Insegnamento mutuato da: B020840 - MOD. 1 MICROECONOMETRICS Corso di Laurea Magistrale in ECONOMICS AND DEVELOPMENT- ECONOMIA POLITICA E SVILUPPO ECONOMICO Curricula ECONOMICS
The multiple linear regression model and ordinary least squares
estimation (OLS) with the notations of linear algebra.
Complements of linear algebra.
Algebraic and statistical properties of OLS,
Gauss-Markov theorem, forecast errors, distribution of linear and
quadratic forms, linear restrictions, restricted least squares, t-test,
F-test, structural change, specification errors.
Linear models for panel data; fixed effects and randiom effects; Hausman test.
Discrete choice models.
1) Calzolari, G. (2012-2019): "Econometric Notes", freely downloadable from Repec/Ideas "Research Papers in Economics"
2) Greene, W. H. (2008): " Econometric Analysis" (6th edition). Prentice-Hall, Inc. Upper Saddle River, NJ.
3) Stock, J. H., and M. W. Watson (2015): "Introduction to Econometrics" (Updated 3rd edition). Boston: Person Education Limited. Edizione italiana a cura di F. Peracchi (2016): "Introduzione all' Econometria" (quarta edizione). Milano: Pearson Italia S.p.A.
4) Computer package: Gretl (free software, open source). For Windows, Linux, Mac http://gretl.sourceforge.net/index.html#dl
Understand the foundations of econometric analysis, learn basic econometrics in some details, learn the theory underlying the most common microeconometric models (panel and discrete-choice), and use econometric packeges.
Genaral mathematical analysis and linear algebra. Basic statistical inference.
Lectures mainly using black-board and chalk; theorems "with proofs" (almost always). Examples using an "open source" computer package.
Type of Assessment
Written/computational exam consisting of:
- theoretical questions requiring derivations (up to 5 questions);
- computations, with numerical answers (up to 50 questions).
Time: 4 or 5 hours.
General introduction, notes, textbooks, software, exams. Elements of linear algebra (refresh and new elements):
vectors, graphical representation, operations, scalar product, orthogonality, linear combinations,
linear dependence or independence, unit vectors.
Sets of linear independent vectors and basis, representation of a vector as a linear combination of the basis-vectors.
Matrices, operations, product of matrices, distributive and associative properties of the product.
Square matrices, symmetric matrices, identity matrix.
Transpose matrix, transpose of the product (with proof).
Maximum number of linearly independent columns and rows in a matrix (without proof that they are equal) and definition of rank.
Inverse matrix (without intruducing determinants).
Existence of a unique "right" inverse and of a unique "left inverse".
Left and right inverses are equal (with proofs, based on the unique representation of a vector as linear combination
of the basis vectors).
Eigenvalues of a real symmetric matrix are real (with proof).
Eigenvectors corresponding to different eigenvalues are orthogonal (with proof).
Spectral decomposition of a square symmetric matrix
(with the diagonal matrix of eigenvalues and the orthogonal matrix of normalized eigenvectors).
Trace of a square matrix and its properties, with proof of the theorem tr(AB)=tr(BA).
The trace is the sum of the eigenvalues (in symmetric matrices).
Inverse of the product (with proof). Quadratic forms.
X?X is always a square symmetric matrix positive semidefinite matrix; positive definite if X has full column rank;
A'X'XA is always positive semi-definite.
A positive definite matrix is non-singular (without proof).
Idempotent matrices: the matrix that produces deviation from the arithmetical mean;
the projection matrices (projection on the plane and projection on the direction orthogonal to the plane).
Trace of an idempotent matrix is equal to rank.
Vector of the first order derivatives of a scalar product.
Explicit formula for a quadratic form (with double sum). Vector of the first order derivatives of a quadratic form.
Inequality and order relationship between positive semidefinite matrices.
Statistical inference: random variables, expectation, variance, covariance, correlation.
Univariate normal distribution (comments on the multivariate normal;
in a multivariate normal, uncorrelation implies independence: without proof).
Random vectors, expectation vector, variance-covariance matrix.
Expectation and variance-covariance matrix of linear combinations of the elements of a random vector.
Expectation and variance-covariance matrix of linear combinations of random vectors
(such as Ax, if A is a constant and x is a random vector).
Independence implies uncorrelation, but not viceversa (with examples), unless the distribution is (multivariate) normal.
Some comments on the multivariate normal distribution (without proofs).
Chi-square, Student?s-t, Fisher?s F.
Argomento: Linear regression model; notation and algebraic assumptions; ordinary least squares estimation;
first order conditions for the minimum with matrix notation (with proof).
The second order conditions for the minimum.
Algebraic properties of OLS residuals, orthogonality between residuals and explanatory variables.
Example with a two-variables model (the arithmetical averages are on the estimated regression line).
R-square in the model with intercept; problems; extreme values of the R-square.
The var-cov matrix is always positive-semi-definite.
Statistical assumptions (still without the assumption of normality for the distribution of the error terms)
and first consequences; unbiasedness of OLS coefficients.
Variance-covariance matrix of a generic linear unbiased estimator; Gauss-Markov theorem (traditional proof).
