A correlated randon effects spatial Durbin model

Transcription

1 A correlated randon effects spatial Durbin model K Miranda, O Martínez-Ibañez, and M Manjón-Antolín QURE-CREIP Department of Economics, Rovira i Virgili University Abstract We consider a correlated random effects specification of the spatial Durbin dynamic panel model that allows the identification of the invididual effects and their spatial spillovers Following Lee and Yu 16, we derive identification conditions under quasi-maximum likelihood estimation We also derive the asymptotic properties of the quasi-maximum likelihood estimator Lastly, we provide illustrative evidence from a growth-initial level equation Our results indicate the existence of spatial contagion in the individual effects Keywords: correlated random effects, Durbin model, economic growth, spatial panel data JEL Classification: C3 This research was funded by grants ECO P Ministerio de Economía y Competitividad and 14FI B31 Agaur, Generalitat de Catalunya Usual caveats apply

2 1 Introduction The spatial Durbin model is a widely used specification in studies using georeferenced data LeSage and Pace, 9; LeSage, 14 In the panel data case, however, its use appears to be more limited Against its appeal as a general framework to analyse spatial relations, concerns have been raised about both the estimation and identification of this model, particularly in its dynamic version Elhorst et al, 1; Elhorst, 1 Yet one may expect that the spatial Durbin dynamic panel model will gain popularity in applied work, since identification conditions for -Stage Least Squares SLS and Quasi Maximum Likelihood QML estimators have recently been provided by Lee and Yu 16 In particular, they analyse a two-way error components model ie, with individual and time effects and provide results for a rather general variance-covariance matrix of the error term In this paper we consider a correlated random effects specification Mundlak, 1978; Chamberlain, 198 of the spatial Durbin dynamic panel model Following Lee and Yu 16, we derive the identification conditions and propose a QML estimator analogous to theirs We differ from them in that our model specification involves an error-components structure with individual effects and their spatial spillovers time effects can easily be incorporated, which results in a concrete albeit involved variance-covariance matrix This makes our identification conditions more specific with respect to the parameters of the variance-covariance matrix of the error term We also make use of this variance-covariance matrix to derive the asymtotic properties consistensy and asymptotic normality of the QML estimator under rather standard assumptions in the spatial econometrics literature 1 Our model specification is inspired by the work of Beer and Riedl 1, who advocate using an extension of the spatial Durbin model for panel data that controls for both the individual effects and the spatially weighted individual effects see also Miranda et al, 17 Ultimately, however, they argue that it is advisable to remove the spatial lag of the fixed effects from the equation as the inclusion of both, [the individual effects] and [their spatial spillovers], leads to perfect multicollinearity p 3 Removing the spatial lag of the fixed effects does 1 See, among others, Kelejian and Prucha 1998, 1; Lee 4; Yu et al 8 and Su and Yang 15 1

3 not generally preclude the consistent estimation of the parameters of the model However, this practice rules out obtaining an estimate of the individual-specific effects net of the spatially weighted effects, which can be critical in certain applications This is the case, for example, in growth models, where a measure of the unobserved productivity of the geographical units under study can be obtained from the estimated individual effects Islam, 1995 Distinguishing the individual effects from their spatial spillovers can thus provide interesting insights about how the unobserved characteristics of the neighbouring territories affect the output of a certain territory To illustrate this point, we estimate a growth-initial level equation using OECD data from Lee and Yu 16 In particular, we differ from previous related studies in that our model specification not only accounts for observable technological interdependences as in Ertur and Koch 7; see also Lopez-Bazo et al 4 but also for unobserved ones through the spatial spillovers of the individual effects As expected, our estimates largely replicate those of Lee and Yu 16, particularly in that we do not find evidence of observable technological interdependences in OECD countries On the other hand, our results point to the existence of unobservable technological interdependences ie, spatial contagion in the individual effects The rest of the paper is organised as follows In Section we present the model and discuss its identification problem In Section 3 we discuss the QML estimation and derive the asymptotic properties In Section 4 we present illustrative evidence from a growth-initial level equation estimated using OECD data Section 5 summarises the main findings of this work Model Specification and Identification conditions In this section we specify the model and discuss the identification conditions In particular, we assume that data is available for i = 1,, N spatial units and t = 1,, T time periods In terms of notation, we distinguish matrices with time-varying elements ie, the endogenous variable and the explanatory variables from matrices with non time-varying elements ie, matrices that are functions of some parameter vector and the mean variables matrix using the

