Econometrics - Lecture 6 GMM-Estimator and Econometric Models Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 2 Hackl, Econometrics, Lecture 6 3 Estimation Methods nLinear model for yt q yi = xi'β + εi, i = 1, …, N (or y = Xβ + ε) qgiven observations xik, k =1, …, K, of the regressor variables, error term εi nMethods for estimating the unknown parameters β: nOLS estimation: minimizes the sum of squared vertical distances between the observed responses and the responses predicted by the linear approximation nML (maximum-likelihood): selects values of the parameters for which the distribution (likelihood function) gives the observed data the greatest probability; coincides with OLS under normality of εt nGMM (generalized method of moments): minimizes a certain norm of the sample averages of moment conditions; knowledge of distribution shape beyond moment conditions not needed; example: IV estimation Jan 16, 2012 Hackl, Econometrics, Lecture 6 4 The Method of Moments nEstimation method for population parameters such as mean, variance, median, etc. nFrom equating sample moments with population moments, estimates are obtained by solving the equations nExample: Random sample x1, ..., xN from a gamma distribution with density function f(x; α, β) = xα-1 e-x/β [βα Γ(α)]-1 nPopulation moments: E{X} = αβ, E{X2} = α(α+1)β2 nSample moments: m1 = (1/N)Σi xi, m2 = (1/N)Σi xi2 nEquating n m1 = ab, m2 = a(a+1)b2 n leads to n a = m12/(m2 - m12) n b = (m2 - m12)/m1 nNeeds knowledge of the distribution Jan 16, 2012 Hackl, Econometrics, Lecture 6 5 Generalized Method of Moments nMoment conditions: vector function f(β; yi, xi) of the model parameters and the data, such that their expectation is zero at the true values of the parameters; moment equations: E{f(β; yi, xi)}= 0 nR: number of components in f(.); K: number of components of β nSample moment conditions gN(β) = 1/N Σi f(β; yi, xi) nEstimates b chosen such that sample moment conditions fulfill the equations gN(b) = 0 are as closely as possible; if more conditions than parameters (R > K): nGMM estimates: chosen such that the quadratic form or norm n QN(θ) = gN(θ)‘ WN gN(θ) n is minimized; WN: symmetric, positive definite weighting matrix Jan 16, 2012 Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 6 Hackl, Econometrics, Lecture 6 7 Generalized Method of Moments (GMM) Estimation nThe model is characterized by R moment conditions and the corresponding equations n E{f(wi, zi, θ)} = 0 n [cf. 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 should fulfill n n nEstimates d are chosen such that the sample moment conditions are fulfilled n Jan 16, 2012 Hackl, Econometrics, Lecture 6 8 GMM Estimation nThree cases 1.R < K: infinite number of solutions, not enough moment conditions for a unique solution; under-identified or not identified parameters n R ≥ K is a necessary condition for GMM estimation 2.R = K: unique solution, the K-vector d, for θ; if f(.) is nonlinear in θ, numerical solution might be derived; just identified or exactly identified parameters 3.R > K: in general, no choice d for the K-vector θ will result in gN(d) = 0 for all R equations; for a good choice d, gN(d) ~ 0, i.e., all components of gN(d) are close to zero n choice of estimate d through minimization wrt θ of the quadratic form or norm n QN(θ) = gN(θ)‘ WN gN(θ) n WN: symmetric, positive definite weighting matrix Jan 16, 2012 Hackl, Econometrics, Lecture 6 9 GMM Estimator nSample moment conditions gN(θ) with R components, functions of the K-vector θ; R ≥ K nEstimates obtained by minimization, wrt θ, of the quadratic form or norm n QN(θ) = gN(θ)‘ WN gN(θ) n with a suitably chosen symmetric, positive definite weighting matrix WN nWeighting matrix WN nDifferent weighting matrices result in different consistent estimators with different covariance matrices nOptimal weighting matrix n WNopt = [E{f(wi, zi, d) f(wi, zi, d)’}]-1 n i.e., the inverse of the covariance matrix of the sample moment conditions nFor R = K : WN = IN with unit matrix IN n Jan 16, 2012 Hackl, Econometrics, Lecture 6 10 GMM Estimator, cont’d nGMM estimator: nMinimizes QN(θ), using the optimal weighting matrix nThe optimal weighting matrix n WNopt = [E{f(wi, zi, d) f(wi, zi, d)’}]-1 n is the inverse of the covariance matrix of the sample moment conditions nMost efficient estimator nFor nonlinear f(.) nNumerical optimization algorithms nWN depends on θ; iterative optimization Jan 16, 2012 Hackl, Econometrics, Lecture 6 11 Example: The Linear Model nModel: yi = xi‘β + εi with E{εi xi} = 0 and V{εi} = σε² nMoment or orthogonality conditions: n E{εi xi} = E{(yi - xi‘β)xi} = 0 n f(.) = (yi - xi‘β)xi, θ = β; moment conditions are the exogeneity conditions for xi nSample moment conditions: n 1/N Σi (yi - xi ‘b) xi = 1/N Σi ei xi = gN(b) = 0 nWith WN= IN, QN(β) = (1/N )2 Σi ei 2 xi xi‘ nOLS and GMM estimators coincide, give the estimator b, but qOLS: residual sum of squares SN(b) = 1/N Σi ei2 has its minimum qGMM: QN(b) = 0 n n Jan 16, 2012 Hackl, Econometrics, Lecture 6 12 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 16, 2012 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 16, 2012 Hackl, Econometrics, Lecture 6 13 Labor Demand Function: OLS Estimation Hackl, Econometrics, Lecture 6 14 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 16, 2012 Specification of GMM Estimation Hackl, Econometrics, Lecture 6 15 GRETL: Specification window # initializations