Multiple Equation GMM with Common Coefficients: Panel Data
|
|
- Monica Francis
- 5 years ago
- Views:
Transcription
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
Econ 582 Fixed Effects Estimation of Panel Data
Econ 582 Fixed Effects Estimation of Panel Data Eric Zivot May 28, 2012 Panel Data Framework = x 0 β + = 1 (individuals); =1 (time periods) y 1 = X β ( ) ( 1) + ε Main question: Is x uncorrelated with?
More informationEconomics 582 Random Effects Estimation
Economics 582 Random Effects Estimation Eric Zivot May 29, 2013 Random Effects Model Hence, the model can be re-written as = x 0 β + + [x ] = 0 (no endogeneity) [ x ] = = + x 0 β + + [x ] = 0 [ x ] = 0
More informationSingle Equation Linear GMM
Single Equation Linear GMM Eric Zivot Winter 2013 Single Equation Linear GMM Consider the linear regression model Engodeneity = z 0 δ 0 + =1 z = 1 vector of explanatory variables δ 0 = 1 vector of unknown
More informationChapter 6. Panel Data. Joan Llull. Quantitative Statistical Methods II Barcelona GSE
Chapter 6. Panel Data Joan Llull Quantitative Statistical Methods II Barcelona GSE Introduction Chapter 6. Panel Data 2 Panel data The term panel data refers to data sets with repeated observations over
More informationPanel Data Models. James L. Powell Department of Economics University of California, Berkeley
Panel Data Models James L. Powell Department of Economics University of California, Berkeley Overview Like Zellner s seemingly unrelated regression models, the dependent and explanatory variables for panel
More informationEconometric Analysis of Cross Section and Panel Data
Econometric Analysis of Cross Section and Panel Data Jeffrey M. Wooldridge / The MIT Press Cambridge, Massachusetts London, England Contents Preface Acknowledgments xvii xxiii I INTRODUCTION AND BACKGROUND
More informationRepeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data
Panel data Repeated observations on the same cross-section of individual units. Important advantages relative to pure cross-section data - possible to control for some unobserved heterogeneity - possible
More informationPanel Data Models. Chapter 5. Financial Econometrics. Michael Hauser WS17/18 1 / 63
1 / 63 Panel Data Models Chapter 5 Financial Econometrics Michael Hauser WS17/18 2 / 63 Content Data structures: Times series, cross sectional, panel data, pooled data Static linear panel data models:
More informationDealing With Endogeneity
Dealing With Endogeneity Junhui Qian December 22, 2014 Outline Introduction Instrumental Variable Instrumental Variable Estimation Two-Stage Least Square Estimation Panel Data Endogeneity in Econometrics
More informationLecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook)
Lecture 9: Panel Data Model (Chapter 14, Wooldridge Textbook) 1 2 Panel Data Panel data is obtained by observing the same person, firm, county, etc over several periods. Unlike the pooled cross sections,
More informationAsymptotics for Nonlinear GMM
Asymptotics for Nonlinear GMM Eric Zivot February 13, 2013 Asymptotic Properties of Nonlinear GMM Under standard regularity conditions (to be discussed later), it can be shown that where ˆθ(Ŵ) θ 0 ³ˆθ(Ŵ)
More informationRecent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data
Recent Advances in the Field of Trade Theory and Policy Analysis Using Micro-Level Data July 2012 Bangkok, Thailand Cosimo Beverelli (World Trade Organization) 1 Content a) Classical regression model b)
More informationZellner s Seemingly Unrelated Regressions Model. James L. Powell Department of Economics University of California, Berkeley
Zellner s Seemingly Unrelated Regressions Model James L. Powell Department of Economics University of California, Berkeley Overview The seemingly unrelated regressions (SUR) model, proposed by Zellner,
More informationPanel Data Model (January 9, 2018)
Ch 11 Panel Data Model (January 9, 2018) 1 Introduction Data sets that combine time series and cross sections are common in econometrics For example, the published statistics of the OECD contain numerous
More informationNonlinear GMM. Eric Zivot. Winter, 2013
Nonlinear GMM Eric Zivot Winter, 2013 Nonlinear GMM estimation occurs when the GMM moment conditions g(w θ) arenonlinearfunctionsofthe model parameters θ The moment conditions g(w θ) may be nonlinear functions
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36 Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao
More informationLecture 4: Heteroskedasticity
Lecture 4: Heteroskedasticity Econometric Methods Warsaw School of Economics (4) Heteroskedasticity 1 / 24 Outline 1 What is heteroskedasticity? 2 Testing for heteroskedasticity White Goldfeld-Quandt Breusch-Pagan
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 30 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Non-spherical
More informationAdvanced Econometrics
Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013 Outline Univariate
More informationEconometrics. Week 6. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 6 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 21 Recommended Reading For the today Advanced Panel Data Methods. Chapter 14 (pp.
