Time Series Models. Stochastic processes Stationarity White noise Random walk Moving average processes Autoregressive processes More general processes. Topics. Stochastic Processes. 1. Time series are an example of a stochastic or random process

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Time Series Models

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Time series are an example of a stochastic or random process A stochastic process is 'a statistical phenomenen that evolves in timeaccording to probabilistic laws' Mathematically, a stochastic process is an indexed collection of random variables Stochastic processes

Inference is most easy, when a process is stationary—its distribution does not change over time This is strict stationarity A process is weakly stationary if its mean and autocovariance functions do not change over time Stationarity

The mean and variance are given by The process is weakly stationary because the mean is constant and the covariance does not depend on t Moving average processes

In order to ensure there is a unique MA process for a given acf, we impose the condition of invertibility This ensures that when the process is written in series form, the series converges For the MA(1) process Xt = Zt + Zt - 1, the condition is ||< 1 Moving average processes

The first order autoregression is Xt = Xt - 1 + Zt Provided ||<1 it may be written as an infinite order MA process Using the backshift operator we have (1 – B)Xt = Zt Autoregressive processes

This is for for some 1, 2, . . . This gives Xt as an infinite MA process, so it has mean zero Autoregressive processes

Conditions are needed to ensure that various series converge, and hence that the variance exists, and the autocovariance can be defined Essentially these are requirements that the i become small quickly enough, for large i Autoregressive processes

The i may not be able to be found however. The alternative is to work with the i The acf is expressible in terms of the roots i, i=1,2, ...p of the auxiliary equation Autoregressive processes

The approach is an iterative one involving model identification model fitting model checking If the model checking reveals that there are problems, the process is repeated Box-Jenkins approach