Systems of simultaneous equations

Transcription

1 Systems of simultaneous equations Jean-Marc Robin January 29

2 References Je rey M. Wooldridge, Econometric Analysis of Cross Section and Panel Data, the MI Press, October 2. ISBN William Green, Econometric Analysis, Prentice Hall, 6th Edition, 28. Russell Davidson and James G. MacKinnon, Estimation and Inference in Econometrics, New York, Oxford University Press, 993. ISBN (For technical details.)

3 Introduction Examples of simultaneous equation models A Vector Autoregressive Model (VAR) of money (M) and GDP (Y): mt = α y t + β m t + γ y t + ε t (M) y t = α 2 m t + β 2 m t + γ 2 y t + ε 2t (Y) () A simple macroeconomic model: 8 < C t = α + βq t + γr t + u t I t = δ + ζr t + v t (2) : Q t = I t + C t + G t A supply-demand equilibrium: yt = a bp t + u t (D) y t = α + βp t + v t (S) (3)

4 Introduction Variable types Dependent vs explanatory: depending on which side of the equations they are Observed vs unobserved: residuals are zero-mean variables which are not observed but have an economic interpretation (consumption shock: taste change, supply shock: innovation in the production process). Endogenous vs exogenous: dependent variables are endogenous but explanatory variables can be also endogenous.

5 Introduction Exogeneity akes the form of a conditional independence restriction on residuals: Model (): past values of dependent variables are exogenous: E (εt jm t, y t ) = E (ε 2t jm t, y t ) = Model (2): interest rate r t and public spendings G t are exogenous: ut E r v t, G t = t Model (3): both y t and p t are endogenous in general, as both depend on demand and supply shocks u t and v t : 8 aβ + bα >< y t = (3) ) b + β + βu t + bv t b + β >: p t = a α b + β + u t v t b + β

6 Introduction Reduced form Puts all endogenous variables to the left: Model (): By substitution, m t = α y t + β m t + γ y t + ε t = α α 2 m t + (α β 2 + β )m t + (α γ 2 + γ )y t + ε t + α ε 2t y t = α 2 m t + β 2 m t + γ 2 y t + ε 2t = α 2 α y t + (α 2 β + β 2 )m t + (α 2 γ + γ 2 )y t + α 2 ε t + ε 2t Hence the reduced form: ( mt = α β 2 +β α α 2 m t + α γ 2 +γ α α 2 y t + ε t +α ε 2t α α 2 y t = α 2 β +β 2 α α α 2 m 2 γ +γ 2 t α α 2 y t + α 2ε t +ε 2t α α 2

7 Model (2): Investment equation is already a reduced form: I t = δ + ζr t + v t but not consumption: C t = α + βq t + γr t + u t = α + β(i t + C t + G t ) + γr t + u t = α + β(δ + ζr t + v t ) + βc t + βg t + γr t + u t ) C t = α + βδ β + βζ + γ β r t + β β G t + u t + βv t β and GDP equations: Q t = I t + C t + G t = δ + ζr t + v t + α + βδ β + βζ + γ β r t + β β G t + u t + βv t β + G t = α + δ β + ζ + γ β r t + β G t + u t + βv t β

8 Model (3): 8 >< >: aβ + bα y t = b + β + βu t + bv t b + β p t = a α b + β + u t v t b + β

9 Introduction Estimation he coe cients of each exogenous variable in the reduced form can be consistently estimated by OLS (the regressors are by de nition uncorrelated with the error terms and they should be linearly independent). hese coe cients are functions of the structural parameters. he structural parameters can be consistently estimated if the relationship between the reduced form parameters and the structural parameters can be inverted. In this case, we shall say that the structural model is identi ed.

10 Seemingly Unrelated Regressions (SUR) De nition N equations indexed by i and observations indexed by t: y it = x it β i + u it, where y it 2 R, x it 2 R K i and β i 2 R K i, with E(u it jx t,..., x Nt ) = Matrix form: with y i = X i β i + u i y i = y i.. y i C A, u i = u i.. u i C A, X i = K i x i.. x i C A

11 Seemingly Unrelated Regressions (SUR) OLS equation by equation Consider the ith equation: y it = x it β i + u it or y i = X i β i + u i OLS estimator, bβ OLS i = (X i X i ) X i y i = = β +! x it xit t=! x it xit t= x it u it t= x it y it t= exists if X i X i non singular or invertible (i.e. regressors are linearly independent. OLS estimator is consistent if u it is uncorrelated with x it for all t : plim x it u it = E(x it u it ) = t=

12 Seemingly Unrelated Regressions (SUR) Correlated residuals In general one will not rule out the possibility that error terms are contemporaneously correlated (think of how reduced forms s residuals mix structural shocks). De ne σ ij = E(u it u jt jx i,..., x Nt ) and Σ = (σ ij ) Residuals could also be autocorrelated. However, we assume iid observations (across t-dimension).

