Introduction to Financial Time Series

From Ruey. S. Tsay’s slides

What is Financial Time Series Analysis Theory and practice of asset valuation over

time. Different from other T.S. analysis?

Close, but with some added uncertainty.

For example, FTS must deal with the ever-changing business & economic environment and the fact that volatility is not directly observed.

Examples of financial time series Daily log returns of GE stock Quarterly earnings of Johnson & Johnson

Seasonal time series useful in earning forecasts pricing weather related derivatives (e.g. energy) modeling intraday behavior of asset returns

US monthly interest rates Relations between the two series? Term structure of interest rates

Exchange rate between US Dollar vs Japanese Yen Fixed income, hedging, carry trade

Size of insurance claims Values of fire insurance claims from 1972 to 1992.

High-frequency financial data: Tick-by-tick data of Boeing stock: December 5, 2005.

Example of FTS

Example of FTS (Cont.)

Objective of the Course

Provide some basic knowledge of financial time series data.

Introduce some statistical tools & econometric models useful for analyzing these series.

Gain empirical experience in analyzing FTS.

Design your own method to predict future FTS.

Asset Returns

From Ruey. S. Tsay’s slides

Asset Returns

Asset Return Example

Asset Returns

Compound Assert Returns

Compound Assert Returns

Compound Assert Returns

Asset Returns

For multi-period returns, we have

Captial Asset Price Model (CAPM): Consider the joint distribution of N return at a single time

index t, Or, study the distribution We will focus on the dynamic structure of individual

asset returns, in other words, we study the distribution of How to describe the joint distribution of

11 ...)( ktttt rrrkr

},...,{ 1 Ntt RR

},...,{ 1 iTi RRTtitR 1}{

Asset Returns

For discrete case:

For continuous case:

parameter of vector a is where

),...,|()...|()(

as );,...,(

11,121

1

θ

RRRFRRFRF

RRF

iTiiTiii

iTi

parameter of vector a is andfunction density is where

),...,|()...|()(

as );,...,(

11,121

1

θf

RRRfRRfRf

RRf

iTiiTiii

iTi

Marginal Distribution of Asset Returnd When asset returns have weak empirical serial

correlation, their marginal distributions are close to their conditional distributions. Thus the question is how to estimate marginal distributions.

Several statistical distributions have been proposed for marginal distribution of asset returns: Normal distribution Lognormal distribution Stable distribution Scale-Mixture Normal distribution

Linear Time Series Analysis

From Ruey. S. Tsay’s slides

Linear Time Series (TS) Models

Basic Concepts

Basic Concepts (Cont.)

Basic Concepts (Cont.)

Basic Concepts (Cont.)

Univariate TS Analysis Purpose:

Example of Linear Time Series

Linear Financial Time Series

AR Model

AR Model (Cont.)

AR Model (Cont.)

AR Model (Cont.)

AR(2) Model

Building an AR Model

Moving-Average (MA) Model

MA(1) Model (Cont.)

MA(2) Model

Building MA Model

Mixed ARMA Model

Mixed ARMA Model (Cont.)

Building an ARMA (1, 1) Model

Three Model Comparisons:

Seasonal Time Series

Examples of Seasonal Time Series (1)

Figure 1: Time plot of electricity demand of an industrial sector: 15th day of each month from1972 to 1993.

Example of Seasonal Time Series (2)

Example of Seasonal Time Series (3)

Figure 3: Time plot of quarterly logged earnings of Johnson and Johnson: 1960-1980

Seasonal Difference Model