and roust MD estimators. ACKNOWLEDGEMENTS This research was party funded y the Medica Research Counci (MC_ UU_12013/9). I woud ike to thank a referee, the co-editor Victor Chernozhukov, Steve Pischke, Mark Schaffer, Chris Skees and Jon Tempe for hepfu comments. REFERENCES Angrist, J. D. (1991). Grouped-data estimation and testing in simpe aor-suppy modes. Journa of Econometrics 47, The Author. The Econometrics Journa puished y John Wiey & Sons Ltd on ehaf of Roya Economic Society.

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