ECONOMETRICS II (ECO 2401) Victor Aguirregabiria (March 2017) TOPIC 1: LINEAR PANEL DATA MODELS

Transcription

1 ECONOMETRICS II (ECO 2401) Victor Aguirregabiria (March 2017) TOPIC 1: LINEAR PANEL DATA MODELS 1. Panel Data 2. Static Panel Data Models 2.1. Assumptions 2.2. "Fixed e ects" and "Random e ects" 2.3. Estimation with exogenous individual e ects

2 2.4. Estimation with endogenous individual e ects 2.5. Test of exogeneity of individual e ects 2.6. Standard errors 2.7. Within-Groups with measurement error 2.8. Applications 3. Dynamic Panel Data Models 3.1. Introduction 3.2. Inconsistency of OLS-FD 3.3. Inconsistency of WG

3 3.4. Anderson-Hsiao (JE, 1982) 3.5. Arellano-Bond (REStud, 1991) 3.6. Blundell-Bond (JE, 1998) 3.7. Speci cation tests

4 1. PANEL DATA We have panel data when we observe a group of individuals (e.g., households, rms, countries) over several periods of time. Data = (x it : i = 1; 2; :::; N; t 2 f1; 2; :::; T g) x it : K 1 vector of observed variables i : subindex for individuals t : subindex for time N = # of individuals T i = # of periods of time the individual i is observed

5 Balanced Panels: When T i = T for every i: Unbalanced Panels: When T i can be di erent each i. Unbalanced panels can be such that individuals have di erent starting periods. i = 1 ` a i = 2 ` a i = 3 ` a i = 4 ` a

6 Remark 1: - We consider panels with large N and T i relatively small (asymptotics as N! 1). This is typically the case in micro-panels of households, individuals, or rms. - For static panel data models (i.e., strictly exogenous regressors) all the results that we present in these notes apply to the reverse case with N small and T large (but swapping T and N in the asymptotic formulas). Remark 2: - A panel dataset is much more than a sequence of cross-sections over time. In a sequence of cross-sections we do not observe the same individuals over time. In panel data, we follow individuals for several periods of time. - This important feature makes it possible to control for (potentially endogenous) unobserved individual heterogeneity.

7 Panel data (PD) have several advantages relative to cross-sectional (CS) and time-series (TS) data to control for spurious correlations due to omitted / unobservable variables. (1) Two sources of sample variation (time and cross-sectional) we can control for the potential endogeneity bias induced by unobservables that vary only over time but not over individuals (aggregate time e ects) or that vary over individuals but are constant over time (time-invariant individual e ects). (2) We can apply the concept of Granger Causality to individual lave data with asymptotics in N. [Granger causality: X i causes Y i if Y it and X it controlling for the previous history X i and Y i.] 1 are partially correlated after

8 Example 1. Production function CS data from N rms producing the same product. Douglas production in logarithms: Consider the Cobb- y i = l i + 2 k i + " i " i represents the amount of unobserved inputs, e.g., manager ability, land quality, rainfall, etc. Unobserved inputs can be correlated with labor and capital: cov(l i ; " i ) > 0 and cov(k i ; " i ) > 0. we expect OLS estimation will provide inconsistent estimates of 1 and 2.

9 Example 1 (cont.) Suppose that we have PD such that: y it = l it + 2 k it + " it And we assume that: " it = i + t + u it where: i represents time-invariant unobserved inputs, e.g., quality of land; t common shocks for all the rms; u it is a rm-speci c productivity shock.

10 Example 1 (cont.) Under some conditions we can use PD to control for endogeneity bias generated by the unobservables i and t. The model in rst di erences is: y it = 1 l it + 2 k it + t + u it If regressors are not correlated with u it (e.g., rainfall), the OLS estimator of the equation in rst di erences is consistent. Using PD we can also obtain instruments [exploit Granger causality] for the consistent estimation of the model even when (l it ; k it ) are correlated with u it.

11 Example 2 (Dynamic labor demand). Sargent (JPE, 1978) estimates the following dynamic model of labor demand (Euler equation) for a representative rm. where: L t = L t W t + 3 Y t L t + t + u t L t is employment change; W t is the wage rate; Y t is output; u t is an expectational error orthogonal to information at t or before; t is a component of the marginal pro t unobserved to the researcher but observable to the rm;

12 Example 2 (cont.) The error term in this model is: " t t + u t. While u t is not correlated with the regressors, the error t could be. OLS estimation will be inconsistent. Suppose we have rm level PD fl it ; W it ; Y it g. And suppose that we assume that where u it is not serially correlated. " it = i + t + u it The Euler equation in di erences: 2 L it = 1 2 L it W it + 3 Y it L it + t + u it We can estimate consistently 0 s using an IV/GMM method.

13 An General PD Model Consider the model: where: y it = f (x it ; " it, ) f (:) is a known function; is a vector of unknown parameters. x it is a vector of observable explanatory variables; " it is unobservable; A standard speci cation of the unobservable is: " it = i + t + u it

14 INCIDENTAL PARAMETERS PROBLEM In PD model with an error structure " it = i + t + u it, we can treat s and s as parameters to estimate. We can include as explanatory variables (N 1) individual dummies associated to s and (T 1) time dummies associated to s. The number of parameters s increases at the same rate as N as N! 1. We cannot estimate these parameters consistently. Inconsistency of our estimator of s may contaminate / bias our estimation of the parameters of interest.

15 INCIDENTAL PARAMETERS PROBLEM [more generally] Given a model with vector of parameters and a random sample with size n, we say that the model has an incidental parameters problem if: As n! 1, we have that dim()! 1 Neyman and Scott (Econometrica, 1948) show that in general, there is not a consistent estimator of the whole vector. Suppose that = (; ), such that as n! 1, dim()! 1, but dim() = K for any value of n Can we obtain a consistent estimator of despite our estimation of will be always inconsistent?

16 Two relevant classi cations of PD Models There are two classi cations of PD models that have important implications on the properties of di erent estimators (e.g., xed e ects estimators). (A) Classi cation according to the additivity of the unobservables: Additive separable unobservable: y it = h (x it ; ) + " it Non additive separable unobservable: e.g., y it = maxfx 0 it + " it ; 0g (B) Classi cation according to Static and Dynamic PD models.

17 Additive vs. Nonadditive Unobservables The Incidental Parameters Problem has a very di erent nature depending on whether unobservables are additive or non-additive models. The properties of some estimators are very di erent in linear and in nonlinear PD models. For instance, the xed-e ects or within-groups estimator is consistent in static linear PD models but it is inconsistent in most nonlinear static PD models.

