Fin250f lecture 8 1 spring 2010 reading brooks chapter 5 1 5 7
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Fin250f: Lecture 8.1 Spring 2010 Reading: Brooks, chapter 5.1-5.7. Time Series Basics. Outline. Linear stochastic processes Autoregressive process Moving average process Lag operator Model identification PACF/ACF Information Criteria. Stochastic Processes. Time Series Definitions.

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Time Series Basics

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Fin250f lecture 8 1 spring 2010 reading brooks chapter 5 1 5 7

Fin250f: Lecture 8.1

Spring 2010

Reading: Brooks, chapter 5.1-5.7

Time Series Basics


Outline

Outline

  • Linear stochastic processes

  • Autoregressive process

  • Moving average process

  • Lag operator

  • Model identification

    • PACF/ACF

    • Information Criteria


Stochastic processes

Stochastic Processes


Time series definitions

Time Series Definitions

  • Strictly stationary

  • Covariance stationary

  • Uncorrelated

  • White noise


Strictly stationary

Strictly Stationary

  • All distributional features are independent of time


Weak or covariance stationary

Weak or Covariance Stationary

  • Variances and covariances independent of time


Autocorrelation

Autocorrelation


White noise

White Noise


White noise in words

White Noise in Words

  • Weakly stationary

  • All autocovariances are zero

  • Not necessarily independent


Time series estimates

Time Series Estimates


Ljung box statistic

Ljung-Box Statistic


Linear stochastic processes

Linear Stochastic Processes

  • Linear models

  • Time series dependence

  • Common econometric frameworks

  • Engineering background


Autoregressive process order 1 ar 1

Autoregressive Process, Order 1:AR(1)


Ar 1 properties

AR(1) Properties


More ar 1 properties

More AR(1) Properties


More ar 1 properties1

More AR(1) properties


Ar 1 zero mean form

AR(1): Zero mean form


Ar m order m

AR(m) (Order m)


Moving average process of order 1 ma 1

Moving Average Process of Order 1, MA(1)


Ma 1 properties

MA(1) Properties


Time series basics

MA(m)


Stationarity

Stationarity

  • Process not exploding

  • For AR(1)

  • All finite MA's are stationary

  • More complex beyond AR(1)


Ar 1 ma infinity

AR(1)->MA(infinity)


Lag operator l

Lag Operator (L)


Using the lag operator mean adjusted form

Using the Lag Operator (Mean adjusted form)


An important feature for l

An important feature for L


Ma 1 ar infinity

MA(1) -> AR(infinity)


Ma ar

MA->AR


Ar s and ma s

AR's and MA's

  • Can convert any stationary AR to an infinite MA

  • Exponentially declining weights

  • Can only convert "invertible" MA's to AR's

  • Stationarity and invertibility:

    • Easy for AR(1), MA(1)

    • More difficult for larger models


Combining ar and ma arma p q more later

Combining AR and MA ARMA(p,q) (more later)


Modeling procedures box jenkins

Modeling ProceduresBox/Jenkins

  • Identification

    • Determine structure

      • How many lags?

      • AR, MA, ARMA?

    • Tricky

  • Estimation

    • Estimate the parameters

  • Residual diagnostics

  • Next section: Forecast performance and evaluation


Identification tools

Identification Tools

  • Diagnostics

    • ACF, Partial ACF

    • Information criteria

    • Forecast


Autocorrelation1

Autocorrelation


Partial autocorrelation

Partial Autocorrelation

  • Correlation between y(t) and y(t-k) after removing all smaller (<k) correlations

  • Marginal forecast impact from t-k given all earlier information


Partial autocorrelation1

Partial Autocorrelation


For an ar 1

For an AR(1)


Ar 1 0 9

AR(1) (0.9)


For an ma 1

For an MA(1)


Ma 1 0 9

MA(1) (0.9)


General features

General Features

  • Autoregressive

    • Decaying ACF

    • PACF drops to zero beyond model order(p)

  • Moving average

    • Decaying PACF

    • ACF drops to zero beyond model order(q)

  • Don’t count on things looking so good


Information criteria

Information Criteria

  • Akaike, AIC

  • Schwarz Bayesian criterion, SBIC

  • Hannan-Quinn, HQIC

  • Objective:

    • Penalize model errors

    • Penalize model complexity

    • Simple/accurate models


Information criteria1

Information Criteria


Estimation

Estimation

  • Autoregressive AR

    • OLS

    • Biased(-), but consistent, and approaches normal distribution for large T

  • Moving average MA and ARMA

    • Numerical estimation procedures

    • Built into many packages

      • Matlab econometrics toolbox


Residual diagnostics

Residual Diagnostics

  • Get model residuals (forecast errors)

  • Run this time series through various diagnostics

    • ACF, PACF, Ljung/Box, plots

  • Should be white noise (no structure)


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