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Time Series Basics

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Fin250f: Lecture 8.1

Spring 2010

Reading: Brooks, chapter 5.1-5.7

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

- Strictly stationary
- Covariance stationary
- Uncorrelated
- White noise

- All distributional features are independent of time

- Variances and covariances independent of time

- Weakly stationary
- All autocovariances are zero
- Not necessarily independent

- Linear models
- Time series dependence
- Common econometric frameworks
- Engineering background

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

- 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

- Identification
- Determine structure
- How many lags?
- AR, MA, ARMA?

- Tricky

- Determine structure
- Estimation
- Estimate the parameters

- Residual diagnostics
- Next section: Forecast performance and evaluation

- Diagnostics
- ACF, Partial ACF
- Information criteria
- Forecast

- Correlation between y(t) and y(t-k) after removing all smaller (<k) correlations
- Marginal forecast impact from t-k given all earlier information

- 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

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

- 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

- Get model residuals (forecast errors)
- Run this time series through various diagnostics
- ACF, PACF, Ljung/Box, plots

- Should be white noise (no structure)