4 subindex t Also, we denote with the subindex the true parameters of the model 1 Model specification Let us consider the following spatial autoregressive model with spatially weighted regressors and spatially weighted fixed effects: Y t = λ Y t 1 + ρ W N Y t + X t β + W N X t γ + µ + W N α + ε t 1 where, Y t = y 1t, y t,, y Nt is an N dimensional vector of dependent variables at time t, W N is the exogenous spatial weight matrix which describes the spatial arrangement of the units in the sample, X t = x 1t, x t,, x Nt is a N K matrix of regressors ie, x it is a row vector of 1 K and ε t is the N dimensional vector of disturbances at time t This model specification critically differs from alternative specifications of the spatial Durbin dynamic panel data model see eg Elhorst 1 and Lee and Yu 16 in the inclusion of both the individual effects µ and their spatial spillovers α Distinguishing the individual effects from their spatial spillovers may be of great interest in certain applications However, this is generally not possible because 1 is observationally equivalent to a model that only includes individual effects To address this issue, we follow Miranda et al 17 in using a correlated random effects specification to identify the spatial contagion in the individual effects In particular, we make use of the following correlation functions: µ = l N c + Xπ µ + υ µ α = Xπ α + υ α, where l N is the unit vector of dimension N 1, π µ and π α are K 1 parameter vectors and c is the constant term ie, we are implicitly assuming that the individual effects correspond to deviations from the constant term Notice also that X = X 1, X,, X N are composed For the sake of simplicity and without loss of generality, the model does not contain time effects Notice, however, that these can easily be incorporated into the model by eg including time dummies among the regressors, as we illustrate in the empirical application of Section 3 3

5 by the period-means of the regressors, X i = 1 T T x it Lastly, the error terms υ µ and υ α are assumed to be random vectors of dimension N with υ µ, σ µ I N and υ α, σ α I N However, υ µ and υ α are not assumed to be independent, the covariance, σ µα, being such that Eυ µ υ α = σ µα I N with E denoting the mathematical expectation Plugging equations into the model 1 we obtain t=1 Y t = l N c + λ Y t 1 + ρ W N Y t + X t β + W N X t γ + Xπ µ + W N Xπ α + η t 3 where η t = υ µ + W N υ α + ε t = ϕ + ε t Thus, ϕ is the composed error term of the individual effects and their spatial spillovers Notice also that the variance-covariance matrix of this model is given by Ω η = Ω ϕ + σ ε I N with Ω ϕ = σ µ I N + σ µα W N + W N + σ α W N W N Identification under QML estimation To derive the identification conditions of the model in 3, we make use of the following notation: Y t = λ Y t 1 + ρ W N Y t + X t δ + Xπ + η t 4 where X t = X t W N X t, X = l N X W N X, δ = β, γ and π = c, π µ, π α Assumption 1 W N is a non-stochastic matrix with zeros in the diagonal Also, W N is row-normalized, so that W N l N = l N 1 1 Assumption λ < 1 and ρ,, where ω min denotes the smallest and ω max ω min ω max denotes the largest characteristic root of the spatial weight matrix W N ω min <, ω max > Our identification proof follows from Lemma 1 in Lee and Yu 16 This means that we need to derive the concentrated likelihood function with respect to the parameter σε However, the derivation of the quasi maximum likelihood function of the complete sample is not immediate, since we assume that Y could be endogenous and the process started at 4

6 t = m periods ago 3 To derive the probability function of Y 1, we follow Bhargava and Sargan 1983 in considering the following stochastic approximation to the original model in 4 see Appendix B for details: S Y = Z θ + U 5 where Y = Y 1,, Y T, θ = τ, λ = b, π, δ, λ, U = U 1, U,, U T and Z = H X X X Y 1 X X T Y T 1 with A = λ I N ρ W N 1 = λ S 1 N, H = X WN X and X = X 1 X X T Notice that H is used to approximate Y Also, the variance-covariance matrix of this model is given by see Appendix C for details Σ = σ ζ I N + P + σ ε M l T 1 Ω ϕ A j l T 1 A j Ω ϕ J T 1 Ω ϕ + σ ε I 1 6 where σζ is the variance of the error term of the stochastic approximation, J T 1 = l T 1 l T 1, P = A j Ω ϕ A j and M = A j A j 3 Notice that this includes the case where Y is exogenously given, which corresponds to m = 5