18 Static and Dynamic Panel Data Models The classi cation Static vs Dynamic PD models depends on the relationship between the observable variables fx it g and the transitory shock fu it g. In a static model regressors are "strictly exogenous" in the sense that do not include lagged endogenous variables: E (u it x it+s ) = 0 for any s 2 f:::; 2; 1; 0; +1; +2; ::::g Dynamic models include predetermined or lagged endogenous variables. For instance, y it 1 is included in x it. In these models, x it+1 includes y it, and therefore E (u it x it+1 ) 6= 0

19 The properties of some estimators are very di erent in static and dynamic PD models. Estimators which are consistent and e cient in static models, are not even consistent in dynamic models.

20 2. STATIC PANEL DATA MODELS Consider the model: y it = x 0 it + " it where: x it is a K 1 of regressors. " it is the error term. Most PD models assume the following structure for the error term: " it = i + t + u it t is a common aggregate e ect; i represents persistent unobserved heterogeneity is called "individual e ect" or "unobserved heterogeneity" u it is called "transitory shock" or "time-variant unobservable",...

21 Time dummies We can incorporate parameters = f 1 ; 2 ; :::; T g using time dummies. Let D (s) t be the time-dummy for period s such that: D (s) it = 1 if t = s; and D (s) it = 0 if t 6= s Then, t = T P D (s) s=1 it s = [D (1) it ; :::; D(T ) it ] = D 0 it Therefore, the model: # y it = [x it ; D it ] 0 " + i + u it

22 Time dummies (cont.) For notational simplicity, unless state otherwise, I represent the model as: y it = x 0 it + i + u it But it should be understood that x it includes the time-dummy variables, and includes the time-e ect parameters. This notation applies to large N and T small. In this case, and are not subject to the incidental parameters problem [in cotrast to ]. When T is large and N is small we can include individual dummies in x it. In that case, and are not subject to the incidental parameters problem [in cotrast to ].

23 More notation... We can see a PD model as a system of T equations (an eq. for t = 1, for t = 2,...) such that we have a CS of N observations for each of these T equations. y i1 = x 0 i1 + i + u i1 y i2 = x 0 i2 + i + u i2.. y it = x 0 it + i + u it We can represent this system in vector form as: where: Y i = X i + 1 i + U i Y i and U i are T 1 vectors; and 1 is a T 1 vector of 1 0 s; X i is T K matrix.

24 For unbalanced panels we can use a similar notation. Let d it 2 f0; 1g be the indicator of the event "individual i is observed in the cross-section at period t". Then: where: Y i = T d i1 y i1 d i2 y i2. d it y it Y i = X i + d i i + U i ; X i = T K d i1 x 0 i1 d i2 x 0 i2. d it x 0 1T ; d i = T 1 The expressions of the di erent estimators we will see below apply to this de nition of Y i, X i, and d i d i1 d i2. d it 3 7 5

25 Assumption: Strict exogeneity (Static Model) E[x it u is ] = 0 for any (t; s) 2 f1; : : : ; T g Comment: This assumption does not hold for models where the set of regressors includes predetermined endogenous variables:.e.g., y i;t 1 ; y i;t 2 : For instance, suppose that y i;t 1 x it. The model establishes that y i;t 1 depends on u i;t 1. Therefore, it is clear that: E[x it u i;t 1 ] 6= 0

26 FIXED EFFECTS vs RANDOM EFFECTS One of the most important issues in the estimation of panel data models is endogeneity due to correlation between the regressors and the individual e ect. E (x it i ) 6= 0 There are two main approaches to control for this endogeneity problem: (1) the Fixed e ects approach; (2) the Correlated Random E ects approach.

27 Fixed e ects (FE) models / methods This approach does not impose any restriction on the joint distribution of (x i1 ; x i2 ; :::; x it ) and i. CDF ( i j x i1 ; x i2 ; :::; x it ) is completely unrestricted. In this sense, the FE model is nonparametric with respect the distribution CDF ( i j x i ). Typically, xed e ects methods are based on some transformation of the model that eliminates the individual e ects, or that make them redundant in a conditional likelihood function.

28 Correlated Random E ects (CRE) models / methods The CRE model imposes some restrictions on the distribution CDF ( i j x i1 ; x i2 ; :::; x it ). The stronger restriction is that i is independent of (x i1 ; x i2 ; :::; x it ) and iid(0; 2 ). Some textbooks de ne RE in this restrictive way. However, there are more general RE models. For instance, Chamberlain s CRE model: i = 0 + x 0 i1 1 + ::: + x 0 it T + e i where e i is independent of (x i1 ; x i2 ; :::; x it ). Based on this assumption, we estimate the parameters and 0 s. It is a parametric approach because it depends on a parametric assumption on the distribution of fx i1 ; x i2 ; :::; x it g and i.

29 Relative advantages and limitations of FE and CRE models (a) FE is more robust because it does not depend on additional assumptions. If the assumption of the CRE is not correct the CRE estimator may be inconsistent. (b) The FE transformation may eliminate sample variability of the regressors that is exogenous and useful to estimate the model. Therefore, the FE estimator may be less precise or e cient than the CRE estimator (provided the CRE assumption is consistent). (c) For some models (e.g., some nonlinear dynamic models) there is not a consistent FE method. Chamberlain (ECMA, 2010 on dynamic probit models).

30 ESTIMATION WHEN E[x it i ] = 0 We start with the simpler case in which the individual e ect is not correlated with the regressors. In this model, the OLS estimator in the equation in levels is consistent: ^ OLS = 0 NX i=1 t=1 x it x 0 it 1 A 1 0 NX i=1 t=1 x it y it 1 A = NX X 0 i X i 1 A 1 NX X 0 i Y i 1 A i=1 i=1 This estimator is also called the Pooled OLS estimator because we pool together all the observations as if we had pure cross-sectional data.

31 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) The pooled OLS estimator is not e cient because the error term " it is serially correlated: E h i " it " i;t j = E [i + u it ] i + u i;t j = E 2 i = 2 The asymptotically e cient estimator is the GLS. To obtain the e cient GLS estimator of it is convenient to write the model in matrix form: Y = X + "

32 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) where: Y = NT Y 1 Y 2. Y N ; X = NT K X 1 X 2. X N ; " = NT " 1 " 2. " N Let V " be the variance matrix of ". We know that the GLS estimator can be de ned as the OLS estimator of the following transformed model: V" 1=2 Y = V" 1=2 X + V" 1=2 " We now derive the expressions of V ", V" 1=2, and of the transformed variables V" 1=2 Y and V" 1=2 X.

