Chapter 6: Endogeneity and Instrumental Variables (IV) estimator

Chapter 6: Endogeneity and Instrumental Variables (IV) estimator Advanced Econometrics - HEC Lausanne Christophe Hurlin University of Orléans December 15, 2013 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 1 / 68

Section 1 Introduction Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 2 / 68

1. Introduction The outline of this chapter is the following: Section 2. Endogeneity Section 3. Instrumental Variables (IV) estimator Section 4. Two-Stage Least Squares (2SLS) Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 3 / 68

1. Introduction References Amemiya T. (1985), Advanced Econometrics. Harvard University Press. Greene W. (2007), Econometric Analysis, sixth edition, Pearson - Prentice Hil (recommended) Pelgrin, F. (2010), Lecture notes Advanced Econometrics, HEC Lausanne (a special thank) Ruud P., (2000) An introduction to Classical Econometric Theory, Oxford University Press. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 4 / 68

1. Introduction Notations: In this chapter, I will (try to...) follow some conventions of notation. f Y (y) F Y (y) Pr () y Y probability density or mass function cumulative distribution function probability vector matrix Be careful: in this chapter, I don t distinguish between a random vector (matrix) and a vector (matrix) of deterministic elements (except in section 2). For more appropriate notations, see: Abadir and Magnus (2002), Notation in econometrics: a proposal for a standard, Econometrics Journal. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 5 / 68

Section 2 Endogeneity Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 6 / 68

2. Endogeneity Objectives The objective of this section are the following: 1 To de ne the endogeneity issue 2 To study the sources of endogeneity 3 To show the inconsistency of the OLS estimator (endogeneity bias) Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 7 / 68

2. Endogeneity Objectives in this chapter, we assume that the assumption A3 (exogeneity) is violated: but the disturbances are spherical: E ( εj X) 6= 0 N 1 V ( εj X) = σ 2 I N Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 8 / 68

2. Endogeneity The reasons for suspecting E ( εj X) 6= 0 are varied: 1 Errors-in-variables 2 Jointly endogenous variables: the usual example is running quantities on prices to estimate a demand equation (supply also a ects the determination of equilibrium). 3 Omitted variables: one or more columns in X cannot be included in the regression because no data on those variables are available estimation will be altered to the extent that the missing variables and the included ones are correlated Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 9 / 68

2. Endogeneity 1. Error-in-variables 1 Consider the regression model: where E ( ε i j x i ) = 0. y i = xi > β + ε i 2 One does not observe (y, x ) but (y, x) y i = y i + v i x i = x i + w i with E (v i ) = E (v i ε i ) = E (v i yi ) = E w i > xi = 0 E (w i ) = E (v i w i ) = E (w i ε i ) = E (w i y i ) = E (v i x i ) = 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 10 / 68

2. Endogeneity 1. Error-in-variables (cont d) 1 The mismeasured regression equation is given by: with η i = ε i v i + w > i β. y i = xi > β + ε i () y i = x > i β + ε i v i + w > i β () y i = x > i β + η i 2 The composite error term η i is not orthogonal to the mismeasured independent variable x i. E (η i x i ) 6= 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 11 / 68

2. Endogeneity 1. Error-in-variables (cont d) Indeed, we have: As a consequence: E (η i x i ) = E (ε i x i ) η i = ε i v i + w > i β. E (v i x i ) + E w i > β x i = E w i > β x i E (η i x i ) 6= 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 12 / 68

2. Endogeneity 2. Simultaneous equation bias Consider the demand equation q d = α 1 p + α 2 y + u d where q d, p and y denote respectively the quantity, the price and income. Unfortunately, the price p is not exogenous or the orthogonality condition E (u d p) = 0 is not satis ed! E (u d p) 6= 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 13 / 68

2. Endogeneity 2. Simultaneous equation bias (cont d) Indeed, the supply/demand system can be written as: q d = α 1 p + α 2 y + u d q s = β 1 p + u s q d = q p where E (u d ) = E (u s ) = E (u s u d ) = E (u s y) = E (u d y) = 0. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 14 / 68

2. Endogeneity 2. Simultaneous equation bias (cont d) Solving q d = q p, the reduced-form equations, which express the endogenous variables in terms of the exogenous variables, write: Therefore p = α 2y β 1 α 1 + u d u s β 1 α 1 = π 1 y + w 1 q = β 1 α 2y β 1 α 1 + β 1 u d α 1 u s β 1 α 1 = π 2 y + w 2 E (u d p) = σ2 u d β 1 α 1 6= 0 This result leads to an overestimated (upward biased) price coe cient. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 15 / 68