7 Interestingly, Σ can be rewritten as follows: σ ε σ 1I N + P + M l T 1 Ω ϕa j l T 1 A j Ω ϕ JT 1 Ω ϕ + I 1 = σε Σ 7 where Ω ϕ = σ I N + σ 3W N + W N + σ 4W N W N, Ω η = Ω ϕ + I N, P = with σ = σ 1, σ, σ 3, σ 4 = 1 σ ε σ ζ, σ µ, σ µα, σ α Thus, the conditional maximum likelihood function of the model is: A j Ω ϕ A j ln Lφ, σε = ln S ρ lnπ ln σε 1 Σ ln λ, ρ, σ 1 [ 1U U σε τ, λ, ρ Σ λ, ρ, σ ] τ, λ, ρ 8 where φ = τ, λ, ρ, σ, denotes the determinant of a matrix, and the brackets denote functions or matrices evaluated at the parameters to be estimated and not at the true parameter values, eg, S ρ = I ρw = I ρ I T W N Finally, the expected concentrated likelihood function is see Appendix D for details Qφ = 1 ln S ρ 1 ln π 1 ln σ εφ 1 Σ ln λ, ρ, σ 1 9 = 1 ln π ln S ρ 1 ln σ εφ 1 Σ ln λ, ρ, σ 1 with τ τ τ τ σ εφ = λ λ Hλ, ρ, σ λ λ ρ ρ ρ ρ [ + σ ε tr S [ ] ρs 1 1S Σ λ, ρ, σ ρs 1 Σ ] 11 6

8 and where Hλ, ρ, σ = 1 Z [ Σ λ, ρ, σ ] 1 Z, and, Z = Z W S 1 Z θ Lemma 1 Σ λ, ρ, σ is definite positive Proposition 1 The vector of parameters θ, ρ are identified if B = X W N X W N X ln and l N X W N X WNX Y 1 W N Y 1 X W N X WNX D = l N X T W N X T WNX T Y T 1 W N Y T 1 X W N X WNX have both full column rank and b + ρ b 1, γ + ρ β, π α + ρ π µ Also, if Σ λ, ρ, σ and Σ are linearly independent, then the parameters of the variancecovariance matrix, σ = σ1, σ, σ3, σ4, are identified These are sufficient conditions for identification analogous to those derived by Lee and Yu 16 Corollary Matrices Σ λ, ρ, σ and Σ are linearly independent if the matrix J T 1 I N J T 1 W N + W N J T 1 W N W N I 1 has full column rank model This is a necessary and sufficient condition for the identification of all the parameters of the 3 QML estimation: Likelihood function and asymptotic properties In this section we derive the quasi likelihood function of the spatial Durbin dynamic panel model with correlated random effects We also study the consistency and asymptotic normality of the associated QML estimator 7

9 31 QML estimator To make the estimation procedure less burdensome, we assume that Y is exogenous 4 Let us then define η = η 1,, η T and X = l Y 1 X X, with Y 1 = Y, Y 1,, Y T 1, X = X 1,, X T 1, and X = X N,, X N We can thus rewrite the model in 4 as to include all observations: S ρy = X θ + η ρ, θ 31 where θ = c, λ, δ, π This means that the quasi likelihoood function of the model in 4 can be written as ln L ρ, θ, σε, σ = ln S ρ lnπ lnσ ε 1 ln Γ σ 1 η σε ρ, θγ 1 σ η ρ, θ 3 where E [η η ] = Γ σ = J T Ω ϕ σ + I N with σ = σ 1, σ, σ 3 = 1 σ ε σ = 1 σ µ, σ µα, σα σ ε Maximising this function yields the QML estimator of the model, ψ = ρ, θ, σ ε, σ 5 3 Asymptotic Properties In order to analyse the asymptotic properties of the QML estimator of the model, ψ, we employ the following assumptions: Assumption 3 The available observations are y it, x it, i = 1,, N and t = 1,, T, with T fixed and N Also, all the elements of x it are independent across i, and have 4 + ɛ moments for some ɛ > 4 We leave the endogenous case for future ongoing research 5 To facilitate the search for the maximum when using Newton-Raphson algorithms, we also provide the first and second derivatives of 3 in Appendix E 8

10 Assumption 4 The elements of the disturbance vector ε it are iid for all i and t, with E ε it =, V ar ε it = σ ε and E ε it 4+ɛ < for some ɛ > Similarly, υ iµ, υ iα are iid with E υ iµ = E υ iα =, V ar υ iµ = σ µ, V ar υ iα = σ α, Cov υ iµ, υ iα = σ µα and E υ iµ 4+ɛ <, E υ iα 4+ɛ < for some ɛ > Lastly, ε it and υ jµ, ν jα are i mutually independent, and ii independent of x ks for all i, j, k and s Assumption 5 The elements of W N, w ijn, are at most of order h 1 N, uniformly in all i and j with h N /N as N Assumption 6 The matrix S N ρ is nonsingular for all ρ Furthermore, is a compact and ρ is in the interior of Assumption 7 The sequence of matrices W N and S N ρ are uniformly bounded in both row and column sums and uniformly in ρ in These assumptions are commonly used in the spatial panel data literature In particular, Assumption 3 is standard for linear panel data models with large N and small T where Y is exogenous 6 The first part of Assumption 4 is also largerly standard in random-effects panel data models What is not that common is the part that refers to the bivariate random vector υ iµ, υ iα, which is justified by the existence of spatial spillovers in the individual effects of our model As for the last three assumptions, they are widely used in spatial econometrics models In particular, Assumption 5 can be found in eg Lee 4 and Su and Yang 15, and it is always satisfied if {h N } is a bounded sequence Assumptions 6 and 7 parallel Assumptions 3 and 5 of Yu et al 8 In particular, while Assumption 6 guarantees that Y can be expressed exclusively in terms of Z and U, Assumption 7 essentially limits the spatial correlation Assumption 7 was first employed by Kelejian and Prucha 1998, 1, but is also found in eg Lee 4 and, as previously pointed out, Yu et al 8 Notice that we initially assumed a row-normalized spatial weight matrix Assumption 1, which means that each of the rows of W N sum 1 Thus, Assumption 7 imposes further constraints on the spatial correlation 6 The case of large N and large T and the case of Y endogenous are left for future ongoing research 9

