Chapter 1. Linear Panel Models and Heterogeneity

Transcription

1 Chapter 1. Linear Panel Models and Heterogeneity Master of Science in Economics - University of Geneva Christophe Hurlin, Université d Orléans Université d Orléans January 2010

2 Introduction Speci cation tests and analysis of covariance The outline of the chapter: 1 Speci cation tests and analysis of covariance Fixed-E ects methods: Least Squares dummy variable approach 5 6 Heterogeneous panels: random coe cients models

3 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Section 1. Speci cation tests and analysis of covariance

4 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance A linear model commonly used to assess the e ects of both quantitative and qualitative factors is postulated as y i,t = α i,t + β 0 i,t x i,t + ε i,t where 8 i = 1,.., N, 8 t = 1,.., T α it and β = (β 1it, β 2it,..., β Kit ) are 1 1 and 1 K vectors of parameters that vary across i and t, x it = (x 1it,..., x Kit ) is a 1 K vector of exogenous variables, u it is the error term.

5 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Three aspects of the estimated regression coe cients can be tested: 1 the homogeneity of regression slope coe cients 2 the homogeneity of regression intercept coe cients. 3 the time stability of parameters (slopes and constants). We will not consider this issue (not speci c to panel data models) here.

6 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance We assume that parameters are constant over time, but can vary across individuals. y i,t = α i + β 0 i x i,t + ε i,t Three types of restrictions can be imposed on this model. Regression slope coe cients are identical, and intercepts are not (model with individual / unobserved e ects). y i,t = α i + β 0 x i,t + ε i,t Regression intercepts are the same, and slope coe cients are not (unusual). y i,t = α + β 0 i x i,t + ε i,t Both slope and intercept coe cients are the same (homogeneous / pooled panel). y i,t = α + β 0 x i,t + ε i,t

7 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance De nition An heterogeneous panel data model is a model in which all parameters (constant and slope coe cients) vary accross individuals. De nition An homogeneous panel data model (or pooled model) is a model in which all parameters (constant and slope coe cients) are common

8 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Speci cation Tests

9 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Figure: Hsiao (2003) 1 TestH 0: α i =αβ i =β i, [ 1N] H 1 0 rejetée H 1 0 vraie [ 1N] TestH 2 : β i =β i, 0, =α+β ' + ε y i t xi, t i, t H 2 0 rejetée H 2 0 vraie = TestH 3 0: α i =α i [ 1, N], α +β ' + ε y i t i i xi, t i, t H 3 0 vraie H 3 0 rejetée, =α+β ' + ε y i t xi, t i, t, =α +β' + ε y i t i xi, t i, t

10 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Lemma Under the assumption that the ε it are independently normally distributed over i and t with mean zero and variance σ 2 ε, F tests can be used to test the restrictions postulated

11 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance First step (homogeneous/ pooled assumption) Let us consider the general model y i,t = α i + β 0 x i,t + ε i,t The hypothesis of common intercept and slope can be viewed as a set of (K + 1)(N 1) linear restrictions: H 1 0 : β i = β α i = α 8 i 2 [1, N] H 1 a : 9 (i, j) 2 [1, N] / β i 6= β j ou α i 6= α j

12 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Under the alternative H 1, there are NK estimated slope coe cients for the N vectors β i (K 1) and estimated N constants. Under H 1, the unrestricted residual sum of squares S 1 divided by σ 2 ε has a chi-square distribution with NT N (K + 1) degrees of freedom.

13 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance De nition Under the homogeneous assumption H 1 0, H 1 0 : β i = β (K,1) (K,1) α i = α 8 i 2 [1, N] the F statistic, denoted F 1,and de ned by: F 1 = (RSS 1,c RSS 1 ) / [(N 1) (K + 1)] RSS 1 / [NT N (K + 1)] has a Fischer distribution with (N 1) (K + 1) and NT N (K + 1) degrees of freedom. RSS 1 denotes the residual sum of squares of the model and RSS 1,c the residual sum of squares of the constrained model:

14 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Under H 1, the residual sum of squares is equal to the sum of the N residual sum of squares associated to the N individual regressions: with S yy,i = RSS 1 = N N RSS 1,i = Syy,i Sxy 0,i Sxx,i 1 S xy,i i=1 i=1 T t=1 (y i,t y i ) 2 with x i = 1 T S xx,i = S xy,i = T t=1 x i,t and y i = 1 T T (x i,t x i ) (x i,t x i ) 0 t=1 T (x i,t x i ) (y i,t y i ) 0 t=1 T y i,t t=1

15 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Under H0 1, the model is : y i,t = α + β 0 x i,t + ε i,t the least-squares regression of the pooled model yields parameter estimates bβ = S 1 xx S xy S xx = S xy = N i=1 N i=1 T t=1 T t=1 (x i,t x) (x i,t x) 0 with x = 1 NT (x i,t x) (y i,t y) 0 with y = 1 NT N i=1 N i=1 T t=1 T t=1 x i,t y i,t

16 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Under H0 1, the overall RSS is de ned by SCR 1,c = S yy S 0 xy S 1 xx S xy with S xx,i = S xy,i = S yy = N T i=1 t=1 N T i=1 t=1 N T i=1 t=1 (y i,t y i ) 2 (x i,t x i ) (x i,t x i ) 0 (x i,t x i ) (y i,t y i ) 0

17 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Second step (individual/unobserved e ects) Let us consider the general model y i,t = α i + β 0 i x i,t + ε i,t The hypothesis of heterogeneous intercepts but homogeneous slopes can be reformulated as subject to (N 1)K linear restrictions (non restrictions on α i ). H0 2 : β i = β 8 i = 1,..N

18 y i,t = α i + β 0 C. Hurlin Panel x i,t Data + εeconometrics i,t Speci cation tests and analysis of covariance Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance De nition Under the assumption H 2 0, H 2 0 : β i = β 8 i = 1,..N the F statistic, denoted F 2, and de ned by: F 2 = (RSS 1,c 0 RSS 1) / [(N 1) K ] RSS 1 / [NT N (K + 1)] has a Fischer distribution with (N 1) K et NT N (K + 1) degrees of freedom under H0 2. RSS 1 denotes the residual sum of squares of the model and RSS 1,c 0 the residual sum of squares of the constrained model (model with individual e ects):

19 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Under H0 2, the residual sum of squares is: RSS 1,c 0 = N S yy,i i=1! 0 N S xy,i i=1! 1 N S xx,i i=1! N S xy,i i=1 S yy,i = T t=1 (y i,t y i ) 2 with x i = 1 T S xx,i = S xy,i = T t=1 x i,t y i = 1 T T (x i,t x i ) (x i,t x i ) 0 t=1 T (x i,t x i ) (y i,t y i ) 0 t=1 T y i,t t=1