2. Endogeneity 3. Omited variables Consider the true model: y i = β 1 + β 2 x 1i + β 2 x 2i + ε i with E (ε i ) = E (ε i x 1i ) = E (ε i x 2i ) = 0. If we regress y on a constant and x 1 (omitted variable x 2 ): y i = β 1 + β 2 x 1i + µ i µ i = β 2 x 2i + ε i If Cov (x 1i, x 2i ) 6= 0, then E (µ i x 1i ) 6= 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 16 / 68

2. Endogeneity Question What is the consequence of the endogeneity assumption on the OLS estimator? Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 17 / 68

2. Endogeneity Consider the (population) multiple linear regression model: where (cf. chapter 3): y = Xβ + ε y is a N 1 vector of observations y i for i = 1,.., N X is a N K matrix of K explicative variables x ik for k = 1,., K and i = 1,.., N ε is a N 1 vector of error terms ε i. β = (β 1..β K ) > is a K 1 vector of parameters Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 18 / 68

2. Endogeneity The OLS estimator is de ned as to be: 1 bβ OLS = X X > X > y If we assume that Then, we have: E ( εj X) 6= 0 E bβols 1 X = β 0 + X > X X > E ( εj X) 6= 0 E bβols = E X E bβols X 6= β 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 19 / 68

2. Endogeneity Theorem (Bias of the OLS estimator) If the regressors are endogenous, i.e. E ( εj X) 6= 0, the OLS estimator of β is biased E bβols 6= β 0 where β 0 denotes the true value of the parameters. This bias is called the endogeneity bias. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 20 / 68

2. Endogeneity Remark 1 We saw in Chapter 1 that an estimator may be biased ( nite sample properties) but asymptotically consistent (ex: uncorrected sample variance). 2 But in presence of endogeneity, the OLS estimator is also inconsistent. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 21 / 68

2. Endogeneity Objectives We assume that: plim 1 N X> ε = γ 6= 0 K 1 where γ = E (x i ε i ) 6= 0 K 1 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 22 / 68

2. Endogeneity Given the de nition of the OLS estimator: 1 bβ OLS = β 0 + X > X X > ε We have: Or equivalently: plim β b 1 1 1 OLS = β 0 + plim N X> X plim N X> ε plim b β OLS = β 0 + Q 1 γ 6= β 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 23 / 68

2. Endogeneity Theorem (Inconsistency of the OLS estimator) If the regressors are endogenous with plim N 1 X > ε = γ, the OLS estimator of β is inconsistent where Q = plim N 1 X > X. plim b β OLS = β 0 + Q 1 γ Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 24 / 68

2. Endogeneity Remark The bias and the inconsistency property is not con ned to the coe cients on the endogenous variables. Consider a case where all but the last variable in X are uncorrelated with ε: 0 1 0 plim 1 N X> ε = γ = B 0 C @.. A γ Then we have: plim b β OLS = β 0 + Q 1 γ There is no reason to expect that any of the elements of the last column of Q 1 will equal to zero. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 25 / 68

2. Endogeneity Remark (cont d) plim b β OLS = β 0 + Q 1 γ 1 The implication is that even though only one of the variables in X is correlated with ε, all of the elements of b β OLS are inconsistent, not just the estimator of the coe cient on the endogenous variable. 2 This e ects is called smearing; the inconsistency due to the endogeneity of the one variable is smeared across all of the least squares estimators. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 26 / 68

2. Endogeneity Example (Endogeneity, OLS estimator and smearing) Consider the multiple linear regression model y i = 0.4 + 0.5x i1 0.8x i2 + ε i where ε i is i.i.d. with E (ε i ). We assume that the vector of variables de ned by w i = (x i1 : x i2 : ε i ) has a multivariate normal distribution with with = @ w i N (0 31, ) 0 1 0.3 0 0.3 1 0.5 0 0.5 1 It means that Cov (ε i, x i1 ) = 0 (x 1 is exogenous) but Cov (ε i, x i2 ) = 0.5 (x 2 is endogenous) and Cov (x i1, x i2 ) = 0.3 (x 1 is correlated to x 2 ). 1 A Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 27 / 68

