Econometrics - Lecture 6 GMM-Estimator and Econometric Models Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 2 Hackl, Econometrics, Lecture 6 3 Assumption (A7): E{xt εt} = 0 for all t nLinear model for yt q yt = xt'β + εt, t = 1, …, T (or y = Xβ + ε) qgiven observations xtk, k =1, …, K, of the regressor variables, error term εt n(A7) E{xt εt} = 0 for all t, i.e., no contemporary correlation nGuaranties unbiasedness and consistency of the OLS estimator nIn reality, (A7) not always fulfilled nE{xt εt} ≠ 0: biased, inconsistent OLS estimator nExamples of situations with E{xt εt} ≠ 0 qRegressors with measurement errors qRegression on the lagged dependent variable with autocorrelated error terms qEndogeneity of regressors qSimultaneity n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 4 Instrumental Variables nThe model is n yt = xt‘β + εt n with V{εi} = σε² and n E{εt xt} ≠ 0 nInstrumental variables zt 1.Exogenous: E{εt zt } = 0: zt uncorrelated with error term 2.Relevant: Cov{xt , zt } ≠ 0: zt correlated with endogenous regressors Jan 7, 2011 IV Estimator nBased on the moment conditions n E{εi zi} = E{(yi – xi‘β) zi} = 0 nSolution of corresponding sample moment conditions n 1/N Σi(yi – xi‘β) zi = 0 nIV estimator based on the instruments zt n n nIdentification requires that the KxK matrix Σtztxt’ = Z’X is finite and invertible; instruments zt are relevant when this is fulfilled n Jan 7, 2011 Hackl, Econometrics, Lecture 6 5 Hackl, Econometrics, Lecture 6 6 IV Estimator: Properties nIV estimator is nConsistent n(Asymptotic) covariance matrix n n nEstimated covariance matrix: σ² is substituted by n n nThe asymptotic distribution of IV estimators, given IID(0, σε²) error terms, leads to the approximate distribution n n with the estimated covariance matrix n Jan 7, 2011 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 7 The General Case nR: number of instrument variables and of components of zi nThe R moment equations are n n 1.R = K: one unique solution, the IV estimator; identified model n 2.R < K: Z’X has not full rank, is not invertible; infinite many solutions fulfill moment equations, but no consistent estimator; under-identified or not identified model 3.R > K: more instruments than necessary for identification; over-identified model; a unique solution cannot be obtained such that all R sample moment conditions are fulfilled; strategy for choosing the estimator among all possible estimators Jan 7, 2011 Hackl, Econometrics, Lecture 6 8 The GIV Estimator nFor R > K, in general, a unique solution of all R sample moment conditions cannot be obtained; instead: nGeneralized instrumental variable (GIV) estimator n n uses best approximations for columns of X nThe GIV estimator can be written as n nGIV estimator is also called “two stage least squares” (TSLS) estimator: 1.First step: regress each column of X on Z 2.Second step: regress y on predictions of X n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 9 GIV Estimator and Properties nGIV estimator is consistent nThe asymptotic distribution of the GIV estimator, given IID(0,σε²) error terms εt, leads to the approximate distribution n nThe (asymptotic) covariance matrix of is given by n n nEstimated covariance matrix: σ² is substituted by n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 10 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 11 The Generalized IV Estimator nFor R > K, in general, no unique solution of all R sample moment conditions can be obtained; instead: nThe weighted quadratic form in the sample moments n n with a RxR positive definite weighting matrix WN is minimized nGives the generalized IV estimator n nFor each positive definite weighting matrix WN, the generalized IV estimator is consistent nGIV estimator: special case with WNopt (see below) nFor R = K, the matrix Z’X is square and