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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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
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
white noise in words
White Noise in Words
  • Weakly stationary
  • All autocovariances are zero
  • Not necessarily independent
linear stochastic processes
Linear Stochastic Processes
  • Linear models
  • Time series dependence
  • Common econometric frameworks
  • Engineering background
stationarity
Stationarity
  • Process not exploding
  • For AR(1)
  • All finite MA's are stationary
  • More complex beyond AR(1)
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
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
partial autocorrelation
Partial Autocorrelation
  • Correlation between y(t) and y(t-k) after removing all smaller (
  • Marginal forecast impact from t-k given all earlier information
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
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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