ECONOMETRICS - 2011/12

Giorgio Calzolari

Lecture 1

Introduction. Complements of linear algebra, idempotent matrices, projection matrices, trace of the projection matrix Mx. The idempotent matrix that produces deviations from the arithmetical average. model, compact notation using vectors and matrices, algebraic assumptions that lead to estimation (OLS). X'X is a positive definite matrix. Minimization of the sum of squared errors, estimated coefficients, residuals. Orthogonality between OLS residuals and explanatory variables. Sum of residuals is zero for a model with intercept. R^2 in a model with intercept. Limitations of the R^2 concept.

Lecture 2 Inequality between positive semidefinite matrices. Statistical assumptions for the classical linear regression model. Expectation and -covariance matrix of OLS coefficients. Unbiasedness of OLS estimator. Constructing other linear unbiased estimators of coefficients. Gauss-Markov theorem, with 2 different proofs: traditional textbook proof and proof based on Schwarz inequality. Unbiased estimation of sigma^2, with proof. Forecast (conditional) and variance of the forecast error. The additional assumption of normality of the error terms. Standardization of a random variable; standard normal distribution. Test of a simple hypothesis on a coefficient using the standard normal (supposing Sigma^2 known). “Operational” introduction to the “t-test” (two-sided and one-sided), without proofs involving the multivariate normal and the chi^2 distribution (that will be explained in the third lesson on Statistics next week)

Lecture 3 Some properties of eigenvalues and eigenvectors of symmetric matrices and idempotent matrices; the orthogonal matrix containing the normalized eigenvectors; the spectral decomposition of a symmetric matrix. Distributions of linear and quadratic forms, when the vectors have a multivariate normal distribution, with proofs.. Independence between OLS coefficients and residuals, under hypothesis of normality. Distribution of the estimated sigma^2. Test of an hypothesis on a single coefficient based on the Stuednt’s-t distribution, if sigma^2 is estimated. Multiple linear restrictions on coefficients and hypothesis testing using Fisher’s-F distribution, when sigma^2 is estimated.. Fisher’s F test of a multiple restrictions hypothesis, using the quadratic form. Restricted least squares, Lagrangian function and solution of the first order conditions (trace of proof). Residuals of restricted least squares and Fisher’s F test based on the sums of squared residuals, with proof.

1 Lecture 4 Expectation and variance-covariance matrix of restricted least squares coefficients. Specification error: omitted and/or not relevant explanatory variables, bias and inefficiency. and Chow test. Some comments on recursive residuals, when Chow test cannot be applied in its standard way. Introduction to simultaneous equations macro-econometric models. Klein-I model (used as a typical example). Exogenous, endogenous and lagged endogenous variables. Structural form and reduced form. Notation for dynamic models. Forecast one-step-ahead and multi-steps ahead. Impact and delay multipliers and their use for economic policy actions.

Lecture 5 One-step ahead (static) solution and dynamic solution “in sample”; RMSE, Theil’s inequality coefficient. Economic policy targets, instruments and Tinbergen’s problem. Structural forms observationally equivalent (necessary and sufficient condition for their existence); exclusion restrictions and their representation in matrix form. Admissible transformation. Rank condition for identification (necessary and sufficient); order condition (only necessary); demand-supply model, unidentified equations.

Lecture 6 Identification of demand inserting an additional exogenous regressor into supply equation (with proof of the rank condition). Just-identified and over-identified equations. Numerical techniques for solving linear and nonlinear systems of equations; Gauss-Seidel and Jacoby methods (without proof of the conditions for convergence). Law of Large numbers for sequences of i.i.d, i.ni.d. and ni.ni.d. random variables: historical notes, without proofs; some examples dealing with independent but not identically distributed random variables (Chebishev), and with non-independent variables (Markov). Small sample and asymptotic properties of OLS estimator: unbiasedness and consistency problems in 4 cases. Instrumental variable estimator, consistency. Some “intuition” behind the choice of instruments that lead to asymptotic efficiency (proof will be given later) and operational introduction to 2SLS (for the homework).