20 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Last step If H0 2 is not rejected, one can also apply a conditional test for homogeneous intercepts (N 1 linear restrictions). H 3 0 : α i = α 8 i = 1,.., N given β i = β Under the null, the model is homogeneous (pooled) and the restricted residual sum of squares is SCR 1,c.. Under the alternative, the model is y i,t = α i + β 0 x i,t + ε i,t, and there is NT K N degrees of freedom

21 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance De nition Under the assumption H0 3, H0 3 : α i = α 8 i = 1,.., N given β i = β the F statistic, denoted F 3, and de ned by: F 3 = (RSS 1,c RSS 1,c 0) / (N 1) RSS 1,c 0/ [N (T 1) K ] (1) has a Fischer distribution with N 1 and N (T 1) K degrees of freedom under H0 2. RSS 1,c 0 denotes the residual sum of squares of the model with individual e ects and SCR 1,c the residual sum of squares of the pooled model previously de ned.

22 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Application: Strikes in OECD

23 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Example Let us consider a simple panel regression model for the total number of strikes days in OECD countries. We have a balanced panel data set for 17 countries (N = 17) and annual data form 1951 to 1985 (T = 35). General idea: evaluate the link between strikes and some macroeconomic factors (in atiion, unemployment etc..) s,t = α i + β i u,t + γ i p it + ε it 8 i = 1,.., 17 s i,t the number of strike days for 1000 workers for the country i at time t. u itt the unemployement rate p i,t, the in ation rate

24 Speci cation tests Application: strikes in OECD Figure: Spéci cation tests with TSP 4.3A

25 Speci cation tests Application: strikes in OECD

26 Speci cation tests Application: strikes in OECD

27 Speci cation tests Application: strikes in OECD Speci cation tests and analysis of covariance Conclusion of section 1. Fact It is possible to test the heterogeneity / homogeneity of the parameters under some speci c assumptions (normality of residuals). More generaly, the assumption of heterogenity / homogeneity of the parameters (slope coe cients and constants) has to be evaluated with an economic interpretation. Example Example: It is reasonnable to assume that the slope parameters of the production function are the same accros countries? what does it imply? Should I impose a common mean for the TFP for France and Germany?

28 Section 2. Linear unobserved/individual e ects panel data models

29 2.1.

30 We now consider a model with individual e ects and common slope parameters. De nition A linear unobserved / individual e ects panel data model is de ned as follows: y it = α i + β 0 x it + ε it where 8 i = 1,.., N, 8 t = 1,.., T α i and β = (β 1, β 2,..., β K ) are 1 1 and 1 K vectors of parameters, x it = (x 1it,..., x Kit ) is a 1 K vector of exogenous variables, ε it is the error term, assumed to be i.i.d., with 8 i = 1,.., N, 8 t = 1,.., T E (ε it ) = 0 E ε 2 it = σ 2 ε

31 De nition There are many names for α i : (1) unobserved e ects, (2) individual e ects, (3) unobserved component, (4) latent variable (for random e ects models), (5) individual heterogeneity. De nition The ε it are called the idiosyncratic errors or idiosyncratic disturbances because these change accross t as well as accross i.

32 Writting in vector form. Let us denote y i = (T,1) 0 y i,1 y i,2... y i,t 1 C A X i = (T,K ) 0 x 1,i,1 x 2,i,1... x K,i,1 x 1,i,2 x 2,i,2... x K,i, x 1,i,T x 2,i,T... x K,i,T 1 C A Let us denote e a unit vector and ε i the vector of errors: ε i,1 e = B 1 C A ε i = B ε i,2 (T,1)... A 1 ε i,t

33 De nition For a given individual i, the linear unobserved / individual e ects panel data model is de ned as follows: y i = eα i + X i β + ε i 8 i = 1,.., N E (ε i ) = 0 E ε i ε 0 i = σ 2 ε I T E ε i ε 0 j = 0 (T,T ) if i 6= j

34 Example Let us consider the case of a Cobb Douglas production function in log, as de ned previously, for the case T = 3. We have: y i,t = α i + β k k i,t + β n n i,t + ε i,t 8 i, 8 t 2 [1, 3] or in a vectorial form for a country i : y i,1 1 k i,1 n y i,2 A 1 A α i k i,2 n i,2 y i,3 1 k i,3 n i,3 1 A βk β n 0 ε i,1 ε i,2 ε i,3 1 A

35 It is also possible to stackle all this vectors/matrix as follows Y = (TN,1) 0 y 1 y 2... y N 1 C A Y = eeα + X β + ε 0 1 X 1 X = B X 2 C (TN,K A X N where 0 T is the null vector (T, 1). 0 1 e 0 T... 0 T e = B 0 T e... 0 T C (TN,N 0 T A 0 T 0 T... e ε = (TN,1) eα = (N,1) 0 0 α 1 α 2... α N ε 1 ε 2... ε N 1 C A 1 C A

36 Example Consider the case of the production function with T = 3 and N = y 1,1 k 11 n 11 y 1,2 y k 12 n 12 1,3 C B C α1 B k 13 n 13 C βk y 2,1 y 2,2 y 2,3 = C B C A α 2 + () Y = eeα + X β + ε k 21 n 21 k 22 n 22 k,3 n 23 C A β n 0 + ε 1 ε 1 ε 1, ε 2, ε 2, ε 2,

37 2.2.

38 Especially in methodological papers, but also in applications, one often sees a discussion about whether α i will be treated as a random e ect or a xed e ect. De nition In the traditional approach to panel data models, α i is called a random e ect when it is treated as a random variable and a xed e ect when it is treated as a parameter to be estimated for each cross section observation i.

39 Fact For Wooldridge (2003), these discussions about whether the α i should be treated as random variables or as parameters to be estimated are wrongheaded for microeconometric panel data applications. With a large number of random draws from the cross section, it almost always makes sense to treat the unobserved e ects, α i, as random draws from the population, along with y it and x it. This approach is certainly appropriate from an omitted variables or neglected heterogeneity perspective.

40 Fact As suggested by Mundlak (1978), the key issue involving α i is whether or not it is uncorrelated with the observed explanatory variables x it, for t = 1,.., T. Mundlak Y. (1978), On the Pooling of Time Series and Cross Section Data, Econometrica, 46, 69-85

41 De nition In modern econometric parlance, random e ect is synonymous with zero correlation between the observed explanatory variables and the unobserved e ect: cov (x it, α i ) = 0, t = 1,.., T

42 Actually, a stronger conditional mean independence assumption, E ( α i j x i1,.., x it ) = 0 will be needed to fully justify statistical inference. In applied papers, when α i is referred to as, say, an individual random e ect, then α i is probably being assumed to be uncorrelated with the x it.

