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Background Course in Probability and Statistics (ECO-CO-PROBSTATS)

ECO-CO-PROBSTATS

Purpose

The aim of this course is to review core concepts in probability theory and in the basic theory and practice of classical statistical inference, that are most important for econometrics and economics.

The course will also provide an introduction to programming with R and basic computational methods will be reviewed using R.
There will be 5 2-hour lectures (by a senior lecturer), 5 sessions in R (by TA), and 3 TA sessions for review and problem sets.

Learning outcomes:    
By the end of this module, students will be able to:
•    Understand and apply basic concepts of probability theory, including the behavior of discrete and continuous random variables, as well as multivariate distributions.
•    Interpret and solve problems related to statistical inference, including point estimation, hypothesis testing, and properties of estimators.
•    Maximum likelihood estimators and method of moments.
•    Evaluate the properties of estimators, including unbiasedness, efficiency, and consistency.
•    Understand and apply asymptotic theory to assess the asymptotic efficiency and normality of estimators.
•    Conduct hypothesis testing using asymptotic tests, and likelihood-based tests.
•    Apply the Delta method and interpret p-values in the context of hypothesis testing and statistical analysis.
•    Use R for statistical computing, including random variable generation, Monte Carlo methods, and numerical optimization techniques.

Assessment    
•    Final exam (40%)
•    Problem Sets (60%): There will be a total of 3 problem sets, distributed throughout the course, with each having an equal weight of 20%. 
 
The R-sessions will provide an introduction to:
•    User-defined functions in R;
•    Introduction to programming with R;
•    Random Variable Generation;
•    Monte Carlo Methods;
•    Bootstrap in R;
•    Introduction to numerical optimization methods.

Module structure

WEEK 1
Foundations of Probability Theory and Random Variables
Topics:
•    Basic probability theory,
•    Discrete and continuous random variables,
•    Exponential Families,
•    Transformations of random variables,
•    Multivariate random variables,
•    Joint, marginal and conditional probabilities,
•    Law of Iterated Expectations,
•    Bivariate and multivariate normal densities,
•    Conditional normal densities

WEEK 2
Statistical Model, Sampling and Estimation
Topics:
•    Parametric statistical models,
•    Sampling from infinite population,
•    Statistics,
•    Sampling distributions,
•    Sum, mean and variance for random samples,
•    Sufficiency and Likelihood functions,
•    Point Estimation,
•    Estimators: method of moments estimators, maximum likelihood estimators.

WEEK 3
Properties of Estimators and Asymptotic Theory
Topics:
•    Properties of estimators:
o    Unbiasedness,
o    Efficiency.
o    Fisher information,
o    Cramér-Rao inequality.
•    Asymptotic theory:
o    Consistency,
o    Estimators with Normal asymptotic distributions,
o    Asymptotic efficiency,
o    Properties of Likelihood estimators.

WEEK 4
Hypothesis testing 
Topics:
•    Hypothesis testing,
•    Formulation of the hypothesis testing problem,
•    The Neyman-Pearson Theorem and most powerful tests,
•    Uniformly most powerful tests, Unbiased tests,
•    Asymptotic tests,
•    Important examples based on the likelihood.

WEEK 5
Delta method and p-value
Topics:
•    The Delta method,
•    P-value: definition and interpretation

Bibliography and further readings

Main References:
•    Casella, G. and Berger L. R. (2002). Statistical Inference. Second Edition. Duxbury Press.
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