11 For the QML estimators of ψ = ρ, θ, σ ε, σ, ψ, to be consistent, we need to ensure that ψ is identifiable Conditions for the identifiability of ψ are given in Proposition 1 We also require some limits on the Hessian and the variance covariance matrix of the score vector to exist and, in the case of the Hessian, to produce a positive definite matrix This is the aim of the following assumption: Assumption 8 Let H N ψ = [ E ψ ln L N ψ ] ψ ln L N ψ ] H = lim N matrix 1 [ E H N ψ ψ ψ ln L N ψ be the Hessian matrix and G N ψ = the variance covariance matrix of the score vector and G = lim N Both 1 G N ψ exist Also, H is a positive definite Notice that this is also a rather standard assumption when proving the consistency of an estimator see eg Su and Yang 15 Theorem 3 Under assumptions 1 to 8 and the conditions of Proposition 1, ψ is globally identified and d ψ ψ N Also, if the disturbances are normally distributed, then, H 1 G H 1, d ψ ψ N, H 1 4 Empirical application In this section we provide empirical evidence on a growth-initial level equation see eg Islam 1995 and Elhorst et al 1 using the correlated random effects specification of the spatial Durbin dynamic panel model presented in this paper The principal aim of this illustrative empirical exercise is to show that i we can largely replicate the results obtained by Lee and Yu 16 using a standard spatial dynamic Durbin model our benchmark; and ii our model specification not only provides an estimate of the individual-specific effects but also of their spatial spillovers 1

12 To this end, we use the data and basic model specification of Lee and Yu 16 The dataset covers 8 OECD countries over the period 197 to 5 in time intervals of 5 years The dependent variable, Y t, is the real GDP per capita ie, effective labour in year t As for the explanatory variables, N t + 5 proxies the sum of the annual average working-age population growth over the last 5 years N t, the exogenous technical progress rate and the capital depreciation rate in the model see Ertur and Koch 7 for details; S t is the average investment share in GDP; and Y t 1 is the real GDP per capita 5 years before [Insert Table 1 about here] The first column of Table 1 reports the results obtained by Lee and Yu 16 using a weighting matrix W N defined by the geographical distance between the capital of the countries The second column provides the estimates of our model Notice that the parameter λ measures the effect of the time-lagged real GDP Y t 1 on the dependent variable, whereas ρ does that of the intensity of its contemporaneous spatial interactions W N Y t Also, the βs measure the effect of the regressors β 1 is the coefficient associated with N t + 5 and β is the coefficient associated with S t, whereas the γs measure the intensity of the spatial contagion between the OECD countries γ 1 and γ are the counterparts of β 1 and β Lastly, µ and α provide estimates of the unobserved effects and their spatial spillovers, respectively The first thing to notice is that our results largely concur with those of Lee and Yu 16 This means in both cases the coefficients of the working-age population growth rate β 1 are negative and statistically significant at standard confidence levels, while the coefficients of the savings rate β are positive and statistically significant Notice also that the intensity of the contemporaneus spatial interactions of Y t is not statistically significant In contrast, the parameter associated with the time lagged real GDP λ is positive and statistically significant It is also interesting to note that we find evidence of spatial spillovers in the weakly significant individual effects In particular, the coefficient associated with the spatial contagion of the saving rates π α1 is negative and statistically significant With the estimation of π α in hand, we can then distinguish spillovers on and from the neighbours, since while α provides an 11