43 De nition In microeconometric applications, the term xed e ect does not usually mean that α i is being treated as nonrandom; rather, it means that one is allowing for arbitrary correlation between the unobserved e ect α i and the observed explanatory variables x it.

44 Wooldridge (2003) avoids referring to α i as a random e ect or a xed e ect. Instead, we will refer to α i as unobserved e ect, unobserved heterogeneity, and so on. Nevertheless, later we will label two di erent estimation methods random e ects estimation and xed e ects estimation. This terminology is so ingrained that it is pointless to try to change it now.

45 Example (Production function) Let us consider the simple example of the Cobb Douglass production function. y i,t = β i k i,t + γ i n i,t + α i + v i,t In this case, α i corresponds to the unobserved e ect on TFP due to scountry speci c omitted factor (climate, institutions, organiszation etc..). In this case, we might expect that the the level of factors are positivily correlated with this component of TFP: the more a country is productive, the more it invests in capital for instance. cov (α i, k i,t ) > 0 cov (α i, n i,t ) > 0

46 Example (Patents and R&D) Hausman, Hall, and Griliches (1984) estimate (nonlinear) distributed lag models to study the relationship between patents awarded to a rm and current and past levels of R&D spending. A linear version of their model is: patents it = θ t + z it γ + δ 0 RD it + δ 1 RD i,t δ 5 RD i,t 5 + α i + v i,t where RD it is spending on R&D for rm i at time t and z it contains other explanatory variables. α i is a rm heterogeneity term that may in uence patents it and that may be correlated with current, past, and future R&D expenditures. cov (α i, RD i,t k ) 6= 0 8k

47 De nition The Hausman s test (1978), is a test of the null hypothesis cov (x it, α i ) = 0, t = 1,.., T and is generally presented as test of speci cation ( xed or random) of the unobserved e ects (see later, section 5.).

48 2.3.

49 Let us consider the following model: y i = eα i + X i β + ε i 8 i = 1,.., N where α i is a constant term, β 0 = (β 1 β 2...β K ) 2 R K and: y i = ε i,1 ε i,2... ε i,t 0 (T,1) ε i = ε i,1 ε i,2... ε i,t 0 (T,1) e = (T,1) 0 x 1,i,1 x 2,i,1... x K,i,1 X i = B x 1,i,2 x 2,i,2... x (T,K ) x 1,i,T x 2,i,T... x K,i,T 1 C A

50 Assumtions (H1) Let us assume that errors terms ε i,t are i.i.d. 8 i 2 [1, N], 8 t 2 [1, T ] with: E (ε i,t ) = 0 σ 2 E (ε i,t ε i,s ) = ε t = s 0 8t 6= s, or E (ε i εi 0) = σ2 ε I T where I t denotes the identity matrix (T, T ). E (ε i,t ε j,s ) = 0, 8j 6= i, 8 (t, s), or E ε i εj 0 = 0 where 0 denotes the null matrix (T, T ).

51 Theorem Under assumption H 1, the ordinary-least-squares (OLS) estimator of β is the best linear unbiased estimator (BLUE). De nition In this context, the OLS estimator bβ is called the least-squares dummy-variable (LSDV) of Fixed E ect (FE) estimator, because the observed values of the variable for the coe cient α i takes the form of dummy variables.

52 OLS estimators of α i and β and are obtained by minimizing o nbα i, bβ LSDV arg min S = fα i,βg N i=1 = N εi 0 ε i i=1 N (y i eα i X i β) 0 (y i eα i X i β) i=1

53 FOC1 (with respect to α i ): bα i = y i bβ 0 LSDV x i with x i = 1 T T t=1 x i,t y i = 1 T T y i,t t=1

54 Given the second FOC (with respect to β) and the previous result, we have: De nition Under assumption H 1, the xed e ect estimator or LSDV estimator of parameter β is de ned by: bβ LSDV = " N T i=1 t=1 " N (x i,t x i ) (x i,t x i ) 0 # 1 T i=1 t=1 (x i,t x i ) (y i,t y i ) #

55 1 The computational procedure for estimating the slope parameters in this model does not require that the dummy variables for the individual (and/or time) e ects actually be included in the matrix of explanatory variables. 2 We need only nd the means of time-series observations separately for each cross-sectional unit, transform the observed variables by subtracting out the appropriate time-series means, and then apply the leastsquares method to the transformed data.

56 The foregoing procedure is equivalent to premultiplying the i th equation y i = eα i + X i β + ε i by a T T idempotent (covariance) transformation matrix (within operator) Q = I T 1 T ee0 to sweep out the individual e ect α i so that individual observations are measured as deviations from individual means (over time).

57 Qy i and QX i correspond to the observations are measured as deviations from individual means : 1 Qy i = I T T ee0 y i 1 = y i e T e0 y i 0 1 y i,1 = B y i,2 C A T y i,t T t=1 0! y i,t C A

58 1 QX i = X i T ee0 X i 0 = x 1,i,1 x 2,i,1... x K,i,1 x 1,i,2 x 2,i,2... x K,i, x 1,i,T x 2,i,T... x K,i,T 1 T C A C A T t=1 x 1,i,t T t=1 x 2,i,t... T t=1 x K,i,t

59 Finally, when the transformation Q is applied to a vector of constant (or a time invariant variable), it lead to a null vector. 1 Qe = I T T ee0 e 1 = e T ee0 e = e e = 0 since 0 e 0 e = A = T

60 So, we have: y i = eα i + X i β + ε i () Qy i = Qeα i + QX i β + Qε i () Qy i = QX i β + Qε i

61 De nition Under assumption H 1, the xed e ect estimator or LSDV estimator of parameter β is de ned by: bβ LSDV = " N X 0 i=1 i QX i # 1 " N i=1 X 0 i Qy i # where Q = I T 1 T ee0

62 Example Let us consider a simple panel regression model for the total number of strikes days in OECD countries. We have a balanced panel data set for 17 countries (N = 17) and annual data form 1951 to 1985 (T = 35). General idea: evaluate the link between strikes and some macroeconomic factors (in atiion, unemployment etc..) s,t = α i + β i u,t + γ i p it + ε it 8 i = 1,.., 17 s i,t the number of strike days for 1000 workers for the country i at time t. u itt the unemployement rate p i,t, the in ation rate

63

64

65 Theorem The LSDV estimator bβ is unbiased and consistent when either N or T or both tend to in nity. Its variance covariance matrix is: V b β LSDV = σ 2 ε " N bβ LSDV p! NT! β # 1 Xi 0 QX i i=1

66 Theorem However, the estimator for the intercept, bα i, although unbiased, is consistent only when T!. bα i p! T! α i

67 2.4.

