Chapter 2. Dynamic panel data models

Transcription

1 Chapter 2. Dynamic panel data models School of Economics and Management - University of Geneva Christophe Hurlin, Université of Orléans University of Orléans April 2018 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

2 1. Introduction De nition (Dynamic panel data model) We now consider a dynamic panel data model, in the sense that it contains (at least) one lagged dependent variables. For simplicity, let us consider y it = γy i,t 1 + β 0 x it + α i + ε it for i = 1,.., n and t = 1,.., T. αi and λ t are the (unobserved) individual and time-speci c e ects, and ε it the error (idiosyncratic) term with E(ε it ) = 0, and E(ε it ε js ) = σ 2 ε if j = i and t = s, and E(ε it ε js ) = 0 otherwise. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

3 1. Introduction Remark In a dynamic panel model, the choice between a xed-e ects formulation and a random-e ects formulation has implications for estimation that are of a di erent nature than those associated with the static model. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

4 1. Introduction Dynamic panel issues 1 If lagged dependent variables appear as explanatory variables, strict exogeneity of the regressors no longer holds. The LSDV is no longer consistent when n tends to in nity and T is xed. 2 The initial values of a dynamic process raise another problem. It turns out that with a random-e ects formulation, the interpretation of a model depends on the assumption of initial observation. 3 The consistency property of the MLE and the GLS estimator also depends on the way in which T and n tend to in nity. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

5 Introduction The outline of this chapter is the following: Section 1: Introduction Section 2: Dynamic panel bias Section 3: The IV (Instrumental Variable) approach Subsection 3.1: Reminder on IV and 2SLS Subsection 3.2: Anderson and Hsiao (1982) approach Section 4: The GMM (Generalized Method of Moment) approach Subsection 4.1: General presentation of GMM Subsection 4.2: Application to dynamic panel data models C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

6 Section 2 The Dynamic Panel Bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

7 2. The dynamic panel bias Objectives 1 Introduce the AR(1) panel data model. 2 Derive the semi-asymptotic bias of the LSDV estimator. 3 Understand the sources of the dynamic panel bias or Nickell s bias. 4 Evaluate the magnitude of this bias in a simple AR(1) model. 5 Asses this bias by Monte Carlo simulations. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

8 2. The dynamic panel bias Dynamic panel bias 1 The LSDV estimator is consistent for the static model whether the e ects are xed or random. 2 On the contrary, the LSDV is inconsistent for a dynamic panel data model with individual e ects, whether the e ects are xed or random. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

9 2. The dynamic panel bias De nition (Nickell s bias) The biais of the LSDV estimator in a dynamic model is generaly known as dynamic panel bias or Nickell s bias (1981). Nickell, S. (1981). Biases in Dynamic Models with Fixed E ects, Econometrica, 49, Anderson, T.W., and C. Hsiao (1982). Formulation and Estimation of Dynamic Models Using Panel Data, Journal of Econometrics, 18, C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

10 2. The dynamic panel bias De nition (AR(1) panel data model) Consider the simple AR(1) model y it = γy i,t 1 + α i + ε it for i = 1,.., n and t = 1,.., T. For simplicity, let us assume that α i = α + α i to avoid imposing the restriction that n i=1 α i = 0 or E (α i ) = 0 in the case of random individual e ects. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

11 2. The dynamic panel bias Assumptions 1 The autoregressive parameter γ satis es jγj < 1 2 The initial condition y i0 is observable. 3 The error term satis es with E (ε it ) = 0, and E (ε it ε js ) = σ 2 ε if j = i and t = s, and E (ε it ε js ) = 0 otherwise. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

12 2. The dynamic panel bias Dynamic panel bias In this AR(1) panel data model, we will show that plim bγ LSDV 6= γ n! dynamic panel bias plim bγ LSDV = γ n,t! C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

13 2. The dynamic panel bias The LSDV estimator is de ned by (cf. chapter 1) bα i = y i bγ LSDV y i, 1 bγ LSDV = n T i=1 t=1 (y i,t 1 y i, 1 ) 2! 1 n T i=1 t=1 (y i,t 1 y i, 1 ) (y it y i )! x i = 1 T T t=1 x it y i = 1 T T t=1 y it y i, 1 = 1 T T y i,t 1 t=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

14 2. The dynamic panel bias De nition (bias) The bias of the LSDV estimator is de ned by: bγ LSDV γ = n T i=1 t=1 (y i,t 1 y i, 1 ) 2! 1 n T i=1 t=1 (y i,t 1 y i, 1 ) (ε it ε i )! C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

15 2. The dynamic panel bias The bias of the LSDV estimator can be rewritten as: bγ LSDV γ = n i=1 T t=1 n i=1 (y i,t 1 y i, 1 ) (ε it ε i ) / (nt ) T t=1 (y i,t 1 y i, 1 ) 2 / (nt ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

16 2. The dynamic panel bias Let us consider the numerator. Because ε it are (1) uncorrelated with α i and (2) are independently and identically distributed, we have plim n! 1 nt n i=1 T t=1 1 = plim n! nt y i,t 1 ε it t=1 i=1 {z } N 1 T T n (y i,t 1 y i, 1 ) (ε it ε i ) n T 1 plim n! nt y i,t 1 ε i t=1 i=1 {z } N plim n! nt y i, 1 ε it + plim t=1 i=1 n! nt y i, 1 ε i t=1 i=1 {z } {z } N 3 N 4 T n n C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