13 estimate of the spillover of each countries individual effect on its first ring neighbours, W N α provides an estimate of the spillover of the first ring neighbours individual effects on each country 7 5 Conclusions In this paper we consider a correlated random effects specification of the spatial Durbin dynamic panel model that allows the identification of the invididual effects and their spatial spillovers Obtaining an estimate of the individual-specific effects net of the spatially weighted effects can be critical in certain applications, such as growth models in which a measure of the unobserved productivity of the geographical units under study can be obtained from the estimated individual effects We illustrate this point by estimating a growth-initial level equation using OECD data and providing evidence of unobservable technological interdependences ie, spatial contagion in the individual effects Following Lee and Yu 16, we derive the identification conditions of the model under quasi-maximum likelihood estimation We differ from them in that our identification conditions employ a less involved variance-covariance matrix Another major difference with respect to their work is that while they provide Monte Carlo evidence on the behaviour of their QML estimator, we derive its asymptotic properties consistency and normality under rather standard assumptions in the spatial panel data literature 7 We will provide an analysis of the cuantiles of these spillovers in future versions of this paper 1

14 Table 1: QML estimates Parameters Lee and Yu 16 Our model ρ λ β β γ γ π µ π µ π α π α 4 65 σ µ 1 σ α σ µα 1 σ ε 4 5 Note: p-value<1; p-value<5; p-value<1 13

15 A Appendix: Proofs of lemmas and propositions Proof of Lemma 1 We start by noticing that Σ λ, ρ, σ can be expressed as a sum of semipositive and positive definite matrices Lemma 144, Harville 8, ie, Σ λ, ρ, σ = E 1 + E + E 3 + E 4 + E 5 + E 6, with σ E 1 = diag 1 4 I N, 1 5 I 1 E 3 = diag Mλ, ρ, 1 5 I 1 σ 1 4 I N l T 1 A j λ, ρω ϕσ, σ3, σ4 E 5 = 1 5 I 1 Since σ 1, E 1 is semi-definite positive E = diag P λ, ρ, 1 5 I 1 1 E 4 = diag 4 σ 1I N, J T 1 Ω ϕσ, σ3, σ4 E 6 = E 5 If P λ, ρ = semi-positive definite A j λ, ρω ϕσ, σ 3, σ 4A j λ, ρ is semi-definite positive, then E Since λ 1 Aλ, ρ has full column rank by assumption and Ω ϕσ, σ 3, σ 4 is a variance-covariance matrix, P λ, ρ is a sum of semi-positive definite matrices E 3 is definite positive because Mλ, ρ = A j λ, ρa j λ, ρ = I N + is a sum of positive definite matrices given that Aλ, ρ has full rank is Aλ, ρa j λ, ρ E 4 is semi-positive definite if J T 1 Ω ϕσ, σ 3, σ 4 is semi-definite positive see collorary 149, Harville 8 Given that l T 1 I N has full column rank and Ω ϕσ, σ 3, σ 4 is a variance-covariance matrix, J T 1 Ω ϕ = l T 1 I N σi N + σ3 W N + W N + σ4w N W N l T 1 I N = l T 1 I N Ω ϕσ, σ 3, σ 4 l T 1 I N is semi-definite positive 14

16 Since E 5 is an upper-triangular matrix, its eigenvalues are the κ values such that N E 5 κi = Given that E 5 κi = 4 σ 1 κ 5 κ, the eigenvalues are κ 1 = 1 5 and κ = 1 4 σ 1 Therefore E is semi-definite positive if σ 1 By analogy with E 5, E 6 is semi-positive definite if σ 1 Proof of Proposition 1 We need to proof that Qφ Qφ < for all φ φ Since, at the true parameters, σ εφ = σ ε, we have, after some basic operations, that Qφ Qφ = 1 ln S ρs 1 1 σ ln ε φ Also, by the properties of determinants, Therefore, σ ε S ρs 1 = 1 ln Σ λ, ρ, σ [Σ ] 1 S ρs 1, so that 1 ln S ρs 1 = 1 ln S ρs ln S ρs 1 = 1 ln S ρs 1 S ρs 1 Qφ Qφ = 1 ln S ρs 1 S ρs 1 1 σ ln ε φ Thus, Qφ Qφ < if only if 1 Σ ln λ, ρ, σ [Σ ] 1 = 1 ln S ρs 1 1S Σ λ, ρ, σ ρs 1 σ εφ σε σ ε Σ 1 < S ρs 1 1S Σ λ, ρ, σ ρs 1 Σ 1 < σ εφ σ ε A1 15