68 De nition The random speci cation of unobserved e ects corresponds to a particular case of variance-component or error-component model, in which the residual is assumed to consist of three components y i,t = β 0 x it + ε i,t 8 i8t ε i,t = α i + λ t + v i,t

69 Assumptions (H2) Let us assume that error terms ε i,t = α i + λ t + v i,t are i.i.d. with 8 i = 1,.., N, 8 t = 1,., T E (α i ) = E (λ t ) = E (v i,t ) = 0 E (α i λ t ) = E (λ t v i,t ) = E (α i v i,t ) = 0 σ 2 E (α i α j ) = α i = j 0 8i 6= j σ 2 E (λ t λ s ) = λ t = s 0 8t 6= s σ 2 E (v i,t v j,s ) = v t = s, i = j 0 8t 6= s, 8i 6= j E α i xi,t 0 = E λt xi,t 0 = E vi,t xi,t 0 = 0

70 De nition As suggested by Wooldridge (2001), the " xed e ect" speci cation can be viewed as a case in which α i is a random parameter with cov α i, xi,t 0 6= 0, whereas the "random e ect model" correspond to the situation in which cov α i, xi,t 0 = 0. De nition The variance of y it conditional on x it is the sum of three components: σ 2 y = σ 2 α + σ 2 λ + σ2 v

71 De nition Sometimes, the individual e ects αi are supposed to have a non zero mean, with E (α i ) = µ, then we can de ned a individual e ects α i with zero mean. A linear unobserved e ects panel data models is then de ned as follows: y i = eµ + X i β + ε i ε i = e α i + v i (T,1) (T,1)(T,1) (T,1)

72 De nition The vectorial expression of the individual e ects model is then: y i = ex i (T,1) γ (T,K +1) (K +1,1) + ε i (T,1) with ε i = e α i + v i (T,1) (T,1)(T,1) (T,1) ex i = (e : X i ) and γ 0 = µ : β 0

73 De nition Under assumptions H 2, the variance-covariance matrix of ε i is equal to: V = E ε i ε 0 i = E (αi e + v i ) (α i e + v i ) 0 = σ 2 αee 0 + σ 2 v I T Its inverse is: V 1 = 1 σ 2 v I T σ 2 α σ 2 v + T σ 2 α ee 0

74 The presence of α i produces a correlation among residuals of the same crosssectional unit, though the residuals from di erent cross-sectional units are independent V = 0 V = E ε i ε 0 i = σ 2 α ee 0 + σ 2 v I T σ 2 α + σ 2 v σ 2 α... σ 2 α σ 2 α + σ 2 v... σ 2 α... σ 2 α σ 2 α + σ 2 v 1 C A

75 However, regardless of whether the α i are treated as xed or as random,the individual-speci c e ects for a given sample can be swept out by the idempotent (covariance) transformation matrix Q Qy i = Qeµ + QX i β + Qeα i + Qv i Since Qe = I T T 1 ee 0 e = 0, we have Qy i = Qeµ + QX i β + Qv i

76 Theorem Under assumptions H 2, when α i are treated as random, the LSDV estimator is unbiased and consistent either N or T or both tend to in nity. However, whereas the LSDV is the BLUE under the assumption that α i are xed constants, it is not the BLUE in nite samples when α i are assumed random. The BLUE in the latter case is the generalized-least-squares (GLS) estimator.

77 Fact Moreover, if the explanatory variables contain some time-invariant variables z i, their coe cients cannot be estimated by LSDV, because the covariance transformation eliminates z i.

78 Let us consider the model y i = ex i γ + ε i 8 i = 1,.., N where ε i = α i e + v i, ex i = (e, X i ) and γ 0 = µ, β 0. The variance covariance matrix V = E (ε i εi 0 ) is known. De nition If the variance covariance matrix V is known, the GLS estimator of the γ vector, denoted bγ GLS, is de ned by: bγ GLS = " N i=1 ex 0 i V 1 ex i # 1 " N i=1 Under assumptions H 2, this estimaor is BLUE. ex 0 i V 1 y i #

79 De nition Following Maddala (1971), we write V as: V 1 = 1 Q σ 2 + ψ 1T ee0 v where Q = (I T ee 0 /T ) and where parameter ψ is de ned by: σ ψ = 2 v σ 2 v + T σ 2 α

80 Given this de nition of V 1, we have: bγ GLS = bγ GLS = " N i=1 " N i=1 ex i Q 0 + ψ 1T # 1 " N ee0 ex i ex i Q 0 + ψ 1T # ee0 y i i=1 ex 0 i Q ex i + ψ 1 T with ex i = (e X i ) and γ 0 = # 1 " N N ex i 0 ee 0 ex i ex i 0 Qy i + ψ 1 i=1 i=1 T µ β 0 # N ex i 0 ee 0 y i i=1

81 bµgls bβ GLS = ψnt ψt N x i i=1 ψnt y ψt N N X 0 i=1 x 0 i i=1 N Xi 0Qy i + ψt N x i y i i=1 i=1 i QX i + ψt N x i x 0 i i= Using the formula of the partitioned inverse, we can derive bβ GLS.

82 De nition If the variance covariance matrix V is known, the GLS estimator of vector β is de ned by: bβ GLS = " 1 T " 1 T N X 0 i=1 N X 0 i=1 i QX i + ψ i Qy i + ψ N i=1 N i=1 (x i x) (x i x) 0 # 1 (x i x) (y i y) # where ψ is a constant de ned by ψ = σ 2 v σ 2 v + T σ 2 1 α

83 This estimator can be expressed as a weigthed average of the LSDV (OLS) estimator and the between estimator. De nition The between-group estimator or between estimator,denoted bβ BE, corresponds to the OLS estimator obtained in the model: bβ BE = y i = c + β 0 x i + ε i " N # 1 " N (x i x) (x i x) 0 i=1 8 i = 1,.., N i=1 (x i x) (y i y) The estimator bβ BE is called the between-group estimator because it ignores variation within the group #

84 De nition The pooled estimator,denoted bβ pooled, corresponds to the OLS estimator obtained in the model: y it = α + β 0 x it + ε it 8 i = 1,.., N 8 t = 1,.., T bβ pooled = " T N t=1 i=1 " T t=1 (x i,t x) (x i x) 0 # 1 N (x it x) (y it y) i=1 #

85 Theorem Under assumptions H 2, the GLS estimator bβ GLS is a weighted average of the between-group bβ BE and the within-group (LSDV) estimators bβ LSDV. bβ GLS = bβ BE + (I K ) b β LSDV where denotes a weigth matrix de ned by: = ψt " N " N Xi 0 QX i + ψt i=1 # (x i x) (x i x) 0 i=1 # 1 N (x i x) (x i x) 0 i=1

86 1 If ψ! 1, then GLS converges to the OLS pooled estimator. bβ GLS p! ψ!1 bβ pooled 2 If ψ! 0, the GLS estimator converges to LSDV estimator. bβ GLS p! ψ!0 bβ LSDV

87 Fact The constant ψ measures the weight given to the between-group variation. In the LSDV (or xed-e ects model) procedure, this source of variation is completely ignored. The OLS procedure corresponds to ψ = 1. The between-group and within-group variations are just added up.