13 Seemingly Unrelated Regressions (SUR) GLS on the pooled system Write y t N X t N (K +...+K N ) hen, for all t, = = y t u t C B. A, u t C. A, N y Nt u Nt x t x2t C.. A, β = xnt β.. β N C A y t = X t β + u t

14 Seemingly Unrelated Regressions (SUR) GLS on the pooled system Residuals orthogonalisation: Let W be a (triangular) matrix such that Σ = W W (Cholesky decomposition). hen, W y t = W X t β + W u t (4) has orthogonal residuals: Var(W u t ) = W (Var u t )W = WW W W = I N. GLS is OLS on model (4): bβ GLS (Σ) = = = β +! X t W W X t t=! X t Σ X t t=! X t Σ X t t= X t W W y t t= X t Σ y t t= X t Σ u t t=

15 Consistency requires that Seemingly Unrelated Regressions (SUR) Asymptotic properties plim! t= X t Σ X t be non singular and plim! t= X t Σ u t = E X t Σ u t = (true if E(u t jx t ) = as E X t Σ u t = E X t Σ E(u t jx t ) ) Asymptotic normality: p b β GLS p β N, Var as bβ GLS where means converges in distribution and where Var as p bβ GLS = plim! with Var(u t jx t ) = Σ.! X t Σ X t t= Feasible GLS (FGLS) replace Σ by a consistent estimator like the empirical variance of OLS residuals: bσ = (bσ ij ) with bσ ij = t y it xit b β OLS i y jt xjt b β OLS j

16 Seemingly Unrelated Regressions (SUR) Kronecker product Let A = (a ij ) and B be matrices. hen A B = a ij B = a B a 2 B a L B a 2 B a 22 B a 2L B a M B a M 2 B a ML B C A. Properties: (A B) = (A B) = A B A B (A B) (C D) = (AC ) (BD) (if A and B are non singular and if A and C and B and D are conformable).

17 Seemingly Unrelated Regressions (SUR) GLS = OLS when same regressors in each equation Suppose that x it = x t for all i and X i = (x t ) = X. hen, xt xt X t = xt and, for example, C A = I N xt X t Σ = (I N x t ) (Σ ) = (I N x t )(Σ ) = Σ x t

18 It thus follows that:! bβ GLS = X t Σ X t X t Σ y t t= t= 2! 3 = 4I N x t xt x t 5 y t t= t= t= x t xt t= x t y t = B. A t= x t xt t= x t y Nt = bβ OLS

19 Suppose u t N (, Σ) Density of u t is (2π) N /2 jσj Seemingly Unrelated Regressions (SUR) Maximum Likelihood and normal errors /2 exp 2 u t Σ u t where jσj is the determinant of Σ. Log likelihood: `(y jx, β, Σ) = N 2 ln(2π) 2 ln jσj 2 (y t X t β) Σ (y t X t β) t= Di erentiating wrt to Σ (see e.g. Davidson and McKinnon) yields bσ(β) = t= (y t X t β)(y t X t β) which is the usual estimating formula. A consistent estimator is obtained by replacing β by bβ OLS.

20 Concentrated likelihood: `c (y jx,β) = N 2 [ln(2π) + ] 2 ln bσ(β) as t= = r (y t X t β) bσ(β) (y t X t β) " # bσ(β) (y t X t β)(y t X t β) = r(i N ) = N t= he trace operator (sum of diagonal entries of a matrix) is linear and such that r(ab) = r(ba) if matrices comute. It is usually a small trick that works well to identify a matrix to its trace.

21 Di erentiating ln Σ(β) b wrt β yields the FOC: or X t Σ(bβ) b (y t X t bβ) = t= bβ =! X t Σ(bβ) b X t t= X t Σ(bβ) b y t t= Iterating FGLS thus yields the ML estimator. Using bσ(bβ OLS ) yields an asymptotically equivalent estimator. FGLS estimator is therefore e cient.

22 Simultaneous Equations Model Structural form Add endogenous regressors in each equation i =,..., N: y i = Y i γ i + X i β i + u i (FC) = Z i α i + u i, where Y i is a selection of N i columns of Y = (y,..., y N ) and N γi Z i = (Y i, X i ), α i = x (N i +K i ) β i Notation. Let Y = (y,..., y N ), U = (u,..., u N ) and let X be N N K the matrix of all exogenous variables..