More informationInstrumental Variables, Simultaneous and Systems of Equations
Chapter 6 Instrumental Variables, Simultaneous and Systems of Equations 61 Instrumental variables In the linear regression model y i = x iβ + ε i (61) we have been assuming that bf x i and ε i are uncorrelated
More informationInstrumental Variables and GMM: Estimation and Testing. Steven Stillman, New Zealand Department of Labour
Instrumental Variables and GMM: Estimation and Testing Christopher F Baum, Boston College Mark E. Schaffer, Heriot Watt University Steven Stillman, New Zealand Department of Labour March 2003 Stata Journal,
More informationEconometrics. Week 4. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 4 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 23 Recommended Reading For the today Serial correlation and heteroskedasticity in
More informationA Primer on Asymptotics
A Primer on Asymptotics Eric Zivot Department of Economics University of Washington September 30, 2003 Revised: October 7, 2009 Introduction The two main concepts in asymptotic theory covered in these
More informationSpecification testing in panel data models estimated by fixed effects with instrumental variables
Specification testing in panel data models estimated by fixed effects wh instrumental variables Carrie Falls Department of Economics Michigan State Universy Abstract I show that a handful of the regressions
More information1 Outline. 1. Motivation. 2. SUR model. 3. Simultaneous equations. 4. Estimation
1 Outline. 1. Motivation 2. SUR model 3. Simultaneous equations 4. Estimation 2 Motivation. In this chapter, we will study simultaneous systems of econometric equations. Systems of simultaneous equations
More informationLinear Panel Data Models
Linear Panel Data Models Michael R. Roberts Department of Finance The Wharton School University of Pennsylvania October 5, 2009 Michael R. Roberts Linear Panel Data Models 1/56 Example First Difference
More informationLecture 8 Panel Data
Lecture 8 Panel Data Economics 8379 George Washington University Instructor: Prof. Ben Williams Introduction This lecture will discuss some common panel data methods and problems. Random effects vs. fixed
More informationApplied Economics. Panel Data. Department of Economics Universidad Carlos III de Madrid
Applied Economics Panel Data Department of Economics Universidad Carlos III de Madrid See also Wooldridge (chapter 13), and Stock and Watson (chapter 10) 1 / 38 Panel Data vs Repeated Cross-sections In
More informationFixed Effects Models for Panel Data. December 1, 2014
Fixed Effects Models for Panel Data December 1, 2014 Notation Use the same setup as before, with the linear model Y it = X it β + c i + ɛ it (1) where X it is a 1 K + 1 vector of independent variables.
More informationORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT
ORTHOGONALITY TESTS IN LINEAR MODELS SEUNG CHAN AHN ARIZONA STATE UNIVERSITY ABSTRACT This paper considers several tests of orthogonality conditions in linear models where stochastic errors may be heteroskedastic
More informationPANEL DATA RANDOM AND FIXED EFFECTS MODEL. Professor Menelaos Karanasos. December Panel Data (Institute) PANEL DATA December / 1
PANEL DATA RANDOM AND FIXED EFFECTS MODEL Professor Menelaos Karanasos December 2011 PANEL DATA Notation y it is the value of the dependent variable for cross-section unit i at time t where i = 1,...,
More informationPanel Data Econometrics
Panel Data Econometrics RDB 24-28 September 2012 Contents 1 Advantages of Panel Longitudinal) Data 5 I Preliminaries 5 2 Some Common Estimators 5 21 Ordinary Least Squares OLS) 5 211 Small Sample Properties
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 4 Jakub Mućk Econometrics of Panel Data Meeting # 4 1 / 26 Outline 1 Two-way Error Component Model Fixed effects model Random effects model 2 Hausman-Taylor
More informationEcon 836 Final Exam. 2 w N 2 u N 2. 2 v N
1) [4 points] Let Econ 836 Final Exam Y Xβ+ ε, X w+ u, w N w~ N(, σi ), u N u~ N(, σi ), ε N ε~ Nu ( γσ, I ), where X is a just one column. Let denote the OLS estimator, and define residuals e as e Y X.