33 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) We have V " = E[" " 0 ] = E " 1 " 0 1 " 1 " 0 2 : : : " 1 "0 N " 2 " 0 2 : : : " 1 "0 N.... " N " 0 N Notice that: E[" it " is ] = 8 >< >: E[( i + u it ) 2 ] = u if t = s E[( i + u it )( i + u is )] = 2 if t 6= s

34 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) Therefore, E[" i " 0 i ] = u 2 : : : u : : : u = 2 u I T where 1 is a T 1 vector of ones.

35 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) Thus, V " = : : : 0 0 : : : = I N = I N 2 u I T The matrix V 1=2 " is: V 1=2 " = (I N ) 1=2 = I N 1=2 And it is simple to verify that: 1=2 = 2 u I T =2 = IT T 110

36 GLS ESTIMATOR WHEN E[x it i ] = 0 (cont.) where: = 1 v u 1 t 1 + T 2 2 u The transformed variables are: Y = (I N 1=2 )Y = Y Y Y N 3 7 5

37 And: 1 2 Y i = I T T 110 Y i = y i1 y it. y i y i 3 7 5

38 Similarly, we get that: X = (I N 1=2 )X = X X X N And: 1 2 X i = I T T 110 X i = x i1 x it. x i x i Therefore the GLS estimator is equivalent to the OLS estimator in the transformed model: (y it y i ) = (x it x i ) 0 + " it

39 Comments on the GLS transformation: (y it y i ) = (x it x i ) 0 + " it with = 1 v u 1 t 1 + T 2 2 u When 2 = 2 u = 0: we have that = 0. OLS estimator in equation "in levels" y it = x it 0 + " it is e cient because " it is not serially correlated. When 2 = 2 u = 1: we have that = 1. The e cient estimator is OLS in the Within-Groups equation (y it y i ) = (x it x i ) 0 + " it.

40 Feasible GLS estimator. We do not know and therefore the GLS is unfeasible. However, we can obtain a FGLS estimator based on a consistent estimate of : To estimate we should estimate 2 u and 2. We can use OLS residuals to estimate these parameters. Let f^" it g be the OLS residuals. Then, the estimators of 2 u and 2 are: ^ 2 1 P u = Ni=1 P Tt=1 (^" it " i ) 2 N(T 1) ^ 2 = 1 P Ni=1 P Tt=1 (^" it ) 2 ^ 2 u NT It is possible to verify that as N goes to in nity (and T xed) ^ 2 u and ^ 2 converge in probability to 2 u and 2, respectively.

41 The estimator of is ^ = T ^ 2 =^ 2 u 1=2. The FGLS estimator is the OLS estimator of the transformed model: (y it ^ yi ) = (x it ^ xi ) 0 + " it This is also called Balestra-Nerlove estimator.

42 GLS and FGLS when u is not i.i.d. So far we have assumed that u it is i.i.d. Now, consider that this transitory shock can be autocorrelated and heteroscedastic over time. That is, V " = I N where for any individual i the variance-covariance matrix is an unrestricted. = E " 2 i1 " i1 " i2 : : : " i1 " it " 2 : : : " i2 i2 " it.... " 2 it The GLS estimator is equivalent to the OLS estimator of the transformed model: (I N 1=2 )Y = (I N 1=2 )X + " 3 7 5

43 This FGLS estimator can be obtained without any restriction on the structure of the variance matrix. Let b" i = (b" i1 ; b" i2 ; :::; b" it ) 0 be the vector of OLS residuals for individual i. Then, a consistent estimator of is: b = 1 N P Ni=1 b" i b" 0 i The FGLS estimator uses this estimator of.

44 ESTIMATION WHEN E[x it i ] 6= 0 Now, the Pooled OLS estimator is inconsistent because E[x it ( i +u it )] 6= 0: We now examine the following estimators: (a) OLS in rst di erences (b) Within-groups (Fixed E ects) estimator (c) Least squares dummy variables estimator (d) Chamberlain s CRE

45 (a) OLS in rst-di erences. The OLS-FD estimator is a FE estimator based on a rst-di erencing transformation of the model: y it = x 0 it + u it where represents a rst time di erence, e.g., y it = y it y i;t 1. Strict exogeneity of x it implies that E[x it u it ] = 0: E[x it u it ] = E [x it u it ] E h x it u i;t 1 i E h x i;t 1 u it i + E h xi;t 1 u i;t 1 i = Therefore, OLS in FD is a consistent estimator. ^ F D = P N TP i=1 t=2 x it x 0 it! 1 NP TP i=1 t=2 x it y it!

46 The estimator is not e cient (unless u it is a random walk), but we can de ne a FGLS-FD estimator.

47 (b) Within-Groups (or FE) estimator In general, Fixed E ect estimators of are based either on a transformation of the model or on some su cient statistic that eliminates the "incidental" or "nuisance" parameters f i g. The WG estimator is a FE estimator based on the WG transformation of the model: (y it y i ) = (x it x i ) 0 + (u it u i ) The WG estimator is just the OLS estimator of the WG transformation. ^ W G = P N TP i=1 t=1 (x it x i )(x it x i ) 0! 1 NP TP i=1 t=1 (x it x i )(y it y i )!

48 Consistency of WG estimator in Static-Linear PD This estimator is consistent if there is not perfect collinearity in (x it x i ) and E ((x it x i ) (u it u i )) = 0. Note that: E ((x it x i ) (u it u i )) = E [x it u it ] E [x it u i ] E [x i u it ] +E [x i u i ] = Strict exogeneity of the regressors imply that all these expectations are zero and therefore the WG estimator is consistent. However, the estimator of i is not consistent for xed T : p lim N!1 c i = 1 T TP t=1 y it x 0 it p lim N!1 b = 1 T TP t=1 yit x 0 it = i + u i 6= i

49 (c) Least squares dummy variables (LSDV) estimator. Suppose that we treat the individual e ects f 1 ; 2 ; :::; N g as parameters to be estimated together with. We can write the model in vector form as: Y = X + D + U = (X. D)! + U D is a NT N matrix of dummy variables, one for each individual. The i th column of matrix D contains the observations of the dummy variable for individual i, i.e., 1 if the observation belongs to i and 0 otherwise. is the vector of "parameters" ( 1 ; 2 ; :::; N ) 0.

