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Econometrics

A.Y. 2021/2022

Learning objectives

The aim of the course is to provide students with the basic principles of econometrics. All the aspects of econometric models treated during the course will be investigated through modern empirical applications in order to motivate students and respond to important problems coming from the real world with appropriate and specific numerical answers. Specifically, the first aim of the course is to extend the simple linear regression model, already thought in the course of Statistics, in different directions: extend the number of regressors, consider potential departures from the standard assumptions of the model, develop a theoretical framework for making inference on the parameters of the model, both for small sample and asymptotically. The second specific aim, concerns the introduction to non-linear regression models like models for binary dependent variables or non-linear specifications among the regressors.

Expected learning outcomes

At the end of the course students will have received the introductory notions of econometrics. In particular, they will be able to specify a linear regression model, estimate the coefficients and perform tests of hypothesis on them. Moreover, students will be able to read and critically comment on the results of econometric analyses based on linear regression models or on regression models presenting some nonlinearities, like logit and probit ones. These expected outcomes should help students in understanding empirical analysis introduced in different courses, as well as provide them with quantitative tools for the development of the final thesis.

**Lesson period:**
Second trimester

**Assessment methods:** Esame

**Assessment result:** voto verbalizzato in trentesimi

Course syllabus and organization

### Single session

Responsible

Lesson period

Second trimester

**Course syllabus**

- Economic questions and economic data

quantitative economic questions

causal effects and ideal experiments

data: sources and types

- Basic notions of probability

stochastic variables and probability distributions

expected value and variance

bivariate distributions: independence, covariance and correlation

Normal, chi-squared, Student-t and F distributions

law of large numbers and central limit theorem

- Basic notions of statistics

estimation of the mean of a population

hypothesis testing about the mean of a population

confidence intervals for the mean of a population

scatterplot, sample covariance and correlation

- Linear regression model with one single regressor

the linear regression model

estimation of the coefficients of the linear regression model

goodness of fit

assumptions of the linear regression model

OLS estimator and its sample distributions

California test score dateset (Appendix)

derivation of the OLS estimator (Appendix)

sample distribution of the OLS estimator (Appendix)

Formulas for the standard errors of the OLS estimator (Appendix)

- Linear Regression with Multiple Regressors

Omitted Variable Bias

The Multiple Regression Model

The OLS Estimator in Multiple Regression

Measure of Fit in Multiple Regression

The Least Squares Assumptions in Multiple Regression

The Distribution of the OLS Estimators in Multiple Regression

Multicollinearity

- Hypothesis Tests and Confidence Intervals in Multiple Regression

Hypothesis Tests and Confidence Intervals for a Single Coefficient

Tests of Joint Hypotheses

Testing Single Restrictions Involving Multiple Coefficients

Model Specification for Multiple Regression

Analysis of the Test Score Data Set

- Nonlinear Regression Functions

A General Strategy for Modeling Nonlinear Regression Functions

Nonlinear Functions of a Single Independent Variable

Interactions Between Independent Variables

Nonlinear Effects on Test Scores of Student-Teacher Ratio

- Assessing Studies Based on Multiple Regression (to read only)

Internal and External Validity

Threats to Internal Validity of Multiple Regression Analysis

Internal and External Validity when the Regression is Used for Forecasting

Example: Test Scores and Class Size

- Regression with a Binary Dependent Variable

Binary Dependent Variables and the Linear Probability Model

Probit and Logit Regression

Estimation and Inference in the Logit and Probit Models

Some applications

- Instrumental Variable Regression

The IV Estimator with a Single Regressor and a Single Instrument

The General IV Regression Model

Checking Instrument Validity

Where Do Valid Instruments Come From?

Appendix 2: Derivation of the Formula for the TSLS Estimator

Appendix 3: Large-Sample Distribution of the TSLS Estimator

- Introduction to Time Series Regression and Forecasting

Using Regression Model for Forecasting

Introduction to Time Series Data and Serial Correlation

Autoregressions

Time Series Regression with Additional Predictors and ADL Model

Lag Length Selection Using Information Criteria

Nonstationarity: Trends

Detecting Stochastic Trends: Testing for a Unit AR root (to read only)

- Estimation of Dynamic Causal Effects

Dynamic Causal Effects

Estimation of Dynamic Causal Effects with Exogenous Regressors

Dynamic Multipliers (to read only)

- The Theory of Linear Regression with One Regressor (to read only)

The Extended Least Squares Assumptions and the OLS Estimator

Fundamentals of Asymptotic Distribution Theory (basic notions only)