More informationIncreasing the Power of Specification Tests. November 18, 2018
Increasing the Power of Specification Tests T W J A. H U A MIT November 18, 2018 A. This paper shows how to increase the power of Hausman s (1978) specification test as well as the difference test in a
More informationEcon 583 Final Exam Fall 2008
Econ 583 Final Exam Fall 2008 Eric Zivot December 11, 2008 Exam is due at 9:00 am in my office on Friday, December 12. 1 Maximum Likelihood Estimation and Asymptotic Theory Let X 1,...,X n be iid random
More informationShort T Panels - Review
Short T Panels - Review We have looked at methods for estimating parameters on time-varying explanatory variables consistently in panels with many cross-section observation units but a small number of
More informationA Course in Applied Econometrics Lecture 4: Linear Panel Data Models, II. Jeff Wooldridge IRP Lectures, UW Madison, August 2008
A Course in Applied Econometrics Lecture 4: Linear Panel Data Models, II Jeff Wooldridge IRP Lectures, UW Madison, August 2008 5. Estimating Production Functions Using Proxy Variables 6. Pseudo Panels
More informationNon-Spherical Errors
Non-Spherical Errors Krishna Pendakur February 15, 2016 1 Efficient OLS 1. Consider the model Y = Xβ + ε E [X ε = 0 K E [εε = Ω = σ 2 I N. 2. Consider the estimated OLS parameter vector ˆβ OLS = (X X)
More information1 Estimation of Persistent Dynamic Panel Data. Motivation
1 Estimation of Persistent Dynamic Panel Data. Motivation Consider the following Dynamic Panel Data (DPD) model y it = y it 1 ρ + x it β + µ i + v it (1.1) with i = {1, 2,..., N} denoting the individual
More informationLecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16)
Lecture: Simultaneous Equation Model (Wooldridge s Book Chapter 16) 1 2 Model Consider a system of two regressions y 1 = β 1 y 2 + u 1 (1) y 2 = β 2 y 1 + u 2 (2) This is a simultaneous equation model
More informationPanel Data. March 2, () Applied Economoetrics: Topic 6 March 2, / 43
Panel Data March 2, 212 () Applied Economoetrics: Topic March 2, 212 1 / 43 Overview Many economic applications involve panel data. Panel data has both cross-sectional and time series aspects. Regression
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot October 28, 2009 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationEC327: Advanced Econometrics, Spring 2007
EC327: Advanced Econometrics, Spring 2007 Wooldridge, Introductory Econometrics (3rd ed, 2006) Chapter 14: Advanced panel data methods Fixed effects estimators We discussed the first difference (FD) model
More informationA Course in Applied Econometrics Lecture 7: Cluster Sampling. Jeff Wooldridge IRP Lectures, UW Madison, August 2008
A Course in Applied Econometrics Lecture 7: Cluster Sampling Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. The Linear Model with Cluster Effects 2. Estimation with a Small Number of roups and
More informationMissing dependent variables in panel data models
Missing dependent variables in panel data models Jason Abrevaya Abstract This paper considers estimation of a fixed-effects model in which the dependent variable may be missing. For cross-sectional units
More informationEconometrics. Week 8. Fall Institute of Economic Studies Faculty of Social Sciences Charles University in Prague
Econometrics Week 8 Institute of Economic Studies Faculty of Social Sciences Charles University in Prague Fall 2012 1 / 25 Recommended Reading For the today Instrumental Variables Estimation and Two Stage
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 2 Jakub Mućk Econometrics of Panel Data Meeting # 2 1 / 26 Outline 1 Fixed effects model The Least Squares Dummy Variable Estimator The Fixed Effect (Within
More informationTopic 10: Panel Data Analysis
Topic 10: Panel Data Analysis Advanced Econometrics (I) Dong Chen School of Economics, Peking University 1 Introduction Panel data combine the features of cross section data time series. Usually a panel
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 3 Jakub Mućk Econometrics of Panel Data Meeting # 3 1 / 21 Outline 1 Fixed or Random Hausman Test 2 Between Estimator 3 Coefficient of determination (R 2
More informationA Practitioner s Guide to Cluster-Robust Inference
A Practitioner s Guide to Cluster-Robust Inference A. C. Cameron and D. L. Miller presented by Federico Curci March 4, 2015 Cameron Miller Cluster Clinic II March 4, 2015 1 / 20 In the previous episode
More informationDynamic Panel Data Ch 1. Reminder on Linear Non Dynamic Models
Dynamic Panel Data Ch 1. Reminder on Linear Non Dynamic Models Pr. Philippe Polomé, Université Lumière Lyon M EcoFi 016 017 Overview of Ch. 1 Data Panel Data Models Within Estimator First-Differences Estimator
More informationEconomics 308: Econometrics Professor Moody
Economics 308: Econometrics Professor Moody References on reserve: Text Moody, Basic Econometrics with Stata (BES) Pindyck and Rubinfeld, Econometric Models and Economic Forecasts (PR) Wooldridge, Jeffrey
More informationLecture 10: Panel Data
Lecture 10: Instructor: Department of Economics Stanford University 2011 Random Effect Estimator: β R y it = x itβ + u it u it = α i + ɛ it i = 1,..., N, t = 1,..., T E (α i x i ) = E (ɛ it x i ) = 0.