17 2. The dynamic panel bias Theorem (Weak law of large numbers, Khinchine) If fx i g, for i = 1,.., m is a sequence of i.i.d. random variables with E (X i ) = µ <, then the sample mean converges in probability to µ: or m 1 p m X i! E (Xi ) = µ i=1 plim m! 1 m m X i = E (X i ) = µ i=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

18 2. The dynamic panel bias By application of the WLLN (Khinchine s theorem) N 1 = plim n! 1 nt n i=1 T t=1 y i,t 1 ε it = E (y i,t 1 ε it ) Since (1) y i,t 1 only depends on ε i,t 1, ε i,t 2 and (2) the ε it are uncorrelated, then we have E (y i,t 1 ε it ) = 0 and nally N 1 = plim n! 1 nt n i=1 T t=1 y i,t 1 ε it = 0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

19 2. The dynamic panel bias For the second term N 2, we have: N 2 = plim n! = plim n! = plim n! = plim n! 1 nt 1 nt 1 nt 1 n n i=1 T t=1 y i,t 1 ε i n T ε i y i,t 1 i=1 t=1 n i=1 n ε i y i, 1 i=1 ε i T y i, 1 as y i, 1 = 1 T T y i,t 1 t=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

20 2. The dynamic panel bias In the same way: N 3 = plim n! 1 nt n i=1 T t=1 y i, 1 ε it = plim n! 1 nt n T 1 y i, 1 ε it = plim i=1 t=1 n! n n y i, i=1 1 ε N 4 = plim n! 1 nt n i=1 T t=1 y i, n 1 1 ε i = plim n! nt T y i, i=1 n 1 1 ε i = plim n! n y i, i=1 1 ε i C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

21 2. The dynamic panel bias The numerator of the bias expression can be rewritten as plim n! = 0 {z} N 1 1 nt = plim n! n i=1 T t=1 (y i,t 1 y i, 1 ) (ε it ε i ) n 1 plim n! n ε i y i, 1 i=1 {z } 1 n n y i, i=1 N 2 1 ε i n 1 1 plim n! n y i, 1 ε i + plim i=1 n! n y i, 1 ε i i=1 {z } {z } N 3 N 4 n C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

22 2. The dynamic panel bias Solution The numerator of the expression of the LSDV bias satis es: plim n! 1 nt n i=1 T t=1 (y i,t 1 y i, 1 ) (ε it ε i ) = plim n! 1 n n y i, i=1 1 ε i C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

23 2. The dynamic panel bias Remark bγ LSDV γ = n i=1 T t=1 n i=1 (y i,t 1 y i, 1 ) (ε it ε i ) / (nt ) T t=1 (y i,t 1 y i, 1 ) 2 / (nt ) plim n! 1 nt n i=1 T t=1 (y i,t 1 y i, 1 ) (ε it ε i ) = plim n! 1 n n y i, i=1 If this plim is not null, then the LSDV estimator bγ LSDV is biased when n tends to in nity and T is xed. 1 ε i C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

24 2. The dynamic panel bias Let us examine this plim plim n! 1 n n y i, i=1 1 ε i We know that y it = γy i,t 1 + α i + ε it = γ 2 y i,t 2 + α i (1 + γ) + ε it + γε i,t 1 = γ 3 y i,t 3 + α i 1 + γ + γ 2 + ε it + γε i,t 1 + γ 2 ε i,t 2 =... = γ t y i0 + 1 γt 1 γ α i + ε it + γε i,t 1 + γ 2 ε i,t γ t 1 ε i1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

25 2. The dynamic panel bias For any time t, we have: y it = ε it + γε i,t 1 + γ 2 ε i,t γ t 1 ε i1 + 1 γt 1 γ α i + γ t y i0 For y i,t 1, we have: y i,t 1 = ε i,t 1 + γε i,t 2 + γ 2 ε i,t γ t 2 ε i1 + 1 γt 1 1 γ α i + γ t 1 y i0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

26 2. The dynamic panel bias y i,t 1 = ε i,t 1 + γε i,t 2 + γ 2 ε i,t γ t 2 ε i1 + 1 γt 1 1 γ α i + γ t 1 y i0 Summing y i,t 1 over t, we get: T y i,t 1 = ε i,t γ2 t=1 1 γ ε i,t γt 1 1 γ ε i1 (T 1) T γ + γt + (1 γ) 2 αi + 1 γt 1 γ y i0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

27 2. The dynamic panel bias y i,t 1 = ε i,t 1 + γε i,t 2 + γ 2 ε i,t γ t 2 ε i1 + 1 γt 1 Proof: We have (each lign corresponds to a date) T y i,t 1 = y i,t 1 + y i,t y i,1 + y i,0 t=1 = ε i,t 1 + γε i,t γ T 2 ε i1 + 1 γt 1 +ε i,t 2 + γε i,t γ T 3 ε i1 + 1 γt ε i,1 + 1 γ1 1 γ α i + γy i0 +y i0 1 γ α i + γ t 1 y i0 1 γ α i + γ T 1 y i0 1 γ α i + γ T 2 y i0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

28 2. The dynamic panel bias Proof (ct d): For the individual e ect αi, we have = αi 1 γ + 1 γ γ T 1 1 γ αi T 1 γ γ 2.. γ T 1 1 γ T αi 1 γ = T 1 γ 1 γ = α i T T γ 1 + γ T (1 γ) 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

29 2. The dynamic panel bias So, we have y i, 1 = 1 T = 1 T T y i,t 1 t=1 ε i,t γ2 1 γ ε i,t γt 1 + T T γ 1 + γt (1 γ) 2 α i + 1 γt 1 γ y i0 1 γ ε i1! C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