23 Simultaneous Equations Models Example: model (2) Let y = (C t ), y 2 = (I t ), y 3 = (Q t ), Y = (C t, I t, Q t ), X = (, r t, G t ). 3 3 First equation: C t = α + βq t + γr t + u t, (C t, I t, Q t ) Second equation: I t = δ + ζr t + v t, (C t, I t, Q t ) hird equation: Q t = I t + C t + G t, (C t, I t, Q t ) Hence, Y Γ = XB + U, with u v B U C A, Γ A = (, r t, G t ) A = (, r t, G t ) δ ζ A = (, r t, G t ) A, α γ A + v A α δ γ ζ A + u t A.

24 Simultaneous Equations Models Reduced form In general, there exist Γ and B containing the parameters γ i and β i N N K N (together with some ones and zeros) such that: Y Γ (y Y γ,..., y N Y N γ N ) = (X β + u,..., X N β N + u N ) XB + U his form is also called the structural form. A reduced form exists if only Γ is non singular, in which case: Y Γ = XB + U ) Y = XBΓ + UΓ his is the constrained reduced form. he unconstrained reduced form is Y = X Π + V Identi cation requires that 8Π, 9!(Γ, B) : Π = B Γ. K N K N N N

25 Simultaneous Equations Models wo Stage Least Squares equation by equation Equation i has N i endogenous regressors and K i exogenous ones. he K K i excluded exogenous variables present in all other equations can be used to instrument Y i. Identi cation requires K K i N i (order condition). Necessary but not su cient condition for identi cation. If K K i < N i the equation is under-identi ed, if K K i > N i the equation may be over-identi ed. Let by i = P X Y i, where P X = X (X X ) X, be the orthogonal projection of Y i on the vector space spanned by all exogenous variables. If the order condition is satis ed and if ( by i, X i ) has linearly independent columns (rank condition), then 2SLS provide a consistent estimator of α i. For example, use G t to instrument Q t in the consumption equation of model (2).

26 Instrumental Variables Quick reminder - 2SLS 2SLS = OLS regression of y i on bz i = P X Z i = Y b i, X i, where by i = P X Y i is the projection of the columns of Y i on instruments (prediction from instrumental regression). Root-N consistent and asymptotically normal: p bα 2SLS i α i N, σ ii Zi P X Z i Estimate σ ii as mean square error: bσ ii = (y i Z i bα 2SLS i ) (y i Z i bα 2SLS i ) Caution: using an OLS software to implement 2SLS gets you the wrong estimator of σ ii as mean square error of second-step regression: eσ ii = (y i bz i bα 2SLS i ) (y i bz i bα 2SLS i )

27 Exogeneity test: use the control function approach, i.e. regress y i on Z i and residuals: V i = Y i by i, by OLS. est the nullity of the coe cients of V i using Wald or Fisher tests e.g. under the null H : Y i is exogenous wrt to u i, the Fisher statistic SSR SSR df SSR df df χ 2 df (N i ), where df df when is large. Note that under H the variances computed by OLS softwares are consistent. est of overidentifying restrictions: regress residuals bu i = y i Z i bα 2SLS i on X or on X i, the columns of X which are not columns of X i. he statistic R 2 χ 2 (K K i N i ) under the null that variables X are exogenous.

28 Simultaneous Equations Models hree Stage Least Squares 2SLS not e cient. Additional e ciency can be obtained by pooling equations. 3SLS is the GLS estimator on the system of equations: y i = bz i α i + ε i, i =,..., N, treated as a SUR model with Eε it ε jt = σ ij. If Σ = (σ ij ) is singular (as in model (2)) this means that a linear combination of the equations is an accounting equation that allows to substitute one dependent/endogenous variable out of the system. A feasible estimator can be obtained by replacing σ ij by the consistent estimator: bσ ij = (y i Z i bα 2SLS i ) (y j Z j bα 2SLS j ). 3SLS is required to test parameter restrictions across equations.

29 Simultaneous Equations Models Maximum Likelihood with Normal errors Full Information ML (FIML) is similar to ML for SURs: one rst estimates the reduced form model, which is a SUR model, and impose the structural constraints in a second step. Note that these constraints are nonlinear. If FIML is done in one single step on the structural model: Y Γ = XB + U then there is a Jacobian term of the form ln jγj which is non di erentiable at all Γ such that Γ has less than full rank. hese values delimit regions where di erent local maxima can be found (see Davison and MacKinnon). 3SLS and FIML are asymptotically equivalent. Hence 3SLS is e cient and FIML is consistent even if residuals are not normal.

30 Limited Information ML (LIML) is a way of estimating the parameters of each equation separately: 8 < : y i = Y i γ i + X i β i + u i N i K i Y i = X K Π i + V i (regression of interest) (intrumental regressions) where the rows of (u i, V i ) are iid and normally distributed. 2SLS and LIML are asymptotically equivalent but LIML has better nite sample properties. In particular with many (K >> N i ) or weak instruments (low R-square of the instrumental regressions) (Stock, Wright, Yogo, JBES, 22).