Discussion of the assumptions and of the thesis of Gauss-Markov theorem.
Unbiased estimator of sigma^2 and of the variance-covariance matrix of OLS coefficients;
standard errors of coefficients. Prediction (forecast), prediction error;
unbiasedness of (conditional) prediction; variance of prediction error.
The "additional" assumption of normality and its consequences:
OLS coefficients and residuals have a multivariate normal distribution.
Introduction to the software package GRETL.
Download and install the software (open source, free software).
Read or import data files. Series of data, descriptive statistics and graphical representation;
introduction of new series or of new data into a series (for forecast).
OLS coefficients, residuals, variance, standard errors.
Examples with cross-section data and with time-series data.
Verify some results using the algebra formulae, like (X?X)^-1 X?y.
Some theorems on the distributions of linear and quadratic forms involving multivariate normal vectors
Independence between OLS coefficients and residuals.
Standardization of "one" OLS coefficient and transformation into a Student?s-t random variable;
t-test (double tail). Significance test using Student?s-t.
Linear restrictions on coefficients and matrix notation.
Test of an hypothesis that involves more than one linear restriction.
Chi-square distribution of the quadratic form obtained from restrictions and OLS coefficients;
transformation into Fisher?s F of the test statistic when the "true" sigma-2 is replaced by its OLS estimate.
Student's t-test (one-sided).
Test of an hypothesis that involves more than one linear restriction
(example using a Keynesian model, where disposable income is disaggregated into many components,
like wages, profits, pensions, etc., with different marginal propensities).
Test of the hypothesis of constant returns to scale in a Cob-Douglas function.
Lagrangean function for minimization of the sum of squared residuals ?under constraints? (or restrictions).
First order conditions form a system of linear equations (same number of equations and unknowns,
so "presumably" with one solution only; without the details of the proof).
Restricted least squares coefficients and residuals; restricted residual sum of squares;
the alternative form of the Fisher?s F test statistic.
How to simplify the practice, computing the restricted least squares estimates
without resorting to the Lagrangean function and its derivatives.
Examples (the Keynesian consumption function, the t-test as a particular case,
testing the equality of two coefficients, testing the constant return to scale in a Cob-Douglas function,
Testing an hypothesis on the "equal expected" value of consumptions for two different families using the F-test.
Testing an hypothesis on the "expected" value of consumptions
for a family with given values of income and number of members.
Structural break and Chow test (splitting the matrix).
Dummy variables, and structural break and Chow test.
Remarks on the difficulties and limitations of the Chow test
(too short subsamples, specification of the exact point of the break).
Dummy variables and intercept: the trap of the dummies:
Test based on Fisher?s F, where the distribution of the test statistic in finite samples is not
"exactly" a Fisher?s-F (only asymptotically the distribution has a well defined form; without proof):
the specification error test Reset (Ramsey); Breusch-Godfrey test on uncorrelation versus autocorrelation;
Breusch-Pagan test on homoscedasticity versus heteroscedasticity (without proofs; a good description of the test
procedures is available in Wikipedia).
Computer Lab: examples of multicollinearity (and quasi collinearity;
the matrix X?X is nearly singular: some remarks on the consequences, without proof).
Linear restrictions, test t and F using the "icons". Test F computing the sums of squared residuals.
Impose restrictions from the Gretl-consolle. Chow test on structural stability-versus-break and definition of
a dummy variable. Lists of commands using Gretl page.
Prediction (or forecast), standard error of the prediction error and confidence interval.
RESET test (linear specification error).
Breusch-Pagan test (heteroskedasticity).
Breusch-Godfrey test (autocorrelation).
All the tests are performed as Fisher?s F test.
expected value of restricted least squares estimates; bias when restrictions are not valid.
Specification error and bias due to omission of relevant explanatory variables.
Variance-covariance matrix of restricted least squares coefficients
(with main steps of the proof),
and inefficiency due to inclusion of not-relevant explanatory variables.
Specification error due to omitted "qualitative" or "unobserved" variables and repeated observations
on the same units over different time periods.
The example of car accidents as in the textbook of Stock and Watson
(some pages of the book have been distributed to the students).
The linear model for panel data;
introducing the individual dummy variables (fixed effect model);
OLS estimation of the fixed effect model,
transforming variables as deviations from the arithmetic averages.
Models for discrete choice
(binary dependent variable).
The linear probability model; the example of the probability
of a successful request of a mortgage loan from a bank
(taken form the book of Stock and Watson).
The logit model: probability, likelihood, log-likelihood.
Just mentioned that maximization of the log-likelihood
Computer Lab: Estimation of a linear model for panel data, with fixed effects
(example on car accidents, taken from the textbook of Stock and Watson).
Estimation of a Logit model
(example of a successful request of a mortgage loan from a bank
taken form the book of Stock and Watson;
example of probability of success in exams or "concorsi").
The law of large numbers (history).
Applications: OLS estimation of an AR(1) model is biased but consistent;
the White-type (or robust) estimate of the variance covariance matrix of OLS coefficients
in case of heteroskedasticity.