50 LSDV estimator (cont.) The LSDV is simply the OLS estimator of b LSDV b LSDV!! in this model: = h (X. D) 0 (X. D) i 1 h (X. D) 0 Y i = " X 0 X X 0 D D 0 X D 0 D # 1 " X 0 Y D 0 Y # This way of implementing this estimator can computationally demanding if we the panel contains many individuals. We nee to keep in memory and to invert a matrix (N + K) (N + K) e.g., panel from the CPS with N = 120; 000 individuals...

51 LSDV estimator (cont.) Fortunately, we can obtain the LSDV estimator without having to invert "directly" (or by "brute force") this matrix. We can use properties of partitioned matrix together with the particular structure of the matrices D 0 D and D 0 X. By Frish-Waugh Theorem, the OLS estimator of can be written: b LSDV = h X 0 M D X i where M D is the idempotent matrix: 1 h X 0 M D Y i M D = I NT D(D 0 D) 1 D 0 = (I N I T ) = I N 1 T 1 I T T 110 I N 11 0

52 LSDV estimator (cont.) When we pre-multiply Y and X by M D, these variables are tramsformed in deviations with respect to individual means: M D Y = Y and M D X = X : y it = y it y i ; x it = x it x i and y i = T 1 P t t=1 y it and x i = T 1 P t t=1 x it. Since M D is an idempotent matrix, we can write: b LSDV = h X 0 X i 1 h X 0 Y i Therefore, the least squares dummy variables estimator of is EQUIV- ALENT to the OLS estimator of the transformed model: (y it y i ) = (x it x i ) 0 + " it

53 LSDV estimator (cont.) The LSDV and the Within-Groups are the numerically equivalent (i.e., the same estimator). The WG approach is the e cient algorithm, from a computational point of view, of obtaining the LSDV. It is simple to show that the estimator of i is: \ i;lsdv = T 1 T X t=1 yit x 0 it ^

54 Is the LSDV estimator consistent? Incidental parameters problem In general, the LSDV is NOT consistent as N! 1 and T is xed. The reason is the incidental parameters problem: the number of parameters in grow at the same rate as N, and we have only T observations for each i. Therefore, \ LSDV is NOT consistent. In general, the inconsistency of \ LSDV a ects the estimation of and implies that LSDV \ is also inconsistent. However, we show below that in STATIC & LINEAR PD models, LSDV \ consistent. is In general, the consistency property of LSDV \ PD models and to nonlinear PD models. does NOT extend to dynamic

55 Remark: It is interesting that we can get a consistent estimator of despite our estimator of is inconsistent as N goes to in nity and T is xed. In the LSDV estimator we try to control for the endogeneity of the unobservables i by introducing individual dummies. However, the parameter estimates associated with these dummies are asymptotically biased (as N goes to in nity and T is xed) such that these dummies are not fully capturing the individual e ects i. Then, how is it possible that we get a consistent estimator of? The answer is that the estimator of i, though inconsistent, is such that the estimation error (^ i i ) is not correlated with the regressors when these are strictly exogenous. Notice that asymptotically ^ i is equal to i +u i. Therefore, the estimation error ^ i i is equal to u i. When the regressors are strictly exogenous this estimation error is not correlated with x it. When the transitory shock u it is i.i.d., the WG estimator is also asymptotically e cient. WHY?

56 ROBUST STANDARD ERRORS Consider the static linear PD model: ey it = ex it + ~" it ey it, ex it, ~" it are transformations of the original variables y it, x it, " it. This representation includes as particular cases: - Model in levels: ey it = y it - Model in FD: ey it = y it - Within Groups: ey it = y it y i - Balestra-Nerlove: ey it = y it y i

57 ROBUST STANDARD ERRORS (cont.) In vector form: fy i = X f i + e" i (T 1) (T K) (K 1) (T 1) The OLS estimator in this transformed model is: b = N P i=1 fx 0 i f X i! 1 NP i=1 fx i f Y i! Depending on the transformation, the estimator can be the Pooled-OLS, the WG, the OLS-FD, the FGLS,... Under the assumption that the vectors e" i are independent over individuals, the variance matrix of this estimator is: V( ) b P = N! 1 fx 0 i X f NP i fx 0 i E h!! e" i e" 0 i j X f i 1 f NP i X i fx 0 i X f i i=1 i=1 i=1

58 ROBUST STANDARD ERRORS (cont.) Therefore, the panel-robust (or clustered over individuals) estimator of V( ) b is: cv( ) b P = N! 1!! 1 fx 0 i X f NP i fx 0 i " b i b" 0 i X f NP i fx 0 i X f i i=1 where b" i = f Y i f X i b. i=1 i=1 Note that cov(u it ; u is ) and var(u it ) are completely unrestricted to vary over i, t, s. Observations within individual i are correlated in an unrestricted way. These standard errors are denoted Clustered-over individuals Standard Errors. With panel data, it is very important to correct standard errors for serial correlation.

59 ROBUST STANDARD ERRORS: And Example (Labor Supply) We have a sample of 532 males in US over 10 years: (from Ziliak, JBES 1997). The labor supply equation is: log H it = log W it + i + u it H it represents the individual s annual hours of work W it represents the individual s after-tax hourly wage We are interested in the estimation of the elasticity parameter. Using cross-sectional data, estimates of are very small. We are concerned with cov(w it ; i ) 6= 0.

60 And Example (Labor Supply) [Cont.] cov(w it ; i ) > 0 [upward biased in ] taste for working positively correlated with wages. cov(w it ; i ) < 0 [downward biased in ] omitted sources of wealth and nonlabor income (with negative e ect on hours) can be positively correlated with wage. Main results: - Substantial downward bias in because cov(w it ; i ) 6= 0; - Very important to use robust s.e.; - Substantial loss of precision in FE estimation

61 Estimates Labor Supply Dependent variable: log(annual Hours of Work) Parameter Pool OLS Within FD FGLS Robust s.e. (0.030) (0.085) (0.084) (0.052) Default s.e. (0.009) (0.020) (0.021) (0.014) u Number Obs. 5,320 5,320 4,788 5,320

62 Some issues with FE estimators in static PD models Under the strict exogeneity assumption, FE estimators are consistent and robust because the consistency does not depend on any assumption of the joint distribution of i and x. However, there are two relevant issues with FE estimators: (a) Variance of FE cab be substantially larger than Pooled-OLS; (b) FE estimator may exacerbate measurement error bias.

