1 / 44

Time Series Basics

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.

annora
Download Presentation

Time Series Basics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Fin250f: Lecture 8.1 Spring 2010 Reading: Brooks, chapter 5.1-5.7 Time Series Basics

  2. Outline • Linear stochastic processes • Autoregressive process • Moving average process • Lag operator • Model identification • PACF/ACF • Information Criteria

  3. Stochastic Processes

  4. Time Series Definitions • Strictly stationary • Covariance stationary • Uncorrelated • White noise

  5. Strictly Stationary • All distributional features are independent of time

  6. Weak or Covariance Stationary • Variances and covariances independent of time

  7. Autocorrelation

  8. White Noise

  9. White Noise in Words • Weakly stationary • All autocovariances are zero • Not necessarily independent

  10. Time Series Estimates

  11. Ljung-Box Statistic

  12. Linear Stochastic Processes • Linear models • Time series dependence • Common econometric frameworks • Engineering background

  13. Autoregressive Process, Order 1:AR(1)

  14. AR(1) Properties

  15. More AR(1) Properties

  16. More AR(1) properties

  17. AR(1): Zero mean form

  18. AR(m) (Order m)

  19. Moving Average Process of Order 1, MA(1)

  20. MA(1) Properties

  21. MA(m)

  22. Stationarity • Process not exploding • For AR(1) • All finite MA's are stationary • More complex beyond AR(1)

  23. AR(1)->MA(infinity)

  24. Lag Operator (L)

  25. Using the Lag Operator (Mean adjusted form)

  26. An important feature for L

  27. MA(1) -> AR(infinity)

  28. MA->AR

  29. 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

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

  31. 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

  32. Identification Tools • Diagnostics • ACF, Partial ACF • Information criteria • Forecast

  33. Autocorrelation

  34. 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

  35. Partial Autocorrelation

  36. For an AR(1)

  37. AR(1) (0.9)

  38. For an MA(1)

  39. MA(1) (0.9)

  40. 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

  41. Information Criteria • Akaike, AIC • Schwarz Bayesian criterion, SBIC • Hannan-Quinn, HQIC • Objective: • Penalize model errors • Penalize model complexity • Simple/accurate models

  42. Information Criteria

  43. 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

  44. 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)

More Related