17 To derive our proof of A1, we make use of two main results First, τ σ εφ τ = σε λ λ ρ ρ + 1 tr Hλ, ρ, σ σ ε [ S [ ρs 1 Σ λ, ρ, σ τ τ λ λ ρ ρ ] 1S ρs 1 Σ ] A Second, the inequality of arithmetic and geometric means implies that S ρs 1 1 tr [ ] 1S Σ λ, ρ, σ ρs 1 [ S [ ρs 1 Σ λ, ρ, σ Σ 1 ] 1S ρs 1 Σ ] A3 This means that inequality A1 holds if either i Hλ, ρ, σ is definite positive or ii Hλ, ρ, σ is semidefinite positive and inequality A3 strictly holds In particular, notice that inequality A3 strictly holds if Σ λ, ρ, σ and Σ are linearly independent Notice further that i is a sufficient condition for identification see Lee and Yu, 16: it guarantees the identification of θ but not σ since it does not guarantee that Σ λ, ρ, σ and Σ are linearly independent Lastly, notice that ii does not guarantee the identification of all the parameters in θ Thus, we first discuss conditions for Hλ, ρ, σ to be definite positive Since Σ λ, ρ, σ is definite positive Lemma 1, these sufficient identification conditions correspond to the conditions for Z to have full column rank see Harville 8, theorem 149, page 15 Next we discuss under which conditions Σ λ, ρ, σ and Σ are linearly independent These provide the necessary and sufficient conditions for the identification of all the parameters of the model θ and σ Since W S 1 = S 1 W, then Z can be expressed as Z = S 1 S Z W Z θ 16

18 Thus, we consider the following linear combination: l N l Ṇ l N c + W N H b + W N X W N X W N X π + W N X W N X T δ + W N Y 1 W N Y T 1 λ a 1 + l N l Ṇ l N 1 ρ a + I N ρ W N H d + I N ρ W N X I N ρ W N X I N ρ W N X e+ I N ρ W N X I N ρ W N X T f + I N ρ W N Y 1 I N ρ W N Y T 1 a 3 where a 1, a, and a 3 are real numbers and d = d 1, d, e = e 1, e and f = f 1, f are vectors of the appropriate dimension Also, since H = X WN X, X = X W N X, X t = X t W N X t, δ = β, γ and π = π µ, π α, after combining and reordering terms we obtain 17

19 X d 1 + W N X b 1 a 1 + d ρ d 1 + W N X b a 1 ρ d + l N l Ṇ l N c a ρ a + X X T f 1 + W N X W N X T β a 1 + f ρ f 1 + W NX W NX T γ a 1 ρ f + Y 1 Y T 1 a 3 + W N Y 1 W N Y T 1 λ a 1 ρ a 3 + X X X e 1 + W N X W N X W N X π µ a 1 + e ρ e 1 + W NX W NX W NX π α a 1 ρ e A4 Notice that the coefficients of the linear combination in A4 are zero if only if the following matrix has full rank: X W N X W N X l N X W N X W N X X W N X W N X Y 1 W N Y 1 l N X W N X W N X X T W N X T W N X T Y T 1 W N Y T 1 l N X W N X W N X = B 1 D 1 D A5 Therefore, an identification condition is that B 1 and D 1 have both full column rank see Harville 8, Lemma 4511, page 45 In particular, if A5 has full column rank the coefficients in A4 define the following system of equations: 18

20 d 1 = A6 γ a 1 ρ f = A1 b 1 a 1 + d ρ d 1 = A7 a 3 = A13 b a 1 ρ d = A8 λ a 1 ρ a 3 = A14 c a ρ a = A9 e 1 = A15 f 1 = A1 π µ a 1 + e ρ e 1 = A16 β a 1 + f ρ f 1 = A11 π α a 1 ρ e = A17 We start by noticing that, if d 1 = A6, then d = b 1 a 1 A7 and, by replacing it into A8, we obtain that a 1 = if b + ρ b 1 Notice also that if a 1 =, then d =, and, from equations A9 and A13, a = a 3 = Similarly, if f 1 = e 1 = A1 and A15, a 1 = if and only if γ + ρ β, and, π α + ρ π µ These, together with the rank conditions on B 1 and D 1, are sufficient identification conditions Proof of Colorary Next we consider the linear independence between Σ λ, ρ, σ and Σ Notice that if θ is identified, then we can assume that λ, ρ = λ, ρ Thus, A1 becomes [ Σ σ ] 1Σ 1 1 [ < tr ] 1Σ Σ σ A18 This inequality holds if the following linear combination: [ ] 1Σ Σ σ is not proportional to I Thus, we consider aσ σ + bσ = aσ 11 σ + bσ 11 aσ 1 σ, σ3, σ4 + bσ 1 aσ 1 σ, σ3, σ4 + bσ 1 aσ σ, σ3, σ4 + bσ where a and b are scalars In particular, we need to prove that the only solution to the resulting system of equations, aσ σ + bσ = [], is a = b = To this end, we start by noting 19