30 2. The dynamic panel bias Finally, the plim is equal to plim n! = plim n! 1 n 1 n n i=1 n i=1 y i, 1 T 1 ε i ε i,t γ2 1 γ ε i,t γt 1 1 γ ε i1! + T T γ 1 + γt (1 γ) 2 α i + 1 γt 1 γ y i0 1 T (ε i ε it ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

31 2. The dynamic panel bias Because ε it are i.i.d, by a law of large numbers, we have: plim n! = plim n! 1 n 1 n = σ2 ε 1 T 2 n i=1 n i=1 y i, 1 T 1 ε i ε i,t γ2 1 γ ε i,t γt 1 1 γ ε i1! + T T γ 1 + γt (1 γ) 2 αi + 1 γt 1 γ y i0 γ 1 γ + 1 γ2 1 γ γt 1 1 γ T T γ 1 + γ T T 2 (1 γ) 2 = σ2 ε 1 T (ε i ε it ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

32 2. The dynamic panel bias Theorem If the errors terms ε it are i.i.d. 0, σ 2 ε, we have: plim n! = plim n! = 1 nt 1 n n i=1 T t=1 n y i, i=1 (y i,t 1 y i, 1 ) (ε it ε i ) 1 ε i σ 2 ε T T γ 1 + γ T T 2 (1 γ) 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

33 2. The dynamic panel bias By similar manipulations, we can show that the denominator of bγ LSDV converges to: = 1 plim nt n! n i=1 σ 2 ε 1 γ 2 1 T t=1 1 T (y i,t 1 y i, 1 ) 2 2γ (1 γ) 2 T T γ 1 +! γt T 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

34 2. The dynamic panel bias So, we have : plim (bγ LSDV γ) n! = plim n! = 1 nt n i=1 1 nt σ 2 ε γ 2 T T t=1 n i=1 (y i,t 1 y i, 1 ) (ε it ε i ) T t=1 (y i,t 1 y i, 1 ) 2 σ 2 ε (T T γ 1+γ T ) T 2 (1 γ) 2 2γ (T T γ 1+γT ) (1 γ) 2 T 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

35 2. The dynamic panel bias This semi-asymptotic bias can be rewriten as: = = plim (bγ LSDV γ) n! 1 γ 1+γ T 2 (1 γ) T 2 T T γ 1 + γ T 2γ T (T T γ 1 + γ T ) (1 γ) 2 (1 + γ) T T γ 1 + γ T T 2γ (T T γ 1 + γ T ) (1 γ) 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

36 2. The dynamic panel bias Fact If T also tends to in nity, then the numerator converges to zero, and denominator converges to a nonzero constant σ 2 ε / 1 γ 2, hence the LSDV estimator of γ and α i are consistent. Fact If T is xed, then the denominator is a nonzero constant, and bγ LSDV and bα i are inconsistent estimators when n is large. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

37 2. The dynamic panel bias Theorem (Dynamic panel bias) In a dynamic panel AR(1) model with individual e ects, the semi-asymptotic bias (with n) of the LSDV estimator on the autoregressive parameter is equal to: (1 + γ) T T γ 1 + γ T plim (bγ LSDV γ) = n! (1 γ) T 2 T 2γ (1 γ) 2 (T T γ 1 + γ T ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

38 2. The dynamic panel bias Theorem (Dynamic panel bias) For an AR(1) model, the dynamic panel bias can be rewriten as : plim (bγ LSDV γ) = n! 1 + γ T γ T T 1 γ 2γ (1 γ) (T 1) 1 1 γ T 1 T (1 γ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

39 2. The dynamic panel bias Fact The dynamic bias of bγ LSDV is caused by having to eliminate the individual e ects αi from each observation, which creates a correlation of order (1/T ) between the explanatory variables and the residuals in the transformed model 0 B (y it y i ) = i,t 1 y {z} i, 1 + it depends on past value of ε it 1 ε {z} i A depends on past value of ε it 1 C A C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

40 2. The dynamic panel bias Intuition of the dynamic bias with cov (y i, (y it y i ) = γ (y i,t 1 y i, 1 ) + (ε it ε i ) 1, ε i ) 6= 0 since cov (y i, 1, ε i ) = cov = cov 1 T 1 T T t=1 y i,t 1, 1 T T t=1 y i,t 1, 1 T! T ε it t=1! T ε it t=1 = 1 T 2 cov ((y i y it 1 ), (ε i ε it )) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

41 2. The dynamic panel bias Intuition of the dynamic bias (y it y i ) = γ (y i,t 1 y i, 1 ) + (ε it ε i ) with cov (y i, 1, ε i ) 6= 0 If we approximate y it by ε it (in fact y it also depend on ε it 1, ε t 2,...) then we have cov (y i, 1, ε i ) = 1 T 2 cov ((y i y it 1 ), (ε i ε it )) ' 1 T 2 (cov (ε i,1, ε i,1 ) (cov (ε i,t 1, ε i,t 1 ))) ' (T 1) σ2 ε T 2 6= 0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

42 2. The dynamic panel bias Intuition of the dynamic bias (y it y i ) = γ (y i,t 1 y i, 1 ) + (ε it ε i ) with cov (y i, 1, ε i ) 6= 0 If we approximate y it by ε it then we have cov (y i, 1, ε i ) = (T 1) σ2 ε T 2 By taking into account all the interaction terms, we have shown that n 1 plim n! n y i, 1 ε i = cov (y i, 1, ε i ) = σ2 ε (T 1) γ 1 + γ T i=1 T 2 (1 γ) 2 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

