Time Series Basics

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Time Series Basics

• Fin250f Lecture 3.1
• Fall 2005
• Reading Taylor, chapter 3.1-3.3

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Outline

• Random variables
• Distributions
• Central limit theorem
• Two variables
• Independence
• Time series definitions

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Random Variables Discrete
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Random Variables Continuous
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Random Variables Continuous
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Random Variables Continuous
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Important Distributions

• Uniform
• Normal
• Log normal
• Student-t
• Stable

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Normal/Gaussian
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Normal Picture Sample 2000
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Normal Exponential Expectations
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Why Important in Finance?

• Central limit theorem
• Many returns almost normal

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Log Normal
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Log Normal

• Not symmetric
• Long right tail

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Log Normal Histogram (Sample 5000)
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Chi-square
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Student-t
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Student-t Moments

• All moments gt r do not exist

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Stable Distribution

• Similar shape to normal
• Infinite variance
• Sums of stable RVs are stable

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Central Limit Theorem (casual)
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Consequence of CLT and continuous compounding
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Two Variables
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More on Two Variables
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More Two Variables
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Independent Random Variables
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More than Two RVs
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Multivariate Normal
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Independence
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Independent Identically Distributed

• All random variables drawn from same distribution
• All are independent of each other
• Common assumption
• IID
• IID Gaussian

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Stochastic Processes
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Time Series Definitions

• Strictly stationary
• Covariance stationary
• Uncorrelated
• White noise
• Random walk
• Martingale

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Strictly Stationary

• All distributional features are independent of
  time

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Covariance Stationary

• Variances and covariances independent of time

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Uncorrelated
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White Noise

• Covariance stationary
• Uncorrelated
• Mean zero

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Random Walk
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Geometric Random Walk
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Martingale
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Autocovariances/correlations
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Outline

• Random variables
• Distributions
• Central limit theorem
• Two variables
• Independence
• Time series definitions