Time Series Basics (2) Fin250f: Lecture 3.2 Fall 2005 Reading: Taylor, chapter , 3.9(skip 3.6.1)

Outline  Linear stochastic processes  Autoregressive process  Moving average process  Lag operator  Forecasting AR and MA’s  The ARMA(1,1)  Trend plus noise models  Bubble simulations

Linear Stochastic Processes  Linear models  Time series dependence  Common econometric frameworks  Engineering background

AR(1) Autoregressive Process, Order 1

AR(1) Properties

AR(m)

Moving Average Process of Order 1, MA(1)

MA(1) Properties

MA(m)

AR->MA

Lag Operator (L)

Using the Lag Operator

An important feature for L

MA -> AR

Forecasting the AR(1)

Forecasting the AR(1): Multiperiods

Forecasting an MA(1)

The ARMA(1,1): AR and MA parts

ARMA(1,1) with L

Forecasting 1 Period

ARMA(p,q)

Why ARMA(1,1)?  Small, but persistent ACF’s  Comparing the AR(1) and ARMA(1,1)

AR(1) ACF’s

ARMA(1,1) ACF’s

Adding an AR(1) to an MA(0) (Trend plus noise)

Why Is This Useful? (Taylor 3.6.2)  Returns follow a combination process  Sum of: Small, but very persistent trend Independent noise term

Trend Plus Noise

Parameter Example  A small   big  A = 0.02, 

Trend Plus Noise ACF

Temporary Pricing Errors Bubbles(3.6.1)

AR(1) Difference

Variance Ratio

Return Autocorrelations

An Example

Bubble Price Simulation

Return ACF

Outline  Linear stochastic processes  Autoregressive process  Moving average process  Lag operator  Forecasting AR and MA’s  The ARMA(1,1)  Trend plus noise models  Bubble simulations