43 2. The dynamic panel bias Remarks plim (bγ LSDV γ) = n! 1 + γ T γ T T 1 γ 2γ (1 γ) (T 1) 1 1 γ T 1 T (1 γ) 1 When T is large, the right-hand-side variables become asymptotically uncorrelated. 2 For small T, this bias is always negative if γ > 0. 3 The bias does not go to zero as γ goes to zero. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

44 Semi asymptotic bias 2. The dynamic panel bias Dynam ic panel bias T=10 T=30 T=50 T= C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

45 semi asymptotic bias semi asymptotic bias semi asymptotic bias semi asymptotic bias 2. The dynamic panel bias 1 T=10 1 T=30 True value of True value of plim of the LSDV estimator plim of the LSDV estimator T=50 1 T=100 True value of plim of the LSDV estimator 0.9 True value of plim of the LSDV estimator C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

46 relative bias (in %) 2. The dynamic panel bias 0 Dynam ic bias for T=10 (in % of the true value) T=10 T=30 T=50 T= C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

47 2. The dynamic panel bias Monte Carlo experiments How to check these semi-asymptotic formula with Monte Carlo simulations? C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

48 2. The dynamic panel bias Step 1: parameters Let assume that γ = 0.5, σ 2 ε = 1 and ε it i.i.d. N (0, 1). Simulate n individual e ects αi once at all. For instance, we can use a uniform distribution αi U [ 1,1] C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

49 2. The dynamic panel bias Step 2: Monte Carlo pseudo samples Simulate n (typically n = 1, 000) i.i.d. sequences fε it g T t=1 value of T (typically T = 10) for a given Generate n sequences fy it g T t=1 for i = 1,.., n with the model: y it = γy i,t 1 + α i + ε it Repeat S times the step 2 in order to generate S = 5, 000 sequences n o T for s = 1,.., S for each cross-section unit i = 1,..., n y (s) it t=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

50 2. The dynamic panel bias Step 3: LSDV estimates on pseudo series For each pseudo sample s = 1,..., S, consider the empirical model y s it = γy s i,t 1 + α i + µ it i = 1,.., n t = 1,...T and compute the LSDV estimates bγ s LSDV. Compute the average bias of the LSDV estimator bγ LSDV based on the S Monte Carlo simulations av.bias = 1 S S bγ s LSDV s=1 γ C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

51 2. The dynamic panel bias Step 4: Semi-asymptotic bias 1 Repeat this experiment for various cross-section dimensions n: when n increases,the average bias should converge to plim (bγ LSDV γ) = n! 1 + γ T γ T T 1 γ 2γ (1 γ) (T 1) 1 1 γ T 1 T (1 γ) 2 Repeat this this experiment for various time dimensions T : when T increases,the average bias should converge to 0. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

52 2. The dynamic panel bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

53 2. The dynamic panel bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

54 2. The dynamic panel bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

55 2. The dynamic panel bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

56 Number of simulations 2. The dynamic panel bias 350 Histogram of the LSDV estimates for=0.5, T=10 and n= h at C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

57 2. The dynamic panel bias Click me! C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

58 2. The dynamic panel bias Theoretical semi asymptotic bias MC average bias Sample size n C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

59 2. The dynamic panel bias Question: What is the importance of the dynamic bias in micro-panels? Macroeconomists should not dismiss the LSDV bias as insigni cant. Even with a time dimension T as large as 30, we nd that the bias may be equal to as much 20% of the true value of the coe cient of interest. (Judson et Owen, 1999, page 10) Judson R.A. and Owen A. (1999), Estimating dynamic panel data models: a guide for macroeconomists. Economics Letters, 1999, vol. 65, issue 1, C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

60 2. The dynamic panel bias Finite Sample results (Monte Carlo simulations) n T γ Avg. bγ LSDV Avg. bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

61 2. The dynamic panel bias Finite Sample results (Monte Carlo simulations) n T γ Avg. bγ LSDV Avg. bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

62 2. The dynamic panel bias Fact (smearing e ect) The LSDV for dynamic individual-e ects model remains biased with the introduction of exogenous variables if T is small; for details of the derivation, see Nickell (1981); Kiviet (1995). y it = α + γy i,t 1 + β 0 x it + α i + ε it In this case, both estimators bγ LSDV and bβ LSDV are biased. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

63 2. The dynamic panel bias What are the solutions? Consistent estimator of γ can be obtained by using: 1 ML or FIML (but additional assumptions on y i0 are necessary) 2 Feasible GLS (but additional assumptions on y i0 are necessary) 3 LSDV bias corrected (Kiviet, 1995) 4 IV approach (Anderson and Hsiao, 1982) 5 GMM approach (Arenallo and Bond, 1985) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

64 2. The dynamic panel bias What are the solutions? Consistent estimator of γ can be obtained by using: 1 ML or FIML (but additional assumptions on y i0 are necessary) 2 Feasible GLS (but additional assumptions on y i0 are necessary) 3 LSDV bias corrected (Kiviet, 1995) 4 IV approach (Anderson and Hsiao, 1982) 5 GMM approach (Arenallo and Bond, 1985) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

65 2. The dynamic panel bias Key Concepts Section 2 1 AR(1) panel data model 2 Semi-asymptotic bias 3 Dynamic panel bias (Nickell s bias) 4 Monte Carlo experiments 5 Magnitude of the dynamic panel bias C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