63 (a) Variance of FE can be substantially larger than Pooled-OLS We can decompose the variance of the variables y it and x it in the sum of Within Groups and Between Groups variance: NP T i P i=1 t=1 (x it x) 2 = N P = N P T i P i=1 t=1 T i P i=1 t=1 ([x it x i ] + [x i x]) 2 (x it x i ) 2 + N P i=1 T i (x i x) 2 SS T OT AL SS W G + SS BG Tyypically, in micro-level datasets of individuals or rms, most of the variability is BG: individuals are very heterogeneous and this heterogeneity is very persistent over time: SS BG >> SS W G

64 (a) Variance of FE can be substantially larger than Pooled-OLS Note that for in the Pooled OLS estimator we are using SS T OT AL of the regressors, and this typically implies a very small variance. In contrast, in the WG estimator (and similarly in the OLS-FD) we exploit only the SS W G of the regressors. Typcally, this represents a small fraction of SS T OT AL. Therefore, Var(WG) >>> Var(Pooled-OLS).

65 (b) FE estimator may exacerbate measurement error bias When regressors are measured with error the WG estimator still eliminates the bias associated with the individual e ect i, but it may amplify the bias associated with the measurement error. To illustrate this, consider the following simple model: y it = x it + i + u it where x it satis es the strict exogeneity assumption, but it is correlated with i, cov(x it ; i) = > 0. There researcher does not observe the "true" regressor x it x it that is measured with error: but the variable x it = x it + e it

66 (b) FE estimator may exacerbate measurement error bias The model in levels is: where " it = i + u it equation is: y it = x it + " it e it. The asymptotic bias of the OLS estimator of this Bias(OLS) = cov(x; ") var(x it ) = 2 e SS T OT AL =NT The model in WG transformation is: (y it y i ) = (x it x i ) + (" it " i ) where (" it " i ) = (u it e it ) (u i e i ). The asymptotic bias of the WG estimator of this equation is: Bias(W G) = cov((x it x i )(" it " i )) var((x it x i )) = 2 e SS W G =NT

67 WG AND MEASUREMENT ERROR (cont.) Bias(OLS) = 2 e SS T OT AL =NT Bias(W G) = 2 e SS W G =NT Remark 1: The sign of the bias may go from positive to negative. Remark 2: If most of the variability in x is cross-sectional (i.e., SS T OT AL >> SS W G ), then the absolute value of the bias can be larger with the WG.

68 Remarks Given these issues, it is clear that there is a "price" the researcher should pay for the robustness of the FE estimator. In this context, it is useful to have: (a) A test of the null hypothesis E(x it i ) = 0. We do not want to get rid of the BG variation of the regressots if this variation is not correlated with i. [Hausman test] (b) An estimator that allows for E(x it i ) 6= 0 but does not eliminate the whole BG variability of the regressors (trades-o robustness for e ciency) [Chamberlain RE estimator].

69 HAUSMAN TEST OF ENDOGENOUS INDIVIDUAL EFFECTS H 0 : E (x it i ) = 0 and H 1 : E (x it i ) 6= 0 Let ^F GLS be the optimal (FGLS) estimator under the assumption that E (x it i ) = 0. Under [H 0 : E (x it i ) = 0] ) ( ^F GLS is consistent and e cient ^ W G is consistent (but not e cient) Under [H 1 : E (x it i ) 6= 0] ) ( ^F GLS is inconsistent ^ W G is still consistent

70 HAUSMAN TEST OF ENDOGENOUS INDIVIDUAL EFFECTS Therefore, under H 0 : plim (^F GLS ^W G ) = 0 and it is possible to prove that: H = (^F GLS ^W G ) 0 [V ar(^f GLS ^W G )] 1 (^F GLS ^W G ) a 2 K And, under H 1 : plim (^F GLS ^W G ) 6= 0 and H a noncentral 2 Note that the covariance between an e cient estimator and an ine cient estimator is equal to the variance of the e cient estimator: Cov(^eff ; ^ineff ) = V ar(^eff ) Therefore, H = (^F GLS ^W G ) 0 [V ar(^w G ) V ar(^f GLS )] 1 (^F GLS ^W G )

71 HAUSMAN TEST OF ENDOGENOUS INDIVIDUAL EFFECTS This is a very useful test in PD econometrics. However, note that the potential problem of not enough time-variability in the regressors appears also here and a ects the power of this test. If x s do not have enough time variability then V ar(^w G ) is "large" relative to V ar(^f GLS ) such that, and therefore the statistic H can be small even when k ^W G ^F GLS k is quite large. In other words, if there is not enough time variability in x the test may have low power and we may not be able to reject the H 0 even if it is false

72 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR Consider the PD model y it = x 0 it + i + u it Our main concern is the correlation between i and the regressors x. Suppose that we make the following assumption about the joint distribution of i and x: i = x 0 i1 1 + ::: + x 0 it T + e i where e i is independent of x i (x i1 ; x i2 ; :::; x it ). e i? x i

73 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR Based on this assumption, we have: y it = x 0 i1 1 + ::: + x 0 i1 [ t + ] + :::x 0 it T + e i + u it = [x 0 i1, x0 i2, :::, x0 it, :::, x0 it ] t +. T e i + u it = x 0 i t + u it where x 0 i (x0 i1 ; x0 i2 ; :::; x0 it ) is 1 KT vector of regressors, t is a KT 1 vector of parameters, and u it = e i + u it.

74 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR Then, we have a system of T equations with the same regressors but di erent dependent variables and parameters (a SURE): y i1 = x 0 i 1 + u i1 y i2. = x 0 i 2 + u i2. y it = x 0 i T + u it We can estimate each of these T equations separately by OLS (or by FGLS to account for the serial correlation in u it ) to obtain consistent estimates of the parameters But we want to estimate, not s.

75 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR Note that we can represent the relationship between s and s and as a linear regression-like equation: or in a compact form: 1 2. T = 6 4 I K I K 0 ::: 0 I K. 0. I K. ::: 0. I K 0 0 ::: I K = A " where A is a T 2 K (T + 1)K matrix of 0 0 s and 1 s. # T 3 7 5

76 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR Given the estimate of c, we can obtain a consistency estimator of s and as: " b b # = A 0 A 1 A 0 c We can use the estimated variance of, c i.e., V( c ), c to obtain a more e cient estimator (the Optimal Minimum Distance estimator): " # b = A b 0 V( c ) c 1 A 1 A 0 V( c ) c 1 c This estimator is asymptotically e cient under the assumptions of the Correlated RE model.

77 CHAMBERLAIN S CORRELATED RANDOM EFFECTS ESTIMATOR A very interesting property of the CRE model/estimator is that we can include time-invariant regressors. Consider the model: y it = 1 x it + 2 z i + i + u it Under the CRE assumption: i = x 1 x i1 + ::: + x T x it + z z i + e i It is simple to show that we can estimate/identify the parameter 2 + z.