invertible; the IV estimator is (Z’X)-1Z’y for any WN n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 12 Most Efficient IV Estimator nWeighting matrix WN nDifferent weighting matrices result in different consistent generalized IV estimators with different covariance matrices nOptimal weighting matrix: n WNopt = [1/N(Z’Z)]-1 qCorresponds to the most efficient IV estimator n n with qCoincides with the GIV (or TSLS) estimator n n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 13 Consistency of the Generalized IV Estimator nWith a RxR positive definite weighting matrix WN, minimizing the weighted quadratic form in the sample moments n n results in a consistent estimator for β nSample moments converge asymptotically to the corresponding population moments nThe population moments are zero for the true parameters nMinimizing the quadratic loss function in the sample moments results in solutions which asymptotically coincide with the true parameters nThis idea is basis of the generalized method of moments estimator n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 14 Generalized Method of Moments (GMM) Estimator nGMM generalizes the IV estimation concept nEstimates of model parameters are derived from moment conditions which are not necessarily linear nNumber of moment conditions at least as large as number of unknown parameters n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 15 Hackl, Econometrics, Lecture 6 16 Generalized Method of Moments (GMM) Estimator nThe model is characterized by R moment conditions n E{f(wi, zi, θ)} = 0 n[generalization of E{(yi – xi‘β) zi} = 0] nf(.): R-vector function nwi: vector of observable variables, exogenous or endogenous nzi: vector of instrumental variables nθ: K-vector of unknown parameters nSample moment conditions n n 1.R = K: unique solution for θ; if f(.) is nonlinear in θ, numerical solution might be derived 2.R < K: parameters not identified n Jan 7, 2011 Hackl, Econometrics, Lecture 6 17 GMM Estimator 3.R > K: minimization, wrt θ, of the objective function, i.e., the quadratic form n QN(θ) = gN(θ)‘ WN gN(θ) n WN: symmetric, positive definite weighting matrix n GMM estimator corresponds to the optimal weighting matrix n n the inverse of the covariance matrix of the sample moments, n and is the most efficient estimator n For nonlinear f(.) nNumerical optimization algorithms nWN depends on θ; iterative optimization Jan 7, 2011 Hackl, Econometrics, Lecture 6 18 Example: The Linear Model nModel: yi = xi‘β + εi with E{εi xi} = 0 and V{εi} = σε² nMoment or orthogonality conditions: n E{εt xt} = E{(yt - xt‘β)xt} = 0 n f(.) = (yi - xi‘β)xi, θ = β, instrument variables: xi; moment conditions are exogeneity conditions for xi nSample moment conditions: n 1/N Σi (yi - xi ‘b) xi = 1/N Σi ei xi = gN(b) = 0 nWith W = I, QN(β) = [1/N Σi ei xi ]2 nOLS and GMM estimators coincide, but for the estimators qOLS: residual sum of squares SN(b) = 1/N Σi ei2 has its minimum qGMM: QN(b) = 0 n n Jan 7, 2011 Hackl, Econometrics, Lecture 6 19 Linear Model with E{εt xt} ≠ 0 nModel yi = xi‘β + εi with V{εi} = σε², E{εi xi} ≠ 0 and R instrumental variables zi nMoment conditions: n E{εi zi} = E{(yi - xi‘β)zi} = 0 nSample moment conditions: n 1/N Σi (yi - xi‘b) zi = gN(b) = 0 nIdentified case (R = K): the single solution is the IV estimator n bIV = (Z’X)-1 Z’y nOptimal weighting matrix WNopt = (E{εi2zizi‘}) -1 is estimated by n n nGeneralizes the covariance matrix of the GIV estimator to White‘s heteroskedasticity-consistent covariance matrix estimator (HCCME) n n Jan 7, 2011 Example: Labor Demand nVerbeek’s data set “labour2”: Sample of 569 Belgian companies (data from 1996) nVariables qlabour: total employment (number of employees) qcapital: total fixed assets qwage: total wage costs per employee (in 1000 EUR) qoutput: value added (in million EUR) nLabour demand