21 that aσ σ, σ 3, σ 4 + bσ = aσ + bσ J T 1 I N + aσ 3 + bσ 3 J T 1 W N + W N + aσ 4 + bσ 4J T 1 W N W N + a + bi 1 Then, if the matrix J T 1 I N J T 1 W N + W N J T 1 W N W N I 1 has full rank, we have that aσ + bσ = A19 aσ 3 + bσ 3 = A aσ 4 + bσ 4 = A1 a + b = A The last equation implies that a = b and, by replacing it into the other equations, we have that aσi aσi = aσi σi = for i =, 3, 4 Since σi σi, then a = and b = This means that, given that λ, ρ are identified, the identification of σ, σ3, σ4 derives from the sub-matrix J T 1 Ωϕ + I 1 To address the identification of σ1, we need to prove that, if σ1 σ1, the only solution to the system aσ 11σ 1 + bσ 11 = A3 a + bσ 1 = A4 a + bσ 1 = A5 a + bσ = A6 is a = b = Then, from A6, a + b = given that Σ By replacing a = b into equation A3 we obtain that aσ 1 + bσ 1I N + a + bp + a + bm = aσ 1 σ 1I N = Thus, if σ 1 σ 1, then a = and b =

22 B Stochastic approximation of the model for QML estimation From equation 4 we have that I N ρ W N Y t = l N c + λ Y t 1 + X t δ + Xπ + η t B1 or Y t = I N ρ W N 1 l N c + λ Y t 1 + X t δ + Xπ + η t B Then, we repeatedly lag equation B by one period For m, we obtain Y t m = I N ρ W N 1 l N c + λ Y t m+1 + X t m δ + Xπ + η t m B3 Therefore, by continuous substitution of Y t 1, Y t,, Y t into B1, we get I N ρ W N Y t = A j ι N c +λ A Y t m + A j X t j δ + A j Xπ + A j η t j B4 where A = λ I N ρ W N 1 For t = 1, I N ρ W N Y 1 = notice that A j η 1 j = η 1 + A j l N c +λ A Y 1 m + A j η 1 j = ϕ + A j X 1 j δ + A j ϕ + ε 1 + A j Xπ + A j ε 1 j A j η 1 j B5 B6 1

23 Therefore, I N ρ W N Y 1 = ϕ + A j l N c + λ A Y 1 m + A j ϕ + ε 1 + A j ε 1 j A j X 1 j δ + A j Xπ + B7 Bhargava and Sargan 1983 suggest predicting λ A Y 1 m + A j X 1 j δ, by all non-stochastic variables and an additional error term In maths, H = X WN X, X = X 1 X X T and b = b 1, b Notice that b 1 is the parameter vector associated to X and b is the parameter vector associated to W N X As for the additional error term, ζ, we assume that ζ, σζ I N and uncorrelated with ϕ+ A j ϕ and ε 1 + A j ε 1 j then, I N ρ W N Y 1 = Let U 1 = ζ + ϕ + A j l N c + Hb+ A j ϕ + ε 1 + A j Xπ +ζ + ϕ + A j ε 1 j, therefore, A j ϕ + ε 1 + A j ε 1 j B8 I N ρ W N Y 1 = A j l N c + Hb + A j Xπ + U 1 B9 C Variance-covariance matrix of the stochastic approximation of the model for QML estimation To derive the variance-matrix covariance of the disturbances, Σ = E [U U ], we proceed by first deriving E [U 1 U 1], then E [U 1 U t] for t, and finally E [U t U t] for t

24 Firstly, since the elements of U 1 are independent, E [U 1 U 1] = σ ζ I N + V ar ϕ + A j ϕ + V ar ε 1 + A j ε 1 j C1 Also, V ar ϕ + A j ϕ = Ω ϕ + = Ω ϕ A j + A j Ω ϕ + A j Ω ϕ A j = P A j Ω ϕ A j and so that V ar ε 1 + A j ε 1 j E [U 1 U 1] = σ ζ I N + = σε I + σε A j A j = σε A j Ω ϕ A j + σε A j A j = σ ε M, A j A j C = σ ζ I N + P + σ ε M C3 Secondly, since U t = η t = Ω + ε t for t, we get And, finally, E [U 1 U t] = Ω ϕ + A j Ω ϕ = E [U t U t] = Ω ϕ + σ ε I N A j Ω ϕ C4 C5 3