More informationSingle Equation Linear GMM with Serially Correlated Moment Conditions
Single Equation Linear GMM with Serially Correlated Moment Conditions Eric Zivot November 2, 2011 Univariate Time Series Let {y t } be an ergodic-stationary time series with E[y t ]=μ and var(y t )
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationEconomics 583: Econometric Theory I A Primer on Asymptotics
Economics 583: Econometric Theory I A Primer on Asymptotics Eric Zivot January 14, 2013 The two main concepts in asymptotic theory that we will use are Consistency Asymptotic Normality Intuition consistency:
More informationA Course in Applied Econometrics Lecture 18: Missing Data. Jeff Wooldridge IRP Lectures, UW Madison, August Linear model with IVs: y i x i u i,
A Course in Applied Econometrics Lecture 18: Missing Data Jeff Wooldridge IRP Lectures, UW Madison, August 2008 1. When Can Missing Data be Ignored? 2. Inverse Probability Weighting 3. Imputation 4. Heckman-Type
More informationA Course on Advanced Econometrics
A Course on Advanced Econometrics Yongmiao Hong The Ernest S. Liu Professor of Economics & International Studies Cornell University Course Introduction: Modern economies are full of uncertainties and risk.
More informationG. S. Maddala Kajal Lahiri. WILEY A John Wiley and Sons, Ltd., Publication
G. S. Maddala Kajal Lahiri WILEY A John Wiley and Sons, Ltd., Publication TEMT Foreword Preface to the Fourth Edition xvii xix Part I Introduction and the Linear Regression Model 1 CHAPTER 1 What is Econometrics?
More informationA Robust Test for Weak Instruments in Stata
A Robust Test for Weak Instruments in Stata José Luis Montiel Olea, Carolin Pflueger, and Su Wang 1 First draft: July 2013 This draft: November 2013 Abstract We introduce and describe a Stata routine ivrobust
More informationNotes on Panel Data and Fixed Effects models
Notes on Panel Data and Fixed Effects models Michele Pellizzari IGIER-Bocconi, IZA and frdb These notes are based on a combination of the treatment of panel data in three books: (i) Arellano M 2003 Panel
More informationCluster-Robust Inference
Cluster-Robust Inference David Sovich Washington University in St. Louis Modern Empirical Corporate Finance Modern day ECF mainly focuses on obtaining unbiased or consistent point estimates (e.g. indentification!)