2. Endogeneity Example (Endogeneity, OLS estimator and smearing (cont d)) Write a Matlab code to (1) generate S = 1, 000 samples fy i, x i1, x i2 g N i=1 of size N = 10, 000. (2) For each simulated sample, determine the OLS estimators of the model y i = β 1 + β 2 x i1 + β 3 x i2 + ε i Denote β b > s = b β 1s bβ 2s bβ 3s the OLS estimates obtained from the simulation s 2 f1,..sg. (3) compare the true value of the parameters in the population (DGP) to the average OLS estimates obtained for the S simulations Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 28 / 68

2. Endogeneity Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 29 / 68

2. Endogeneity Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 30 / 68

2. Endogeneity Question: What is the solution to the endogeneity issue? The use of instruments.. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 31 / 68

2. Endogeneity Key Concepts 1 Endogeneity issue 2 Main sources of endogeneity: omitted variables, errors-in-variables, and jointly endogenous regressors. 3 Endogeneity bias of the OLS estimator 4 Inconsistency of the OLS estimator 5 Smearing e ect Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 32 / 68

Section 3 Instrumental Variables (IV) estimator Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 33 / 68

3. Instrumental Variables (IV) estimator Objectives The objective of this section are the following: 1 To de ne the notion of instrument or instrumental variable 2 To introduce the Instrumental Variables (IV) estimator 3 To study the asymptotic properties of the IV estimator 4 To de ne the notion of weak instrument Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 34 / 68

3. Instrumental Variables (IV) estimator De nition (Instruments) Consider a set of H variables z h 2 R N for h = 1,..N. Denote Z the N H matrix (z 1 :.. : z H ). These variables are called instruments or instrumental variables if they satisfy two properties: (1) Exogeneity: They are uncorrelated with the disturbance. E ( εj Z) = 0 N 1 (2) Relevance: They are correlated with the independent variables, X. for h 2 f1,.., Hg and k 2 f1,.., K g. E (x ik z ih ) 6= 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 35 / 68

3. Instrumental Variables (IV) estimator Assumptions: The instrumental variables satisfy the following properties. Well behaved data: plim 1 N Z> Z = Q ZZ a nite H H positive de nite matrix Relevance: plim 1 N Z> X = Q ZX a nite H K positive de nite matrix Exogeneity: plim 1 N Z> ε = 0 K 1 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 36 / 68

3. Instrumental Variables (IV) estimator De nition (Instrument properties) We assume that the H instruments are linearly independent: E Z > Z is non singular or equivalently rank E Z > Z = H Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 37 / 68

3. Instrumental Variables (IV) estimator Remark The exogeneity condition E ( ε i j z i ) = 0 =) E (ε i z i ) = 0 with z i = (z i1..z ih ) > can expressed as an orthogonality condition or moment condition E z i y i x i > β = 0 The sample analog is N 1 N z i y i i=1 x i > β = 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 38 / 68

3. Instrumental Variables (IV) estimator De nition (Identi cation) The system is identi ed if there exists a unique β = β 0 such that: E z i y i x i > β = 0 where z i = (z i1..z ih ) >. For that, we have the following conditions: (1) If H < K the model is not identi ed. (2) If H = K the model is just-identi ed. (3) If H > K the model is over-identi ed. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 39 / 68

3. Instrumental Variables (IV) estimator Remark 1 Under-identi cation: less equations (H) than unknowns (K)... 2 Just-identi cation: number of equations equals the number of unknowns (unique solution)...=> IV estimator 3 Over-identi cation: more equations than unknowns. Two equivalent solutions: 1 Select K linear combinations of the instruments to have a unique solution )...=> Two-Stage Least Squares 2 Set the sample analog of the moment conditions as close as possible to zero, i.e. minimize the distance between the sample analog and zero given a metric (optimal metric or optimal weighting matrix?) => Generalized Method of Moments (GMM). Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 40 / 68

3. Instrumental Variables (IV) estimator Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 41 / 68

3. Instrumental Variables (IV) estimator Assumption: Consider a just-identi ed model H = K Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 42 / 68

3. Instrumental Variables (IV) estimator Motivation of the IV estimator By de nition of the instruments: plim 1 N Z> ε = plim 1 N Z> (y Xβ) = 0 K 1 So, we have: or equivalently plim 1 N Z> y = β = plim 1 N Z> X plim 1 1 N Z> X plim 1 N Z> y β Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 43 / 68

