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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.

Time Series Basics

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#### Presentation Transcript

Fin250f: Lecture 8.1

Spring 2010

### 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 (<k) correlations

• 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)