88 Fact The procedure of treating α i as random provides a solution intermediate between treating them all as di erent and treating them all as equal.

89 Given the de nition of ψ, we have: σ ψ = 2 v σ 2 v + T σ 2 α lim ψ = 0 T! Fact When T tends to in nity, the GLS estimator converges to the LSDV estimator: bβ GLS! T! bβ LSDV This is because when T!, we have an in nite number of observations for each i.therefore, we can consider each α i as a random variable which has been drawn once and forever, so that for each i we can pretend that they are just like xed parameters.

90 Fact Computation of the GLS estimator can be simpli ed by introducing a transformation matrix P such that P = hi T 1 ψ 1/2 (1/T ) ee 0i We have V 1 = 1 σ 2 v P 0 P Premultiplying the model by the transformation matrix P, we obtain the GLS estimator by applying the least-squares method to the transformed model (Theil (1971, Chapter 6)).

91 This is equivalent to rst transforming the data by subtracting a fraction 1 ψ 1/2 of individual means y i and x i from their corresponding y it and x it, then regressing y it 1 ψ 1/2 y i on aconstant and x it 1 ψ 1/2 x i.

92 We can show that: var b β GLS = σ 2 v " N # 1 Xi 0 N QX i + ψt (x i x) (x i x) 0 i=1 i=1 Because ψ > 0, we see immediately that the di erence between the covariance matrices of bβ LSDV and bβ GLS is a positive semide nite matrix. For K = 1, we have: var b β GLS var b β LSDV

93 De nition If the variance components σ 2 ε and σ 2 α are unknown, we can use two-stepgls estimation. 1 In the rst step, we estimate the variance components using some consistent estimators. 2 In the second step, we substitute their estimated values into bγ GLS = " N i=1 ex i 0 bv 1 ex i # 1 " N i=1 # ex i 0 bv 1 y i or its equivalent form.

94 Lemma When the sample size is large (in the sense of either N!,or T! ), the two-step GLS estimator will have the same asymptotic e ciency as the GLS procedure with known variance components. Lemma Even for moderate sample size [for T 3, N (K + 1) 9; for T 2, N (K + 1) 10], the two-step procedure is still more e cient than the covariance (or within-group) estimator in the sense that the di erence between the covariance matrices of the covariance estimator and the two-step estimator is nonnegative de nite

95 Noting that y i = α i + β 0 x i + ε i and (y it y i ) = (x it x i ) + (v it v i ), we can use the within- and between-group residuals to estimate σ 2 ε and σ 2 α by bσ 2 v = N i=1 T t=1 bσ 2 α = h (y i,t y i ) bβ 0 i 2 LSDV (x i,t x i ) N (T 1) K N y i bβ 0 LSDV x i i=1 N K 1 2 bσ 2 v

96 Then, we have an estimate of ψ and V 1 bσ 2 v! bψ = bσ 2 v + T bσ 2 α bv 1 = 1 Q bσ 2 + bψ 1T ee0 v

97 Speci cation tests and analysis of covariance Example Let us consider a simple panel regression model for the total number of strikes days in OECD countries. We have a balanced panel data set for 17 countries (N = 17) and annual data form 1951 to 1985 (T = 35). s,t = α i + βu,t + γp it + ε it 8 i = 1,.., 17 s i,t the number of strike days for 1000 workers for the country i at time t. u itt the unemployement rate p i,t, the in ation rate

98 Figure: Random e ects method

99 2.5.

100 Fact Whether to treat the e ects as xed or random makes no di erence when T is large, because both the LSDV estimator and the generalized least-squares estimator become the same estimator: bβ GLS! T! bβ LSDV

101 When T is nite and N is large, whether to treat the e ects as xed or random is not an easy question to answer. It can make a surprising amount of di erence in the estimates of the parameters.

102 Example For example, Hausman (1978) found that using a xed-e ects speci cation produced signi cantly di erent results from a random-e ects speci cation when estimating a wage equation using a sample of 629 high school graduates followed over six years by the Michigan income dynamics study. The explanatory variables in the Hausman wage equation include a piecewise-linear representation of age, the presence of unemployment or poor health in the previous year, and dummy variables for self-employment, living in the South, or living in a rural area.

103 Figure: Hausman (1978)

104 Fact In the random-e ects framework, there are two fundamental assumptions. One is that the unobserved individual e ects α i are random draws from a common population. The other is that the explanatory variables are strictly exogenous. That is, the error terms are uncorrelated with (or orthogonal to) the past, current, and future values of the regressors: E ( ε it j x i1,.., x ik ) = E ( α i j x i1,.., x ik ) = E (v it j x i1,.., x ik ) = 0

105 What happens when this condition is violated? E ( α i j x i1,.., x ik ) 6= 0 or E α i x 0 i,t 6= 0 1 The Mundlak s speci cation (1978) 2 The Hausman test

106 a. The Mundlak s speci cation

107 Mundlak (1978) criticized the random-e ects formulation on the grounds that it neglects the correlation that may exist between the e ects α i and the explanatory variables x it. There are reasons to believe that in many circumstances α i and x it are indeed correlated. Mundlak Y. (1978), On the Pooling of Time Series and Cross Section Data, Econometrica, 46,

108 The properties of various estimators we have discussed thus far depend on the existence and extent of the relations between the X s and the e ects. Therefore, we have to consider the joint distribution of these variables. However, α i are unobservable. =) Mundlak (1978a) suggested that we approximate E (α i x i,t ) by a linear function.