Lecture 7 Instrumental variable estimator: asymptotic variance-covariance matrix for a generic choice of the instruments. Central limit theorem: historical notes, without proofs; comments on sequences of i.i.d. and ni.ni.d random variables and the need of suitable versions of the theorem in econometric applications. Efficient choice of the instruments: expected values of regressors (with proofs). Efficient choice of the instruments: conditional expectation of regressors (without proofs). The (theoretical) efficient matrix of instruments for a structural form equation: exogenous and lagged endogenous variables are unchanged, while current endogenous are replaced by their conditional expectation. Efficient matrix in practice; use a preliminary consistent estimate of the reduced form coefficients; asymptotic equivalence with the method using the theoretically efficient matrix. Two stage least squares (2SLS); second stage as instrumental variables or as least squares; limitations in case of large scale models. Limited information instrumental variables efficient (LIVE). Iterative instrumental variables (IIV). Linear regression model with heteroskedasticity: OLS coefficients and White’s robust estimator of the variance-covariance matrix (HCE matrix, or sandwich matrix).

2 Lecture 8

Likelihood, log-likelihood, score, Fisher’s information, information matrix. Cramer-Rao inequality for regular estimators (unbiased estimators, in particular; with proof for the single parameter case, without proof for multidimensional case). Maximum likelihood; consistency (with proof for the single parameter case); asymptotic normality and asymptotic variance-covariance matrix. Alternative estimators of the information matrix and of the variance-covariance matrix (Hessian and Outer Product); some discussion of the misspecification problem and motivation of the robust variance- covariance estimator (sandwich matrix, or White’s type matrix). Numerical methods to maximize the likelihood: Newton-Raphson (NR) and Berndt, Hall, Hall, Hausman (BHHH).

Lecture 9

Algebra: Kronecker product and properties. Linear regression model, generalized least squares estimator (GLS), Aitken’s theorem. Seemingly unrelated regression equations (SURE), infeasible GLS and feasible GLS and asymptotic equivalence; Zellner’s example; particular cases where GLS=OLS (only trace of proof); iterative GLS and maximum likelihood (without proof). Simultaneous equations and full information instrumental variables; infeasible and feasible efficient instrumental variables; three stage least squares (3SLS: operational steps); full information instrumental variables efficient (FIVE: only mentioned as a full information countepart of LIVE). iterative FIVE and maximum likelihood (Durbin’s method, without proof).

Lecture 10

Simulation based estimators: simulated maximum likelihood and indirect inference. Examples of application: multilevel logit model with random effects (or logit model for panel data with random effects); discrete time stochastic volatility model; MA estimated as AR model. Possible applications (just mentioned) on latent factor models and Tobit models with Arch errors.

References :

- Fair, R. C. (1986): “Evaluating the Predictive Accuracy of Models”, in Handbook of Econometrics, ed. by Z. Griliches and M. D. Intriligator. Amsterdam: North-Holland Publishing Company, Vol. III, 1979-1995.

- Greene, W. H. (2008): Econometric Analysis (6th edition). Prentice-Hall, Inc. Upper Saddle River, NJ. (Sec. 2, 3, 4, 5, 6.1, 6.2, 6.4, 7.1, 7.2, 8.1, 8.4, 8.5, 8.6, 19.7, App. A).

- Hausman, J. A. (1983): “Specification and Estimation of Simultaneous Equation Models”, in Handbook of Econometrics, ed. by Z. Griliches and M. D. Intriligator. Amsterdam: North-Holland Publishing Company, Vol. I, 391-448.

- Hsiao, C. (1983): “Identification”, in Handbook of Econometrics , ed. by Z. Griliches and M. D. Intriligator. Amsterdam: North-Holland Publishing Company, Vol. I, 223-283.

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