78 Hausman tests of endogenous heterogeneity using Correlated RE Suppose that the SS W G variation in the data is small such that the WG estimator is imprecise and the Hausman test of endogenous heterogeneity based on b F GLS has very little power. b W G We can use the correlated RE estimator to construct a more powerful test. H = (^F GLS ^CRE ) 0 [V ar(^cre ) V ar(^f GLS )] 1 (^F GLS ^CRE ) Of course, we need to assume that CRE estimator is always consistent under the null and under the alternative. Note that under the null, the FGLS is more e cient than the CRE because it imposes the (correct) restrictions = 0.

79 APPLICATION: Crime in Norway We have data on crime for 53 districts and 2 years (1972 and 1978) in Norway. We are interested in the estimation of the following model: log(crime it ) = dum78 t + avgpsolv it + i + u it crime it is the crime rate in district i at year t; dum t is the dummy for year 1978; avgpsolv it is the percentage of crimes which were solved in district i during the previous two-years. We are particularly interested in the estimation of the deterrence e ect of solving crimes (i.e., parameter ).

80 OLS in levels OLS in levels Dependent variable: log(crime) Standard errors robust of heterocedasticity Regressor Estimate Std. error constant dummy avgpsolv Number obs. 106 R-square 0.47 According to these estimates, police s resolution of crimes has a very signi - cant deterrence e ect. If the proportion of solved cases increases in 1 percentage point, the crime rate decreases 3.6%. This e ect is statistically signi cant.

81 However, the consistency of this OLS estimator is based on the assumption that E(avgpsolv it * i ) = 0. Suppose that it is easier to solve crimes in communities where citizens have a low propensity to criminal activities. For instance, good citizens can be more collaborative with police in the solution of crimes. If that is the case, we have that E(avgpsolv it * i ) < 0 and the OLS estimator of will be downward biased. Therefore, we should be cautious to interpret this OLS estimate as a pure causal e ect of avgpsolv on crime.

82 Within-groups estimator Within-Groups Dependent variable: log(crime) Standard errors robust of heterocedasticity Regressor Estimate Std. error constant dummy avgpsolv Number obs. 106 R-square (within) R-square (all) ^ ^ u The deterrence e ect is still signi cantly di erent to zero, but much smaller in magnitude. Though we should make a formal test, it seems that i is correlated

83 with the regressors. There may be unobserved district characteristics (i.e., i ) which have a positive e ect on crime and are negatively correlated with the percentage of cases solved. That is, there are "good districts" (i.e., low i ) where crime is low and most crimes are easy to solve, and "bad districts" (i.e., high i ) where crime is high and crimes are more di cult to solve. If we do not control for these unobserved district characteristics, we get a downward bias estimate of, i.e., an estimate of the deterrence e ect that is partly spurious.

84 FGLS (Balestra-Nerlove) estimator FGLS Estimates Dependent variable: log(crime) Regressor Estimate Std. error constant dummy avgpsolv Number obs. 106 R-square (within) R-square (all) ^ ^ u This FGLS estimate of is between the OLS and the WG. Still, it seems that i is correlated with the regressors and OLS overestimates the deterrence e ect.

85 Hausman test of the null hypothesis that the regressors are not correlated with i. The Hausman statistic is equal to 14:85. Under the null hypothesis that E( i X i ) = 0 it is distributed as Chi-square with 2 degrees of freedom (i.e., number of regressors, other than the constant term). The p-value of the test is 0:0006, which means that we can clearly reject the null hypothesis under the typical choice of signi cance level (e.g., 5%, 1%) and even with much smaller signi cance levels.

86 Which estimate of do you nd more reasonable? There is clear evidence that the unobserved district e ect i is correlated with the regressors. Therefore, both the OLS and the Balestra-Nerlove estimators are inconsistent. The most reasonable estimator in this context seems the Within-Groups.

87 3. DYNAMIC PANEL DATA MODELS 3.1. INTRODUCTION Dynamic models: lagged values the variables (dependent and/or explanatory) have a causal e ect on the current value of the dependent variable. Multiple sources of dynamics: adjustment costs, irreversibilities, habits, storability, etc. Short-run and long-run responses are di erent. The assumption of strict exogeneity of the regressors does not hold.this has very important implications on the properties of estimators.

88 Example 1: Firms dynamic labor demand. Consider the dynamic labor demand equation (Sargent, JPE 1978; Arellano and Bond, REStud 1991): n it = n i;t w it + 3 y it + t + i + u it n = ln(employment); w = ln(w age); and y = ln(output); t represents aggregate shocks, and i represents rm speci c and time invariant factors that a ect productivity and or input prices and which are unobservable to the econometrician. The unobservable u it represents idiosyncratic shocks. The parameter 1 is associated with labor adjustment costs. In this model, it is clear that n i;t 1 is not strictly exogenous with respect to u it. It is not correlated wit u it or with future values of u, but by construction n i;t 1 depends on u i;t 1 ; u i;t 2 ; :::

89 Parameter 1 has important economic implications. Short-run elasticity = 2 : Long-run elasticity =

90 Example 2. [Cont] Employment has a strong positive serial correlation (after controlling for wages and output). Two possible explanations. (A) Structural state dependence. Labor adjustment costs. (B) Persistent unobserved heterogeneity. i represents rm heterogeneity in the steady-state optimal level of employment (due to productivity, other labor costs). i generates positive serial correlation in employment. The two explanations have very di erent economic implications (e.g., LR elasticity).

91 Example 2 (Investment in R&D): Consider the following dynamic model for rm s investment in R&D: RD it = RD it 1 + " it RD it is the rm investment during a year, and RD it investment at previous year. 1 is the same rm The term 1 RD it 1 captures the fact that previous investment in R&D increases the marginal pro t (e.g., reduces the marginal cost) of investment today, e.g., know-how. OLS estimation of this equation typically provides a positive, large, and signi cant estimate of the parameter 1.

92 Two possible explanations for the OLS estimate of 1. (A) Structural state dependence. Previous investment in R&D causes an increase in the marginal pro t of investment today. (B) Persistent unobserved heterogeneity. " it represents rm heterogeneity in the pro tability of investment in R&D. If " it is persistent over time, then rms with higher values of " it tend to have also higher past levels of R&D: cov(rd it 1 ; " it ) > 0. OLS estimation will provide an upward biased estimate of 1.