109 De nition Let us assume that the individual e ects satisfy: α i = x i 0 a + αi with a 2 R K and E (αi x it 0 ) = 0. Under this assumption, the model is: y i,t = µ + β 0 x i,t + x i 0 a + ε i,t ε i,t = αi + v i,t

110 Assumptions H 4 The error term ε i,t = αi + v i,t satis es, 8 i 2 [1, N], 8 t 2 [1, T ] E (αi ) = E (v i,t) = 0 E (αi v i,t) = 0 E αi σ 2 α j = α i = j 0 8i 6= j σ 2 E (v i,t v j,s ) = v t = s, i = j 0 8t 6= s, 8i 6= j E v i,t x 0 i,t = E α i x 0 i,t = 0

111 The model can be rewritten as follows: y i (T,1) = ex i γ (T,K +1) (K +1,1) + ε i (T,1) 8 i = 1,.., N with ε i = αi e + v i, ex i = (ex i 0, e, X i ) and γ 0 = a, µ, β 0. E ε i εj 0 = E h(αi e + v i ) αj i 0 e + v j σ 2 = α ee 0 + σ 2 v I T = V i = j 0 i 6= j

112 Utilizing the expression for the inverse of a partitioned matrix, we obtain the GLS of (µ, β, a ) as: bµ GLS = y x 0 bβ BE ba GLS = bβ BE bβ LSDV bβ GLS = bβ BE + (I K ) b β LSDV

113 Then, the GLS estimator bβ GLS is unbiased and we have: bβ GLS! T! bβ LSDV bβ GLS! NT! β

114 Now let us assume that the DGP corresponds to the Mundlak s model α i = x 0 i a + α i and we apply GLS to the initial model: y i,t = µ + β 0 x i,t + ε i,t ε i,t = α i + v i,t

115 In general, we have: bβ GLS = bβ BE + (I K ) b β LSDV In this case, we can show that: E b p β BE = β + a bβ BE! β + a N! E b β LSDV = β

116 Theorem So, if α i = x i 0a + α i, the GLS is biaised when T is xed. More precisely: E b β GLS = β + a bβ GLS p! N! β + a As usual, the GLS is asymptotically unbiased: bβ GLS p! T! β

117 When T is xed and N tends to in nity, the GLS is biased if there is a correlation between the individual e ects and the expanatory variables: plim N! bβ BE + N! = e (β + a) + I K bβ GLS = e plim I K e e β plim N! bβ LSDV = β + e a with e = plim. N!

118 Summary E ( α i j x i1,.., x ik ) = 0 E ( α i j x i1,.., x ik ) 6= 0 LSDV GLS LSDV GLS T Fixed, N! Unbiased Unbiased Biased T! and N! Unbiased BLUE Unbiased Unbiased

119 b. The Hausman s speci cation test

120 Hausman (1978) proposes a general test of speci cation, that can be applied in the speci c context of linear panel models to the issue of speci cation of individual e ects ( xed or random). Hausman J.A., (1978) Speci cation Tests in Econometrics, Econometrica, 46,

121 The general idea of the an Hausman s test is the following.let us consider a particular model y = f (x; β) + ε and particular hypothesis H 0 on this model (parameter, error term etc.).let us consider two estimators of the K-vector β, denoted bβ 1 and bβ 2, both consistent under H 0 and asymptotically normally distributed. 1 Under H 0, the estimator bβ 1 attains the asymptotic Cramer Rao bound. 2 Under H 1, the estimator bβ 2 is biased.

122 By examing the "distance" between bβ 1 and bβ 2, it is possible to conclude about H 0 : 1 If the "distance" is small, H 0 can not be rejected. 2 If the "distance" is large, H 0 can be rejected.

123 This distance is naturally de ned as follows: H = b β 2 0 h i 1 bβ 1 Var b β 2 bβ b 1 β 2 bβ 1 However, the issue is to compute the variance-covariance matrix Var b β 2 bβ 1 of the di erence between both estimators. Generaly we know V b β 2 and V b β 1, but not Var b β 2 bβ 1.

124 Lemma (Hausman, 1978) Based on a sample of N observations, consider two estimates bβ 1 and bβ 2 that are both consistent and asymptotically normally distributed, with bβ 1 attaining the asymptotic Cramer Rao bound so that p N b β 1 β is asymptotically normally distributed with variance covariance matrix V 1. Suppose p N b β 2 β is asymptotically normally distributed, with mean zero and variance covariance matrix V 2. Let bq = bβ 2 bβ 1. Then the limiting distribution [under the null] of p N b β 1 β and p Nbq has zero covariance: E (bβ 1 bq 0 ) = 0 K

125 Theorem From this lemma, it follows that Var b β 2 bβ 1 = Var b β 2 Var b β 1 Thus, Hausman suggests using the statistic H = b β 2 0 h bβ 1 Var b β 2 i 1 Var b β b 1 β 2 bβ 1 or equivalently H = bq 0 [Var (bq)] 1 bq

126 Under the null hypothesis, this statistic is distributed asymptotically as central chisquare, with K degrees of freedom. H H O! N! χ 2 (K ) Under the alternative, it has a noncentral chi-square distribution with noncentrality parameter eq 0 [Var (bq)] 1 eq, where eq is de ned as follows: eq = p lim H 1 /N! b β 2 bβ 1

127 Problem Now, apply the Hausman s test to discriminate between xed e ects methods and random e ects methods. We assume that α i are random variable and the key assumption tested is here de ned as: H 0 : E ( α i j X i ) = 0 H 1 : E ( α i j X i ) 6= 0

128 This test can be interpreted as a speci cation test between " xed e ect methods" and "random e ect methods". 1 If the null is rejected, the correlation between individual e ects and the explicative variables induces a bias in the GLS estimates. So, a standard LSDV approach ( xed e ect method) has to be privilegiated. 2 If the null is not rejected, we can use a GLS estilmator (random e ect method) and specify the individual e ects as random variables (random e ects model).

129 How to implement this test? Let us consider the standard model with random e ects (µ = 0): y i = X i β + eα i + v i 1 Under H 0 (and assumptions H 2 ) we know that bβ LSDV and bβ GLS are consistent and asymptotically normally distributed. 2 Under H 0, bβ GLS is BLUE and attains asymptotic Cramer Rao bound. 3 Under H 1, bβ GLS is biased.

130 According to the Hausman s lemma, we have: cov b β GLS, b β LSDV bβ GLS = 0 () cov b β LSDV, bβ GLS = Var b β GLS Since, Var b β LSDV bβ GLS = Var b β LSDV + Var b β GLS 2cov b β LSDV, bβ GLS We have: Var b β LSDV bβ GLS = Var b β LSDV Var b β GLS

131 De nition The Hausman s speci cation test statistic of individual e ect can be de ned as follows: 0 h i 1 H = b β LSDV bβ GLS Var b β LSDV Var b β b GLS β LSDV bβ GLS Under H 0, we have: H H O! NT! χ2 (K )

132 1 When N is xed and T tends to in nity, bβ GLS and bβ MCG become identical. However, it was shown by Ahn and Moon (2001) that the numerator and denominator of H approach zero at the same speed. Therefore the ratio remains chi-square distributed. However, in this situation the xed-e ects and random-e ects models become indistinguishable for all practical purposes. 2 The more typical case in practice is that N is large relative to T, so that di erences between the two estimators or two approaches are important problems.