25 D Derivation of the concentrated likelihood function First we concentrate the maximum likelihood function in 8 on σ ε Let us define Qφ, σ ε = 1 E [ ln Lφ, σ ε ] and denote by σ εφ the solution to the first order condition: Thus, σεφ + 1 σεφ E 4 Qφ, σ σεφ ε φ = D1 [ [ 1U U δ, λ, ρ Σ λ, ρ, σ ] δ, ] λ, ρ = D σ εφ = 1 [ [ ] 1U ] E U δ, λ, ρ Σ λ, ρ, σ δ, λ, ρ D3 Notice that U δ, λ, ρ = S ρy Z θ can be expressed in terms of the true parameters δ, λ, ρ as U δ, λ, ρ = S ρs 1 S Y Z θ + Z θ Z θ Also, since we have that S Y = Z θ + U, then U δ, λ, ρ = S ρs 1 Z θ + U Z θ + Z θ Z θ = Z θ θ + S ρ S S 1 Z θ + S ρs 1 U D4 D5 Lastly, given that S ρ S = ρ ρw and θ, ρ = τ, λ, ρ, then: U δ, λ, ρ = Z θ θ + ρ ρw S 1 Z θ + S ρs 1 U = Z W S 1 Z θ θ θ + S ρs 1 U ρ ρ τ = Z τ λ λ + S ρs 1 U ρ ρ D6 4

26 Plugging D6 into D3, and making use of Hλ, ρ, σ = 1 Z [ Σ λ, ρ, σ ] 1 Z and [ E = σ ε tr S ρs 1 U [ ] 1 Σ λ, ρ, σ S ρs 1 Σ λ, ρ, σ S ρs 1 U 1S ρs 1 Σ ] we obtain τ τ τ τ σ ε φ = λ λ Hλ, ρ, σ λ λ ρ ρ ρ ρ + σ ε tr S [ ] ρs 1 1S Σ λ, ρ, σ ρs 1 Σ D7 Finally, we take this expression to Q φ, σ ε to obtain the concentrated likelihood function: Qφ = 1 ln S ρ ln π ln σ εφ 1 Σ ln λ, ρ, σ 1 D8 To conclude, we plug equation D3 into equation D8 to obtain expression 1 E Gradient and Hessian of the QML function E1 Gradient The first derivative of the log-likehood function in 3 is ln L ρ, θ, σε, σ ln L = ρ, ln L θ ln L,, σε ln L σ 5

27 with ln L ρ ln L θ ln L σε ln L σκ = tr S 1 ρw + 1 Y σε W Γ 1 σ η ρ, θ = 1 X σε Γ 1 σ η ρ, θ = σ ε + 1 η σε 4 ρ, θγ 1 σ η ρ, θ = 1 tr Γ 1 1 σ B κ + η σε ρ, θγ 1 σ B κ Γ 1 σ η ρ, θ and where, to obtain the last derivative, notice that that for κ = 1,, 3 we have that Γ σ and B 3 = J T W N W N σ κ ln L ln L =, σ σ1 ln L, σ ln L, so σ3 = B κ, with B 1 = J T I N, B = J T W N +W N E Hessian matrix The second derivative of the log-likehood function in 3 is H N = ln L ρ ρ ln L ρ θ ln L ρ σ ε ln L ln L θ θ θ σε ln L σε σ ε ln L ρ σ ln L θ σ ln L σ ε σ ln L σ σ Next we provide detailed results for each row of the Hessian matrix 6

28 First row of the Hessian matrix ln L ρ ρ ln L ρ θ ln L ρ σε ln L ρ σκ = tr S 1 ρw Y W Γ 1 σ W Y = 1 X σε Γ 1 σ W Y = 1 Y σε 4 W Γ 1 σ η ρ, θ = 1 Y σε W Γ 1 σ B κ Γ 1 σ η ρ, θ, κ = 1,, 3 where κ = 1,, 3 and to obtain the last derivative we use ln L ln L ρ σ =, ln L, ln L ρ σ1 ρ σ ρ σ3 Second row of the Hessian matrix ln L θ θ ln L θ σε ln L θ σκ = 1 X σε Γ 1 σ X = 1 X σε 4 Γ 1 σ η ρ, θ = 1 X σε Γ 1 σ B i Γ 1 σ η ρ, θ where κ = 1,, 3 and to obtain the last derivative we use ln L ln L θ σ =, ln L, ln L θ σ1 θ σ θ σ3 Thind row of the Hessian matrix ln L σε σ ε ln L σε σ κ = σ 4 ε 1 η σε 6 ρ, θγ 1 σ η ρ, θ = 1 η σε 4 ρ, θγ 1 σ η ρ, θ where κ = 1,, 3 and to obtain the last derivative we use ln L ln L σε σ =, ln L, ln L σε σ 1 σε σ σε σ 3 Last row of the Hessian matrix ln L σ κ σ κ = 1 [ Γ tr ] 1 σ B κ 1 η σε ρ, θ Γ 1 σ B κ Γ 1 σ η ρ, θ 7

29 while, for κ ϱ, ln L σ κ σ ϱ = 1 tr Γ 1 σ B κ Γ 1 σ B ϱ 1 η σε ρ, θγ 1 σ [ ] B κ Γ 1 σ B ϱ + B ϱ Γ 1 σ B κ Γ 1 σ η ρ, θ 8

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