66 Section 3 The Instrumental Variable (IV) approach C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

67 Subsection 3.1 Reminder on IV and 2SLS C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

68 3.1 Reminder on IV and 2SLS Objectives 1 De ne the endogeneity bias and the smearing e ect. 2 De ne the notion of instrument or instrumental variable. 3 Introduce the exogeneity and relevance properties of an instrument. 4 Introduce the notion of just-identi ed and over-identi ed systems. 5 De ne the IV estimator and its asymptotic variance. 6 De ne the 2SLS estimator and its asymptotic variance. 7 De ne the notion of weak instrument. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

69 3.1 Reminder on IV and 2SLS Consider the (population) multiple linear regression model: y = Xβ + ε y is a N 1 vector of observations y j for j = 1,.., N X is a N K matrix of K explicative variables x jk for k = 1,., K and j = 1,.., N β = (β 1..β K ) 0 is a K 1 vector of parameters ε is a N 1 vector of error terms ε i with (spherical disturbances) V ( εj X) = σ 2 I N C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

70 3.1 Reminder on IV and 2SLS Endogeneity we assume that the assumption A3 (exogeneity) is violated: E ( εj X) 6= 0 N 1 with plim 1 N X0 ε = E (x j ε j ) = γ 6= 0 K 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

71 3.1 Reminder on IV and 2SLS 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 where β denotes the true value of the parameters. This bias is called the endogeneity bias. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

72 3.1 Reminder on IV and 2SLS Theorem (Inconsistency of the OLS estimator) If the regressors are endogenous with plim N 1 X 0 ε = γ, the OLS estimator of β is inconsistent where Q = plim N 1 X 0 X. plim b β OLS = β + Q 1 γ C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

73 3.1 Reminder on IV and 2SLS Proof: Given the de nition of the OLS estimator: We have: bβ OLS = X 0 X 1 X 0 y plim b β OLS = β + plim = X 0 X 1 X 0 (Xβ + ε) = β + X 0 X 1 X 0 ε = β + Q 1 γ 6= β N X0 X plim N X0 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

74 3.1 Reminder on IV and 2SLS Remarks plim b β OLS = β + 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 e ect: the inconsistency due to the endogeneity of the one variable is smeared across all of the least squares estimators. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

75 3.1 Reminder on IV and 2SLS Example (Endogeneity, OLS estimator and smearing) Consider the multiple linear regression model y i = x 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, ) 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 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

76 3.1 Reminder on IV and 2SLS 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 0 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 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

77 3.1 Reminder on IV and 2SLS C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

78 3.1 Reminder on IV and 2SLS C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

79 3.1 Reminder on IV and 2SLS Question: What is the solution to the endogeneity issue? The use of instruments.. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

80 3.1 Reminder on IV and 2SLS 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 jk z jh ) 6= 0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

81 3.1 Reminder on IV and 2SLS Assumptions: The instrumental variables satisfy the following properties. Well behaved data: plim 1 N Z0 Z = Q ZZ a nite H H positive de nite matrix Relevance: plim 1 N Z0 X = Q ZX a nite H K positive de nite matrix Exogeneity: plim 1 N Z0 ε = 0 K 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

82 3.1 Reminder on IV and 2SLS De nition (Instrument properties) We assume that the H instruments are linearly independent: E Z 0 Z is non singular or equivalently rank E Z 0 Z = H C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

83 3.1 Reminder on IV and 2SLS The exogeneity condition E ( ε j j z j ) = 0 =) E (ε j z j ) = 0 H can expressed as an orthogonality condition or moment condition 0 z j y j 1 xj 0 β A = 0 H (H,1) (1,1) (H,1) So, we have H equations and K unknown parameters β C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

84 3.1 Reminder on IV and 2SLS De nition (Identi cation) The system is identi ed if there exists a unique vector β such that: E z j y j xj 0 β = 0 where z j = (z j1..z jh ) 0. 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

85 3.1 Reminder on IV and 2SLS 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 (2SLS) 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). C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

86 3.1 Reminder on IV and 2SLS C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

87 3.1 Reminder on IV and 2SLS Assumption: Consider a just-identi ed model H = K C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

88 3.1 Reminder on IV and 2SLS Motivation of the IV estimator By de nition of the instruments: plim 1 N Z0 ε = plim 1 N Z0 (y Xβ) = 0 K 1 So, we have: or equivalently plim 1 N Z0 y = β = plim 1 N Z0 X β plim 1 1 N Z0 X plim 1 N Z0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

89 3.1 Reminder on IV and 2SLS De nition (Instrumental Variable (IV) estimator) If H = K, the Instrumental Variable (IV) estimator b β IV of parameters β is de ned as to be: bβ IV = Z 0 X 1 Z 0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

90 3.1 Reminder on IV and 2SLS De nition (Consistency) Under the assumption that plim N 1 Z 0 ε = 0, the IV estimator β b IV is consistent: p! β bβ IV where β denotes the true value of the parameters. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

91 3.1 Reminder on IV and 2SLS Proof: By de nition: bβ IV = Z 0 X Z 0 y = β + N Z0 X N Z0 ε So, we have: plimβ b IV = β + plim 1 1 N Z0 X plim 1 N Z0 ε Under the assumption of exogeneity of the instruments plim 1 N Z0 ε = 0 K 1 So, we have plim b β IV = β C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

92 3.1 Reminder on IV and 2SLS De nition (Asymptotic distribution) Under some regularity conditions, the IV estimator β b IV is asymptotically normally distributed: p N bβiv d! β N 0K 1, σ 2 QZX 1 Q ZZ QZX 1 where Q ZZ K K = plim 1 N Z0 Z Q ZX K K = plim 1 N Z0 X C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