133 Example Let us consider a simple panel regression model for the total number of strikes days in OECD countries. We have a balanced panel data set for 17 countries (N = 17) and annual data form 1951 to 1985 (T = 35). s,t = α i + β i u,t + γ i p it + ε it 8 i = 1,.., 17 s i,t the number of strike days for 1000 workers for the country i at time t. u itt the unemployement rate p i,t, the in ation rate

134

135 Variable coe cient model Mixed xed and random coe cient model Section 3.

136 Variable coe cient model Mixed xed and random coe cient model There are cases in which there are changing economic structures or di erent socioeconomic and demographic background factors that imply that the response parameters may be varying over time and/or may be di erent for di erent crosssectional units.

137 Variable coe cient model Mixed xed and random coe cient model 3.1. Variable coe cient model

138 Variable coe cient model Mixed xed and random coe cient model When data do not support the hypothesis of coe cients being the same, a single-equation model in its most general form can be written as: y it = K β kit x kit + v it k=1 i = 1,.., N and t = 1,.., T where, in contrast to previous sections, we no longer treat the intercept di erently than other explanatory variables and let x 1it = 1.

139 Variable coe cient model Mixed xed and random coe cient model Fact We assume that parameters do not vary with time (the time stability issue is not speci c to panel data econometrics) β kit = β ki y it = K β ki x kit + v it k=1 i = 1,.., N and t = 1,.., T

140 Variable coe cient model Mixed xed and random coe cient model In the case in which only cross-sectional di erences are present, this approach is equivalent to postulating a separate regression for each cross-sectional unit y i = X i β i + v i i = 1,.., N or y it = β 0 i x it + v it i = 1,.., N and t = 1,.., T where β i = (β 1i, β 2i,..., β Ki ) is a 1 K vector of parameters, and x it = (x 1it,..., x Kit ) is a 1 K vector of exogenous variables.

141 Variable coe cient model Mixed xed and random coe cient model De nition Alternatively, each regression coe cient can be viewed as a random variable with a probability distribution (e.g., Hurwicz (1950); Klein (1953); Theil and Mennes (1959); Zellner (1966)).

142 Variable coe cient model Mixed xed and random coe cient model 1 The random-coe cient speci cation reduces the number of parameters to be estimated substantially, while still allowing the coe cients to di er from unit to unit and/or from time to time. 2 Depending on the type of assumption about the parameter variation, it can be further classi ed into one of two categories: stationary and nonstationary random-coe cient models.

143 Variable coe cient model Mixed xed and random coe cient model De nition Stationary random-coe cient models regard the coe cients as having constant means and variance covariances.namely, the K 1 vector of parameters β i is speci ed as: β i = β + ζ i for i = 1,.., N, where β is a K 1 vector of constants, and ζ i is a K 1 vector of stationary random variables with zero means and constant variance covariances.

144 Variable coe cient model Mixed xed and random coe cient model For this type of model we are interested in 1 Estimating the mean coe cient vector β, 2 Predicting each individual component β i, 3 Estimating the dispersion of the individual-parameter vector β i (variance covariance matrix ) 4 Testing the hypothesis that the variances of ξ i (and so β i ) are zero.

145 Variable coe cient model Mixed xed and random coe cient model Fact Because of the computational complexities, variable-coe cient models have not gained as wide acceptance in empirical work as has the variable-intercept model. However, that does not mean that there is less need for taking account of parameter heterogeneity in pooling the data.

146 Variable coe cient model Mixed xed and random coe cient model De nition When regression coe cients are viewed as invariant over time, but varying from one unit to another, we can write the model as y it = K (β k + ξ ki ) x kit + v it i = 1,.., N and t = 1,.., T k=1 where β = (β 1, β 2,..., β K ) can be viewed as the common mean coe cient 1 K vector, and ξ i = (ξ 1i, ξ 2i,..., ξ Ki ) as the individual deviation from the common mean.

147 Variable coe cient model Mixed xed and random coe cient model 1 If individual observations are heterogeneous or the performance of individual units from the data base is of interest, then ξ i are treated as xed constants. 2 If conditional on x kit, individual units can be viewed as random draws from a common population or the population characteristics are of interest, then ξ i are generally treated as random variables having zero means and constant variances and covariances.

148 Variable coe cient model Mixed xed and random coe cient model De nition (Fixed-coe cient model) When β i are treated as xed and di erent constants, we can stack the NT observations in the form of the Zellner (1962) seemingly unrelated regression model y 1. y N 1 0 A X X N 1 0 β 1. β N 1 0 A where y i and v i are T 1 vectors (y it,..., y it ) and (v it,..., v it ), and X i is the T K matrix of the time-series observations of the i th individual s explanatory variables with the t th row equal to x it. v 1. v N 1 A

149 Variable coe cient model Mixed xed and random coe cient model 1 If the covariances between di erent cross-sectional units are not zero, e.g. E (v i v j 0) 6= 0, the GLS estimator of β 0 1,..., β0 N is more e cient than the single-equation estimator of i for each cross-sectional unit. 2 If X i are identical for all i or E (v i v i 0) = σ 2 i I T and E (v i v j 0) = 0 for i 6= j, the GLS estimator for β 0 1,..., β0 N is the same as applying least squares separately to the time-series observations of each cross-sectional unit.

150 Variable coe cient model Mixed xed and random coe cient model 3.2.

151 Variable coe cient model Mixed xed and random coe cient model De nition We now assume that all the coe cients β i are random, with common mean β. y it = K (β k + ξ ki ) x kit + v it i = 1,.., N and t = 1,.., T k=1 where β = (β 1, β 2,..., β K ) can be viewed as the common mean coe cient 1 K vector, and ξ i = (ξ 1i, ξ 2i,..., ξ Ki ) as the individual deviation from the common mean. Let us denote β ki = β k + ξ ki

152 Variable coe cient model Mixed xed and random coe cient model 1 The vector x i = (x 1i..x Ki ) includes a constant term.the parameter β ki associated to this constant term correspond to an individual e ect 2 An alternative notation is: y it = α + K k=2 (β k + ξ ki ) x kit + α i + v it E (α i ) = 0

153 Variable coe cient model Mixed xed and random coe cient model We will consider a set of assumptions used in the seminal paper of Swamy (1970) Swamy P.A. (1970), E cient Inference in a Random Coe cient Regression Model, Econometrica, 38,

154 Variable coe cient model Mixed xed and random coe cient model Assumptions (Swamy, 1970) Let us assume that E (ξ i ) = 0, E (v i ) = 0 E ξ i ξj 0 i = j = 0 8i 6= j E x it ξj 0 = 0, E ξ i v 0 = 0, 8 (i, j) E (v i v j 0) = σ 2 i I T 0 j t = s, i = j 8t 6= s, 8i 6= j

