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Generalized Method of Moments Estimator

Generalized Method of Moments Estimator. Lecture XXXI. Basic Derivation of the Linear Estimator. Starting with the basic linear model

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Generalized Method of Moments Estimator

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  1. Generalized Method of Moments Estimator Lecture XXXI

  2. Basic Derivation of the Linear Estimator • Starting with the basic linear model where yt is the dependent variable, xt is the vector of independent variables, 0 is the parameter vector, and  is the residual. In addition to these variables we will introduce the notion of a vector of instrumental variables denoted zt.

  3. Reworking the original formulation slightly, we can express the residual as a function of the parameter vector • Based on this expression, estimation follows from the population moment condition

  4. Or more specifically, we select the vector of parameters so that the residuals are orthogonal to the set of instruments. • Note the similarity between these conditions and the orthogonality conditions implied by the linear projection space:

  5. Further developing the orthogonality condition, note that if a single 0 solves the orthogonality conditions, or that 0 is unique that Alternatively,

  6. Going back to the original formulation • Taking the first-order Taylor series expansion

  7. Given that this expression implies

  8. B. Given this background, the most general form of the minimand (objective function) of the GMM model can be expressed as • T is the number of observations, • u(0) is a column vector of residuals, • Z is a matrix of instrumental variables, and • WT is a weighting matrix.

  9. Given that WT is a type of variance matrix, it is positive definite guaranteeing that • Building on the initial model In the linear case

  10. Given that WT is positive definite and the optimality condition when the residuals are orthogonal to the variances based on the parameters

  11. Working the minimization problem out for the linear case

  12. Note that since QT() is a scalar, are scalars so

  13. Solving the first-order conditions

  14. An alternative approach is to solve the implicit first-order conditions above. Starting with

  15. The Limiting Distribution • By the Central Limit Theory

  16. Therefore

  17. Under the classical instrumental variable assumptions

  18. Example: Differential Demand Model • Following Theil’s model for the derived demand for inputs • The model is typically estimated as

  19. Applying this to capital in agricultural from Jorgenson’s database, the output is an index of all outputs and the inputs are capital, labor, energy, and materials.

  20. Rewriting the demand model • Thus, the objective function for the Generalized Method of Moments estimator is

  21. Initially we let WT = I and minimize QT(). This yields a first approximation to the estimator

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