## TIME SERIES AND ECONOMETRICS

## Syllabus

### Updated A.Y. 2019-2020

**Prerequisites**

Linear algebra (operations with matrices and vectors, trace of a matrix, properties of symmetric matrices, quadratic forms, eigenvectors, eigenvalues, diagonalization).

Calculus.

**Syllabus **

The following syllabus is to be considered temporary and may change in the future.

**Econometrics **

Simple linear regression model:

- OLS estimators: derivation through first order conditions.
- Definition and interpretation of the coefficient of determination.
- Unbiasedness of OLS estiamtors: theory and practice (with Matlab).
- Conditional variance of OLS estiamtors: theory and practice (with Matlab).
- Unbiased estimator of error variance.
- Statistical inference: hypothesis testing and t-statistic.
- Statistical inference: the Capital Asset Pricing Model and the beta of a stock.

Multiple linear regression model:

- Recap of matrix algebra and gradient of a function.
- OLS estimators: derivation through first order conditions.
- Unbiasedness of OLS estiamtors.
- Conditional variance-covariance matrix of OLS estiamtors.
- Unbiased estimator of error variance.
- Multicollinearity.
- Blueness of the OLS estimator: the Gauss-Markov theorem.
- Multiple hypothesis testing.
- Maximum Likelihood Estimator.
- Model comparison.
- Omitted and irrelevant variables.
- Measurement errorrs.
- OLS asymptotics.

**Bibliography**

Wooldridge J. M. (2016). Introductory Econometrics: A Modern Approach.

Brooks C. (2014). Introductory Econometrics for Finance.

**Time Series **

- Introduction to time series (trends, seasonality, co-integration), log-returns.
- Convergence to equilibrium of an AR(1) process.
- Strongly stationary processes: definition and properties.
- White noise.
- Covariance-stationarity: definition and main properties.
- Random walk.
- Gaussian processes.
- Non-stationary processes and order of integration. Consequences of non-stationarity on regression models.
- Mean-ergodic and variance-ergodic processes. Sufficient conditions for ergodicity. Ergodicity of gaussian processes.
- Memory of a process and the test for the autocorrelation coefficient.
- Lag operator. Polynomials of the lag operator.
- ARMA, AR and MA processes: definition. Invertibility and causality: definition.
- Mean, variance, co-variance and stability condition for AR(1)-processes.
- Causality and explosiveness of AR(1)-processes.
- Stability region for AR(2) processes. Variance, Co-variance, autocorrelation and partial autocorrelation of AR(p) processes.
- Selection of the autoregressive order: ACF vs PACF.
- Moving Average Processes: stationarity, invertibility, ACF and PACF.
- Moving Average vs. Autoregressive.
- Invertibility and Causality of AMRA(p,q).
- ARIMA models
- The Box-Jenkins procedure.
- Vector Auto-Regressive Models: structural, reduced-form and companion representation.
- Stationarity conditions of VAR models.
- Triangularisation and identification of structural VAR.
- Wold Theorem and the Impulse response function.
- Co-integration: definition and examples. Vector Error Correction Models: estimation and interpretation.

**Bibliography**

Wei (2006). Time Series Analysis: Univariate and Multivariate Methods. Addison-Wesley.

Brockwell P.J. and Davis R.A. (2002). Introduction to Time Series and Forecasting, Springer.

Brockwell P.J. and Davis R.A. (1991). Time Series: Theory and Methods. Springer.