93 3.1 Reminder on IV and 2SLS 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 asy bβiv = bσ 2 Z 0 X 1 Z 0 Z X 0 Z 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

94 3.1 Reminder on IV and 2SLS Remarks 1 If the system is just identi ed H = K, the estimator can also written as Z 0 X 1 = X 0 Z 1 Q ZX = Q XZ bv asy bβiv = bσ 2 Z 0 X 1 Z 0 Z Z 0 X 1 2 As usual, the estimator of the variance of the error terms is: bσ 2 = bε0 bε N K = 1 N K N y i x 0 b 2 i β IV i=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

95 3.1 Reminder on IV and 2SLS 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 Z0 ε = 0 K 1 2 A growing literature has argued that greater attention needs to be given to the relevance condition plim 1 N Z0 X = Q ZX a nite H K positive de nite matrix with H = K in the case of a just-identi ed model. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

96 3.1 Reminder on IV and 2SLS Relevant instruments (cont d) plim 1 N Z0 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

97 3.1 Reminder on IV and 2SLS 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 Z0 X ' 0 H K C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

98 3.1 Reminder on IV and 2SLS 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 0 X ' 0 H K, the estimated asymptotic variance covariance is also very large since bv asy bβiv = bσ 2 Z 0 X 1 Z 0 Z X 0 Z 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

99 3.1 Reminder on IV and 2SLS Assumption: Consider an over-identi ed model H > K C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

100 3.1 Reminder on IV and 2SLS Introduction If Z contains more variables than X, then much of the preceding derivation is unusable, because Z 0 X will be H K with rank Z 0 X = K < H So, the matrix Z 0 X has no inverse and we cannot compute the IV estimator as: bβ IV = Z 0 X 1 Z 0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

101 3.1 Reminder on IV and 2SLS Introduction (cont d) The crucial assumption in the previous section was the exogeneity assumption plim 1 N Z0 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

102 3.1 Reminder on IV and 2SLS 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: bx = Z Z 0 Z 1 Z 0 X With this choice of instrumental variables, bx for Z, we have bβ 2SLS = = 1 X b 0 X X b 0 y X 0 Z Z 0 Z 1 1 Z X 0 X 0 Z Z 0 Z 1 Z 0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

103 3.1 Reminder on IV and 2SLS 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 0 X X b 0 y where bx = Z Z 0 Z 1 Z 0 X corresponds to the projection of the columns of X in the column space of Z, or equivalently by bβ 2SLS = X 0 Z Z 0 Z 1 Z 0 X 1 X 0 Z Z 0 Z 1 Z 0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

104 3.1 Reminder on IV and 2SLS Remark By de nition Since bβ 2SLS = 1 X b 0 X X b 0 y bx = Z Z 0 Z 1 Z 0 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 0 Z = P Z. So, we have bβ 2SLS = = X 0 P 0 1 Z X X b 0 y X 0 P 0 1 Z P Z X X b 0 y = 1 X b 0 bx X b 0 y C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

105 3.1 Reminder on IV and 2SLS 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 0 bx X b 0 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

106 3.1 Reminder on IV and 2SLS 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 kj = α 1 z 1j + α 2 z 2j α H z Hj + v j Step 2: Compute the OLS estimators bα h and the tted values bx kj bx kj = bα 1 z 1j + bα 2 z 2j bα H z Hj Step 3: Regress the dependent variable y on the tted values bx ki : y j = β 1 bx 1j + β 2 bx 2j β K bx Kj + ε j The 2SLS estimator b β 2SLS then corresponds to the OLS estimator obtained in this model. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

107 3.1 Reminder on IV and 2SLS 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

108 3.1 Reminder on IV and 2SLS 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. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

109 3.1 Reminder on IV and 2SLS Key Concepts SubSection Endogeneity bias and smearing e ect. 2 Instrument or instrumental variable. 3 Exogeneity and relavance properties of an instrument. 4 Instrumental Variable (IV) estimator. 5 Two-Stage Least Square (2SLS) estimator. 6 Weak instrument. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

110 Subsection 3.2 Anderson and Hsiao (1982) IV approach C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

111 3.2 Anderson and Hsiao (1982) IV approach Objectives 1 Introduce the IV approach of Anderson and Hsiao (1982). 2 Describe their 4 steps estimation procedure. 3 Introduce the rst di erence transformation of the dynamic model. 4 Describe their choice of instruments. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

112 3.2 Anderson and Hsiao (1982) IV approach Consider a dynamic panel data model with random individual e ects: y it = γy i,t 1 + β 0 x it + ρ 0 ω i + α i + ε it α i are the (unobserved) individual e ects, x it is a vector of K 1 time-varying explanatory variables, ω i is a vector of K 2 time-invariant variables. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

113 3.2 Anderson and Hsiao (1982) IV approach Assumption: we assume that the component error term v it = ε it + α i E (ε it ) = 0, E (α i ) = 0 E (ε it ε js ) = σ 2 ε if j = i and t = s, 0 otherwise. E (α i α j ) = σ 2 α if j = i, 0 otherwise. E (α i x it ) = 0, E (α i ω i ) = 0 (exogeneity assumption for ω i ) E (ε it x it ) = 0, E (ε it ω i ) = 0 (exogeneity assumption for x it ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

114 3.2 Anderson and Hsiao (1982) IV approach The K 1 + K parameters to estimate are y it = γy i,t 1 + β 0 x it + ρ 0 ω i + α i + ε it 1 γ the autoregressive parameter, 2 β is the K 1 1 vector of parameters for the time-varying explanatory variables, 3 ρ is the K 2 1 vector of parameters for the time-invariant variables, 4 σ 2 ε and σ 2 α the variances of the error terms. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

