Time Series Basics Fin250f: Lecture 3.1 Fall 2005 Reading: Taylor, chapter

Outline  Random variables  Distributions  Central limit theorem  Two variables  Independence  Time series definitions

Random Variables: Discrete

Random Variables: Continuous

Important Distributions  Uniform  Normal  Log normal  Student-t  Stable

Normal/Gaussian

Normal Picture: Sample = 2000

Normal Exponential Expectations

Why Important in Finance?  Central limit theorem  Many returns almost normal

Log Normal

 Not symmetric  Long right tail

Log Normal Histogram (Sample = 5000)

Chi-square

Student-t

Student-t Moments  All moments > r do not exist

Stable Distribution  Similar shape to normal  Infinite variance  Sums of stable RV’s are stable

Central Limit Theorem (casual)

Consequence of CLT and continuous compounding

Two Variables

More on Two Variables

More Two Variables

Independent Random Variables

More than Two RV’s

Multivariate Normal

Independence

Independent Identically Distributed  All random variables drawn from same distribution  All are independent of each other  Common assumption  IID  IID Gaussian

Stochastic Processes

Time Series Definitions  Strictly stationary  Covariance stationary  Uncorrelated  White noise  Random walk  Martingale

Strictly Stationary  All distributional features are independent of time

Covariance Stationary  Variances and covariances independent of time

Uncorrelated

White Noise  Covariance stationary  Uncorrelated  Mean zero

Random Walk

Geometric Random Walk

Martingale

Autocovariances/correlations

Outline  Random variables  Distributions  Central limit theorem  Two variables  Independence  Time series definitions