Multiple Equation GMM with Common Coefficients: Panel Data

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1 Multiple Equation GMM with Common Coefficients: Panel Data Eric Zivot Winter 2013

2 Multi-equation GMM with common coefficients Example (panel wage equation) 69 = = Note: common coefficients across two equations and ( 1 2 ) 6= 0

3 The general multi-equation model with common coefficients δ is = z 0 δ + =1 ; =1 1 1 [z ] 6= 0 forsomeelementof Themodelforthe equation is Note y 1 = Z δ 1 + ε 1 =1 δ 1 = = δ = δ

4 The stacked system (across equations) is y 1 = Z 1 δ + ε 1 y = Z δ + ε y 1 y = Z 1 Z δ + ε 1 ε This is in the form of a giant regression (with common coefficients) y 1 = Z δ + ( )( 1) ε 1

5 Assume each equation has instruments x satisfying [x ]=0 Very often the same instruments x are used for each equation The moment conditions are g (δ) 1 = x 1 1 x = x 1 ( 1 z 0 1 δ) x ( z 0 δ) There are a total of = P moment conditions

6 Identification [g (δ)] = 0 for δ = δ 0 6= 0 for δ 6= δ 0 Identification implies that or [x 1 1 ] [x ] σ 1 Σ [x 1 z 0 1 ] [x z 0 ] δ 0 ( )( 1) = 0 1 δ 0 = 0 For δ 0 to be the unique solution requires the rank condition rank(σ )=

7 Remarks: In the model with common coefficients, some equations may be individually unidentified but identified within the system Asufficient condition for identification is that [x z 0 ] have full rank forsomeequation

8 The moments g (δ) are a MDS with [g (δ 0 )g (δ 0 ) 0 ]=S = S 11 S 1 S 1 S S = [x x 0 ] Efficient GMM estimation min (δ Ŝ 1 ) = g (δ) 0 Ŝ 1 g (δ) g (δ) = S S δ S = Straightforward algebra gives S 1 1 S S = S 1 1 S δ(ŝ 1 )=(S 0 Ŝ 1 S ) 1 S 0 Ŝ 1 S

9 Special cases 1 3SLS with common coefficients conditional homoskedasticity x 1 = x 2 = = x = x S = S 3 = Σ [x x 0 ] 2 SUR with common coefficients (Random effects) conditional homoskedasticity x 1 = x 2 = = x = x x = union(z 1 z ) S = S = Σ [z z 0 ]

10 Random Effects Estimator (Efficient GMM with common coefficients) The SUR model with common coefficients is the giant regression y 1 = Z ( ) [εε 0 ] = Σ I Σ = δ + ε Z = ( 1) Z 1 Z The FGLS estimator is the random effects (RE) estimator and has the form ˆδ =(Z 0 (ˆΣ 1 I )Z) 1 Z 0 (ˆΣ 1 I )y

11 The elements of ˆΣ are usually estimated using the pooled OLS estimator of the giant regression and forming ˆδ =(Z 0 Z) 1 Z 0 y ˆ =(y Z ˆδ ) 0 (y Z ˆδ ) Note: Hayashi uses a somewhat non-standard version of the RE estimator He assumes that Σ is unstructured Most other treatments of the RE estimator assume a special structure for Σ based on an error components representation for In this case, the FGLS estimator uses a different estimator for Σ

12 Simplifying the RE estimator It turns out that the RE estimator may also be derived from an alternative representation of the giant regression In the alternative representation, the system is stacked by observations instead of by equations We can do this because δ is the same across equations Define y 1 = 1 Z = z 0 1 z 0 ε 1 Then, the model for the individual across all equations is y 1 = Z [ε ε 0 ] = Σ [ε ε 0 ] = 0 δ + ( )( 1) = 1 ε =1 1

13 Example: 2 period panel wage equation Ã! Ã y = Z 80 = δ = ( ) 0 ε =( 1 2 ) 0 Ã! 11 Σ = !

14 The giant regression stacked by observations has the form or y 1 y = Z 1 Z δ + ε 1 ε y 1 [εε 0 ] = Z = = δ + ( )( 1) ε 1 [ε 1 ε 0 1 ] [ε 1ε 0 ] [ε ε 0 1 ] [ε ε 0 ] Σ Σ = I Σ

15 TheFGLSorREestimatoris Now ˆδ =(Z 0 (I ˆΣ 1 )Z) 1 (Z 0 (I ˆΣ 1 )y Z 0 (I ˆΣ 1 )Z = h Z 0 1 Z0 = i Z 0 ˆΣ 1 Z ˆΣ ˆΣ 1 Z 1 Z