function n labour = b1 + b2*output + b3*capital Jan 7, 2011 Hackl, Econometrics, Lecture 6 20 Labor Demand Function: OLS Estimation Hackl, Econometrics, Lecture 6 21 In logarithmic transforms: Output from GRETL Dependent variable : l_LABOR Heteroskedastic-robust standard errors, variant HC0, coefficient std. error t-ratio p-value ------------------------------------------------------------- const 3,01483 0,0566474 53,22 1,81e-222 *** l_ OUTPUT 0,878061 0,0512008 17,15 2,12e-053 *** l_CAPITAL 0,003699 0,0429567 0,08610 0,9314 Mean dependent var 4,488665 S.D. dependent var 1,171166 Sum squared resid 158,8931 S.E. of regression 0,529839 R- squared 0,796052 Adjusted R-squared 0,795331 F(2, 129) 768,7963 P-value (F) 4,5e-162 Log-likelihood -444,4539 Akaike criterion 894,9078 Schwarz criterion 907,9395 Hannan-Quinn 899,9928 Jan 7, 2011 Specification of GMM Estimation Hackl, Econometrics, Lecture 6 22 GRETL: Specification of function and orthogonality conditions for labour demand model # initializations go here matrix X = {const , l_OUTPUT, l_CAPITAL} series e = 0 scalar b1 = 0 scalar b2 = 0 scalar b3 = 0 matrix V = I(3) Gmm e = l_LABOR - b1*const – b2*l_OUTPUT – b3*l_CAPITAL orthog e; X weights V params b1 b2 b3 end gmm Jan 7, 2011 Labor Demand Function: GMM Estimation Hackl, Econometrics, Lecture 6 23 In logarithmic transforms: Output from GRETL Using numerical derivatives Tolerance = 1,81899e-012 Function evaluations: 44 Evaluations of gradient: 8 Model 8: 1-step GMM, using observations 1-569 e = l_LABOR - b1*const - b2*l_OUTPUT - b3*l_CAPITAL estimate std. error t-ratio p-value -------------------------------------------------------------------------- b1 3,01483 0,0566474 53,22 0,0000 *** b2 0,878061 0,0512008 17,15 6,36e-066 *** b3 0,00369851 0,0429567 0,08610 0,9314 GMM criterion: Q = 1,1394e-031 (TQ = 6,48321e-029) Jan 7, 2011 Hackl, Econometrics, Lecture 6 24 Linear Model: MM Estimator nModel n yi = xi‘β + εi n with V{εi} = σε² and E{εi xi} ≠ 0 and R instrumental variables zi nOver-identified case (R > K): GMM estimator from n minβ QN(β)= minβ gN(β)’WN gN(β) nFor WN = I, the first order conditions are n n n method of moments estimator n bMM = [(X’Z)(Z’X)]-1 (X’Z)Z’y n bMM coincides with the IV estimator if R = K n n n Jan 7, 2011 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 25 Hackl, Econometrics, Lecture 6 26 GMM Estimator nModel with R moment conditions n E{f(wi, zi, θ)} = 0 nSample moment conditions n nOver-identified case (R > K): GMM estimator from n minθ QN(θ)= minθ gN(θ)’WN gN(θ) n WN: symmetric, positive definite weighting matrix nThe GMM estimator is consistent for any choice of WN nOptimal weighting matrix n n the inverse of the covariance matrix of the sample moments, gives the most efficient estimator n For nonlinear f(.) nNumerical optimization algorithms nWN depends of θ; iterative optimization Jan 7, 2011 Hackl, Econometrics, Lecture 6 27 GMM Estimator: Properties nUnder weak regularity conditions, the GMM estimator is nconsistent (for any W) nmost efficient if W = nasymptotically normal: n where V = D Wopt D’ with the KxR matrix of derivatives n n nThe covariance matrix V-1 can be estimated by substituting in D and Wopt the population moments by sample equivalents evaluated at the GMM estimates n Jan 7, 2011 Hackl, Econometrics, Lecture 6 28 GMM Estimator: Calculation 1.One-step GMM estimator: Choose a positive definite W, e.g., W = I, optimization gives (consistent, but not efficient) 2.Two-step GMM estimator: use the one-step estimator to estimate V = D WNopt D‘, repeat optimization with W = V-1; this gives 3.Iterated GMM estimator: Repeat step 2 until convergence nIf R = K, the GMM estimator is the same for any W, only step 1 is needed; the objective function QN(θ) is zero at the minimum nIf R > K, step 2 is needed to achieve efficiency Jan 7, 2011 