More informationAn overview of applied econometrics
An overview of applied econometrics Jo Thori Lind September 4, 2011 1 Introduction This note is intended as a brief overview of what is necessary to read and understand journal articles with empirical
More informationGeneralized Method of Moment
Generalized Method of Moment CHUNG-MING KUAN Department of Finance & CRETA National Taiwan University June 16, 2010 C.-M. Kuan (Finance & CRETA, NTU Generalized Method of Moment June 16, 2010 1 / 32 Lecture
More informationApplied Microeconometrics (L5): Panel Data-Basics
Applied Microeconometrics (L5): Panel Data-Basics Nicholas Giannakopoulos University of Patras Department of Economics ngias@upatras.gr November 10, 2015 Nicholas Giannakopoulos (UPatras) MSc Applied Economics
More informationDynamic Panel Data Workshop. Yongcheol Shin, University of York University of Melbourne
Dynamic Panel Data Workshop Yongcheol Shin, University of York University of Melbourne 10-12 June 2014 2 Contents 1 Introduction 11 11 Models For Pooled Time Series 12 111 Classical regression model 13
More informationHeteroskedasticity in Panel Data
Essex Summer School in Social Science Data Analysis Panel Data Analysis for Comparative Research Heteroskedasticity in Panel Data Christopher Adolph Department of Political Science and Center for Statistics
More informationHeteroskedasticity in Panel Data
Essex Summer School in Social Science Data Analysis Panel Data Analysis for Comparative Research Heteroskedasticity in Panel Data Christopher Adolph Department of Political Science and Center for Statistics
More informationLecture 12 Panel Data
Lecture 12 Panel Data Economics 8379 George Washington University Instructor: Prof. Ben Williams Introduction This lecture will discuss some common panel data methods and problems. Random effects vs. fixed
More informationxtseqreg: Sequential (two-stage) estimation of linear panel data models
xtseqreg: Sequential (two-stage) estimation of linear panel data models and some pitfalls in the estimation of dynamic panel models Sebastian Kripfganz University of Exeter Business School, Department
More informationOrdinary Least Squares Regression
Ordinary Least Squares Regression Goals for this unit More on notation and terminology OLS scalar versus matrix derivation Some Preliminaries In this class we will be learning to analyze Cross Section
More informationDynamic panel data methods
Dynamic panel data methods for cross-section panels Franz Eigner University Vienna Prepared for UK Econometric Methods of Panel Data with Prof. Robert Kunst 27th May 2009 Structure 1 Preliminary considerations
More informationSpatial Regression. 13. Spatial Panels (1) Luc Anselin. Copyright 2017 by Luc Anselin, All Rights Reserved
Spatial Regression 13. Spatial Panels (1) Luc Anselin http://spatial.uchicago.edu 1 basic concepts dynamic panels pooled spatial panels 2 Basic Concepts 3 Data Structures 4 Two-Dimensional Data cross-section/space
More informationEcon 583 Homework 7 Suggested Solutions: Wald, LM and LR based on GMM and MLE
Econ 583 Homework 7 Suggested Solutions: Wald, LM and LR based on GMM and MLE Eric Zivot Winter 013 1 Wald, LR and LM statistics based on generalized method of moments estimation Let 1 be an iid sample
More informationSimultaneous Equations with Error Components. Mike Bronner Marko Ledic Anja Breitwieser
Simultaneous Equations with Error Components Mike Bronner Marko Ledic Anja Breitwieser PRESENTATION OUTLINE Part I: - Simultaneous equation models: overview - Empirical example Part II: - Hausman and Taylor
More informationmrw.dat is used in Section 14.2 to illustrate heteroskedasticity-robust tests of linear restrictions.
Chapter 4 Heteroskedasticity This chapter uses some of the applications from previous chapters to illustrate issues in model discovery. No new applications are introduced. houthak.dat is used in Section
More informationEconometrics of Panel Data
Econometrics of Panel Data Jakub Mućk Meeting # 1 Jakub Mućk Econometrics of Panel Data Meeting # 1 1 / 31 Outline 1 Course outline 2 Panel data Advantages of Panel Data Limitations of Panel Data 3 Pooled
More informationα version (only brief introduction so far)
Econometrics I KS Module 8: Panel Data Econometrics Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: June 18, 2018 α version (only brief introduction so far) Alexander
More informationInstrumental Variables Estimation in Stata
Christopher F Baum 1 Faculty Micro Resource Center Boston College March 2007 1 Thanks to Austin Nichols for the use of his material on weak instruments and Mark Schaffer for helpful comments. The standard
More informationGeneralized Method of Moments: I. Chapter 9, R. Davidson and J.G. MacKinnon, Econometric Theory and Methods, 2004, Oxford.