16 and Therefore, Z 0 (I ˆΣ 1 )y = h Z 0 1 Z0 = ˆδ = i Z 0 ˆΣ 1 y ˆΣ ˆΣ 1 Z 0 ˆΣ 1 Z 1 y 1 y Z 0 ˆΣ 1 y

17 Asymptotic Properties of the RE estimator Consistency of the RE estimator follows from the usual manipulations ˆδ δ = as long as 1 X 1 X Z 0 ˆΣ 1 Z 1 Z 0 ˆΣ 1 ε 0 1 X Z 0 ˆΣ 1 ε the RE estimator will be consistent This will occur, for example, if Z is uncorrelated with (no endogeneity)

18 Asymptotic normality also follows from the usual manipulations: (ˆδ δ) = 1 X Under the SUR model assumptions Z 0 ˆΣ 1 Z X Z 0 ˆΣ 1 ε so that X X Z 0 ˆΣ 1 Z [Z 0 Σ 1 Z ] Z 0 ˆΣ 1 ε (0 [Z 0 Σ 1 Z ]) (ˆδ δ) (0 [Z 0 Σ 1 Z ] 1 )

19 Pooled OLS again or y 1 y y 1 = = Z Z 1 Z δ + δ + ( )( 1) ε 1 ε ε 1 The pooled OLS estimator may also be computed using the giant regression stacked by observations: ˆδ = (Z 0 Z) 1 Z 0 y = Z 0 Z 1 Z 0 y The asymptotic properties of the OLS estimator are straightforward to derive

20 For asymptotic normality, (ˆδ δ) = 1 X 1 X 1 2 X Z 0 Z Z 0 Z Z 0 ε [Z 0 Z ] X Z 0 ε (0 [Z 0 ε ε 0 Z ]) Note, by iterated expectations and conditional homoskedasticity [Z 0 ε ε 0 Z ] = [ [Z 0 ε ε 0 Z Z ] = [Z 0 [ε ε 0 ]Z ] Z ] = [Z 0 ΣZ ]

21 Therefore, Equivalently, (ˆδ δ) [Z 0 Z ] 1 (0 [Z 0 ΣZ ]) (0 [Z 0 Z ] 1 [Z 0 ΣZ ] [Z 0 Z ] 1 ) avar(ˆδ d ) = ˆδ (δ 1 avar(ˆδ d )) 1 X 1 X Z 0 Z 1 1 Z 0 Z Question: WhenispooledOLSefficient? 1 X Z 0 ˆΣ 1 Z

22 Introduction to Panel Data = z 0 δ + = 1 (individuals); =1 (time periods) y 1 = Z ( ) δ + ε {y Z } is iid ( 1) Main question: Is z uncorrelated with? 1 If yes, then we have a SUR type model with common coefficients Under conditional homoskedasticity, the appropriate estimator is the RE estimator 2 If no, then we have a multi-equation system with endogenous regressors We need to use an estimation procedure to deal with the endogeneity In the panel set-up, under certain assumptions, we can deal with the endogeneity without using instruments using the so-called fixed effects (FE) estimator

23 Error Components Assumption = + = unobserved fixed effect [z ] = 0 [ε ε 0 x ] = Σ x = union(z 1 z ) The fixed effect component (which is actually an unobserved random variable) captures unobserved heterogeneity across individuals that is fixed over time With the error components assumption, the RE and FE models are defined as follows: RE model: [z ]=0 FE model: [z ] 6= 0 Note: Hayashi s treatment of error components model is slightly non-standard

24 Example: Panel wage equation 69 = = [ ] 6= 0 Here captures unobserved ability that is correlated with Itisassumed that ability does not vary over time Note: The existence of guarantees across equation error correlation: [ ] = [ ³ + 69 ³ + 80 ] = [ 2 ]+ [ 80 ]+ [ 69 ]+ [ ] 6= 0

25 Remark: With panel data, the endogeneity due to unobserved endogeneity (ie, [z ] 6= 0) can be eliminated without the use of instruments To see this, consider the difference in log-wages over time: =( )+ ( ) + ( )+( )+ ³ = ( )+ ( )+ ³ Hence, we can consistently estimate and by using the first differenced data!