93 Example 3: AR(1) Panel Data Model. It is the simplest example of dynamic PD model. y it = y i;t 1 + i + u it For the moment, we maintain assumption u it iid over (i; t) (0; 2 u ): We will use this model as a benchmark. We will derive simple analytical expressions for the asymptotic bias and variance of di erent estimators of in this model. These expressions will help us to understand the problems and merits of di erent estimators.

94 Solving backwards, we have that y it = i ( :::) + u it + u i;t u i;t 2 + ::: If 6= 0 : (1) The regressor y i;t 1 is correlated with i : OLS is never consistent. (2) The regressor y i;t 1 is correlated with u i;t 1 ; u i;t 2 ; :::; and therefore it is not strictly exogenous. Point (2) has important implications on the properties of di erent estimators. More speci cally, for dynamic PD models the WG estimator and the OLS estimator in rst di erences are not consistent

95 3.2. INCONSISTENCY OF OLS-FD Consider the model in FD: y it = y i;t 1 + u it OLS estimation of this equation is consistent if Cov(y i;t 1 ; u it ) = 0: However, Cov(4y i;t 1 ; 4u it ) = E(y i;t 1 ; u it ) E(y i;t 1 ; u i;t 1 ) E(y i;t 2 ; u it ) + E(y i;t 2 ; u i;t 1 ) = 0 2 u = 2 u 6= 0

96 Exercise: Show that V ar(4y i;t 1 ) = 22 u 1 +. Therefore p lim b OLS F D = + Cov(4y i;t 1; 4u it ) N!1 V ar(4y i;t 1 ) = 1 2 (1+) And the bias is 2. It is clear that the bias of this estimator can be very large. For instance, when the true is +0:5 the estimator converges in probability to 0:25. Note that the bias does not depend on T. Therefore, the bias does not go to zero as T goes to in nity. The bias of the OLS-FD is as bad for T = 3 as for T = 500.

97

98 Remark 1: What if u it is serially correlated? Suppose u it AR(1): u it = u it 1 + a it with a it not serially correlated and jj < 1. Then, it is possible to show that Bias ^OLS F D = Cov(4y i;t 1 ; 4u it ) V ar(4y i;t 1 ) = This expression does not apply to = 1 (i.e., u it is a random walk). For = 1, ^OLS F D is consistent.

99 Remark 2: Bias reduction If u it is not serially correlated, we have that b OLS F D! p 1 2 This provides a very simple approach to correct for the bias and obtain a consistent estimator of. De ne the estimator: ^ = 2 ^OLS F D + 1 For this model, ^ is consistent and asymptotically normal (and very simple to calculate). Can we generalize this approach to more general DPD models?

100 Remark 2: Bias reduction (2) In general, the models that we have in empirical applications include more explanatory variables (not only y it 1 ) and the error term u it may be serially correlated. In that model, we do not have a simple expression for the bias (it depends on the joint stochastic process of the exogenous regressors) and on unknown parameters in the stochastic process for u it (e.g., see the previous example for the bias when u it follows an AR(1)). Still in the absence of better methods (e.g., in dynamic nonlinear PD models), bias reduction techniques can be useful.

101 3.3. INCONSISTENCY OF WG Consider the WG transformed equation: (y it y i ) = y i;t 1 y i( 1) + (uit u i ) The WG estimator is consistent if Cov((y i;t 1 y i ); (u it u i )) = 0: However, this covariance is Cov((y i;t 1 y i ); (u it u i )) = E(y i;t 1 ; u it ) E(y {z } i;t 1 ; u i ) E(y {z } i ; u it ) {z } + E(y i; u i ) {z } = 0 6= 0 6= 0 6= 0 < 0

102 Nickell (Econometrica, 1981) derived the expression for the asymptotic bias of the WG estimator in an AR(1) panel data model (for xed T ): where: bias ^W 1 2 h(t; ) G = (T 1) 2 h(t; ) h(t; ) = T! T (1 ) (1) For jj < 1, it is always negative. (2) It increases in absolute when goes to one (though there is a discontinuity at = 1 where it is zero). (3) It decreases with T and becomes zero as T goes to in nity. (4) If is not small, the bias can be important even for T greater than 20.

103

104 Remark: WG is consistent as T! 1 In contrast to the case of the OLS-FD, the bias of the WG estimator declines monotonically with T. Why? As we have seen, W G is numerically equivalent to LSDV. In the LSDV, we use T observations to estimate each individual e ect. As T increases, the bias in the estimates of the individual e ects becomes smaller and this contributes to reduce the bias of. In contrast, the bias of the OLS-FD comes from Cov(4y i;t 1 ; 4u it ) 6= 0, and this source is not a ected by T! 1.

105 Remark: Bias reduction Again, we have obtained an expression: plim^w G = p(; T ), where the function p(:) is known, and it strictly monotonic in, and therefore it is invertible. De ne the estimator: ^ = p 1 (^W G ; T ) where p 1 (:) is the inverse function. By the continuous function theorem, ^ is consistent and asymptotically normal. Again, a practical problem with this approach is that when we generalize the DPD model, the expression of p(; T ) becomes complicated and depends on many unknown parameters.

106 3.4. ANDERSON-HSIAO ESTIMATOR (JoE, 1982) Consider the equation in FD: y it = y i;t 1 + u it Suppose that u it is iid. Then, (a) 4u it is NOT correlated with y i;t 2 ; y i;t 3 ; :::. (b) If 6= 0; then 4y i;t 1 is correlated with y i;t 2 ; y i;t 3 ; ::: Therefore, under these conditions, we can use y it 2 (and also further lags of y) as an instrument for y i;t 1 in the equation in FD.

107 ANDERSON-HSIAO ESTIMATOR (cont) For the AR(1)-PD model without other regressors, the Anderson-Hsiao estimator is de ned as: ^ AH = TX t=3 2 TX 4 t=3 i=1 2 NX 4 y i;t i=1 NX 3 2 y it 5 3 y i;t 2 y i;t 1 Notice that we need T 3 to implement this estimator. 5 In the AR(1)-PD model that includes also exogenous regressors, Anderson- Hsiao estimator is an IV estimator in the equation in FD where the FD of the lagged endogenous variable is instrumented with y it 2.

