Time Series Basics Fin250f: Lecture 8.1 Spring 2010 Reading: Brooks, chapter

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

Stochastic Processes

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

Autocorrelation

White Noise

White Noise in Words  Weakly stationary  All autocovariances are zero  Not necessarily independent

Time Series Estimates

Ljung-Box Statistic

Linear Stochastic Processes  Linear models  Time series dependence  Common econometric frameworks  Engineering background

Autoregressive Process, Order 1:AR(1)

AR(1) Properties

More AR(1) Properties

More AR(1) properties

AR(1): Zero mean form

AR(m) (Order m)

Moving Average Process of Order 1, MA(1)

MA(1) Properties

MA(m)

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

AR(1)->MA(infinity)

Lag Operator (L)

Using the Lag Operator (Mean adjusted form)

An important feature for L

MA(1) -> AR(infinity)

MA->AR

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)

Modeling Procedures Box/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

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 Autocorrelation

For an AR(1)

AR(1) (0.9)

For an MA(1)

MA(1) (0.9)

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

Information Criteria

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)