26 Fixed Effects Estimation Key insight: With panel data, δ can be consistently estimated without using instruments There are 3 equivalent approaches 1 Within group estimator 2 Least squares dummy variable estimator 3 First difference estimator

27 Within group estimator To illustrate the within group estimator consider the simplified panel regression with a single regressor = + + [ ] 6= 0 [ ] = 0 Trick to remove fixed effect : First, for each average over time = + + = 1 = 1 = 1

28 Second, form the transformed regression or = ( )+( )+ = + The FE estimator is pooled OLS on the transformed regression (stacked by observation) ˆ = ( 0 ) = 0 0

29 Remarks 1 If z does not vary with (eg z = z ) then z = 0 and we cannot estimate δ 2 Must be careful computing the degrees of freedom for the FE estimator There are total observations and parameters in δ so it appreas that there are degrees of freedom However, you lose 1 degree of freedom for each fixed effect eliminated So the actual degrees of freedom are = ( 1)

30 Hayashi Notation for Within Estimator Consider the general model (assume all variables vary with and ) Stack the observations over giving = z 0 δ + + Define y 1 = Z δ + ( )( 1) 1 (1 1)( 1) + η 1 Q = I 1 (1 0 1 ) = I P P = 1 (1 0 1 ) =

31 Note P 1 = 1 Q 1 = 0 P y = 1 (1 0 1 ) y = 1 Q y = (I P )y = y 1 = ỹ

32 The transformed error components model is then Q y = Q Z δ + Q 1 + Q η ỹ = Z δ + η The giant regression (stacked by observation) is or ỹ 1 ỹ = Z 1 Z δ + η 1 η ỹ 1 = Z δ + ( )( 1) η 1 Note: Unless [ η η] =σ 2ˆ I ˆδ = ˆδ is not efficient

33 The FE estimator is again pooled OLS on the transformed system ˆδ = = = since Q is idempotent 1 Z 0 Z (Q Z ) 0 Q Z Z 0 Q Z 1 Z 0 ỹ 1 Z 0 Q y (Q Z ) 0 Q y

34 Least Squared Dummy Variable Model Consider the general model Stack the observations over giving = z 0 δ + + y 1 Now create the giant regression or y 1 y 1 y = = Z Z 1 Z δ δ + = Z δ η D ( )( 1) ( )( 1) D = I 1 α 1 + η 1 η η = Zδ +(I 1 ) α + η ( 1)

35 Aside: Partitioned Regression Consider the partitioned regression equation y 1 = X 1 1 β X 2 2 β The LS estimators for β 1 and β 2 can be expressed as + ε ˆβ 1 = (X 0 1 Q 2X 1 ) 1 X 0 1 Q 2y Q 2 = I P X2 ˆβ 2 = (X 0 2 Q 1X 2 ) 1 X 0 2 Q 1y Q 1 = I P X1 where P X1 = X 1 (X 0 1 X 1) 1 X 0 1 P X 2 = X 2 (X 0 2 X 2) 1 X 0 2

36 Result: The FE estimator is the OLS estimator of δ in the giant regression Now, Therefore, ˆδ = (Z 0 Q Z) 1 Z 0 Q y Q = I P P = D(D 0 D) 1 D 0 D = I 1 P = (I 1 ) h (I 1 ) 0 (I 1 ) i 1 (I 1 ) 0 = (I 1 )[I ] 1 (I 1 ) 0 = (I 1 )[I ³ ] ³ I 1 0 = I P Q = I P = I (I P ) = I Q

37 As a result ˆδ = (Z 0 Q Z) 1 Z 0 Q y = (Z 0 (I Q ) Z) 1 Z 0 (I Q ) y 1 = Z 0 Z Z 0 ỹ which is exactly the result we got when we transformed the model by subtracting off group means

38 Recovering Estimates of α With the LSDV approach, it is straightforward to deduce estimates for By examining the normal equations for α in the Giant Regression, one can deduce that ˆ = z 0 ˆδ = 1 z = 1 z

39 FE as a First Difference Estimator Consider the error components model = z 0 δ + + [η η 0 ] = 2 I Note: The spherical error assumption is made so that GLS estimation is possible Take 1st differences over : 2 = 2 1 = z 0 2 δ + 2 = ( 1) = z 0 δ + Notice that differencing removes the fixed effect

40 In matrix form the model is or (in Hayashi notation) C 0 y ( 1) 1 = C 0 Z δ + C 0 η ŷ = Ẑ δ + ˆη where ŷ = C 0 y Ẑ = C 0 Z ˆη = C 0 η and Note that C 0 ( 1) = [ˆη ˆη 0 ]= [C0 η η 0 C]=C0 [η η 0 ]C = 2 C 0 C