108 Asymptotic Properties of Anderson-Hsiao Estimator To derive this asymptotic variance, notice that: p N ^AH P h Tt=3 N 1=2 P i N i=1 y i;t 2 u it = P h Tt=3 N 1 P i N i=1 y i;t 2 y i;t 1! d (as N! 1) N 0 1 V ar y i;t 2 u it E 2 y i;t 2 y i;t 1 C A Therefore, V ar ^AH 1 V ar y i;t 2 u it = N (T 2) E 2 y i;t 2 y i;t 1

109 Asymptotic Variance Anderson-Hsiao Estimator Excercise: Show that E y i;t 2 y i;t 2 1 = u 1 + and V ar y i;t 2 u it = 2 2 u V ar (y it ). Given these expressions, we have that the asymptotic variance is: V ar ^AH 2 2 u 1 = N (T 2) 2 (1 ) u u (1 + ) 2! = 2 N (T 2) 1 + 1! !! 2 2 u

110 Asymptotic Variance AH Estimator [Cont] Comment 1: The variance increases more than exponentially with. It can be pretty large if is close to 1. For = 1, the variance is in nite, i.e., the instrument y i;t with y i;t 1. 2 is not correlated

111

112 Comment 2: The variance also increases with 2 = 2 u. Typically, in many PD models with household or rm level data, the variance of the individual e ect is large relative to the variance of the transitory shock. Therefore, the AH estimator can be quite imprecise. Example: Suppose that: = 0:6 ; = u = 6 ; N (T 2) = 3600 You can verify that for this example sd ^AH = 0:567, and the 95% con - dence interval for is: 95% CI for = [ 0:511 ; + 1:711] That basically does not provide any information about the true value of.

113 Comment 3: There are several reasons for the ine ciency of this estimator. First, it does not exploit all the moment conditions that the model implies. Relatedly, for every instrument that we include (i.e., y it 2, y it 3,...) we lose a complete cross-section of data. The Arellano-Bond GMM estimator deals with these problems. Second, even if we exploit all the moment conditions and all the data in an e cient way, lagged levels y can be weakly correlated with current y when is close to one. The Arellano-Bover and the Blundell-Bond estimators deal with this second problem.

114 3.5. ARELLANO-BOND GMM (RESTUD, 1991) The AH estimator is based on the moment condition: P E T! P y i;t 2 u it = E T y i;t 2 (y it y i;t 1 ) t=3 t=3 The estimator pools together T 2 cross-sections.! = 0 But the model implies other moment conditions which are not exploited by the AH estimator. Let s represent the model in FD as a system of linear equations, one for each time period. 8 >< >: Equation for t = 3 : y i3 = y i2 + u i3 Equation for t = 4 :. y i4 = y i3 + u i4. Equation for t = T : y it = y it 1 + u it

115 If fu it g is iid, we have the following moment conditions: For t = 3 : E[y i1 4 u i3 ] = 0 For t = 4 : E[y i1 4 u i4 ] = 0 E[y i2 4 u i4 ] = 0 For t = 5 : E[y i1 4 u i5 ] = 0 E[y i2 4 u i5 ] = 0 E[y i3 4 u i5 ] = 0 The total number of moment conditions is: q ::: + (T 2) = (T 2)(T 1) 2

116 We can represent these q moment conditions in vector form as: E y i1 4 u i3 y i1 4 u i4 y i2 4 u i4 : : : = E h Z i 4Yi 4 Y i( 1) i where 4Y i and 4Y i( 1) are (T 2) 1 vectors 4Y i = (4y i3, 4 y i4,..., 4 y it ) 0 and Z i is the q (T 2) matrix of instruments.

117 And Z i is the q (T 2) matrix of instruments: Z i = y i1 0 0 ::: 0 0 y i1 0 ::: y i1 0 y i2 0 ::: y i2 ::: y i y it

118 To understand the structure of the q (T 2) matrix of instruments Z i, note that: - Each row of Z i represents an instrument (moment condition) for one of the T 2 equations; - For an instrument (moment condition) in equation t, we set equal to zero all the elements of that row except the element corresponding to period t. - For instance, row h y i1 0 0 ::: 0 i represents y i1 as instrument in the rst equation, t = 3. - Row h 0 0 y i1 ::: 0 i represents y i1 as instrument in the third equation, t = 5. - Row h 0 y i2 0 ::: 0 i represents y i2 as instrument in the second equation, t = 4.

119 Let m N () be the q1 vector of sample moment conditions, i.e., the sample counterpart of E h Z i 4Yi 4 Y i( 1) i Note that we use the T m N () = N 1 N X i=1 Z i 4Yi 4 Y i( 1) 2 cross-sections to construct these moment conditions. The Arellano-Bond GMM estimator is: ^ AB = arg min fg m N () 0 ^ 1 m N () where ^ is a consistent estimator of the optimal weighting matrix, = E Z i U i U 0 i Z i, with Ui = (u i3 ; u i4 ; :::; u it ) 0.

120 Deriving the rst order conditions and solving for ^AB we get: ^ AB = NX i=1 NX i=1 Z i Y i( 1) Z i Y i( 1) 10 A 10 A ^ 1 ^ 1 NX i=1 NX i=1 Z i Y i 1 A Z i Y i( 1) 1 A

121 This estimator is obtained in two steps. In the rst stage, we choose the weighting matrix under the assumption that u it is iid over time. Under this assumption = 2 u E Z i H Z 0 i where H is the (T 2) (T 2) matrix with 2 0 s in the main diagonal, 1 0 s in the two sub-main diagonals, and zeros otherwise. H = ::: ::: 0 0 ::: ::: ::: ::: 0 0 ::: :::

122 Then, the rst stage GMM estimator is: ^ (1) AB = NX i=1 NX i=1 Z i Y i( 1) Z i Y i( 1) 1 1 A0 A0 NX i=1 NX i=1 Z i H Z 0 i Z i H Z 0 i 1 A 1 A 1 1 NX i=1 NX i=1 Z i Y i 1 A Z i Y i( 1) 1 A

123 In the second stage GMM estimator, given ^(1) ^(1) AB we can get the residuals ^u it = y it AB y i;t 1. And we can use these residuals to obtain a consistent estimator of that is robust to heterocedasticity. ^ = N 1 N X i=1 where c U i = (^u i3 ; ^u i4 ; :::; ^u it ) 0. Finally, the two-stage GMM estimator is: ^ (2) AB = NX i=1 NX i=1 Z i Y i( 1) Z i Y i( 1) 1 1 A0 A0 NX i=1 NX i=1 Z i c U i c U 0 i Z0 i Z i c U i c U 0 i Z0 i Z i c U i c U 0 i Z0 i 1 A 1 A 1 1 NX i=1 NX i=1 Z i Y i 1 A Z i Y i( 1) We can generalize this estimator to any PD model with predetermined explanatory variables. 1 A