115 3.2 Anderson and Hsiao (1982) IV approach Remark If the vector ω i includes a constant term, the associated parameter can be interpreted as the mean of the (random) individual e ects y it = γy i,t 1 + β 0 x it + ρ 0 ω i + α i + ε it ω i = (K 2,1) αi = µ + α i E (α i ) = B z i2 A ρ = (K 2,1) z ik2 µ ρ 2... ρ K2 1 C A C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

116 3.2 Anderson and Hsiao (1982) IV approach Vectorial form: y i = y i, 1 γ + X i β + ω 0 i ρe + α i e + ε i ε i, y i and y i, 1 are T 1 vectors (T is the adjusted sample size), X i a T K 1 matrix of time-varying explanatory variables, ω i is a K 2 1 vector of time-invariant variables, e is the T 1 unit vector, and E (α i ) = 0 E α i x 0 it = 0 E α i ω 0 i = 0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

117 3.2 Anderson and Hsiao (1982) IV approach In the dynamic panel data models context: The Instrumental Variable (IV) approach was rst proposed by Anderson and Hsiao (1982). They propose an IV procedure with 2 choices of instruments and 4 steps to estimate γ, β, ρ and σ 2 ε. Anderson, T.W., and C. Hsiao (1982). Formulation and Estimation of Dynamic Models Using Panel Data, Journal of Econometrics, 18, C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

118 3.2 Anderson and Hsiao (1982) IV approach The Anderson and Hsiao (1982) IV approach 1 First step: rst di erence transformation 2 Second step: choice of instruments and IV estimation of γ and β 3 Third step: estimation of ρ 4 Fourth step: estimation of the variances σ 2 α and σ 2 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

119 3.2 Anderson and Hsiao (1982) IV approach The Anderson and Hsiao (1982) IV approach 1 First step: rst di erence transformation 2 Second step: choice of instruments and IV estimation of γ and β 3 Third step: estimation of ρ 4 Fourth step: estimation of the variances σ 2 α and σ 2 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

120 3.2 Anderson and Hsiao (1982) IV approach First step: rst di erence transformation Taking the rst di erence of the model, we obtain for t = 2,.., T. (y it y i,t 1 ) = γ (y i,t 1 y i,t 2 ) + β 0 (x it x i,t 1 ) + ε it ε i,t 1 The rst di erence transformation leads to "lost" one observation. But, it allows to eliminate the individual e ects (as the Within transformation). C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

121 3.2 Anderson and Hsiao (1982) IV approach The Anderson and Hsiao (1982) IV approach 1 First step: rst di erence transformation 2 Second step: choice of instruments and IV estimation of γ and β 3 Third step: estimation of ρ 4 Fourth step: estimation of the variances σ 2 α and σ 2 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

122 3.2 Anderson and Hsiao (1982) IV approach Second step: choice of the instruments and IV estimation (y it y i,t 1 ) = γ (y i,t 1 y i,t 2 ) + β 0 (x it x i,t 1 ) + ε it ε i,t 1 A valid instrument z it should satisfy E (z it (ε it ε i,t 1 )) = 0 Exogeneity property E (z it (y i,t 1 y i,t 2 )) 6= 0 Relevance property C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

123 3.2 Anderson and Hsiao (1982) IV approach Anderson and Hsiao (1982) propose two valid instruments: 1 First instrument: z i,t = y i,t 2 E (y i,t 2 (ε it ε i,t 1 )) = 0 Exogeneity property E (y i,t 2 (y i,t 1 y i,t 2 )) 6= 0 Relevance property 2 Second instrument: z i,t = (y i,t 2 y i,t 3 ) E ((y i,t 2 y i,t 3 ) (ε it ε i,t 1 )) = 0 Exogeneity property E ((y i,t 2 y i,t 3 ) (y i,t 1 y i,t 2 )) 6= 0 Relevance property C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

124 3.2 Anderson and Hsiao (1982) IV approach Remarks The initial rst di erences model includes K regressors. The regressor (y i,t 1 y i,t 2 ) is endogeneous. The regressors (x it x i,t 1 ) are assumed to be exogeneous. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

125 3.2 Anderson and Hsiao (1982) IV approach De nition (Instruments) Anderson and Hsiao (1982) consider two sets of K instruments, in both cases the system is just identi ed (IV estimator): z i = y i,t 2 (K 1 +1,1) (1,1) : (x it x i,t 0 1 ) (1,K 1 )! 0 z i = (y i,t 2 y i,t 3 ) (K 1 +1,1) (1,1) : (x it x i,t 0 1 ) (1,K 1 )! 0 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

126 3.2 Anderson and Hsiao (1982) IV approach IV estimator with the rst set of instruments bγiv = Z 0 X 1 Z 0 y = bβ IV n T i=1 t=2 n i=1 T t=2 (y i,t 1 y i,t 2 ) y i,t 2 y i,t 2 (x it x i,t 1 ) 0 (x it x i,t 1 ) y i,t 2 (x it x i,t 1 ) (x it x i,t 1 ) 0! y i,t 2 (y x it x i,t y i,t 1 ) i,t 1!! 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