41 The transformed Giant Regression is or ŷ 1 ŷ = Ẑ 1 Ẑ δ + ˆη 1 ˆη Notice that [ˆηˆη 0 ] ( 1) ( 1) ŷ ( 1) 1 = = Ẑ δ + (( 1) )( 1) 2 C 0 C C 0 C C 0 C ˆη ( 1) 1 = 2 h I (C 0 C) i

42 Pooled GLS on the transformed regression gives ˆδ = h Ẑ 0 ³ I (C 0 C) 1 Ẑ i 1 Ẑ 0 ³ I (C 0 C) 1 ŷ = = = = Z 0 C(C 0 C) 1 C 0 Z Z 0 P Z Z 0 Q Z 1 Z 0 Z 1 1 Z 0 ỹ 1 Z 0 P y Z 0 Q y Z 0 C(C 0 C) 1 C 0 y

43 Here Ẑ 0 ³ I (C 0 C) 1 Ẑ = h Ẑ 0 1 Ẑ 0 = = = i Ẑ 0 (C0 C) 1 Ẑ (C 0 C) (C 0 C) 1 (C 0 Z ) 0 (C 0 C) 1 C 0 Z Z 0 C(C 0 C) 1 C 0 Z = Ẑ 1 Ẑ Z 0 P Z P = C(C 0 C) 1 C 0

44 The result P = C(C 0 C) 1 C 0 = Q = I 1 (1 0 1 ) follows from the result that C 0 1 = = That is, P = C(C 0 C) 1 C 0 is an idempotent matrix satisfying P 1 = 0 It projects onto the space orthogonal to 1 and this is exactly what Q does

45 The FE estimator is a GMM estimator Using the first-difference transformation of the error components model ŷ = Ẑ δ + ˆη the FE estimator can be viewed as a GMM estimator of the form where S = 1 S = 1 ˆδ(Ŵ) = ³ S 0 0 ŴS 1 S ŴS Ẑ x x = union of ( 1 ) ŷ x Ŵ = (C 0 C) 1 S 1

46 Asymptotic Distribution of FE estimator Assumptions: 1 y = Z δ + ε 2 ε = 1 + η (error components) 3 {y Z } is iid 4 [ ] 6= 0(Endogeneity) 5 [ ]=0for =1 2

47 It follows that ³ˆδ δ = 1 1 Z 0 Z 1 X Z 0 η [ Z 0 Z ] 1 (0 [ Z 0 η η 0 Z ]) (0 avar(ˆδ )) where avar(ˆδ )= [ Z 0 Z ] 1 [ Z 0 η η 0 Z ] [ Z 0 Z ] 1 Therefore, ˆδ (δ 1 avar(ˆδ d )) Remark: The sandwich form of avar(ˆδ ) suggeststhatitisaninefficient GMM estimator

48 General Case: Conditional Heteroskedasticity 1 Z 0 Z [ Z 0 1 Z ] Z 0 η η 0 Z [ Z 0 η η 0 Z ] where η = ỹ Z ˆδ Then a consistent estimate for avar(ˆδ ) is avar(ˆδ d ) = Z 0 Z 1 Z 0 Z 1 Z 0 η η 0 Z 1

49 Remarks The formula for avar(ˆδ d ) is called the panel robust covariance estimate It is not the same as the usual White correction for heteroskedasticity in a pooled OLS regression The White correction does not account for serial correlation It is also not the Newey-West correction for heteroskedasticity and autocorrelation The formula for d avar(ˆδ ) is also known as the cluster robust covariance estimate when the clustering variable is (each individual is a cluster)

50 Special Case: Conditional Homoskedasticity Assume: [ε ε 0 x ]=Σ x = union(z 1 z ) Then it can be shown that (Hayashi page 332) [ Z 0 η η 0 Z ]= [ Z 0 [ η η 0 ] Z ]= [ Z 0 V Z ] V = [ η η 0 ] and ˆV = 1 η η 0 [ η η 0 ] where η =(ỹ Z ˆδ )

51 Then a consistent estimate for avar(ˆδ ) is avar(ˆδ d ) = Z 0 Z 1 Z 0 Z 1 1 Z 0 ˆV Z

52 Special Case: When η is Spherical [η η 0 ]= 2 I [ η η 0 ]= 2 Q This implies that there is no serial correlation (correlation across time) Then [ Z 0 η η 0 Z ]= [ Z 0 [ η η 0 ]] Z ]= 2 [ Z 0 Z ] and avar(ˆδ )= 2 [ Z 0 Z ] 1

53 Then a consistent estimate for avar(ˆδ ) is avar(ˆδ d ) = ˆ 2 ˆ 2 = 1 1 η = (ỹ Z ˆδ ) 1 Z 0 Z η 0 η Remark: In this case the FE estimator is an efficient GMM estimator!