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 29 Hackl, Econometrics, Lecture 6 30 Klein‘s Model 1 n Ct = a1 + a2Pt + a3Pt-1 + a4(Wtp+ Wtg) + et1 (consumption) n It = b1 + b2Pt + b3Pt-1 + b4Kt-1 + et2 (investments) n Wtp = g1 + g2Xt + g3Xt-1 + g4t + et3 (private wages and salaries) n Xt = Ct + It + Gt n Kt = It + Kt-1 n Pt = Xt – Wtp – Tt nC (consumption), P (profits), Wp (private wages and salaries), Wg (public wages and salaries), I (investments), K-1 (capital stock, lagged), X (production), G (governmental expenditures without wages and salaries), T (taxes) and t [time (trend)] nEndogenous: C, I, Wp, X, P, K; exogeneous: 1, Wg, G, T, t, P-1, K-1, X-1 Jan 7, 2011 EViews: n=100; series y1 = nrnd; y2 = y2(-1)+y1 Early Econometric Models nKlein‘s Model nAims: qto forecast the development of business fluctuations and qto study the effects of government economic-political policy nSuccessful forecasts of qeconomic upturn rather than a depression after World War II qmild recession at the end of the Korean War n Jan 7, 2011 Hackl, Econometrics, Lecture 6 31 Model year eq‘s Tinbergen 1936 24 Klein 1950 6 Klein & Goldberger 1955 20 Brookings 1965 160 Brookings Mark II 1972 ~200 Hackl, Econometrics, Lecture 6 32 Econometric Models nBasis: the multiple linear regression model nAdaptations of the model qDynamic models qSystems of regression models qTime series models nFurther developments qModels for panel data qModels for spatial data qModels for limited dependent variables q Jan 7, 2011 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 33 Hackl, Econometrics, Lecture 6 34 Dynamic Models: Examples nDemand model: describes the quantity Q demanded of a product as a function of its price P and consumers’ income Y n(a) Current price and current income to determine the demand (static model): n Qt = β1 + β2Pt + β3Yt + et n(b) Current price and income of the previous period determine the demand (dynamic model): n Qt = β1 + β2Pt + β3Yt-1 + et n(c) Current demand and prices of the previous period determine the demand (dynamic autoregressive model): n Qt = β1 + β2Pt + β3Qt-1 + et Jan 7, 2011 Hackl, Econometrics, Lecture 6 35 Dynamic of Processes nStatic processes: independent variables have a direct effect, the adjustment of the dependent variable on the realized values of the independent variables is completed within the current period, the process seems always to be in equilibrium nStatic models may be unsuitable: n(a) Some activities are determined by the past, such as: energy consumption depends on past investments into energy-consuming systems and equipment n(b) Actors of the economic processes often respond with delay, e.g., due to the duration of decision-making and procurement processes n(c) Expectations: e.g., consumption depends not only on current income but also on income expectations in future; modeling of income expectation based on past income development Jan 7, 2011 Hackl, Econometrics, Lecture 6 36 Elements of Dynamic Models 1.Lag-structures, distributed lags: describe the delayed effects of one or more regressors on the dependent variable; e.g., the lag-structure of order s or DL(s) model (DL: distributed lag) n Yt = a + Ssi=0βiXt-i + et 2.Geometric lag-structure, Koyck’s model: infinite lag-structure with bi = l0li 3.ADL-model: autoregressive model with lag-structure, e.g., the ADL(1,1)-model n Yt = a + jYt-1 + β0Xt + β1Xt-1 + et 4.Error-correction model n DYt = - (1-j)(Yt-1 – m0 – m1Xt-1) + β0D Xt + et n obtained from the ADL(1,1)-model with m0 = a/(1-j) und m1 = (b0+b1)/(1-j) Jan 7, 2011 Hackl, Econometrics, Lecture 6 37 The Koyck Transformation nTransforms the model n Yt = l0SiliXt-i + et n into an autoregressive model (vt = et - let-1): n Yt = lYt-1 + l0Xt + vt nThe model with infinite lag-structure in X becomes a model qwith an autoregressive component lYt-1 qwith a single