Time Series Basics

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# Time Series Basics - PowerPoint PPT Presentation

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

Time Series Basics
Outline
• Linear stochastic processes
• Autoregressive process
• Moving average process
• Lag operator
• Model identification
• PACF/ACF
• Information Criteria
Time Series Definitions
• Strictly stationary
• Covariance stationary
• Uncorrelated
• White noise
Strictly Stationary
• All distributional features are independent of time
Weak or Covariance Stationary
• Variances and covariances independent of time
White Noise in Words
• Weakly stationary
• All autocovariances are zero
• Not necessarily independent
Linear Stochastic Processes
• Linear models
• Time series dependence
• Common econometric frameworks
• Engineering background
Stationarity
• Process not exploding
• For AR(1)
• All finite MA's are stationary
• More complex beyond AR(1)
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 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
• Diagnostics
• ACF, Partial ACF
• Information criteria
• Forecast
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
• 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
• Akaike, AIC
• Schwarz Bayesian criterion, SBIC
• Hannan-Quinn, HQIC
• Objective:
• Penalize model errors
• Penalize model complexity
• Simple/accurate models
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
• Get model residuals (forecast errors)
• Run this time series through various diagnostics
• ACF, PACF, Ljung/Box, plots
• Should be white noise (no structure)