Time Series Basics 2

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Time Series Basics (2)

• Fin250f Lecture 3.2
• Fall 2005
• Reading Taylor, chapter 3.5-3.7, 3.9(skip 3.6.1)

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Outline

• Linear stochastic processes
• Autoregressive process
• Moving average process
• Lag operator
• Forecasting AR and MAs
• The ARMA(1,1)
• Trend plus noise models
• Bubble simulations

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Linear Stochastic Processes

• Linear models
• Time series dependence
• Common econometric frameworks
• Engineering background

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AR(1)Autoregressive Process, Order 1
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AR(1) Properties
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AR(m)
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Moving Average Process of Order 1, MA(1)
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MA(1) Properties
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MA(m)
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AR-gtMA
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Lag Operator (L)
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Using the Lag Operator
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An important feature for L
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MA -gt AR
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MA-gtAR
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Forecasting the AR(1)
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Forecasting the AR(1) Multiperiods
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Forecasting an MA(1)
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The ARMA(1,1) AR and MA parts
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ARMA(1,1) with L
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ARMA(1,1) with L
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Forecasting 1 Period
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ARMA(p,q)
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Why ARMA(1,1)?

• Small, but persistent ACFs
• Comparing the AR(1) and ARMA(1,1)

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AR(1) ACFs
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ARMA(1,1) ACFs
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Adding an AR(1) to an MA(0)(Trend plus noise)
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Why Is This Useful?(Taylor 3.6.2)

• Returns follow a combination process
• Sum of
• Small, but very persistent trend
• Independent noise term

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Trend Plus Noise
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Trend Plus Noise
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Parameter Example

• A small
• ? big
• A 0.02, ??????

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Trend Plus Noise ACF
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Temporary Pricing ErrorsBubbles(3.6.1)
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AR(1) Difference
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Variance Ratio
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Return Autocorrelations
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An Example
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Bubble Price Simulation
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Return ACF
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Outline

• Linear stochastic processes
• Autoregressive process
• Moving average process
• Lag operator
• Forecasting AR and MAs
• The ARMA(1,1)
• Trend plus noise models
• Bubble simulations