3. Instrumental Variables (IV) estimator De nition (Instrumental Variable (IV) estimator) If H = K, the Instrumental Variable (IV) estimator β b IV of parameters β is de ned as to be: 1 bβ IV = Z X > Z > y Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 44 / 68

3. Instrumental Variables (IV) estimator De nition (Consistency) Under the assumption that plim N 1 Z > ε, the IV estimator β b IV is consistent: bβ IV p! β0 where β 0 denotes the true value of the parameters. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 45 / 68

3. Instrumental Variables (IV) estimator Proof By de nition: So, we have: 1 1 1 bβ IV = β 0 + N Z> X N Z> ε plimβ b IV = β 0 + plim 1 1 N Z> X plim 1 N Z> ε Under the assumption of exogeneity of the instruments So, we have plim 1 N Z> ε = 0 K 1 plim b β IV = β 0 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 46 / 68

3. Instrumental Variables (IV) estimator De nition (Asymptotic distribution) Under some regularity conditions, the IV estimator b β IV is asymptotically normally distributed: p N bβiv d! β 0 N 0K 1, σ 2 QZX 1 Q ZZ QZX 1 where Q ZZ K K = plim 1 N Z> Z Q ZX K K = plim 1 N Z> X Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 47 / 68

3. Instrumental Variables (IV) estimator De nition (Asymptotic variance covariance matrix) The asymptotic variance covariance matrix of the IV estimator b β IV is de ned as to be: V asy bβiv A consistent estimator is given by = σ2 N Q ZX 1 Q ZZ QZX 1 bv 1 1 bβiv asy = bσ 2 Z > X Z > Z X > Z Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 48 / 68

3. Instrumental Variables (IV) estimator Remarks 1 If the system is just identi ed H = K, 1 1 Z > X = X > Z Q ZX = Q XZ the estimator can also written as bv 1 bβiv asy = bσ 2 Z > X Z > Z Z > X 1 2 As usual, the estimator of the variance of the error terms is: bσ 2 = bε> bε N K = 1 N K N y i i=1 x > i bβ IV 2 Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 49 / 68

3. Instrumental Variables (IV) estimator Relevant instruments 1 Our analysis thus far has focused on the identi cation condition for IV estimation, that is, the exogeneity assumption, which produces plim 1 N Z> ε = 0 K 1 2 A growing literature has argued that greater attention needs to be given to the relevance condition plim 1 N Z> X = Q ZX a nite H K positive de nite matrix with H = K in the case of a just-identi ed model. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 50 / 68

3. Instrumental Variables (IV) estimator Relevant instruments (cont d) plim 1 N Z> X = Q ZX a nite H K positive de nite matrix 1 While strictly speaking, this condition is su cient to determine the asymptotic properties of the IV estimator 2 However, the common case of weak instruments, is only barely true has attracted considerable scrutiny. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 51 / 68

3. Instrumental Variables (IV) estimator De nition (Weak instrument) A weak instrument is an instrumental variable which is only slightly correlated with the right-hand-side variables X. In presence of weak instruments, the quantity Q ZX is close to zero and we have 1 N Z> X ' 0 H K Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 52 / 68

3. Instrumental Variables (IV) estimator Fact (IV estimator and weak instruments) In presence of weak instruments, the IV estimators b β IV has a poor precision (great variance). For Q ZX ' 0 H K, the asymptotic variance tends to be very large, since: V asy bβiv = σ2 N Q ZX 1 Q ZZ QZX 1 As soon as N 1 Z > X ' 0 H K, the estimated asymptotic variance covariance is also very large since bv 1 1 bβiv asy = bσ 2 Z > X Z > Z X > Z Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 53 / 68

3. Instrumental Variables (IV) estimator Key Concepts 1 Instrument or instrumental variable 2 Orthogonal or moment condition 3 Identi cation: just-identi ed or over-identi ed model 4 Instrumental Variables (IV) estimator 5 Statistical properties of the IV estimator 6 Weak instrument Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 54 / 68

Section 4 Two-Stage Least Squares (2SLS) estimator Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 55 / 68

4. Two-Stage Least Squares (2SLS) estimator Assumption: Consider an over-identi ed model H > K Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 56 / 68

4. Two-Stage Least Squares (2SLS) estimator Introduction If Z contains more variables than X, then much of the preceding derivation is unusable, because Z > X will be H K with rank Z > X = K < H So, the matrix Z > X has no inverse and we cannot compute the IV estimator as: 1 bβ IV = Z X > Z > y Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 57 / 68

