SUBJECT

Analysis of time series

lecture

master

Faculty of Science

Mathematics MSc

3+2

Semesters 1-4

Autumn/Spring semester

Basic notions of stationary processes,weak, k-order, strict stationarity, ergodicity, convergence to stationary distribution. Interdependence structure: autocovariance, autocorrelation, partial autocorrelation functions and their properties, dynamic copulas. Spectral representation of stationary processes by an orthogonal stochastic measure, the spectral density function, Herglotz’s theorem.

Introduction and basic properties of specific time series models: Linear models: AR(1), AR(2) AR(p), Yule-Walker equations, MA(q), ARMA(p,q), ARIMA(p,d,q) conditions for the existence of stationary solutions and invertibility, the spectral density function. Nonlinear models: ARCH(1), ARCH(p), GARCH(p,q), Bilinear(p,q,P,Q), SETAR, regime switching models. Stochastic recursion equations, stability, the Ljapunov-exponent and conditions for the existence of stationary solutions, Kesten-Vervaat-Goldie theorem on stationary solutions with regularly varying distributions. Conditions for the existence of stationary ARCH(1) process with finite or infinite variance, the regularity index of the solution.

Estimation of the mean. Properties of the sample mean, depending on the spectral measure. Estimation of the autocovariance function. Bias, variance and covariance of the estimator. Estimation of the discrete spectrum, the periodogram. Properties of periodogram values at Fourier frequencies. Expectation, variance, covariance and distribution of the periodogram at arbitrary frequencies. Linear processes, linear filter, impulse-response and transfer functions, spectral density and periodogram transformation by the linear filter. The periodogram as useless estimation of the spectral density function. Windowed periodogram as spectral density estimation. Window types. Bias and variance of the windowed estimation. Tayloring the windows. Prewhitening and CAT criterion.