155 Variable coe cient model Mixed xed and random coe cient model Remark : we assume that the error term v i is heteroskedastic: E v i v 0 i = σ 2 i I T

156 Variable coe cient model Mixed xed and random coe cient model De nition The two rst moments of the vector of random parameters β i = β + ξ i are de ned by, 8 i = 1,..N : = E β i (K,K ) β0 i = E E (β i ) (K,1) ξi ξi 0 = = β (K,1) 0 σ 2 ξ 1 σ ξ1,ξ 2... σ ξ1,ξ K σ ξ2,ξ 1 σ 2 ξ 2... σ ξ2,ξ K σ ξk,ξ 1 σ ξk,ξ 2... σ 2 ξ K 1 C A

157 Variable coe cient model Mixed xed and random coe cient model Fact The matrix corresponds to variance covariance matrix of the random parameters β i = (β 1i, β 2i,.., β Ki ) 0, which is assumed to be common to all cross section units: = (K,K ) 0 σ 2 β 1 σ β1,β 2... σ β1,β K σ β2,β 1 σ 2 β 2... σ β2,β K σ βk,β 1 σ βk,β 2... σ 2 β K 1 C A

158 Variable coe cient model Mixed xed and random coe cient model De nition For each cross section unit, we have: with y i = X i β + X i ξ i + v i β i = β + ξ i where the vector X i include a constant term (i.e. the average of random individual e ects, α).

159 Variable coe cient model Mixed xed and random coe cient model De nition This model can be rewriten as follows : y i = X i β + ε i with ε i = X i ξ i + v i = X i (β i β) + v i

160 Variable coe cient model Mixed xed and random coe cient model For a given cross unit, the covariance matrix for the composite disturbance term ε i = X i ξ i + v i is de ned by: Φ i = E ε i εi 0 = E (X i ξ i + v i ) (X i ξ i + v i ) 0 = X i E ξ i ξi 0 X 0 i + E v i vi 0

161 Variable coe cient model Mixed xed and random coe cient model De nition For a given cross unit, the covariance matrix for the composite disturbance term ε i = X i ξ i + v i is de ned by: Φ i = X i X 0 i + σ 2 i I T Stacking all NT observations, the covariance matrix for the composite disturbance term is either block-diagonal and heteroskedastic or diagonal and heteroskedastic.

162 Variable coe cient model Mixed xed and random coe cient model 1 Under Swamy s assumption, the simple regression of y on X will yield an unbiased and consistent estimator of β if (1/NT )X 0 X converges to a nonzero constant matrix. 2 But the estimator is ine cient, and the usual least-squares formula for computing the variance covariance matrix of the estimator is incorrect, often leading to misleading statistical inferences.

163 Variable coe cient model Mixed xed and random coe cient model De nition The best linear unbiased estimator of β is the GLS estimator bβ GLS = N X 0 i=1 i Φi 1 X i! 1 N X 0 i=1 i Φi 1 y i!

164 Variable coe cient model Mixed xed and random coe cient model De nition The GLS estimator bβ GLS is a matrix-weighted average of the least-squares estimator bβ i for each cross-sectional unit, with the weights inversely proportional to their covariance matrices: W i = ( N i=1 bβ GLS = N W i bβ i i=1 h + σ 2 i X 0 i X i 1 i 1 ) 1 h + σ 2 i X 0 i X i 1 i 1 bβ i = X 0 i X i 1 X 0 i y i

165 Variable coe cient model Mixed xed and random coe cient model The covariance matrix for the GLS estimator is: V b β GLS = =! 1 N Xi 0 Φi 1 X i i=1 ( N i=1 h + σ 2 i X 0 i X i 1 i 1 ) 1

166 Variable coe cient model Mixed xed and random coe cient model Swamy proposed using the least-squares estimators bβ i = (Xi 0X i ) 1 Xi 0y i and their residuals bv i = y i X i bβ i to obtain unbiased estimators of σ 2 v and bσ 2 i = = with y i,t = bβ 0 i x i,t + bv i,t. 1 h T K y i 0 I X i Xi 0 i 1 X i X 0 i y i T 1 T K bv i,t t=1

167 Variable coe cient model Mixed xed and random coe cient model For the matrix, we have: b (K,K ) = " N 1 N 1 i=1 N 1 N bσ 2 X 0 1 i i X i i=1! N 1 bβ i N bβ i i=1! 0 # N 1 bβ i N bβ i i=1

168 Variable coe cient model Mixed xed and random coe cient model De nition However, the previous estimator b is not necessarily nonnegative de nite. In this situation, Swamy (1970) has suggested replacing this estimator by: " b = 1 N (K,K ) N 1 i=1! N 1 bβ i N bβ i i=1! 0 # N 1 bβ i N bβ i i=1 This estimator, although not unbiased, is nonnegative de nite and is consistent when T tends to in nity.

169 Variable coe cient model Mixed xed and random coe cient model Swamy proved that substituting bσ 2 i and b for σ 2 i and yields an asymptotically normal and e cient estimator of β. The speed of convergence of the GLS estimator is N 1/2.

170 Variable coe cient model Mixed xed and random coe cient model Summary: how to estimate a random coe cient model? 1 Run the N individual regressions y i,t = bβ 0 i x i,t + bv i,t. 2 Compute bσ 2 i and the Swamy s estimator b as follows " b = 1 N (K,K ) N 1 i=1! N 1 bβ i N bβ i i=1 bσ 2 = 1 h i T K y i 0 I X i Xi 0 i 1 X i X 0 i y i! 0 # N 1 bβ i N bβ i i=1

171 Variable coe cient model Mixed xed and random coe cient model 3. Compute the GLS estimate of the expectation of individual parameters β i! bβ GLS = N i=1 X 0 i bφ i 1 X i! 1 N i=1 bφ i = X i b X 0 i + bσ 2 i I T X 0 i bφ i 1 y i

172 Variable coe cient model Mixed xed and random coe cient model Example Swamy (1970) used this model to reestimate the Grunfeld investment function with the annual data of 11 U.S. corporations. His GLS estimates of the common-mean coe cients of the rms beginning-of-year value of outstanding shares and capital stock are and , with asymptotic standard errors and , respectively. The estimated dispersion measure of these coe cients is b =

173 Variable coe cient model Mixed xed and random coe cient model Predicting Individual Coe cients 1 Sometimes one may wish to predict the individual component β i, because it provides information on the behavior of each individual and also because it provides a basis for predicting future values of the dependent variable for a given individual. 2 Swamy (1970, 1971) has shown that the best linear unbiased predictor, conditional on given β i, is the least-squares estimator bβ i.