4. Two-Stage Least Squares (2SLS) estimator Introduction (cont d) The crucial assumption in the previous section was the exogeneity assumption plim 1 N Z> ε = 0 K 1 1 That is, every column of Z is asymptotically uncorrelated with ε. 2 That also means that every linear combination of the columns of Z is also uncorrelated with ε, which suggests that one approach would be to choose K linear combinations of the columns of Z. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 58 / 68

4. Two-Stage Least Squares (2SLS) estimator Introduction (cont d) Which linear combination to choose? A choice consists in using is the projection of the columns of X in the column space of Z: 1 bx = Z Z Z > Z > X With this choice of instrumental variables, bx for Z, we have bβ 2SLS = = 1 X b > X X b > y 1 1 1 X > Z Z Z > Z X > X > Z Z Z > Z > y Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 59 / 68

4. Two-Stage Least Squares (2SLS) estimator De nition (Two-stage Least Squares (2SLS) estimator) The Two-stage Least Squares (2SLS) estimator of the parameters β is de ned as to be: 1 bβ 2SLS = X b > X X b > y 1 where bx = Z Z Z > Z > X corresponds to the projection of the columns of X in the column space of Z, or equivalently by bβ 2SLS = 1 1 1 X > Z Z Z > Z X > X > Z Z Z > Z > y Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 60 / 68

4. Two-Stage Least Squares (2SLS) estimator Remark By de nition bβ 2SLS = 1 X b > X X b > y Since 1 bx = Z Z Z > Z > X = P Z X where P Z denotes the projection matrix on the columns of Z. Reminder: P Z is symmetric and P Z P > Z = P Z. So, we have bβ 2SLS = = X > P > 1 Z X X b > y X > P > 1 Z P Z X X b > y = 1 X b > bx X b > y Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 61 / 68

4. Two-Stage Least Squares (2SLS) estimator De nition (Two-stage Least Squares (2SLS) estimator) The Two-stage Least Squares (2SLS) estimator of the parameters β can also be de ned as: 1 bβ 2SLS = X b > bx X b > y It corresponds to the OLS estimator obtained in the regression of y on bx. Then, the 2SLS can be computed in two steps, rst by computing bx, then by the least squares regression. That is why it is called the two-stage LS estimator. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 62 / 68

4. Two-Stage Least Squares (2SLS) estimator A procedure to get the 2SLS estimator is the following Step 1: Regress each explicative variable x k (for k = 1,..K ) on the H instruments. x ki = α 1 z 1i + α 2 z 2i +.. + α H z Hi + v i Step 2: Compute the OLS estimators bα h and the tted values bx ki bx ki = bα 1 z 1i + bα 2 z 2i +.. + bα H z Hii Step 3: Regress the dependent variable y on the tted values bx ki : y i = β 1 bx 1i + β 2 bx 2i +.. + β K bx Ki + ε i The 2SLS estimator b β 2SLS then corresponds to the OLS estimator obtained in this model. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 63 / 68

4. Two-Stage Least Squares (2SLS) estimator Theorem If any column of X also appears in Z, i.e. if one or more explanatory (exogenous) variable is used as an instrument, then that column of X is reproduced exactly in bx. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 64 / 68

4. Two-Stage Least Squares (2SLS) estimator Example (Explicative variables used as instrument) Suppose that the regression contains K variables, only one of which, say, the K th, is correlated with the disturbances, i.e. E (x Ki ε i ) 6= 0. We can use a set of instrumental variables z 1,..., z J plus the other K 1 variables that certainly qualify as instrumental variables in their own right. So, Z = (z 1 :.. : z J : x 1 :.. : x K 1 ) Then bx = (x 1 :.. : x K 1 : bx K ) where bx K denotes the projection of x K on the columns of Z. Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 65 / 68

4. Two-Stage Least Squares (2SLS) estimator Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 66 / 68

4. Two-Stage Least Squares (2SLS) estimator Key Concepts 1 Over-identi ed model 2 Two-Stage Least Squares (2SLS) estimator Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 67 / 68

End of Chapter 6 Christophe Hurlin (University of Orléans) Christophe Hurlin (University of Orléans) Advanced Econometrics - HEC Lausanne December 15, 2013 68 / 68