Asymptotic Distribution of the OLS Estimator and t-Statistic

Exact Sampling Distributions When the Errors Are Normally Distributed

Weighted Least Squares (basic notions only)

- The Theory of Multiple Regression

The Linear Multiple Regression Model and the OLS Estimator in Matrix Form

Asymptotic Distribution of the OLS Estimator and t-Statistic

Test of Joint Hypotheses

quantitative economic questions

causal effects and ideal experiments

data: sources and types

- Basic notions of probability

stochastic variables and probability distributions

expected value and variance

bivariate distributions: independence, covariance and correlation

Normal, chi-squared, Student-t and F distributions

law of large numbers and central limit theorem

- Basic notions of statistics

estimation of the mean of a population

hypothesis testing about the mean of a population

confidence intervals for the mean of a population

scatterplot, sample covariance and correlation

- Linear regression model with one single regressor

the linear regression model

estimation of the coefficients of the linear regression model

goodness of fit

assumptions of the linear regression model

OLS estimator and its sample distributions

California test score dateset (Appendix)

derivation of the OLS estimator (Appendix)

sample distribution of the OLS estimator (Appendix)

Formulas for the standard errors of the OLS estimator (Appendix)

- Linear Regression with Multiple Regressors

Omitted Variable Bias

The Multiple Regression Model

The OLS Estimator in Multiple Regression

Measure of Fit in Multiple Regression

The Least Squares Assumptions in Multiple Regression

The Distribution of the OLS Estimators in Multiple Regression

Multicollinearity

- Hypothesis Tests and Confidence Intervals in Multiple Regression

Hypothesis Tests and Confidence Intervals for a Single Coefficient

Tests of Joint Hypotheses

Testing Single Restrictions Involving Multiple Coefficients

Model Specification for Multiple Regression

Analysis of the Test Score Data Set

- Nonlinear Regression Functions

A General Strategy for Modeling Nonlinear Regression Functions

Nonlinear Functions of a Single Independent Variable

Interactions Between Independent Variables

Nonlinear Effects on Test Scores of Student-Teacher Ratio

- Assessing Studies Based on Multiple Regression (to read only)

Internal and External Validity

Threats to Internal Validity of Multiple Regression Analysis

Internal and External Validity when the Regression is Used for Forecasting

Example: Test Scores and Class Size

- Regression with a Binary Dependent Variable

Binary Dependent Variables and the Linear Probability Model

Probit and Logit Regression

Estimation and Inference in the Logit and Probit Models

Some applications

- Instrumental Variable Regression

The IV Estimator with a Single Regressor and a Single Instrument

The General IV Regression Model

Checking Instrument Validity

Where Do Valid Instruments Come From?

Appendix 2: Derivation of the Formula for the TSLS Estimator

Appendix 3: Large-Sample Distribution of the TSLS Estimator

- Introduction to Time Series Regression and Forecasting

Using Regression Model for Forecasting

Introduction to Time Series Data and Serial Correlation

Autoregressions

Time Series Regression with Additional Predictors and ADL Model

Lag Length Selection Using Information Criteria

Nonstationarity: Trends

Detecting Stochastic Trends: Testing for a Unit AR root (to read only)

- Estimation of Dynamic Causal Effects

Dynamic Causal Effects

Estimation of Dynamic Causal Effects with Exogenous Regressors

Dynamic Multipliers (to read only)

- The Theory of Linear Regression with One Regressor (to read only)

The Extended Least Squares Assumptions and the OLS Estimator

Fundamentals of Asymptotic Distribution Theory (basic notions only)

Asymptotic Distribution of the OLS Estimator and t-Statistic

Exact Sampling Distributions When the Errors Are Normally Distributed

Weighted Least Squares (basic notions only)

- The Theory of Multiple Regression

The Linear Multiple Regression Model and the OLS Estimator in Matrix Form

Asymptotic Distribution of the OLS Estimator and t-Statistic

Test of Joint Hypotheses

**Prerequisites for admission**

Basic course of Statistics, including notions of inferential statistics. Basic notions of calculus and matrix algebra.

**Teaching methods**

Lessons and classes, using the econometric software STATA.

**Teaching Resources**

Textbook: "Introduction to Econometrics" by J.H. Stock e M.W. Watson.

**Assessment methods and Criteria**

Written exam.

SECS-P/05 - ECONOMETRICS - University credits: 6

Lessons: 40 hours

Professor:
Fasani Francesco Maria

Professor(s)