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Time Series Analysis - Arima Models - Basic Definitions and Theorems About Arima Models

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V.I.1.a Basic Definitions and Theorems about ARIMA models

First we define some important concepts. A stochastic process (c.q. probabilistic process) is defined by a T-dimensional distribution function.

Time Series Analysis - ARIMA models - Basic Definitions and Theorems about ARIMA models

marginal distribution function of a time series

(V.I.1-1)

Before analyzing the structure of a time series model one must make sure that the time series are stationary with respect to the variance and with respect to the mean. First, we will assume statistical stationarity of all time series (later on, this restriction will be relaxed).

Statistical stationarity of a time series implies that the marginal probability distribution is time-independent …show more content…

A practical numerical estimation algorithm for the PACF is given by Durbin

(V.I.1-29)

with

(V.I.1-30)

The standard error of a partial autocorrelation coefficient for k > p (where p is the order of the autoregressive data generating process; see later) is given by

(V.I.1-31)

Finally, we define the following polynomial lag-processes

(V.I.1-32)

where B is the backshift operator (c.q. BiYt = Yt-i) and where

(V.I.1-33)

These polynomial expressions are used to define linear filters. By definition a linear filter

(V.I.1-34)

generates a stochastic process

(V.I.1-35)

where at is a white noise variable.

(V.I.1-36)

for which the following is obvious

(V.I.1-37)

We call eq. (V.I.1-36) the random-walk model: a model that describes time

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