go here gmm orthog weights params end gmm Jan 16, 2012 Labor Demand: Specification of GMM Estimation Hackl, Econometrics, Lecture 6 16 GRETL: Specification of function and orthogonality conditions for the 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 16, 2012 Labor Demand Function: GMM Estimation Results Hackl, Econometrics, Lecture 6 17 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 16, 2012 Hackl, Econometrics, Lecture 6 18 GMM Estimator: Properties nUnder weak regularity conditions, the GMM estimator is nconsistent (for any WN) nmost efficient if WN = WNopt = [E{f(wi, zi, Ď) f(wi, zi, Ď)’}]-1 nasymptotically normal: n where V = D WNopt D’ with the KxR matrix of derivatives n n nThe covariance matrix V-1 can be estimated by substituting in D and WNopt the population moments by sample equivalents evaluated at the GMM estimates n Jan 16, 2012 Consistency of the Generalized IV Estimator nWith a RxR positive definite weighting matrix WN, minimizing the weighted quadratic form in the sample moment expressions 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 16, 2012 Hackl, Econometrics, Lecture 6 19 Hackl, Econometrics, Lecture 6 20 GMM Estimator: Calculation 1.One-step GMM estimator: Choose a positive definite WN, e.g., WN = IN, 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 WN = V-1; this gives 3.Iterated GMM estimator: Repeat step 2 until convergence nIf R = K, the GMM estimator is the same for any WN, 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 16, 2012 GMM and Other Estimation Methods nGMM estimation generalizes the method of moments estimation nAllows for a general concept of moment conditions nMoment conditions are not necessarily linear in the parameters to be estimated nEncompasses various estimation concepts such as OLS, GLS, IV, GIV, ML n n Jan 16, 2012 Hackl, Econometrics, Lecture 6 21 moment conditions OLS E{xi(yi – xi’β)} = 0 GLS E{xi(yi – xi’β)/σ2 (xi)} = 0 IV E{zi(yi – xi’β)} = 0 ML E{∂/∂β f[εi(β)]} = 0 Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 22 Hackl, Econometrics, Lecture 6 23 Instrumental Variables nThe model is n yi = xi‘β + εi n with V{εi} = σε² and n E{εi xi} ≠ 0 n endogenous regressors xi nInstrumental variables zi 1.Exogenous: E{εi zi } = 0: zi uncorrelated with error term 2.Relevant: Cov{xi , zi } ≠ 0: zi correlated with endogenous regressors Jan 16, 2012 IV Estimator nBased on the moment conditions - or moment equations - 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 zi n n nIdentification requires that the KxK matrix Σizixi’ = Z’X is finite and invertible; instruments zi are relevant when this is fulfilled n Jan 16, 2012 Hackl, Econometrics, Lecture 6 24 The General Case nR: number of instrumental variables, of components of zi nThe R moment equations are n n nR < K: Z’X has not full rank, is not invertible; under-identified or not identified parameters; no consistent estimator; R ≥ K is a necessary condition 1.R = K: one unique solution, the IV estimator; identified model n 2.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 needed Jan 16, 2012 Hackl, Econometrics, Lecture 6 25 The Generalized IV Estimator nFor R > K, in general, no unique solution of all R sample moment conditions can be obtained; application of the GMM concept: nThe weighted quadratic form in the sample moment expressions 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 16, 2012 Hackl, Econometrics, Lecture 6 26 Weighting Matrix WN nDifferent weighting matrices WN 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 qGIV (or TSLS) estimator n n n Jan 16, 2012 Hackl, Econometrics, Lecture 6 27 The GIV Estimator 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 16, 2012 Hackl, Econometrics, Lecture 6 28 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 16, 2012 Hackl, Econometrics, Lecture 6 29 Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 30 Hackl, Econometrics, Lecture 6 31 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 16, 2012 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 16, 2012 Hackl, Econometrics, Lecture 6 32 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 33 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 16, 2012 Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 34 Hackl, Econometrics, Lecture 6 35 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 16, 2012 Hackl, Econometrics, Lecture 6 36 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 16, 2012 Hackl, Econometrics, Lecture 6 37 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 16, 2012 Hackl, Econometrics, Lecture 6 38 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 16, 2012 Contents nEstimation Concepts nGMM Estimation nThe GIV Estimator nEconometric Models nDynamic Models nMulti-equation Models Jan 16, 2012 Hackl, Econometrics, Lecture 6 39 Hackl, Econometrics, Lecture 6 40 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 16, 2012 Hackl, Econometrics, Lecture 6 41 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 16, 2012 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 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 16, 2012 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 43 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 16, 2012 Hackl, Econometrics, Lecture 6 44 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 16, 2012 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 16, 2012 Hackl, Econometrics, Lecture 6 45