Time series (ctd.) Stat701 Fall 1999

Time series continued.

Todays class.

Resources: RiskMetrics Readings and links.

RiskMetrics at J.P.Morgan

#95-19. This link takes you to the pdf version of the paper.
Probability and Statistics Applied to the Practice of Financial Risk Management: The Case of JP Morgan's RiskMetrics
Michael Phelan, August 1995. This is the abstract.

ARMA and ARIMA models (models for time series).

p - the order of the autoregression.

d - the order of the differencing undertaken to achieve stationarity

q - the order of the moving average term

The partial autocorrelation function.

The partial autocorrelation function.

Helpful for accessing the oder of an AR process. Idea: fit an AR(p) model, the last coefficient is , the excess correlation at lag p not accounted for by an AR(p-1) model. Plot p-th partial autocorrelation against p.

For an AR(p) process expect the partial acf to cut off at lag p.

For an MA process the partial acf attenuates. In this sense the partial acf has opposite behavior to the acf.

Acf examples:

The GM87 series.

Strategies:

In a regression context look at the autocorrelation function of the residuals. Ideally a purely random process,

If decreasing lag-k autocorrelations: model the residuals.

Lag-k autocorrelations that do not drop off reasonably quickly or show cyclical behavior: non-stationary series : look to remove more of the systematic component.

Options for the stationary series scenario.

Differencing .

Cochrane-Orcutt (for AR(1) process)

Formally modeling the residuals: ARMA and ARIMA models:

Three part process:

Model specification: p,d,q. Determination of the ARIMA model orders. Use the ts.plot and acf functions.

Model fitting: estimate the parameters in the model. Use the arima.mle function.

Diagnostics and model checking: the residuals from the model are investigated - ideally they form a pure noise process. Use the arima.diag function.