## Lecture 01 (Jan 21, 2022)

First Lecture: Intorduction of autocorrleation for linear time series

This is the note I am taking in class for Stat 556, the advanced time series analysis.

First Lecture: Intorduction of autocorrleation for linear time series

Calculate the variance and convergence rates for autocorrelation sample estimator by central limit theorem. Introduce statistical tests for ACF.

Some introduction of AR model and linear model covariance calculation.

Introduce AR(2) model and disucss the condition of weakly stationary for AR(1) and AR(2) models

Periodicity of AR models with complex roots; Asymptotic statistics of AR(p) models

Introduction to PACF and Information criteria (AIC, BIC)

Introduction to MA model and conditional loglikelihood maximization method. Introdcution to ARMA model

Introduction to ARMA model and its AR and MA representation

Introduction of Karhunen-Leove theorem, the spectral theorem and autocovariance generating function

Spectral density as the Fourier transform of autocovariance, the sample version spectral representation theory and sample periodogram

Spectral representation sample version proof. Variance decomposition into Fourier components. Estimate spectral density

Non parametric estimation of spectral density (Kernel Method)

Introduction to Conditional Variance

Introduction to ARCH model

Estimation of ARCH Model with method of moments

Introduction to Generalized ARCH and model estimation and forecast

Introduction to Non-linear Models including TAR ans SETAR

Introduction to unit root models

Construct statistical test for unit root from samples

Random Walk with Drift term and statistical test for non-zero drift

Statistical Test for The Drift Term

Continuous Time Model

Introduction to Ito’s Process and Ito’s Lemma

Introduction to Ito’s Process and Ito’s Lemma

Application of Ito’s Lemma: Derivation of Black-Scholes Formula

Diffusion Processes with Jumps

Continutation of Jump Diffusion Model and Introduction to Multiple Time Series

Introduction to Vector Autoregressive (VAR) model

Introduction to Cointegration and Vector Error Correction Model

Introduction to Vector Error Correction Model

Introduction to state space model, the Kalman Filterl

Continuation of the discussion of Kalman Filter