Generalized Method of Moments: I References Chapter 9, R. Davidson and J.G. MacKinnon, Econometric heory and Methods, 2004, Oxford. Chapter 5, B. E. Hansen, Econometrics, 2006. http://www.ssc.wisc.edu/~bhansen/notes/notes.htm
More informationDynamic Panel Data Models
Models Amjad Naveed, Nora Prean, Alexander Rabas 15th June 2011 Motivation Many economic issues are dynamic by nature. These dynamic relationships are characterized by the presence of a lagged dependent
More informationEconometrics Honor s Exam Review Session. Spring 2012 Eunice Han
Econometrics Honor s Exam Review Session Spring 2012 Eunice Han Topics 1. OLS The Assumptions Omitted Variable Bias Conditional Mean Independence Hypothesis Testing and Confidence Intervals Homoskedasticity
More informationReview of Classical Least Squares. James L. Powell Department of Economics University of California, Berkeley
Review of Classical Least Squares James L. Powell Department of Economics University of California, Berkeley The Classical Linear Model The object of least squares regression methods is to model and estimate
More informationHeteroskedasticity and Autocorrelation
Lesson 7 Heteroskedasticity and Autocorrelation Pilar González and Susan Orbe Dpt. Applied Economics III (Econometrics and Statistics) Pilar González and Susan Orbe OCW 2014 Lesson 7. Heteroskedasticity
More informationAn estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic
Chapter 6 ESTIMATION OF THE LONG-RUN COVARIANCE MATRIX An estimate of the long-run covariance matrix, Ω, is necessary to calculate asymptotic standard errors for the OLS and linear IV estimators presented
More informationApplied Quantitative Methods II
Applied Quantitative Methods II Lecture 10: Panel Data Klára Kaĺıšková Klára Kaĺıšková AQM II - Lecture 10 VŠE, SS 2016/17 1 / 38 Outline 1 Introduction 2 Pooled OLS 3 First differences 4 Fixed effects
More informationGreene, Econometric Analysis (7th ed, 2012)
EC771: Econometrics, Spring 2012 Greene, Econometric Analysis (7th ed, 2012) Chapters 4, 8: Properties of LS and IV estimators We now consider the least squares estimator from the statistical viewpoint,
More informationEcon 2120: Section 2
Econ 2120: Section 2 Part I - Linear Predictor Loose Ends Ashesh Rambachan Fall 2018 Outline Big Picture Matrix Version of the Linear Predictor and Least Squares Fit Linear Predictor Least Squares Omitted
More information1 Introduction to Generalized Least Squares
ECONOMICS 7344, Spring 2017 Bent E. Sørensen April 12, 2017 1 Introduction to Generalized Least Squares Consider the model Y = Xβ + ɛ, where the N K matrix of regressors X is fixed, independent of the
More informationHeteroskedasticity. Part VII. Heteroskedasticity
Part VII Heteroskedasticity As of Oct 15, 2015 1 Heteroskedasticity Consequences Heteroskedasticity-robust inference Testing for Heteroskedasticity Weighted Least Squares (WLS) Feasible generalized Least
More informationNew Developments in Econometrics Lecture 11: Difference-in-Differences Estimation
New Developments in Econometrics Lecture 11: Difference-in-Differences Estimation Jeff Wooldridge Cemmap Lectures, UCL, June 2009 1. The Basic Methodology 2. How Should We View Uncertainty in DD Settings?
More informationProblem Set 2
14.382 Problem Set 2 Paul Schrimpf Due: Friday, March 14, 2008 Problem 1 IV Basics Consider the model: y = Xβ + ɛ where X is T K. You are concerned that E[ɛ X] 0. Luckily, you have M > K variables, Z,
More informationGreene, Econometric Analysis (5th ed, 2003)
EC771: Econometrics, Spring 2004 Greene, Econometric Analysis (5th ed, 2003) Chapters 4 5: Properties of LS and IV estimators We now consider the least squares estimator from the statistical viewpoint,
More informationExogeneity tests and weak identification
Cireq, Cirano, Départ. Sc. Economiques Université de Montréal Jean-Marie Dufour Cireq, Cirano, William Dow Professor of Economics Department of Economics Mcgill University June 20, 2008 Main Contributions
More informationIV and IV-GMM. Christopher F Baum. EC 823: Applied Econometrics. Boston College, Spring 2014
IV and IV-GMM Christopher F Baum EC 823: Applied Econometrics Boston College, Spring 2014 Christopher F Baum (BC / DIW) IV and IV-GMM Boston College, Spring 2014 1 / 1 Instrumental variables estimators
More informationECON 4160, Autumn term Lecture 1
ECON 4160, Autumn term 2017. Lecture 1 a) Maximum Likelihood based inference. b) The bivariate normal model Ragnar Nymoen University of Oslo 24 August 2017 1 / 54 Principles of inference I Ordinary least
More informationCasuality and Programme Evaluation
Casuality and Programme Evaluation Lecture V: Difference-in-Differences II Dr Martin Karlsson University of Duisburg-Essen Summer Semester 2017 M Karlsson (University of Duisburg-Essen) Casuality and Programme
More informationECON 4551 Econometrics II Memorial University of Newfoundland. Panel Data Models. Adapted from Vera Tabakova s notes
ECON 4551 Econometrics II Memorial University of Newfoundland Panel Data Models Adapted from Vera Tabakova s notes 15.1 Grunfeld s Investment Data 15.2 Sets of Regression Equations 15.3 Seemingly Unrelated
More information