regressor Xt and qwith autocorrelated error terms nEconometric applications qThe partial adjustment model n Example: Kpt: planned stock for t; strategy for adapting Kt on Kpt n Kt – Kt-1 = d(Kpt – Kt-1) qThe adaptive expectations model q Example: Investments determined by expected profit Xe: q Xet+1 = l Xet + (1 - l) Xt Jan 7, 2011 Contents nThe IV Estimator nThe GIV Estimator nThe Generalized Method of Moments nThe GMM Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 7, 2011 Hackl, Econometrics, Lecture 6 38 Hackl, Econometrics, Lecture 6 39 Multi-equation Models nEconomic phenomena are usually characterized by the behavior of more than one dependent variable nMulti-equation model: the number of equations determines the number of dependent variables which describe the model n nCharacteristics of multi-equation models: nTypes of equations nTypes of variables nIdentifiability n Jan 7, 2011 Hackl, Econometrics, Lecture 6 40 Types of Equations nBehavioral or structural equations: describe the behavior of a dependent variable as a function of explanatory variables nDefinitional identities: define how a variable is defined as the sum of other variables, e.g., decomposition of gross domestic product as the sum of its consumption components n Example: Klein’s model I: Xt = Ct + It + Gt nEquilibrium conditions: assume a certain relationship, which can be interpreted as an equilibrium nDefinitional identities and equilibrium conditions have no error terms Jan 7, 2011 EViews: n=100; series u = nrnd; y1 = y1(-1)+u; y2 = 0.1+y2(-1)+u; y3 = 0.2+0.7*y3(-1)+u; Hackl, Econometrics, Lecture 6 41 Types of Variables nSpecification of a multi-equation model: definition of nVariables which are explained by the model (endogenous variables) nVariables which are in addition used in the model n nNumber of equations needed in the model: same number as that of the endogenous variables in the model n nExplanatory or exogenous variables: uncorrelated with error terms nstrictly exogenous variables: uncorrelated with error terms et+i (for any i) npredetermined variables: uncorrelated with current and future error terms (et+i, i ≥ 0) n nError terms: nUncorrelated over time nContemporaneous correlation of error terms of different equations possible Jan 7, 2011 EViews: n=100; series u = nrnd; y1 = y1(-1)+u; y2 = 0.1+y2(-1)+u; y3 = 0.2+0.7*y3(-1)+u; Hackl, Econometrics, Lecture 6 42 Identifiability: An Example n(1) Both demand and supply function are n Q = a1 + a2P + e n Fitted to data gives for both functions the same relationship: not distinguishable whether the coefficients of the demand or the supply function was estimated! n n(2) Demand and supply function, respectively, are n Q = a1 + a2P + a3Y + e1 n Q = b1 + b2P + e2 n Endogenous: Q, P; exogenous: Y n Reduced forms for Q and P are n Q = p11 + p12Y + v1 n P = p21 + p22Y + v2 n with parameters pij Jan 7, 2011 Hackl, Econometrics, Lecture 6 43 Identifiability: An Example, cont‘d nThe coefficients of the supply function can uniquely be derived from the parameters pij: n b2 = p12/p22 n b1 = p11 – b2 p21 n consistent estimates of pij result in consistent estimates for bi nFor the coefficients of the demand function, such unique relations of the pij can not be found nThe supply function is identifiable, the demand function is not identifiable or under-identified n nThe conditions for identifiability of the coefficients of a model equation are crucial for the applicability of the various estimation procedures Jan 7, 2011 Econometrics II 1.ML Estimation and Specification Tests (MV, Ch.6) 2.Models with Limited Dependent Variables (MV, Ch.7) 3.Univariate time series models (MV, Ch.8) 4.Multivariate time series models, part 1 (MV, Ch.9) 5.Multivariate time series models, part 2 (MV, Ch.9) 6.Models Based on Panel Data (MV, Ch.10) Jan 7, 2011 Hackl, Econometrics, Lecture 6 44