54 Hausman Specification Test for FE vs RE The hypotheses to be tested are 0 : [z ]=0(RE estimation) 1 : [z ] 6= 0(FE estimation) Hausman and Taylor (1981, Ecta) considered a test statistic based on ˆq = ˆδ ˆδ in the context of maximum likelihood estimation Newey (1985, see also Hayashi Chapter 3, Analytic Exercise 9) studied the test in the context of GMM

55 Intuition: Under 0 both ˆδ and ˆδ are consistent (but ˆδ is efficient) so that ˆq 0 Under 1 ˆδ is not consistent but ˆδ is consistent so that Therefore, consider the test statistic ˆq 9 0 = ˆq ( avar(ˆq)) d 1 ˆq If is big then reject 0 ; otherwise do not reject 0 Q1: What is avar(ˆq)? d Q2: What is the asymptotic distribution of?

56 To answer these questions consider the general linear GMM model = z 0 δ + =1 [x ] = 0 [x x 0 2 ]=S Let ˆδ 1 (Ŵ 1 ) denote an inefficient GMM estimator with weight matrix Ŵ 1 W 1 6= S 1 Let ˆδ 2 (Ŵ 2 ) denote another inefficient GMM estimator with weight matrix Ŵ 2 W2 6= W 1 6= S 1 The following results are based on Newey 1985 (See also Hayashi, Chapter 3, Analytic Exercise 9)

57 Result 1 ³ˆδ 1 (Ŵ 1 ) δ ³ˆδ 2 (Ŵ 2 ) δ "à 0 0! " A11 A 12 A 0 12 A 22 ## where A 11 = avar(ˆδ 1 (Ŵ 1 )) A 22 = avar(ˆδ 2 (Ŵ 2 )) A 12 = acov(ˆδ 1 (Ŵ 1 ) ˆδ 2 (Ŵ 2 ))

58 Define ˆq = ˆδ 1 (Ŵ 1 ) ˆδ 2 (Ŵ 2 ) Result 2: ˆq (0 avar(ˆq)) avar(ˆq) = A 11 + A 22 A 12 A 0 12 = avar(ˆδ 1 (Ŵ 1 )) + avar(ˆδ 2 (Ŵ 2 )) acov(ˆδ 1 (Ŵ 1 ) ˆδ 2 (Ŵ 2 )) acov(ˆδ 1 (Ŵ 1 ) ˆδ 2 (Ŵ 2 )) 0

59 Result 3: Set Ŵ 2 = Ŝ 1 where Ŝ S Then A 12 = A 0 12 = A 22 =avar(ˆδ 2 (Ŝ 1 )) and avar(ˆq) = A 11 + A 22 2A 22 = A 11 A 22 = avar(ˆδ 1 (Ŵ 1 )) avar(ˆδ 2 (Ŝ 1 )) Remark: Because ˆδ 2 (Ŝ 1 ) is the efficient GMM estimator avar(ˆq) 0

60 Applying Newey s result to ˆq = ˆδ ˆδ gives avar(ˆq) =avar(ˆδ ) avar(ˆδ ) and = ˆq ( avar(ˆq)) d 1 ˆq 2 ( ) Therefore, we reject 0 : [z ]=0at the (100)% level if 2 1 ( ) Remark: Hayashi shows (Appendix to chapter 5) that avar(ˆq) 0 so that avar(ˆq) 1 is well defined

61 Remarks The form of avar(ˆq) = avar(ˆδ ) avar(ˆδ ) depends on assumptions about heteroskedasticity Typical case: Assume conditional homoskedasticity (ˆδ δ) (0 [Z 0 Σ 1 Z ] 1 ) ³ˆδ δ (0 [ Z 0 Z ] 1 [ Z 0 [ η η 0 ] Z ] [ Z 0 Z ] 1 )

62 Then avar(ˆδ d ) = avar(ˆδ d ) = X 1 Z 0 Z 1 Z 0 Z Z 0 ˆΣ 1 Z 1 1 Z 0 ˆV Z

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