127 3.2 Anderson and Hsiao (1982) IV approach IV estimator with the second set of instruments bγiv = Z 0 X 1 Z 0 y = bβ IV n T i=1 t=3 n i=1 (y i,t 1 y i,t 2 ) (y i,t 2 y i,t 3 ) (y i,t 2 y i,t 3 ) (x it x i,t 1 (x it x i,t 1 ) (y i,t 2 y i,t 3 ) (x it x i,t 1 ) (x it x i,t 1 ) 0! yi,t 2 y i,t 3 (y i,t y i,t 1 ) T t=3 x it x i,t 1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

128 3. Instrumental variable (IV) estimators Remarks 1 The rst estimator (with z it = y i,t 2 ) has an advantage over the second one (with z it = y i,t 2 y i,t 3 ), in that the minimum number of time periods required is two, whereas the rst one requires T 3. 2 In practice, if T 3, the choice between both depends on the correlations between (y i,t 1 y i,t 2 ) and y i,t 2 or (y i,t 2 y i,t 3 ) => relevance assumption. Anderson, T.W., and C. Hsiao (1981). Estimation of Dynamic Models with Error Components, Journal of the American Statistical Association, 76, C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

129 3.2 Anderson and Hsiao (1982) IV approach The Anderson and Hsiao (1982) IV approach 1 First step: rst di erence transformation 2 Second step: choice of instruments and IV estimation of γ and β 3 Third step: estimation of ρ 4 Fourth step: estimation of the variances σ 2 α and σ 2 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

130 3.2 Anderson and Hsiao (1982) IV approach Third step y it = γy i,t 1 + β 0 x it + ρ 0 i ω i + α i + ε it Given the estimates bγ IV and bβ IV, we can deduce an estimate of ρ, the vector of parameters for the time-invariant variables ω i. Let us consider, the following equation with v i = α i + ε i. y i bγ IV y i, 1 bβ 0 IV x i = ρ 0 ω i + v i i = 1,..., n The parameters vector ρ can simply be estimated by OLS. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

131 3.2 Anderson and Hsiao (1982) IV approach De nition (parameters of time-invariant variables) A consistent estimator of the parameters ρ is given by bρ = (K 2,1) with h i = y i bγ IV y i, 1 bβ 0 IV x i.! 1 n ω i ωi 0 i=1! n ω i h i i=1 C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

132 3.2 Anderson and Hsiao (1982) IV approach The Anderson and Hsiao (1982) IV approach 1 First step: rst di erence transformation 2 Second step: choice of instruments and IV estimation of γ and β 3 Third step: estimation of ρ 4 Fourth step: estimation of the variances σ 2 α and σ 2 ε C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

133 3.2 Anderson and Hsiao (1982) IV approach Fourth step: estimation of the variances De nition Given bγ IV, bβ IV, and bρ, we can estimate the variances as follows: with bσ 2 α = 1 n bσ 2 ε = 1 n (T 1) T t=2 n bε 2 it i=1 n 2 y i bγ IV y i, 1 bβ 0 IV x i bρ 0 1 z i i=1 T bσ2 ε bε it = (y i,t y i,t 1 ) bγ IV (y i,t 1 y i,t 2 ) bβ 0 IV (x i,t x i,t 1 ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

134 3.2 Anderson and Hsiao (1982) IV approach Theorem The instrumental-variable estimators of γ, β, and σ 2 ε are consistent when n (correction of the Nickell bias), or T, or both tend to in nity. plim bγ IV = γ n,t! plim bβ IV = β n,t! plim bσ 2 ε = σ 2 ε n,t! The estimators of ρ and σ 2 α are consistent only when n goes to in nity. plim bρ = ρ n! plim bσ 2 α = σ 2 α n! C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

135 3.2 Anderson and Hsiao (1982) IV approach Key Concepts SubSection Anderson and Hsiao (1982) IV approach. 2 The 4 steps of the estimation procedure. 3 First di erence transformation of the dynamic panel model. 4 Tow choices of instrument. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

136 Section 4 Generalized Method of Moment (GMM) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

137 4. The GMM approach Let us consider the same dynamic panel data model as in section 3: y it = γy i,t 1 + β 0 x it + ρ 0 ω i + α i + ε it α i are the (unobserved) individual e ects, x it is a vector of K 1 time-varying explanatory variables, ω i is a vector of K 2 time-invariant variables. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

138 4. The GMM approach Assumptions: we assume that the component error term v it = ε it + α i E (ε it ) = 0, E (α i ) = 0 E (ε it ε js ) = σ 2 ε if j = i and t = s, 0 otherwise. E (α i α j ) = σ 2 α if j = i, 0 otherwise. E (α i x it ) = 0, E (α i ω i ) = 0 (exogeneity assumption for ω i ) C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

139 4. The GMM approach De nition (First di erence model) The GMM estimation method is based on a model in rst di erences, in order to swip out the individual e ects α i and th variables ω i : (y it y i,t 1 ) = γ (y i,t 1 y i,t 2 ) + β 0 (x it x i,t 1 ) + ε it ε i,t 1 for t = 2,.., T. C. Hurlin (University of Orléans) Advanced Econometrics II April / 209

140 4. The GMM approach Intuition of the moment conditions Notice that y i,t 2 and (y i,t 2 y i,t 3 ) are not the only valid instruments for (y i,t 1 y i,t 2 ). All the lagged variables y i,t 2 j, for j 0, satisfy E (y i,t 2 j (ε i,t ε i,t 1 )) = 0 Exogeneity property E (y i,t 2 j (y i,t 1 y i,t 2 )) 6= 0 Relevance property Therefore, they all are legitimate instruments for (y i,t 1 y i,t 2 ). C. Hurlin (University of Orléans) Advanced Econometrics II April / 209